bt (W ChE tly Syringe TN. goa: TM, Sd ENS iC ee 4 Hy by a 4 i iat Suuget Srnec % — oe mare * & hi naa Bang mye SS et ath ae i vas und 224 sy a By ee PE wah 28 Be iii ahha kia Abbie ' aut ACH 203 fice af Navel Research fepartwent of the Navy 5 4F2e2?2000 TOEO oO WIV A IOHM/181N AN EN. tM Mf a } \) ) mh j F w iby : d ie 4 i \ L~ fe i i : - Ninth Symposium NAVAL HYDRODYNAMICS VOLUME 1 ee aaa VASE UNCONVENTIONAL SHIPS , 52° QCEAN ENGINEERING peas ast pliet wae ae BAASS i sponsored by the , Cte CE OF NAVAL RESEARCH the MINISTERE D’/ETAT CHARGE DE LA DEFENSE NATIONALE and the ASSOCIATION TECHNIQUE MARITIME ET AERONAUTIQUE August 20-25, 1972 Paris, France R. BRARD A. CASTERA Editors ACR-203 OFFICE OF NAVAL RESEARCH— DEPARTMENT OF THE NAVY Arlington, Va. For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $15.55 Stock Number 0851-00062 PREVIOUS BOOKS IN THE NAVAL HYDRODYNAMICS SERIES “First Symposium on Naval Hydrodynamics,”’ National Academy of Sciences—National Research Council, Publication 515, 1957, Washington, D.C.; PB133732, paper copy $6.00, 335-mm microfilm $1.45. “Second Symposium on Naval Hydrodynamics: Hydrodynamic Noise and Cavity Flow,” Office of Naval Reserach, Department of the Navy, ACR-38, 1958; PB157668, paper copy $10.00, 35-mm microfilm $1.45. “Third Symposium on Naval Hydrodynamics: High-Performance Ships,” Office of Naval Research, Department of the Navy, ACR-65, 1960; AD430729, paper copy $6.00, 35-mm microfilm $1.45. ‘Fourth Symposium on Naval Hydrodynamics: Propulsion and Hydroelasticity,” Office of Naval Research, Department of the Navy, ACR-92, 1962; AD447732, paper copy $9.00, 35-mm microfilm $1.45. “The Collected Papers of Sir Thomas Havelock on Hydrodynamics,” Office of Naval Research, Department of the Navy, ACR-103, 1963; AD623589, paper copy $6.00, microfiche $1.45. “Fifth Symposium on Naval Hydrodynamics: Ship Motions and Drag Reduction,” Office of Naval Research, Department of the Navy, ACR-112, 1964; AD640539, paper copy $15.00, microfiche $1.45. “Sixth Symposium on Naval Hydrodynamics: Physics of Fluids, Maneuverability and Ocean Platforms, Ocean Waves, and Ship-Generated Waves and Wave Resistance,” Office of Naval Research, Department of the Navy, ACR-136, 1966; AD676079, paper copy $10.00, microfiche $1.45. “Seventh Symposium on Naval Hydrodynamics: Unsteady Propeller Forces, Funda- mental Hydrodynamics, Unconventional Propulsion,” Office of Naval Research, Depart- ment of the Navy, DR-148, 1968: AD721180; Available from Superintendent of Docu- ments, U.S. Government Printing Office, Washington, D.C. 20402, Clothbound, 1690 pages, illustrated (Catalog No. D 210.15:DR-148; Stock No. 0851-0049), $13.00; microfiche $1.45. “Eighth Symposium on Naval Hydrodynamics: Hydrodynamics in the Ocean Environ- ment,” Office of Naval Research, Department of the Navy, ACR-179, 1970; AD748721; Available from Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20404, Clothbound, 1185 pages, illustrated (Catalog No. D 210.15: ACR-179; Stock No. 0851-0056), $10.00; microfiche $1.45. NOTE: The above books are available on microfilm and microfiche from the National Technical Information Service, U.S. Department of Commerce, Springfield, Virginia 22151. The first six books are also available from NTIS in paper copies. The catalog numbers and the prices for paper, clothbound, and microform copies are shown for each book. Statements and opinions contained herein are those of the authors and are not to be construed as official or reflecting the views of the Navy Department or of the naval service at large. il PREFACE The Ninth Symposium on Naval Hydrodynamics continues in all aspects the precedent, established by previous symposia in this series, of providing an initer- national forum for the presentation and exchange of the most recent research re- sults in selected fields of naval hydrodynamics. The Symposium was held in Paris, France on 20-25 August 1972 under the joint sponsorship of the Office of Naval Research, the Ministére d’Etat chargé de la Défense Nationale and the Association Technique Maritime et Aeroanutique. The technical program of the Symposium was devoted to three subject areas of current naval and maritime interest. These subject areas are covered in the Proceedings in two volumes: Volume 1 — The Hydrodynamics of Unconventional Ships — Hydrodynamic Aspects of Ocean Engineering Volume 2 — Frontier Problems in Hydrodynamics. The planning, organization and management of a Symposium such as this is an undertaking of considerable magnitude, and many people have made invaluable contributions to the resolution of the myriad of large and small problems which invariably arise. The Office of Naval Research is acutely aware of the fact that the success of the Ninth Symposium is directly attributable to these people and wishes to take this opportunity to express its heartfelt gratitude to them. We are particu- larly indebted to Vice Admiral Raymond THIENNOT, Directeur Technique des Constructions Navales, Ministére d’ Etat chargé de la Défense Nationale, to Professor Jean DUBOIS, Directeur des Recherches et Moyens d’ Essais, Ministere d’' Etat chargé de la Défense Nationale, and to Monsieur Jean MARIE, Président de |’ Association Technique Maritime et Aeronautique, who provided the formal structure which made this joint undertaking possible. The detailed organization and management of the Ninth Symposium lay in the capable and competent hands of Vice Admiral Roger BRARD, President de la Academie des Sciences, and Rear Admiral André CASTERA, Directeur du Bassin d’Essais des Careénes, who were most ably assisted in this endeavor by the charming Madame Jean TATON. Throughout the long days of planning and preparation the experienced and practical counsel of Mr. Stanley DOROFF of the Office of Naval Research provided continuous guidance which contributed in an immeasurable way to the success of the Ninth Symposium on Hydrodynamics. RALPH D. COOPER Fluid Dynamics Program Office of Naval Research MARINE my! REICAL Lk eel SNe BE | ili VOLUME 1 CONTENTS Page Pretace. 455 Revs. LING 8G. 2ST OGRA, Wee) TVR oe ili Introductory} Add Pegs >< POLY HG as Ps, he xv Roger Brard, Chairman, French Organizing Committee OPEB RA GUEeAS LOaaneines Gis bic Mean sie un Le atthe ok maha xvii Jean Blancard, Délégué Ministériel pour 1'Armement Paris, France POLIS Sys aie! Bei Na taiae ba lacie + Aia ain, oe. calla caiiath aes ebbiins cows enc , xix Honorable Robert A, Frosch, Assistant Secretary of the Navy for Research and Development, Washington, U.S.A. UNCONVENTIONAL SHIPS WATER-JET PROPULSION FOR HIGH-SPEED SURFACE SHEERS 10H IY a PPE A PDP tis SER Bay So a ee 3 J. Duport, M. Visconti and J, Merle (SOGREAH), France DISGUSSION ussoue.ods ot. usw. Wdoivesanan we wi. ded 29 William B, Morgan, Naval Ship Research and Development Center, Bethesda, Maryland RUBIA FO DISC USSION 3°. Ee eee Pe See 29 J. Duport, SOGREAH, Grenoble, France THE FORCES ON AN AIR-CUSHION VEHICLE EXECUTING ANUNSTEADY MOTION’. <2. “owl awe. Barge, oes a's aval 35 Lawrence J, Doctors, University of New South Wales, Sydney, Australia PRESS WSBT so ice eins shal et wal ae at eal ter 95 Jorgens Strom-Tejsen, Naval Ship Research and Development Center, Bethesda, Maryland iv REN LO Mots CUSSTON oo. + uel aimless, 2 a sae 96 Lawrence J. Doctors, University of New South Wales, Sydney, Australia A LINEARIZED POTENTIAL FLOW THEORY FOR THE MOTIONS OF AMPHIBIOUS AND NON AMPHIBIOUS AIR GwiSLlON SVE HC TE onbN PAGS BAW I ArY) tas cases SA es SP mates 99 Di ayes Murthy. —ontsmouth Polytechnic, Ui. i, DiS@USSTON i pipe vce speek ware as. oils Sebi ae 2AS s 255 Lawrence J. Doctors, University of New South Wales, Sydney, Australia REE by SL OmisGUSsiION shy . Ate eae... 256 T.K.S. Murthy, Portsmouth Polytechnic, U.K. DISCUSSION ce cs op tu5.3 6 Pe eet Se ES Poi | Roger Brard, Bassin d'Essais des Carénes, Paris, France se OBI) ES CSS LOIN ices co gui ogre ciwiivey eine tenveoyese sips 258 ine oe Murthy. ortsniouth Polytechnic, U.K, DISCUS SON. ots neta teva eee tata ifs EPS DE ara) 258 Paul Kaplan, Oceanics Inc, Plainview, New York LOE Ee NeeO IS CUSSION 3... 6 ss sR ete Beals 259 T.K.S. Murthy, Portsmouth Polytechnic, U.K. ON THE DETERMINATION OF AEROHYDRODYNA MIC Piha ORMANGIS, OF 7NIRICUSHION VWEHIC IBS os 4 1. . 261 S.D. Prokhorov, V.N. Treshchevsky and L.D. Volkov Kryloff Ship Research Institute, Leningrad, U.S.S.R. DiS CU SOLON *o¥ ee nk eee Rt Mb en Mi ie min gies oo 288 Paul Kaplan, Oceanics Inc, Plainview, New York Ree SE Nel@ DiS Gu SSmON es. sss ec isl wince ls tay oe. « 289 Vjacheslav N. Treshchevsky, Kryloff Research Institute, Leningrad, U.S.S.R. HYDRODYNAMICS AND SIMULATION IN THE CANADIAN Page Oi Chie) OG AIM eee yn eee et eet Se Sete ook s 295 R.T. Schmitke and E,A. Jones, Defence Research Establishment Atlantic, Dartmouth, N.S., Canada DES CUS SON An He a RAL NEA, a TROIS, otk es 339 Christopher Hook, Hydrofin, Bosham, Sussex, WK: REBLY. TO DISC USSIOM (on choses a aieaniealk oe 339 Rodney T. Schmitke, Defence Hepeancs Establishment Atlantic, Dartmouth, N.S., Canada DISCUSSION 2 cheer oe ete at fon eo ele ee cae ee 340 Reuven Leopold, U.S. Navy Naval Ship Engineer - ing Center, Hyattsville, Maryland REPLY TO DISCUSSION 4.6 «‘.s. )3 cleus sais 340 Rodney T. Schmitke, Defence Researell Esta- blishment Atlantic, Dartmouth, N.S., Canada DISGUSSION «5 s/s 6's Pee ees ee ee. 341 Reuven Leopold, U.S. Raa Naval Ship paGiber- ing Center, Hyattsville, Maryland REPLY TO DISCUSSION. .- oo a0." 2Grs Ape 341 Rodney T. Schmitke, Defeat Reseda Sta blishment Atlantic, Dartmouth, N.S., Canada BENDING FLUTTER AND TORSIONAL FLUTTER OF FLEXIBLE Hy PROP orl OTRULS.. sis 6 oe bas, 4 a 4 owe bn tn a 343 Peter K. Besch and Yuan-Ning Liu, Naval Ship Re- search and Development Center, Bethesda, Maryland DISCUSSION +554 tate teretele a ee + ate Soe ke kee en 395 Reuven Leopold, U.S. Navy Naval Ship Engineer - ing Center, Hyattsville, Maryland REPLY TO DISCUSSION ca i: 5955 p>. SG lates Cae he a95 Peter K. Besch, Naval Ship eas and De- velopment Center, Bethesda, Maryland DISGUSSTION a ss se 5 oe te oe eo MRCS Te ee . 396 Reuven Leopold, U.S. Navy Naval Ship Engineer - ing Center, Hyattsville, Maryland REPLY TO DISCUSSION ....... 2+ eee ee eeee 396 Peter K. Besch, Naval Ship Research and De- velopment Center, Bethesda, Maryland DISSE USSION 542 sh." see a a eat os eae aan : 397 John P. Breslin, Stevens Institute of Technology, Hoboken, New Jersey REPLY ‘TO DISCUSSION ais “os: 6a. es cee wate 397 Peter K. Besch, Naval Ship Research and De- velopment Center, Bethesda, Maryland DIS GUSSION Neh A Sis Ly tit. Cena Sener aie aa John P, Breslin, Stevens Institute of Technology, Hoboken, New Jersey Rb Pe vn OrDIseUSSION fs). i045 2.7 | oa) Maes Peter K. Besch, Naval Ship Research and De- velopment Center, Bethesda, Maryland DUS EUS SON Tee dee Nh. gee oni a. iyi A John P, Breslin, Stevens Institute of Technology, Hoboken, New Jersey IEP IC MODUS GUISE ION Nes Geonulo Bide bes ob Sd Peter K. Besch, Naval Ship Research and De- velopment Center, Bethesda, Maryland DS WS SEO Ni nerits 22 ae cece eel ster afta nether ea apa Reuven Leopold, U.S. Navy Naval Sth Engineer - ing Center, Hyattsville, Maryland ON THE DESIGN OF THE PROPULSION SYSTEMS WITH "Zz"! Pee oP OR Lew ROnOM SHlUPS 1), 8, ee ee. Feae A.A, Rousetsky, Kryloff Ship Research Institute, Wenimip reads Ooo. HSCS SON Mememteme rte eae a tee ee econ Henry M. Cheng, Office of the Chief of Naval Operation, U.S.A, IRIE IE IL Mh yA O) IDS SG TSISTUCIN ete 5 Ga Hig ol bss ween bes dee ee Vjacheslav N, Treshchevsky, Kryloff Research Institute, Leningrad, U.S.S.R. PUSS SONG re ee tee ee ee ee ee Reuven Leopold, U.S, Navy Naval Ship Engineer - ing Center, Hyattsville, Maryland. DISCUSSION Jorgens Strom-Tejsen, Naval Ship Research and Development Center, Bethesda, Maryland DES CHO SOMONE are eee he et a ek A NOR NS ROR, Sd William B, Morgan, Naval Ship Research and De- velopment Center, Bethesda, Maryland DNSCHE SOMONE Ne tS tPA Pe RP A Bate ease. Horst Nowacki, University of Michigan, Ann Harbor, Michigan 398 395 B15) 299 401 411 412 412 413 415 415 REPLY *TO DING USS LONG: ot oh eet itn! o's te a ee ee te 416 Vjacheslav N, Treshchevsky, Kryloff Research Institute, Leningrad, U.S.S.R. DIS PSESIOIN oa ate a eee Saar ee ee ee ee eee 417 Reuven Leopold, U.S. Navy Naval Ship Engineer - ing Center, Hyattsville, Maryland REPLY !TOWISCUSSION( 4: .* 2%. pal Na eee Eat 417 Vjacheslav N. Treshchevsky, Kryloff Resa Institute, Leningrad, U.S.S.R. HYDRODYNAMIC DEVELOPMENT OF A HIGH SPEED PLANING BULL FOR ROUGH WA tL E Be g.. «ccdue aclousthy = anna pcm beneruenn 419 Daniel Savitsky, Stevens Institute of Technology, Hoboken, New Jersey, John K. Roper, Atlantic Hydro- foils, Inc. Hancock, N.H. and Lawrence Benen, Naval Ship Systems Command, U.S. Navy DISELSSION ves 4 a cera olee Lt (B teh k SUE CSI OE Ek, Brey er 459 Manley Saint-Denis,. University of Hawai, Honolulu DISCUSSION .ueicpies. Cenmeh, .neteeients Ete 460 Reuven Leopold, U.S. Navy Naval Ship iengincadll ing Center, Hyattsville, Maryland REPUY Biv eat eee Carl-Anders Johnsson, Statens Skeppsprovnings - anstalt, Goteborg, Sweden DOO S| ee, ee a ee ee me Marinus Oosterveld, Netherlands Ship Model Basin Wageningen, Netherlands REPAY TO DISCUSSION: ¢ avait > apnwusy domed See Carl-Anders Johnsson, Statens Skeppsprovnings - anstalt, Goteborg, Sweden DISC LISSIGIN: 2. onink dune) Adele ans vals +o Gabe ee Edmund V, Telfer, R.I.N.A., Ewell, Surrey, UR: REPLY EO. DISCUSSION: 4. - apesmief> oth Bype pepe lee Carl-Anders Johnsson, Statens Skeppsprovnings - anstalt, Goteborg, Sweden DISGUSS IGN, 4 city tel? Pedy 20d i wiadts C0 eee Harrison Lacicenbe British Ship Research As- sociation, Wallsend, Northumberland, U.K, REPLY FO.DISCUSSION. «4.32 stand p etd epee es 2 Carl-Anders Johnsson, Statens Skeppsprovnings- anstalt, Goteborg, Sweden 579 580 581 655 656 658 659 660 660 661 661 DISCUSSION ces ie. 6.4, 0. ves) sup ace camel ook oe Pe Finn C, Michelsen, Norges Tekniske Hggskole, Trondheim, Norway REP LY eLOrDIsGUSSION mA e, a e ooee mete Carl-Anders Johnsson, Statens Skeppsprovnings - anstalt, Goteborg, Sweden DISCUSSION 5 ti. 5) Ly ae eee: Gel ade ets John P, Breslin, Stevens Institute of Technology, Hoboken, New Jersey REPRO: DISCUSSION 4 Veils Ons Rh Pe? Carl-Anders Johnsson, Statens Skeppsprovnings - anstalt, Goteborg, Sweden AE SICAUIS SILO IN Gs vrceresiei's snc os) pol (6 5 ave 'ouio cb cms tap Nodh oy sg Erling Huse, Ship Research Institute of Norway, Trondheim, Norway Rene ODIs CUSSION: 4 sucech act ot seat ot ee eee ee Carl-Anders Johnsson, Statens Skeppsprovnings - anstalt, Goteborg, Sweden MOTIONS OF MOORED SHIPS IN SIX DEGREES OF JP LR IE IDB) ODN UREA ce ts eA oie ae oe RG og Oo Se a I-Min Yang, Tetra Tech. Inc., Pasadena, California DLS CW SS LO Nin. wre go er. si choke sete url tele eae tie seers Vhs Manley Saint-Denis, University of Hawai, Honolulu REPLY oO) DISCUSSION wo 51.05 165 See eae eee I-Min Yang, Tetra Tech. Inc, Pasadena, California TS Ci SOUOIN Gy. uecstesniepeeree sta shou Gl se 6 eoLe\ iene) su 180 Sag haume Paul Kaplan, Oceanics Inc., New York RE Pye ©_ DIS@GUSSION:.. 8 bie. eee © I-Min Yang, Tetra Tech. Inc. Pasadena, California ESIC SSO IN po slaty siete ohses! ions Gceiom oncevenns pour eT Ernest O, Tuck, University of Adelaide, Australia OTS OG SS UO Nig ei eles ay ales Merrorce ote, aod be yee eee Grand Lewison, National Physical Lab., Feltham, Middlesex, U.K, 663 664 665 666 668 671 679 680 681 683 684 685 REPLY ££ ©O- DISGUSSIONe . <. 6... «+ tues Bs 685 I-Min Yang, Tetra Tech, Inc, Pasadena, California DISCUSSION. 23.225 5-2-5 Bice ' Ss Re 685 Grant Lewison, National Phisical’ Lab. Feltham, Middlesex, U.K. REPRY. FO IIS CUSSION. a8 SiR ae s Lecce 686 I-Min Yang, Tetra Tech, Inc,, Pasadena, California ANALYSIS OF SHIP-SIDE WAVE PROFILES, WITH SPECIAL REFERENCE,..TO HULL'S SHELTERING ERFEGT. .ocseire 687 Kazuhiro Mori, Takao Inui and Hisashi Kajitani, University of Tokyo, Japan DISC UST ei ao) do. 6) ce of RENE EM MED eh eee : 745 Klaus W. ee Institut fur Schiffbau der Universitat Hamburg, Germany REPLY. TO DISGUSSION. syecoss e6 re ere wet hE eee 746 Kazuhiro Mori, University of Tokyo, Japan DISS VSor OWN pete he i lot's St Ge an s,m ey, My ne a, Ga 746 Roger Brard, Bassin d'Essais des Carénes, Paris, France REPLY ‘TO DISCUSSION: & fi sia! ft oy See eee 749 Kazuhiro Mori, University of Tokyo, Japan DISG USSiOR 4.3 a wes $l wae cht seh were wivgery yates 756 Louis Landweber, Tadeo te of Iowa, Iowa City REPRE TO DIsG Usa Ole sos « sie 1079 A.V. Gerassimov, R.Y. Pershitz, N.N. Rakhmanin, Kryloff Ship Research Institute, Leningrad, U.S.S.R. DISCUSSION 6: 83 so.) co hee ae, ion'es dies SN ees 1106 J-C. Dern, Bassin d'Essais des Carénes, Paris, France REPLY TO. DISG UssSIOW « <.« Gee wemad 6. teens LLOT N.N. Rakhmanin, Kryloff Ship eseee Institute, Leningrad, U.S.S.R. XIV OPENING CEREMONY INTRODUCTORY ADDRESS Roger BRARD Chatrman, French Organizing Committee Ladies and Gentlemen, As an introduction to the Opening Ceremony of the Ninth ONR-Symposium on Naval Hydrodynamics I would first like to say how pleased and honored we felt when Dr. Frosch accepted the invi- tation to attend the said ceremony and to speak on behalf of the United States Navy. The presence of such a high ranking official of the United States Department of Defence is a very unusual privilege. To welcome the Assistant Secretary of the Navy for Research and Development it was only befitting that the French Government representative be Mr. Jean Blancard, Délégué Minis- tériel pour 1'‘'Armement, who ranks immediately after the Ministre d'Etat chargé de la Défense Nationale. As Délégué Ministériel pour l'Armement, Mr. Jean Blancard is responsible for all the armament programs of the French Armed Forces, Army, Navy and Air Force, This includes the pro- curement of all the materials, either from the military establish- ments or from the industry, and also all the activities in Research and Development, In particular, the Direction Technique des Constructions Navales, which is in charge of designing and building the naval ships, comes under his supervision, So does the Direction des Recherches et Moyens d'Essais which deals with fundamental research and eva- luation of new scientific concepts. Mr, Jean Blancard's academic achievements comprise ranking first at both Ecole Polytechnique and Ecole Nationale Supé- rieure des Mines, Before being appointed as Délégué Ministériel pour l'Armement, Mr, Jean Blancard has held very important posi- tions either in Government agencies or Government controlled indus - tries, He has for instance been Directeur des Carburants and, as such,has for several years supervised the French Government con- trolled oil companies, He has also been deputy of the Minister of Defence for the Air Force, Recently he was president of the SNECMA, the major French jet engines manufacturer, The great interest taken by Mr. Jean Blancard in the three themes of our Symposium accounts for his accepting readily to preside over the Opening Ceremony, Thank you for your attention, OPENING ADDRESS Jean BLANCARD Délégué Mintstértel pour L'Armement Parts, France Ladies and Gentlemen, I thank Ingénieur Général Brard for having successfully explained in a few words what the 'Délégation Ministérielle pour 1'‘Armement" is, I am very happy to open the Ninth Symposium on Naval Hy- drodynamics, I would first like to emphasize two particular reasons why it gives me great pleasure : - the first one is that this Symposium be held in France, in Paris, It is traditional that the Symposia take place alternately in the United States and in a foreign country, This is the first time that the Symposium on Naval Hydrodynamics takes place in Paris, and I feel particularly glad for it, - the second reason is the number and quality of the repre- sentatives from the twenty two countries who favoured us with a posi- tive answer to our invitation, This shows the utmost importance attached to the problems of Naval Hydrodynamics, Whatever the positions I have held and that Mr. Brard just recalled, I am not at all a theoretician in Naval Hydrodynamics, From my student time, some forty years ago, I remember that the intricate equations of hydrodynamics were stretching unendingly on the blackboard, and when the professor had finished writing them down, he would add: ''The solutions are not known, so one has to resort to experiment", Looking at your technical program, I can see that a rather important part of it is devoted to what you call frontier problems, a somewhat strange term for the non specialist, This shows that there still exist numerous unsolved problems in naval hydrodynamics, and that scientists still have a wide field of research ahead of them, I would like to make two remarks : - The first one concerns the importance that what you call Ocean Engineering begins to take, For twenty five years I have been an oil prospector, and therefore I have known the beginning of off- shore prospection, The oil prospectors at first behaved like lands- men, that is that they cautiously began by erecting fixed drilling platforms to keep their feet dry. Progressively, through advances in Naval Hydrodynamics, their technique evolved, and did so with a striking rapidity if one recalls that only ten years ago the North Sea had never been crossed by a geophysical ship, In these ten years, through survey of currents and winds and study of seakeeping quali- ties of platforms, all the difficulties were overcome, and this inte- rior european sea has became one of the main sources of oil for Europe, so much so that the supply from the North Sea would com- pensate possible difficulties in the Middle East, This could be obtained thanks to Naval Hydrodynamics, - The second remark I shall make concerns the importance that you give to unconventional ships, It is a difficult problem for the leaders of our Navies, to know at any time what are the tech- niques to be applied to have better naval ships when needed, New techniques are never sufficiently called upon to adapt the existing material to the wars of to-morrow, Even if it is true - and I believe that this was said by an American Admiral - that wars are won with outdated weaponry, a certain balance has however to be kept. No doubt that the leaders of our Navies have the final say in deciding what is needed for to- morrow, but it is the aims of a Symposium such as this one, of scientists such as you are, to give them the necessary elements to define their policy, I would not like to be too long and delay your work any further, I only want to thank Dr. Frosch, Assistant Secretary of the Navy for Research and Development, and Mr. Ralph D. Cooper from Office of Naval Research, whose presence here proves the great interest taken by the American Authorities in your studies, In conclusion I wish you full success in your work as well as a pleasant stay in Paris, Thank you, Ladies and Gentlemen, for your attention, ADDRESS Honorable Robert A, FROSCH Asststant Secretary of the Navy for Research and Development Washington, U.S.A. It is a great pleasure to be here, not only because of my pleasure in assisting at this inaugural ceremony for the Ninth Sym- posium on Naval Hydrodynamics but because it is a particular pleasure for me to be able to share this platform with Mr. Blancard, with whom I have had a number of pleasant and fruitful discussions on various aspects of naval warfare and naval technology. We are very happy to have this Symposium in France not only because oftra- ditional friendships but because of my view of the importance of the French contributions to ocean engineering in general, the French pioneering work with the bathyscaph, the advanced diving experi - ments of the French Navy, of Comex, of the Cousteau group and some of the unique Government and industry relationships, such as Cnexo, for the exploitation and investigation of the oceans, All of these pioneering works make it appropriate to have such a conference here in France, It is excellent to see 12 nations contributing to a programme attended by 22, to exchange information on this particular branch of ocean engineering, which I think is significant and important to ocean engineering in general and to the basic purpose for which we all work in ocean engineering - namely, the careful use and exploitation of the oceans for the benefit of all of us. I speak here as an outsider to naval hydrodynamics and in some sense as a consumer of naval hydrodynamics as a scientific and engineering element in ocean engineering, of naval warfare matters and of contributions to general oceanographic and oceanolo- gical matters, Viewing it as an outsider one can frequently see things that are not apparent to an insider or look different to an insider and it seems to me that we have been going through a period of very im- portant change in naval hydrodynamics, Perhaps it looks more gradual to those who have been working onit, but to those of us who have not been working in detail on the subject it appears as though there is a renaissance or efflorescence of ideas, a great expansion of new things in the subject. I am thinking, for example, of what I might call the super ship, changing very rapidly from ships whose tonnage was measured in tens of thousands to ships whose tonnage is measured in hundreds of thousands, I am thinking also of the fact that the hydrofoil has come from being a curiosity to being a useful vessel not only in a naval warfare sense but, as importantly, in the sense of transpor- tation, I have been in several European cities in which the hydrofoil is beginning to be part of the bus transportation of the city or the region, and I think we shall see this more and more, As I look at my own country and particularly my own locality of Washington DC I know that we have available, already constructed by nature, a number of highways that we do not use, You have begun to use them in Europe with hydrofoils and I think this will spread all over the world so that the waterways may become the autoroutes for many places where it is possible to do this, Along with the hydrofoils we have begun to exploit the sur- face effects vehicles in the same way, first for transportation, but several navies are looking at them as possible advanced warships. As an entirely different trend of development, but also important in the development of ships for the future and growing, as far as I can tell, from the floating oil platform techniques which Mr. Blancard has mentioned and separately coming together from a very old technique, the catamaran, we have now the various versions of what might be called low water plane catamaran or the semi- submerged ship that are being looked at in several countries and will probably have a place for ship transportation somewhere inter - mediate between the monohull of the conventional type and the hydro- foil or surface effect ship, But that, of course, remains for the future and perhaps this is the first international symposium in which that kind of ship will be discussed in some detail, These transportation modes, as 1 might call them, are all obvious contributions not only to naval matters in the specific sense but to ocean matters in the general transportation sense, This is extremely important because for all the improvements in air trans- portation still most of the trade and tonnage of the world moves by sea and it appears to me that the laws of nature are such that this will go on perhaps indefinitely, at least so far as aerodynamics do not permit us to produce an aircraft which can go a long distance and carry enough fuel to get back as well as any cargo, so that all our heavy and bulky materials will certainly continue to move by sea, So the work in naval hydrodynamics as a contribution to im- proved sea transportation remains as important or more important to trade and international relationships as it has ever been, Beyond ships and ship transportation, however, we are entering an era of the use and exploitation of the oceans in which other contributions of hydrodynamic understanding wiil be impor- tant, We are beginning the expansion of our examinations of the oceans to a complete sensor and surveying system so that we may know the properties of the oceans and of the weather over the oceans all over the world. We have the beginnings of the means to do this in various types of buoys and instruments but many of the hydrody- namic problems associated with the long-term mooring and move- ment of buoys in the oceans have only begun to be broached and there will be a good deal of work ahead if we are able to make really permanent stations at sea so that we can understand the influence of the weather on the oceans, the oceans on the weather and both on the entire global environment, As we become more concerned about the degradation of the environment and the problems of pollution, some of which are upon us and some of which we can see coming rapidly, we are looking to new means for unloading the ships that I have discussed earlier and of moving them about in such a way that we can prevent catastrophes and problems from occurring, This has begun to increase interest in offshore terminals, offshore storage and the means for moving equipment from these offshore platforms to the land, These struc- tures and terminals also pose new problems in ocean engineering and new questions of hydrodynamics, of structures, of wave forces, of the movement of sediments on the bottom, These too will pose problems for this branch of the engineering profession, In addition to these problems of manmade equipment and structures we have continuing and increasing human interest in the nature of the coastline itself, in the forces that shape the coastline, either to construct it or to destroy it, and we always have the inte- rest in learning how either to control these forces and movements or to predict them and understand in what says we can live with them, These also are traditional problems of naval hydrodynamics in the broadest sense and problems that are very far from being solved. So it is clear to me that in all these areas of transportation, of terminals, of major structures at sea, of sensors to understand what happens at sea and have an understanding of the very workings of the natural forces themselves, this subject of naval hydrodyna- mics is important and full of problems that are interesting in them- selves, but also of great importance for the human interprise of keeping our lives, our civilisation and our planet together in the best possible way. For these reasons I am delighted to be here to help to open this Ninth International Symposium on Naval Hydrodynamics, UNCONVENTIONAL SHIPS Monday, August 21, 1972 Morning Session Chairman: R. Brard Bassin d'Essais des Carénes, Paris Water-Jet Propulsion for High-Speed Surface Ships Ja upon via Vasconti, Je Viele (SaOUG. RB ALE | Prance) The Forces on an Air-Cushion Vehicle Executing an Unsteady Motion. L.J. Doctors (University of New South Wales, Australia). A linearized Potential Flow Theory for the Motions of Amphibious and Non Amphibious Air Cushion Vehicles in a Seaway. T.K.S. Murthy (Portsmouth Polytechnic, U.K. ) On the Determination of Aerohydrodynamic Performance of Air Cushion Vehicles. S.D. Prokhorov, V.N. Treshchevsky, L.D. Volkov (Kryloff Research Institute, Leningrad, U.S.S.R.). Page 35 99 261 Gowunle Wh, OH, ian 4 ie . 2 *, ois ~ J ion he 5%, omg Somatte mpayt gy Vs ’ vye ~ Pete meas ey We} gay nN Ia tiie iinta. Mik Ray ere! 1. Vey exer ST OF eS raugeora, yebaoM 7 m ‘pote se? a wiceno Mv as 4 ip best A 2 mactvstado ited 4008789 Gob. siseeds'h rien wal aget | , aqit® sostive boeg?-dgilt 10? aoiaisgort 16b- $e after. .L (itnovel¥ MM pieqru Ma (ooner’d. 4H ALT 7.0.0.8) i ‘ Vs pa ' F } os paliooexd oloideV noida ol TiA na oO 89o16% ee. —=—“nenoM yiae 2olaW divo® wot) ta yin vietd) e tage, Lod a Anifertews ; ty emeoeriold of to, yroea tT woltt lan eiaF be oa i teeth { ‘wel mv? ety £2 lon netey Vv wointeru.) clhA Bi oidiiqms — 05115) to’ 0, 20 tp = slighthy higher than one ; the limitation is the ''constant power'', not the cavitat- ion limit. IV. EFFECT OF CAVITATION LIMITS UPON CERTAIN JET PROPULSION UNIT CHARACTERISTICS Certain conditions must be respected to avoid the inception of cavitation within the internal circuit of a jet propulsion unit. These conditions directly affect the efficiency which is to be expected of such units with the limit of efficiency depending, among other things, upon the forward speed of the vessel for which the propulsion unit is designed. 4.1. Z Flow jet propeller (ZFJP) As already stated, these propulsion units are fitted with a scoop and a forward intake followed by an elbow. Vaned elbows are generally used to reduce external dimensions and head losses toa minimum while avoiding cavitation. The critical cavitation condition of an elbow is expressed by the Thoma parameter : 11 Duport, Vtscontt,and Merle NPSH 2 Ve /2¢ NPSH being the net positive suction head at the elbow intake, Ve is the mean velocity in the intake section of the elbow. To the best of our knowledge, even with an extremely good vane profile, o must be at least 0,35. If this condition is to be respected then a diverging section is required at high speed so that the velocity at the elbow intake is lower than forward speed. This limit has a direct effect upon the central cross-section of the scoop and upon strut thickness. Scoop drag (friction and wave resistance) depends upon : the shape of the cross-section of the scoop and the intake pipe, the fairing and dimensions of the scoop and the strut, dimensions. In order to calculate simply the incidence of the non-cavitation condition of the elbow upon propulsion unit performance, we will suppose that the external centre section for optimised shapes of scoop and strut is proportional to the cross-section of the elbow in- take. Scoop and strut drag D may thus be expressed as: p See P Se v- zz x where K is the form factor C,. is the coefficient of drag p is water density Se is the elbow cross-section Vis forward speed Water-Jet Propulsion for High-Speed Surface Ships Moreover the gross thrust of the propulsion unit may be expres- sedas: r =e PHO (yal) & where and Vy is jet velocity. The critical cavitation condition may be used to calculate Se: Seca Oat ee a \ 2g (NPSH) where gg is the "critical" value ofa From these equations it is possible to calculate the non-dimensional parameter : 2 ie, Meo ane oy Boe Anie rie CeNESH D KOE V To This equation shows that, for a given forward speed (and thus a given NPSH) TG/D increases with w, which is evident since an increase in w for a given thrust leads to a decrease in the rate of propulsive flow. This effect though beneficial upon scoop drag, reduc- es the theoretical drive efficiency which is equal toe — 2) hws will not be expanded in this discussion since the optimisation of w also involves head losses in the circuit and the weight balance. It should also be noted that ae decreases when the forward speed of the ship increases, Moreover the non-cavitation condition of external flow and the external streamlining of the scoop will increase coefficients K and Cx. Beyond a speed of approximately 50 knot , sub-cavitational flow can not be maintained around the scoop and supercavitational conditions of external flow would lead to an increase in KCx. Sigg (SDS The non-cavitation condition for the external circuit may not, in this case, be expressed so simply as for the scoop elbow of the Zale us Duport, Vtseontt,and Merle The critical conditions for the inception of cavitation in the pump involve the pump characteristics : LW ae Cli To gee aiapes in which : Q_ is the rate of propulsive flow Sr is the cross-section of the ian eed H is the head generated by the pump H y= u’ /2g in which : u is the peripheral velocity of the impeller. Finally for the given value of Cm and wy the pump cavitation limit is expressed by the Thomas parameter : NPSH H c= A complete examination of this question, which will not be given here, reveals the following principal considerations : In propulsion units for high forward speeds : a diverging section before reaching the pump inletis, in every case, necessary : for example at speeds of 50 knots pump intake speed has had to be reduced by approximately 80% of the forward drive speed. As with the ZFJP scoop, increase in speed to slightly over 50 knots results in a slight increase of coefficient K and, beyonda certain limit speed, super-cavitating external flow is required. Finally, it may be seen quite readily that available pump a for a given w decreases as velocity increases, this may be compen- sated by: - either reducing w by discarding the optimum values result- ing from compromise between theoretical efficiency and nacelle drag, 14 Water-Jet Propulston for High-Speed Surface Ships -— or by adopting a 2 stage pump. V. HYDRODYNAMIC STUDIES OF JET PROPELLERS 5.1. Advantages and limitations of separating the study of internal and external circuits For long circuit water-jet propulsion units of the 4 flow- type, separate examination of the hydraulics of the external and internal circuits, and particularly the pump, is clearly well-founded and advantageous. The internal and external flux have to be examined conjointly only in the study of the immersed scoop. In the design studies of nacelle type propulsion units of the "straight flow'' type we considered that the same separate theoretic- al and experimental approach was also of great interest for the fol- lowing reasons : theoretical analysis of the internal flow and circuit design is much simplified if itis considered separately from the extern- al circuit. This is particularly significant when applied to the pump design. experimental approach also is very much facilitated. For example the test rig allocated for internal circuit study may be used for measuring directly all characteristics of the internal flux (rate of flow, momentum, thrust, cavitation limits, etc. ) without any interference of the external flow. High enough Reynolds number and a proper cavitation simul- ation can be obtained with a reasonably small test rig as the one described below. If equivalent limits were to be attained in a hydrodynamic tunnel then the vein size would have to be “at least 1,2 metres in diameter with a vacuum of 0,5m absolute, a flow speed of 12m/s and discharge of 14m? /s so that the facility would be considerably larger than that describ- ed below whose discharge is limited to 0,5 m/s. Moreover, in such a tunnel facility, cavitation around the nacelle would limit investigation of the cavitation limits of the internal cir- cuit. Nevertheless separate study of the two flux gives rise to certain difficulties since the internal flow is influenced by the external flow as is clearly shown, for instance in theories concerning ducted propellers. However in the case of "straight flow"’ propulsion units of the type which we have developped for rapid surface ships, this effect of nS Duport, Vtscontt,and Merle the external flow upon the internal flow is relatively low and can be approximately taken into account in the ''separate flow'' approach. For instance the transversal distribution of approach velocity into the pump may be simulated by properly adjusting the profile of the intake bell-mouth of the model. This adjustment is based upon calculation (perfect fluid and boundary layer) and upon smaller scale tests of the complete propulsion unit in a hydrodynamic tunnel. As regards the discharge nozzle, the absence of external flow in the test rig described below, slightly modifies the jet contraction compared with the prototype nozzle. Therefore we carried out model tests of the pump with various nozzle diameters and finalised the nozzle diameter adjustment during the prototype tests in the TOULOUSE high-speed towing -tank. 5.2. Methods of approach applied to the design studies of jet propel- lers The methods mentionned hereafter are the ones we applied for finalising the hydraulic design of the straight flow 50 knots jet propel- ler dealt with in § 7. However the same general way could be follow- ed for 4 flow jet propeller design, with some adaptation. a) Intake mouth of the nacelle Analysis of flow in this part of the machine requires the con- sideration of both internal and external flow. Two main methods have been used : - potential flow axisymmetric computation with a special at- tention towards the cavitation limits of the circular leading edge area, - experimental study on a hydrodynamic tunnel where the in- ternal flow was separately controlled (see fig. 3). These approaches could well be adapted to the design studies of intake scoop of a # flow jet propeller. b) Driving pump A special test rig had to be developped for that purpose as will be explainedin § 6. 16 Water-Jet Propulsion for Htgh-Speed Surface Ships c) Nacelle body and rejection nozzle — conventionnal friction resistance computations were ap- plied to predict the nacelle external friction drag. . experimental study of a complete small scale model of the propulsion unit (fig. 4) was carried out, having in view the determination of cavitation limits of the external flow and the lift coefficient of the propeller. During these tests the internal rate of flow was simulated with help of a small internal pump similar to that of the prototype but no shaft power measurement was made since the impeller Reynolds number and bearing friction torque were not proper for that purpose, d) Integral test of the entire propulsion unit Since the effect of mutual interaction of external and internal flows is of the same order of magnitude as the one of scale effects, we decided to run integral accurate tests only on the full-size prototype unit. As already mentionned this implies that the hydraulic design of some parts of the propulsion units (mainly the rejection nozzle) has to be finalised during the prototype test. This is presently carried out as explained in So Ge VI. TEST RIG FOR HIGH SPECIFIC SPEED PROPELLER-PUMP DRIVE UNITS 6.1. Description of the test-rig If centrifugal or mixed flow pumps are involved, conventional test rigs may be used to perfect the hydraulic design of pumps used in water-jet drive units. The only significant difference between these and normal pumps is the relatively higher velocity at the volute outlet. However the test rigs normally used for propeller pumps are not suitable for solving the problems associated with high specific speed propeller pumps, used in the SFJP. Firstly the head generated by these pumps is often relatively low in comparison with the kinetic head corresponding to approach velocity and to discharge velocity so that accurate measurement of the generated head is difficult. Moreover pump efficiency, as a pro- peller, depends upon both generated head and also upon the transversal Lig Duport, Vtseontt, and Merle velocity distribution at the nozzle outlet, i-e the momentum commu- nicated to the jet. For these reasons we have found it necessary to construct a specialized test-rig in which the momentum transmitted to the propulsive flux is measured directly and not solely the incre- ment of energy transmitted to the said flux. The design of this test-rig also allowed for the necessity of simulating head (captation energy) due to the forward drive speed of the ship. Thus a circula- tory pump had to be used. The simulation of cavitation conditions is achieved by the con- trol of absolute pressure on both sides of the model pump. The figures 5 and 6 show the basic arrangement of the test rig. Maximum impeller diameter is 300 mm. The model, placed between two tanks, allows visualisation of the impeller, distributor and diffuser. Water circulation is ensured by a 520 1/s pump with 9 metre head at 835 rpm driven by a DC thyristorised motor at variable speed between 0 and 2 500 rpm. A 1/3 reduction unit is mounted between the pump and the motor. Motor speed stability is controlled to within 1/100. The 500 mm diameter piping is fitted with two vaned elbows, a manually controlled 500 mm diameter valve, a 350 mm diameter vertical turbine flowmeter with calibrating piping. The 2.2. metre diameter upstream tank was specially design- ed to ensure correct feed to the pump: uniform flow distribution, absence of vortex, etc. The water level is regulated by a capacitance sensor acting upon rotation velocity of the circulation pump. The water level is Zp = 1,8 metres above the plane of the impeller. Absolute air pres- sure above the water surface may vary between 0,05 and 1,7 atmospheres, The upper tank also supports the two water-floating bearings - diameter 360 mm and 160 mm of the stator casing. Internal stiffeners avoid displacement between these bearings. The tank also supports the thrust-balance and the torque balance. The lower tank, diameter 2,5 metres, was specially design- ed to dampen jet energy without sucking unwanted air to the pump while ensuring a stable level. The facility may be operated with the nozzle either drowned or not. Absolute pressure above the water 18 Water-Jet Propulston for High-Speed Surface Shtps level may also be varied between 0,05 and 1,4 atmospheres. Water head between the two tanks is measured accurately by a dif- ferential manometer (mercury weighing), Air pressure in the upper tank is weighed in the same manner. Model rotation is ensured by an asynchronous motor, between 300 and 1 750 rpm whose speed is varied through a frequency converter. Resultant stability is to within 1/1000 of rotation speed. The whole body of the jet propeller (i.e. intake bell-mouth, pump and nozzle) is rigidly connected to a main stator structure which is vertically guided in two self-centering guide-bearings fed with water under pressure. The vertical resultant of weight and hydraulic thrust acting on this structure is measured through a weighing -balance to which it is connected through a oil pressure frictionless thrust bearing. Since the total weight of the balanced body, and pressures on both sides of each guide-bearings are known, it is possible to derive from the balance measurement the net thrust due to the jet effect. Because of the absence of friction in the guide and thrust bearings, it is also possible to measure the stator torque (which corresponds to the jet rotational momentum). The stator of the driving motor is mounted in the above- mentionned stator body with help of self-centering guide and thrust bearings fed with oil under pressure, thus allowing for accurate driving torque measurement, then accurate shaft-power. determina - tion. The friction torque of the shaft guide bearings situated in the propeller body is not directly measured but is taken into account through a precalibration procedure. A pressure sensor in the oil floating shaft thrust bearing is used to measure rotor thrust after calibration. The 360 mm diameter water floating bearing, mounted bet- ween the two tanks serves as a water seal. Its leakage rate is ca- librated and is approximately 0,151/s. The 160 mm diameter upper floating bearing serves as an air seal, Its leakage is also taken into account. This rig is equipped with centralised remote control and mea- surement equipment, The flow, head and pressure characteristics of the rig are such that it proves very useful for perfecting propeller pumps for SFJP type drive, covering the forward speed range from 0 (fixed point propellers) up to approximately 100 knots. 19 Duport, Vtscontt,and Merle 6.2. The following figures can be directly derived from the measure- ments made with a propeller of diameter D on the test rig described above : the head H applied between the external limits of the pro- pulsion unit model ; this head simulates the effect of kinetic energy due to the forward speed V of the ship. the rate of mass-flow M passing through the propeller. the component of the jet momentum ” in the axial direc- tion, the shaft power P. the rotational speed N the net positive suction head NPSH. From these figures, it is possible to determine : The equivalent ship speed: V = 2gH The gross thrust: T ae tte ry g (i.e. the gross thrust obtainable at speed V from a propulsion unit operating under the same internal hydraulic conditions as the model) Mino Gb The gross efficiency Sha = G £ z The thrust coefficient C py eerie with S_ = ae aly alld beavis R 4 > S) IRL Pe, ND The rpm coefficient u = er The cavitation coefficient ALA = ee Vv /2g The dimensionless parameters defined above can be plotted in a diagram as the one represented in fig. 7 and then applied through similarity considerations to the determination of a prototype propul- sion unit of given dimension and speed. 20 Water-Jet Propulston for Htgh-Speed Surface Shtps VII. TESTSON THE 50 KNOT SOGREAH PROPULSION UNIT - SFJP TYPE IN THE CEAT (Centre d'Essais Aéronautique de Toulouse) HIGH-SPEED TOWING TANK ' After completion of tests carried out on the internal circuit, using the test rig already described, and on the external circuit in the tunnel, we designed and constructed a 800 kw prototype unit for complete testing in the CEAT high-speed towing tank. Impeller diameter is 0,772 m for the prototype compared with 0,268 for the impeller of the model while the ratio of power consumed under cruising conditions was 800/7Kw for the two types of test. The graph 8 shows the comparison of efficiency obtained on the test rig for the scale model and in the tank for the prototype. Curve 1 represents the net efficiency obtained with the 0,772 diameter prototype. This efficiency takes account of the nacelle drag. Curve 3 is derived from curve 1 by subtracting the external drag losses of the nacelle and therefore represents the gross ef- ficiency as defined above. Curve 2 represents gross efficiency obtained with the scale model, diameter 0,268 m. Comparison of curves 2 and 3 shows that the variation of efficiency with the thrust coefficient is properly predicted from the model. The difference in efficiency between model and prototype is of order of magnitude which can be anticipated from scale effect consideration, Figure 9 shows the net thrust Tn versus V characteristics - the limiting curves are those derived from the model test results. The points represent the results obtained to date on the 0,772 m prototype with operation near to the diagram limits. The test program will be continued for even closer approach to the operating limits. It must be noted here that the drag losses are likely to be reduced if the trailing edge profile of the nozzle is mo- dified which will help to improve overall efficiency. The following characteristic parameters may be noted from both figures : rag Duport, Vtseontt,and Merle =. Me to ‘ g 4H =) (02655 tp = je 50 Vp = @.°55 VIII CONCLUSION A general method of approach has been briefly described, this applying to the hydraulic design studies of jet propellers for high speed surface ships. The validity of this method where internal and external flows are toa large extent separately considered had to be proven, at least for application to short ducted straight flow jet propellers. The tests of a 50 knot , 800Kw prototype propulsion unit in the high speed towing tank of CEAT in Toulouse has shown a good accordance between the prototype performances and the predicted characteristics which had been derived from the proposed approach method. The descrepancies are within the limits of the expected scale effect which is normally observed when dealing with reduced model approachs, We have also proposed some simple dimensionless parameters for the sake of comparison of various types of propellers applicable to rapid surface-ships. The prototype test proved the high perfor- mances of the Straight-Flow-Jet Propeller unit, and its potential interest in this field of application. Ee Water-Jet Propulston for High-Speed Surface Shtps Subcavitating Propeller Net efficiency Straight-Flow- Jet Propeller Supercavitating Propeller 4 PROPULSIVE COEFF. Dp Tur bofan 30 40 50 (SO) 748) 80 90 100 DESIGN VELOCITY — KNOTS Fig. 1 Practical propulsive coefficient comparison Cavitation limit Constant shaft power Cruise conditions Fig. 2 Definition of peak thrust ratio ZS Duport, Vtseontt,and Merle Fig. 3 Scale model used in tunnel test on flow conditions at the inlet of an SFJP propulsion unit Fig. 4 Scale model test in tunnel of SFJP propulsion unit external circuit 24 SOGREAH GRENOBLE Water-Jet Propulston for High-Speed Surface Shtps Test-rig for high specific speed propeller—pump NPSH= H+ Zp-Hy ly, Fr (@=F,- EAP) (Tg = ™M-MV) ( Pg= NTr) Fig. 5 SFJP propulsion unit test rig zZ5 Duport, Vtscontt,and Merle — LZ Space | py SS s ‘ oS i bers aK + Vl , wihG foo I.“ iy WA JN | \ = E Be Fig. 6 SFJP scale model used in tests BISON BED-H¥IIDOS 26 Water-Jet Propulston for High-Speed Surface Ships na Qc Ct Fig. 7 SFJP Rig tests results 27 Duport, Vtseontt,and Merle S. F.J.P. propulsion unit 3. NG of the internal circuit deduced i from 800 KW prototype tests (D = 0.772 m) 2. NG of the internal circuit deduced from test platform results (D = 0.268 m) 1. pn obtained from tests on the 800 KW prototype (D = 0.772 m) Cruising speed + a2 -4 -6 -8 Fig. 8 Comparison of the test results r Power limit Cavitation limit Immersion constant at axis: 1.5 D kN Test point for 800 KW Cruising speed point prototype (D = 0.772 m) 20 SOGREAH GRENOBLE to) 5 10 15 20 25 V (m/s) Fig. 9 Range of characteristics of the 800 kw prototype (D = 0.772 m) deduced from test rig results (D = 0.268 m) 28 Water-Jet Propulston for High-Speed Surface Ships DISCUSSION William B. Morgan Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. The propulsion device discussed by the authors would more appropriately be called a ducted propeller. The term water-jet is usually reserved for a propulsion device where a pump is located in a long pipe and the pump does not induce a drag around the duct. The device shown in Figure 4 is a ducted propeller of the decelerating - flow type. Since the authors do not give any references it is not clear whether they are familiar with the vast amount of literature which is available. I thought that it has been conclusively shown that it is pos- sible to consider the duct separately from the impeller and to perform tests on the inlet, Figure 3, and the interior flow, Figure 6, separa- tely. In the duct shown in Figure 4, the impeller would induce a cir- culation about the duct which, in case of the decelerating -flow, would cause a drag increase. I do not believe it is possible to separate the internal and external flow as you have done. In Figure 1, you have purportedly shown a comparison bet- ween the efficiencies of various propulsion devices. This figure was not valid when it was published nor is it valid today. Also one should consider the partially-submerged propeller. This propeller is not applicable to hydrofoil craft but can be a very efficient means of pro- pulsion where there is a flat bottom at the stern. REPLY TO DISCUSSION Jacques Duport SQGREAH Grenoble, France Thank you, Dr. Morgan for your comments, About the dis- cussion on denominations, I think we have also to find names in 29 Duport, Viscontt,and Merle French, for instance for '"'Z flow'' or ''straight flow'' propeller units. Regarding the information contained in Figure 1 we know that it may be obsolete to some extent, since we mention the date of the source (1965). This discussion might be a good opportunity to collect recent characteristics of propeller devices applicable to fast surface ships. These informations should include not only the efficiency at cruise condition but also some other parameters as the one we have proposed (or equivalent) dealing with the ''intermediate-speed" per- formance and the weight-balance of the propeller unit. We agree with Dr. Morgan that the "partially submerged propeller'' should be included in the comparison, while keeping in mind that it is suitable only when no important relative moveme nt between the ship body and the water surface can occur, The authors are well aware of the literature available on ducted propellers, even if they are not familiar with all the methods used in this field. For the sake of simplicity of experimental approach they decided to carry out a ''separate flow approach" as it has been described in the paper. They agree that this is justified only if the drag of the duct is properly taken into account, which is the case with the method used. However they recognize that the justification has not been clearly and sufficiently explained in the paper and they plan to complement the present reply to Dr. Morgan in an appendix to their paper. APPENDIX TO THE PAPER BY J.DUPORT, M. VISCONTI AND J. MERLE “Water-Jet Propulston for Htgh-Speed Surface Shtps" This appendix is complementing the reply to Dr. J. W. Morgan's discussion. The authors also take the opportunity of bringing comple- mentary information on the test of the 800 kW prototype of the 'Isére"' propeller. The authors have written that ''the hydrodynamics of intern- al flow can be to a large extent analysed separately from the external 30 Water-Jet Propulston for High-Speed Surface Ships flow analysis''. This does not mean that each flow can be considered as independent from the other ; particularly the circulation around the duct and the negative thrust on this part of the propeller are properly taken into account, The authors were probably wrong not to explain clearly enough that the so-called ''separate flow approach" is preceded by a ''complete flow approach (i.e. dealing with both flows together) which serves the purpose of determining the respective "limit conditions" (i.e. bound- ary conditions) of each flow. Before coming back to this ''complete flow approach ''which, in the case of the "Isere'! design studies, was a simplified one - we feel it advisable to emphasize : (i) that either internal or external flow is entirely determined as soon as the "limit conditions" are fixed. For the internal flow, these conditions are (see Figure 10) : the ''flow tube'' separating both flows, upstream as well as downstream of the propeller, and the internal solid boundaries ; pressure (which is equal to pw) at each remote cross- section Sl and Sj; velocity distribution in each of these sections. Each flow can then be analysed from these limit conditions (as soon as they are known) without further reference to the other part of the complete flow. (ii) predetermination of the ''complete flow'' need presetting of - the circulation around the duct (or equivalently the intern- al rate of flow) - the radial load distribution of the impeller (or equivalent- ly the normal velocity distribution in section Sj. Once these figures are preset, the profiles (and r.p.m.) of the impeller (stator and rotor) have to be designed in accordance with them ; it is enough for that to apply to the impeller design the limit conditions proper to the internal flow. (iii) in a "inviscid flow approach" as the one considered present- ly, the thrust of the propeller (including the duct) can be derived from the internal flow description alone. This is an obvious result of the momentum equation applied to the complete flow. a4 Duport, Vtscontt,and Merle iS [= 32, Water-Jet Propulston for Htgh-Speed Surface Shtps Asa result of this : - presetting of the figures mentionned in (ii) here above takes account directly of the required thrust ; - the propeller thrust can be derived from the thrust measur- ed on the "internal flow'' model as described in paragraph VI of our paper. As already mentionned the ''complete flow approach" was a simplified one which appeared to be applicable due to the reduced contraction between the nozzle outlet and the jet diameter, (Diameter ratio 95). | According to this the downstream separating flow tube was assumed to be the same as the "'free-jet'' flow surface. As indicated in the paper, the slight discrepancy might be corrected through a li- mited nozzle diameter adaptation if required. It must be noted here that the ''complete flow'' test on the prototype did not give any evidence of such effect, and no correction had to be applied. The determination of the "upstream separating surface was made through the ''potential flow approach" described in paragraph V.2.a and illustrated in Figure 11 attached to this annex. To conclude we would like to insist on the fact that the ac- cordance between the "'expected performances"! (for instance thrust versus r.p.m. etc) and the measured performances of the prototype is quite satisfactory in the whole range of parameters. This, as well as the relative simplicity of the approach, is, may be, the best justi- fication for the way we followed. The tests in Toulouse have been continued through an im- provement of the trailing edge of the nozzle (which was formerly un- duly thick) the efficiency has been raised by more than 1%, the net efficiency is then now a little more than 66,5% . 33 < evods area (2) al beantolaean’ rt Ms “| qemidt heriepet, eck Ro. geri biro32s andes -¢racacs teoxdt of) mtott bevize® el ‘Kas ‘geared Putas aru < Hqaseet a mt bedixoeob @& lobomt yy FR Saazetei” eds cobs sTaxpeg 400 10 We & zew “dosotqge well oieiqua te hel benmotiesin hinle ode 3 booubys +03, of sub ofdapilegqe od of betseqqe doidw ono "949 t.6202) Piped toad pen A cn oth 2 Wad rot aa ee nana. oe : fen eee, | nl 4 al ‘ - ‘ ‘~ esw edit woll.gerisrages. His 3b igh My-atay oF gaitnsa29 a Ia bewsnibnd eA svelte wold “Jor-go1t” af? ab otras odd ode «il a dgywesd Setoes 709 od Idgion yousgss.eih tdgiie ents eted beten od geawr Ji bow iepes ti cotiatrabs seforrelb Sonebive ys aviy toa billy Neer olert oft co teed ‘wot ee . Deatlaqa ed ot bad moitsex109 on Ln& yro8 a aéw oreiive gtitetages misortequ" edt to Bol RELA IOIEP, oft Agat gated at bedinszeoh .“dyso 1qqs Ww 1} Laiigetog™ 93 dp west .xormna wird oO Setoaesin if 9 niga a ot Deis isa ii} hit d con of 24d2 156) watt no Jelent of etl) bivett.aw s5BLINeg ot aa jez2dt .annasen? rot) “ee 208907 eto g DOIS TITS" \-apshd. st DRPSE f eqpysororg siz Toes 55) 7 OLisdg Boxasecocn~ i} Bie dgie ae tiew 2a. ‘alat 22 5ia45 $v TRY TO ny RX “al boxky pay mes rion eae ~h3eut: Seod-cri? cud te. ae aft aotqas say" fy ovsen higex +8, Vile ' bawolloi +o ses “add Cc mit cond rip he non sacl q 4 " , maneh £58 iquody bd UNS Matin Igd aes ~otoneiga ta FT ‘ni dest ost ae “in (sreurron Owes Aw) eee 2it 5 wabe gniltar? P tuck OHI Gal elt a tecn yg, LFTB AT eal. 4 DATH > 2h hat. iF PA—8d, nadl STORE BIT 4 eres Fig. ti abt THE FORCES ON AN AIR-CUSHION VEHICLE EXECUTING AN UNSTEADY MOTION Lawrence J. Doctors Untverstty of New South Wales Sydney, Australta ABSTRACT This paper treats the theoretical problem of an air-cushion vehicle (ACV) travelling over water of uniform finite or infinite depth, with an arbitrary Monon vu nemeNGv) TIS smTVodelleds ‘by “a ~~ pressure distribution applied to the free surface of an invis- cid incompressible fluid and the boundary condi - tions on the free surface are linearized. Numerical results are presented firstly for accele- rated motion from rest. In deep water, the hump condition is delayed to a higher Froude number, while in finite depth, the hump resistance is appreciably reduced - even by moderate levels of acceleration. The effect of tank walls on these results when Carrying, out model Vtestsnis next examined. “ihe side walls of the tank alter the resistance by less than ome per cent in accelerated motion, if the tank width is greater than four times the model. beam. Finally, calculations of the side force acting ona yawing ACV are presented. For super-hump speeds, thie isidentorce Pus) Ot ithe same) order as) the wa — ve resistance and favorably aids the turn. It is al- so shown that the steady-state forces are realized when the craft has travelled a distance of less than two vehicle lengths after a manoeuver. 35 Doctors I. INTRODUCTION I. 1. Background Havelock (1909, 1914 and 1926) was the first to study the theoretical problem of the wave resistance of a pressure distribu- tion. His interest lay in a desire to represent the disturbance from a ship. Asa result, the pressure distributions that he chose to ana- lyze were very smooth and were not typical of the pressure under- neath an air-cushion vehicle (ACV). However, later on, Havelock (1932) derived the general expression for a pressure distribution travelling at a constant speed of advance. In this paper, he also showed that, under the assumption of a small disturbance, the action of the pressure was equivalent to a source distribution on the undis- turbed free surface. The relation was : Swe” Pg Ox (1) where o and p are the source intensity and pressure at the same point, c is the velocity in the x direction, p is the density of the fluid and g is the acceleration due to gravity. Lunde (195la) extended the theoretical treatment to include the case of an arbitrary distribution moving over finite depth. Nume- rical calculations which are directly applicable to the ACV have been carried out by other workers. For example, Newman and Poole (1962) considered the two cases of a constant pressure acting over a rectangular area, and over an elliptical area. The most striking feature of their results is the very strong interaction between the bow and stern wave systems. The resistance curves displayed a se- ries of humps and hollows - particularly for the rectangular distri- bution (where the interaction would be greater). A hump is produced when the bow and stern wave systems are in phase and combine to give a trailing wave of a maximum height. A hollow occurs when the two wave systems are out of phase by half a wavelength so that the combined amplitude is a minimum. The interference effects are found to be stronger for large beam to length ratios, as would be expected from this argument, since the wave motion becomes more nearly two-dimensional. The humps are found to occur at Froude numbers given approximately by Bo ry N(2n - 1) fara = loa oe oe (2) 36 Forces on an A.C.V. Executing an Unsteady Motton and the hollows occur approximately when ime Nae fOr ay = late our ema (3) In the limit of infinite beam (a two-dimensional pressure band), the humps and hollows are given precisely by Eqs (2) and (3). This re- sult was found by Lamb (1932). In water of finite depth, the main, or "last", hump (n = 1) is shifted to a lower Froude number, and for sufficiently shallow water, occurs at a depth Froude number, fy ,» equal to unity (that is, at the depth critical speed). Similar calculations were given by Barratt (1965). Newman and Poole also considered the effect of a restricted waterway, such as a canal. In such a case, the wave pattern is cons- tituted from wavelets of discrete frequencies only, which can exist in the tank, whereas in laterally unrestricted water a continuous distribution of frequencies exists. As the speed is increased past the critical speed, the transverse wave component can no longer occur, and as a result there is a discontinuity in the wave resistance curve. This sudden drop in resistance is given by 2 AR 3W : (4) 2 pgBd where W is the weight of the ACV, B is the width of the tank and d is the water depth. Havelock (1922) also presented some results for a very smooth pressure distribution moving over water of finite depth. These too, showed the shift of the main hump and the increase in its magni- tude in shallower water. Havelock's curves displayed only the main hump. The secondary (n= 2) and other humps did not occur because of his choice of pressure distribution. In recent years, a number of experiments have been carried out in order to check the above-mentioned theoretical results. These have been performed in particular by Everest (1966a, 1966b, 1966c and 1967) and Hogben (1966). The fundamental question pointed out in these papers is the resolution of the total drag on the ACV into its components. Apart from the wave resistance, the forces acting on the craft are aerodynamic drag and wetting drag. Shih Doctors In these experiments, the total drag was usually measured with a dynamometer. The aerodynamic drag was then estimated from the drag coefficient on the model, and the momentum drag was obtain- ed from the mass flow into the cushion. The agreement between theory and experiment was found to be best at speeds greater than hump. At lower speeds, nonlinear and viscous effects become important and there was a large scatter in the data. To avoid the troublesome wetting drag, Everest (1966a) attempted to eliminate it using a thin polythene sheet floating on the water surface. The resistance breakdown is further discussed by Hogben (1966). The experiments only indicated the presence of the first two (n=1, 2) and possibly there (n = 3) humps. Hogben (1965) showed that this result fitted in with the idea that the maximum ratio of wave height to length is about one seventh. That is, wave breaking prevents the occurrence of the additional humps. Further experimental work by Everest, Willis and Hogben (1968 and 1969) dealt with an ACV at an arbitrary angle of yaw. This problem was also studied numerically by Murthy (1970). In the expe- riments, the wave resistance was measured directly from a wave pattern survey, using the transverse cut method. There was consi- derably less scatter in the data using this method, since the rather doubtful technique of estimating the wetting drag was eliminated. In- deed, the agreement was found to be somewhat better, particularly for the lower of the two cushion pressures tested. In an attempt to get better agreement with experiments at lower speeds, Doctors and Sharma (1970 and 1972) used a pressure distribution which essentially acted on a rectangular area but had a controlled amount of smoothing - both in the longitudinal (x axis) and in the transverse (y axis) direction. The distribution used was : pbx syd iz = Py tem {a(x + a)} - tanh {a(x - a)} x x [tant 8 Oy +b)t - tanh{B (y - »)}| : (5) where pg, is the nominal cushion pressure, and a and b are the nominal half-length and half-beam respectively. The rate of pressure fall-off at the edges is determined by the parameters a and 6. This function is shown in Fig. 1. Asa special case, a , 8B > @ is equivalent to a uniform pressure acting ona rectangular area 2ax2b. In practice, of course, the pressure at the edge of an ACV 38 Forces on an A.C.V. Executing an Unsteady Motton does fall-off at a finite rate, corresponding approximately toaa =Ba = 40. Nevertheless, it was found that only by selecting aa =fBa = 5, could the humps inthe resistance curve above the third be essentially eliminated. Thus clearly, viscosity and non-linearity are important at low speeds. Some of these calculations are reviewed in this paper. Another problem of practical interest is the wave resistance during accelerated motion, as one is frequently concerned with the ability of a heavily-laden craft to overcome the hump resistance in order to reach the cruising speed. The problem of accelerated motion for a ship has been treat- ed by Sretensky (1939), Lunde (1951b, 1953a and 1953b) and Shebalov (1966). Wehausen (1964) made numerical calculations for a particu- lar motion of a ship model starting from rest. His results consisted of asymptotic expressions valid for large values of the time after a steady speed was obtained. Djachenko (1966) derived an expression for the resistance of an arbitrary pressure distribution moving with a general accele- ration pattern in deep water. He also presented some results for a two-dimensional distribution. Doctors and Sharma found that the main effect of accelerat- ed motion on an ACV is to shift the main hump to a higher Froude number, and in finite depth to reduce the magnitude of the hump quite significantly. These results are partially presented in this paper. I.2. Present Work The basic theory for the wave resistance of a time varying pressure distribution will first be given. Then the results will be applied to the case of an ACV executing rectilinear motion in a hori- zontally unrestricted region. The work will then be extended to the case of an ACV moving along the centerline of a rectangular channel. From these calcula- tions it is possible to determine the effect of the tank walls on the wave resistance. Finally, the case of a yawing ACV will be examined. In particular, the induced side force acting on the craft will be deter- mined, so that its importance during a manoeuver may be assessed. 39 Doctors ila BASIC ILA BOmRY, II. 1. Problem Statement We represent the ACV by a pressure distribution p(x, y, t) acting on the free surface, and travelling with the speed of the craft. Two right-handed coordinate systems (reference frames) will be used, and are shown in Fig. 2. A third coordinate system that rotates with the craft during a yawing motion will be introduced later. The axis system féyz is fixed in space, and the system xyz moves with the craft, z being vertically upwards while x and & are in the direc- tion of the rectilinear motion. The relation between the coordinates is then given by (6) where s is the distance that the craft has moved. The pressure in the stationary reference frame is denoted by p*(é, y, t). The velocity potential in the stationary frame, @ (such that the velocity is its positive derivative), satisfies the Laplace equation, so that iT] str ! o—, (o) -,, 4 — jor 4 vo =< 'Q (7) The kinematic boundary condition on the free surface requires that a particle of fluid on the surface remains on it (for example, see Stoker (1957)), so that D — Dt | - Cesy tls Pa where ¢ is the elevation of the free surface. Now we have the sub- stantial derivative : D Dt 0 Ce] Q Ot sis aaa els Oz re) = ee pe at so that the exact kinematic condition becomes = = - = 0 8 +. trad, yy [ar Br (8) The linearized kinematic condition is obtained by dropping the second 40 Forees on an A.C.V. Executing an Unsteady Motton order terms, and then writing the remaining terms as a Taylor expansion about the point z=0. After dropping the higher order terms again, we obtain simply l*,| z=) vy as (9) The dynamic condition on the surface - the Bernoulli equa- tion - in the stationary frame is meee a 2 S P ws Dee ae eke apace: =f, (10) where f is an arbitrary function of time, which may be put to zero without loss of generality. Eq. (10) is now linearized to give Ss Pp aw +. esr a eS =O (11) The combined free surface condition is obtained from Eqs. (9) and (11) by eliminating ¢ 1 s lee eae *,| re) 1 io P t (12) The final boundary condition needed states that there should be no flux through the water bed : |. | Be nba (13) The solution of this set of equations can be obtained by an application of the double Fourier transform pair : II. 2. The Potential co F(w dx dy f(x, y) exp { -i(wx + uy) ' and He We dw du F(w, u) exp {i(wx - uy) } 5 ane lA’) [o"a) - CO —-co foo) [o,0) wd 20 co —-co and the Laplace transform pair : Doctors and : 6+1 00 hs ft). = _ fue} exp(qt) dt (15) 5-100 6 being a positive constant. The Fourier transform of Eq. (7) is first taken, giving ag, kaa (16) ZZ where @ is the Fourier transform of @¢ , and k* = w) ao (17) The solution of Eq. (16) subject to the transformed bed condition, Bq ’ and z y = gke tanh(kd) (20) We now take the Laplace transform of Eq. (19) : ue + Be Z {a} = -— sech(kd) [2 L4 P(w, u;t) Fs tw, ui0) The inverse Laplace transform is taken, using the convolution theo- rem on the first term : 42 Forces on an A.C.V. Executing an Unsteady Motton t b(w, u;Z; t) F SS ik = [ro u;T ) ° cos{y (t - dr 0 sin (y t) ay es P(w, u;0) We express FP by means of the Fourier transform, and then the inverse Fourier transform is taken: bop" aol VAL es ave sii Zeth= < i 5 [fos fof fare o 4m p s! 0 yoccr nes : a Seas Nei eaaied) (t= 7)} sso dee? = §) +4 uly - rb ffs | a ete A co S' See sin(V gk *tanh(kd)° t) gk *tanh(kd) exp{i(w(— - &') + uly - y'))} (21) where p = p (€',y', + ) , defined over the area S' , while &' and y' are dummy variables in the stationary reference frame. Eq. (21) is the potential for an arbitrary time-varying pressure distribution starting at t =0 . Thus the solution for the general motion of an ACV is obtained. In the following sections, we shall consider special motions of a pressure distribution which is non-time varying with respect to axes rigidly attached to the vehicle. III. RECTILINEAR MOTION IN HORIZONTALLY UNRESTRICTED WATER IlI,l. The Potential We now consider motion of a craft starting from rest at t = 0. The expression for the potential, Eq. (21), may be simpli- fied by partial integration of the five-fold integral with respect toT: 43 Doctors t fore) fore) 1 @(E Sy, 2,t)'= 5 feof +f w | du p. CE yiee 4m p 4 cosh{ k(z + a} cauaivat (t-7) cosh(kd) OE. exp {i(w(é - é') + uly - y'))} (22) The pressure distribution, Pp » as measured in the stationary reference frame is a function of time. It is related to the pressure in the moving frame, p , by the following equation : p(x, y) p (£,y,t) p(é - s(t), y) (23) III. 2. The Wave Resistance The resistance of the pressure distribution is defined as the longitudinal component of the force acting on the free surface, and is therefore given by R(t) ef p (£ ,y,t) f dé dy (24) - Substituting Eq. we obtain z s robbers .e| 0 Pa /*) dé dy The Aah eat term in this expression contributes nothing to the integral providing the pressure drops to zeroat £& =too. The result for ¢ , Eq. (22), is now used. If one expresses the pressure in terms of the moving frame by Eq. (23), then the wave resistance becomes : aafieyire [oefnforonn exp en) gains + s(t) - s(r ) + uly - y'))} The real part of the integrand is now expanded. Then it is simplified by invoking properties of even and odd functions. The final result is : 44 Forces on an A.C.V. Executing an Unsteady Motton = taf aia eae ; (p* - Q*) é Done Pg cos{ a tanh-kd)* (t -7)}» cos { w(s(t) = s(7))}, (25) where Ae [po y) re (wx + uy) dx dy (26) $ The range of the u integration in Eq. (25) may be halved for a pres- sure symmetric about the x axis. Eq. (25) is similar to that for a thin ship obtained by Lunde (1951b). His formula included an additional integral which was sim- ply proportional to the instantaneous acceleration. This extra term is zero if the singularity distribution (Eq. (1)) lies on the free surfa- ce - as for a pressure distribution. The steady-state wave resistance can be derived from Eq. (25) by allowing the velocity of the craft to be constant for a long time. If the velocity oS aaa established at a value c_ , then one obtains = ofan pe ney - ne if sin{(¥ set y+we Y- we 47 aoe As to , the oscillations in the integrand increase so that there is only a contribution from the second term, and this occurs when y-we = 0 (27) This is the relationship between the transverse and longitudinal wave numbers for free waves travelling at the speed of the craft. The analysis is simplified if we use polar coordinates : w =kcos 6 and tee en sanal a (28) where k is the circular wave number and @ is the wave direction. The limit process is carried out for a similar case by Havelock 45 Doctors (1958), and the final result is - 6, nf2 3 Sipe 1 a k cos 6 27pg 1 - kd. sec? @.sech@(kd) ° -W2 6, 2 {p" (k cos 6,k sind) + Q° (k cos #,k sin @)} dé (29) in which 2 ke =pie Je (30) and k is the non-zero solution of Eq. (27), that is, of 2 k= kK sec 0° tanh(kd) = 0 (31) The lower limit for @ is takenas 6, , the smallest positive value of @ satisfying Eq. (31) fora real k . Itis given by: 6, =. for kod >1l (subcritical speed) arccosV kod for kd <1 (supercritical speed) (32) D> iT] III. 3. Results Some results previously published (Figs. 3 to 7) are now presented to show some of the effects of the choice of pressure dis- tribution given by Eq. (5). For this choice, it was shown that 7° sin(aw) n* sin(bu) P(wiu) = Po Gesinh(w/2a) "> sinh( ru/2B) and Q(w,u) = 0 (33) while the weight supported by the pressure is just Ww = 4 Py ab (34) For convenience the wave resistance is expressed as a dimensionless coefficient : 46 Forces on an A.C.V. Executing an Unsteady Motton Ey eR RET are! BEAN (35) Fig. 3a shows the wave resistance of a distribution with a beam to length ratio of 1/2. The variable used for the abscissa is ee 1/2 F° . This has the effect of expanding the horizontal scale at low Froude numbers. Curve 1, with ga = ga = 0 , displays the unrealistic low-speed oscillations which are characteristic of the sharp-edged distribution and were referred to previously. It is seen, that with increasing degrees of smoothing (smaller values of @a and Ba), the low-speed humps and hollows may be eliminated. The case with aa = Ba =5 results in only about three humps, more in keep- ing with experiments. Fig. 3b presents results for finite depth water for three different distributions. The chief difference now is that the main hump is shifted to the right and occurs near the critical depth Froude number. It is seen that Curve 2 has smoothing applied only at the bow and stern -equivalent to a sidewall ACV. The result is similar to the case for smoothing all around, showing that the wave pattern is essentially produced by the fore and aft portions of the cushion and not the sides. The resistance in the region of the main hump is hardly affected by the smoothing. The result of varying the depth of water is displayed in Fig. 4. The peak resistance increases in magnitude as the depth decrea- ses, and occurs in each case at a depth Froude number slightly less than unity. The location of the other humps is also affected, but to a lesser degree. Beam to length ratio is varied in Fig. 5. The general effect of increasing the beam is to increase the maxima and to decrease the minima in the wave resistance curve. This is due to the transverse waves assuming greater importance as the two-dimensional case is approached. A secondary effect is a shift in the locations of the oscil- lations to the right, so that in the limit of infinite beam, they occur at Froude numbers given precisely by Eqs (2) and (3). We now turn to the effect of constant levels of acceleration of the craft from rest. Fig. 6a applies to a smooth (aa =5) two- dimensional pressure band moving over deep water. A general dis- placement of the oscillations to higher Froude numbers occurs. This shift is greater for the higher acceleration. In addition, most of the low-speed oscillations apparent in steady-state motion do not occur in accelerated motion. The resistance of a smooth band over finite depth water is shown in Fig. 6b. Here the reduction of the peaks is 47 Doctors even more dramatic than in deep water. More striking, however, is that for this and for all other two-dimensional cases studied, the wave resistance becomes negative beyond a certain velocity in finite depth. The resistance then asymptotically approaches zero. (The ordinate in this figure is plotted on an arsinh scale. ) The depth Froude number at which the negative peak resistance occurs in shal- low water has been found to be Brio ava dazddl anes (36) d The resistance of an accelerating three-dimensional pressu- re distribution is shown in Fig. 7a (deep water) and Fig. 7b (finite depth). In deep water, the wave resistance shows similar, but less marked, effects due to acceleration as does the corresponding two- dimensional case (Fig. 6a). In finite depth, there is again a strong reduction in the main peak as well as an elimination of nearly all the low-speed oscillations. However, there is no region of negative wave resistance -thus indicating the influence of the diverging wave pattern. IV. RECTILINEAR MOTION IN A TANK IV.1. The Potential We now consider the problem of an ACV moving along the centerline of a rectangular tank of length L and width B . The initial distance at t = 0 between the starting end of the tank and the coordinate axes xyz fixed to the model is takenas o . This pro- blem is crucial to the testing of models, as one must know the effect of tank walls. For instance, during steady motion in an infinitely long tank, Newman and Poole showed that the effect of tank width in the neighborhood of unit depth Froude number to be importance (see Eq: (4)). We utilize Eq (22) for the potential in a horizontally unbound- ed region, and satisfy the additional condition of no flux through the four tank walls, by employing a system of image ACVs as shown in Fig. 8. We consider first only the array of distributions on the tank centerline, and later on apply the boundary condition on the tank sidewalls. The potentials for the individual distributions, ¢” » are related to the primary potential, @ , as follows: g (g py, zytp? = sg (>€= mL, y; z; t) for n even, 48 Forces on an A.C.V. Executing an Unsteady Motton an Geet Calas tals I eon zt) tor mi odd. We add ¢ to f , d” to g?) , o? to g?) ,» and so on. This only alters the exponential factor in Eq. (22), which now beco- mes: factore= "2 exp{i(w/(- o -&')+ uly - y'))} 3 cos{w( ao+é iS exp(2inwL) n=— 2° The integral with respect to w of this factor in Eq. (22) can be simplified using the Poisson so ial a Peeemla to give dll & 45a Au eS raaffe fo 8 fst Cee yaa) acai K (2 + a} sin{¥(t - 7 )} eee al) i 5 cos{wi(o + £)} n n . exp {i(w (-o -§') + uly - y'))} ’ where = Ik wa) an / 2 Z 2 k = uy +u n n and Yin = gk - tanh(k d) (37) n n We now satisfy the condition on the side walls of the tank by including the image ACVs on lines parallel to the tank centerline. The procedure is similar to that just carried out, and if we assume that the pressure distribution is Speer about the x axis, then = ' € p> t ' o( &,y,z,t) = ffir f = en m P (ey 7)... sare! K an cc + d)} sinjy _(t = 2 )} Se dena te leigbne meow, Shyr | * exp ji(w (-o - &’) (y-y'))} (38) where 29 Doetors a ie p2yem/B yy (39) m k ¢ = w F + @ 5 (40) mn n 9 as et ea Cn a (41) Y man PNG ay : and ers f2), ec=d for n>0O fe) n IV.2. The Wave Resistance The method of obtaining the wave drag is the same as in the previous section and utilizes Eqs (24) and (38). After some algebra, one obtains : R “pene | ) dnd ty 2. yn w ° cos} Vek tanh(k 4) *(t t a) je 0 | {cos(w_ (s(t) - s(7))) - cos(w (s(t) + s(7) +2 a ))} + $ eer {cos(w (s(t) - s(7))) + cos(w (s(t) + s(r)+20)) + widicdisae eas sin(w (s(t) + s(r) +20 » | ' (42) where a A =e i PA, Fag and Q - Qibwo head) n m It is clear that the fluid motion in the tank consists only of wavelets whose wavenumbers are given by Eqs (37) and (39), and that in the limit of L—e and Bo , the result for a longitudinally and laterally unbounded region is recovered. The terms containing ¢ are due to reflections off the starting end of the tank, andas go +o they contribute nothing to the wave resistance. 50 Forces on an A.C.V. Executing an Unsteady Motton The wave resistance for steady motion in an endless tank may be obtained from Eq. (29) by setting up a laterally disposed array of images. The result, derived by Newman and Poole, in the present notation, is 2 Ble ty k ‘tanh(k d) {P ~+Q wind BS m m m m Pe ai 0 aint Seah tanh d)p=1k le ds Beene d) m .e) m m o m in which u_, is given by Eq. (39) and sth OD The circular wave number, k » 1s the solution of m 2 k anki ik itank(kewd)it=" a (44) m m oO m m (The value of ky, When m=0 is distinct from, and generally not equal to, k, , the fundamental wave number. ) IV.3. Results The wave resistance of a smoothed rectangular distribution moving in a tank is shown in Fig. 9. In deep water (Fig. 9a), itis seen that the effect of the walls is small for B/a=2 . For B/a> 4 (tank width greater than four times model width), the resistance coef- ficient differs from the infinite width value by less than 0.01. It may be pointed out here that for the special case of B/a =P leew tnatalse. ithe tank width equal to the nominal beam of the model, the pressure carries approximately 7% of the weight of the ACV beyond the tank walls. However, it can be shown that this case is mathematically equivalent to a two-dimensional pressure band spanning the width of the channel. In finite depth (Fig. 9b) the influence of the tank walls in the region of unit depth Froude number is considerably greater, as was shown by Newman and Poole. The drop in wave resistance (Eq. (4)) at the critical speed does not depend on smoothing. Even when B/a = 64 , so that the tank width is sixty-four times the model beam there is a discontinuity in resistance coefficient of 0.188 . Thus steady-state experiments in this speed range are difficult. 5t Doetors The effect of side walls of an endless tank on the wave resis- tance of an accelerating ACV is displayed in Fig. 10. Two different levels of acceleration in both deep water and finite depth were calcul. ated. In all cases the wave resistance is a smooth function of the tank width. For the low-speed range, increasing tank width generally decreases the wave resistance. On the other hand, this trend is re- versed for high speeds (greater than the hump speed). The case of infinite tank width is not plotted, in order to avoid confusion with the case of B/a =4 , with which it is almost identical. This difference in wave resistance coefficient for the cases calculated is less than 0.01 , so that one might consider that a tank width equal to four times the model beam to be essentially infinite. Even in finite depth there is no sudden change in resistance as the model accelerates through the critical depth Froude number. (A depth Froude number of unity is passed when tyg/a = 14.14 if c/g = 0.05 , and when tg/a = 7.07 if s¢/g)= 001 4.) This sharply contrasts the case of steady motion, in which the drop or discontinui- ty in wave resistance coefficient when d/a = 0.5 and B/a =4 is 3.0 | The effect of the tank end walls was found to be slightly greater in finite depth, and thus only the former is shown, in Fig. 11. The case of an infinitely wide tank is presented in Fig. lla for a/a = 1,2 and «”~.°In the region near t= 0, thére is a slight in= crease in the resistance when o/a=1 only. Incidentally, when g/a=1 , part of the pressure ''extends'' beyond the starting end wall, so one must expect some interference. When a/a ="2 bile clearance from the starting end wall is half a craft length and there is no noticeable interference. The two curves for the finite values of o were calculated for a tank length L/a = 20 . There is no perceptible effect from the far end walluntil the model ''passes'' through its image - as indicated by one or two oscillations in the curves near tyle/a = 20. The case of B/a=1 (that is, a two-dimensional pressure band) is shown in Fig. 1lb. For the case of no nominal separation of the craft from the starting end wall at t=0 , there is now a slight- ly greater effect on the wave resistance. 52 Forees on an A.C.V. Executing an Unsteady Motton V. FORCES ON A YAWING ACV V.1l. The Potential We now consider the special case of an ACV travelling for a long time in the longitudinal or x direction. The craft is either fixed in a steady yaw position, or it starts a yawing motion after initial transients have died away. We may therefore use Eq. (21) for the potential, and drop the second term which will approach zero as t— oo V.2. The Forces The wave resistance is defined by Eq. (24), and the side for- si)= [fr (ey be dg dy (45) S Thus the side force is the positive force to port (the y direction) required to hold the craft ona straight course. ce by The analysis for the two forces now continues, as in the case for rectilinear unyawed motion in horizontally unrestricted wa- Homuunc tOnees ane : see ook Jp p(x, y,t vaso y's7) a foe! f at aye eo * exp {i(w (x - x! + s(t) - s(7) + uly - y'))}. And after some simplification: dr] dw day (4 47 esi (t -7)t° 7 Oe ag { oe iy (46) oP. .cos{ 7))} +(PP'+QQ') sin{w/(s(t) - s( ))}| ’ in which IPs JE), Wig tell A OE-TOR Gy ny bee Tee IE) (N75 wis )) 59 Doetors and Qr=QO (w,u,7) It is convenient to calculate the P and Q functions using an axis system x y z that rotates with the craft rather than the xyz system, in which the x axis lies in the direction of motion. This is illustrated in Fig. 12. The yaw angle e(t) is taken positive for clock- wise rotation of the craft, when looking down on it. If w* and u* are the induced wavenumbers relative to these craft axes, then w" (t) = w cos{e (t)}- u sin {e (t)} =k cos {9 + € (t)} and u* (t) = w sin fe (t)} + u cos{e(t)} =k sin{ @ + e (t)} : (47) Then it may be shown that Pw, ust) | / BF (a ay” ) SO wt xt + ut yt) at ayy AR) S analogous to Eq. (26). For the pressure distribution given by Eq. (5), it immediately follows from Eq. (33) that ad msin(aw” ) m+sin(bu") P(w t,t) = Po a*sinh(tw/2a) B*sinh(ru*/2B and Olw; ua, t) = (49) We now consider a craft travelling at a constant velocity at a fixed angle of yaw from time -T to O , and then allowed to yaw up totime t . The +r integral in Eq. (46) for just the first phase of the motion is 0 ref sinfo sat) (QP - PQ') cos {we(t -7)}+ ay oT + (PP' + QQ') sin {we(t -T dt | dr 54 Forces on an A.C.V. Executing an Unsteady Motton {Q(w, u, t) P(w,u, -0) - P(w, u, t) Q(w,u, -0)}° Si |= (y+we)t} ‘ cos{(yt+we)(t+T)}, cos{y-we)t} pesmi lace) ytwe ytwe Y-we Y-we as ; +{P(w, u, t) P(w, u, -0) + Q(w, u, t) Q(w, u, HO). sin{(y+wc)t} sin{ (y+we)(t+T)} sin{(y-wc)t} Fi sulle) ; Ytwe Ytwe y-wce Y-we We consider first the case when t=0 and To (that is, a steady state). The four terms containing the cosine factors, and the first and third sine factors are zero. The fourth sine term is the only one that gives a non-zero result in the wu integral of Eq. (46) as T—oo . The steady-state forces may be obtained in the same manner as the limit of Eq. (25) for large time : ale cos sist Re "ie 7 Ree ciel ONS Enea nate ein Seen SCRE Tear EELS WEA 2 ee 1 - k d-sec ao seehe(ed) - 1/2 2 } Z, ; -{P" (k cos 0, k sin #)+Q = (kcos 0, k sin @)} d§, (50) where k , ko and 46, are given by Eqs (30), (31) and (32). If we now assume that the ACV starts yawing at t=0 , then as T— o , the second and fourth cosine terms, and the second sine term contribute nothing to the wu integral in Eq. (46). The expres- sion for the forces after t =0 becomes : - 6 2 J wf ke emake A | + | ub cba fa Sa deal eal eg 5 s2rpe 2 Z 3 _ a2 é 1 - kod sec @ + sech (kd) *{P(w, u, t) P(w,u, -0) + Q(w, u, t) Q(w, u, -0)} the tune 1s (cont'd over) 55 Doctors ytwe y-wce + Q(w,u, t) Q(w, u, -0y}| Sede | aa i: w , pakke dr dw du (ere sin{ y(t- 7 )te 2m pg 0 0 5? -[foww, u, t) P(w,u, 7) - P(w,u, t) Q(w,u,r)} ecos}w(s(t) -s(r))} +... oe {P(w,u, t) P(w,u,7 ) + Q(w, u, t) Q(w,u,7)} *sin{w(s(t) - s(r))}]+ Veosu Results The (steady-state) wave resistance of a yawed ACV is shown in Fig. 13. Fig. 13a indicates the marked effect of smoothing the pressure fall-off on a rectangular cushion, for a Froude number of unity. This is accentuated for yaw angles in the neighborhood of 10° and 85° . The peaks would seem to be caused by interference bet- ween short wavelets - as short wave components are not produced by a smoothed distribution. The slopes of the curves are zero at yaw angles of 0° and 90° - as required by symmetry. The variation of wave resistance of a smoothed distribution with yaw angle for a series of different Froude numbers is displayed in Fig. 13b. At super-hump speeds, yawing the vehicle increases the effective Froude number so that the resistance drops a little. On the other hand, yawing at a sub-hump speed (for example, F = 0.4) can bring the craft onto the hump (at constant speed of advance), and thereby increase the resistance. The wave-induced side force is shown in Fig. 14. It is non- dimensionalized in the same manner as the wave resistance in Eq. (35) 56 Forces on an A.C.V. Executing an Unsteady Motton The effect of smoothing on side force (Fig. 14a) is seen to be even more vivid than on resistance (Fig. 13a). Increase in sharp- ness has a considerable effect on the side force for very small, or for very large, yaw angles - even at this relatively high speed. At the same Froude number, the effect of sharpness on unyawed wave resistance (Fig. 3a) was considerably less. The linear theory pre- dicts a peak dimensionless side force of 2.63 in contrast to a dimen- sionless wave resistance of 0.73 at zero yaw angle. It seems that nonlinear and viscous effects would preclude the development of such faemerside tOrces ih practice: Different Froude numbers are considered in Fig. 14b. The side force (for aa = Ba = 5) is seen to be positive for super-hump speeds, and therefore favorable during a coordinated turn. It reaches a maximum ata yaw angle of about 30° . Thus there is an optimum sideslip angle for generating the maximum side force. For sub-hump speeds, there is a range of yaw angle in which the side force is nega- tive. Unsteady yawing motion is now considered. The side force for different rates of constant rotational speed after travelling at zero yaw angle for a long time is presented in Fig. 15. The abscissa is the yaw angle, and is proportional to the time after the initiation of the manoeuver. The general effect of increasing the yaw rate is to decrease the available side force. However, as typical average yaw EAbes abe in tie wicinity,ot 5) pet Unit time, datas clear that the un— steady influence is of secondary importance. The side force qualita- tively follows the same trends at the two speeds considered, namely WD = Ono Ghia lSey) ayers sige aihs Om Gastg el Sis))p Finally, in Fig. 16, a manoeuver is studied, in which the yaw angle is instantaneously changed from zeroto 5°, 10°, 15” and 20°. The distance the ACV must travel before the steady-state side force is achieved is slightly greater for larger manoeuvers. Nevertheless, this effect is small. Almost the full steady-state side force is generated after the vehicle has moved one craft length at Ee Ooch tonelon\sand atten lo25 crac lengths at. & — I 0) (big. 16b). A favorable side force is developed immediately after this sudden yaw manoeuver,and then increases slowly at first. It may be shown that for a small jump in yaw angle, the initially generated side force is just one half of the final steady-state side force. This feature is evident in the curves, particularly for the smaller ma- noeuvers. Sil Doetors VI. CONCLUDING REMARKS VI. 1. Present Work Turning firstly to the case of Rectilinear Motion in a Tank, it is clear that the problem of interference from the side walls during accelerated motion in finite depth water is considerably less than that during steady motion. Model tests under such unsteady conditions would be much easier to perform as a tank width equal to four times the model beam essentially simulates the laterally unrestricted case. With regard to the yawing ACV, the great dependence of side force at super-hump speeds on smoothing was an unexpected result. So much so, that it would be unrealistic to model the pressure under the craft with a sharp distribution. Even assuming practical values of aa = Ba = 40 (which has a negligible effect on unyawed wave resis- tance) reduces the maximum predicted induced side force by almost one half. A study of the expression for the steady-state forces, Eq. (50), reveals that this difference is due to the high frequency oscilla- tions in the integral for @ justless than 7/2 . The effect is worst for a yawed sharp distribution when the oscillations decay very slowly and is further emphasized in the integral for side force which con- tains a sin @ factor, rather, than the integral for wave resistance which contains a cos @ factor. A particularly large number of sub- divisions in the integration is therefore required under these condi- tions. This probably explains the small discrepancies found at small non-zero yaw angles and yaw angles just below 90” , when attempt- ing to verify the theoretical wave resistance calculated by Murthy (1970) and Everest (1969). In practice, these high frequency wavelets probably break due to excessive theoretical steepness, and other practical effects such as cushion air flow. The induced side force has nevertheless been found to be significant, being of similar magnitude to the wave resistance. It clearly plays a role in the control of ACVs. This force has been experienced by drivers of air-cushion vehicles, who usually refer to it as ''keel effect"'. During a typical manoeuver, it has been found that the induc- ed side force is almost equal to the steady-state value at the same instantaneous yaw angle. 58 Forees on an A.C.V. Executing an Unsteady Motton VI.2. Future Work It would be interesting to verify some of the above-mention- ed theoretical results by experiment. In particular, one would like to know how accurately the induced side force is predicted - or what the equivalent smoothing would be. Such an experiment would have to take into account aerodynamic and momentum side forces as well as skirt contact, which might be significant. Numerical work can be extended in various areas. Further test cases, including the effect of finite depth might be examined. Incidentally, many manoeuvers are carried out in finite depth near the terminals. This aspect is therefore important. Possibilities for theoretical work include an investigation into the yawing moment acting on the vehicle about the vertical axis. Some experiments by Everest indicated that the craft is generally stable in yaw. ACKNOWLEDGEMENTS The writer is grateful to the Office of Naval Research, Washington for their support of part of this work under Contract No. N00014-67-A-0181-0018 Task No. NR 062-420, which was carried out during 1969 and 1970 in the Department of Naval Architecture and Marine Engineering at the University of Michigan in Ann Arbor, Michigan. This work is briefly covered in the section on Rectilinear Motion in Horizontally Unrestricted Water. For a more detailed account, the reader is referred to Doctors and Sharma (1970 and 1972). The Section on Rectilinear Motion in a Tank represents some calculations performed for research supported by the Australian Research Grants Committee during 1972. The writer also wishes to acknowledge valuable suggestions pertaining to this paper made by Professor P. T. Fink, Dean of the Faculty of Engineering at the University of New South Wales in Sydney. 59 Doctors NOMENCLATURE a half length of craft A 1/2F" b half beam of craft B width of tank e velocity of craft d water depth F Froude number = c/ 2ga Fa depth Froude number = c/ Eg g acceleration due to gravity k circular wave number = w t+ a kK fundamental wave number = Vice L length of tank n, m_ indices for longitudinal and transverse wavenumbers in a tank p cushion pressure measured in the moving reference frame xyZ p- cushion pressure measured in the stationary frame yz Be nominal cushion pressure P, Q- functions defined by Eq. (26) or (48) R wave resistance Ro ‘wave resistance.coefficient defined by Eq. (35) Ss distance travelled by craft iS) induced side force, or area of pressure distribution 60 Forees on an A.C.V. Executing an Unsteady Motton S. side force coefficient, analogous to R. iE time w, u induced longitudinal and transverse wavenumbers W weight of craft x,y,z coordinate system travelling with craft, but not rotating with it a longitudinal cushion pressure fall-off parameter B transverse cushion pressure fall-off parameter y V gk’ tanh (kd) € yaw angle of vehicle relative to x axis (see Eq. (47)) i? free surface elevation 6 wave direction with respect to the x axis & longitudinal coordinate in the stationary reference frame p water density o initial position of model in the tank T dummy time variable ce) velocity potential in the stationary reference frame, such that the velocity is its positive derivative Superscripts variable reffered to axis system moving and rotating with craft dummy variable time differentiation 61 Doctors REFERENCES Barratt, M.J. : ''The Wave Drag of a Hovercraft", J. Fluid Mecha- nics, 22, Part 1 pp 39-47 (1965) Djachenko, V.K. : ''The Wave Resistance of a Surface Pressure Distribution in Unsteady Motion", Proc. Leningrad Ship- building Inst. (Hydrodynamics and Theory of Ships Division) English Translation : Dept. Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan, Report 44, 12 pp (1966) Doctors, L.J. and Sharma, S.D. : ''The Wave Resistance of an Air-Cushion Vehicle in Accelerated Motion", Dept. Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan, Report 99, 104 pp + 92 figs (1970) Doctors, L.J. and Sharma, S.D. : ''The Wave Resistance of an Air Cushion Vehicle in Steady and Accelerated Motion", J. Ship Research, 16,4, pp. 248-260 (1972). Everest, J.T. : ''The Calm Water Performance of a Rectangular Hovercraft", National Physical Laboratory (Ship Division), Report 72, 12 pp + 29 figs (1966) Everest, J.T : 'Shallow Water Wave Drag of a Rectangular Hover- craft", Ibid., Report 79, 8 pp + 19 figs (1966) Everest, J.T. : ''Measurements of the Wave Pattern Resistance of a Rectangular Hovercraft", Ibid., Tech. Mem. 147 (1966) Everest, J.T. and Hogben, N : ''Research on Hovercraft over Calm Water", Trans. Royal Inst. Naval Architects, 109, pp 3ll- 326 (1967) Everest, J.T. and Willis, R.C. : 'Experiments on the Skirted Hovercraft Running at Angles of Yaw with Special Attention to Wave Drag", National Physical Laboratory (Ship Division) Report 119, 8pp + 15 figs (1968) Everest, J.T. and Hogben, N. : ''A theoretical and Experimental Study of the Wavemaking of Hovercraft of Arbitrary Plan- form and Angle of Yaw'', Trans. Royal Inst. Naval Archi- tects, 111, pp 343-365 (1969) Havelock, T.H. : ''The Wave-Making Resistance of Ships : A Theo- retical and Practical Analysis", Proc. Royal Soc. London, Series A, 82, pp 276-300 (1909) 62 Forces on an A.C.V. Executing an Unsteady Motton Havelock, T.H. : "Ship Resistance : The Wave-Making Properties of Certain Travelling Pressure Disturbances',Ibid., 89, pp 489-499 (1914) Havelock, T.H. : ''The Effect of Shallow Water on Wave Resistance", Ibid., 100, pp 499-505 (1922) Havelock, T.H. : "Some Aspects of the Theory of Ship Waves and Wave Resistance'', Trans. North-East Coast Inst. Engineers and Shipbuilders, 42, pp 71-83 (1926) Havelock, T.H. : 'The Theory of Wave Resistance'', Proc. Royal Soc. London, Series A, 138, pp 339-348 (1932) Havelock, T.H. : ''The Effect of Speed of Advance upon the Damping of Heave and Pitch", Trans. Royal Inst. Naval Architects, 100, pp 131-135 (1958) Hogben, N. : ''Wave Resistance of Steep Two-Dimensional Waves", National Physical Laboratory (Ship Division), Report 55, 9 pp + 5 figs (1965) Hogben, N. : "An Investigation of Hovercraft Wavemaking", J. Royal Aeronautical Society, 70, pp 321-329 (1966) Lamb, H. : Hydrodynamics, New York, Dover Pubs., 738 pp (1945). Originally : Cambridge, Cambridge University Press (1932) Lunde, J.K. : 'On the Linearized Theory of Wave Resistance fora Pressure Distribution Moving at Constant Speed of Advance on the Surface of Deep or Shallow Water", Skipsmodelltan- ken, Norges Tekniske Hggskole, Trondheim, Medd. 8, 48pp, in English (1951) Lunde, J.K. : ''On the Linearized Theory of Wave Resistance for Displacement Ships in Steady and Accelerated Motion", Trans. Soc. Naval Architects and Marine Engineers, 59, pp 25-85 (1951) Lunde, J.K. : 'The Linearized Theory of Wave Resistance and its Application to Ship-Shaped Bodies in Motion on the Surface of a Deep, Previously Undisturbed Fluid", Skipsmodelltan- ken, Norges Tekniske H¢gskole, Trondheim, Medellelse 23 (1953). Translation : Soc. Naval Architects and Marine Engineers, Tech. and Research Bulletin 1 - 8, 70pp (1957) Lunde, J.K. : 'A Note on the Linearized Theory of Wave Resistan- ce for Accelerated Motion'', Skipsmodelltanken, Norges Tekniske H¢ggskole, Trondheim, Medellelse 27, 14 pp (1g53)) 63 Doctors Murthy, T.K.5. : "The Wave: Retstance, of a Drifting Hovercra. Hovering Craft and Hydrofoil, 9, pp 20-24 (1970) Newman, J.N. and Poole, F.A.P. : ''The Wave Resistance ofa Moving Pressure Distribution in a Canal", Schiffstechnik, 9, pp 21-26, in English (1962) Shebalov, A.N. : ''Theory of Ship Wave Resistance for Unsteady Motion in Still Water"', Proc. Leningrad Shipbuilding Inst. (Hydromechanics and Theory of Ships Division). English Translation : Dept. Naval Architecture and Marine Engineer- ing, University of Michigan, Ann Arbor, Michigan, Report 67, 14 pp (1966) Sretensky, L.N. : 'On the Theory of Wave Resistance'' Trudy Tsentral. Aero-Gidrodinam. Inst., 348, 28 pp, in Russian (1939) Stoker, J.J. : ''Water Waves", 4 of Pure and Applied Mathematics, Interscience Publishers Inc. , New York, 567 pp (1957) Wehausen, J. V. : 'Effect of the Initial Acceleration upon the Wave Resistance ofShip Models", J. Ship Research, 7,3, pp 38- 50 (1964) He Hot 64 Forees on an A.C.V. Executing an Unsteady Motton ESO h PiGURES Mowe tescsune Distribution UW sed 2. The Two Coordinate Systems 3. Wave Resistance for Different Amounts of Smoothing (a) Deep Water f (b) Finite Depth 4. Wave Resistance for Different Depths 5. Wave Resistance for Different Beam to Length Ratios 6. Unsteady Two-Dimensional Wave Resistance (a) Deep Water (b) Finite Depth 7. Unsteady Three-Dimensional Wave Resistance (a) Deep Water (b) Finite Depth 8. Image System Used to Represent Tank Walls 9. Wave Resistance in an Endless Tank (a) Deep Water (b) Finite Depth 10. Unsteady Wave Resistance for Different Widths of an Endless Tank (a) Deep Water, c/g = 0.05 (b)) Deep Water,, c/g = 0.1 (c) Finite Depth, c/g = 0-05 (a) hinite Depths c/s = 0. 1 11. Unsteady Wave Resistance for Different Locations of Tank Ends in Finite Depth (a) Infinitely Wide Tank (b) Two-Dimensional Pressure Band 12. Axis System Fixed to Craft 13. Wave Resistance in Deep Water while Yawed (a) For Different Amounts of Smoothing (b) For Different Froude Numbers 14, Side Force in Deep Water while Yawed (a) For Different Amounts of Smoothing (b) For Different Froude Numbers 15, Unsteady Side Force in Deep Water while Yawing (a) = 1056 (5) Sl © 65 OR ey. RSD WB 9.18 inte GORA 2S 2% DHhwI oA) ye Giada 10 We alan’ ne ino > han eee . ; » age spi No 16. Unsteady Side se in Deep Water isla a step i Angle (a) F (b) F 66 | = ‘ 1.28 ven one sore ”) Pe * % perh ae 4 o + nt { ? Jit ‘ rt re Forces on an A.C.V.~ Executing an Unsteady Motton pesn uoT NQIA}SsSIp ernsserq “{T 24nst [{ (q-A) d}uuez - { (+4) d }yUes]e[ { (P-x) O}yUeZ — { (e+x) 0} Yue} ] e 0) q@LZ_ (K'xyd To ge eP— 67 Doctors suroysAs 21 eUTp1009 om} OU, ‘2 BINS 3 (a7 APS) 2 (q/A‘x)d 68 rayem doop (e) ‘sutyJoouIs jo syuNOUe JUSTOJJIP TOF QOURISTSOAT OAeCM “FS GANSTA 24¢/T Forces on an A.C.V. Executing an Unsteady Motton 69 Doctors 9T vT eT yydep eytutz (q) 242/T OT ( *3u09) °€ o1n3dTy 70 Forces on an A.C.V. Executing an Unsteady Motton SUjJdep JUSTOFJIp TOF 9OURISTSOI OAC M 24Z/T 9 S Vv € ‘ph earnest a? Doctors SOMeL YISUST OJ ULeSq JUSTOTJIP TOF OOURISTSeT OAC M zdc/T °G oanstz 2 Forces on an A.C.V. Executing an Unsteady Motton royem doop (ke ) ‘20ULISTSOI SACM TCUOTSUSUITp-oMm} Apeojisup 242/T ‘9 eandtgy 73 Doctors yydep eqturz (q) 242/T ( "yU0D ) ‘9 «ernst yz 74 Forces on an A.C.V. Executing an Unsteady Motton rojyem doop ( e) ‘QOURISTSOI SAM [RUOTSUSUTTp-9914} Apeojsuy aaa ‘2 oansty tS Doctors yidep aztury (q) z4c/T ("3u09) "2. eansty 76 Forces on an A.C.V. Executing an Unsteady Motton ae Image system used to represent tank walls Figure 8. Doctors rozem doop (te) ‘yue] sso[pue ue UT DOUeISTSOT CACM 24z/T *'6 ernst 78 Forces on an A.C.V. Executing an Unsteady Motton yydep e3tutz (q) 24Z/T (ENBES)) "6 oanst 7 9 Doctors O€ G0‘0 = 3/29 ‘aeyem doop (e) ‘Ue, sseTpus ue JO SUPTM JUITIJJIP TOF OOueISTSeI oAeM Apeojsuy eA 02 SE OT Sc OT eansta 80 Forees on an A.C.V. Executing an Unsteady Motton 0c 8T ‘rezem doop (q) e/Bs/Q al OT ("au6D) voi einst 7 81 Doctors 0€ S0 °0 SZ 38/2 02 ‘yydep aytutz ( 2/6 ST ro) ) ( 305) OT ‘OI = eansty 82 Forces on an A.C.V. Executing an Unsteady Motton 0c 8T ao) 9T 8/2 vT ‘yjydop oytuty (Pp) P/BAX GIL OT (+3409) ‘OI DING iT 83 Doctors yuez opt AToyrurzur (e) ‘yjdep ayz1ury ur spue yuey JO SUOTJEDOT JUSTOFJIP TOF OOULISTSOI sAeM Apeoisuy /oy 9 vT eT OT 8 8T 9T “TI eansty 84 Forces on an A.C.V. Executing an Unsteady Motion 0z pueq einsseid [euotsusurtp-omy (q) eps 8T 9T vT as OT ( *qU0 9) "IT eansty 85 Doctors (uoTJOU 47eID) eID 0} pexty ureysAs stxy Ssop A 2° 3 ute. oe * x 3 Urs Af 5 Sco) ox * * x x x "21 eansty 86 Forces on an A.C.V. Executing an Unsteady Motion SuryJOoUIs Jo syuNOUIe JUSTOZTIp 10; (e) ‘pomed oTTyM 103em deop ut souRIsIseI OAc M FEL eins 87 Doctors Stoquinu epnoi qT JUeTezIp oz (q) (“3u09) "€l 4eansta 88 Forces on an A.C.V. Executing an Unsteady Motton SutyJOOULS Jo sjuNowe JUSTaZIp 103 (ke) ‘pomed o[tymM 193em doop ul 90.107 opts otal OINnST 7 89 Doctors SsIoquinu epnodr gy, jUsIEFJIp TOF (q) 0? (+3409) oot ‘pl eansty 90 Forees on an A.C.V. Executing an Unsteady Motton O€ 9°0 = a (e) ‘surmed oTTyM 103emM doep ut 90.107 opts Apeojsuy 3 Gc oe onllé oot fe) “SI oInst iT 91 Doetors oot SZ aI (4) 3 ("7105) ost "GT eansty oot 92 Forees on an A.C.V. Executing an Unsteady Motton Scat 9°0 = a (e) ‘oTsue mef ut aSueyo days e r0}7e 194eM doop UT 9d410F Opts Apeojsuy ez/s G2 Tr 0°T SL°0 S*0 93 Doctors S2°T a O°T I (4) ez/s (-3u09) SL*0 ‘QT eansty S*0 Sz°0 T°0 (an) £20) 7°0 S*0 94 Forces on an A.C.V. Executing an Unsteady Motton DISCUSSION Jorgens Strom-Tejsen Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. I should like to take the opportunity to ask Dr Doctors a few questions about the work he has carried out. Dr Doctors has shown in his paper that by smoothing the pressure distribution some of the very pronounced humps and hollows in the resistance curve disappear. This is the case in particular at the lower Froude numbers, whereas at the higher Froude numbers the last hump seemed to have been af- fected very little. I would like to ask Dr Doctors if he has any similar experience when rounding the corners of the surface of a big shift ? One would expect that rounding the corner, actually using the right angle, a pressure platform would be round the corner, and ina simi- lar way the humps and hollows at the lower Froude numbers would disappear. This brings me to another question. Actually, the linearised wave resistance theory being used, we can only expect it to be accu- rate at the highest Froude numbers, whereas going down to the lower Froude numbers, on the assumptions we make, namely, that the wave slope is low is not valid any longer. Has it any importance at all, this smoothing or rounding away of the corner ? If this is not so at the lower Froude numbers, the wave resistance theory would not hold at all any how. We saw some rather interesting figures showing the effect when the surface effect shift was at the yaw angle or drift angle. In some cases the side force was negative and in other cases it was po- sitive. This means that in certain cases the surface effect shift would heel to one side and in other cases it would heel to the other side. I wonder if Dr Doctors could clarify this sine convention, and give us an idea about whether it is heeling away, or in what way it is heeling, relative to the drift angle he has been using. Finally, I would like to indicate that at the Naval Ship Research and Development Center we have made some somewhat similar com- putations, using pressure distribution, not smooth but actually using a staircase variation so that the pressure distribution is varied from 95 Doctors its full value down to a zero value in a staircase fashion rather than in a smooth way. We found, in the same way as in Dr Doctors' case, a certain effect at lower Froude numbers and very little effect at higher Froude numbers. Again, it appeared that it is only in the case of using a very large smoothing area that it had any real effect. I wonder if Dr Doctors could comment on what the alpha and beta really mean in physical terms ? REPLY TO DISCUSSION Lawrence J. Doctors Untverstty of New South Wales Sydney, Australta I thank Dr. Strom-Tejsen for his questions. In answer to the first question regarding the shape of the planform, I have not made numerical studies of this parameter myself. Other workers, mention- ed in the paper, have studied shapes which include elliptical and triangular ones. At high Froude numbers (i. e. in the region of the hump speed, and above), smoothing of the planform shape has a simi- lar effect to decreasing the rate of pressure fall-off at the edges. That is, the wave resistance is reduced a little. These two types of smoothing have, however, different effects at low Froude numbers. For example, the sharp-edged rectangular distribution has a wave resistance coefficient which oscillates between 0 and 2 at asymptotically low speeds, while the smooth rectangular distribution has a resistance which approaches zero (see Fig. 3). On the other hand, the wave resistance coefficient of a sharp-edged circular area appears to have a behaviour somewhat intermediate to these two (see Barratt (1965)). With regard to the second question, the limitation to the ap- plication of the linear theory is basically the slope of the waves ge- nerated by the pressure distribution. Thus, for typical ACV's, only the main or secondary hump have been realised in practice. On the other hand, by using models with a smaller nominal pressure (more precisely : P,/ega), more low speed humps and hollows may be measured. My thesis supervisor at the University of Michigan, Dr. S.D. Sharma, initially suggested the method of smoothing in the manner described. We found that smoothing in the transverse direc- 96 Forces on an A.C.V. Executing an Unsteady Motion tion has very little effect, and that smoothing in the longitudinal di- rection, simulated by aa=5, gave reasonable agreement with ex- perimental results. Since this value of aa does not produce an ex- cessively steep wave system (for the usual range of P,/pga), linear theory would be valid in this case. On the other hand, actual measure- ments of the pressure drop-off indicate that aa = 40 (i.e. eight times as smooth) would be more realistic. Thus, we found that nonlinearity and viscous action was the main cause of the discrepancy at low speeds. However, we pursued the use of such large amounts of smoothing, since it gave us an adequate model at low speeds, and allowed us to compute the unsteady wave resistance of an ACV starting from rest. Finally, with regard to the sign convention during yaw motion; side force on the craft and yaw to starboard were considered positive. In most cases the side force had the same sign as the yaw angle, indicating that this effect favourably aids the turn. In a couple of in- stances at low speeds, principally subhump speeds, the opposite was true. SIT Pinte pi eases rt oat Se Sales R ina beng, tape, aR wee ae abebttts Fae \yae Xa: ele ie Pa lgkiaciisioncabepte One { Th Py wee rmttadeto: Kaa) 0b ce areas 4 Bat ae: Viitnesil apm at brave em, 81 oe sh Bax aoc od ‘Rboode wot 18 yoneqet Sib edi 'to eaney hee edt enw avin aoe a otre to alavorma yr al dope bo hee ear Bowe ted oer 294 of ob bewoils bus ,abeoqge wol 36 lehonr a BUPODE XB Vis vag tis Ve" Seen ctott gatixetea VOA o6 to soastelnet evew 75 BG DRatty eit ean an REPLY TO DISCUSSION x : . wmoeRicn: Way gainuh aetasynoo agis oi? 01-7 ages tw yet ae ioVisie ag boxebieroo view br sot teleod way Diss Mato welt me oro | eidnks V By oi) aw ogre ari bans Bo} Oe: i ode shic oN} eon ea I LHe Be adeutes « al caret ay wh ii vices rgvet lak. 3! 98 Wit? terkd yo line sc haw oticodad wih .sbougqe qeamiddse yi pees bheage wel Ig is . 718 P 4 i , { , C 4 be \ ee ry (4 ii a — 2 4¢ i] + ' vv . ‘ J GA : i" 2 “¢ rT i ri é a bi ‘ : J a ai i r¢ ‘|i ‘4 6 , . 4 . - ee TAR, z i ui sh wit (iy Med aed « 5g 8G ‘ Ss tw aa : | vee: wry A rity «i tly ih ~* Ri ‘ a 7 ; tation te cheng F ait ine i hy Sane itt ‘ ye he ow a: . Mei Om Vir Ue loin at chee PC as OE bry Wii Be ew rye Jittn. “thie. Yer toe eee ‘ ' E hr ‘ Lat . wae y : / hee. $b, ti ru “Ek: ‘ eng . Cite raeniia?e¢ its ae ty Le. On: : eee hate. bar with 5 ehtlley noses URES p Prerieey oo <_. Soe Dik yaad bottews : at Sure 3 ‘Ws . ya ‘ 1 chive p ; j a Sisk Wo) ‘ OS, AS Neaetaht’: } < > } 0 ire oe Pe a , a . : Age ‘ Shiw te: ’ Higieetion thd ry PLD oe Mohair Geacrabed: Wo fac ce Wis erinatig » hy tae orang ae A LINEARIZED POTENTIAL FLOW THEORY FOR THE MOTIONS OF AMPHIBIOUS AND NON AMPHIBIOUS AIR CUSHION VEHICLES IN A SEAWAY Dr. T.K.S. Murthy Portsmouth Polytechnic Portsmouth, U.K. ABSTRACT The problem of the motions ina seaway of an ACV supported by an air cushion which is bounded by flexible extensions at the front and the rear and by rigid hulls immersed in the water along the sides is first formulated in the most general sen- se just to show how impossible it is to obtaina solution without some form of acceptable lineari- zation. Four perturbation parameters are there- fore selected relating to the cushion pressure, the width of the side hulls, the amplitude of the oscil - lations and the slope of the incident waves. The velocity potential for the motion of the side hulls is derived in the form of.an integral representa - tion, but it has been found possible to derive on- ly an integral equation for the potential due to the motion of the air cushion. This could, however, be reduced to an integral representation under certain additional assumptions, such as that the side hulls are slender. The steady motion and forced oscillation of the ACV in calm water are first discussed. The ex- pressions for the forces and moment show clear- ly the separate effects of the air cushion and the side hulls together with the interaction between the two which may enable the optimization of the overall configuration to be made. An investigation of the free oscillation in regular waves yields expressions for the response functions which may 99 Murthy be used to obtain an estimate of the motions of the ACV in irregular seas within the limits of the theory of linear superposition. I - INTRODUCTION AND SUMMARY. The purpose of this study is to develop ultimately a compre- hensive hydrodynamic theory for the general motion of an Air Cushion Vehicle (denoted, in short, by ACV and alternatively referred to as hovercraft) in an arbitrary seaway. This general non-linear problem will be formulated presently and, as may be expected, it will soon become apparent that the solution will have to be carried through in various successive stages, with some form of acceptable lineariza- tion adopted at each stage in order to render the mathematical solu- tion tractable and to keep the algebraic work within reasonable bounds. Practical results can, however, be obtained from calculations based on the lower order theory which can be relatively simple and a com- parison can then be made with the results of full-scale trials and model test data so that any differences pointing perhaps to a deficien- cy in the theory may possibly be reconciled by invoking the higher order theory. In a previous work (1970) the author has considered the case of an ''ideal''hovercraft as a starting point for the larger study. This amphibious craft is completely separated from the water surfa- ce during its motions and oscillations and was assumed to be travell- ing under a constant longitudinal thrust at a uniform speed in a uni- directional seaway composed of regular waves with their long crests normal to the direction of motion. Although the hovercraft was assum- ed to be clear of the water surface, practical expressions for the wa- ve resistance and side force in longitudinal and drifting motion over calm water, the restoring forces and moments due to forced oscilla- tion over calm water and the response functions for free oscillation in a regular seaway have been derived. The effects of the compart- mentation of the cushion and the overall cushion stiffness on the mo- tion have also been presented. The mean increased resistance over waves and the added mass and damping of water can also be calculat- ed. In order to keep the algebra simple, the hovercraft was assumed to undergo coplanar motion in the longitudinal plane with freedom in pitch, heave and surge only. This restriction of the motion to a plane, although not a strict requirement of the linearized theory, was considered as the only ty- pe of motion which was likely when the craft was operating for a long 100 Lineartzed Potential Flow Theory for ACVs in a Seaway time (long enough for the transients to have died away) ina regular seaway with a uniform speed in a direction normal to the wave crests and also capable of showing the essential features of a more general type of motion. The extension of the theory to longitudinal or drifting motion in a direction oblique to the regular seaway with six degrees of freedom is straightforward and no major revision of the theory is required as the beam/length ratio of present day hovercraft is of the order of unity and the disturbance of the water surface due to the mo- tion of the craft in the longitudinal or beamwise direction may be considered to be of similar order providing that no water contact takes place. The situation therefore is quite different from the case of con- ventional displacement vessels. Also, the extension of the theory to motion in an arbitrary course such as that during manoeuvering, to accelerated motion in starting from rest and to motion in shallow and restricted off-shore coastal waters can all be undertaken with suita- ble modification of the results. The prediction of the motion in an irregular, multi-directional, seaway can also be made by the method of spectral analysis on the basis of the theory of linear superposition. The amphibious hovercraft free from water contact may be considered as a special case of a more general type of ACV which we take up as the subject of our present study. The ACV is now assumed to be borne on air cushion contained by peripheral skirts at the bow and the stern and by the side hulls which extend below the hard struc- ture along the sides of the craft and which remain permanently im- mersed in the water during the motion and oscillations of the craft (see fig. 1). It is however, assumed that the flexible extensions do not contact the water surface during the motions and oscillations of the ACV, but an extension of the present theory to take into account skirt contact is straightforward if it is assumed that the flexible extensions are rigid enough to retain their shape when contacting the water. A later extension would be to cover the case of compliance to the pres- sure of the water. It is assumed that the air cushion is bounded by thin hulls along the sides and the air jets (or plenum air escape) at the front and the rear. The theory can also be suitably revised to cover the case of hulls (or skegs) which are located inboard of the lateral boun- dary, the whole air cushion then being enclosed within peripheral skirts. This configuration is sometimes adopted when water propul- sion is used. The side hulls are assumed to be "'thin'' with different "semi-widths'' on either side. A vertical plane is sometimes used on the inboard side of the hulls because of the relative simplicity in 101 Murthy production, but we have covered the possibility of having different off-sets on the two hull surfaces on either side of a longitudinal plane. We are, however, assuming that the surfaces on the outer sides of the two hulls and those on the inner sides are respectively of the same shape in order to have lateral symmetry very essential to the motion in a straight line we shall be considering. The ACV is considered as a freely hovering (but partially floating) rigid body in motion under the action of given external forces (such as those due to wind, propeller thrust, etc.) together with the hydrostatic and hydrodynamic forces arising out of the "ground effect" of the air cushion in depressing the water surface and from the immers- ed part of the side hulls. The equations of motion for the most general type of motion in six degrees of freedom will include in addition to the external forces and the forces due to ground effect, some types of internal forces peculiar to ACVs, such as momentum drag, forces arising from the uneven escape of momentum due to the leakage of the air cushion through the air curtain at the front and the rear and, pos- sibly, even through the troughs of the induced waves which may make part of the side hulls run dry unless the hulls are of suitable draught. There is also the pneumatic effect of the ''wave pumping" of the air in the cushion due to the passage through progressive waves. We shall assume, however, that the only force which enables the uniform translation of the ACV is the longitudinal thrust, leaving due account to be taken of all the other factors when the occasion arises. In the earlier study, the hovercraft was replaced by its equi- valent mathematical model, namely a ''travelling pressure disturban- ce'' with a basic "hull form" for the craft dictated by the planform of the air cushion and the two-dimensional distribution of pressure on the water surface consituting the lower boundary of the cushion. All the results were derived on this basis and without enquiring into the actual mechanism employed for the generation and retention of the air cushion, i.e. whether a peripheral jet system or plenum chamber with or without compartmentation was used. This model will be retain- ed for the present study with a separate examination of the effect of the side hulls and the possible interaction between the two. It may be taken for granted that hull design has arrived ata stage of perfection due to the efforts of naval architects over the cen- turies, but a basic requirement for developing the hydrodynamic theo- ry of the motions of the composite ACV, i.e. with the air cushion enclosed along the sides by the hulls is a knowledge of the hull form of the air cushion which plays usually the major role in supporting the ACV above the water with a small contribution from the buoyancy of the side hulls. It is commonly assumed for want of a more precise 102 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway knowledge that the pressure in the cushion at all points is uniform and that the pressure on the water surface is also uniform and of the same value. This kind of stepped cushion with an abrupt drop between the pressure within the cushion and the ambient pressure outside will have a hull in the form of a right cylinder with vertical sides and, in our case will resemble a rectangular box, It is obvious that this type of hull will be totally unsuitable for a fast planing type of displace- ment vessel. It will appear during the course of this study that the mathematical work is considerably simplified if it can be assumed that the pressure is diffused continuously from a maximum value at the centre in such a manner that it becomes zero at the boundary and, preferably, with a zero gradient in the direction of motion. The pur- pose will be equally served if the pressure is uniform in the main part of the cushion and diffused over an annular region close to the boundary. It is interesting to note that this type of diffusion of pres- sure selected with the object of mathematical expediency in obtaining an easier solution of the problem on hand gives a hull shape for the air cushion with an aesthetic appeal and with a reasonable dead rise and flare at the bow and at the stern which may be considered by the naval architect as very acceptable for planing motions in the dis- placement mode. The actual mechanism by which the above pressure distribu- tion may be generated in the case of practical ACVs considered as hardware is merely an engineering matter, although in the present state of the art not much progress has been made in this direction, presumably because no investigation as so far been made as to the direction in which to proceed or, wether it is necessary to proceed in any direction at all towards perfecting a ''tailor made'' cushion, This could probably be achieved by suitable compartmentation of the cushion and by introducing auxiliary flows in the (supposedly) quies- cent air in the cushion, It is only a question of accentuating the en- trainment of air and trapped vertices known to exist in the vicinity of the boundary. Assuming that such a manipulation of the pressure distribution is feasible (and there is no reason to doubt this), we are indeed in a fortunate position with hovercraft for without altering the deck space, it seems possible to give the ACV an arbitrary cushion hull shape by a simple manipulation of cushion aerodynamics. The analytical methods used in this study are essentially those first used in the classic work of Peters and Stoker “/ and later fol- lowed by Newman @)@ » Joosen (5) and others. These works con- stitute a rationalapproach to the unsteady motion of ships, but a solu- tion has so far been obtained only for thin ships and slender ships. 103 Murthy Peters and Stoker @) have indeed considered a flat planing type of hull and a yacht type of hull which is a combination of a thin vertical hull and a thin horizontal hull. However, the utmost that has been achieved in these cases is the derivation of integral equations for the potential with singularities at the edges of the hull. No method of solu- tion of the singular integral equations, or even the possibility of a solution has been indicated, as the equations contain singular kernels and are therefore not of the classic Fredholm type. In the case of an amphibious hovercraft, however, we had managed to derive an explicit integral representation for the poten- tial in the form of a source singularity distribution over the free sur- face directly below the cushion opening together with a distribution of line sources and line doublets along the boundary of this region. This happy position had come about because the two boundary condi- tions for this boundary value problem for amphibious hovercraft free from water contact were of identical nature, both relating to the pres- sure on the free surface, and therefore constituting a Dirichlet pro- blem. In the case of bodies floating on the water surface there is a pressure condition on the free surface not occupied by the floating body, namely that the pressure is constant (taken as zero for conve- nience) and a velocity condition on the immersed part of the hull, na- mely that the normal velocity of the hull and of the contiguous water particles are equal. In other words, the flow is tangential to the hull when boundary layer effects are ignored. There are, of course, the usual conditions at infinity and at the ocean bottom. This is therefore a Neumann problem. In the case of the ACV we are now considering, having an air cushion of the type previously studied but with the addition of a pair of parallel side hulls of arbitrary immersion, the boundary conditions are of a mixed nature. The two pressure conditions for a freely hover- ing air cushion are still present together with the normal velocity condition for floating bodies just discussed. It will be seen presently that an explicit integral representation for the potential due to the hulls is possible on the assumption that they are ''thin" (a common and necessary assumption in the theory of ship motions) and witha sufficiently large separation so that the effects of mutual interference may be ignored. However, it has been found possible only to derive an integral equation for the potential due to the air cushion with the ker- nel containing the ''jumps'' in the potential across the boundary. Al- though the presence of the air cushion does not appear to affect the potential for the motion and oscillations of the side hulls in calm wa- ter, the influence of the side hulls on the potential of the air cushion 104 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway cannot be ignored by simply setting the parameter representing the width of the hulls equal to zero, for although the wave-making effects of the hulls are thereby eliminated, they nevertheless provide a ver- tical barrier for the fluid flow. However, on the assumption that the immersion of the side hulls is of a small order (say, of the same order as their width) the integral over the longitudinal planes of the hulls can be ignored and if the cushion pressure is also assumed to be diffused in such a manner that it is zero at the front and rear of the cushion where air leakage occurs, itis possible to obtain a simple integral representation for the potential. This procedure enables practical results to be derived pending a rigorous solution of the in- tegral equation. The problem is first formulated in the most general terms in Section 2, just to show how impossible it is to obtain a general solu- tion. If the problem is difficult to solve in the case of displacement ships, it will certainly be more so in the case of ACVs, where the laws of cushion aerodynamics relating to ground effect enter with an extremely complicated relationship between the pressure distribution and the relative distance between a point on the hemline of the flexi- ble skirts at the bow and stern and the elevation of the water surface directly below. It is therefore clear that the problem has to be linearized in a suitable manner if its solution is to be rendered mathematically tractable. The usual method of solution in problems of this nature is the assumption of a basic slenderness parameter representing geo- metrical restrictions on the body. Thus, for example, in "'thin ship" theory, the slenderness parameter is the beam/length ratio which is assumed to tend to zero. Similarly, in "flat ship'' theory it is the draught/length ratio and in ''slender body" theory both the beam and the draught are assumed to be small compared with the length. These restrictions are necessary for the validity of the linearized theory which assumes that the ship reduces to a thin vertical or horizontal disc or a thin straight line and that it can then have a translatory mo- tion with finite velocity parallel to the plane of the disc or along the longitudinal axis without creating waves of finite amplitude. The squa- res of the perturbation velocities of the water particles can then be neglected and the problem becomes linear. This then is the objective, namely, that the wave making of the vessel in steady motion shall be negligible. In the case of an amphibious ACV, it would appear that the geometrical dimensions are not directly relevant to the problem so long as the craft is not immersed in the water. It is the cushion pressure, i.e. the total weight of the craft divided by the cushion KOS Murthy area which determines the wave making at any particular speed. A large planform area is therefore desirable (for a given weight) and as the popular value for beam/length ratio of present-day hovercraft is about 2/3, it appears that the geometrical dimensions are to be unrestricted for hovercraft in order so that the theory may be success- fully applied. In the case of a rigid sidewall ACV, the total weight of the craft is usually supported mainly by the air cushion with a smaller contribution of the order of 10% from the buoyancy of the immersed side hulls. We shall select 6 as the small parameter representing the thin width of the side hulls and B to denote the smallness of the cushion pressure, We shall not make any a priori assumptions as to the fractional weights supported by the air cushion and by the hulls so that we shall not stipulate the relative orders of magnitude of 6 and B . 0 in the case of deep water of infinite extent or in the domain dulce, x) = Z,- = 10 in the case of shallow water of infinite extent. In the moving co-ordinate system the velocity potential may be written Oey, 2.: t)7= OX, +xcosa-ysina, Y + zsina + ycos a , z;t) G = (x, y> Zisie)ls say. It is then easy to derive from (1-1) the following equations for the transformation of various derivatives between the two systems : O-=@ cosa-@ sina, O. = @ Coeok Bos sin cosat® win eh se y xx xy yy xX a a wile 3 2 @ =6 sin®+@ cos®, @ = sin® +2@ sina cosa+@ cosa ay: x y YY xx xy yy %, Fi 2 ®77 e ve (i z3) so that 2 2 Ivo CXAYe Zt) | = [Ve (x, y> z;t) | (4 -4) and 2 2 7 On Owe Zt) a= Van blon cr: zat) (1-5) It follows that @ is a harmonic function in the (x,y,z) system in the same way as @ isunthe (X,Y,Z) system. Il. 3 Bernoulli's Equation The compressibility of water may be ignored even at the high speeds attained by ACVs at present and we may write Bernoulli's law as Murthy 1 2 - - gZ +>[vo ] + Q, = constant (1 -6) where the constant on the right-hand side is independent of the space variables and, as is usually done, may be set equal to zero, it being understood that Q, is suitably adjusted. Now, @,4 can be expressed in a manner similar to the other derivatives in (1-3)as co) _ = (wy -V)o, —9 Cees Seat (1-7) where the speed V of the ACV in its course is an arbitrary function of the time for accelerating motion. The relation between the pressure p (x,y, z;t) and the velocity potential (x,y,z;t) may therefore be written a 1 PA = Se = = = Ave : gz + : (Vb) + (wy - V(t) ) = wx 2, + a 0 (1-8) II, 4 Conditions on Boundary Surfaces i P(e Ye Z st) oS Ee are = 0 is a boundary surface, which may be fixed or moving, the kinematic condition on such a surface is ee i dt =O, Fy +O, Fy +O,F, +F, Ss Using the relations (1-3) and (1-7) the corresponding condition in the (x, y, z) system becomes f - - f = 0 1- Boys Eni fier Robes ikea Midis g ahptatek, (1-9) The free surface of water given by the equation Zar Ce. yst) = 0 is a boundary surface, fluctuating with respect to time, and the kine- matic condition on this surface may therefore be written pari es [soe to laa fy -wx ee f= 0 (1-10) Et2 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway to be satisfied for z = ¢. This condition applies both to the external free surface (EFS) and the internal free surface (IFS) defined and discussed in the Introduction. The dynamic conditions on the free surface z = § are obtained from (2-8) by setting p=0 for the EFS and p=p (x,y), the surface rn : : s : pressure applied by the air cushion on the IFS, respectively. The kinematic condition (1-9) is also applicable to the instant- aneous position of the moving (and oscillating) side hulls of the ACV and to the lower edges of the flexible extensions immersed below the surface of water. Il. 5 The General Non-Linear Problem The strict formulation of a very general type of ACV problem would be on the following lines. A rigid body in the form of an ACV is supported above the water surface partly by the air cushion (contain- ing air at a pressure higher than atmospheric) and partly by the buoyancy of the immersed part of the side hulls. In the position of "static hovering", i.e. at zero speed ahead, the steady pressure ap- plied by the air cushion to the IFS may be assumed to have a distribu- tion of the form | pt ola Ds (x, y) over a region S_ of the water surface which is the vertical projection of the cushion opening on the water surface. This region is therefore bounded by the inner sides of the side hulls and the curves representing the vertical projection of the hemline of the skirts at the bow and at the stern. The "cushion hull form!" is thus determined by the plan form of So and the pressure distribution thereon. It may be assumed as an approximation that the latter is unaltered during steady forward motion in the horizontal plane. However, the water surface will now be disturbed due to the generation of surface waves by the air cushion and by immersed side hulls (with perhaps a complicated kind of cou- pling between the two as will be shown later). The steady disturbance will travel with the same speed as the ACV but will cause a steady variation of the shape of the IFS i.e. of the cushion hull form. If the ACV now performs oscillations during steady translation, which may be forced oscillations in calm water or wave excited oscillations ina seaway, the pressure distribution on the water surface will no longer be steady or of the basic form. This is because the region S over which the pressure is apllied is now a fluctuating domain, as itis the instantaneous position of the vertical projection of the cushion opening ll 3} Murthy on the water surface and its shape and location in space will be there- fore dependent on the oscillations of the craft in all modes expect, perhaps, in heave. The actual pressure distribution during oscilla- tions will also be different from the basic distribution on account of the cushion and peripheral jet (or plenum chamber) characteristics peculiar to hovercraft which dictate the pressure in the cushion in terms of the local clearance between any point of the periphery and the elevation of the water surface directly below it. In our case, the pressure variations will be initiated in annular regions adjacent to the bow and stern skirts, but the perturbation pressures will no doubt be transmitted to the interior due to induced flows and entrain- ment of external air with the result that the distribution over the enti- re region may be substantially altered. The basic problem is essentially that of determining the velo- city potential @ (x, y,z;t) as a harmonic function satisfying Laplace's equation in the domain z>§(x,y;t) for all time t > 0 when the ini- tial position and velocity of the ACV and of the water particles are prescribed at time t=0 . A singularity has to be accepted for the solution of @ at the boundary of the region S if the applied pressure is discontinuous there, i.e. if the pressure is different from atmos- pheric. The velocity potential can be used to calculate the elevation and slope of the IFS on which the pressure is applied by the air cushion. The forces and moments on the ACV considered as a rigid body are in part due to the action of the applied pressure on the IFS which is the cushion hull and can therefore be determined in terms of the ap- plied pressure and the slope of the disturbed water surface. It is appreciated that in common with other surface wave problems the elevation and slope at individual points of the region S cannot be de- termined accurately from the potential due to interference effects although the evaluation will be corrected at some distance away from the pressure field. However, we only require the total integral effect of the applied pressure and for this purpose the potential can be used to obtain practical results. The other contribution to the forces and moments on the ACV arises from the action of the pressure of the water particles acting on the instantaneous position of the immersed portion of the side hulls. The boundary conditions dictate that the relative velocity of the water particles at each point in a direction normal to the instantane- ous position of the oscillating side hulls is zero. The pressure on the free surface is also prescribed as zero. But both the immersed hulls and the free surface of water are moving boundaries of the domain in which the velocity potential is to be determined. A coupling between the motion of the side hulls and that of the water therefore exists. As Linearized Potenttal Flow Theory for ACVs in a Seaway stated above, this coupling between the motion of the ACV and that of the water introduces an additional complication in the case of the air cushion due to the laws of cushion aerodynamics. Also, the IFS is a moving boundary on which the potentialis to be determined. This complicated situation is illustrated in the block diagram below uodn spuedap eaie 2esoyM s{[Ny apts eyj jo adeFIns pejjeM 9y} 12aA0 SaTOTZAed 19}eM 9y} JO eA1nssead rz011903uT 24} pue [[TNy uoTYysnd 9y} uo eainssoad eoejins 94} jo uoWOe 24} 0} Onp esoyy Bsutpntout ‘JI uo dutjoe sqyusurou pue S9910F 9y} [Te Aq u1opseajz jo so00I13ep xts ut Apoq praia e S® ADV 24} Jo uoTjour jo uodn spusdep vere osoym STINY Spts ey} fo soezans poem 24} 1TdA0 seToyaed 193eMm 9} Jo eanssead rot193uT oy} pue IIny uotysno oy} uo eanssoad BdeFINS ay} FO uote 9y} 0} Onp asoy} Surpnpout ‘41 uo Bunoe SJuUSsUWUIOUI pue sadr0F 94} [Te Aq 2 WIOpeerj Jo s9ea3eap xis ut Apo Sua SPasts eer eiaye ate < res ¢ Ee eat ysnory} pexty sutpurey ayy jo = “Tray PIStI © Se ADV ey} Jo uoTjour p ; : 8 uoTYs nd uotjour ay} uodn spuedeq s ey} Aq aoejains deaf jo suot} mba [enuesezzIp ape éiy 44 pesodutt ey} JO UOT} eUIIOFap ey} Ysnory} pexiz sTiny ASAI auy Aq aoeyins ey} uodn spuedap SuieO WOT OUtou;ruodTspuadap s 4 @ “) Be Se 2 eg Re eee ee eee I 22e1F OY} JO uOT}eUILOFJOp ey} uodn spuedep aoeyins 1ezeM 9y} 07; ptedser yzIM ouTTUTeY 3y} | jo Jutod yove jo uorj1sod eAT}ETAI ay} uodn spuedag | TINY 24} 07 TeNUedUe} ST MOT} *9 ‘T ioe ee E oy aoezins *(uoTsuay voezIns t aoezains saToyaed 193eM jusdel[pe p2332M ay} Jo yutod yore ye SuljOeTJeu) aoezrIns 9y} uO Ieyem uo uoTYsNd ate jo AJIOOTAA TeuLION S[[NY Opts Jo AZIDOTAA TeUIION seToWa1ed 19}eM Jo oinssoig 4q pesoduit asinssoig STINH ePts ey} uo suoT{IpuoD Arepunog uIINH vuotysnyd,, 243 uo suot}IpuoDy Arepunog STTOH ddIs NOTHSND UIv 116 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway It is clear from the above that the situation in respect of the ACV leads to a very complicated non-linear problem. The position of the free surface is not known a priori and the velocity potential has therefore to be determined in a fluctuating and unknown domain. The boundary conditions are of a mixed type. These relating to the air cushion are stipulated in terms of pressure and those relating to the side hulls in terms of velocity. Appropriate boundary conditions at infinity may be imposed from physical considerations in order to obtain a unique solution of the problem, but it is doubtful whether this general non-linear problem will be mathematically tractable. It will be observed that the motions of the ACV are not given in advance (except in the case of forced oscillations of a pre-determined kind), but are unknown functions of the time to be determined as part of the solution. The motions vary the pressure distribution on the IFS and therefore dictate the appropriate elevation slope of the water surface which together with the pressure distribution determine the forces and moments on the ACV due to the air cushion, Similarly, the mo- tions vary the wetted surface of the side hulls and the pressure of the water particles both of which determine the forces and moments due to the side hulls. This highly non-linear problem has therefore to be linearized in a suitable manner if a practical solution is to be achieved and this we shall endeavour to do in the next section, Ill. THE LINEARIZED PROBLEM As the general non-linear problem has been shown to be highly complicated, we shall not attempt the solution of the single boundary value problem, but consider instead a sequence of linear boundary- value problems which result when all the relevant physical variables relating to the motions of the ACV and of the water are expanded in terms of say, four small perturbation parameters, 6 , B ,a@ and e , describing respectively the orders of magnitude of the width of the side hulls, the cushion pressure, the amplitude of the unsteady motions and the slope of the incident waves in the regular seaway. The reasons for the choice of these parameters are explained in the Introduction. The solution of the sequence of linear boundary-value pro- blems is relatively simple and all relevant quantities determining the motion are obtained in the form of a power series in the pertur- bation parameters. An approximate solution of the general non-linear problem can thus be obtained up to any desired degree of accuracy. Murthy We shall now consider the problem of an ACV that has been operating at sea for a long time under a constant propulsive thrust. This problem is simpler and more of practical interest than that where the ACV starts from rest and moves over water under the action of given forces, such as these due to wind, control setting, etc., in an arbitrary seaway. After the lapse of a sufficiently long time, all the transients would have disappeared, and if the propulsi- ve thrust is the only force acting on the ACV, it would be moving at a steady speed of translation. The linear displacement of the centre of gravity of the ACV from its equilibrium position of steady transla- tion may be represented by components along three axes fixed in the craft (this system of axes will be described presently), namely, sur- ge, sway and heave. Similarly, the angular displacements of the craft may be represented by components along these axes, namely, roll, pitch and yaw. It is expected that each component of displace- ment will consist of two terms, one independent of time and repre- senting the steady displacement that would exist during motion with uniform velocity in calm water and the other an oscillatory term sim- ple harmonic in the time due to the excitation by the incident waves coming from infinity. If the ACV is symmetrical about a longitudinal axis, it may be expected that the motion in calm water will produce non-zero displacements only in pitch and heave (and, possibly, in surge, which is trivial, since the steady surge displacement can be absorbed in the forward motion). The complete solution of our linea- rized problem depends then on the determination of the forward speed for a given thrust (or thrust required for a given forward speed), the steady pitch and heave displacements (usually known as trim and sin- kage) and the six oscillatory components of displacement. An irregular, but long crested, seaway may be assumed to be composed of a system of simple harmonic progressive waves, each of a given frequency. Any irregular wave train may therefore be ex- panded as a Fourier series with respect to time. In the linearized theory, we may assume that the ACV responds to each wave compo- nent as though it existed independently of the others. By the theory of linear superposition the motion of the ACV will be composed of the same Fourier components. Similarly, in the case of forced oscillation in calm water, any arbitrary type of oscillation may be represented by a Fourier series with respect to time. It is therefore only neces- sary to consider a single sinusoidal component for our solution, The results can then be generalised by spectral analysis. Lineartzed Potenttal Flow Theory for ACVs tn a Seaway We shall presently be expanding all the physical variables describing the motion of the ACV and that of the water in powers of the perturbation parameters. Taking the velocity potential of the water as an example, the correct expansion would be Boom Ve ee kod SP ei opt (x, y, z) (x, y, 236; B3; a; €; t) = RED) 6 Cot a k,1, m,n, p Corresponding to excitation by waves with frequencies 0, 20, 30, ------ However, as the algebra will become extremely complex, the whole solution can first be carried through for one frequency component with say, p=1. The final result can then be extended to any number of Fourier components in the wave system. It may be observed in this connection that in simulating an irregular seaway in a towing tank a finite number (of the order of ten) Fourier components is usually selected. In this case, when we desire a verification of the theory from experimental results, the solution should cover the same number of Fourier components. Ill. 1 Coplanar Mction The analysis will be restricted to a study of the ACV moving in a longitudinal plane. This is by no means a requirement of the linearized problem, but this simpler study will reveal clearly the general features of arbitrary motion in all six degrees of freedom. IlI.2 Body-fixed Axes The third co-ordinate system mentioned in section [I. 1 is the (x', y', z' ) system fixed in the ACV. The origin o' coincides with the origin o of the moving system when there are no oscillations. Also, the z' -axis (like the z-axis) contains the C.G. (on the negati- ve side) and the x', z' -plane (like the x, z-plane) is the fore-and-aft plane of lateral symmetry of the ACV. It is clear that the x', y' -plane is the load waterplane (LWP) of the side hulls when the ACV is on its air cushion, All the three systems of axes are illustrated in Figure 2. 119 Murthy III.3 Transformation of Co-ordinates It is easy to derive the following equations for the transforma- tion of co-ordinates between the moving system (x, y, z) and the body- fixed system (x', y',z' ). x=x+x'cos@ + (z' - Zc) sin @ x! = (x - x) cos@ + (z + 2Q) sin 0 = y' y'=y < 1 zZz=Zzt+z_+ (z'-2-) cos@ -x'sin@ z'= (x - x)sin®@ - (z+z_-z)cos® tz G G (2-1) where x, Z and 9 are the surge, heave and pitch displacements G III,4 Perturbation Expansions We will now expend some of the physical variables represent- ing the motion of the ACV and that of the water in powers of four perturbation parameters §6,8,a and e¢ defined below. (he Es SES ratio of the side hulls, length B= SrauEne ratio of the air cushion (i.e. A length pg L R H amplitude of small motion (and oscillation) pli : a € = amplitude ratio of the incident waves. length The first two parameters have been selected from the require- ment of the linearized theory that the amplitude/length ratio of the waves induced by the side hulls and the air cushion due to the motion of the ACV should be small. The amplitudes of the induced waves may be assumed to be proportional to the beam of the side hulls and the cushion pressure (in head of water) and the length is proportional to Bye . It is therefore clear that the speed of the ACV should be suffi- ciently large. The linearized theory is therefore inapplicable to very low speeds on account of the unacceptable steepness of the induced wa- ves, 120 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway The third parameter need not be specifically defined at this stage, but one criterion is that this parameter should be sufficiently small so that the water contact of the flexible extensions does not take place at the bow and at the stern. It will be shown later that this parameter is of the same order as that of the incident wave. The fourth small parameter (like the first two) ensures that the orbital velocities of the water particles in the incident wave are sufficiently small so that the squares of the perturbation velocities may be neglected in the linearized theory. All the four parameters are assumed to be sufficiently small to ensure the convergence of the perturbation series which follow. Assuming that the motion is periodic of frequency 5 , we may write the following perturbation expansions for the motion of the wa- ter and of the ACV considered as a rigid body. It will be assumed that the unsteady flow of water is produced by the periodic forced oscillation of the craft. Since we shall only consider the linearized problem, the motion for arbitrary periodic oscillations may be dedu- ced by the method of Fourier transforms. Basic ''Hull Form!" of the air cushion Po Gry a= Bp. (x, y) Hull surface of the side hulls S Starboard side of starboard hull y' - b =6 h, Gorn lee S,_ Port side yo b= dh, Gs) S,, Starboard side of port hull y'+b=6 h, Ge zs) S, Port side y'tb=-6 h, Gu, Sz) Surface pressure distribution due to the air cushion are " TOT I Spedleyereaie £0) p, (&yitsdsBsase) = Relie 5 Bae Phimn ¥) kim; n Velocity potential 5g la mn ® (x, y, z3t; 6; B ,a,€) hans a € Oiaei Grog, 2) ECA Murthy Water elevation 6 (x, ysti6,B,a,€) = Reva; e 6 Baie $ Surge displacement zs ror et me az ((63 ds. ;a)) = Re Ds at Tes B a” x it hax klm Heave displacement Z (t3;d;Bsa) = Radas ‘ge at at 53 Ze ke, 1, m as Pitch displacement Giligaoh edt. =i Reward + og gee O90 kK, 1).m (2-2) The frequency o is that of forced oscillation in calm water or equal to the encountered frequency o, of the waves ina regular seaway. The displacement parameters are not expanded in powers of ¢« for, by definition, a is the order of the displacement due wave excitation of the order of forced displacements in the absence of waves. The quantities relating to the motion of the water contain, however, all the four parameters. A phase lag between the displace- ments is not explicitly shown in the above expressions, However, if for example, zZ and @ are considered as complex quantities a phase lag is pea between heave and pitch. The steady displacements of the type Z and om ann) 110 10 relate to motion in calm water and the oscillatory displacements 1m 44cm aoe Och are restricted to the first order in @,i.,.e. with m=1 since we shall be evaluating forces and moments up to a maximum order of GF we, 5Ba€ and B2ae and for this purpose displacement of the first £22 - Lineartzed Potenttal Flow Theory for ACVs tn a Seaway order will be sufficient. In a similar manner the surface pressure during oscillations will also be restricted to the same order. We may now write down the expansions we shall actually be using in the subsequent work. (x,y, z3t;6;B;a;e) = 8% 090 + Béo100 + 8B) 159 + ict TES of [ 80%, 91 BAG 1197 €PoQ0) Z is Pee + Bee is ote Meee arco) 1001 0101 aes Be Ae aL OP e ee _— M(t O.b5a) = 0x a +Re.ae Es +6X) 9) Hea 100 L? 2 aig 2 + 66%), +0(6,8 , dBa,a) ANE Os Psa) = 62. + Bz ee “Le qe OTA, Oe ap [Ee | ign oP oon ee | pole” 10d, COL) + OBZ + 0 (So. ea, a 110 ict @ (t35; Bsa) = 60 +B, + Re.ae | Goo * 88102 2%) | 0 (a2 6 abe Fy a) (2-3) The linearization of our problem is achieved by substituting the above expansions in Laplace's equation and in the boundary con- ditions and collecting terms of the same order. The result is a se- quence of linear boundary value problems for the potentials $ Having derived the potentials, the pressure of the water particles on the side hulls and the shape of the cushion hull can be calculated. The forces and moments on the ACV considered as a rigid body can thus be evaluated. IV. DEVELOPMENT OF BOUNDARY CONDITIONS In the case of a conventional displacement vessel there are two types of boundary conditions for the velocity potential t23 Murthy 6 (x, y, z;t). These are (i) the kinematic and dynamic conditions on the free surface of water on all sides exterior to the immersed part of the hull and (ii) the conditions on the immersed part of the hull itself. The latter condition takes the form of the equality of the nor- mal velocity of the fluid and that of the hull, i.e. that the flow is pu- rely tangential to the hull surface when boundary layer effects are ignored, This is therefore a Neumann problem. Also, when dealing with an ideal amphibious hovercraft as in the previous study (1970) which was assumed completely separated from the water surface at all times both conditions relate to the free surface of water, one on the external free surface (EFS) and the other on the internal free surface (IFS) which is the vertical projection of the cushion opening (i.e. of the hemline of the skirts) on the water surface directly below the craft. Both these conditions relate to the pressure on the free surface giving a Dirichlet problem. In the case of the general type of ACV now under considera- tion, there will be three types of conditions : (i) The kinematic and dynamic conditions on the EFS (ii) A normal velocity condition on the immersed parts of the side hulls which separate the EFS from the IFS (iii) A pressure condition on the IFS which forms the lower boundary of the cushion. This is therefore a mixed boundary condition problem, IV. 1 Conditions on the External Free Surface (EFS) The kinematic free surface condition applicable to the EFS and IFS has been derived in Section II in the form ereulag dang sae Magslegiye PBA ROME SIE” 0 (4-1) on z= $. On the EFS the pressure is zero and substituting p= 0 in Bernoulli's equation (1 -8) we derive l 1 2 fs al , bey - "VP Stele s+ (Ve) (4 -2) also on z2=¢ 124 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway As the position of the free surface denoted by z= ¢ is un- known we will endeavour to eliminate the derivatives of § from (4-1) by using (4-2) in order to derive an explicit equation for ©. Denoting the terms on the right-hand side of (4-2) by R, for the time being, and noting that Ris a function of z and t and that its value is to be taken when z = ({, the derivatives of { are evaluated and substituted in (4-1) giving the condition ) 3 b. (spe ag) hr oly Rye- x Ro) hive. VR -@ =D (4 -3) where Ris, of course, 1 | 1 *] = - - 4 — R 2h Vb + w (ym x®@ ) + 7) (Ve) and eliminating the derivatives of R, we derive finally, 3 3 2 : e eS ole 2 eV eV Coe ae Ut y|(ve) | =g2 = 5 (4-4) where S is an algebraic expression of a complicated nature each term of which has, however, w or wasa factor. This condition can therefore be simplified when w is set equal to zero and in the case of our coplaner motion in a straight course, S ae 1 ra ee ae A i) ® - ge + 2V8. V(b, - Ve) +3ve.9| (ve) | = 0 (4-5) This is the exact free surface condition. We have made no approximations so far, but the condition is only applicable on the actual free surface z = $, Although we have elimanated { from the equation itself, we are still in some difficulty as we do not know the position of the free surface for the application of the condition the- reon, We will therefore attempt to derive a condition, even if itis an approximate one, which can be imposed on the known surface z= 0. This, in effect, is the first stage in the linearization of the problem. Assuming that the potential may be continued analytically from the actual free surface z= ¢ tothe plane z= 0, we may expand it in the form of Taylor's series ad Sele vs, = (0) Pic 2st) = h(x, y,o5t) + § 25 Murthy The expansion in series may safely be expected to be convergent for @ is of small order [0 (6, B) | and the derivatives of ® may al- so be assumed to be of the same small order. It is clear from (4-2) that § is also of small order and may therefore be treated asa small constant for the purpose on the expansion, Substituting the above series in (4-5), we get 3 ae re) P) ee (pn a) oh ti Siparenl eeaen Winey Oi eR, rules 3 a +2V(b+S@ ) vle,- Va, +655 (@, - ve) | + 0(#) = 0 on z= 0, Now, from (4-2) with w = 0 2 f= (@ - Ve) +0 (#) fal g so that we have an approximate condition 2 5 « +2V¢. aia 5. 2Vo: +V Box go. 6.7 ( ® ) 1 +—(o ai e) z 3 iV oe i +VO@_ - = S ® ) Sa (e. 2Ve@. ge) +0(@)=0 (4-6) on z= 0. IV. 2 Conditions on the Internal Free Surface (IFS) If there is a distribution of surface pressure p_(x,y;t) on the IFS, we have from Bernoulli's equation (1-8) with w = 0, P, (x, y3t) 1 1 2 t= tle, vo, +3 (va) | + bee Z= with x, y in the IFS. The kinematic condition is the same as that on the EFS and given by (4-1) with w= 0, 126 Lineartzed Potential Flow Theory for ACVs tn a Seaway Gy = = ( V)S +6 € -@6¢ +S 0 also on Z= é . Eliminating £ as before, we derive 3 3 \2 1 2 Ss Tee b- gb +2Ve. V(b - Ve) +v6.9| (V9) p= | lS > VS ; ’ Z: ’ = [Se Vg) Py) #80, Gov tap, Goy)] =o x y This is the exact free surface condition on z=f§ and the approximate condition on z = 0 is obtained as before by a Taylor expansion of @, giving the final result 2 a z Vo. = oe 2 Vee +V OG Eb + 2V@ re. ee) ag . S- (6, - Ve) +0 (#) =O) (4-7) IV. 3 Conditions on the Hull Surfaces Setting w= 0 in (1-9) the kinematic condition applicable to the wetted hull surfaces is r= WV) ® Hee Ho +b 8H OPH = "0 x x Vi bay, Zio Z t ise. ve.VH+(S--vS)H =) 0 where the hull surface is given in the moving co-ordinate system by an equation of the form eG. yi425t) = 0 The above condition stipulates that the normal velocity of the hull at each point is equal to that of the contiguous fluid particle, i.e. that bed Murthy the flow is tangential when viscous effects are ignored. Now, the fluid velocity normal to the hull surface is sae on p> Ve where Nf is the unit normal vector drawn into the hull given by He one A en piss - _VH n= _ 1 = Exteel IVHI| x y Z so that oH oH See Se a. ive | IVHI (<8) Now, the equation of the hull surfaces is naturally given in the body fixed system (x', y', z' ) in the form Fl (; y> z;t) = bh, fae! 5 ae ) - (y' - b) = 0 on Si4 = bh, (x 2" ) ly’ ffe, @ xn) - p (r' x n) dS C H S $, 15, (5-2) The surface integrals are taken over the displaced position of the effective hull, S, of the air cushion, i.e. the instantaneous position of the IFS and the instantaneous wetted surfaces S, and S, of the side hulls. r' is the position vector of an element of area dS with o' as origin and the unit normal n is taken out of the water and into the cushion hull and the side hulls. Also, p, is the surface distri- bution of pressure acting on the cushion hull (z = § ) and p the pressure of water at an interior point (z a) of the immersed sur- face of the side hulls. The rigid body force and moment are F. = mR Goeth aka (5-3) Mp ie Plex (ai - a) dm (5-4) where r' is the position vector of an element of mass dm of the ACV with absolute velocity U , and the triple integral is taken throughout the volume of the ACV contained by matter. and These are the forces and moments acting at and about the C.G. of the ACV due to its inertia and the pressure of the water on the effective hull of the air cushion and on the side hulls. The above expressions are written partly in the moving (x, y, z) system and partly in the fixed (x' ,y' ,z' ) system, but itis obviously to be preferred that we should study the motion of the craft in the steady (x,y, z) system particularly in view of the fact that the pressure distribution on the water surface and the motion of the water are given in this system. The detailed derivation of the forces and moments are carried out in Appendix, II, III, and IV where the forces and moments on the cushion hull, those on the side hulls and the rigid body forces and RSt Murthy moments are separately evaluated. The forces and moments of the same order may be added together and if, following Newman's notation we write the total forces and moment in the form : 2 = + = RSM Pp lt oe Oe nee | aoe Cc H 2 + OPK 199 +8 Xoo 09 * 80% 1 919 + EX 99) + B¢X qi 19 + = + Be Le (5-5) with similar expansions for Z and M, we have the following results : Longitudinal Force oy sy Ux, coats @ OTs Bota Sel 0 es he =-2pv ff @ Ge! .b, of ) 22 la 2.000 P 1000) Cees Ps Bh) 5 oe S x 1 oO if * to $e (x,y,0) dxdy a ane Se Rete If», 1000 = xX S fe) h jh Ge 22") 50 “T , 2 se : 2ev If | eor60 Fa Sty gy 3! okra A ne S x x x 1 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway 2 i iot EN) on TS Goden eae eee Gee 3x! 7001 Sz! Si 1 BOE TCs By ee come te gar et Wie + (x +h_ 0 ) {Flop +p WAT 2 © 001 G 001 roo Oo Po *0100 xx x xx is + og i Re e 2 FN Orig woe © ott. Ore +V 1%) a= +z a2 O24 (alae ou2 - x! ae )} 7 001 dz! 001 Guy Oz! P0100. hi 0 2 AEE AAV oun Va! aati) se | ede m 2P. P1900 + Po P1000 Moe ‘i [ e wi wil! P.O. @.< 133 Murthy a|a Qu * Qu N int [I]. = e - aaa O23) Rigow A Pes [fv=e.oo1 Y¥o001_,) lo 1 io.t = e = (<9) pa 0101 g° ee OP eF ange. "nope ) Pee So ae (5-6) Vertical Force te 0 (6) Pao ape ff (x' ,z' ) dx'dz' (4, = 0 ly ard oer 2100" “ff. ax ay -() ae 2, ws 005) 23099 * rf 2 BO +2pef(Z9 x'8) 59) b (x! : os r pore. h OH yoo 2ev fis 0100 an ** o100.. oa" ee + 2a fi (Zo10 = x 8510) h (x' , 0) dx' - 2 (190 L. +h, OT eB, dxdy = eS is 2 a 0 (8B) 25599 = - 2 Fo19 t+ 8G *oi0) ff Po, ee I [e) Lineartzed Potenttal Flow Theory for ACVs tn Seaway (a, rs iot _ 2 vail 1 1 ! 2pge f 01 x'809)) x h (x! ,o) dx' + E ud lot 0 (Ba) Moiio = 2 “01 * ®c ®001) © sce rot 3) Diss i) 2 iot AG Bee “Goa 1.2 ool, a. \esey cuore * Zi a) iot = 0(6¢)M,,,,=-2pe A [Ay Ce *Pa®o01) P1000, 5. 1, SN onias ool 1000 2 ov 10ior A104 x Z x Oh oh ] pve _a Ls ! ! 1 1 ee (2! sn yet) ayn ee dx'dz x ict at : oh es RES {fv Zooat 4% ®o01) P1000» Sz" CN ae L mE 2 a 1 1 1 1 x'8) 91) h (x fo) | x'dx [I hoe 9) I pull 2 2, iot ws Sao ge (ic Z901°100. 8° 101 is ®% 011 . 2 tort = = = Bil hee Maan ongas ile (*910%001 * 0018010 * 27011 Fowl So ©, 1 Po*001 p Po Net 1 eT oLevaol Po 7 Bowe g x DO .4 x tan 2 _vye@ emer} lionel e v"0100 | pay AG he @ (11 - 2901 10) I 2 Wort: SC i 9 8011 () g et & ae) ff 1 Murthy 2 ict = a ORCS Sa Wh Gags #2 7h ®901) - 0100 , 15 x'x a 6 \@t ot ee & - x! = Pp Vl go1 — 2) oor “Gree, PP ratiy (ae x's x h ah fz! Sion: —_ 1) 5 fees” SS een PO" hes 4 ax" az 001 G Oo) “(pee xXx X — = ! ai op. ip = enon = oo grea elie x x ah, dh, 1——— _ ! —— ! ! ! ! ! (z ae a1) +8 9511 2 h (x' ,z' )] dx'dz' + i by ¥ ya 4 e pbs x (Fi 998001 * To01%100 * 27101 fou + 25) 907001 ( e pa Vp 2,0" dociyyOt ; ee g Po 001 1000 ey ee xX xx x 2 q 2 Sy Ce : F425) he |} @ 101 ~ 2001 *100) ~ 8101 Ge eee (2 2 8 2A 42 ath A a) = i] vy Pa fat NC ca + (ST bg J 11 - 201 8010) - F011 A a ah ict ict = : + 1 én [fv (2991 ~ *'89,) @ 0100, az! ‘: L oe Lo al = a teat 1 Taaet gino gar? 2 ae 9) < x iat = (=, i ® = eC Moat a I Ge, 001 0001, ) S dh () ¥ (z' ax! - x! eae, dx'dz' 0(Be) M 0 (5-8) Lineartzed Potenttal Flow Theory for ACVs in a Seaway VI. STEADY MOTION IN CALM WATER When the ACV moves at a uniform speed in a longitudinal direction under the action of a constant propulsive thrust the motions of the craft and of the fluid are independent of time. The waves indu- ced by the air cushion and by the side hulls travel with the same speed as the craft and there is therefore no periodic disturbance, Although it is not expected that the ACV would develop periodic oscillations, it is quite conceivable that it will takeupa steady state trim onaccount of the steady disturbance of the water surface appropriate to the forward speed, Let us assume that the steady displacements of the ACV are: Surge Fe X10 + Xo10 + X110 heave BS 82 59 t Bz 19 t 982119 i = 60 6B 0 pitch 6 foGm oe como f B 110 (6-1) These are the displacements at and about the C.G. of the vehicle. The first set of terms denote the displacement due to the motion of the side hulls, the second due to the air cushion and the third due to the interference between the motions of the air cushion and of the side hulls. The only force acting on the ACV apart from its weight is the thrust T which may be assumed to act in a direction parallel to the deck of the ACV ata height h., above the C.G. This is on the assump- tion that air propulsion is employed. In the case of water propulsion, the thrust line will be below the C.G. and also, possibly, oblique to the deck surface, but the principle of the discussion which follows will be the same. The thrust is adjusted in such a manner that it is just suffi- cient to overcome wave resistance as this is the only horizontal for- ce (apart from skin friction, which is not considered in this study, for the fluid has no viscosity) to enable uniform progression. When the craft takes up a pitch trim the components of thrust along the x and z axes will be T cos@ and -T sin@ respectively, so that these should equal respectively the longitudinal and vertical forces at the C.G. of the ACV arising out of the action of the fluid pressure due 19 Murthy to the uniform motion and steady displacements. Also, the moment of the thrust about the C.G. - ho, - must be equal to the moment due to the fluid pressure. Using the expansion (5-5) for the longitudinal force and similar ex- pansions for the vertical force and pitching moment, we may write ee 2 = 6 = T cos@ ji +0(3 Te ae a) | T 6X, p00 +t 8*oi00 +6 X5 000 2 2 2 + oo eine Oe Bap) (6-2) *no= 160 +B6 5 B@ i sua pens 100 010° ito | * “°°? 1000 *.°46196n 2 2 2 Z + 6 Z5000 + B Z 9200 + BZ, 1 0 +0(6°B , 6B ) (6-3) h T=68M + BM ub we Agee + Tae 1000 0100 2000 0200 + 66M + 0 ( 5°B 58°) (6-4) 1100 It will be noted that we have not used terms containing a or € as we are considering steady motion in calm water. Referring to (5-6) and (5-7) we see that aa TT ee eae ST Salles and, similarly, 2 o0pm - pod: 200.” from conditions of equilibrium in the hydrostatic case. Similarly, during static hovering, i.e. at zero speed ahead, the total moment about the C.G, of the pressure on the cushion hull and that on the side hulls should be zero for equilibrium i.e. 6M + BM 1000 s 0100 — 140 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway Wen) =2up aS Oly (ex! gpa) dx’ dz" AL Bp. dxdy =I O S 0) = 0 which specifies the distance of the centre of buoyancy of the side hulls and that of the centre of pressure of the air cushion from the C.G. If these distances are x B and as respectively a result which is obvious. We may therefore delete the lowest order terms of 0( 6,8) from (6-2) (6-3) and (6-4). The first equation then shows that T is of 0 ( 5°. g*, 5g) z 2, = + = Ips (ODS + B 00 8BX, 9 (6-5) Orr eZ, ZS and since the left-hand side of (6-3) now becomes of 0( 6 ,6 6,68 ,8°) we conclude that 2 2 6 + Ly = in 27000 * ” Zo200 * 8871100 a G6) and we may also write = Alaeal, ign iM + B°M + 6BM (6-7) as 2000 0200 1100 The wave resistance is given exactly by (6-5) for T = - Ry and this will be discussed in detail presently. Some general remarks can be made without the actual solu- tion of the surface integrals for the forces and moment in (6-5), (6-6) and (6-7). Let us consider the integrals } uf L § Pp, Gxdy J Pp, dy Ss Ps pa S x Po dxdy = f x Po dy - JP, dxdy S x L iS) fo) Cc fo) 141 Murthy u 1 and If Po Py dxdy aa Sy 52 Leo 2 fo) % where the contour L_, is the boundary of the cushion C = T Le Ln, th, t+tbl,t 52 discussed in silane V. Since L,_ and L,, differ from the longi- tidinal planes y = +b, by the semi eedth of the hulls i.e. by 0(¢ ) we may indeed set dy =0 + 0(35) along L,_ and L, and the line integral may be taken over the bow and stern sections L, and Le only. In the case of a uniform cushion with p_ = constant throughout the cushion, the first and last line integrals vanish. The line inte- grals also vanish in the case of a non-uniform cushion with the pres- sure reduced to a zero or non-zero uniform value at the front and rear boundaries and generally, in the case of any non-uniform cushion with fore-and-aft symmetry both in the pressure distribution and in the plan form of the cushion. The line integrals will only survive when the pressure along the front and rear boundaries have different values, say, in the case of compartmented cushions. VI.1 Sinkage and Trim Let us now consider (6-6) which shows that each term should be separately equal to zero as the three terms are of different or- ders and substituting for these terms from (5-7). f L. <<" dx'dz' + rb te = 2 Oye + 906 ig ie gr ee Ty: JE x09 =F 008% om (x".'0) dx" =.0 (6-8) 2 a0 ae a Py dxdy = 0 (6-9) Sy x @) and 142 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway ff f dh, 4 dh, = ————_ —————— 1 1 2eVIJ (%1 99 Gat)” ony agi © Taxidzt + S, xe x o + 2pg JSG, - x'9 510) hos" jo) dx" = L -2 (E09 th, 00 Py dxdy = 0 (6-10) S x o In the case of an arbitrary non-uniform cushion without fore- and-aft symmetry and with different pressures along the front and rear boundaries th Py dxdy S) fe) x will have a non-zero value as has just been established. In this case, therefore, (6-9) gives = a i Og! ae Ol In the case of other types of cushion just discussed i.e. in the case of a uniform cushion or non-uniform cushion with fore-and-aft symmetry, the surface integral in (6-9) vanishes and the set of three equations becomes degenerate. In the general case, however, we have three equations abo- ve for the six unknowns determining the surge displacement, sinkage and trim of the ACV during steady motion, namely xX x Zz Z 6 *iao? “oly loo" treu0; roo: “= 7010 The other three equations will be provided by the equations for the moment which we shall now consider. Substituting for T on the left-hand side of (6-7) from (6-5) and equating terms of the same order given by (5-6) and (5-8) we have the three equations : 143 Murthy S 9h oh 9 -2p V(z' Baty” x! Oar? ?) 000 +g 100 z' h(x' ,z' )] dx'dz! = i m a 2 fe Seat?) ! ! ! - (52) ; 2pg | 2199 x 100) = h (x',0) dx 5 / en @ FS ee | cMaod balls dx'dz' 5 i Ox ie) If ) 2 Kyig the Might) *P, axdy - (2) e bg F419 So x 1 1 wes ef NT piive gag E pp p 0 ae ° 0100 oo) — x xx (6-12) ee |e 3 i ene x . 02 02 o Ox! oe Iz") Po100 iis pe es az) 0 6 ! ! ! e100, | Phe Se" Y aan L va a PSD ey hag a *io0) *, ony (3) 100 * m °o h _2\¢ & J a1 esate er krat 4 le oo _ x! OX dx'dz' - a 3 a2 + | dx'dz' + =i ff (x, y,0) dxdy 0100, 9. "1000, (6-13) We may simplify equations (6-8) to (6-13) using the following rela- tions : oe ge o) dx' = A 28 f x h(x" 0) dae! = AP 50 2a f x hy (&o) vax! =" dy er )-dx'dz’ = i m 144 Lineartzed Potential Flow Theory for ACVs tn a Seaway We thus have the set of ice We ; yo fe) ae ie b; Zz oh la Z 1100) Ter A aUOOwe | ot P00. ESTO Sc a J 6 = as X10 + 26 010 0 “ff Po di dy =7 20 (6-15) ve 2a a°2 = = —=— 1 ! Z@1i0 wt As OAi0 av ff 0100. , a © ooo. , apn eee os ee a he eC )f, By Waaccelah (6216) peA 100)! G00 O. So Zz i Y Ud NGo eri ath of iain) as got fle (eee ee “100 pA xX, 10020 can ape! dh ssi edad oni ileal S x at 1000, dx'dz (6-17) m 1 g f x als 2 (199 tg “100! Nee a ego eG A0e ( ) S B h uy ee i] pe?) eo —h = 2 pg *o Sey Pele 0100 ces enue) C m ioe 00) f* PEG Ne gg Ne. mae 2) 2 h dl Ong = ! eres fs baie 20 ff|v (z + ho) 5 x 32! ? 5100 3 +: xe 6G) @ 2 - ! ed 1 ! 1 +V{ (z ie ea) ai aa 0100, |= dz' + +— be ff Po 1000 (x,y,0) dxdy (6-19) So 145 Murthy It will be readily observed from the above equations that when x =O0i.e. when the C.G. of the waterplane area of the side hulls lies below the C.G. of the ACV, the pitch and heave displace ments are uncoupled. The above equations are applicable to arbitrary non-uniform cushions, From (6-15) and (6-18) we readily obtain ho 2 B25 9 °F *- Be yg “E> oe [vr dy 2pg m, Lath, Vh,, ee aa a [[ 22.) or0 ) dxdy (6-20) oO so that the steady surge and pitch displacements of 0 ( 8) are zero of the thrust line passes through the C.G. The other four displacements can be obtained from the four equations (6-14), (6-16), (6-17) and (6-19). The displacements in the lowest order X19 + FX o10 82199 + B2o16 and 2150+ P10 have thus been solved. The higher order displacement of 0 ( 68) can also be obtained by considering the higher order forces and moment, In the case of uniform cushions or non-uniform cushions with fore-and-aft symmetry, If Po dxdy = 0 S) fe) 146 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway and equation (6-15) does not exist. We could obtain x ae iz 5 and. 6 100 from (6-14), (6-17) and (6-19), but there are on eee two remaining equations for solving 910° 2010 and 10 . We could, however, make the additional assumption that x 1 and = are both zero, for the steady surge displacement is not a use- ful quantity and can, in any case, be absorbed in the co-ordinate system, In this manner the sinkage (heave displacement) and trim (pitch displacement) can be calculated. VI. 2 The Steady Potential It is shown in Appendix V that the potential in steady motion can be derived in the form Bhs ® (x, y, z) ©1000 * PPoi00 where I ce) Nd oh. E (x, Y> 4; 24519), oo) TG (x, Ys 2; Seen ie y)arras and V aPG Solan eee at Gia. 256 5 2st) dé dwt... 2 So ie 4 rpg P5100 > +} FS oé > 3 ? 3 > a BS IFS ales F100 (97 eee G (x,y, 238.7, o) | dn- “tfc ide rah eas apes ei aa, ei S. ) a dG (Gey Wisizs ae Wess, | dé’af’ dn’ ; 07° aes we 0 (6-21) As indicated in Appendix V, we have only derived an integral equation for ® 0? although ® has been explicitly solved in the form of an integral representation on the assumption that the separa- tion between the side hulls is sufficiently large for the interference effects between the hulls to be considered negligible. In this case there will be no 'jump"' in the potential between the two sides of each hull 147 Murthy and an explicit solution is possible. The velocity potential due to the air cushion now derived can be shown to be of the same general form as that previously obtained for the acceleration potential of an amphibious ACV in Reference 1. There is, however, an additional term now in the form of a surface integral over the longitudinal planes of the hulls which provide a ver- ticalbarrier along the lateral sides of the cushion, The Green's function to be used for steady motion is 7 Ye = 2 2 2 Gt. ¥,2:.0 5) = (x -[) +(y-n) +(z -r)| wa ~1 4g V2 “1 [ix -£)7 + (y -1)" + (2 +) | 7) ee - p(z+f) + ip (x-&) cosé@ II e cos p(y - 7) siné aide (6-22) re OM g-pVvV epnkig where M is the contour along the real axis of the complex p-plane passing above the point p=p_= Beene oO aa The integrand in (6-22) is complex, but as we are interested only in the real part of the integral, we must seek the contribution from the real part along the real p-axis and that of the imaginary part along the semi-circle above the simple pole p = k sec @ where Ko = el vel We thus obtain after evaluating the residue -1/2 Eleysstnt)= [ese ty-1+@-FP] = -1/2 - [le -8) +n tees) - Ty -k_ (z+¢ ) eH e = 4 k Le 3 sin lk, (x-&) sec a | 148 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway "2 Z 4ko cos [iss (y-7) see ete | sec 6 dé - secté dé ee) 0 era) cos | p(x- & ) cad) cos hee ") sin | 2 (6-23) wnere f indicates that the Cauchy principal value is to be taken. VI. 3 Wave Resistance The wave resistance is given by Te el oe Wig WwW 2000 ote: 0200 1100 and substituting for the longitudinal forces from (5-6), we may write 2 oh = cc) ! ! ee ' im ee Ry, = 2pVs HE 1000. (EI eee act isle 1 B2 ° ] -— —_— - V h we dy + g oe Po Po Po 0100 Oy °) rey & x xX h h 071 a2 + Ca) ! i> @ ! pet ! ZpV lf idea ve b+; 2! ) Ae + paae oe bea Zz SS =) dx'dz' + Jf. Sane 0 bone 0) dxdy The first term on the RHS, is the wave resistance due to the side hulls in calm water. It is assumed that the separation between the side hulls is sufficiently large to avoid the necessity of evaluating the potential on the two separate sides of S, due to possible inter- ference between the two hulls. The second em gives the wave resis- tance of the air cushion and the two remaining terms represent the interference effects of the air cushion on the side hulls and of the si- de hulls on the air cushion respectively. These terms may be evalua- ted separately and then arded together to give the combined wave re- sistance of the entire system moving in calm water. (6-23A) 149 Murthy Considering first the hull resistance and substituting for the potential from (6-21), we have 1p lo Gy (et bs 258d, F) +G, Get bea Bb, £"D] (6-24) Now aG(x' ,b, z'; Ae Pa 5 l= Fy? + (e-r )?] “~— a x = E' 1/2 2 -k (z'+¢') snap 3 ae er e cos oe (x'- —') sec | sec 0 dé + , W7 co 4k =p(2' fit yi ~- f ) ia of pe sin | p(x- —) cos@ 0 0 3 -k seen e fo) (6-25) and CLO Ne gr ds ling en f" . ss + ax (x! - V2 ey a 7] 3/2 rr a 2 3/2 [(x"- gly? + 4b fe 2 -k (z'+S) eee 3 Meeks ean? cos [i (x'- t')seco| cos [2k bsec ) sin gf 0 1/o es 4k + (ates) | alc nnd sec¥dé +—°} secoaspe® _._Sin[p(x' “)cos@Jcos|2pbsindy, 7 (=) 0 p-k sec” 8 * (6-26) Lineartzed Potenttal Flow Theory for ACVs tn a Seaway We may write (6-24) in the form iy R 20 [force 2") dx'dz' pee (§'8") ae dé: 6 W hulls . 9 ! lg ¢30 ' dpe @: le, Goma nee did Mh ez ay my | where = and f, are given by (6-25) and (6-26) respectively. The above equation may also be written — 2 1 pV dh( e764) , ahi" ,.z") aut R = en ' 1 ! a mati ih 0g’ oof Ox' L2C2 S S. 15 lo lf (35 osx! Jz! ) 4 a Ce eee jz! | by interchanging x' with &' and z' with S$’ An Ox! S14 ee toe HOELE « we EZ a) ht A ies 6 is 6 (8, 03x, 21) by the addition of the above two expressions and taking half the value. It will be seen that the first, second and fourth terms on the RHS of (6-25) and (6-26) are odd functions of (x' - £') so that they cancel respectively with each other when we take the sum fie (Gale ouz new ec: Shetty (Cee aye ez) and feat nee Si Detbicl wa babs oeel 36' eazeke) but the third term is an even function and gives a contribution to the sum by doubling itself in each case so that we have E51 Murthy ape seach ic a Ry : A? farce 2 cxtae'ff UE) abiat 6 hulls z ‘ 2 I 1p 5) ™ 0 [o (z'+f"') OP a ear [« (x'- €) sec a eo oO 0 2 ; 3a E +cos2k) b sec 9 sin i sec 0d0 (6-27) If we use the expansions cos E (x'- &') sec 6] Rw = _ / (Py oe) geoos dé (6-30) 5 hull 7V E ? where 2 P +iQ sails (-k_z' sec 6 + ik x' sec@) dx'dz' fo) fo) fo) rf) which is the fame Michell integral for the steady state wave resist- ance of a single hull. The constant 2 is usually given as 4 in Michell's formula, but we have taken h to be the total width of the hull and not the semi-width. We will now consider the wave resistance of the air cushion 1 1 i ee pr eco | fe 25 of 33) d Beri Nh aE Sel Po 100 Y¥ | aay cusnion x xx 5, where the potential is given by (6-21). This is an integral equation and attempts are being made to solve this, but we can obtain a sim- ple integral representation for the potential of the form V ee) %100)° dang TC.a8. dey (Gel Gre sii VARS a) oh) didn if we make the following assumptions. (i) The pressure in the cushion is diffused in such a manner that it becomes zero at the front and rear boundaries where the plenum air escape with air entrainment from the atmosphere and the generation of trapped vortices will probably ensure that this is so in practice. If this assumption is valid, the ''jumps"' in the potential and in the longitudinal velocity of the water particles will vanish and the line integral in (6-21) may be ignored. (ii) Although a discontinuity in the potential may be assumed not to exist at the front and rear boundaries, a discontinuity will cer- tainly be present across the vertical barriers imposed by the side hulls as there is no air escape across these boundaries to alleviate a discontinuity of the pressure. However, if the depth of immersion of the side hulls is small and of the same order, say, as that of the hull width, the surface integral over S, will be of 0 (6) higher 1 fe) 153 Murthy than that of the potential and may therefore be ignored, As stated above, the above assumptions may be made pending a rigorous solution of the integral equation for the potential so that practical results can be achieved even if they are approximate ones, Also, under assumption (i) above, 2 Ifo. dxdy $x dy = 0 x S L 0 and we may write simply 1 Vv 2 Rw Te i B cushion 1 => K an ot ates H 1 ga | < a on ra os — So So ra Q << H 1 < ia?) S Te} mS hae | x| x ) Q * a — Q ue) a]/o Caam > Q Jwtr Qu 3 Anpg 0 fe (xiv; on8, n, 0) (6-31) This is almost exactly of the same form as the expression (6-24) for the hull wave resistance with Py instead of h, Now a | 8G (x.y, 05 £050) 0; $050) == ae [ cos [x (x-&) sec | Ox fe) ° Wf, 2 z ” 3 a cos [ (ym) sec 0 sin 0| sec 6 dO + —° sec 6d0@ 0 154 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway p 2 p- Kk) sec 0 fp sin ce £) cos | cos hee n) sin | d 0 Substituting in (6-31) and noting that the second term on the RHS makes no contribution to the double surface integrals, we have 1 —R Z i "2 y) [|- (€ 7) = eam Tee OKOVMNC are Sy B W cushion we zs JS ue 50 yy S 72 ) 2 3 cos | Kol — ) sec a|cos Exe n) sec @ sing] sec 9 q9 It is easy to reduce this as before to the form 1/2 1 tales = < (Gee Q* (0) | Bee dé (6-32) Dior WW: : a fas B cushion ;zpV where now P (6) op. (&, y)) {eos i ax fk, (xcos@ + ysin@) sec i dxdy Q(e)} 2 50 sin (6233) These expressions may be compared with (6-28) and (6-29) for the hull wave resitance. It is clear from (6-33) that we may write by the use of Stokes! theorem, P (6) cos > = Dp. (x, y) [k, (xcos@ + ysin@) sec A dy - Q (6) sin -sin b 2 a ko sec off (x, y) lk, (xcos g + ysing)sec a | dxdy cos °0 The line integral vanishes and we have 2 Tle k os oe 2 [& (9) + O° (8) | sec @ dé (6-34) ZrO, : 2 1 i B cushion mpV i where Murthy cos > olf (x, y) [k, (xcos6@ + ysing) sec 4 dxdy Q (6) i ie (6-35) These results agree exactly with those given by Havelock [6] for a surface pressure distribution which is continuous and is zero at the outer boundaries. We now come that part of the wave resistance which is due to the interference between the air cushion and the side hulls. Referring to (6-23A) we shall not attempt to evaluate the inte- gral over S]_ for under the assumption made habeas <2) (i.e. with small side hull immersion) the integral will be of 0 (6°@). The in- terference of the air cushion on the side hulls may therefore be neglec- ted: The effect of the interference of the side hulls on the air cushion is given by : I oR =—TI||/p © (x,y,0) dxdy = 66 Me ee eiamoace 8 S , = aff off dé' df’ late, vo: ©, Be Sn +G fo: 776: & +3 )] (6-36) It does not appear that this double surface integral could be reduced to the simple form of a single integral as in the case of the hull resistance and the cushion resistance. VI. 4 Drifting Amphibious ACV The case of an amphibious hovercraft drifting in calm water has been discussed in section 7 of Reference 1 and a solution for the wave resistance in closed form has been obtained in the form 12 Tip = R =< [ben + 07(0, 8) sec” 9 dé 2. OW : 2 B cushion 27pV - 1/5 156 Lineartzed Potential Flow Theory for ACVs tn a Seaway with P(g; B) a? By pow i nic (:cos(e+8) + ysin(o+e)) sec | dxdy (e=am where Bis the angle of drift on the right-hand side. The side force F in drifting motion has also been obtained in Refe- S rence l 1 i 1 fe) 2 2 6 = F, =—°~]Ip" (0, 8) + Q° (8, B)| sec 8 sin (B+ g) dé cee oh ae It is not worth attempting to find a solution for a drifting ACV with side hulls since the lateral motion will induce a disturbance of the water which will not be consistent with the basic assumptions of this linearized theory. Vil. FORCED OSCILLATION IN CALM WATER Let us now consider the case of an ACV which is forced to oscillate in calm water during steady translation. Obviously, we have in mind the forced oscillation of an ACV model during towing tank tests. We will now derive the oscillatory forces and moment acting on the craft when the motion and oscillations are confined to the lon- gitudinal plane. Such a motion can be deliberately imparted to the model by a mechanical oscillator such as the Planar Motion Mecha- nism (PMM). The added mass and damping of water can be determined by experiments of this nature in calm water. In the case of an amphibious ACV free from water contact, the discussion of the motion in the longitudinal plane can be applied direct- ly to motion in the lateral plane for the disturbance of the water would be comparable in both cases, the beam/length ratio of present day hovercraft being of the order of unity. The discussion of surge, heave and pitch in this section will then apply also to sway, heave and roll in beamwise motion, IRS y7 Murthy VII. 1 The Unsteady Potential The potential can be expressed in the form Pakep 2 ih steg (x, y, z3t; 6;8;a@) * 000 35 B00 + ict bab is & i010." P*?o110* | The displacements will consist of the steady terms constitut- ing the trim of the ACV in calm water together with the oscillatory Lenms: : lot igt- + +édae xX 1 Bae XO1] ‘i pair gags Fl = iogt. = =x + BX, + ae X01 10 with similar expressions for z and @. In the above expansions g is the frequency of forced oscilla- tion. The discussion in this section will apply equally to free oscilla- tion due to wave excitation in which case o will be the frequency of encounter of the waves. This is discussed in the next section. The steady potentials Pio and P10 have been discussed in the preceding section. The osci idtary paveuriate } and 75110 are derived in Appendix V with an explicit integral representation for the former and an integral equation for the latter which could, however, be simplified and an explicit solution obtained under certain assumptions similar to those outlined in the last section. VII. 2 Lowest Order Restoring Forces and Moments The lowest order restoring forces and moments are obtained from (5-6), (5-7) and (5-8) and after simplification reduce to: iot I pee iot e (8aX) 919 *BaXy 119) = - 001 scaadee GE 52 +BaZ ) =ae A gE Ott a 9 t 1010 E (299) -*a “poy? 2 Sgt ek BERS ile 9 i 2001 ~ 901 * 8G 001) |] Po, pains! 0110 158 Linearized Potenttal Flow Theory for ACVs tn a Seaway iot ( i 1010 2 a: + ( pel, -M,82Zp - % I) 9001 - PBZ 00) Ax, + + 2 (x 4a eer 0) ante) Bxp dxdy Dolan oor ff O. | ee 5 These expressions give the restoring forces and moment acting on the craft when it is given oscillatory displacements in sur- ge, heave and pitch of a forced nature. The longitudinal force contains the inertia term only, but the vertical force and pitching moment contain in addition, the contributions due to hydrostatic pressure. The velocity potential of the water does not enter in the forces and moment of this order and we cannot therefore expect to find such effects as the added mass and damping of water which are of a hydro- dynamic nature and of a higher order. VII. 3 Higher Order Restoring Forces and Moment Added Mass and Damping Effects The higher order forces and moment can also be written down from (5-6), (5-7) and (5-8) in the same manner as (7-1). These second order forces and moment include the added mass and damping effects of water as the potentials ® and ® enter into these expressions in addition to the steady potentials and the steady displa- cements as the steady disturbance of the water persists during the oscillations. The pitch and heave stiffness of the air cushion also play their part in determining the forces and moments of this order. It is clear that the steady potentials ® and ©® (which are real) will be in phase with the displacements and will not there- fore yield any damping ee. The only contribution will be from the unsteady potentials and ® which have a real part and an imaginary part, so 1949 when the product with et, as taken there will be components of force in phase and in quadrature with the displacements giving rise to added mass and damping effects. It will have to be remembered that we are discussing only hydrodynamic effects here. The damping due to aerodynamic and pneumatic effects in the air cushion will also have to be taken into account, B59 igt baM aig BoM), 10) = ae | hg (o x - E4_9}) + Murthy VIII. FREE OSCILLATION IN WAVES We will now consider the case of an ACV moving at a constant speed in a seaway composed of regular waves with their crests nor- mal to the course of the craft. It is assumed that the craft has been operating for a long time so that all the transients would have subsid- ed. The craft would then undergo periodic oscillations in pitch, hea- ve and surge with the same frequency as that at which the progressi- ve waves are encountered. It is assumed for the purpose of this ini- tial study that the seaway is composed of a single system of simple harmonic waves of a particular frequency. An extension to motion in an irregular seaway can then be made by using the methods of spectral analysis. The velocity potential of the water will now be composed of several terms as follows : Ai Be = 6 ig, t oy Sets 6, pie) 1000 * B00 * Re-e [#0001 = Sere rors | where ® and ® are the steady potentials for motion in 1000 0101 : ; : calm water discussed in Section 6; P0001 is the potential of the incident wave; f is the potential of the wave diffracted from the side 1001 hulls; Po101 is the potential of the wave representing the disturban- ce of the incident wave by the air cushion; and ae is the encountered frequency of the regular waves. VIII. 1 Incident Wave Potential It has been shown in Reference 1 that the velocity potential of a regular wave of amplitude a 3 circular frequency o and progres- sing along the negative x-axis may be written in the form a >= = (=) exp [-K2 +ifkx to t+ 7 | where the wave length 160 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway eee k ' and the wave number Ik = oe & The frequency of encounter Cie ey iO etki, e and y is the phase angle of the wave. As we have denoted the amplitude/length ratio of the incident waves by the small perturbation parameter ¢€, we may write p= € Eee exp |-k2 + i(lex + ott+y | ak e so that in our notation GG | : | P0001 Ripe: XP -kz + i(kx +y ) (8-1) VIIl. 2 Lowest Order Exciting Forces We are now in a position to compute the lowest order exciting force in the longitudinal direction from (5-6) = idet = oh Ido! bX oo) +BeXq 194 = 2éepe [feo*e001 Y¥o001,,) ax! dx'dz : S. 1 ice t 10 sVOlE zi : ae 2 (io e*0001_ VP001 ) Pp, dxdy So and substituting for the derivatives of P5001 from (8-1) we get 4nripg icet = §¢ tUiPg Xi 001 tP&Xo101 jie Sea wee -kz' + i(kx +y 5 a dx'dz' = Sh Dar Bee: ve ffon [ Ec + ¥ ) Py dxdy (8-2) The vertical force and ae moment due to wave excitation arise mainly from the side hulls as the air cushion provides no contribu- tion and we have 161 Murthy 4ri eizet d« Z001 =de —= cae lf - kz' + ilex +7) 3% dx'dz' (8-3) and be ee nt = be aniPE se -kz' + i(kx +y)] lz" a - x! | Sita! (8-4) It will be noted that these forces and moment are purely of a hydrostatic nature. VIII. 3 Free Oscillation When the ACV responds freely in waves taking up displace- ments appropriate to the excitation, restoring forces are developed on account of the displacements. It is assumed, of course, that there are no external forces acting on the craft apart from the constant thrust. It has been shown earlier in Section 6 that the small displa- cement in pitch will not provide a restoring force in the longitudinal or vertical directions although the thrust line is displaced in space. There will also be no restoring moment available in pitch on this account as the contribution is of a higher order. As the ACV is in equilibrium under the action of the restoring forces due to the dis- placements caused by the exciting forces, we may set Force Force Restoring + Exciting = 0 Moment Moment D4 x i.e. 62/7} + BaiZ)} + higher order restoring forces + M M x x + 6€1Z + Be{Z} + higher order exciting forces = 0 M M It is clear from the above system of equations that the order of the displacement caused by the incident wave has to be the same as the order «€ of the waves themselves. Using the expressions derived in Section 7 for the restoring forces and moment, we can now write the following set of equations : 162 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway m : i} 2 Be! lget 4ripg iget tig 1001 7 -{ 5 ) ce 00” mien. [fore[xe +100 1] 38 dxitdz/ = 0 S lg m 1 h 25 eroet 3 iget 2 OO Boies + Soo Xo110 X0101 ( B (0) iget A fo) eA eZ = = ee Bi ae ei, ONO MOOT ar. les =} Zo01 =a “oo1) ( 2). Z01| ti Ari ioet +e el” |e - kz' + i (kx rv dx'dz' = 0 ali(er( Biype. ( : + = os alo, Otol 1. sont G Oot) m IN 2 loet _ I. dxdy i F. 2991° = 0 So x i loet Za ect, = eTteuomid siGol 5 Yn ( 7%) ~ $801) * C2) (B)on- les rasgy Cho} ne 3/°'B “ce \8/\ 001 - og 20, ay A 4m7ipg iocet : ‘ iit , oh pa Wait sania anus Bee +i(kx +7 ) Blea ah een dx'dz = A M +M Soles ee lie. "te eee dady + 0110 0001 DOde- 7G OL 2 o So x m I 1 oe: Lye 2 iget AE he ( #001 7 2%01) (Ae 01 ag (8-5) The lowest order displacements in surge, heave and pitch are given by aX, gisele oe vbaeisebir and oe ey and it may ap- pear from (225) that there are six equations oe these three unknown quantities. However, these should be reduced to a set of three equa- tions, by setting the total longitudinal force, vertical force and pitch- ing moment equal ve o. This step is suggested by the fact that 6 although (+) and , the masses supported by the air cushion 163 Murthy and side hulls respectively, are known from design considerations, it is not possible to separate the total inertia I il 1 =A) +8 (72 B 6 I I , 1 2 into separate components fe and i) : 6 We thus obtain the set of equations : (ge mo. Xoo] = = arf BP, expi(kx +y ) dxdy - ry p | sdf -kz' +i (kx +y |e seslis 2')] dx'dz' 4nripg Aes - pgA - xe y = e k 001 001 A, OO) ! ! Jew |-kz" +i (kx +y ) elener.2' D] dx'dz' + 5, / + (0) + ha ®n031? | Pip dxdy = 0 x te) mh (9. X91 ~ 8%) + (rel, - mA 8%, - Pee 4 4 PBZ 501 Ax, wi + peaffe| Ex +i (kx +7 | O[sh(x' ,z' )] _, af6h(x' ,z') ; . ET aber le alone 52, dz! + +2 (%_, +h, ®01 ff ex, dxdy = 0 g . (8-6) So The first of the above equations gives the oscillatory surge displacement explicitly. The left-hand side is actually the surge acce- leration and thisisindependentof speed for the term o, which depends upon the speed of motion is absent on the right-hand side. The other two equations enable the solution of the heave and pitch oscillatory displacements. As before, these displacements are 164 Lineartzed Potential Flow Theory for ACVs tn a Seaway uncoupled if Nai OM, When we set the parameter 6 denoting the thickness of the side hulls equal to zero and also the water plane area A equal to ze- ro we get the results for an amphibious ACV which are discussed in great detail in Section 9 of Reference 1. Having calculated the oscillatory displacements and thereby the accelerations in the threee modes we can estimate the ride com- fort in waves by combining these accelerations in the appropriate forms at various locations in the ACV. The higher order displacements can also be derived by consi- dering the higher order forces and moment. The response functions in surge pitch and heave can be com- puted for a specific ACV configuration and motion predictions in an irregular seaway can be made by the use of spectral analysis within the limits of the theory of linear superposition. IX . DISCUSSION AND CONCLUSIONS The assumptions underlying this theoretical investigation of the motions of an ACV in a seaway and some of the results obtained here have been discussed in the Introduction and Summary. Attention is confined in this study to coplanar motion in the longitudinal plane with freedom in surge, pitch and heave only. Results for the amphi- bious ACV free from water contact can be obtained from the general results by setting the hull parameter 6= 0 and omitting the surface integral over the longitudinal plane of the hull and the line integral over the waterline occurring in the integral representation for the potential due to the air cushion. These results can then be applied equally well for beamwise motion in the lateral direction as the beam/length ration is generally of the order of unity and the distur- bance of the water due to longitudinal, drifting or purely lateral mo- tion will be of the same order. Obviously, this does not apply to an ACV with immersed side hulls. The primary results are contained in the horizontal and ver- tical forces and for the pitching moment derived at the end of Sec- tion 5. As may be expected, the lowest order forces and moment are purely of inertial and hydrostatic nature. The hydrodynamic pressure of the water does enter in the higher order and it is possi- ble to calculate the added mass and damping of water, the mean in- creased resistance due to forced oscillation in calm water and due 165 Murthy to the free oscillation in waves etc. The evaluation of these quantities and the derivation of the higher order potentials such as those due to the interference between the air cushion and the side hulls, the diffrac- tion of the incident wave by the side hulls and the disturbance of the incident wave by the air cushion are not carried out here, but the simpler results suchas the steady trim taken up by the craft during uniform translation in calm water and the expression for the wave resistance which combines the well-known results of Michell and Havelock and also introduces an additional term representing the ef- fects of interference between the air cushion and the side hulls show that our method of approach to the solution of the problem is a prac- tical one. The expression for the side force on a drifting amphibious ACV and the response functions for the amphibious and non-amphi- bious ACV (for which expressions have been derived although not explicitly solved here) will also have practical applications. The ride comfort in waves can also be estimated by combining the levels of acceleration in surge, pitch and heave in an appropriate manner depending on the location in the ACV. It would, however, be prema- ture to suggest that these response functions can be used for the pre- diction of the performance of the ACV in an irregular seaway by the application of the theory of linear superposition in the absence of experimental results confirming the linearity of the motions in waves of small amplitude. The method of. solution presented here can also be used in the case of the water contact of the flexible extensions and even in the case of immersion in water if the flexible extensions are assu- med to be of a fixed shape. A later extension could cover the case of flexible extensions compliant to the water pressure. It cannot be stressed too highly that the theory presented here must be used with a certain amount of caution when applied to the actual operation of an ACV over water. The underlying assumptions for the linearized theory are that the cushion pressure is small, that the hull is "'thin'' and that the speed of translation is moderate or lar- ge. The slope of the induced wave may then be considered to be small. Also, the oscillatory displacements and the slope of the incident wave should also be small quantities. It is needless to add that the theoretical results derived here should be confirmed (or corrected) by experimental work such as that with a mechanical oscillator of the PMM type and also by full scale trials so that the scale effect can also be established. Apart from the configuration of the side hulls, which can no doubt be per- 166 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway fected by the naval architect, the ''hull form" of the air cushion appears to play an important part in the performance of the ACV. It is com- monly assumed for want of more precise knowledge that the pressureis uniform within the cushion, It is obvious that this cannot to so unless the flexible extensions are also immersed in the water in the same manner as the side hulls. There seems to be some evidence to show that the pressure is reduced to atmospheric at the front and rear boundaries of the cushion where there is leakage of air. This renders the mathematical work slightly easier and enables practical results to be obtained, This also gives the ACV cushion a "hull form" with an acceptable deadrise and flare which may be considered by the naval architect as very suitable for fast planing motions. It is also necessary to point out that we have gone into some detail to study the hydrodynamic (including hydrostatic) effects on the motions of the ACV. It is assumed for this purpose that the aero- dynamic effects are known including, in particular, the stiffness and damping of the peripheral-jet or plenum type of cushion. It may be thought that such effects as that due to ''wave pumping'"' should have been taken into account, but this is purely a pneumatic effect as the compression (or rarefaction) of the air cushion is due to the form of the progressive wave underneath it and not due to the pressure of the disturbed water and as such has not been treated here. No doubt this important aspect will have to be taken into account along with other aerodynamic effects in arriving at an overall picture of the © motions of the ACV in a seaway. The higher order expressions for the forces and moment deri- ved in this study are extremely complicated but their solution by nu- merical methods with the aid of present-day high speed digital com- puters need not present any serious problems. The finite element method (FEM) which is receiving increasing attention in recent years may prove to be valuable and powerful tool for the solution of pro- blems of this nature, 167 Murthy xX. ACKNOWLEDGEMENTS The author is indebted to the President of Portsmouth Poly- technic for the facilities provided for the preparation of this study and is personally grateful to Dr. D.C. Chandler, Head of Depart- ment of Mechanical Engineering and Naval Architecture for his interest and encouragement of this work, Special thanks are also due to Professor G.M. Lilley, Head of the Department of Aeronautics & Astronautics, University of Southampton, for his continuing interest in this development of the previous work carried out at the University. The assistance provided by Mrs. Ann Martyn in the patient and careful preparation of the manuscript is also very much appre- ciated. 168 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway XI LIST OF SELECTED REFERENCES ] Murthy T.K.S. Peters A.S, and Stoker J.J Newman J. N. Newman J.N. Joosen W.P.A. Havelock T.H. A Linearized Potential Flow Theory for the Motions of Air Cushion Vehicles in a Seaway Univeristy of Southampton, Department of Aeronautics & Astronautics, A.A.S.U. Report N° 299, June 1970, (240 pages, including 5 appendices; 12 figs) The Motion of a Ship, as a Floating Rigid Body, in a Seaway. Comm, Pure and App. Maths., vol. 10, N° 3, 1957, pp 399-490 The Damping and Wave Resistance of a Pitching and Heaving Ship. Journal of Ship Research, vol. 3, N° 1, June 1959, pp 1-19, A Linearized Theory for the Motion ofa Thin Ship in Regular Waves. Journal of Ship Research, vol. 5, N° 1, June 1961, pp 34-55, Slender Boby Theory for an Oscillating Ship at Forward Speed. Fifth Symposium on Naval Hydrodynamics. U.S. Office of Naval Research, Department of the Navy, ACR .- 112, pp 167-183. The Theory of Wave Resistance. Proc. Roy. SOC Av uvOlen 136, ULO22 sop 349-5 46- A complete list of 40 references will be found in Reference 1. 169 Murthy APPENDIX I DISTRIBUTION OF SURFACE PRESSURE DURING OSCILLATIONS Let the pressure distribution on the IFS due to the air cushion be of the basic form PSs PR, vee during ''static hovering", i.e. at zero speed ahead. This distribu- tion determines the basic "hull form" of the air cushion. We may assume that the same distribution prevails even during steady trans- lation, The motion of the side hulls through the water will generate an induced wave, but this will only cause a pressure variation in the interior of the fluid leaving conditions on the surface unaltered. This will also be true of the incident waves whose surface pressure is equal to that of the atmosphere, but the action of the waves in altering the cushion pressure due to ''wave pumping" should be taken into account, This requires separate treatment. In other words, the sur- face pressure distribution is dictated only by the motion and oscilla- tions of the air cushion and will remain of the basic form if the rela- tive vertical and angular positions of the air gap at the bow and stern are the same as those when the ACV is stationary. We are, however, allowing for oscillatory displacements in surge, heave and pitch during uniform translation. The surge displa- cements will not directly affect the pressure distribution, but the displacements in pitch and heave will certainly cause a variation of the design "daylight clearance"! along the bow and stern sections of the air cushion and a different distribution of pressure on the water surfa- ce will result. The situation is fully discussed in Appendix II of Refe- rence 1 where the result for the revised pressure distribution is deri- ved in the form i 24 fea y) df (x',y') : Pe (x,y) = Po (x, y) + [x +h. @ + 0(8) ee ae ant ee Of, tx ay" ) 2 Bg o8 Pr Pa pe 06%, 2] aoe a x= X ap. (x,y) ef (x',y') : +3 [m+n 9 +0 (0")|? —— 9 hr 2 G Ox xt” 170 Iineartzed Potential Flow Theory for ACVs in a Seaway 2 def ceny) es 3 +Z wine) oe Oe »z )| + 0 (x + h.9) Ox! x'=x where h_ is the height of the C.G. above the undisturbed water sur- face andf, and f, are the pitch and heave stiffness functions of the air cushion, It will be seen from (2-2) that the basic hull form is assumed to be of 0(8). The pitch and heave stiffness functions are therefore also of 0(8) and we may write Bp, (x.y) for p. (x, y) Bf, Gc" ye) fox f (cee yi) and 1 ! ! ! ! Bi, (Sh si ) for £, Cry) we may also substitute from (2-3) x = 6X +BXx oa DGit ops ie eo 100 *?*o10 * 98 X19 t+ Re-we = (qq) + 5%) 9)+B%11) + 2 + 0 ( s* B 5B a, a”) with similar expansions for zZ and 6 and derive finally [Ae = lot_ Beto) ep. eat 8 pollbno oe toate eae ees (F)90%001 * T001°100 * Oo “oo1™ 100% _)* eSgerea t Be Hgts x Se Beh 2: + + 1B we” (For ofool 001 “O10 > stor “0107 001%) aes As pwith + +¥ + + + 88 (Fs 998010 * Fo10%100 “110%. *1007010% ? Piaget i es 1001002 100% )+ >. O.< Murthy + 0 ( 6°, 8°B, 68a we | (I-1) It may be as well to point out that the terms of 0( 5“ B58" Ba ) are slightly in deficit due to the forcible linearization implied in writing a Z = 's x- x xt+h.9+0(6) where the 0 (6”) terms cannot be explicitly written down. We have written above = & 4 i “kim = “kim” “G %kim for the coupled displacement parameter and Skim = ‘1. “kim °2 for the coupled stiffness function. It will be seen that the pitch and heave stiffness functions enter only in the higher order terms. Also, if the cushion is truly uniform in the longitudinal direction such that p =o", Dp = 0 the variation of pressure distribution due to the oscillations is of a very high order. itZ Lineartzed Potential Flow Theory for ACVs tn a Seaway APPENDIX II PRESSURE FORCES AND MOMENTS ON THE CUSHION HULL The force due to the action of the surface pressure on the cushion hull is form (5-1) where S is the instantaneous position of the IFS and is the unit normal drawn out of the water surface and into the lower boundary of the cushion, The equation of the IFS is given by Gcsy) =z = 0 so that the outward drawn normal is NA NA A Ns % G+ Sy 5 -k = ee ee fi + Sx" + Sy | 1/2 and Z dS = af + s + ‘| Wye dxdy where the negative sign has been used on the right-hand side since dS is an element of the cushion hull positive on the upper side of the free surface, whereas in our co-ordinate system the element of area dxdy is positive along the z-direction which is vertically downwards. We therefore have A J A A K rals)) = elute Gas SH ty k) dxdy and {£ is given on the IFS by Bernoulli's equation (1-8) with w=0 Pp, (* yst) ps oS where the potential has the value on z =, i.e. with the argument (x, y, §) but may be continued analytically from the surface to the E73 Murthy plane z=0 inthe form oP (x,y, §;t) = &(x,y,0;t) + £ ve a Z=0 Since £ may be assumed to be of the same order as @ , we may retain only the first term in the above expansion and write P. (x, yst) Pg We therefore have im I (¢ 4 ¢. See tov ta Re Se Sao al jst ee aay So and substituting for the derivations of ¢ Fv Ms (eaiAes icen, <> een ee p Pate} xt mG ee ae PB 36 1 1 ae — ( - V +t 3 + A A 4 g ( yt a ve Vey p nae oh * | dxdy where the potential has now the argument (x, y, 0) and the integration has thus been reduced to So which is the instantaneous position of the part of the plane z=0 contained within the vertical projection of the hemline of the skirts at the bow and the stern and between the inner surfaces of the side hulls. The components of the pressure force are therefore 1 1 x ff. ie - Ve + Vb. VO, : are Pa? dxdy (e) (® Vb +6. Ve ee ) dxd Pg fy gk i EE cara y S, ¥ 2 = -|f P. dxdy (II. 1) 174 Po g mi Shoat Po g and Lineartzed Potenttal Flow Theory for ACVs in a Seaway From the symmetry of the motion with displacements in the longitudinal plane it may be concluded that the lateral force Yp is zero. This could in fact, be established as follows : Cc dé mee = ll Ee “dy dxdy So and Pp, may be assumed to be an even function of y as indeed P, no doubt is, as may be seen from (I.1) and ¢ is also an even func- tion of y (i.e. df/dy an odd function), there are surface element pairs on oppposite sides of the longitudinal axis where the integrand is of equal but opposite sign. The surface integral therefore vanishes on account of the symmetry of the domain of integration about the longitudinal axis. Since We are therefore left only with the longitudinal force and the vertical force. We may substitute for @ from (2.3) and for P, from (I. 1) ot i P= 5P) 999 + BPQ1 09 * 884,199 + Re-e (dab,919 * +Babyii9 + €®o991 + 8€%1 99) * BE%Q1 0, ? iot- + Ba + — EE O01 ae Pp = Bp + Be + 66r S 5 “OG 46 100 Po x x igt ,- = 3 i ae (T190%001 + 7001*100 + Oo) : "100% 001P0_) + 6Bae lot _- (7910%001 + 70018010 * *o11P 0 i ono soot le . >< 2 +Boae 2 Fane +68 + "71008010 * 7010%100 * "110Po_* *100"010%o x Z Dao 3 As ae eM iociom 2 t100 Po.) xx The integrals in (II. 1) are to be taken over the instantaneous position LTS Murthy So and if this domain is subdivided into SDE el ab Oo fe) fe) where ee is the equilibrium configuration and S4 is the oscillating strip, it is shown in Appendix V that an integral of the form If: (x,y) dxdy S fe) is equivalent to an integral over the known steady surface If: (x,y) + (x + he) se dxdy a Substituting for @ and P, in (II. 1) and making the above correction for the domain of integration we derive finally after simplification the following results. Longitudinal Force. 1 1 Hf axes | 28(-vp, ® 000 ) + 55 = a5 2 + Brote.P, Pie VPS Fo100_) + 6B a7BS P1100, * (@) (@) 2S + PL (VF 999° Y8o100 * Y?1 000, - VEo1 99) - x = V (X19 * BG % 10) (PS Pi000 + Po*io00 ) = x xx xxxX = Vi 99 * BG 8100) (2P5 0100 +P, P0100 } bs = xxx + 2 5 B IP, Ve 000° Y# 1000, - V (199 + BG 8109) (2P, F900 + Po P1000 i * oe xx xxxX ict 5 a : + eaee Py (ivi 919 - VPig19 ) x xx 176 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway - V (Xq9, + BG %01) 2P, Fi900 * P, Pi000 i a x xx xxx Zz igt : ne ee , vig) 7 Yon, a x xx Pie, tenn totes IN, hws Rae 001 G 001 p o°O fe) xx x - iot : Boe IP. (i791 - Y8o001 } y x xx iot : ‘ay eas IP. (16% 991 - Y®i001 ? x xx + p (Ve A ENO Yoo01_ MVE V® LOCO. > 0001 x + * — (=) i=) + ja @ D> — (=) 2s (=e) iy) ue) {e) GaSe Q o (jo) (=) So — * < ae (>) i=) — re + + io) ca Q ie, (ep) (eo) [o) a < ich (=) (=) = v4 * a nin < ot Gen) Ss ee’ + Veqoo | ’oloome? Y x x + ( h = 6 2 ; ie 2510 9eGe010) P, (e899) 0001 x x xx +P, fe€99) - Y8o001 if as xx xxx Ziget*™ = : DAR jg Bage bee + h8o9)) bee (67% 991 ~ Vo01 ? x x Doe +P. (eh 99) : "0001, ]f] (II. 2) LETT Murthy Lateral Force Vertical Force Se x 35 ) Z5 [fas be, + 268 (X, 49 9, o9) Bo G a et + +28 (X19 Ma Pots. fe é “ 3 abe + ae ones “o10°100 * “*110Po_ * °*100*010%o } B€ 2 wae. Paes aaa 2; = OF a= 6 ee +P Rett saat martes od igt ,- + Bae (Too) P. ) x ey ee A ea & ae oe Se 100°001 | *001°100 101P 0. *100" 00Po,, igt ,- 1 ge EE ORL RE Ee 010 001 001 010 2 a 7011 Po x + 22 070g Py ] (Ir. Sy xx It is of course understood that the real part of the complex quantities on the right-hand side are to be taken, although the symbol "Re'' , has not been explicitly indicated. But, in view of the possible confusion in the case of terms with e°'?t *as a factor, the following convention may be established. The factor e'7%' occurs both with respect to the displacements which are assumed to be simple harmonic (and of the same frequency as the wave) and with respect to the wave potential which is also simple harmonic. As the displacements and the potentials could be complex, the real part of these quantities multiplied by e'?%t is to be taken. In the case of terms containing e2ig t*asa factor, the correct procedure would be to assign a factor of e!%t to the displacement and take the real part, assign another 178 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway factor of e '%'t to the terms containing the potential and again take the real part. The two real parts are then multiplied together. This procedure is implicitly understood when a factor of e2ig t* is indi- cated by an asterisk. It will be seen that the cushion stiffness terms denoted by the parameter 51,),,enters in the higher order vertical force of 0( 62 8,582, 68a, Ba) although the longitudinal force of these orders is free from this effect. In actual fact, the effect of stiffness is contained in the longitudinal force of 0(édBae, B*a6) although these forces have not been written down. Also, it may appear a little surprising that the dynamic pressure of the water obtained from the potential @ does not enter in the expression for the vertical force at all, whereas the horizon- tal force contains the potential in all orders. This is due to the approximation contained in our expression A ace cae + ct - k) dxdy which is valid only for an infinitesimal slope of the water surface in the x- and y-directions . It appears, therefore, that the actual shape of the IFS determined by @ is irrelevant as far as the vertical force is concerned so long as the slope is very small. The situation is therefore much the same as over wavy ground as the hydrodynamic properties of the water surface do not seem to matter for small slo- pes. On the other hand, in the case of the horizontal force, which is proportional to the actual slope, however small, the dynamic pressure determined by the motion of the water is very relevant. This also applies to the moment as will be seen presently. Turning now to the moment due to the action of the surface pressure on the cushion hull — > — A M -- ffs (eis nds 12 s & S and since the position vector with respect to o' of an element dS of the IFS at P (see sketch) may be written ys) fee ae A aad Veta ae fe (2, - 2.) where x = Xa - Zo =x+ hv, sin 6 aa Zur Fah - ae cos 69 = a te - cos6) and if (x,y, $) are the co-ordinates of P Yr’ = (x - x - h-.siné) Dey Sale 2+ hg (i - cose)| " and therefore ee = , A A M = Pp |(x -x-h_sin@)i + yj Po s G fe) “ A A war. E z+h (1 - cosé)| | & + el - x dxdy where the cross denotes the vector product A j . ae ~ it i.e. My, = P. fe yo | ee wen - cos) | Cc So Sy + x-Zohgsinos] F- 84 RG (1 - cone] a + 2( x - h_sin6) ie, barca x - x - h-sin sae | “ge! xdy 180 Lineartzed Potenttal Flow Theory for ACVs in a Seaway we may now linearize this by setting 2 sin@ =8@ and cos @= 1 +0(6 ) bearing in mind that P. is of 0(B), {is of 0(6,8) and 6? is of 0( 6°,68 ,B*, ba ,Ba,a*) . The linearization will certainly be valid for the moments ofsthe order we are considering and therefore a Ife a ee Oty eat - 30 + Jog! Sy -y hed & | dxdy As the motion is confined to the longitudinal plane we may conclude as before that for this symmetrical motion p. , $ ands, are even functions of y whilst Sy is an odd function so that the integrals of all the terms containing by may be set equal to zero and for the same reason | P, y dxdy = 0 So I Be Se veces dy 30 SOA The i and k integrals therefore vanish and we are left only with the j component or the pitching moment oe - ff, |* Se gate na Giz) | dxdy So This result is obvious as there cannot be a rolling moment or yawing moment in symmetrical motion in the longitudinal plane normal to the crests of the waves. and Substituting for pgand for § in terms of ® and carrying out the correction implied in replacing S, by S, as before, the final result is: fe} 181 Murthy ca eu |[e=* lpxp, + 26B (X) 49 + A910) XP. C SS x So +287 { + *910 * G4 0) os 2 : : + s +2 +58 * (7) 998910 * 7910100 Pit es yas le +22) 997 Q10Po ) - 271007010 Po xx x 2 ae +°-> P1000 for0n * “iano. Poipa? g x xx xx x Vv 1 a Bed @ eT 2 Po Poon Pe toon eee Ps Aree pg xx x x x a) ig ap oe ie z Po ‘100 0100 010 ®1000 xx xX + 8° Bdx (7 ‘igh ae. yao AON TOO" 2 100 Fo 160, a xX x 2 pe SG * pt Se 2 1000 ® pf. 100 *1000 g i x “pinay “ene = evidence Y Uz) + hon ® FE *001 '°G 001 = gd igt - = 7 + Be a i 00001 * O01 100 *7101P 0 192 > Somerree? 1} 280. fue ott oldie 100 001l°o 100 O0Ol'o gp “oa 001 1008 xX x xx 2 iot - - Ba + * . *(79 198909) * T991%010 011 Po 227. > fsa aie 3To107001 Po) - 2% 0107001 Po xx x 182 Lineartzed Potential Flow Theory for ACVs in a Seaway at ee 1 Bet _ ve cil taal. oil o100,.)t iget V ; + piae ae | "40001 sii ir ia abs g x xx x — + Ge 99) - YFq001 ) P1000 | x xx - — ie 2700 (iog - V@ & 0001 0001 } x xX z iget W. ; eS ie in 2h Pro [800 -Y®o001_) €o100 x xx x g #6 @ 0501 7 T0001.) 20100 x xx ES ve ) = o ps =o (010) 5 2000) 0001 x xx Po be Ti) EB oat fay) = %SuGq42) pg x ager Po coor 8 Fo001 ani (II. 4) 183 Murthy ~~. APPENDIX III PRESSURE FORCES AND MOMENTS ON THE SIDE HULLS The force on the ACV due to the action of the pressure of the fluid on the immersed side hulls is from (5. 1) —_ eS | p fds H 34+ So where the pressure of the water particles on and below the surface of the fluid on either side of each hull is given by Bernoulli's equa- tion (l- 8) with w = 0 1 2 p(x, y. 2st) =.- .p | +, - ve, + : (Ve ) - e2| The pressure is therefore given in the moving (x,y, z)-sys- tem , but the domain of integration, namely, the instantaneous posi- tion of the wetted surfaces of the moving and oscillating hulls is however, given in the body-fixed (x',y',z') system by the hull func- tions (4-9) below the free surface z={¢. This difficulty did not arise in the case of the cushion hull since the domain of integration in that case was that part of the plane z=0 lying within the instantaneous boundary of the cushion. The unit normal into the hulls is discussed in section (IV. 3) and we may write AT: Il A A (HL 1 + HJ + H. Q ) dxdz tl ("ae ao ts ! : a'a * | > (eo) ° n SS - ag | > wn - t= | Sd ~ - h h A By Ss io A - 3 + s( zi cos@ - am a) k dx'dz' This applies to the hull surface Say . In the case of the other hull surfaces the above expression has to be modified slightly. The scheme is set out below. 184 Lineartzed Potential Flow Theory for ACVs tn a Seaway Hull Surface Hull function j-component It will be noted that we have replaced the element of area dxdz by dx'dz' since the Jacobian of the transformation, _O (x, Zz) 3 (x', z') is unity. Also we have altered the domain of integration from the hull surface to the longitudinal planes and we will therefore have to express the pressure in the (x', y',z') system and use its value on both sides of the longitudinal planes in view of the possibility of a discontinuity across the planes. This could arise because of the singularity of the potential discussed in Appendix V. We may now replace the surface of S, by o+ O- where S is the side of the plane y'=b facing the tive y' - axis and Si._ "is the side facing the -ive y' - axis with a similar nota- tion for the other hull, It will be remembered that these surfaces which are both sides of the 185 Murthy longitudinal planes of the hulls are to be taken in their instantaneous position when evaluating the surface integral for the pressure force. The pressure of water may be assumed to be an even function of y and therefore of y' and and in integrating over S, and S, we may combine the integrals over and o- ot where the hull functions are the same for each group. The j-compo- nent is however, of a different sign in each group and therefore the lateral force vanishes. We thus have ~~ ao) iG iit 1 lye) = *) ys oS po ue) + ala * a -|— + ! | | P “ee re) ° wn D> N ac) HIE, Il i] NM a Ze es) + a a Ny] o> — + rs) 1 oY) N | > (e) (e) n > 186 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway h h : (>" saa +p =) sno] dx'dz' (III. 1) x where we have combined all the four sides of the two longitudinal planes whose immersed parts are geometrically similar, except that the profile of the disturbed water surface will be dissimilar on the inner and outer sides of each plane. This wave elevation, however, introduces a very high order correction, as willbe seenpresently, and for our present purposes we may indeed consider that all the four sides are equivalent. The integration is now over one side of a longitudinal plane, but the different pressures on the two sides are to be taken into account. We will now have to express the pressure on both sides of the longitudinal plane of a hull in the (x', y',z') system. The pressure is given in the (x,y,z) system in terms of the potential in the form 1 2 . =e Ca Ni ee —— a Pu Gceyienzis tc) p A V2. 5 (Ve) «| and inserting the expansion for i ict bap P= 99) 09 + FSi 90 + FPF 199 + © ( 1010 + BAS) 119 + €Syq91 +9190) * P8101) we have ie 5 p(x, y, 2;t) = dpVe Haile OT : Be(ve 1000 1100 x oe 28 {1 VP 000 - 79100) - Bo. > PV®o100 - YPo100 ot iapeee |. Ve ) Tae Foe us Vou x ist ,. t MERe Aeon Soaie » x WORE A. =| €pe (ie) 91 veono1.) 187 Murthy be AT bed Vv + ) "ub "T Sala Fy ant F001, V®i000 ° Y®o001 ict (i Ve b ) edie aA TIT RO 0101, ¥®o100° YPo001 Zicot * nite sees Y®ioio: YPoo01 2iot* - Baepe Toe8 Awe et eee (III. 2) where the argument to be used for the potentials is (x, b+,z) and (x,b-,z) for pt and p~ respectively. To obtain the pressure in the (x',y',z') system we may expand the potentials in the form of a Taylor series. In view of the singularity of the potential at y'=b , separate expansions will have to be used for the two sides. Thus, in the case of S o+ OF ny Joel ' bt, 2") + (x-x' ' Bree P+) p90 (& ¥? 2) © c00 (x', b+, z') er) thbos bt, z') Es ! 1 + (y-b) 000 ot ees) gst ! ! + (z 2") #1000, (x', b+, z') where = Rapes ' = lot x-x'=x+(z +h) 6 85) 99 TF 19 + O01 y-b=y'-b= dh, (x', z') and Petre an a ' ! iot 1 Z-Z z-x'@ br" og + FF'oig +. 2° antl but the normal derivative "4 1000 y’ i.e. the velocity across the plane may be considered equal to zero, so that 188 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway + »Y> = 6 BEV io 99 (Xv 2) pv?) 000! | (c', bey zt) x x oe + Re "91071000, Sager 1000,’ s Sore & 1 Piece ger aNles Sigg. xf ool tooo.’ x x x Z when 0( 6¢) is neglected. will be The expansion on the inner side i.e. on S16 identical but the argument of the potential will now be (x', b-, z') Similar expansions can be written down for the other ‘terms in (III. 2) giving finally - »y, Z;t) = 4 + i Yi iaa0) 7 scam RAY 70100) 2 i & tr! @ B SY (ono LOO, ee Oe ORO, ) x xX x Z i U ) or "To 106 **or00| ! a ee UGee on saith 010° 1000, +r! : : (oo.0100 =e 71100 , ) z x * *100%0100_,.., ? V2) 000 ° Foro} + 6 a Vv ® aera et Ne ool 1000) 2). 0011000... x xX x Z +o 1010, io}, 510 + B wit P<. V } +2" ® : ae r ; 7 Col -olo0. 9 ool oloo.. x xX x 2Z@ 189 Murthy ict : - ‘ . : Po110 ) *Fo110f ee = Plt Soog1.5 “Muon ? x x ! t = p : : a 10 (0 5001 VO 001_, ) x x xX ! ; Z rc) * = 409 ($5001. oe 001...) +i - Ve + v ee 1001, , Vv’? 000 *oo01 | Bee" (i ve ) nr is P S01 Dw od OG0lenctancQOl. ot: x a ok +r! - T'510 @%%q001 , - YFo001_, |) Zz x ° m ) ) eft “Po101 *VPo100° v'Po001! Ziat* - aee °} Foo (85001 oP a001,, ) x x x ! + wz * F001 (68550, VF001_, ) Zz Mae + pg ); + z'cos @ - x'sing - ha (1 - cost (III. 3) The argument of the potentials in the above expression is (x', b+, z') for p+ and (x',b-,z') for p . The corresponding potentials may then be denoted by @+ and @- We may now substitute for p* and p in (III. 1), but before doing so we will reduce the domain of integration from the unknown to a known region with a correction term. We may write 190 Lineartzed Potenttal "lo Theory for ACVs in a Seaway = S + 1 + W “i 1 = FA fo) fe) fo) fe) where S ; ah Bk a ve 1g is the equilibrium position below the load water plane z' = 0 which is known from design considerations, S,’ is the strip between z'=0 and the undisturbed water surface z=0 and S40 is the additional strip between z= 0 and the actual free surface Za © Let us first consider the contribution of the last term in (III. 3) which represents the hydrostatic pressure of the water to the horizontal and vertical forces given by (III. 1). This pressure is the same on both sides of the longitudinal plane so that we may combine h, “and h, , and considering first the horizontal force 1 = 26 p offi + z' cos@ - x' siné - he (i= cosé)| S ra Te lo h dee: cos@ + pls sing] dx'dz' x Zz which may be transformed by Stokes' theorem into -2 Opg conog Tela 30) E + z' cos @ -x'sin@ - hg (1-cos@) | dz' Cc - 25p g sin@cosé | h(x', z') dx'dz' S 7 -26pg sin (a2 E + z' cos@ - x'sing - ho (1-cos D dx' Cc +26Pg sing cose ff m0 29 dx'dz' — / 7a 2, where the line integrals are taken along the boundary of the region S1, + §S, , 1.e. the line of intersection of the plane y = b with the plane z=0 and along a line from the stem to the stern running along the keel. Along the upper boundary, i.e. along the waterline, z+z'cos@ - x' sing -h Ines) i127 .— 0 e | 191 Murthy and along the lower boundary bh (x, 2") = 0 as the hull may be assumed to have a pointed keel. The line integrals along C are therefore both zero and the surface integrals cancel with each other so that the net result is zero. Now, on the strip Si the integral may be written fo) Za -26pg [ «| = cose + SE siné) dz' Z=0 and expanding the integral about its value when z' = 0 the inner integral becomes z= oz | oh Oh ! pee — + es + ! iE Sa! ies cos@ Sa! sin @ dz and since ne = cos@=1 + 0( 6°) Zz the integral becomes fe, | cote SB ant] SC] z=0 z=0 on linearizing with respect to 9 and noting that z feeb + 0(0) . z' The surface integral over Sig therefore becomes tog | Du beez es dx' L Ox! 192 Linearized Potenttal Flow Theory for ACVs tn a Seaway and writing { = 58) 49 POE GuOT Oa. ache cool” the result is iot 2 ¢ ; dh : - 2 pge | (: oo bene rene Sault foro) va dx te the integration being along the load waterline L i.e. along the x' axis. The contribution is therefore of a much higher order. Similarly the contribution of the hydrostatic pressure to the vertical force may be evaluated. In this case, as h - seal | + z'cos 0 - x'sin6 - ho (1-cos0)| Por cos # = 28 sind] aces" S +5" lo lo =- 26pg cos f h(x'z') | # + wcos9- x'sind = ng (1-co84) | dx' Z + 26pg cos | h(x', z') dx'dz' S45, 'o 6 + 26p mot! h(x', z') E + z'cos 6 - x'sin 6 - hg(l-c080) dz' Cc 2 + 26pgsin 0 If f(z) dato! oerS, ails As before, the line integrals are each zero and the surface integrals combine together to give 25P ¢g | A(x! z!)) dx'diz! S445, lo lo which is simply the displacement of the two hulls below z=0 . We may write the above integral in the form 2 ail h(x', z') dx'dz' + 2areff h(x', z') dx'dz! f) lo on 193 Murthy The first integral is the known force due to the displacement below the load waterline i.e. m,g . The second integral will however, have to be evaluated. This is discussed below. As regards the strip S46 , the integral becomes by expand- ing the integrand about z'=0 and carrying out the integration as before iot 2 oh -2pge tf (s “ Sito” S Dol _ Bes O19 f aay eS. i L We will now evaluate the correction for the hydrodynamic pressure terms when substituted in (III. 1). We will replace the integration Over Sy by that on the known surface a together with a correction denoting the integral over Si, . The evaluation of the integral over Sj) will be unnecessary as this raises the order by de or Be The integrals are all of the form |f coe ao dx'dz' “16 or If (x', z') ae dx'dz' “is Now, z=06 [[ ee = dx'dz' - fox f(s tb omt) a, dz' Ss’ Z=6 lo and when z = 0 so that in view of the smallness of the range of integration, we may replace the integrand by its value when z'=0 giving x ues z—- x'"'@ ft") 2" : dx' [i ) (x', 2) |. 1.94 Linearized Potential Flaw Theory for ACVs in a Seaway which becomes an integration by parts E Gols z') (z CN 6) ff en) stem stern stem ‘j h(x', 0)-—2 | = x'6) £ (x', 2) | dx' stern 2*=6 The first term vanishes at both ends on the waterline, and the second integral is taken along the tives x' direction so that this term becomes Stern | h(x", 0) ae [@-x 9) £629 | dx Stem To this integral may be added the integral of zero value along the keel from the stern back to the stem as h(x',z') is zero everywhere along this line giving Prix, z') a E =e) 0) of Max, | dx! which is taken around the boundary of S, fo) This may be transformed by Stokes' theorem into Oz! Cc S. lo 195 Murthy The line integral is obviously zero, since h Gee se = 0 fou). zits’ 0 and we may combine the second surface integral with the first giving the result [for evar ff — ees 2")| Sh (z-x'8) = dx'dz' S, S fo) lo ~aigrio| Oh Ofyephpet oh phe ae {fe of Sreaiaced alniate a ) si 540 In the case of | f(x', z') a8 dx'dz' S lo the integral may be written 720 fox | ie", 2") — dz' 72 SO) which can be approximated as before to It does not seem easily possible to convert this into a surfa- ce integral for merging with the main integral over S, as has just been done in the previous case. . Substituting for pt and p in terms of the potentials and reducing the integration to the known surface Sy after correcting for the strips S46 and S' we have the following results for the longitudinal force and the vertical force. It is assumed here that the potentials Piggg and Pipi will be continuous across the longitudinal planes for these are the potentials of the side hulls in the absence of the air cushion. The 196 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway wave potential @go9; and the hydrostatic pressure will, of course, be the same on both sides. The hull functions h, and h, have therefore been combined in these cases to give the total width h In the case of the potentials involving the cushion pressure namely, ®o100 and ®-o101 , the two sides are considered separately. The final result is : 2 = Oh - oh oe) eee ol vt 50 Sx’ | “001 Oz! oh Oh ! See cSta =s i] von Ge = Oe } P1000 x'x! oh ; W010, a alot sae |- h h 1 onal + + ete , ! Ojon 7 Me maaan, aS 3100, h + Chan! & = Oy 0110, oy RG e omne + + + + + + Murthy i 3h oie (oo) ~ “#001 ) ‘Ox! 2 iot = oh = oh 2 O24 a shia hoe 3x!” 7100 dz! . =e 4+ f / (4,50 "#1001, eee , , oh 9,00 YP1000° YPo001 “dz! iot| )- dh - dh =o iee [Foro “Sx! * 7010 “Sa! + + 1+ 3 : Ve (4 01 “Foao1 |. ofbd 5001) ° a A = pe we , (ie85101 "®o101_, P0100 Voo01) ) oh 8010 Y?i000° Y?o001 | reece Oh) = Oh A ) OO 4x’ 001 Oz! oh Pe ie Pooh Pigs vapeies, Fo haga oh Oe | Lineartzed Potenttal Flow Theory for ACVs tn a Seaway 2 | dh , = 2:6 62 2 ie ci01 Tae oo L -26 aco | S oxo ySoory satel | oo E i 2: oh _ ! ! ne —— Z5 = ic pg h(x',z!') - 26 P V® ooo Se) H J x! > ro) h ie + Chal - m2 3 & a — 2380 (F109. ae * P0100, Ox! 2 iot = : p 6 Bae E 10 Sone 001) *1000_ a "a 1 (259) - ¥'8 991)? + 1000. i "21010. ——— oh oh = 010 | ee Mean *1000_, Sx! ot i : Fe er + - 26 Bae AL ©} 0 +z + he 8 55) P0100. s + if as ! * (2591 ~ *'8901) ®o100_, t y2onloo.. DSars x can a aise on 7 2%p 100 | (azt” | Con. g?0' Olay, Oe x +z'+h_6 o. |v 1S eae ea oay 0100... me = of z ® * (299) ~ *'8 01) Po100_, if NOT00 , x Z x id, 32 Vid ee. ot 0100 Aza, | COAG GALI F933) 19g Up (III. 4) Murthy ald Wee - Ve es i suis 0001 0001,” dz mt: 2 ae eve eee f ee ee [00 21 +h) go) Foor ie ag (eo ’ * (21 99 ~ ¥'8 199) F991 - VFoo01_ , Zz Ses) ; oh z + es —— TIO O01 *1 001, Viggo: YPooo1 = oh - F199 FF®o 901 7 YFo001 | 5x! kN Peg (x +z'+h_@ ) (ice - Vb p} 2 \*o10 G 010 0001 , 0001 x x - dh - ! = * (2519 ~ *'8 19) F#F001 | - Y®o001_, D Sa! Z x Z h + - y+ / 0-1 ; : * (ie® 19) ~ V®q191 +%%o101° Y®0001) Om x + (iob) . - Veo + Vb, Ve ) an OVO I 0101, 0101” 000100 Sz" oh =n i = —— o10 7 p91 YFo001_,) Ox! Zig te - oP ee ay + 26aee o[ ooo1 +h 9597) - oi = — Bt es V®o01 _.) * o01 7 *'o01? Pee, Ca $901 - Yoo01 | ) Sz! Z x Z 200 Lineartzed Potenttal Flaw Theory for ACVs in a Seaway dh - 6 - ie aaa ! ! 001 * F001 yoo” 5 a deez + line integrals which follow (III. 5) Line integrals to be added to surface integrals for vertical force Z5 H 2 = Me: pg V2 aya - ¥'9, 59) b (x!,0) ie 256 pg (z + = 1 ! 010 7 * 9o19) B &&'e) ae ine) fa") D RQ (@) bh: a=) fees GQ os Ni i=) — = * D (jo) e — — De a » fo) — A ’ 7001 001 h eet seid Oz! * 0100. Oz! } 2 iot - ; ; Oh Pe Be Eo ag | Sooo t *Fo001 Oz! Ahi SS 00) “om Se lot - dh = ' 1 = ——_ “pad Bas Nor *' 919) PF) YFo001_,) Sz! oh flay OHO) ah OO ane Drier ek ae. 4, SRN 4 ts Vo eee ee P\7901 001’ “7%o001 7 0001, dz" (III. 6) 201 Murthy We will now turn our attention to the moment on the ACV due to the action of the pressure of the water particles on the hulls given by (5. 2) —- M. | p (r' x n) ds Pu $45 i Mane and as "i may be written in terms of the components along the 4 ae » k axis. roces : A A , A r' = (x' cos6+ y' sin®) i+ y'j + (z' cos@ - x' sin@) k and h h Ow Aa h fas = | (3 ; 'eos 0 + PS sind) 4s ($ ;, cosé h oie siné) k dx'dz' we have h h ae. ot Q 10® A (r' x n) dS = [fer a cos6 - Sr sin#) t(z' cos@ - x' sino | i h h + 6 pe ee 1 O11 s ox jz)? h h + | Fatcoee + z' sin@) - dy' (= cos 6+ ae sind | dx'dz' the upper and lower signs referring to the hull surfaces S and ing S3_ with h, replacing h, in the case of S,; and S,_ The integration is now performed on the outer sides of the longitudinal planes of the hulls and combining S164 and S35 toge- ther as before, the integration may be confined to ai Or with the integrands added together. : A : The combined i-moment on the two hull surfaces is there- af) “2 eee oy - Py Sat cosé@ - at sin#8 dx'dz at 202 fore Lineartzed Potenttal Flow Theory for ACVs in a Seaway re : which vanishes as >= and ——— are even functions of y' Pp Ox oz! bf A : ee ; The k-component also vanishes and both components similarly vanish on the inner hull surfaces for the same reason. We are therefore left only with the pitching moment h h h ~ ay aie! - o2 a2 = z! = US es ho Be yee ! ! al er +p (z Seay ay )| dx'dz and using the expression for pt and p given in (III. 3) we derive finally the following result for the pitching moment where the inte- gration is reduced as before to S, incorporating the correction terms where necessary. 2 Il: ZO Pex his". z") > fe) Lon ol LDS cee 1G ee ) + 88,592 h(x 2 h h o ermal eh "Cl 2460) V [9500 2 oe Sr! 1 N oY) ) oO xo) GQ D> G 001. ~ 1000 Sciex! = = ! 2901 ~ *'8% 01) P1000_, |" x Z oh oh = Ne ene = gis, 4 ea peo ioe, mee Oe aay) eee + 203 Murthy Biam git (sha) | iot - a af 25Bae ay coe + 2! 4 "6001 P0100, Vy! (z Sactg ) a + Ve, 001 001’ “0100 , , 0110 x'z x h h eer 5. i On Wh) x +z! ) db” fv 01 Za % BP ani? 0100 VW! x'9 )@ z. Z % + (2901 001 9100, 145 0110. a GQ > oO —_ i N D> — ra N YK _——) lot . ! oh ! oh lhc RT aI ot cn Y®o001_,) oe az! 2 tant - Siar rr) 6 : é6€ € L600 + 2'+ he 100) seg ator "yoo a is Ss) i = (2199 ~ ¥'91 90) MEP oqo: bh: ) se Oh oh < 2 2, i | eee ©, © ae © 16%) 001 ‘Sroot, + VP 000 | 001 I Sgt ce 32 ! igt - . £26 + 2) +B i = Gee ol} B10 2! + hq 10) Mae cat, VP 001 A x 204 Lineartzed Potential Flow Theory for ACVs in a Seaway Z - x! - h h + (Zenon 2 =e O10 2” Toobly. copadeY B) (2B 6. 4.59) z! x'z! Ox! oz! + + + E + VW Serena | 0101,, F100 0001) Ch oh (z' +— -x' a + (iod” Ve_ ax! Oz! 0101 ONO: oh oh +V, Vb Gz a A ey DMOCas OOO Sx! Sa! 2iot™ - ; ; + 26ae e | Goo dp gail dt he %1) OO ae | = = _ ol Yona tw) (Zany = “oo 5 i 1 ! Conan Vp o01,,. | dx'dz | + line integrals which follow (III. 7) Line integrals to be added to surface integrals for pitching moment iot - ‘ ; : 25a e pg (2504 2D na) haGcl0)) 2 lot - oh 5 p Pree nes fice ai Y (2991 - 1804) *1000_, Sa. a g(Z 6) - x'8)5,) 4 0) | 205 Murthy R yy ae ) (& ee "100, oem lot - ‘ eH Caer 1 YG ~ x ot 0100, - 8(24), - ¥'053,) h an e 2 igt oh - 1 ads al oer 100 £001 Sai 7 100 7 *' 8100! Sh F001 ~ YFo001_) = ict oh - : Boo es bes oro book Saat “016° * oe | ie ae Ge $00) > pagal oe ‘ -2 baccit p(z - x'@ ) (ioe - V@ oe x'|dx'! 001 001 0001 0001, Oz' APPENDIX IV RIGID BODY FORCES AND MOMENTS The rigid body force due to the weight and inertia of the ACV is from (5. 3) Fe [ffi - oe dm V where U is the absolute velocity of an element of mass dm contain- ed within the body V of the ACV. It is easy to show that Ife - ie ens = m(U, - oe V 3 where m is the total mass of the ACV and v. the absolute velocity of its C.G. Lineartzed Potenttel Flow Theory for ACVs tn a Seaway and this reduces to 7 wen ay | Gristse ab) avi for the coplanar motion we are considering. We therefore have tsar taraeer de jae and substituting for the displacements, we derive for the components of the rigid body force R (py q, + 8X19, + 8x,,,) and (IV. 1) 2 Roat YH Fe) aemavo, ae + 62 fo f8) (2 9) 101 2911) As regards the moment, we have from (5. 4) —=5 > an M, = a, SLU a eae ala where the absolute velocity U at the point (x,y,z) of the body is ly =) (Gigs 32) Hype caeee reducing to oe ees for coplanar motion and since x = x+x! cos0+ (z' + ho) sin 6 and z= z+ 2z'cos0-x'! sin? - ho (1 - cos 6) we have Seyler -2 SoS) [(z! ais ho) cos@ - x! sin 6 | 6 - [x! cos 0+ (z' + ha) sin 6 | 7] 207 Murthy and fase [x' cos@ + (z' + ho) sing ]6 + [x' sing - (z' + ha) cos 9 16° Also, since ris (x' cos@+ y' sin6) 3+ yi f+ (z' cos@ - x! sind) k we derive after substitution and carrying out the vector product M, «fet gt z -[x' cos 6 + (z' + he) sind] 6 A +[x' sin@ - (z' + ho) cos 6 | 6 1 + Ke cos 6 - x' sin 6)[x fee + ho) cos 6 - x' snahi re he cos@+ (z' + ho) sin6 ti - (x' cos6+z' giné) [- g + a rf cos@ + (z' + ha) sine ° A + (x! sin@ - (z' + ho) cosé | 7 -y' {3 +[(z' + ha) cos §- x' sing | + [x' cos@ - (z' + ho) sing | i‘| 4 dm Now, we have the following results : i ' = = (i) z' dm m Zz G! G V (ii) Ill x dm. =m xO SEU) \ since the z'-axis passes (daa) If. ax 2 vs ently through the C. G. V (iv) xia! ding .=-' OQ, y'z'dm = 0. due to the lateral sym- ff metry 208 Linearized Potential Flow Theory for ACVs in a Seaway where I ‘y! is the moment of inertia of the ACVinpitch about a late- ral axis through o' When these results are used, the { ana & integrals in the above expression vanish (a result not unexpected) and after simplifi- cation the final result for the pitching moment is Mo = ver = mh. E cos@ + (g - z) sin | where is the moment of inertia of the ACV about a lateral axis through the CG: ‘ This expression may be linearized for small values of 0 and introducing the perturbation expansions for : ie +Vh @ (1) - 29019010) ~ 811 7 ) is mou |* m 2 Galt 2 oe ” 5 we z 4 snp & Ino} (X91 - 29019100) atliaet 0 ,f£ > 0 (CS a singularity of the type l/r , > with [=e swe? | {) is a harmonic function except near the point z= ) where it behaves like a unit source with a ae Z The surface integral and normal derivatives are taken with respect to the dummy variables ($,7,{) which have the same disposition as (x,y, z) The integration is performed over an arbi- trary closed surface ys which completely surrounds the point (x,y,z) at which the potential is to be determined and the derivatives are evaluated in the direction of the normal out of Sy 3 The potential @ is assumed to be of various orders (see (2.3)) : OGL 7 oe) = oS har (x,y,z) + By, 09 (x, y, z) iogt 1 6 + dae ®) 010 (x,y,z) + Bae ny lot co) ( z)+ 6 i ee 0001 Vis ee i ot e + Bee CON (x, y, 2) where b is the potential due to the LOOOs in calm water, roy is the potential due to the 0100 over calm water, P1100 composed of potentials + 6B 1100 %&¥? 2) + ot %o119 ¥2) + at e ® 1001 & ¥+ 2) + motion of the side hulls motion of the air cushion is the potential due to the interference between the side hulls and the air cushion in calm water, 213 Murthy ® 010 and ® are the potentials due to the forced oscillation of the side hulls and the air cushion respectively in calm water, ® is the potential of the incident wave, 0001 and P1001 and Py are the potentials denoting the interfe- rence between the side hulls and the incident wave (the diffracted wave) and between the air cushion and the incident wave (the disturbed wave) respec- tively. We shall only derive the lower order potentials » ® and ® ® re 1010 0110 1000 0100 in this study as these will be sufficient to evaluate the forces and moment of low order. The method of derivation of the interference potential @1j99 will also be briefly indicated without actually carrying out the solution. The potential of the incident wave is readily written down. The diffracted wave potential and that of the wave of disturbance are only required in the higher order theory. Boundary Conditions. The boundary conditions satisfied by the potential are : 2 (i) Rae ae This applies to potentials of all orders. (ii) On the EFS (z=0) the condition (4. 6) is 2 22 + Pe 25, V ® ss + 2V®, - V + t x ge ¥ (, ® ) x nie: Ane 2, (@ -2Vb@ +Vo® xt = = 169 Z tt xX Be) 1 +— 8 (2, which reduces to the following conditions : 38) sae 0 1000 ~ 8% 000 xx Zz 214 Linearized Potenttal Flow Theory for ACVs tn a Seaway 2 0(8) V P5100 & £100 ¢ x Z 0(3%) ve She We Jie oe =p 1010 1010 1010 1010 xx x Z (fa) ve Baca 5 SRG * OIE CL TY ve ae On KO 0110 ~ 8%110 oe x Z 0168) ve Sb 2V.( ve i $ 1100 1100 1000 ° V29100 <>. Z + V® 100° Y¥i900 ) + V ene i 8000 32 (VY P1099 x x - gb )+@ 3 (V.3) It will be noted that the first four equations are homogeneous, whereas the equation for the interference potential is an inhomoge- neous one. If we denote by W the time-independent part of the oscilla- tory potential @ (i.e. without the factor e iat) , the first four of the above equations reduce to the form Z z V - 2i0V -7 yy .- = V VY 1g VW gv a 0 (V. 4) In the case of the steady potentials o is, of course, set equal to zero. (iii) On the IFS (z=0) the condition (4-7) is 2 - 2 + ~ 5 V - V® b Ve. Vv Cee 2V®@ (2 V P ) + p 1 s 3 2 +— - +—— - ~ = : Se as ara) spelta ao ee ge ) 215 Murthy x 1 ) ) opm & mes ga° Oy DEE, Rr | : ) pg Ss OZ (f, = Ye. =0@ The surface pressure distribution is given by (I. 1) in the form o iot Pe Gyie7 Bb Poh Pio ho Ph Pore! where Po10 ® Po (x, y) = fr = (x ) Piro = "100°Po &¥) = (gq + he 190) Pa &Y¥) x x and = r = - rr] Pol] “001 an (x, y) (oq) + BE %o)) reg (x, y) and the conditions satisfied by the various potentials are 2 Be VS eae 8 Sagan . xXx Zz 2 Vv Be re * 9100, EPo100 ~*~ p Polo olbay Vos EG fob =o 1010 7 “* “Sipe tore *S*eie* xx x Z 2 ' 2@ - OURO) Na ie OTe i te sy ig = Beet a oe xX x Zz ee (Vp ie p 011 011 216 Linearized Potenttal Flow Theory fur ACVs in a Seaway 0( 8B) V®i 99 ~B%1 199 =2V(V% g99°%Fo100 *%Fo100° M1000. xX Z x x Vv 2 a 3 (V ® } ) BANE acs =e g 1000 dz 0100 0100 x xx Z 1 1 ote ee - Ve g (| Poio 0100 ) ) + ee ee 1000, zs ® pe Poo 100 en oe. x x x Z 1 p Saye cht eee x x +1999 Poo ? y y The first four of the above equations may be written in the form 2 : 2 ia oy eave ad peer Sk Zio WN Cy REM ENE Ze Oe - io p.) (V. 6) where om is the time-independent part of Pp. (iv) On the side hulls the condition (4. 10) is O® . . : 2 Ses sb Toe. era Not ! a i = cos 6 : sin@+ (z' + ha) é + V coool i - oh oh =1V 2 2 Wines ae 1 | Nea Ox! pars Gear ) 27 Murthy which reduces to 0( 8) 2% 000 mag on, on Ox' Ifo 10 0( 6B) —s = 0 Condition does not apply. Om oh oh LOUD g .: >, 1 0( 6a ) on ake Fan ee + (ior 001 * Y %001)—sar rok 0( Ba) an wa no condition o@ oh TTBGi A | 1 Dice ea clas AM hg (Vv. 7) These conditions relate to the outer hull surfaces. In the case of the inner surfaces the hull function h, should be replaced by h, The first four conditions may be represented in the form h oh OV 1 tee 1 ek, 6 (aigry 9, + V) ee + a(ior 0017 V8 oy) Sa! (Vv. 8) (v) Radiation and Boundedness Conditions. We may assume that the potentials of all orders satisfy the conditions @-—>0 , i.e vy —0 and SP 5 ge PS ee as z—»oo at the bottom of the ocean. We may also assume that the potentials and their derivatives satisfy suitable radiation conditions as 218 Lineartzed Potenttal Flow Theory for ACVs in a Seaway 2 2 x + y— > These conditions will be specifically stipulated in due course. Singularity of the Potential: It will be seen from (V. 3) and (V.5) that the potentials ® i500 and 94);, relating to the steady motion and forced oscilla- tion of the side hulls of an ACV in the full displacement mode (i. e. without an air cushion) in calm water satisfy identical conditions both on the EFS et IFS. However, they have to satisfy additional boundary conditions on the hull surfaces given by (V. 7). On the other hand, the potentials o100 «= aNd PoyQ #«relating to the air cushion of an amphibious ACV (without the side hulls) satisfy different condi- tions on the EFS and on the IFS unless the basic pressure distribution is truly uniform in the longitudinal direction such that = 0 and = 0 roe roe, throughout the length of the cushion. However, this would imply a discontinuity in the pressure at the boundary along the bow and the stern where the pressure drops suddenly from the uniform cushion pressure to the atmospheric. A discontinuity in the value of the potential at the boundary is therefore to be expected. It is also to be noted that the cushion potentials ® o10 and Po119 do not have to meet any specific conditions on the hull surfaces and that the interference potential ?., 9 satisfies different boundary conditions on the EFS and IFS and also a condition on the hull surface. The Green's Function. Let us nowchoose a Green's function G(x, y,z; §,7,{) such that 2 V Siyek ae coe oe teams mmeaeas 6 (x - —) d(y- 7) 6(z -¢ ) where 64 is the Dirac delta-function. This ensures that G isa harmonic function in z >0 witha singularity of the type oh. at oS E a Si ps 1 = c Let G also be such that 219 Murthy v" GeE+ 2ia VGE - eG Eo. ee es” (ve It will be observed that this free surface condition we have stipulated for G is the same as that satisfied by the potentials in (V.4) with the difference that V is replaced by -V _ so that the second term on the left hand side is of a different sign. The reason for this will be apparent presently. We may also assume that and [2 ) f tener ha A suitable radiation condition is also imposed on G for large (¢2 + 65) and fixed § in order to obtain a unique solution of the problem. The radiation conditions for ¢ and G are fully discussed in Appendix V of Reference 1. The Domain of Integration. We may subdivide the closed surface » into the following separate regions : a. a ee where ae. is a surface of small depth below the undisturbed water surface which just encloses the immersed part of the side hulls of the ACV in its interior and which intersects the EFS ina closed curve L, (see Figure 3). This curve will therefore contain in its interior the actual boundary L, of the ACV on z=0 , i.e. the closed curve formed by the intersections of the outer surfaces of the hulls on z= 0 and the vertical projection thereon of the hemline of the skirts at the bow and stern. 2 is the lateral surface and 2s the base of a large circu- 220 Lineartzed Potenttal Flow Theory for ACVs in a Seaway lar cylinder with its axis along the z-axis and extending downwards to the bottom of the deep ocean. The radius and depth of this vertical cylinder are assumed to be very large. If the intersection of this cylinder with the plane z=0 is the circle L, , it is obvious that L, (and therefore L,, ) will be contained well within it. Also, >> is the ring-shaped domain on z= 0 lying between the circle L, and the closed curve L, We may now apply Green's theorem to the closed domain in z > 0 bounded by 5, i.e. by, ey Da desta with. (x,.7,,2) lying within ) > W(x, y, z soa ae c=. = ds (V. 10) Dare “ s ° 1 2 3 The boundary conditions satisfied by W are given by (V. 4), (V. 6), and (V. 8). Considering first the integral over ae the base of the large cylinder, the integrand tends to zero in view of the assumed behaviour of @ (and therefore of W )andof G as { —»o, The integral over the lateral surface » -is also zero as the radius r—»w _ since the radiation conditions are specially selected (and considered physically reasonable) to ensure that this is so. A full discussion of this matter will be found in Appendix V of Reference 1. We are therefore left only to deal with the integrals over eR and ig . As regards the later, : Bh phlei hoe 4-ff + 7-6 dé dn and on substituting for WV c from (V.9) the Je ath becomes, = 5 a 2 4 7g if cv Gg¢ + 2i0VG¢E 20 7G) = G(V Wee 2idgVW;-o w) dédn = a dt WGre - G¥,,) + 2ioV (WG_ + oy) | di dy eae = “=f =| Vv" (WGe - chai 2ievac | a EAA Murthy ime $ |v? (Gz - Gly) + 2iev vc dn + 2 : ae $ | Vv (YG; - GV;) + 2ie vac | dn 4 where the line integrals are taken in the clockwise direction (viewed from above) along L, and in the counter-clockwise direction along L, in order to keep the domain of integration to the left, bearing in mind that the positive side of the element of area dédy is along the Z-axis i.e. below the free surface. This is in accordance with the usual convention. It is important to note that the above substitution for We from (V.4) is only valid for potentials of 0(5,8,éaandga) . Itis not valid for the potential of 0(66). It can be shown that the line integral along L, also vanishes under the assumed radiation conditions. We have then, 1 WAxay Zick pz Ire oY -y 88 ) dS... .,+ WG) dn (V. 11) 4ng L V We may now contract the surface pe surrounding the ACV on the water surface to the actual boundary of the craft composed of the immersed hulls S, and S, and the internal free surface S The curve L, will then tend to the curve Ly, in the limit. 222 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway Lo SES + L, vi La+ » 7 Se ey LS Thus we may write, 1 OG Ww Gy, 2) eGR ae ie) dS + 5, +%> Gy ee ou Aull (v iE ac Se ) d&dnt S v2 > Zio + ee it (4G, - Ges 7 WG) dn ‘ 4 (VW. Tz) as where we have denoted by S, the part of the plane z=0 contained within the IFS. It will be recalled that Ly is the boundary of the ACV on the water surface. This curve may be sub-divided as follows : where L,,*+and L,- are the intersections of the outer hull surfaces 220 Murthy with the plane z=0 and L, and L, are the vertical projections of the hemlines of the skirts at the bow and the stern. All these curves are to be traversed in the same direction as L, » 1.e. in the counter- clockwise direction. The surface integral over S, may be transformed by the application of Stokes'theorem : eff WGe dtd | vv’ Grp eie ¥ Gy = a Gy'abay ae So 2 = Wy: $y G¢dn - vif, d¢ dy + be So + Zier V $y G dn -2io vfs didn - Lo Sg ‘If - ¢ WG dédn hi So and since If G, didy= § vpcar - [Isc didn So 3 ne So we may combine all these integrals together and write 1 3G OW BPS! ; at ffi 2%) ata eff a -e VY, > So v2 z gVe )G didn an (WG; - Lo Zio 2s ae ¥.G + v G ) dn where, now, Le is the boundary of the IFS 2 + - Lo Be : 2+ = i L, and L,, being the intersections of the inner hull surfaces with z=0 and La and L, as defined previously in connection with Ly. 224 Lineartzed Potential Flow Theory for ACVs tn a Seaway All these curves are to be traversed in the clockwise direction in order to keep the positive side of S, (which is below the free surfa- ce) to the left. We may now write (V.12) in the form G W(x, y, 2) ) = IIo — - vee eds: l Re a 2 wer (Vive, = 2i0eVY = Be ee dzdy ae) V 2io ee z ries $ (WG, CT oe oe) Oe Lo ve re $ Ex aCe ee) dn 4ng L,—>L, i E SB 4 Now, the function of W and its derivatives in the integrand of the surface integral over S, is given exactly by the free surface condi- tion (V. 6) on the IFS applicable to potentials of 0(6, B,6a,Ba) , namely, - ric - icp.) Substituting this value and combining the line integrals, we may write W (x, y, 2) fe ~ - S© jas + stall SuGp ) G didn + = at 3) | wc, - GY, + Z2e5 + va] dn (Vo13) Murthy It will be noted that the contour L, and L, have in common the curves L, and L, at the bow and stern which are taken in different directions. Choosing the clockwise direction for the integration, the combined line integral may be written 2 V 2i¢g re ${[¥] G;-G [v,] i [v] c | dn + Letts + (line integrals along the hull intersection L,, ’ Lio L and L 2+ 2- ) where [v] and [wv r] denote the "jumps" in these functions across the cushion boundary in crossing from a point on the EFS outside to a point on the IFS just within. These jumps will exist due to the singularity of the potential at the boundary of the cushion indicated earlier. We are, of course, assuming that G is continuous across the boundary. The line integrals along the hull surfaces need not be discussed in detail as their order will be 0( 6) higher than that of the potential under investigation. This is because the total width of each hull is of 0( 6) and the line integral is taken with respect to 7 on the hull surfaces. On the other hand, the line integrals along Ly and L will be of the same order as the potential as 7 can now take a value up to the semi-width of the ACV on either side. However, the line integrals around the hulls will have to be taken into account when evaluating Vito by including the contribution of 0(68 ) arising from the integration of the potential V9 of 0(f). It may be added that the line integral along the cushion boundary will vanish if there is no discontinuity of ¥ at the bounda- ry. As the discontinuity arises mainly because of a pressure distri- bution within the IFS higher than atmospheric, a suitable distribution of pressure will remove the discontinuity and the need for evaluating the line integral. This will be discussed presently. The surface of integration So is that part of the plane z = 0 contained within the instantaneous position of the IFS, which is a fluctuating region oscillating about the steady state position S, say. As the instantaneous position S, is unknown and has to be solved as part of the problem, we will reduce the surface of integration to the known region S, 226 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway The instantaneous surface S, may be assumed to be composed of the steady surface Sy together with an additional strip of area So 2ris- ing out of the oscillations. The strip corresponding to an element of area dzdy extends from the point L onthe boundary of Sto L' on the boundary of So If § and —1 are the longitudinal co-ordinates of L and L! respectively, in the undisturbed condition when L and L' coincide. In the disturbed condition points on S. are obtained by the vertical projections of points lying in the En plane within the displaced, position of the cushion boundary in that plane. Thus, setting f=0 in equation (3. 1) for transformation of co-ordinates, we obtain after linearization with respect to 0 + (x+h_)6 Er, G) i} since the geometry of the cushion boundary in the §& 1’ -plane is unchanged by the displacement. We may now write ff e natan= ff £( En ) didn |f £(& ) dédy So Sa C g The integral over 2 may be written Ey fo [Secon EL and as the length of the strip is small, we may replace the inner integral by firs cents fotaléon Bt 220 Murthy so that the surface integral over oe becomes the line integral § Pe Ca ea wi Lo = (x +h. 6) § £( Bond idan Lc which may be re-written as a surface integral by Stokes' theorem (x+h. 6) He didn We have, therefore, generally, [[ pene = | [iC 2) + Ge ng) 2a) dé dn So fe) It is obvious that there is no additional correction term required when f(£&,7) is uniform throughout So or when it has a zero value at the boundary. Applying this result to the integral over So in (V. 13) we may write tl av dG W(x, y, Z) = A | (G on - saa dS a 5, +S, 4 vif (Pe, - iop.) G+ (x + hy 9) So 5 & [(Vp_ -iop )cljaeay + a: s a: sa $ [tv G, - Gly] + = Ly] c| dn Lgptle (V. 14) We will now proceed to discuss the integral over the hull surfaces. Considering first 228 Lineartzed Potenttal Flow Theory for ACVs in a Seaway since where a , the unit normal out of >, i: €. “into the hully us piven by wot, AVERY sca re we may write, e, ds aon ds oh oh Z 1 OG — OG = E x cosé + Nz sin 0) Se ae. + oh Oh + 6( Wa cos 6 - oly sin 6) sae 4 dea’ on the outer hull surfaces Sa4 and So as discussed in Section IV,3 In the case of the inner hull surfaces S54 and S,- we replace h, by h, On linearizing with respect to 6 , 5G -[ 3 : Sip ide ae oh, dh, 0] ara 5 I Gah Dg E i) olew Ce NENG Now, the terms containing 6 have 6 asa factorandas @ is of 0(5,8,a, 5a,Ba) these terms will actually be of 0(8°,68 , 6a), Also ZA9 Murthy 6 : —~— has W asa factor and these terms will therefore become ofa much higher order. We may therefore ignore the terms containing 6 and write simply — + SS SE" BE x ar) dh dG as = &f;-96 i (ee i 0G It should be noted that the derivatives of G on the actual hull surface should be used although we have reduced the domain of integration to the longitudinal plane of the hulls. We will therefore have to expand the derivatives from the hull surface to the longitudinal plane. Let where G, is the regular part of G and r -[ EPG eer] La Then, using Taylor's theorem for the expansion of regular functions aG G dh dG oh 3G 2) = - as ft Li pgs jéa a -§ On -b=dhe's) da pr at of ot ae b 2 oh, 3G, : dh, , dé! dESn ag’ = dna Wane si) ah, 2 a eT Fares eT: rag sc - h, (é'¢) =| + 0(6 7) 1 on? Linearized Potential Flow Theory for ACVs tn a Seaway This is on the outer hull surface Say . In the case of the other surfa- ce S5_ the expansion would be oer Ki Bi re i her Pe aut | on Tog oe OE Gi Gimmeeronen ad Pl n+b=- 6h, (E°eE") (2a. 2 n= -b ' Pa) oe Z. R Lg h : 0 gh, (E°5") eo Sg ee cing) ie + ee wes 4 op Ein - On Rae: SO es demir 2 Th ee | a hy (& a) a2 n= -b Similar expansions with h, instead of h, will apply to the two inner hull surfaces S, + and S4- The normal derivarive has thus been reduced to the derivative in the lateral direction across the longitudinal planes with the addition of 0( 6 ) terms which are only required when we have to evaluate the 0(58) potential. Let us now consider the singular part of G , namely l/r As in the case of G. Dg aril om os lays ya oa Ole Ie & ee OG ae If the point (x,y,z) at which the potential is to be determined is far removed from (&,7, ¢) which is confined to the hull surfaces for the purposes of integration, there is no difficulty, for l/r is then regular and may be considered as part of G, in the above expansion. We shall, however, be actually interested in the case when (x,y, z) also lies on the hull surfaces for the determination of the potential and thereby the pressure thereon which causes the forces and moment. A closer examination of the normal derivative is therefore necessary in this case in view of the possible singularity at (x =, y=7, Ais he Bal Murthy Now Oo ply, ee Ok eee 1 z -¢ 5a ae Samra are Wes ee! a 3 r ic r so that oh dh ) Le a eet pap ar 1 mee 1 San yabe sting. we? Lda patadae’ the Gaueke we and considering first Sy with = oy Spy oS i (oe we have dh, . Oh, cae \ -¥ th ¢ oh (s a oe ge ees cate (2 35) =e On 2 3/2 r 2 send be 2 Say [mPa tye m (E59 yeaa a Assume now that (x,y,z) lies just outside Ss, so that yi Bit She C2 Le where ¢€ is a small positive quantity < 6 Then A OS to 7 El bh, (8 S{)-+)6 (x -5) Sgr 8 (2 -S) se On r a 3/2 c ae | n, (sla*) 4+ ¢ & 5m (807) |? + (z - | which is of 0(6) as ¢€—»+0 , unless x—>» & and z—» ¢ when ) itis of 0( > € In the limiting case when 6 also—»0 it can be shown that Retort | bl] 3 7’ On’ n=b n’=-b se at ee =) ) 001 EO 7)’ df'3n n=b n=-b s*G 5 2 +e G dé dg 2 001 Sys | 2S, | | n=b i-—5 I! 0 G(x, y> Z>3 E. 1,0 ) dG(x, y> zit, nO ) . = of =soxC¥Ih an ; Age Sa dn ‘ n’=b n=b n=-b v0 (v.18) 238 Linearized Potenttal Flow Theory for ACVs tn a Seaway We may now use (V. 18) in (V. 14) to derive an integral representation for VW , but before doing so the integrand in the integral over S, in (V. 14) will be re-written by using the expansion for P, rae Po (t,9) = Bp, CE ) + Bae” (Ko) + hE qq) Wt): i.e. oo a SP yl 001 EG 8001) Pes ee: ) so that the integral over S, becomes I] bre. © #Paligg, + Nooo} 2VP G+p, (Vo_ is) of ]atay ee E et ; EVE We therefore have finally, W(x,y,z aff ao) GG yinzses ba) Glocsy 7 2. bee, 2) + h Pee eS ae “O01 2 (io + tee (G) Gr te tes dé Neue e a°G 3*c -_—- WV = ? } 1 U ‘ft BSI, ae Von (se ae lo 2 2 aG ome | ag SSS SSS did a TO ey eres : anos | | sa EER SD OGx, y, 2; E, 1 ,0 ) OG(x,y, z3&,7 ,0 ) POC A te ge, gia |: E L n=b nab n=-b l =o i icalfler GF Pe oo” "Ge Pool So 2 p VINE ¥ p - (Vee 10 of | dédn + NEE 2 ee o| : 239 Murthy 2 (v4 G, -G [y] 4eie Wc] dn (v.19) The potentials of various orders may now readily be written down by equating terms of the same order on both sides of (V. 19) with the assumption that the "jumps" in the potential relate only to the cushion and not to the side hulls. This has been discussed above. We thus derive the following expressions for the potentials : 0(5)%, 59 (x, y, z) afl E | Ge, y, 23&,b,¢) + G(x, y, 23, ».t)| dé dt (54) 516 (x,y,z ff 3 (f+he ) roof Se 3 e - he Sh Ps) , y) + ioor = £45) 28]fe +v2e] [oye] ae Vv Met. (x, y> z) rol: Po, G(x, Y, 2; ae nsO ) dé dn + 2 Oo a aa [*oro0t®"»2) | G, (x,y, 23.7.0) - Lptle 3 , E tprool 2) (x,y,Z:&,,0) dn - =, (E,p-,f)- (Fs bt, aa) An 0100 0100 ‘S, 3G(x, y, z3&57', £) OG(x, y, 23 3’, 8) avae On’ : On te n=-b (x +h_@ ) 001 G 001 : 0( 8) ol y+ 2) = eee 2| (ve, ten, ee teem ie} + Vp ae (ts ¥s Ze ah OD didn 240 Ltneartzed Potenttal Flow Theory for ACVs in a Seaway © gE 5 ) + ae § |[*orr0! the 0) | [Gy by 2 i Lgrle. G (ey,2i8+n.0) |- ona (en-0) | Gtery tit 90) | dn - aE 0110 ] , ) 1 aa If 0100 (8, b», 6) - ? 0100 (£5 b+.S } 6 = / h 6 estan ny f) i ange sat (as eer GX? cot! sae wee Bean: n=b Sees z;t,n50) = , 5 ES ) (ears Stee n=-b 2 , : , , ’ 2 A 7 7) 7 é G(x, y,23&,7,$) a re) G(x, y, 23 & 57 ON, Ono ft’ On’ oo’ f nab n=-—b {4 aig (Eisbziatee 2h ae (Erbe) 0 G(x, y, Z; E, n> i) | a G(x, a Z ms ) acto nsf DH |« ag - hhaetar n’=b ’=-b 1 - 2 py E! ie ms Tras | F001 te LE 100 | ot L ) aG(x, Y> Z;5 ES 130 ) . oe (2, p+, 0)] | Seew-s ae) = i L= 241 Murthy OG(x, y, z3 E,n',0) On| |e’ (V. 20) It will be observed that we have derived an intepral repre- sentation for the hull potentials in the form ofa source singularity distribution over the two longitudinal planes of strength equal to the normal velocity in the case of steady motion (the classical Michell potential) and a source distribution of density equal to the normal oscillating velocity together with a distribution of doublets oriented longitudinally and of strength equal to the product of the normal oscillating displacement and the forward velocity. In the case of the air cushion, we have derived integral equations for the potentials and an explicit integral representation is only possible under some additional assumptions. During steady motion there is a primary distribution of sources over the steady position of the lower boundary of the cushion with a strength equal to the longitudinal gradient of the basic pressure distribution with an additional line distribution of sources, doublets and quadru- poles along the bow and the stern,the strength being equal respecti- vely to the ''jump'' in the longitudinal velocity of the water particles and the ''jump" in the potential itself. As these jumps arecaused by the discontinuity in the pressure it may be assumed that the line distribution will vanish if the cushion pressure is such that it is diffused to a zero value at the boundary and with a zero value of the longitudinal gradient there. In addition, there is a distribution of doublets oriented laterally over the longitudinal planes of the side hulls of strength equal to the jump in the potential across the planes. The oscillatory potential is given by a similar distribution of sources and doublets over the IFS, along the boundaries of the cushion, over the longitudinal planes, and along the waterline. These potentials are discussed in further detail in Sections 6 and 7. The Interference Potention ? 100 The derivation of the interference potential@ ,,) is slight- ly more involved as all terms of 0(68) have now to be taken into account . The method depends upon finding first a particular solution of Laplace's equation satisfying the inhomogeneous equation repre- sented by the boundary condition (V. 3). The homogeneous function denoting the difference between the actual potential @,,,, and this 242 Lineartzed Potenttal Flow Theory for ACVs in a Seaway particular solution will then satisfy the homogeneous form of (V...3) and can be solved in the same manner as ® 1000 °F P oi00 The Green's Function. The potential of a source of maximum unit strength and pulsating with frequency o« while moving with uniform velocity V along the x-axis ata depth { below the undisturbed water surface satisfies the conditions stipulated for the Green's function in connec - tion with this problem. This function is given in different forms by various authors, but we shall use the representation derived by Peters and Stoker (2) -1/2 G(x, y,236; 9,0) = Joe - 5) $y 2 ae oF | 3 -1/2 . le-t+ ae ee ey ' 7 00 -p(zt+$) + ip(x-&) cos 4 ‘ ; aff pe cosp (y-7) sin#@ dnd + ig gp - (7+ pV cosé ) ou) T7/2 26 ogee 6) + ip(x-&)cos ee a Bag a a eM we oa oo pe - (o +pV cos@) + Ms Sp = (io +p 2g oe ea) = aes) Ped compli) sin§ i aca, PRT eT dpd é Paik gp - ( + pV cos@) where Sie 0 ! if l <= < Lineartzed Potential Flow Theory for ACVs in a Seaway Ee eit SKIRT —— asian) es See ER » < hi) myo U rr 'U OQ ean ayy oo" o-oo i 0a air cushion vehicle. external free-surface - free surface of water of infinite extent outside the immersed part of the side hulls and the vertical projection of the flexible extensions at the front and rear of the air cushion. internal free surface - free surface of water contained between the inner sides of the immersed part of the side hulls and the vertical projection of the flexible extensions at the front and rear of the air cushion. total waterplane area of the two side hulls at zero speed ahead. amplitude of the regular incident waves. one half of the separation between the two side hulls. external force vector acting at the C.G. of the ACV. pressure force vector. pressure force vector (cushion hull). pressure force vector (side hulls). rigid body force vector. side force on an amphibious ACV in drifting motion. pitch stiffness function of the air cushion. heave stiffness function of the air cushion. steady Green's function. unsteady Green's function. equilibrium position of C.G. during steady translation. instantaneous position of C.G. during oscillations. acceleration due to gravity. hull function in (x,y,z) system. total width of side hulls at (x', z') hull function in (x',y',z') system on the starboard/port side of Sy and port/starboard side of S, 248 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway ha height of C.G. above undisturbed water surface at zero speed ahead. hy, height of the thrust line above C.G I moment of ee of ACV about lateral axis through C.G =] + h ( Vay m a I, moment of inertia of waterplane area of both side hulls about lateral axis through the origin o' ety! moment of inertia of ACV about lateral axis through o' A aa k unit vectors along axis in (x,y,z) - system. A A Ae ee! unit vectors along axes in (x',y',z') - system. k,1,m,n as superscripts refer to the powers of perturbation parameters in perturbation expansions. as subscripts refer to the component terms of the pertur- bation expansions. wave number of incident wave ( = o°/g Ns a related to speed of translation ( = g/Vve yi: L,M,N, components of moment vector. L load waterline on the longitudinal planes of the side hulls. L, vertical projection of hemline of the flexible extension at the front of the cushion on the plane z = 0 Le lower boundary of air cushion on the undisturbed water surface. Ly boundary of ACV on undisturbed water surface. Le vertical projection of the hemline of the flexible extension at the rear of the cushion on the plane z = 0 L, 5 L,, L,, Ly, as defined in Appendix V. M moment vector. = M, pressure moment vector. beer 2 M, pressure moment vector (cushion hull). HAE G: My, pressure moment vector (side hulls). H =a . . M, rigid body moment vector. m total mass of ACV. 249 Murthy m, partial mass of ACV supported by air cushion. m, partial mass of ACV supported by the buoyancy of the side hulls. 4 unit normal vector. origin of co-ordinate system fixed in space. fe) origin of co-ordinate system translating in space with uni form speed V ina straight line. Oo origin of co-ordinate system fixed in the ACV. p variable of integration. Pp; a cushion pressure (excess over ambient). Pi y) basic hull form of the air cushion. p surface pressure on IFS. s om time-independent part of P, Rw wave resistance r distance between (x,y,z) and (t,7,$). — by position vector of element of mass of ACV or element of area of IFS with o' as origin. ; " ; "idm coupled displacement parameter “am + (z' + ha) Ot : 3 +4 r) Be coupled displacement parameter 8 on he Peas S15 | surfaces of integration. S) instantaneous position of IFS during oscillations. So instantaneous position of the part of the plane z=0 contain- ed within IFS. S ae fo) steady position of So : s' oscillating strip denoting the difference between S and S, fo) Sy starboard hull. 2 port hull. me s, starboard/port side of starboard hull. B85 starboard/port side of port hull. Sy - S, longitudinal planes of Sy a in their instantaneous posi- fe) Oo tion. 250 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway ig ne starbord/port side of mis ; eh S, S, steady positions of S, ; Ss, below the load waterplane fe) fo) z -0 ° fo) S, . Sa strip of the longitudinal planes between z'=0 and z= 0: fo) fo) s,” soe strip of the longitudinal planes between z=0 and z =f fo) fo) ss Sti coupled stiffness function = zs ai = De Zim f forward thrust of ACV propulsion system. t time Uv absolute velocity of element of mass of ACV Vv mean forward speed of ACV. Pe) 2 rectangular co-ordinate system fixed in space. 5 Z. components of force vector. xy Yp: Z5 components of pressure for e vector. x, : oe : Z, components of pressure force vector (cushion hull). c C c x, : Y, ‘ a. components of pressure force vector (side hulls). h h h Xp? YR? Zn components of rigid body force vector. Bey 5 rectangular co-ordinate system translating in space with uniform velocity V x',y',z' rectangular co-ordinate system fixed in the ACV. x surge displacement. Z heave displacement. Xn x-coordinate of C.G. of waterplane area of the side hulls. Xp Zp x, Z-coordinates of the centre of buoyancy of the side hulls. a x-coordinate of the centre of pressure of the air cushion. a motion perturbation parameter denoting the small order of the amplitude of the motions due to forced oscillation or due to wave excitation. B cushion pressure perturbation parameter denoting the small order of the cushion pressure. Y the phase angle of the incident waves. 254 Murthy ) side hulls perturbation parameter denoting the small order of the semi-width of the side hulls on either side of the longitudinal plane. € incident wave perturbation parameter denoting the small order of the wave slope (ratio of the amplitude to the length of the wave). — 7,§ dummy co-ordinate system having the same disposition as the (x, yz) system -source point co-ordinates. @ angular co-ordinate ; variable of integration. 0 pitch displacement. d wave length of the incident wave. p fluid density. * control surface (comprising di, d pa and pee ) Ds Ly Ds Z; as defined in Appendix V. Oh peas a frequency of incident wave relative to space. Ge encountered frequency of incident wave ( = o + kV ) ® velocity potential of water in frame of reference fixed in space. ® velocity potential of water in frame of reference moving in space with velocity V. v time-independent part of velocity potential. f elevation of water surface. w angular velocity of ACV about a vertical axis. A dot denotes differentiation with respect to time. Derivatives are noted by subscripts when not written in explicit form. Special Note. In the case of terms containing elot asa factor, it is naturally understood that the real part of the complex quantity is to be taken. This exponential factor denoting harmonic variation with respect to time occurs both in the case of the oscillatory displace- ments and in the case of the unsteady potential. When a factor of 252 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway ante is given it is implied that a factor of e ist should be taken with the displacements and a similar factor with the potential. The real parts of each are separately taken and then combined together. This convention is specifically indicated by the asterisk. PAYS) Murthy DISCUSSION Lawrence J. Doctors Untverstty of New South Wales Sydney, Australta I find this paper very interesting, because the response of an aircushion vehicle to a seaway is an important factor in the opera- tion of these craft. Violent motions are to be avoided from the points of view of both the passengers, and of the craft structure, i.e. safety. In addition, it is desirable to maintain a minimum operating speed, which is not always possible in waves of a large amplitude. I would like to ask some questions about the analysis used by the author. Firstly, I can see no place in the paper, where the fan characteristics are involved. Surely, the interaction between fan pres- sure and volume flow would affect the dynamics of the craft. In other words, this is an additional parameter in the coupling between the motion of the ACV and the response of the water surface. Secondly, one would expect the timewise varying air gap be- tween the skirt hemline and the water to be an important factor. Has this in fact been considered, for the air gap affects the pressure drop under the skirt ? I am also interested in the way the pitch stability of the am- phibious craft is modelled. For example, does the author compart- ment the air cushion with a transverse skirt ? Incidentally, this would introduce a nonlinearity under certain conditions. My final question is this : Has Dr. Murthy carried out any numerical calculations, or does he consider that computer time would be too excessive ? 254 Litneartzed Potenttal Flow Theory for ACVs tn a Seaway REPLY TO DISCUSSION fai Ss. Murthy Portsmouth Polyteehnte Portsmouth, U.K. Dr. Doctors is quite right. I missed mentioning these things in my hurried talk. I would refer you to page 167 where I have clearly stressed that we have gone into some detail to study the hydrodyna- mic, including hydrostatic, effects on the motions of the ACV. Itis only a short paragraph and I will read it: "It is assumed for this purpose that the aerodynamic effects are known including, in particular, the stiffness and damping of the peripheral- jet or plenum type of cushion. It may be thought that such effects as that due to ''wave pumping'' should have been taken into account...'"' Of course, that includes the question of fan characteristics as well, These are assumed to be known and they can be fed into the equations, I am only assuming that the ACV is moving under the ac- tion of a constant thrust. I am assuming that the pressure distribu- tion is known. The expressions involve the pitch and heave stiffness of the air cushion itself, which are assumed to be known ; they are the functions f] and f2 given in the text of the paper, which take into account the air gap of the cushion and the stiffness and damping of the air cushion due to variations in the gap. The third point is concerned with stability. In my work Iam not concerned with the actual cushion ; I only want to know what the pressure distribution is. People think, rather naively, that the cushion is uniform. It is very easy to construct a uniform cushion within a rectangular boundary but one can see very easily that this type of cushion will be unsuitable for high speed motion because if you trans- late it into shipping terms a uniform rectangular cushion will have a hull form in the shape of a vertical cylinder with a vertical bow and stern, and no naval architect would use this type of hull form for a fast bearing planing vessel. It is also made clear in the expressions that the distribution of pressure, particularly the diffusion of the pressure - what Dr. Doctors has been calling ''smoothing" at the boundaries - is very 255 Murthy important as far as the motions are concerned. So one has to looka bit more deeply into the actual pressure distribution in the cushion and we can, it appears, have a tailormade cushion for each job we have in mind, whether we want a cargo ACV ora very fast ACV. It depends on the role the craft has to play and you can design a suitable distribution of pressure. As to how it is done, you can accentuate the entrance entrainment of the air from the atmosphere and produce a non-uniform cushion of a suitable shape for the role the vehicle has to play in actual operation. The last point raised was whether a numerical computation was made. That was meant as my third apology whenI started. No results are given here. The work was done purely as a private ven- ture, It was not sponsored by any Government or any agency. Calcu- lations involve the use of a computer, which costs money. But the ex- pressions are available here and can be applied to any specific confi- guration, and quite realistic results can be obtained, I am also con- sidering the feasibility of applying the finite element method which may be the only answer to problems of this kind, because some of these integral equations are singular and there may not be any easy way of solving them without the use of the FEM technique. This I am studying now. So, once again, Iam sorry I have no results here be- cause results cost money, but they can be obtained probably at some later stage. 256 Lineartzed Potenttal Flow Theory for ACVs tn a Seaway DISCUSSION Roger Brard Basstn d'Essats des Carénes Paris, France You just said that you deal with singular equations. I suppose you mean equafions whose kernel is singular ? REPLY TO DISCUSSION a. Ko. Murthy, Portsmouth Polytechntec Portsmouth, U.K. Yes. Zant Murthy DISCUSSION Paul Kaplan Oceantes Ine. Platnvtew, New-York, U.S.A. I certainly agree as regards the importance of pressure. This is a craft which by definition is an air-cushion vehicle with 90 to 100 per cent of its weight supported by air pressure. A deviation in the equilibrium pressure of 50 per cent, which could easily occur and could be computed for simple ''wave pumping'' by waves commen- surate with the length of the craft (without such great height that would violate linearity) will show changes of pressure of the order of 50 per cent of the equilibrium pressure. Therefore accelerations of !/2g immediately are possible. In order to account for this you do not just say that the equation contains a damping coefficient and stiffness coef- ficient that represent pressure effects. Pressure is a degree of free- dom which must be added into any equation system. Therefore you have heave, pitch, surge, and pressure is also a state variable ; it couples with the others and it is the most important element. That in- troduces the fan as well as the Plenum geometry and the wave pumping. The end result of this is that if you neglect anything with the air pres- sure in the manner shown here, then motion responses are erroneous. I would suggest that this paper is very useful in many respects for determining hydrodynamic forces, but the title is somewhat erroneous. I would suggest it be changed somewhat and there it would still main- tain utility. The paper should be called ''A linearised potential flow theory for the hydrodynamic forces associated with the motions of air-cushion-vehicles in a sea-way", Now here are some positive contributions to balance what I have said before. The aspect of linearity is important and the result- ing ability to predict motions in irregular seas even more so. There has been published recently some results for motions in regular waves which include equations which also have this pressure degree of free- dom. The agreement with experiments, while not perfect, is more than adequate. The predictions based on the application of spectral analysis to predict RNS values of accelerations in irregular seas agrees quite well also. Therefore, not only is there an aspect of utility of the theory for getting hydrodynamic forces ; there also is a virtue in linearity, so that is a benefit as well. 258 Litneartzed Potenttal Flow Theory for ACVs tn a Seaway REPLY TO DISCUSSION rok. o.)Murthy Portsmouth Polytechntec Portsmouth, U.K. I quite agree with Dr. Kaplan, I have made it clear that lam only concerned with the hydrodynamic part of the study and I have put various riders in the results saying that these are only the hydrody- namic effects. These appear throughout the paper in various places. I have mentioned cushion pumping and cushion stiffness and compart- mentation. These are naturally to be added to the results here, I agree with what Dr. Kaplan has said. steps OalGkt 259 — > noe I tect nee bration egy av aris) + opedepepioedietbersens sot igg pip oes inal A boty hetrac ot2\4 of ¢fad (odartetphosby eat. day feed Lov GaSe dis de ve wtrar wemsenpe Jule qpenkypate tant kets 9 greet ies 476 be) Qtek Dice! pega. 2 dont Sl geos Bteta omit it a8 5, coed Sorrma ke byrs ore Goivdie tr tims’ 'gaagarpoed ots eta roach ‘Sqexba @nlerdenesd et bol hon fd ot ties gta, ame 2° oa tgh nearity) will dhe rgb beet igaet solani Seta X ‘ : L 17 & ™ *| ; Oe 4 Bey 1@ << AL< ec Ys # of n 4 Peyi eye! Beeld re possible. ‘ ts i> Acre wt thas oe do r " vhs othe s ‘ ~~ " Po * bh XD. Gg yn tar Co 216 & Gas a4 Lent ith atit as * * . - ’ NOGut that vepregent pri gaa “ te, Pressure ie 6 dege PT Wehhic mY cat ra 'Cloe shite > Te ’ ' , rete Sa Oy eee « Di . = ‘8 : rs. sy ‘ Sate fo) with the oo re and j Ms AS AST S a Dacca 1 » \¥ , % ; , tia ‘ » & >i -: SS va% ™ tire Wve, “) a ‘eae! i way eeics UY! Moy Wite Coe ae . . 4 a ee. : i ’ At10 St Leese ’ / : 9 ipoOnseer are @ Wit ake? pany is wa is im eatiy Wy " 7, Ee og ro 5 z 1 Mera 1. wie n pape Ao : Lita iged potent oOo f tt amic lovces aseotiated with the bbrecty shion enicie: a G28 y Z Now liere ere it DOs « contrib on (> of Lance, nF mave gaid betore, The rank of linieawity te important @ adhe # ing ebility-to predipt motions in irregular wesw even more. eocee he ton published pecentiy vorane results for trottons in fag Ce i Jaa lade evraxtionsg which aiac ave tie § re eeure degrem inry ee SgyYooment with experiments, while sot perfect, Gal ” MAW eHeyuate. Che precichans based on Us appiimetton of or Lived & ~ previci R a<ue sere iccelea rations iz irbeguday.” eT Pe Mm wes 1 inereiore, nol omly de theme an Aepmee a? the tpeey jetting by 4trodywarmic forces > thers dled 1. $c) Jim eete; ab Abe treriadie ap wrest. .! | nd thee <9 ft ’ “Pats ¥ ON THE DETERMINATION OF AEROHYDRODYNAMIC PERFORMANCE OF AIR CUSHION VEHICLES 5...) Prokhorov, V.N. Treshchevsky, Ib. D. Volkov Kryloff Shtp Research Institute Leningrad, U.S.S.R. ABSTRACT Experimental and theoretical methods of determi- nation of hydro-and-aerodynamical characteristics necessary for air cushion vehicles dynamic calcu- lations are discussed. The influence of ship form on mentioned characteristics corresponding to stea- dy and unsteady motion is analysed. The estimation of propulsive performance and manoeuverabi- lity characteristics of ACVs while designing these vehicles can be based on tests carried out with self-propelled models and simulation of various operational conditions for these models. It is obvious that the model behaviour is governed in this case by the total aerohydrody- namic loads which are different in nature. At the same time separate definition of aero-and-hydrodynamic forces of various nature is of interest, specifically, for the analysis of the effect some components of these forces have upon the ACVs performance and dynamics. The forces acting on ACVs can be classified into the follo- wing categories : a) aerodynamic forces due to the oncoming flow around the hull, flexi- ble skirt, stabilizing fins and controls; b) aerodynamic forces due to the air cushion and interaction between the latter and counter air flow; c) hydrodynamic forces due to the contact of the flexible skirt with water surface and interaction between the air cushion and water sur- face; d) hydrodynamic forces due to the effect of waves. The cruising conditions of ACVs are characterized by the active interaction of the structural elements with both water and air 261 Prokhorov, Treshchevsky,and Volkov environments. The difficulties in simulating the aerohydrodynamic effect and the necessity of analysing the data obtained make one carry out the experiments in the towing tanks and wind tunnels independent- ly. In. the last few years the central place in such experiments belon- ged to obtaining the steady and non-steady characteristics necessary for the calculation of transient processes and estimating the ACV sea- keeping qualities. This paper gives a brief description of the most typical, in the authors' opinion, methods of defining the above characteristics. The description of these methods is illustrated by the measurement results concerned with schematized models of ACV s with different bow form. I. DEFINITION OF AERODYNAMIC CHARACTERISTICS As is known, the character of ACV movement in the longitu- dinal plane depends mainly on the air cushion parameters and also on hydrodynamic interaction between the flexible skirt and water surface. The ACV movement in the horizontal plane depends, toa great extend, on aerodynamics of external flow around the vehicle There are two main problems relating to the study of exter- nal flow around the ACV, viz., the decrease in air resistance and the achievement of predetermined maneuverability and stability. The first problem can be solved only on the basis of the rational choice of the hull form. The second one is usually solved by mounting the system of stabilizers. For the ACV s with the length-to-breadth ratio of 1.5 to 2.5 the resistance as well as side forces are determi- ned by the distribution of normal pressure over the perimeter of the hull; so the studies of both problems are closely connected with each other. The experimental investigation of the aerodynamic forces and moments which are due to the external flow is generally carried out on rigid models because slight deformations of the flexible skirt have little effect on its aerodynamic characteristics. A six-compo- nent balance is used for this purpose Water surface is simulated by a flat ground board, The nature of aerodynamic factors can be revealed from the results of the experiment with schematized models carried out for the study of the effect the hull form has the ACV s aerodynamic cha- racteristics. Table 1 lists the values of geometric characteristics of the models tested, their resitance factors and the derivatives of 262 Determinatton of Aerohydrodynamte Performance of ACVs side force and yawing moment with respect to the yaw angle A for B = 0 which are necessary for the calculation of ACV movement in the horizontal plane. The aerodynamic forces are related to the pro- duct of the velocity head f end centerplane area S, while the moment is related togSL where Lis the model length. The moment is calcu- lated with respect to the middle of the hull length. As is seen from the Tablel, the resistance is mainly influen- ced by the forebody form. With the change of relationship LH from 0.5 to 2 the resistance factor decreases by 30-40 per cent. "The in- fluence ofthe stern formon the resistance is less important, which is due to separation effects and the formation of the dead zone at the stern. The angle of run ranging from IK = 0 to 2 leads to a decrea- se in resistance only by 10 per cent. It is known that the heave stability of the ACV s mainly de- pends on the sign and value of the derivative m 8. The minimum va- lue of the derivative m 8 which is favourable ftom the viewpoint of stability of motion occurs for the models 7 and 8 with the smallest values of angles of run and entrance. As is seen from the Table, the bow elongation results in increasing the destabilizing moment m_ , so the requirements of minimum values for the coefficients of resis- tance and yawing moment are rather inconsistent. Since the required value of the derivative m can always be obtained due to the fitting of vertical stabilizers Sour noticeable increase in resistance, the hulls with elongated bows and blunt sterns appear to be the most ad- vantageous. For the approximate estimates of the ACV s aerodynamic characteristics at the initial stages of designing the theoretical me- thods are of interest. The method is developed for the calculation of both the total and distributed aerodynamic characteristics of ACV hulls with flat sterns at different yaw angles and angular velocityW y depending upon the forebody forms and paramenters 7 and are The method is based on replacing the hull of the ACV bya vortex surface which extends beyond the hull for modelling the vor- tex trace effect (figure 1). The free water surface is simulated by an image body so that in fact consideration is given to a model witha double height H. The transverse vortices are directed parallel to the lines forming the flat stern contour. The longitudinal vortex which leaves the body from stern contour corners has the density equal to the difference between the densities of vertical and horizontal vorti- ces replacing the contour. Longitudinal vortices arranged in the deck plane are parallel to the longitudinal axis of the hull. Calculations 263 Prokhorov, Treshchevsky,and Volkov were carried out for the finite number of discrete vortices spaced equally from the control points where the boundary condition is sa- tisfied. The density of all vortices is defined from the system of equations characterizing the boundary condition on the body surface at control points, For the proper choice of the value of circulation around the body, as with the known Joukowski condition in the wing theory, a supplementary condition is introduced, viz. the density of the trace vortices in all the points of the vortex sheet is the same and equals to the density of the vortex shedding from the contour of separation at the stern. Then the system of equations for defining the densities [,f,,[,...f, takes the form : ee Ne | n ea a Sib =P (Rio) a koi i Pp where F, and F = induced velocities corresponding to transverse and longitudinal vortices, F = =normal component of the free stream velocity, w = angular velocity of yaw. Y Figure 2 shows the pressure distribution over the contour of intersection of the model and the basic plane obtained by the calcu- lation method for model 2 with the angle B = 10°. The test results on defining the pressure distribution at the same section of the model are also plotted in the same figure. As is seen, there is a fairly good agreement between the theory and experiment. As noted above, the aerohydrodynamic forces due to the air cushion are the decisive factors for the longitudinal motion of the ACV. One of the methods used for defining the steady and non-steady aerodynamic forces is based on the measurement of forces acting upon the model which performs the harmonic oscillations. When car- rying out these tests in a wind tunnel the forces are evaluated which are connected with the qualities of an air cushion and deformation of flexible skirts above the ground board simulating the water surface. In some cases the results of such tests can be used, for the calcula- tions of the ACV movement above the ice surface. Non-steady aerodynamic characteristics of ACV models with built-in fans and flexible skirts are determined by using the ex- perimental plant shown in figure 3. The principle of operation of this plant consists in generating the definite harmonic oscillations for the model and measuring the loads acting on this model with the con- sequent determination of the lift and lateral moment derivatives ac- cording to the kinematic parameters of motion, The plant is equipped 264 Determtnatton of Aerohydrodynamte Performance of ACVs with a mechanism of compensating for the model inertia forces and with an electric harmonic analyzer for automatically defining and recording the signal constant components which are proportional to the required rotational and translational derivatives. The model is mounted upside-down (figure 3) on two supporting pillars which make reciprocating oscillations with arbitrary shift in phases with respect to each other, Each pillar is supplied with a strain gauge which ser- ves as a connecting link between the model and the oscillating pillar. The distances to the ground board h; the trim angle ¥ and their first-order and second-order time derivatives h, y , h, WY are adopted as kinematic parameters defining the model aerodynamic characteristics in the longitudinal plane. In linear approximation the expansion of the vertical force or longitudinal moment as a series in kinematic parameters has the form (h oY) mets (hs v8) . ht Ruin 2, Cap eeat Re (ho )h ae: (2) R=R bf h ste ea (( he tek (h RV(h HYPER! (ho) b+ RF, Y) SG where h, Y, are mean values of height and trim in respect of which the values h and w are changed, The tests are carried out for two types of motion : transla- tory and angular harmonic oscillations of the model where, with the results of the model static tests also used, all the derivatives entering into equation (2) can be determined. The values of rotatory and trans_ latory derivatives of the vertical force and the longitudinal moment are determined and they are transformed to a dimensionless form through dividing these by the model weight G or by the product GL. AVE . Mz <. G Me TAGE (3) As the dimensionless kinematic parameters the following factors are used : fo) sail Tw aan Qe » Ton we Sree’ ISS Tay ; ' ne (4) Wee eA ahs Joka rie PQ‘ 265 Prokhorov, Treshehevsky,and Volkov where Q. = air flow rate corresponding to the parameters bys, (m/sec) PA g = acceleration of gravity (m/sec ) The value sh,=- 80 should be adopted as the dimensionless crite- rion of similarity which is similar to the known Strouhal number (WW = angular frequency of oscillations). Then the dimensionless va- lues of parameters h ; rc) j r : ® in equations (4) are the products of the dimensionless amplitude of vertical and angular oscillations and Strouhal number Sh, or Sh2Z Using the procedure described the tests were carried out for three schematized models of ACV s with built-in fans. The scheme of the sectional flexible skirt mounted on the models is shown in the sketch (figure 3), The geometric characteristics of models are gi- ven in Table 2. Figures 4-5 show some of the results obtained. The curves in figure 4 illustrate the character of changing the factors Cy ; m! E mip versus the dimensionless parameter Sh,at different values of pressure factor Kp =fe (Pp , Pn = pressures in the re- ceiver and air-cushion, respectively) and air flow rate Q. It is typi- cal that the external air velocity effect on the non-steady characteris- tics is practically absent in the range of the examined actual relation_ ships of the contrary velocity head and the cushion pressure ( Fig. 5). The curve in Figure 5 shows the influence of the bow form at the fixed values of frequency and pressure factor upon the same charac- teristics. It is obvious that the bow form influences mainly the mo- ment characteristics, The non-steady aerodynamic characteristics necessary for the study of the ACV maneuvering in the horizontal plane are also de- fined in the wind tunnel by the method of harmonic oscillations of the model around its vertical axis and measuring the yaw damping moment influencing the model. In this case the procedure of measurements is similar to that described above. The experiments show that the principal role in generating the damping moment belongs to the ver- tical stabilizers and the hull effect on this moment is not significant. It is observed that in some cases the damping moment on the hull- stabilizer system decreases due to the adverse effect of the hull on the stabilizers. II. DETERMINATION OF HYDRODYNAMIC CHARACTERISTICS In the towing tanks the tests are carried out to determine 266 Determtnatton of Aerohydrodynamic Performance of ACVs the main hydrodynamic characteristics of ACV s. In this case the mo- del is usually subjected to the action of both the hydro-and-aerodyna- mic forces. Then, depending on the test conditions, account is made of either the results directly obtained by measurements, or the aero_ dynamic components are tobe excluded with the use of data on the blo- wings af the model in the wind tunnel. The most typical tests carried out in the towing tanks are those with the towed models exhibiting the freedom of heaving and trim; during these tests the resistance and kinematic parameters are mea- sured in the longitudinal plane in calm water and in waves, The pur- pose of such tests is not only in obtaining the propulsive performance data but it is largely connected with the evaluation of dynamic proper- ties of these vehicles. Thus the measurements carried out at diffe- rent positions of the centre of gravity along the model make it possi- ble to plot the positional curves against the trim angle and to judge about the static stability depending upon the conditions of motion. The same type of towing tests is the basis for determining the regions of steady motion in the longitudinal plane defined by the influence of the waves and speed. Such experiments were carried out specifically on a series of models with the particulars given in Table 2. In this case the re- sistance and kinematic parameters were changed up to the critical conditions preceding the development of plough-in. The curves in figure 6 show the effect of the bow planeform on the relative resis- tance = depending upon the running trim angle which is defined by a given position of the center of gravity. It is seen from the curve plotted for a cruising regime that the effect in question is observed only with trim by the bow; in this case model N° 2 appears to be pre- ferable. The advantages of a semi-round bow planeform manifest themselves in waves too, as is seen from the curve of figure 7 where the relative gain in resistance is presented for all three models in waves. The air flow rate effect examined on model N° 2 is typical for this experiment. The curve in figure 8 shows the effect of the dimensionless factor of air rate upon the relative resistance in waves Sn 22Pn where Q =air rate, m/sec; Sn = area of the air cushion, m 2 Pn = pressure in the cushion, kg/m 267 Prokhorov, Treshchevsky,and Volkov I tis seen that practically in all the cases the increase in the flow air rate results in a decrease of resistance; in this case the supply of air into the forward part of the air cushion, is the most favoura- ble. The curve of figure 9 serves as an example of plotting the regions of stable motion. The relationships shown are the result of processing the model N® 2 test data according to the evaluation of the limiting regimes when plough-in is developing with the consequent loss in sta- bility. The curve shows the favourable influence of increase inthe air flow rate at the bow centering, making it possible to delay the set- ting of the critical regimes. The definition of the non-steady hydrodynamic characteristics which are necessary specifically for carrying out the calculations of ACV heaving and pitching is based on the same methods used ina si- milar case for displacement vessels, The linear character of the restoring forces Y h.- £Y and moments M}- oMz defined experi- mentally in the working range of the flying heights h and trim angle p for ACV with a flexible skirt gives grounds as a first approxima- tion to proceed from the linear theory premises while defining the non-steady characteristics, The tests are carried out on a plant which makes is possible to perform in calm water the forced heaving and pitching motions of the model; the plant is equipped with strain gauges and provides the recording of kinematics of motion, To define, for example, the coefficients of inertia and damping forces by the test results, the equation of the forced heaving motions is written in the following way : ee wt e a 0 (M+yY")y +b y + z y = C(t Cos G t-y), dh where M = model mass, C = rigidity of spring, 4 = amplitude of disturbances, 6, = frequency of disturbing force Having experimentally defined the parameters of the forced motions of the model in the form i (6, t-6 where ae = amplitude of oscillations of the model centre of gravity, a = phase shift between the translation of the model and the disturbing force, Determinatton of Aerohydrodynamtce Performance of ACVs one can find the coefficients y" andy" and similarly the moment coef_ ficients characterizing the pure pitching motions M,Y and M_¥. The definition of factors characterizing the influence of pitching upon hea- ving is possible provided that the vertical translations of the center of gravity are recorded. The experimental plant scheme and some of the results ob- tained are given in figures 10-11. Depending upon the type of rope- and-block connections provision is made for the translational verti- cal ( var.1) or angular ( var.2 ) motions of the model and for re- cording kinematic parameters on the oscillograph tape. In figure 11 the coefficients of inertia forces obtained by the above method are plotted against frequency. It is necessary to note that the aerodynamic characteristics corresponding to flight over the water surface are different from si- milar characteristics of the model over the ground board. The diffe- rence is obviously due to the influence of the water surface deforma- tions and mass forces; it minimizes as the Froude number increases. The experimental plant described was also used for the defi- nition of damping and inertia characteristics which manifest themsel- ves at non-steady motion of the model along the longitudinal axis; the plant is switched on according to variant 3. As the test results show, the forces determining the above loads are negligible for ACV models. In the cases when during the tests in a towing tank the aero- dynamic components are so important (in comparison to hydrody- namic forces) that they cannot be neglected, it is necessary to consider that the aerodynamic effect upon the model tested is not fully simulated. This introduces some infinity in the results ob- tained due to both the distorbed aerodynamic action and the effect of this action upon the position of the freely towed model and consequently upon its hydrodynamic characteristics in the total forces and moments measured. In such cases the aerodynamic components are excluded from tests in the towing tank and are de- termined in a wind tunnel. The procedure of carrying out the ex- periments of such kind is as follows : the model is rigidly fasten- ed to the dynamometer which measures the lift, drag and longi- tudinal moment of the model at the fixed values of height and trim of the model, During the tests the measurements are carried out in a prescribed regime, Then according to the same program the tests are conducted on a model equipped with a working fan over the ground board fixed under the model in a close vicinity to water 269 Prokhorov, Treshehevsky,and Volkov surface and transported together with the model. The aerodynamic forces to be excluded are determined as the difference between the results of the measurements carried out over the board both underway and at a speed equal to zero in flight. The specific behaviour of the ACV with a flexible skirt makes difficult in some cases the use of traditional methods in calculating the maneuvering qualities, dynamic stability and so on in terms of the solution of equations of motion. The complexity and considerable amount of tests necessary for defining the coefficients of the equa- tions makes one use other methods of study. The determination of transfer functions according to the required parameters in terms of the experimentally defined frequency characteristics is considered to be reasonable. These functions make it possible, as is known, to cal- culate normal maneuvers of the object according to linear theory. Besides it is important to have the possibility of directly evaluating the behaviour of models in certain conditions, specifically, in dama- ge situations. The tests with the both aims in view are carried out on the experimental plant making it possible to simulate, in the main, the conditions of the model free movement and in some cases to eli- minate the necessity of carrying out the expensive tests with self- propelled models. The basic diagram of the plant is shown in figure 12. During the tests the model is towed along the towing tank; it displays five de- grees of freedom, i. e. vertical emergence, side displacement, hee- ling, yawing and trimming angles; all the kinematic parameters are recorded. In case of side displacement the model is relieved of iner- tia and friction forces in movable units of the plant by means ofa special servo-system. The significant element of the plant is the sys- tem bringing the towing force into coincidence with the model centre line irrespective of the position of model relative to the tank axis. Finally, in case it is difficult to arrange the drives of controls on the model, the electric systems are provided for the plant which are ca- pable to imitate the action of the controls, particularly, side force controls, by prescribing the side force, yawing moment and heeling moment in accordance with the required law. Some test data obtained on the plant described are given be- low. The studies were carried out on the ACV schematized model N° 2 for the purpose of evaluating its course stability (with vertical stabilizers mounted) and checking its other dynamic characteristics. Figure 13 shows the relationship between the amplitude frequency characteristic of the model for yawing angle at different speeds. It 270 Determtnatton of Aerohydrodynamic Performance of ACVs is seen that with the increase in Froude number the response of the model increases up to the values Fr = 1.10. The same figure shows the variation of the relative amplitude of the model yawing angle at fixed frequency and speed as dependent on the pressure factor of the air cushion uy Some damage situations with the same model were studied, particularly, the transition process at first instants after suddenly applying the yawing moment, which may be the consequence of a spon- taneous reverse orfailure of propellers at one side of the model. It is seen from figure 14 how kinematic parameters of motion are chan- ging after the instantaneous application of the rolling moment (induced for example, by the breaking of the flexible skirt along the side) until the steps are taken to keep the ship in the upright position. This expe- rimental plant makes it possible to simulate the maneuver of course- keeping in this condition by applying the counter yawing moment imi- tating the action of controls. It is typical for this case to set the mo- del in a steady motion with a drift to the inclined side. The condition that the towing carriage speed should always be constant makes it impossible to simulate to the full extent the full scale performance of the vehicle. However, this restriction is no bar- rier to solving a wide variety of practical problems; in this case the error is directed to the safe side. Zia Prokhorov, Treshehevsky,and Volkov Table 1 1 2 3 4 5 6 7 8 9 YP YFP YP YP FY Wn Saw. © | =o Paz Determinatton of Aerohydrodynamtc Performance of ACVs Table 2 Particulars Designation =o Planeform of air cushion Length of the same ‘Breadth of the same Area of the same Coefficient of air cushion area 23 Prokhorov, Treshchevsky,and Volkov 7 Yr ee 1 | moe en a PE Ge ee 2 Figure 1 Vortex system of ACV's hull. 274 Determtnatton of Aerohydrodynamice Performance of ACVs | ° oo ° ee u iT] fo) Sel z es WF ne a) Oo a ae ; u N n | ® 2 a 4 a, 255 suction side, theory experiment loo me) Prokhorov, Treshchevsky,and Volkov Figure 3 Scheme of the plant for determining the aerodynamic performance of ACV models. - model - ground board - strain gauges mechanism of compensating for inertia forces - electric motor - phase shift regulation coupling am kh WN i 276 Determination of Aerohydrodynamtic Performance of ACVs u oO Fe 2) S * G Ww n SP 2 u Sane > MIMI MIs S SF i =] (Model N° 1) Figure 4 Coefficients Cr ,m", S Ve 8 | | 8 Pr Ve > Si Sy ee ~ ly ley, Q95 a9 O85 Prokhorov, Treshehevsky,and Volkov or PCC Wha ale ALCS oe Let Be 278 Figure 5 Influence of fullness coefficient / upon coefficients 3 Q, = 0.73 m/sec Shy = 0.1 Determination of Aerohydrodynamte Performance of ACVs a ee - lam | | en ese See -3 -2 -4 0 il 2 aa, Figure 6 Relative resistance versus running trim angle in calm water aes V cts © @e@ model N° 1 15 [Se AAA model N®* 2 000 model N° 3 279 Prokhorov, Treshchevsky,and Volkov Figure 7 Influence of the running trim angle upon the gain in resistance in waves ( wave length \ = 2.2 m, wave height h = 0.08 m ) == model Nz | = -medel Ni 2 122) medal Ne. 5 280 Determtnatton of Aerohydrodynamte Performance of ACVs 0,10 O04 Q02 005° «Gg Figure 8 Relative resistance in waves against the air flow rate for different methods of the additional air supply (wave length 4 = 3.1m, wave height h = 0.08m) Fr = 1.10 ooo additional air supply into the forward part 999 air supply into the after part eee air supply along the perimeter of air cushion 281 Prokhorov, Treshchevsky,and Volkov Figure 9 Influence of the relative air flow rate upon the longitu- dinal stability of motion ( Fro, ~- onset of plough-in of the flexible skirt) for two positions of the center of gravity along the model length (wave length A = 3.1m wave height h = 0.08 m; G = 80 kg; Be = 1.25) oo): —— =0: 01 eee GL 0.012 282 Determtnatton of Aerohydrodynamic Performance of ACVs “STX® [TCUIPNATSUOT 944 Suore uotjour ArojeyTsues} Jo Apnys oy} 10x uO BuTyo IIMs jo ouTeyoS - ¢ fsotpnys BSutyoztd AoF uO BuTYydIIMS Jo out -ayos - 7 {sotpnjys SuLAeoYy TOF UO BSuTYOIIMS Jo suTSYOS - | ‘oueTd [euUIpNyIsUOT UT 9DDUeUITOJI0d OTuUreuUApOApAY Apeoj}s-uou fo uoTJeuTUITZJOp oy} TOF yueTd TequoUITIOdxy OQ o41ndT ZT PETE A 283 Prokhorov, Treshchevsky,and Volkov c"T = 4am 8¢®8 0°0 = 71 O00 ADV Jo Tepout PEZTeUIBYOS OY} TOF SUOTIETIIOSO [eOTIIOA FO Aouenbery snsi9A SODIOF CTJAOUT JO JUSTOTFFIOD TI eanstg 284 Determtnatton of Aerohydrodynamtc Performance of ACVs Figure 12 Experimental plant for dynamic tests with the ACV towed models in the towing tank. 1- system of unloading the movable units; 2 - system of brin- ging the towing force into coincidence with a model centre line 3- system of imitating the external forces; 4- strain gauges for recording the kinematic parameters. 285 Prokhorov, Treshechevsky,and Volkov Figure 13 Relationship between the amplitude-frequency characteristic of the model for yawing angle at different speeds and pressure factors of the air cushion, 286 Determination of Aerohydrodynamic Performance of ACVs *JUOULOUL BUTT[TOI 94} FO uotyeot{dde snoourjuejsur oy} Je Ssod0ad uoT}ISuea], PI OANSTT baer, w 298 2. fee WE eee 09¢ f 287 Prokhorov, Treshchevsky,and Volkov DISCUSSION P. Kaplan Oceantes Ine. Plainvtew, New York, U.S.A. I think my comments with regard to experiments contain some helpful information and also provide a warning with regard to any expectation of prediction of dynamic performance of air cushion vehicles from ordinary experiments in towing tanks. For one thing, referring to the particular case, in which you used forced oscilla- tion, the equation given on page 12 deals with the single degree of freedom of heave motion. There is a pressure variation here and if you actually wrote the equations involving both pressure and heave motion coupled together, you would find that there is an effective mass entering the system. The effective mass is dependent upon the geometry of the plenum system and certain aspects of the pressure ratios, etc. The net result is that you have some reduction in the effective mass. That is why you find what people are referring toas a negative aided mass for ACVs, which may not be quite so. There is a real physical mass, and there may be some hydrodynamic ef- fective mass since the frequency dependence in Fig. 11 may be in- dicative of that, but it also depends on how you look at the water sur- face behaviour at low frequency and at high frequency and also at low speed and at high speed. But that is not the point. The main thing is that the pressure is influential, and that it should be ac- counted for ; it should be measured and the equations for the system should include it. Analyse that and then look at your resulting inertia and the other terms. Correlate the results, and that is the way you will understand what is happening with these vehicles. Secondly, with regard to model tests, let us look at how you scale. All naval architects are accustomed to scaling dynamic phe- nomena in terms of Froude scaling and that is because certain para- meter scalesinthatway. What is the most fundamental parameter that determines the behaviour of any vehicle in response to a seaway disturbance ? There are two, mainly : the natural frequency and the damping. The natural frequency can be derived for any type cushion system for an air cushion vehicle. It involves pressure ratios, i.e. atmospheric to normal pressure that supports the system, and the geometric height of the plenum. 288 Determinatton of Aerohydrodynamtc Performance of ACVs Pressure scales linearly, and the height scales linearly ; atmospheric pressure is not scaled in any facility, at least it is not scaled at speeds appropriate for ACV. The situation is such that if you consider this effect, the natural frequency of the system is scaled linearly while for Froude scaling it varies as the square root of your scale ratio. Therefore you have shifted natural frequency coincidence well out of proportion, depending upon your scale ratio. In addition, the relative damping, that is the matter of so-called relative to critical damping, also changes due to scaling because of the operation pressure system. The resulting question is, what con- sequence does this have upon the motion ? Some results have been obtained based on analysis and simu- lation work, and have been published in the open literature, on this particular point. It shows the following. You can predict the vertical acceleration responses from tests in waves at model scale at atmos- pheric pressure. Unfortunately the prediction isunconservative. You think you are getting by with a certain acceleration, and if you work out the equations which you validated by your original procedure to correlate with your testing at non-scaled atmospheric pressure, you then scale the atmospheric pressure to simulate real life and your accelerations are larger. However I do not say that there is no busi- ness for towing tanks in getting information for ACV. I do say there is business for theoreticians, together with towing tanks, to get the right information. REPLY TO DISCUSSION Vjacheslav N. Treshchevsky Kryloff Research Instttute Leningrad, U.S.S.R. Iam grateful to you for your comment. I would say that the problems of scale effect which you mention are taken into acount and special experiments have been and are being made in order to exclude wide errors connected with the scale effect in various types of experiments. You mentioned the negative value of the term corresponding to mass which is on one of the illustrations. It is difficult to explain it quite correct, I think the reason of the phenomena is in the inter- 289 Prokhorov, Treshchevsky,and Volkov action between the ACV hull, the air cushion and water surface, I may add that this is stable experimental data and I have seen the cor- relating results in a German paper. I can add that frequency characteristic shown on Figure 11 being derived experimentally includes the influence of all the factors mentioned by Mr Kaplan (variable pressure, ACV's geometry). The possibility of exact theoretical calculation of this characteristic seems to be doubtful. The use of model data in ACV dynamics isn't the subject of the paper. Nevertheless the assumption of the linear character of the natural frequency versus scale dimension is not obvious. As for similarity criterion, the Froude number and dimensionless frequency were used among others. The natural frequency and damping are ACV's characteristics, determined by flight conditions. + t+ 290 UNCONVENTIONAL SHIPS Monday, August 21, 1972 Afternoon Session Chairman: R. Leopold Naval Ship Engineering Center, U.S.A. Page Hydrodynamics and Simulation in the Canadian Hydro- foil Program. 293 R. T. Schmitke, E.A. Jones (Defence Research Establishment Atlantic, Canada). Bending Flutter and Torsional Flutter of Flexible Hydro- foil Struts. 343 P.K. Besch, Y.N. Liu (Naval Ship Research and Development Center, U.S.A.). On the Design of the Propulsion Systems with "Z" Drives for Hydrofoil Ships. 401 A.A. Rousetsky (Kryloff Research Institute, Leningrad; U.S. S. R. ). Hydrodynamic Development of a High Speed Planing Hull for Rough Water. 419 D. Savitsky (Stevens Institute of Technology), J.K. Roper (Atlantic Hydrofoils, inc. ), L. Benen (Naval Ship Systems Command), (U.S.A.). Motion and Resistance of a Low-Waterplane Catamaran. P.C. Pien, C.M. Lee (Naval Ship Research 463 and Development Center, U.S.A.) 291 wo relating resoltein a Geran pape. lion betas dha ACY tal, the ai tind Watag oa a RAS thane tidy ig.ate vie csperiareaniat Mt and T Rare wel >, lL cat aad sage" bebay derived “Kp mats | mentioned by Mr Kaplan Haciabte prety re, A cv %y yeornely ¥} posvibility af exact thevretical ca low Weston 4 M thin charat torial seermea to be dukbttal SY@r TS ster sir Webad hi 7 ” ath , poses, gaperar? A ‘ ita ® Amid 2 rT eyuarT ce a lao: SVeEre *s wrote of the | . Vie watiradi Ipeceah Seren ae Gatny lon La Hot wu i ou, Oirrila rity itor208, the Proude sriiote@ad dix Apa sd Peg Wier LORCA SoBe eae) oibreeatigam® dis MuGuen ond de eS paca i . ops! 7 Lad otivii neibeaa. edt ai gol sfuris2 ba $4 Aa'r#en0 8 sonaieG) @esz0l .A a at a ree. is o1bvil o! i'l iat p! gs eao5 < Yi3 ie j see Pe ‘ ; tw gorsleyve sojeluqost aiff Yo wail 1Ob . id2 liolethyi Stutijen? dvikees loiva > tgiesudst JA. . | &2.2.0 ,betgokee sp. paice!S Seeqe igi es }o toeomaqoleved sign Ri» 198 W ‘0 T Yo slutiignl easves 3) vhalived:,; A pet jaliotoubyH Stinely A) 1sq0R TE (SunctrrieD acrstee2 gift level) menset Re (a25g . neremsic.) sesittsstaW-wel # lo sonetetees Be _ ta dors068 8 gidt iavelt) oot M.D me TOS LA &.U sen rneariqgolsvoed Be (vgoloadss 196 HYDRODYNAMICS AND SIMULATION IN THE CANADIAN HYDROFOIL PROGRAM R.T. Schmitke and E.A. Jones Defence Research Establtshment Atlantte Dartmouth, N.S. Canada ABSTRACT Hydrodynamic aspects of the Canadian hydrofoil program are discussed, with particular reference to HMCS BRAS D'OR. Associated simulation stu- dies are described and key results of BRAS D'OR seakeeping trials are presented. I. INTRODUCTION For more than a decade the Canadian hydrofoil program has been synonymous with the design, construction and trials of HMCS BRAS D'OR(') (Figure 1). The recent suspension of trials and sub- sequent mothballing of the ship make this an appropriate time for a comprehensive review of design experience in the light of trials ob- servations. This paper is particularly concerned with the hydrodyna- mics of the hydrofoil system and associated simulation studies. Factors governing foil system configuration and hydrodynamic design are described, with some details of development experience at the model scale and a thorough discussion of full scale trials obser- vations. The mathematical models used in simulation studies are then presented, and predictions of steady state and dynamic performance are compared with trials data. The paper concludes with an analysis of BRAS D'OR seakeeping trials. Il, FOIL SYSTEM HYDRODYNAMICS System Configuration Since its inception the Canadian hydrofoil program has been 293 Sehmitke and Jones based on passively stable surface-piercing hydrofoil systems, there- by complementing U.S. effort on automatically controlled fully-sub- merged systems. The principal relative merits of each type are list- ed below 2) , Fully-submerged systems offer : a) a smoother ride in moderate to heavy seas, b) higher lift-drag ratio, c) lower foil system weight, d) greater foilborne manoeuvrability, e) retraction capability, Surface-piercing systems offer : a) inherent stability, b) a wider range of foilborne speeds, c) better sea-keeping at hullborne speeds, d) higher potential for remaining foilborne in extreme seas, e) greater tolerance to off-design loads, such as those im- posed by towed sonar. Successful contouring of large waves requires that the bow foil respond rapidly to changes in immersion depth ; for satisfactory following sea operation, the bow foil must also be reasonably insensi- tive to wave orbital velocities. Together these requirements dictate that the bow foil combine high rate of change of lift with draft, st ‘ with low rate of change of lift with angle, The after foil, on the other hand, must have high a to pro- vide adequate damping of seaway-induced motions. Furthermore, 2+ must be lower than at the bow foil in order that downward heave dis- placements cause upward trim. Since foil efficiency generally in- creases with te canard configuration is the logical result, with the bow foil carrying as little weight as dynamically feasible. ; A secondary but significant advantage of the canard configura- tion is that it lends itself to a hull with very fine bow lines. This is necessary for reduced pounding due to wave impact when foilborne and is particularly important for the Canadian réle with its emphasis 294 Canadtan Hydrofotl Program. Hydrodynamtes and Stmulatton on good hullborne performance and seakeeping. The Sub-Cavitating Main Foil In addition to providing adequate heave damping through high = , the main foil unit must provide roll stiffness without undue heave stiffness. The BRAS D'OR unit (Figure 2) accomplishes this by use of anhedral and dihedral surface-piercing elements at either end of a fully-submerged main foil unit. The latter serves as the pri- mary lifting element and, combined with the dihedrals, provides the required characteristics with high efficiency. The upper anhedral panels are highly cambered and twisted to develop high lift at take- off. The anhedral tips are incidence-controlled to augment roll sta- bility at low foilborne speeds and improve turning performance. The theoretical tools required for hydrodynamic design of a subcavitating foil system have been obtained by adding free surface corrections to methods borrowed from subsonic aerodynamics. The prevention of cavitation is the chief hydrodynamic constraint in sec- tion design, and cavitation-free operation above 40 knots necessi- tates a departure from conventional low speed aerofoils to delayed- cavitation sections such as illustrated in Figure 3. The design of the particular sections used in BRAS D'OR is described in Roe They are highly efficient and provide an approximately uniform pressure distribution when operating over a wide range of angle-of-attack in close proximity to the free surface. The practical upper limits of the delayed-cavitation regime are approximately 60 knots in calm water and 50 knots in rough seas - the design speeds for BRAS D'OR. Full scale trials showed that in general the BRAS D'OR main foil unit has successfully met its design requirements for high effi- ciency, low and high aa The only significant hydrodynamic problems encountered were associated with emergence of the anhe- dral-dihedral foil intersections at about 45 knots, which resulted in lateral jerkiness even in calm water. There were two specific pro- blems. The first was caused by intermittent ventilation of the dihe- dral foils and anhedral tips and was countered by installing addition- al anti-ventilation fences. The second was fundamental to the main foil geometry and was more difficult to solve. It was due to increas- ed roll and heave stiffness below the intersections, when both the anhedral tips and the dihedral foils become surface-piercing. It was alleviated by reducing the mean angle-of-attack of the anhedral tips “by 2°. This increased main foil immersion and delayed the emerg- ence of the anhedral-dihedral intersections to approximately 50 knots. The net result was a significant increase in riding comfort in 295 Sehmttke and Jones small to moderate waves at speeds up to 50 knots. Main foil cavitation was never observed during calm water trials, even at 62 knots, indicating that section design met all ex- pectations in this very important respect. No problems arose from hydrodynamic interference between individual foil elements. How- ever, both full scale trials and model tests at the National Physical Laboratory showed that bow foil wake reduces main foil lift by ap- proximately 10%. The Super-Ventilated Bow Foil Operating conditions for the bow foil are demanding. At foil- borne speeds in rough seas the bow foil is subject to wide and rapid changes of both immersion and angle-of-attack. The hydrofoil system is wholly area-stabilized longitudinally and the lightly-loaded depth- sensitive bow foilis the primary source of control, so that smooth lift vs. immersion and lift vs. angle-of-attack characteristics are es- sential. Sub-cavitating hydrofoil sections are prone to ventilation in rough water and the resulting sharp losses in lift at the primary lon- gitudinal control element cause an unacceptable diving tendency Superventilated sections are therefore used for the bow foil, despite their lower efficiency. In this case, occasional suppression of ven- tilation gives sharp lift increases, but unlike the converse situation with subcavitating sections this is an inherently safe effect. The BRAS D'OR bow foil is of diamond configuration (Fi- gure 4) with a sub-cavitating centre strut and super-ventilated di- hedral and anhedral elements. Tulin Two-Term lower surfaces! >) were chosen for the super-ventilated sections (Figure 5) because these appeared to offer the best compromise between hydrodynamic efficiency and structural strength. Design incidence is nominally 5° above zero lift (as established by model tests), and rake angle of the unit is adjustable in flight to permit operation at optimum incidence for the prevailing sea condition. Little information was available on the practical operation of surface-piercing super-ventilated hydrofoils, so that extensive ex- perimental development was necessary. Model size had to be as large as practical to minimize scale effects : consequently, the bulk of the work was done at quarter scale, taxing the limits of available towing tank facilities. The same bow foil was also used as part of a complete quarter scale manned model of the system. A great strength of the development program lay in the ability to test the same model both in the controlled environment of towing tanks and as a functional unit in realistic seaways. 296 Canadtan Hydrofotl Program. Hydrodynamtes and Stmulatton A major concern of the experimental program was upper sur- face design, with the objective of inhibiting and controlling intermit- tent flow reattachment, The leading edge was made as fine as practic- able and, to enforce reattachment to occur in stages and hence re- duce the severity of accompanying lift increases, two additional break points were incorporated in the upper surface, at 66% and 87% chord. A major problem of the initial manned model trials was that the an- hedral foils served as fences to inhibit the spread of ventilation down the dihedrals, leading to cyclic pitching at speeds close to intersec- tion emergence, This was overcome by adding another large upper surface spoiler to the anhedral sections in the neighbourhood of the intersection. The most comprehensive set of quarter scale towing tank data was obtained at the National Physical Laboratory (NPL) under Froude-scaled conditions, providing good definition of bow foil cha- racteristics, (Figures 6 and 7). Data points have been coded to show the spanwise extent of leading edge ventilation down the dihedral foils from the upper surface. (For all test conditions of interest, the ca- vities behind the midback spoilers remained consistently ventilated. ) Spanwise extent of leading edge ventilation is indicated by the degree of openness of the points, e.g. 100 % fully open 0 50 % 50% open 0 0% fully closed @ Figure 9 shows that the lift-curve slope decreases gradually with in- creasing rake angle as ventilation spreads down the leading edge of the dihedrals. The most interesting quarter scale tank tests took place at the Lockheed Underwater Missile Facility (LUMF) , where both ca- vitation and Froude numbers were scaled. Cavitation scaling was found to have no significant effect on flow state, lift or drag. An equally significant finding was that lift values obtained at NPL were much higher than at LUMF (Figure 8). These differences were later shown to be due in large part to deterioration of the foil surfaces and leading edges during the time interval between the two series of tests. It was possible to estimate full scale bow foil lift characteris- 297 Sehmttke and Jones tics by direct observation of depth of immersion. Figure 9 shows the steady state lift coefficient (based on horizontally projected immersed area) of the bow foil unit over the foilborne speed range. The full scale lift coefficient falls within limits established by quarter scale model tests at NPL and LUMF except at low speeds. For these, leading edge ventilation extended only partially down the span of the dihedral foils at quarter scale but was complete at full scale, giving lower lift but more stable flow. The inhibiting effect of the outboard intersections was clear at quarter scale. The dihedral foils ventilated from the mid-back spoilers under most conditions, but the intersections, acting as fences, prevented the initiation of leading edge ventilation until they emerged. It almost invariably occurred on one foil at a time since the associat- ed loss of lift caused the intersections to re-immerse, inhibiting the second dihedral more strongly. At full scale, initial establishment of leading edge ventilation on the dihedrals still appeared to be associated with emergence of the intersections, but invariably occurred simulta- neously on both dihedral foils. Effects were less clear but full scale ventilation certainly occurred more readily and more strongly than indicated even by LUMF quarter scale tests at correctly scaled Froude and cavitation numbers. This was due at least in part to the fact that full scale trials were seldom held under really calm condi- tions, the practical limit for ''calm'' water being set at waves 3 feet in height. Occasionally, during take-off in exceptionaly smooth seas, leading edge ventilation was delayed until ship speed approached 40 knots ; during this interim period, pitch angles of up to 9° were ob- served. This situation was easily overcome by increasing speed or bow foil incidence until leading edge ventilation occurred. Bow foil rake angle optimization trials showed that a strong and persistent ventilated cavity was achieved in calm water at the design rake angle setting (0°). In rough water optimum rake angle varied with heading to the sea ; suitable flow and ship motion charac- teristics were generally achieved in head, beam and following State 5 seas at rake angles of -1°, 0° and 11/2° respectively. In short, steep seas, flow re-attachment sometimes occurr- ed on the bow foil dihedrals during deep immersion at the face of larger waves. The resulting discontinuous increase in lift gave added impetus to bow up pitch motion. As depth of immersion decreased at the rear wave slope, ventilated flow was re-established, accompanied by a sudden decrease in lift. Oscillograph records of vertical accele- ration during these periods exhibit a sharp positive and negative spike followed by a return to normal acceleration levels as the ship encout- ered waves of more typical size and normal ventilated flow was re- 298 Canadtan Hydrofotl Program. Hydrodynamtes and Simulatton established. Because these vertical acceleration spikes were an im- portant source of motional discomfort, an objective of future develop- ment must be to improve ventilation stability at extreme depths of immersion. Foilborne sea time has not been sufficient to enable firm and quantitative conclusions to be drawn regarding the suitability of bow foil and 9t for rough water operation. However, the ship never experienced difficulty in following seas, while pitch response to head and bow seas was high. It seems probable that a reduction in OL of about 20% would result in lower vertical accelerations without com- promising stability. A high drag penalty is paid for using super-ventilated sec- tions. The philosophy adopted during BRAS D'OR design was that this condition could be tolerated since the bow foil is primarily a con- trol element carrying only 10% of total ship weight. However, trials and model test data indicate that approximately 30% of total foilborne drag is due to the bow foil. In addition, the high bow foil drag makes fuel consumption, and hence range and endurance, extremely sensi- tive to longitudinal C.G. location. BRAS D'OR sea trials have therefore specified three objec- tives for future super-ventilated bow foil development : reduction of drag, optimization of and for rough sea operation, and im- provment in ventilated flow stability. These are important conside- rations, but are secondary to the demonstrated success of the super- ventilated bow foil unit in stabilizing, controlling and steering the ship over a wide range of speed and sea conditions. Ill. SIMULATION Hydrofoil simulation in Canada began with the extensive and comprehensive studies carried out by the DeHavilland Aircraft of Canada Ltd. in support of BRAS D'OR design!®) . These studies were subsequently supplemented at the Defence Research Establish- ment Atlantic (18) » with the objective of achieving simple methods applicable to all surface-piercing hydrofoil systems. It is largely upon the latter work that this section of the paper is based. Four topics are treated : the general equations of motion for surface-pierc- ing hydrofoil vessels, prediction of steady state performance, analysis of calm water stability and analog simulation of random seas. Because of the close similarity to aircraft practice, descriptive material is kept to a minimum. 299 Sehmttke and Jones Equations of Motion for Surface-Piercing Hydrofoil Ships The equations of motion listed below are written with respect to the axis system illustrated in Figure 10. The origin is at the C.G. and in the reference condition of steady symmetric flight in calm water at speed Uy, the x-axis is directed horizontally forward, the z-axis vertically upward and the y-axis to port. The pitch angle is positive for downward rotation of the bow. Pitch : 1.6 = > As toa he cos. - D.z, + M. cosf) - it i i 2 i i (1) oul (emigsinhonsy 2..4Cas OL al Heave: m ce - U8) = ZL; cos r. + Tsiny- W (Zz) Sia aan (o-oo, = T cosy. - =D, (3) Sideslip: m (v + Ur + gd) =- 2D L. sinl (4) Roll : I¢ = ZL, ty, cos r, to Bs, sin!) (5) Yaw : por sxe |S" 1,e sie So RE ere) (6) Zz ay uk id i i i where summation of lift (L;), drag (Dj) and pitching moment (Mj) is over all foil and strut elements, individually located at (x;, yj, 24). W is all-up-weight and the line of action of thrust (T) passes through (xp , zp). The following sign convention is adopted for dihedral and anhedral angles: fora por dihedra oil o angle I I = Me f : . P EF r or a stbd. dihedral foil of angle DS ? Ir. = DS 300 Canadtan Hydrofotl Program. Hydrodynamtcs and Stmulatton i | fy iy = = Ry for a port anhedral foil of angle AP’ i AP : r is r for a stbd. anhedral foil of angle Png ; i AS and M; may be evaluated by the methods of References [9], Go fe bata oad 2g oe Steady State Performance In the steady state, equations (1) to (3) become =(L.x, cosT, + D,z, - M, cos r.) + (x. cotY - Zz.) (8) a = Oo i ples cost, + cot y2 DD. = |W (9) i 1 r Since dynamic pressure is constant Lj; , Dj and Mj are functions of immersion depth (h) and angle-of-attack (a) alone. For an all-fixed foil system, furthermore, knowledge of h and a for a single foil element enables all other h's and a's to be de- termined. Hence (8) and (9) contain only two unknowns: h anda of a reference foil element. Because of the non-linear nature of these equations the solution must be obtained by an iterative technique. Figure 11 illustrates the accuracy obtainable using the above procedure. Predicted curves of BRAS D'OR trim, keel clearance and weight- drag ratio are presented, along with trials measurements of these quantities. Estimated measurement accuracies are a Ai for trim, + 1 ft. for keel clearance and + 1.5 for W/D. The ac- curacy of the resistance prediction is of particular importance ; the fact that measured drag is higher than predicted is probably due lar- gely to the one to three foot waves encountered during most calm water trials. Calm Water Stability Foilborne stability in calm water is most easily assessed by 301 Sehmttke and Jones solving the linearized equations of ship motion. Linearization of (1) to (6) results in the following two sets of three coupled linear ordi- nary differential equations : Long itudinal (10) PHeh Tee Sowa Se e's Ee ae SG ee see ¥ u Zz w 6 Oo ow ] Heave < me ¢= Zou +. 2.4. + Bom es (2 oh U 20) et oe u Ve, w 6 Oo w 6 - 7 9 et e Be Surge: m (u - g@) Kut X2+ Kz + (X, + UX,,) 6+X-0 6 (12) Lateral Sideslip : m (v + U r+ g¢) = ty + Yr + Yy + ¥4e (13) Rol: I¢ = Kv +K yr + Keg +K yo (14) Yaw : I = Nv + Nt + me + Ny? (15) where t nis | (w - U9) dt (16) re) and the stability derivatives M,, M, etc. are listed in the Appen- dix, ; The longitudinal modes of motion characteristic of passively- stabilized surface-piercing hydrofoil ships consist of a lightly damp- ed oscillation governed by ship pitching characteristics, a heavily damped oscillation related to heave and a simple convergence arising from surge-heave-pitch coupling. These are termed the pitch, heave 302 Canadtan Hydrofotl Program. Hydrodynamics and Simulatton and coupled subsidence modes, respectively. For canard configura- tions, the coupled subsidence mode is always stable, but in airplane configurations instability may result from adverse heave-pitch coupl- ing. Neither the pitch nor the heave modes are significantly influenc- ed by surge. Root locus plots showing the effect on BRAS D'OR's longi- tudinal modes of varying speed are presented in Figure 12. Longitu- dinal dynamics are dominated by the lightly-damped pitch mode in which the damping ratio decreases and natural frequency increases with increasing speed ; this mode's characteristics are a direct result of the bow foil's design, which combined high cae , with low dL Similarly, the characteristics of the heave mode follow from the com- bination of low - L with high oe in the main foil. a Generally speaking, three modes of lateral motion may be distinguished for passively-stabilized surface-piercing hydrofoil ships (Figure 13) : a rapid convergence of little importance, an oscillation governed by ship rolling characteristics, and a slow convergence arising from sideslip-roll-yaw coupling. Simulation of Random Seas In Equations (1) to (6) a seaway acts as a forcing function through the variation of foil immersion depth and angle-of-attack with wave elevation and orbital velocity. The simulation of these seaway variables is now discussed. The Pierson-Moskowitz spectral form is chosen as a model of the sea ''3) | The equation for the wave elevation power spectral density is: 2 4 . 0081 Oe) = See exp =. 74 (= ) (17) w where V is wind speed. Significant wave height is given by : 2 (18) cat) h = 1. 86 (—— for V in knots. 303 Sehmttke and Jones Consider now the case where a hydrofoil ship is travelling at speed U, intoa head sea. The wave elevation spectrum, expressed in terms of frequency of encounter, is @! ( w') 1, wilde lade (19) Pasi g 1 with a similar expression holding for the transformed orbital velocity spectrum, ®|, Gis! Ps Pe eee pipes it 5 (20) is the angular frequency of encounter. #' and 9}, are plotted in Figures 14 and 15 for U, = 50 knots and V = 24 knots (Sea State a). The white noise technique for simulating a random head sea will now be described. The basis of this method is that a signal with a prescribed spectrum can be generated by passing white noise of spectral density , through a linear filter so designed that the square of its frequency response, H(w), has the desired shape. Filter output is, then, > le) =. JH @) (21) In particular, the generation of waves with spectrum %!' can be ac- complished using a filter network consisting of three high pass filters and two integrators ; the approximation to ©' which is thus obtained is shown in Figure 14. Taking the vertical orbital velocity to be the input to the last integrator multiplied by a suitable constant results in the approximation to ®%! shown in Figure 15. Proper phasing bet- ween wave elevation and orbital velocity has been achieved, while also obtaining good approximations to the spectra. In head seas the waves seen by the main foil lag those at the 304 Canadtan Hydrofotl Program. Hydrodynamics and Stmulatton bow foil by 2 A¢ gli (22) g where f is the foil base length. Over the frequency range of interest, A § can be very well approximated by a time lag and constant phase lead (Figure 16) : ig easel Mate lN (23) A block diagram of the head sea simulation systemis shown in Figure 17. Results derived from simulating BRAS D'OR foilborne motions in rough water are presented in the next section ; surge was neglected in this simulation in order to simplify the problem and also because it was felt that neglecting surge should give conservative results. (Consider the case where the bow foil encounters a steep high wave, leading to a rapid increase in bow foil immersion depth. In practice a speed reduction of several knots occurs, and the attendant dynamic pressure decrease results in lower lift and accelerations than on the simulated, constant speed ship. ) IV. BRAS D'OR FOILBORNE SEA-KEEPING TRIALS BRAS D'OR rough water trials were less comprehensive than desired, but enough data were obtained to enable comparison of measured characteristics with predictions and to reach general con- clusions about sea-keeping ability. Data will be presented for three key trials, two in Sea State 4 and one in Sea State 5. The wave eleva- tion power spectral densities measured during these trials are shown in Figure 18 and are compared with the Pierson-Moskowitz theoretical spectra for Sea States 4 and 5 (significant wave heights of 7' and 10'). Wave measurements were made by a buoy equipped to measure vertical accelerations and are admittedly inaccurate, due partly to limitations of the buoy itself and partly to the practical difficulty of making a single measurement representative of rapidly changing con- ditions in a trials area close to the coast. The variation of root mean square values of longitudinal ship motion parameters with heading to the sea is shown in Figure 19 for an average speed of 39 knots. These values do not exhibit the system- atic increase with sea height that one would expect, probably because 305 Sehmttke and Jones of different sea energy distributions, For all three seaways there are similar trends with change of heading : vertical accelerations are high- est in head and bow seas and lowest in following seas, while pitch angle shows the opposite trend. The longitudinal distribution of vertical accelerations is in- dicated in Figure 20, the points representing average root mean square values for the three sea trials. The most interesting feature of these plots is the comparatively small change in vertical acceleration along the length of the ship, illustrating the well-controlled pitch response of this canard configuration, with its special bow foil. Figure 21 compares predicted and measured root mean square vertical accelerations at the bow. The predictions were derived from analog simulation of pitch and heave motions in a theoretical head Sea State 5 with significant wave height of 10.7 feet, while the measure- ments were obtained during head and bow sea runs in Sea States 4 and 5. Predicted acceleration levels are higher than measured, re- flecting the higher theoretical sea state and perhaps to some degree, supporting the intuitive argument that neglect of surge is a conserva- tive simplification of the simulation problem. Figure 22 presents typical power spectral density plots of vertical acceleration at the bow and pitch angle for head sea runs at speeds of 34 and 42 knots. Also given are the corresponding spectra for the encountered seaway, derived from the nominal Sea State 5 plot of Figure 18, There are two dominant seaway frequencies, and al- though the effects of both are clearly apparent in the pitch angle plots, pitching is associated mainly with the lower frequency. Pitch response falls off with increasing speed, particularly at the lower frequencies, Bow vertical acceleration peaks at the higher seaway frequency ; the magnitude of this peak increases with speed and there is also a shift in the energy distribution toward higher frequencies. The response transforms of Figures 23 and 24 quantitatively characterize BRAS D'OR's pitch and heave response to random head seas at speeds of 35 and 40 knots, Although the experimental results are scattered, reflecting inaccuracy of sea state measurement, sys- tem non-linearity and limited statistical confidence, they nevertheless furnish a reasonable indication of mean ship response, as given by the dashed lines in the Figures. Pitch response peaks at approxima- tely .2 Hz, in agreement with the prediction of pitch natural frequency obtained from linear stability analysis. Bow vertical acceleration res- ponse is fairly flat above .2 Hz, but falls off rapidly below this fre- quency. 306 Canadtan Hydrofotl Program. Hydrodynamics and Stmulatton Agreement between simulated and measured response trans- forms is very good. These predictions were obtained by analog simu- lation of pitch and heave motions in regular waves of small amplitude - a procedure equivalent to linearization. The predictions of Fig. 21, on the other hand, were obtained by simulating ship motions in a ran- dom State 5 sea generated as described in Section 3. The regular wave technique was adopted for response transform prediction because it enabled more accurate modelling of the effects of circulation delay and wave orbital velocity on main foil angle-of-attack. Bow foil flow re-attachment and emergence of the main foil anhedral-dihedral in- tersections, both of which occurred occasionally during trials but never in the simulation, are probably the cause of the only notable discrepancy, under-prediction of pitch response below .3 Hz at 35 knots. Roll and sway characteristics are presented in Figure 25 in terms of rms values for lateral acceleration and roll angle, for the three seaways of Figure 18. Lateral accelerations are given at two ship stations, the CG and the Control Information Centre, located comparatively high in the ship in the upper deck superstructure. As with the pitch-heave characteristics given earlier, there seems to be little effective difference between the state 4 and 5 seas. Roll angle is very dependent on direction to the seaway, increasing greatly for seas on and abaft the beam. The lateral accelerations exhibit a much smaller dependence because the roll frequency decreases for beam and stern seas, The effect of heading on the frequency distribution is shown by the power spectral density plots of Figure 26. These show lateral ac- celeration at the CIC for head and following sea runs at 39 knots in Sea State 5. The first peak is at the main rolling frequency and is due to accelerometer tilt. For the head sea, in addition to the increase in frequency of the main acceleration component, there is an increase in levelin the 1 to 2 Hz range. This is significant because lateral "jerkiness'' was considered the most uncomfortable feature of the ride, especially at higher speeds and for higher stations in the ship. The effect of increased speed is shown in Figure 27 by compar- ing lateral acceleration at the CIC for head sea runs of 34 and 42 knots in the same Sea State 5. The effect of increased height within the ship is particularly marked and is illustrated by Figure 28 which compares lateral accelerations at the CG and CIC with the corres- ponding roll angle plot for 39 knots in head Sea State 5. A very small amount of roll angle energy above 0.5 Hz seems responsible for real- ly significant lateral acceleration at the CIC. The problem of lateral acceleration amplification with height is clearly deserving of attention 307 Sehmttke and Jones in large hydrofoil ship design. It is difficult to relate seakeeping data directly to habitability or to compare the capability of different systems, other than subjec- tively. The behaviour of the BRAS D'OR canard system was well up to expectation in seaways encountered, both for straight runs and turns. There was a complete absence of slamming and motions were modest, particularly in pitch and heave. Motions were almost certain- ly greater for BRAS D'OR than for a comparable fully-submerged system but less than for other types of craft with surface-piercing foils. There were no particular problems for the crew when seated. Personnel moving about and standing, tired quickly, mainly because of the roll motions, but this situation would not arise in an operation- al ship. For BRAS D'OR, a deciding factor in the choice of foil sys- tem was the exceptional hullborne seakeeping offered by the canard surface-piercing arrangement. For the tasks envisaged, the habita- bility is more important under hullborne cruise conditions than for short periods of foilborne operation. Experience with BRAS D'OR supports the contention that foilborne motions are acceptable for con- tinuous periods of several hours and has confirmed the promise of exceptionally good hullborne seakeeping (2) V.. CONCLUSION The Canadian hydrofoil program has significantly advanced both the performance and the fundamental understanding of surface- piercing hydrofoil systems designed for open ocean operation ; the development of a successful super-ventilated bow foil unit is especial- ly noteworthy. As regards simulation, the extension of aerodynamic methods into the hydrofoil field has proved reasonably successful, yielding satisfactory predictions of both steady state and dynamic per- formance ; of particular importance is the good characterization of BRAS D'OR pitch and heave response derived from a linear mathe- matical model. NOTATION Cy drag coefficient Ch lift coefficient Cur pitching moment coefficient D drag H frequency response 308 Canadian Hydrofotl Program. Hydrodynamics and Stmulatton *% Se ee < Seen, N Kx €<. (o) gQ mR a 3 Sy emu co fa rolling moment of inertia pitching moment of inertia yawing moment of inertia rolling moment lift pitching moment yawing moment immersed foil area thrust, also time lag ship steady state speed wind speed ship all-up weight surging force swaying force heaving force chord acceleration due to gravity significant wave height foil base length mass dynamic pressure yawing velocity surging velocity swaying velocity heaving velocity foil coordinates heave dihedral angle constant phase lead phase lag spectral density 309 > RQ S ov 3S 8B R Sehmttke and Jones sweep angle angle -of-attack inclination of thrust line to horizontal roll angle wave elevation density pitch angle wave frequency frequency of encounter REFERENCES EAMES, M.C. and JONES, E.A., 'HMCS BRAS D'OR - An Open Ocean Hydrofoil Ship", Journal R.I.N.A., Vol. 1, No, J, Apetl 1971. EAMES, M.C. and DRUMMOND, T.G., ''HMCS BRAS D'OR -Sea Trials and Future Prospects", presented at R.I.N.A. Spring Meeting, April 1972. RICHARDSON, J.R., ''Hydrofoil Profiles with Wide Cavi- tation Buckets'', Engineering Research Associates, Toronto (Prepared for DeHavilland Aircraft of Canada Ltd.) Septem- ber 1961. JONES, E.A., ''Rx Craft, A Manned Model of the RCN Hydrofoil Ship ''BRAS D'OR''"', J. Hydronautics, Vol. 1, No. 1, July 1967. TULIN, M.P. and BURKART, M.P., ''Linearized Theory for Flow about Lifting Foils at Zero Cavitation Number", David Taylor Model Basin, Report C-638, February 1955. OATES, G.L., and DAVIS, B.V., ''Hydrofoil Motions in a Random Seaway, '' 5th Symposium on Naval Hydrodynamics, Bergen, August 1964. SCHMITKE, R.T., ''Longitudinal Simulation and Trials of the Rx Hydrofoil Craft", CAS Journal, Vol. 16, No. 3, March 1970. 10 11 12 13 Canadtan Hydrofotl Program. Hydrodynamics and Stmulatton SCHMITKE, R.T., ''A Computer Simulation of the Perfor- mance and Dynamics of HMCS BRAS D'OR (FHE-400)", GAS Journal, Vol, 17, No. 3, Maren d971, DIEDERICH, F.W., ''A Plan-Form Parameter for Corre- lating Certain Aerodynamic Characteristics of Swept Wings", NASA Tech. Note 2335, April 1951. WADLIN, K.L. and CHRISTOPHER, K.W., "A Method for Calculation of Hydrodynamic Lift for Submerged and Planing Rectangular Lifting Surfaces''", NASA Tech. Report R-14, 195 9° AUSLAENDER, J., ''The Linearized Theory for Supercavi- . tating Hydrofoils Operating at High Speeds Near a Free Sur- face", Hydronautics Technical Report 001-5, June 1961. JOHNSON, V.E., " Theoretical and Experimental Investi- gation of Supercavitating Hydrofoils Operating Near the Free Water Surface!'! NASA Tech. Report R-93, 1961. PIERSON, W.J. and MOSKOWITZ, L., ''A Proposed Spec- tral Form of Fully Developed Wind Seas Based on the Simi- larity Theory of S.A. Kitaigorodski." J. of Geophysical Re- search, Vol. 69, November 1964. Sehmttke and Jones APPENDIX STABILITY DERIVATIVES Stability derivatives are listed below for a single foil element, using the sign convention of equation (7) . Summation of a given ex- pression over all foil elements gives the total ship stability derivative. Longitudinal x = ea U sc De ae os ce) D oC es D Anatol abSiyd ABET 36 batt aqO’ ee _ XS —52 Sicosod spe =b C,) aC i as D Seneca, tps Gyros oc Goa? raat Xp Ri oerag eater tat cee - C,) Z = pU SC, cos [T dC 0s Zz = ean (GY ah + § 2 cos [ pu. =r eae r Z 5 S (Cc) cos ~ Cy) a dC : as L Z, = ¢( Cc) 5 eS ah ix? eons pu. Ze = 5 S j2e 2. cosa ra(cy coer +e] Canadtan Hydrofotl Program. Hydrodynamics and Stmulatton Mano e SC a. eee aC M d(cS) = = = iF M, Xz. ZX, q( cS aes Cur ah ) cos M ENG i, - ZX w wi wi oC af a (cS) My = Xpz, 29%, + qx, (e%S aya o ah )cosT pu. e = = T Bs = e M5 5 Ss 2 Cazes Cas ok c,)] Z. Zax, where the derivation of xX, and Xp assumes constant thrust horse- power. Lateral Au fo) m2 ee Maar S (Cc, sin T +C)) a pur N ° a 2 = A YG 5 S Cer wi sinT + C5; CLY; tan A ) a pu. ae = 5 S (2 Cy sin T -C, x, sin IT - Cpx,) a oC 0s L Yp = -@ sin TP (C) AGEL aa Ree ee ake owe (iG tanA +C sin I Vv yal 2 Li L ei a cos I) pU 156 ry AA aN AN - Ss) t + ¢ Yy 23 = eae + Cyy z tan A y E Say aes cos ) Sehmttke and Jones pU RS ea eee . = S ew tan A ee cos T (Cc. x. pi DP astice c,y,)| a aC os L — T pees == Ky Yori + [yy coe VOiae tae ) ay, + SC La¢ 0,0) ge YX; - 5 Sy, ae PS ee a ° pu. A Ng = Yop X; + Ss Y3Y; res ee ) elt, N. = YS - 5 S y; Ke, x, sin [ +2059, a as aa Be Nyy oe ag, ca ae where S is immersed foil area, A is sweep angle and ran 5 , = . cost +°52° sin F i Yi i Canadtan Hydrofotl Program. Hydrodynamics 35 and Stmulatton HMCS BRAS D'OR (Canadian Forces Photo) Figure 1 Sehmtitke and Jones Figure 2 BRAS D'OR main foil unit Canadian Hydrofotl Program. Hydrodynamics and Stmulatton Delayed cavitation section Figure 3 Suh Sehmitke and Jones 318 BRAS D'OR bow foil unit Figure 4 Canadtan Hydrofotl Program. Hydrodynamtes and Stmulatton uoT}DeS poye[TyUeATedng Gg eanstg Y3110dS MOVE - CIN WNLVG W004 WLNOZIYOH kg SYMBOL O 2 Zyq (FT*) Figure 6 Sehmttke and Jones DRAFT — INS =2 0 3 4 RAKE (DEGREES) Quarter scale lift data (NPL) 320 2 %q (FT ) Canadtan Hydrofotl Program. Hydrodynamics and Stmulatton SYMBOL DRAFT - INS fe) 6.6 3.9 S25 [6.5 pores WEST SREEDETO ERS -6 4 -4¢— -2 O 2 4 6 RAKE (DEGREES) Figure 7 Quarter scale drag data (NPL) 321 Sehmttke and Jones TEST SPEED 40 FPS NPL h= e307 LUMF h=16.5" -6 -4 -2 O 2 4 6 8 RAKE (DEGREES) Figure 8 Comparison of NPL and LUMF lift data $22 Canadtan Hydrofotl Program. Hydrodynamics and Stmulatton 24 16 — L\ N b = oo = CALCULATED FROM FHE 400 NPL ¥, SCALE TESTS TRIALS oO > —| 8 CALCULATED FROM LUMF ¥, SCALE TESTS 30 40 50 60 SPEED (KNOTS) Figure 9 BRAS D'OR bow foil lift coefficient 323 Sehmttke and Jones Figure 10 Axis system 324 Canadian Hydrofotl Program. Hydrodynamics and Stimulation © TRIALS DATA —— PREDICTION 10) oO (DEG) nm TRIM (FT) Mele (CL, 30 40 50 60 SPER Dans) Figure 11 BRAS D'ORtrim, rise and W/D S25 Sehmitke and Jones ° 8 ~ Qa = pitcH TT} oT 50 3 6 35 COUPLED HEAVE SUBSIDENCE 3 50 iO 3 6 8 4 0 x (SEC™') Figure 12 BRAS D'OR longitudinal root loci, 35-60 K 40 WW ” ~N {2} FOLLOWING i?) = ao f°) 40 80 120 LONG. POS'N ART. ORs..hsR Cia) Figure 20 Longitudinal distribution of vertical accelerations, 39 knots —— PREDICTION © TRIALS DATA © = =) uJ O Oo = ¢ 6 oO = ao uw > wn z ec 0 30 35 40 45 SPEED (KT) Figure 21 Vertical acceleration at bow - predicted and measured 331 Sehmttke and Jones -=5° = Sie gkns 42 KTS 0.1 BOW VERTICAL ACCELERATION N 2S a~ On 105 0 is 3 a3 we PITCH ANGLE ac a oO Lu a 10 : N = ENCOUNTERED WAVE. Ap = CaN ee fo) 5 Lo f (HZ) Figure 22 Power spectral densities, sea State 5 332 Canadtan Hydrofotl Program. Hydrodynamtes and Stmulatton SIMULATED ie — — FAIRED CURVE THROUGH DATA — © Fa uJ a | iJ 5) oO q | q oO | or J > = ro} rea} fj re “N © Ww a uJ =] (i) Zz q = oO Le a f (HZ) Figure 23 BRAS D'OR head sea response transforms, 35K 333 Sehmttke and Jones ——— SIMULATED — — FAIRED CURVE THROUGH DATA BOW VERTICAL ACCEL’'N (G/FT) PITCH ANGLE (DEG/FT) f (HZ) Figure 24. BRAS D'OR head sea response transforms, 40 K 334 Canadian Hydrofotl Program. Hydrodynamics and Stimulation O SEA STATE 5 Q SEA STATE 4 (A) RMS LATERAL ACCEL'N (6) (DEG) RMS ROLL ANGLE HEADING TO SEAWAY Figure 25 Roll angle and lateral accelerations, 39 knots 335 Sehmitke and Jones —— HEAD SEA --- FOLLOWING SEA SPECTRAL DENSITY (G7/HZ) f (HZ) Figure 26 CIC lateral acceleration spectra, sea State 5 336 Canadian Hydrofotl Program. Hydrodynamics and Stmulatton Ol — 42 KNOTS --- 34 KNOTS SPECTRAL DENSITY (G7HZ) f (HZ) Figure 27 CIC lateral acceleration spectra, head sea State 5 38% Sehmitke and Jones LATERAL ACCELERATION AT Cl.C. 004 = LATERAL ACCELERATION AT CG. a™ oun — ae w za lJ a ol | amg [aa = oO LJ a iép) ROLL ANGLE ~ 4 N aq we Oo LJ [Sj 18) f (HZ) 2 Figure 28 Lateral acceleration and roll spectra, head sea state 5, 39K 338 Canadtan Hydrofotl Program. Hydrodynamics and Stmulatton DISCUSSION Christopher Hook Hydrofin Bosham, Sussex, U.K. I listened to the two papers by MichaelEames given recently at the Royal Institute of Naval Architects in London and I make no apology for repeating here the specific question I put to Eames at the end of his second Paper. In the discussion of hydrofoils in England there has been a lot of talk about resonance and I asked Mr. Eames specifically if in the course of the development work on the Canadian boat they had ever encountered any resonance phenomena, because whether the damping system be a diamond foil working basically on the Grunberg principle or a mechanically highly damped system, the resonance effects are going to be directly connected with the amount of damping. If the speaker would enlarge on this I think it would be of interest to all the listeners. REPLY TO DISCUSSION Rodney T. Schmitke Defence Research Establishment Atlantte Dartmouth, Nova Seotta, Canada Figures 23 and 24 show that BRAS D'OR's pitch response peaks at approximately .2Hz, the natural frequency of the ship in pitch. Response is fairly broad, however, and the peak is not pro- nounced (low Q), so there is no resonance problem. 389 Schmttke and Jones DISCUSSION Reuven Leopold U.S. Navy Naval Shtp Engtneering Center Hyattsville, Maryland, U.S.A. I have some comments and questions. Could a marginally stable configuration be predictable to be stable by not accounting for surge and thus lead to a bad foil configuration design choice ? The Paper states on page 16: "... surge was neglected in this simulation in order to sim- plify the problem and also because it was felt that neglect- ing surge should give conservative results, "' In a paper titled ''The effect of surge, added mass and un- steady lift on the motion of a hydrofoil boat in a seaway", which I wrote about 10 years ago, I showed that foil configurations which were stable became unstable once the surge was taken into account. REPLY TO DISCUSSION Rodney T. Schmitke Defence Research Establtshment Atlantic Dartmouth, Nova Seotta, Canada In answer to the question, the surge was taken into account in doing stability analysis and the configuration was found to be stable (Figure 12), Furthermore, in all my work on this type of configura- tion I find that the only contribution of surge is a very slow subsidence and it does not seem to couple into the pitch and heave motions signi- ficantly. 340 Canadian Hydrofotl Program. Hydrodynamics and Simulation DISCUSSION Reuven Leopold U.S. Navy Naval Shtp Engineertng Center Hyattsville, Maryland, U.S.A. Page 2 states that the bow foil, as the main control mecha- nics, leads to high 0L/dh for contouring of large waves ; however, high lift versus depth slope for rough seas results in high vertical ac- celeration. For the purpose of not losing momentary control in rough seas and cause diving, the superventilating bow sections are used, which in turn introduce additional drag. The question is this : while reduction of drag and the improvement in ventilated flow stability are definite objectives in all operating conditions, the requirements for QL/dh are different at low frequency - that is calm seas - than at high frequency - that is high sea states. The gain on OL/dh is fixed for a constant speed and could not be modulated unless automatically controlled surfaces were introduced. How does the author propose to optimize dL/dh for rough seas without compromising calm seas ope- rations ? REPLY TO DISCUSSION Rodney T. Schmitke Defence Research Establishment Atlantte Dartmouth, Nova Seotta, Canada The essential compromise is not between rough sea and calm water operation, but rather between head seas and following seas. Op- timization of dL/dh for head sea operation must of course be subject to the constraint of providing adequate stability in following seas. This constraint indeed limits the extent to which dL/dh may be altered, but we feel that at least a slight reduction is feasible and will result in improved motions. We should point out that dL/dh at a given speed may be adjusted, albeit not greatly, by changing bow foil incidence. Thus in head seas the bow foil is trimmed down, lowering dL/dh = 341 Sehmttke and Jones while in following seas incidence is increased, increasing JL/dh. This technique was employed quite successfully during BRAS D'OR sea trials. 342 BENDING FLUTTER AND TORSIONAL FLUTTER OF FLEXIBLE HYDROFOIL STRUTS Peter K. Besch and Yuan-Ning Liu Naval Shtp Researeh and Development Center Bethesda, Maryland, U.S.A. ABSTRACT A large body of experimental and theoretical flutter results for hydrofoil struts were analyzed to deter- mine significant characteristics, Flutter was found to occur in two different structural mode shapes, cor- responding toa predominantly bending mode and a pre- dominantly torsional mode, respectively. The flutter mode shape was related to the vibration mode shapes and the generalized mass ratio of the strut at zero speed. The behavior of the hydroelastic modes of ty- pical struts as a function of speed was investigated using a strip theory with three-dimensional loading mo - difications, Flutter predictions for struts which under - went flutter inthe torsional mode were usually conser- vative and predicted the correct mode shape. However; flutter predictions for struts which underwent flutter in the bending mode were unreliable in predicting the mode of flutter because of an extreme sensitivity to the loading modification used. Strut-foil systems of the inverted-T configuration typical of full-scale hydro- foil craft appear to undergo either bending flutter or torsional flutter, depending on pod and foil character- istics. I. INTRODUCTION The high speeds associated with many unconventional ships will require a better understanding of flutter and other hydroelastic phe- nomena than has been available for design of existing ships. 343 Beseh and Ltu Prominent among unconventional ships are hydrofoil craft and surface effect ships. Flutter is a recognized problem for the strut-foil systems of hydrofoil craft. The rudders contemplated for surface effect ships may be similarly vulnerable. Much research has been done on the flutter of strut models analogous to the above systems. The initial demonstration of strut flutter was made by Hillborne|[1] in 1958.Further experimental work has often been accompanied by difficulties, including models that wouldn't flutter, models that were destroyed by flutter or divergence, and facil- ity limitations. Numerous theoretical analyses have been produced, but none has been successful in predicting all experimental results conser- vatively. Out of these efforts have come many clues to the nature of strut flutter. By combining previous results with some recent experimental and theoretical work we have produced a concept of flutter involving two different flutter regions. This paper will discuss existing flutter data from the standpoint of two flutter regions, and will present cal- culations which indicate the origin of the two regions. The expected ac- curacy of flutter speed predictions within each region will be described. Existing data deals with a large number of simple struts, anda small number of struts with tip pods, some with foils forming an in- verted-T configuration. A sample configuration is shown in Figure l. All tested configurations have been small-scale models. Most discus- sion will be devoted to simple struts and struts with pods. One strut with foils has been included. All struts were cantilever supported from an effectively rigid foundation, so that the structural characteristics of the system were those of a cantilever beam in which both bending and twisting could occur. Because of the relatively high aspect ratio and thin profile of the struts, bending consisted of displacements perpendicular to the plane of the strut, while twisting occurred about a spanwise elastic axis, Vibration modes of the struts consisted of a series of modes which could usually be identified as predominantly bending or predo- minantly twisting or torsion. The mode shapes of the struts at flutter inception could also be characterized as predominantly bending or torsion. In most cases, struts displayed either bending or torsional oscillations at flutter. [1] References are listed on page 393 344 Flutter of Flextble Hydrofotl Struts This was the basis for dividing flutter phenomena into two regions. Flutter in one region occured in a predominantly bending mode shape, and will be referred to as bending flutter. Flutter in the other region occurred in a predominantly torsional mode shape, and will be referr- ed to as torsional flutter. It appears that all hydrofoil struts, including those with pods and foils attached, undergo either bending flutter or torsional flutter. The type of flutter characteristic of a given strut can be determined by examining its vibration modes, exceptina transition region where strong coupling of structural modes occurs. Most available data can be readily placed into the appropriate flutter regions. Experimental results from each flutter region were examined separately. The two flutter regions corresponded to two ranges of ge- neralized mass ratio. In the bending flutter region, struts had low va- lues of generalized mass ratio, while struts in the torsional flutter region had high values of generalized mass ratio. Flutter speed varied differently in each region as a function of mass ratio or strut submer- gence, a related parameter. Calculated flutter characteristics show substantial qualitative agreement with observed characteristics, Flutter was found to occur in a different hydroelastic mode in each flutter region. Predicted flutt- er inception speeds for torsional flutter were conservative for most struts, with many predictions being overconservative. Unfortunately, flutter speed predictions for bending flutter were not usable because two flutter modes were often predicted to be unstable in the bending flutter region, with the wrong mode predicted to be the least stable. This discrepancy was related to an extreme sensitivity of the flutter calculation to hydrodynamic loading modification in the bending flutter region. i EXPERIMENTAL FLUTTER CHARACTERISTICS II.1. Bending-Type and Torsion-Type Struts The flutter mode of a strut is strongly correlated with the nature of the vibration modes of the strut in air or in water. It is therefore convenient to define a method for classifying struts according to im- portant differences in vibration modes, Strut mode shapes are those of a cantilever beam, with bending displacements perpendicular to the plane of the strut and torsional rotations about a spanwise elastic axis. Mode shapes are designated by their similarity to the uncoupled mode shapes of a cantilever beam. Some uncoupled mode shapes are shown in Figure 2, numbered in order of increasing frequency. 345 Besch and Ltu All struts exhibit a fundamental (lowest frequency) vibration mode shape resembling first bending. Struts show a marked differ- ence in their second modes, however, permitting struts to be divided into two groups. The second mode of any strut will consist of a second bending mode coupled with a first torsion mode, with one usually pre- dominating. Predominance is determined by the relative linear dis- placements produced by bending and torsion, which provide an indicat- ion of nodal line characteristics. If the second vibration mode is pre- dominantly second bending, the strut is a bending-type strut. If the second vibration mode is predominantly first torsion, the strutisa torsion-type strut. Struts having little or no tip weighting are usually bending -type struts. Struts having relatively heavy pods are usually torsion-type struts. A transition region exists in which the second vibration mode of a strut is equally due to a second bending mode shape and a first torsion mode shape, with neither predominating. Struts in this tran- sition region have moderately weighted pods or medium to large foils. The effect of foils in coupling second bending and first torsion is very pronounced when the foils are submerged due to the large rotary iner- tia effect at the tip of the strut. When such strong coupling occurs, it is impossible to classify the strut as bending-type or torsion-type. In most cases the third vibration modes of bending-type struts are first torsion, while torsion-type struts have a third vibration mode resembling second bending. This observation indicates that a change in strut type usually involves a reversal in the order of the second and third mode shapes. Most struts have the same mode order in air and in water. If there is a difference, the mode order in water should be used for classifying a strut. Either measurement or calculation can be used to determine the required mode shapes, II.2. Flutter Mode Shapes The flutter mode shapes of bending-type struts are radically different from those of torsion-type struts. Bending-type struts under- go flutter in a predominantly first bending mode shape, while torsion- type struts undergo flutter in a predominantly first torsion mode shape. In accordance with the flutter mode shapes, flutter of bending - type hydrofoils will be referred to as bending flutter, and flutter of torsion-type hydrofoils will be referred to as torsional flutter. The two types of flutter mode shapes have not been quantita - tively measured, but were discovered because the very striking dif- 346 Flutter of Flextble Hydrofotl Struts ferences in mode shape were visually observed. Differences in flutt- er mode shapes were reported by Huang [2] as a result of flutter test- ing a strut with and without a heavy pod. The bending amplitude of the strut alone was reported to be considerably larger than the torsional amplitude which was also present. When the pod was added, the tor- sional amplitude became larger than bending. A similar result was obtained in an experiment performed at the Naval Ship Research and Development Center (NSRDC) in which a bending-type strut and a torsion-type strut were flutter tested. Both struts had been previously tested but mode shapes were not re- ported. Motions of the struts were visually observed and recorded on video tape. The bending-type strut, Model A of Reference 3, under- went large first bending oscillations with little evident twisting. In constrast, the torsion-type strut, Model 2T of Reference 4, displayed first torsion oscillations with no visible bending. In addition to a change in mode shape, a change in frequency would be expected when the flutter mode changes. Several pod confi- gurations for Model 2T were flutter tested| 4 | , and a significant change in flutter frequency occured when the strut changed from bending - type to torsion-type. Flutter data for this strut are plotted in Figure 3. As the pod mass was increased and the pod center of gravity was moved aft, an abrupt increase in frequency occured between pod con- figurations A and B. Vibration modes calculated in water identify pod configuration A as a bending-type model, while pod configuration B gives strongly coupled second bending and first torsion modes for both its second and third modes and therefore falls in the transition region between bending-type and torsion-type struts. Pod configur- ation C was a torsion-type strut. Although mode shapes have been observed in only a small number of cases, other aspects of flutter data exhibit a dual nature corresponding to differences in mode shape. The effects of generaliz- ed mass ratio and of strut submergence vary according to the flutter region. These effects will be discussed below. Il. 3. Generalized Mass Ratio Generalized mass ratio is a parameter which indicates the relative importance of structural and fluid inertia in determining the motion of a strut. Both structural and fluid inertia are related to the vibration mode shape (and therefore to the elastic properties) of the strut. This relationship is included in the most general form of the parameter, which can be expressed in terms of matrix elements as 347 Besch and Ltu Myeneralized — This expression reduces to the mass ratio traditionally used in flutter analysis when pure bending motion of uniform amplitude is assumed. A similar simplification occurs when pure torsional motion is assum- ed. These assumed motions provide suitable approximations to the mode shapes of bending and torsional flutter. Exact flutter mode shapes are of course not available. Therefore simplified expressions for mass ratio were used in analyzing experimental flutter results. It was found that bending flutter occurs at low values of mass ratio, and that torsional flutter occurs at high values of mass ratio. Other than this generalization, comparisons involving mass ratio will not be made between struts having different flutter modes. Such com- parisons would require extensive calculations involving exact flutter mode shapes, which are not available, Calculations presented later indicate that bending flutter and torsional flutter involve entirely dif- ferent vibration modes and do not represent different mass ratio ranges of the same mode. Mass ratio will be used as a parameter for comparing flutter results of similar mode. Each flutter region will be discussed separately. Bending Flutter Region In the bending flutter region, generalized mass ratio can be approximated by dividing total strut mass by the mass of a cylinder of water circumscribing the strut. The cylinder of water should have a diameter equal to the strut chord and a length equal to the submerg- ed span of the strut. This cylinder of water approximates the added mass of the strut for the first bending mode shape associated with bending flutter. Bending mass ratio may be written symbolically as mL + Mpod m ry ieee ere bending eho Miod Flutter speeds obtained from bending-type struts are plotted as a function of bending mass ratio in Figure 4. Values of bending mass ratio range from 0.1 to 0.66. The flutter speeds fall into two 348 Flutter of Flextble Hydrofotl Struts groups. The higher flutter speeds correspond to strut models which are geometrically larger by a factor of approximately 4 than strut models represented in the lower group. Within each group, the flutter speeds are sensitive to mass ratio and a sweep parameter x . The sweep paremeter [5, 6 | combines sweep angle and unswept aspect ratio into a single parameter. Numerical values of x are given for the data points in Figure 4. For similar size models, the data can be fairly consistently divided into families based on similar values of k as shown. An increase in sweep angle therefore increases the flutter speed, while an increase in aspect ratio decreases the flutter speed. Lines of constant « value approach zero as mass ratio decreases in a manner which could be approximated by a square root dependence on mass ratio, a relation which has previously been observed [3] for low mass ratio struts. Similar trends have been predicted in the lower mass ratio region when sweep angle was included in the analysis [6, NG, 8 | . Groups of different sized models can be correlated by di- mensional analysis. It has been shown [9] that the flutter speeds are related according to the square root of bending or torsional stiffness. Torsional Flutter Region Generalized mass ratio for torsional motion can be represent- ed by the ratio of the total moment of inertia of a strut and the added moment of inertia of the submerged portion of the strut. In the present work, rotation was assumed to occur about the elastic axis of the strut. The resulting torsional mass ratio may be written eS Ty V torsion 4 2 mob (1/8 +a E410 pod Available flutter speeds for torsion-type struts are plottedas a function of torsional mass ratio in Figure 5. A substantial amount of data is shown which was obtained at NSRDC and has not been pre- viously published. All strut models in this group had pods and were similar in size to the struts described in Reference 4, A complete description of this data will be published in the near future. As shown in Figure 5, torsional flutter has been obtained at values of torsional mass ratio between 0.61 and 6.2. Flutter speeds generally decrease as mass ratio increases, The wide variation in flutter speed results at least in part from wide variation in strut characteristics. In an attempt to adjust flutter speeds for differences in geometric size and torsional frequency, the data has been replotted 349 Besch and Ltu in Figure 6 after normalization by the factor bw ,. This normalizat- ion was successful for values of mass ratio between 2.0 and 6.2, but large variations still exist at values below 2.0. Parameters that differ among the lower mass ratio models include the elastic axis location, profile, sweep angle, and submergence of the struts, and the size and inertial characteristics of pods attached to the struts. The effects of strut profile have recently been investigated at NSRDC, and results for three different profiles are indicated in Figure 6, At speeds high enough to produce ventilation over the entire chord of the strut, a ventilated cavity originating from a blunt leading edge ona strut substantially destabilizes the system. The effects of strut sub- mergence will be discussed later. The reduced flutter speeds for torsional flutter exhibit the characteristics found in classical hydrofoil flutter. The flutter speed parameter gradually decreases to a minimum value as mass ratio de- creases, and then increases rapidly for related series of strut models at lower values of mass ratio. Minimum values occur approximately between mass ratios of 2.0 to 3.0. The effect of mass ratio on torsional flutter speeds is similar to that predicted by classical two- dimensional flutter theory and also to that predicted in the higher mass ratio region in finite sweep angle analyses [6, ty 8] : Il.4. Strut Submergence The effects of strut submergence on flutter speed are closely related to the effects of generalized mass ratio. When the simplified forms of mass ratio are used, the two parameters are inversely pro- portional to one another. The close relationship is evident in experi- mental flutter results in which submergence has been varied without changing other strut characteristics, These results, shown in Fig, 7, constitute a replotting of data contained in Figures 4 through 6 but are given to illustrate the effects of submergence directly. Flutter speeds for bending-type struts decrease as strut sub- mergence increases, with minimum flutter speeds occurring at full submergence, The increase in submerged length produces a decrease in mass ratio and therefore a decrease in flutter speed. Torsion-type struts show a local minimum in flutter speed at approximately 50 % submergence, This local minimum would be expected to occur if the strut configuration passed through intermediate values of mass ratio, and will not necessarily correspond to 50 percent submergence, An increase in flutter speed will of course occur as the submergence becomes very small regardless of the mass ratio. The effect of sub- mergence on the strut model with pod and foils is similar to that ob- served for struts without foils in the bending flutter region and at high 350 Flutter of Flextble Hydrofotl Struts values of mass ratio in the torsional flutter region, Strut submergence also affects the vibration mode shapes of struts, and has a particularly large effect on the second bending mode, Asa result, a strut could change from a bending-type strut to a tor- sion-type strut during changes in submergence. Because of the occur- rence of minimum flutter speeds at different depths for different modes, and the possibility of different flutter modes occuring at differ ent depths, it is conceivable that a strut could undergo bending flutter at one depth and torsional flutter at another depth. Ill. THEORETICAL FLUTTER CHARACTERISTICS The dual nature of experimental flutter results also appears in theoretical results. Bending flutter and torsional flutter correspond to instabilities in different hydroelastic modes. Transition from bend- ing flutter to torsional flutter occurs when the torsional flutter mode becomes less stable than the bending flutter mode. The frequency and mode shape characteristics of the hydro- elastic modes involved in flutter are predicted accurately in the flutter analysis. However, damping characteristics, and, consequently, flutter speeds are not predicted accurately. In the bending. flutter region, flutter speed predictions are not usable because a second mode is also predicted to be unstable which does not correspond to experimental results. Flutter speed predictions in the torsional flutter region correctly indicate the unstable mode but are generally overconservative. Calculated hydroelastic modes of a strut with at- tached foils indicate that the strut underwent torsional flutter ata speed which was overconservatively predicted. iit. Piutter Theory, Understanding of the differences between bending flutter and torsional flutter requires consideration of the behavior of the hydro- elastic modes fi 2 | » or resonances, of the strut systems over a wide range of speeds, and not merely a calculation of each strut's speed of neutral stability. This approach was in fact used ina paper 8 | presented at the Fourth Symposium on Naval Hydrodynamics, This earlier paper described the hydroelastic modes of bending -type struts only. The present paper extends the earlier results to include a des- cription of the hydroelastic modes of torsion-type struts as well. Hydroelastic modes are the vibration modes of the strut-fluid system and correspond to eigenvalues and eigenvectors of the velocity - dependent equations of motion. The equations of motion were generated S51 Beseh and Ltu by assuming a lumped parameter representation for the strut, with elastic and inertial properties lumped at discrete points along a straight elastic axis, This procedure is well established as an accur- ate means of predicting vibration mode shapes and frequencies of elongated structures in air. The hydrodynamic forces on the strut were also lumped at stations along the axis. Values of structural parameters and hydrodynamic forces at spanwise stations were assign- ed by dividing the strut into strips normal to the elastic axis. A nu- merically converged solution was obtained when 10 strips were used. Displacements were assumed to occur in bending normal to the plane of the strut and in torsion about the strut elastic axis. The equations of motion for the entire system written in matrix form are h pa fe} + fe + oso td fel = tg . 1 it ik The hydrodynamic force F was expressed in terms of the physical displacements and their time derivatives, permitting the structural and hydrodynamic expressions to be combined. Further simplificat- ion is achieved by representing strut motion as a series of standing waves in the form ne =~ te and 8, = O.e The resulting system of linear equations for the hydroelastic system is (aot MA eo er ee eo | = {fo} a Solutions to the above equations are the complex eigenvalues of s, which may be written s = -fw +j Lbaiah hac in terms of the damping ratio { and the undamped natural frequency w . Each eigenvalue of s corresponds to a mode of oscillation of the strut-fluid system. Flutter occurs at the lowest speed for which the real part of one of the eigenvalues becomes zero. 352. Flutter of Flextble Hydrofotl Struts Eigenvalues for selected speeds were obtained by a digital computer calculation based on Muller's quadratic method [13]. Flutter speeds were determined by interpolation among damping values across the zero damping axis. HEigenvectors were also obtained, giving the vibration mode shapes in standing-wave form, The most general form of strut motion is composed of travell- ing waves as wellas standing waves. Further calculations were there- fore made to determine whether travelling-wave oscillations were occurring. Travelling waves were found in connection with bending motion and will be described later. IlI.2. Hydrodynamic loading Hydrodynamic loading on discrete sections of the strut was calculated with a strip theory. The theory was formulated to allow spanwise variation of the loading so that the effects of three-dimen- sional flow could be investigated. The lift and moment expressions used were ee : ® -P. = p. mpb. E -V 0.+ Vo, tanA_ +b,a, (0,+V_¢. tanA | 1 1 1 1 Te! nil ea es 1 A: ea -C yer pM = b. C(k) Ww. a,l apap i CUE ee) (Oe Ve, =e tansAjen) i Se a i aon ea 2D . 3 ee é A + Pp, mpb. Vf, 4 Nike tan At p,tpb. a.(h, + Vie tan sie! Zi 2 - -a,b A ey, (eens veg) 2 1 i 2m7pV b == = a eae oat eee + p i > P; (a, a Fit) Ww. In this formulation, spanwise loading variations were introduc- ed separately for circulatory and noncirculatory loading. The loading 355 Besch and Ltu due to circulatory flow was varied by inserting steady values of lift slope C and aerodynamic center a, obtained by a separate cal- culation. This approach was originated by Yates f14] . The noncircu- latory terms were varied by inclusion of a multiplicative factor p. The factor permitted reducing the magnitude of the noncirculatory loading below that associated with two-dimensional flow. This modi- fication was introduced by the authors [15 ] , in accordance with a suggestion by Yates [1 6 |. Spanwise distributions for p will be dis- cussed later. The given expressions correspond to two-dimensional hydro- dynamic loading when a lift slope of 2”, an aerodynamic center lo- cation at quarter chord, and a noncirculatory modification factor of unity are used. Three-dimensional loading requires that appropriate spanwise distributions of these quantities be used. Ina number of flutter calculations presented later, the effects of three-dimensional flow were studied by varying the above quantities but keeping all span- wise values equal. Spanwise distributions of lift slope and aerodynamic center were obtained from lifting surface theory [1 7 | . The distributions were calculated using a uniform angle of attack along the span of the strut, and an antisymmetric loading boundary condition at the free surface. Two different distributions of noncirculatory modification fac- tor were used, one for low frequencies and one for high frequencies a 5 | . At low frequencies, the factor consisted of the three-dimension- al added mass of the strut, expressed asa fraction of the two- dimensional added mass, outboard of the spanwise position being con- sidered. The free surface was treated as a reflecting plane. At high frequencies, the spanwise distribution of added mass ona surface- piercing strut decreases to zero at the free surface. This condition was approximated by assuming the midspan of the submerged portion of the strut to be a reflecting plane and using the low frequency dis- tribution on either side. Values of the nondimensional frequency, Latfe ,» were used to distinguish between low frequency and high frequency conditions, indicating that the generation of gravity waves was involved in the boundary condition. The low frequency condition exists for values of yy) w*/g of 1 or less, while high frequency loading corresponds to values of bLw2/g of 10 or greater. 354 Flutter of Flextble Hydrofotl Struts III.3. Bending Flutter The hydroelastic modes of several bending-type struts were calculated. In general, two unstable modes were predicted for each strut. One of the unstable modes showed fair correlation with ex- perimental flutter occurrences, while the second unstable mode did not correlate well with experimental results. It therefore appears that one unstable mode corresponded to the experimentally observed instabilities for all of the struts, while the other unstable mode was incorrectly predicted to be unstable. The incorrect prediction was, in fact, found to occur only for limited ranges of spanwise loading inputs, suggesting that the prediction was caused by a slightly inac- curate loading formulation in a highly sensitive calculation, The mode in which bending flutter occurred had a first bend- ing mode shape, and had the lowest frequency among the existing modes at the experimental flutter speed. At speeds below flutter, the mode was highly damped. Its damping decreased rapidly in a short speed interval prior to flutter. Values of damping were predicted nonconservatively. These results will be illustrated by presenting detailed cha- racteristics of the hydroelastic modes of a typical bending-type strut, Model 2 of Reference 4, The structural characteristics and three- dimensional loading parameters for Model 2 are given in the Appendix. Several hydroelastic modes calculated for Model 2 are shown as functions of speed in Figures 8 and 9. The damping ratio ¢§ was plott- ed without structural damping because no experimental values were available. Predominant mode shapes are indicated on the frequency curves, Predicted instabilities must be compared with an experiment- al flutter speed of 81 knots anda frequency of 4.1 Hz at that speed. The mode shape at flutter was observed to be predominantly first bending in a motion picture of the experiment. Flutter is predicted to occur at 83 knots in the presence of two-dimensional loading, as shown in Figure 8. The instability occurs in a mode which first appears, fully damped, at a speed of 30 knots and decreases in stability as speed increases until neutral stability is reached at 83 knots. Although the unstable mode appears at a speed near that at which mode 1 damps out, the two modes coexist over a small speed range. Therefore the unstable mode is considered to be a new mode rather than mode 1. The frequency and mode shape of the new mode show good agreement with experiment. Mode 3 is stable and increases in frequency as speed increases. The predicted instabilities are much different, and less 355 Besech and Ltu accurate, when three-dimensional loading modifications are included. Despite the theoretically improved loading expression, the flutter speed predicted for the new mode is a highly nonconservative 147 knots. Mode 3 is unstable over the entire speed range, except at low speeds where inclusion of structural damping would produce a positive damping ratio. Both unstable modes have a first bending mode shape in the vicinity of the experimental flutter speed. The frequency of mode 3 now decreases rapidly with speed, but nevertheless does not decrease sufficiently to agree with experiment at 81 knots . On the other hand, the frequency of the new mode shows fairly good agree- ment with experiment at 81 knots. In this case and in general, frequencies of hydroelastic modes are predicted more accurately than damping characteristics, and are less sensitive to variation in hydro- dynamic loading. It is concluded that flutter occurred experimentally in the new mode, and that mode 3 is not unstable below 81 knots. Each of the loading modification parameters was varied inde- pendently to determine its effect on predicted flutter instabilities. Equal loading was used at all spanwise positions. The resulting flutter speeds for the new mode and mode 3 are shown in Figure 10. Mode 3 becomes unstable when any of the three modification parameters is changed sufficiently from two-dimensional values. A three-dimensional value of lift slope produces greater instability in mode 3 than three- dimensional values of the other parameters. Interactions among para- meters and variations in strut configuration also affect the stability of mode 3, The behavior of hydroelastic modes 1] and 2 was not signifi- cantly affected by the variation of applied loading. The nature of the oscillations experienced by Model 2 at flutter was further investigated in order to determine whether the oscillations consisted of standing or travelling waves. Calculations by Dugundji, et al.[18 | and Prasad, etal. [1 9 | had indicated a bending flutter con- dition occurring in the form of travelling waves for low mass ratio wings. The present complex eigenvalue calculation restricted oscil- lations to a series of standing waves in which nodal lines remained stationary and all displacements in each mode maintained their rela- tive distributions at all times. Travelling waves are characterized by nodal lines which traverse the entire surface of the strut during a cycle of oscillation. A direct solution to the equations of motion was attempted, using a finite difference technique in the time domain [ 20 ] . The method of solution yields a time history of the transient motion following an initial excitation of finite duration. Flutter inception occurs when os- cillations change from decreasing to increasing amplitude. Neutral stability should occur at the same speed using either method of solution, 356 Flutter of Flextble Hydrofotl Struts Calculations were performed by a digital computer program which required that hydrodynamic force expressions be real, Because of this restriction, the imaginary part of the Theodorsen circulation function was omitted from the loading used. Unsatisfactory results were obtained from the direct method of solution. The predicted flutter speed did not agree with that predict- ed by the eigenvalue calculation. Furthermore, values of negative damping above flutter inception were so large that no oscillation oc- curred, Asa result, the presence of travelling waves could not be detected. The discrepancies between the two methods of solution may have resulted from the difference in hydrodynamic loading used. It is evident that further investigation of this method of calculation is re- quired. The hydroelastic mode characteristics of Model 2 are typical of three other bending-type struts that were analyzed. Flutter predic- tions using two-dimensional loading were often fairly accurate. The mode 3 instability appeared in two of the three additional calculations using three-dimensional loading. The flutter inception speeds for the new mode were again nonconservative, and less accurate than those obtained using two-dimensional loading. Frequencies predicted for the new mode agreed well with frequencies observed at flutter, while those for mode 3 did not agree well. Frequency predictions were equal- ly accurate for both types of loading. Predicted mode shapes for the new mode were predominantly first bending. This agrees with mode shapes observed for bending flutter. The damping behavior of the new mode as a function of speed shows qualitative agreement with experimental results. In an expe- riment performed at NSRDC, damping was found to be extremely high for a bending-type strut at all speeds below flutter inception. At flutter inception, damping decreased sufficiently to permit flow-excited os- cillations of large amplitude. This behavior would be expected of an instability occurring in the new mode, which decreases in damping from a highly damped condition at intermediate speeds. In view of the more accurate frequency correlations of the new mode, and its high damping characteristics at intermediate speeds, it is concluded that bending flutter occurs in the mode designated as the new mode. Measurements of damping of strut modes at various speeds are needed to confirm this conclusion. The appearance ofa calculated instability in mode 3 is probably caused by extreme sen- sitivity of the calculated damping to load variation, The present flutter calculation can be used to indicate a possible occurrence of bending flutter, but cannot be used for estimating flutter speeds. Design Sia) Besch and Ltu calculations should be performed with both two-dimensional and three- dimensional loading so that all potential instabilities will be discovered. IlI.4. Torsional Flutter The calculated hydroelastic modes of torsion-type struts exhibit more simple behavior than those of bending-type struts. Only one mode is unstable. It is the mode with the second-lowest frequency, and therefore with a predominantly first torsion mode shape, at zero speed, Low damping is predicted in this mode at all speeds below flutter, in contrast to the high damping predicted in the bending flutter mode. Observed characteristics of torsional flutter correlate well with the characteristics of this hydroelastic mode. A mode analogous to the new mode previously described appears for some torsion-type struts, but it is stable throughout the speed range of interest. Three- dimensional loading modifications have very little effect on the quali- tative characteristics of the hydroelastic modes of torsion-type struts, but do change the predicted flutter speeds. Predicted torsional flutter speeds ranged from 59 percent conservative to 36 percent nonconservative when three-dimensional loading was used. The predicted flutter speeds were nonconservative for struts with extremely heavy pods, and became increasingly con- servative as the struts decreased in mass ratio and approached the bending flutter region. As an example of hydroelastic modes for torsion-type struts, the modes for Model 2T of Reference 4 are shown in Figure 11. The structural characteristics of Model 2T are identical to those of Model 2 except for the addition of a long, slender pod to the strut tip. The pod is described in the Appendix. The damping ratio includes the value of structural damping measured at zero speed. One value of damping ratio and frequency was reported in Reference 4, and the others were measured at NSRDC by deflecting the strut wth an attached line and cutting it during the test runs. A flutter speed of 18.1 knots was obtained at NSRDC, anda frequency of 6.4 Hz was observed at that speed. The vibration mode shape prior to and at flutter was predomi- nantly first torsion. Flutter is predicted to occur in mode 2 at 14,1 knots when two-dimensional loading is used, This prediction is conservative by 22 percent. The calculated damping values are lower than the expe- rimental values, but show a similar variation with speed. Frequencies of the hydroelastic modes remain relatively constant as a function of speed, and agree well with available data. The mode shape of the un- stable mode, mode 2, changes from first torsion to first bending prior 358 Flutter of Flextble Hydrofotl Struts to the predicted flutter inception, so that flutter is incorrectly predict- ed to occur with a first bending mode shape. Slight changes occur in the hydroelastic modes when three- dimensional loading modifications are included. The three-dimension- al loading used for Model 2 was also used for Model 2T. The damping of mode 2 increases, remaining below the experimental values for part of the speed range below flutter inception but yielding a flutter speed of 18.8 knots, which is very close to the experimental value but is slightly nonconservative. The flutter mode shape is predicted to be predomi- nantly first torsion, which is the mode shape that was observed. The good agreement between experimental and theoretical cha- racteristics of mode 2 clearly establishes that the instability has been correctly predicted. Identification of the unstable mode is easier than for Model 2 because sufficient data are available and the modes are unambiguous in predicting instability. The effects of independent variation of the loading modification parameters on predicted flutter speeds for Model 2T are shown in Figure 12, Equal values of loading were used at all spanwise stations. The calculation is conservative and reasonably accurate using two- dimensional loading, and is unaffected by loading modifications except when lift slope is reduced below 70 percent of the two-dimensional value. While the calculation is sensitive only to lift slope for the con- ditions shown, strong interactions occur among the modifying para- meters when they are varied simultaneously. This interaction is de- monstrated by the 18.8 knot flutter speed prediction obtained when three-dimensional values are used for all modifying parameters. No mode corresponding to the new mode described for Model 2 appears in the speed range shown for Model 2T. Such a mode does appear at higher speeds, however, but remains stable at all speeds for which calculations were made. An indication of travelling wave motion was found in the flutter mode of Model 2T. However, a discrepancy in calculated flutter speed similar to that found for Model 2 prevents full confidence in the results. The direct method of calculation yielded increasing and decreasing os- cillations above and below a different flutter speed from that obtained by eigenvalue calculation. Calculated mode shapes in bending and tor- sion at flutter are shown in Figure 13 as functions of time. The bending oscillations are travelling waves, while the torsional oscillations are standing waves. Strut deflections due to torsion were approximately twice as large as those due to bending. Therefore the flutter oscillat- ions were predominantly standing waves, although travelling waves 359 Besch and Ltu were not insignificant. Flutter characteristics calculated for several other torsion- type struts using three-dimensional loading were similar to those of Model 2T. Flutter invariably occurred in mode 2. Calculated flutter speeds, which are compared with experimental values in Figure 14, ranged from 59 percent conservative for very light pods to 36 percent nonconservative for very heavy pods. Frequency predictions showed good agreement with measured values at the experimental flutter speeds. Flutter mode shapes were predicted to be first torsion, with occasionally significant amounts of first bending or second bending. These mode shape predictions agreed with visually observed mode shapes. Examples of predicted flutter characteristics of both bending - type and torsion-type struts, as well asa strut in the transition region (pod configuration B), are shown in Figure 3. Theincreasingly conser- vative torsional flutter speed predictions are evident as pod inertia decreases, until bending flutter occurs with pod configuration A, In view of the good correlation between theoretical and experimental fre- quency and mode shape in the torsional flutter region, it is concluded that torsional flutter is an instability of hydroelastic mode 2 for torsion- type struts. A new mode similar to that of Model 2 and Model 2T also ap- pears in the hydroelastic modes of other torsion-type struts. This mode appears at lower speeds for struts with lighter pods. The stabi- lity of the new mode decreases as strut pods become lighter and strut configurations shift from torsion-type to bending-type. Bending flutter appears to originate when the new mode becomes unstable at a lower speed than the mode which is unstable in torsional flutter. A shift in the mode shapes of the second and third modes occurs as part of this transition. It is perhaps not coincidental that the mode which is un- stable in the torsional region, and the mode which is incorrectly pre- dicted to be unstable in the bending region, both originate as first torsion modes, Calculations made for struts with large pods included an appro- ximate correction for hydrodynamic forces acting on the pod. The correction, added to the tip of the strut, consisted of the linearized lift and moment due to the unsteady motion of a pod-sized slender body, and is described on page 417 of Reference 12. This correction pro- duced much lower flutter speeds than using pod added mass alone, particularly for heavy pods. 360 Flutter of Flextble Hydrofotl Struts III. 5. Struts with Foils Successful flutter analysis of strut-foil systems is of consider- able practical importance, because struts will generally be used in combination with load-bearing foils. Only inverted-T strut-foil confi- gurations have been considered in the present work, in view of the in- terest of the U.S. Navy in such configurations. It is clear that for such systems foils have a sizable effect on flutter characteristics, Flutter speeds obtained experimentally by Huang [2] showed an increase of as much as 146 percent when a pod was replaced by a pod and foil combi- nation of equal mass, The parameters governing the effects of foils on flutter characteristics have only begun to be investigated. An early discovery has been that foil angle of attack is an important flutter parameter Pl : While experimental results are relatively scarce, much can be deduced about the flutter characteristics of struts with foils by con- sidering the structural effects of adding foils. A strut with no tip at- tachment will usually be a bending-type strut, and a heavily tip- weighted strut will be a torsion-type strut. Therefore struts with foils will vary from bending -type to torsion-type, with many configurations being in the transition region, according to the size and weight of the foils. Other parameters will be important to the extent that they pro- duce bending -type or torsion-type characteristics. The rotational inertia of the foils will affect the coupling between the second and third vibration modes, so that large or high aspect ratio foils will produce struts in the transition region, Large or heavy pods tend to produce torsion-type struts. These effects are related to the generalized mass ratio of the strut. Flutter characteristics calculated for a strut with foils were consistent with these deductions. The strut had a large pod and full- sized foils. The calculated instability occurred in hydroelastic mode 2, the unstable mode in torsional flutter. The flutter speed pre- diction was overconservative. The second and third vibration modes at zero speed were composed equally of second bending and first tor- sion mode shapes, indicating that the strut was in the transition re- gion, The flutter analysis performed on the strut-foil model [2] will be described in detail to permit comparison with previous results. Structural characteristics of the model are given in the Appendix. Several approximations were made in obtaining a theoretical repre- sentation for the pod and foils. Structural properties of the pod and the foils were represented by adding equivalent masses and moments of inertia to the tip of the strut. Hydrodynamic loading on the pod and 361 Besch and Ltu foils was represented only by adding their added mass and moment of inertia to the structural components. The effects of the pod and foils on the vibration mode shapes of the submerged strut are shown in Figure 15, It was necessary to use 67 percent of the published values for bending and torsional stiffness to achieve agreement with measured in-air frequencies. These stiff- ness values were used for in-water frequency calculations and hydro- elastic mode calculations as well. The strut alone is a bending -type strut, and the strut with pod is a torsion-type strut. The second and third modes of the strut with pod and foils each exhibit both first tor- sion and second bending oscillations. Strong couplings due to the foils has also produced similar frequencies for these modes. The strut- foil model must be classified in the transition region. The effect of the foils is particularly striking because the pod-foil combination has the same mass as the pod used on the strut-pod model. Flutter was found experimentally to depend on the angle of at- tack of the foil. Two flutter conditions were obtained : at 16. 6 knots with an angle of attack of -4 deg , andat 18.1 knots with an angle of attack of -2 deg . Testing was halted prior to flutter at higher angles of attack because divergent deflections of the strut began to occur, Flutter mode shapes were described as equally large bending and tor- sional deflections. The bending deflections were seen to change from second bending to first bending, while the torsional deflections were consistently first torsion, Structural damping was not determined ex- perimentally. Calculated hydroelastic modes for the strut-foil model are shown in Figures 16 and 17. Both two-dimensional and three-dimen- sional loading yield a flutter instability in mode 2. The predicted flutter speed is overconservative at approximately 6 knots in both cases. An additional unstable mode is found which is different for the two types of loading. Two-dimensional loading yields an instability in the new mode, while three-dimensional loading yields an instability in mode 3. The frequencies predicted using three-dimensional loading for mode 2 at the observed flutter speeds agree well with the experi- mental frequencies, while those predicted for mode 3 do not agree well. On the basis of the usually reliable frequency calculation of three-dimensional loading, it is concluded that flutter occurred ex- perimentally in mode 2, Additional damping data for individual modes is needed to confirm this conclusion. Predicted mode shapes do not agree with observations. Mode 2 consisted of both second bending and first torsion oscillations at low 362 Flutter of Flextble Hydrofotl Struts speeds, but showed almost exclusively first bending oscillations at the experimental flutter speed. Experimental [2] and theoretical results were also obtained for the strut with a pod lighter than, and equal to, the weight of the pod with foils, Torsional flutter occurred in both cases, at 20.1 knots with the smaller pod weight and at 9.5 knots with the larger pod weight. The large decrease in flutter speed when weight was added to the pod would be expected at low values of torsional mass ratio, as may be seen in Figure 5. Converting part of the pod mass into foils has raised the flutter speed, and has hada similar effect to reducing the pod mass. The foils have reduced the generalized mass ratio of the strut, Calculations of generalized mass ratio are required in order to correlate experimental results with values of this parameter. It is apparent that additional experimental and theoretical re- search is needed to adequately understand flutter of struts with foils. Experimental results can provide a reliable indication of the effects of foil-related parameters and can lead to accurate simulation of full- scale systems with reduced-scale models. Theoretical research is needed to improve the accuracy of flutter speed predictions. IV. DISCUSSION The primary deficiency of the present flutter analysis is its prediction of damping. This deficiency results in inaccurate predic- tions of flutter speed for most struts. In the torsional flutter region, the inaccuracy is strongly correlated with the value of torsional mass ratio of the strut. The relationship between experimental and theore- tical flutter speeds has been illustrated in Figure 14, Predictions fol- low a fairly well-defined curve which is overconservative at low mass ratio, quite nonconservative at high mass ratio, and which crosses over experimental values at a mass ratio of slightly less than 2. It should be noted that the single very nonconservative prediction was strongly influenced by the presence of a large pod, and is therefore not strictly representative of flutter characteristics of simple struts. The conservative predictions obtained at low values of torsion- al mass ratio are highly significant. Many previous studies of hydro- foil and airfoil flutter have shown a tendency for predicted flutter speeds to become nonconservative at low values of mass ration, lead- ing to a loss of confidence in flutter predictions in this region. These studies have used two-dimensional loading without accounting for sweep angle. The present analysis, anda similar analysis [4] made previously, showed no tendency for predictions to become nonconser- 363 Besch and Ltu vative at low mass ratio for torsional flutter. It appears that the most significant difference in the two types of calculations is the in- clusion of sweep angle as a parameter which couples structural and hydrodynamic effects. The present calculation should reliably indicate the presence of a flutter instability throughout the mass ratio range shown in Figure 14, Additional comparisons with existing flutter data can help to determine the accuracy of flutter speed prediction to be ex- pected as a function of mass ratio. Improvement in the accuracy of flutter speed prediction will require improvement in the hydrodynamic loading formulation. The sensitivity of calculated damping to small changes in loading, parti- cularly for bending-type struts, suggests that hydrodynamic loading must be very accurately described in order to obtain accurate flutter speed predictions. Possible sources of inaccuracy in the loading for- mulation are the presence of cavitation, real fluid effects involving the boundary layer and wake, and inexact modification of the two-dimension- al loading for three-dimensional flow. Rowe [2 1] has shown that large changes in calculated flutter speed result when the loading applied to struts is modified to simulate cavitation. Available observations are insufficiently detailed to confirm the existence of the assumed distri- butions of cavitation, It has been shown [22] that altering boundary layer characteristics with disturbance wires affects agreement bet- ween theoretical and experimental loading in two-dimensional flow. However, the results of such modification on flutter characteristics have not been investigated. Reliable measurements of three-dimension- al strut loading which could be used to assess the accuracy of the strip theory employed in the present flutter analysis are not available, The existence of two different unstable hydroelastic modes implies that future flutter experiments and calculations must be carri- ed out in sufficient detail to distinguish between the modes. This will require measurement or calculation of hydroelastic mode characte- ristics as a function of speed. Measurements of damping characteristics at zero speed are important, particularly for struts which undergo tor- sional flutter, so that calculated damping can be adjusted to include structural damping. Flutter research will be incomplete until hydroelastic mode characteristics of full-scale strut systems are measured. These mea- surements will provide comparisons with model data and calculations as well as indicate the stability of the actual struts. V. DESIGN PROCEDURES Design of inverted-T strut-foil systems to operate in subcavi- 364 Flutter of Flextble Hydrofotl Struts tating flow can be based on the flutter-free performance of the exist- ing U.S. Navy hydrofoil craft. In order to estimate the effect of va- riations in configuration, it would be helpful to calculate the hydro- elastic mode characteristics and the generalized mass ratio of exist- ing struts for comparison with parametric trends obtained from models, Further model testing may be required to establish stability criteria in areas where theory and present data are inadequate, such as in the presence of cavitating flow. Additional information about hydroelastic stability can be ob- tained by flutter testing a reduced-scale model of a proposed design. The model should be dynamically and geometrically scaled, except for sweep angle. It has been found to be virtually impossible to obtain flutter in a low density strut model at small sweep angles prior to structural failure due to approaching divergence. Instead of testing the model at the small sweep angle usually found on full-scale struts, the model should be tested at several larger sweep angles, decreasing the angle until static deflections indicate proximity to divergence. Flutter speeds must then be extrapolated to the required value of sweep angle. Damping and frequency measurements for individual hydro- elastic modes of torsion-type struts have been readily obtained at NSRDC by implusive excitation. This technique involves inducing os- cillation of the strut at the desired frequency, and determining damp- ing and frequency from the resulting decaying oscillations. Excitation was obtained from a vibration generator rapidly swept over a narrow frequency interval including the desired resonance. The technique can be applied at small speed increments to permit a close approach to the flutter inception speed to be made safely. It has been found, how- ever, that at speeds above flutter inception struts often exhibit ampli- tude-limited flutter over a varying speed range before large negative damping leads to large amplitude oscillations. Because of the differ - ence in damping characteristics, amplitude-limited bending flutter occurs over a narrow speed range while amplitude-limited torsional flutter can occur over a wide speed range. This phenomenon probably resulted in the failure of Model 2T, pod configuration D, shown in Figure 3 at a speed far above flutter inception. Development of flutter testing techniques for full scale craft would permit verification of the stability of a given design. Such tech- niques should be evaluated in models and on existing craft. Future designs could provide for the flutter testing system to be installed in all craft during construction to make underway flutter testing routine for hydrofoil vessels. 365 Besch and Ltu VI. CONCLUSIONS Strut flutter occurs in two different hydroelastic modes, At low values of generalized mass ratio flutter occurs in a predominantly first bending mode shape with the qualitative characteristics of the "new mode'! previously described, At higher values of generalized mass ratio, flutter occurs in a predominantly first torsion mode shape with the qualitative characteristics ''mode 2' described in the text. The flutter mode of a strut can be determined by examining the mode shape of the second vibration mode of a strut in water, exceptina transition region where strong coupling interferes with this identifi- cation. Flutter speed predictions using the present analysis are ge- nerally inaccurate. In the bending flutter region, flutter is often pre- dicted to occur in the wrong mode so that flutter speed predictions cannot be used, In the torsional flutter region, the accuracy of flutter speed predictions is dependent on the value of torsional mass ratio. Predicted mode shapes and frequencies are nearly always accurate when three-dimensional hydrodynamic loading is used. Foils attached to a strut in am inverted-T configuration have a strong effect on the flutter characteristics of the strut. Further in- vestigation of foil effects is needed. NOTATION a nondimensional distance from midchord to elastic axis, measured perpendicular to elastic axis, positive aftas fraction of semichord b a. nondimensional distance from midchord to local aero- dynamic center (for steady flow) measured perpendicular to elastic axis, positive aft as a fraction of semichord b b semichord measured perpendicular to elastic axis [c] ’ damping matrix of strut ee | ' effective damping matrix of the strut-fluid system C(k) complex Theodorsen circulation function Cha local lift slope for a strip perpendicular to elastic axis in steady flow 366 EI GJ Flutter of Flexthle Hydrofotl Struts bending stiffness hydrodynamic force torsional stiffness structural damping coefficient ; also, gravitational ac- celeration amplitude of bending displacement h linear displacement of strut at elastic axis total mass moment of inertia of strut and tip attachments about elastic axis added moment of inertia of pod about elastic axis, ap- proximated by the added moment of inertia of a prolate spheroid. V1 stiffness matrix of strut effective stiffness matrix of the strut-fluid system strut length along elastic axis distance from free surface to tip of strut along elastic axis oscillatory moment about elastic axis per unit span of strut, positive in direction of positive 90 pod mass added mass of pod mass matrix of strut added mass matrix of strut effective mass matrix of the strut-fluid system mass per unit span along elastic axis S67 Besch and Ltu Pp oscillatory lift per unit span of strut along elastic axis, positive in direction of positive h p spanwise modification factor for noncirculatory loading Tie nondimensional radius of gyration s complex eigenvalue [: time V flow speed w downwash ; vertical component of flow velocity on foil, positive in direction of negative h =, nondimensional distance from elastic axis to center of gravity, measured perpendicular to elastic axis, posi- tive aft as fraction of semichord b y spanwise coordinate along elastic axis of strut i damping ratio, giving damping as a fraction of critical damping © amplitude of torsional displacement 9 8 torsional displacement of strut about elastic axis, posi- tive when leading edge moves in direction of positive h k sweep parameter ; (2b tan #3) Vee A na elastic-axis sweep angle, positive for sweepback Hendin approximation to generalized mass ratio for bending e motion Me : generalized mass ratio generalized mM ; approximation to generalized mass ratio for torsional torsion : motion p fluid density o local bending slope of elastic axis dh/dy 368 Flutter of Flextble Hydrofotl Struts t local rate of change of twist along elastic axis 00/dy w circular frequency of oscillation Wo circular frequency of first torsional vibration in air SUBSCRIPTS i subscript to indicate that the parameter is associated with ith strip station on strut n subscript to indicate that the parameter is perpendicular to elastic axis SUPERSCRIPT (Voie) dot over a quantity indicates differentiation with respect to time 369 Besch and Ltu Re Free cece Foil Pod Figure 1. Typical strut-pod-foil system FIRST FIRST BENDING TORSION H @ y y SECOND SECOND BENDING TORSION H i) y » f Figure 2. Uncoupled mode shapes of cantilever beam 370 Ue in Knots fe in Hz Flutter of Flextble Hydrofotl Struts Experiment Ref. 4 O NSRDC (Low Damping - <4 Not Flutter) D (Model Failure During Flutter) Letters give pod designation from Ref. & Numbers give pod weight in Ib Frequencies Predicted at Experimental Flutter Speeds are Shown Pod Center of Gravity Location in In. Aft of Strut Elastic Axis Figure 3, Effect of pod mass and center of gravity location on flutter speed Un and flutter frequency fo 371 Besch and Ltu Tapered Struts, 0 Ref. L.E. Cavitation 9.9533 110 Q Ref. OD Ref. 100 © Ref. O NSROC BaRS acl q) dete < | x = 0.063 (12 In. Chord) in Knots > .08 Numerical Values are for Sweep Parameter Bending Mass Ratio Figure 4, Experimental flutter speed Up as a function of bending mass ratio for bending-type struts 342 Flutter of Flextble Hydrofotl Struts O Ref. 1 O NSRDC Lines Connect Data Obtained from the Same Strut Torsional Mass Ratio Figure 5, Experimental flutter speed Up asa function of torsional mass ratio for torsion-type struts ie) Besch and Ltu Strut Profile NACA = NACA 16-005 B.B. Torsional Mass Ratio Figure 6. Reduced flutter speed Up/b wq as a function of torsional mass ratio for torsion-type struts 374 Flutter of Flextble Hydrofotl Struts 0 0 20 40 60 80 100 Submergeiice in Percent Span Figure 7b. Strut-pod models (torsion-type) Ur in Knots Pa ° Cc x ait < 20 0 20 TY) 60 80 100 a8 Submergence in Percent Span > Figure 7a. ; Strut models (bending -type) 6 | 2b) wo 360° Bor set0o Submergence in Percent Span Figure 7c. Strut with pod and foils Note : Lines connect data obtained from the same strut Comparison of the effect of strut submergence on flutter speed U, for strut models with and without pods and foils J5e eq bh aioe 3@5 Frequency in Hz Besch and Ltu Experiment Speed @ Ref. 4 81 Knots Theory —*—Two-Dim. 83 Knots Figure 8a. Damping ratio ¢ Speed in Knots Figure 8b. Frequency Hydroelastic mode characteristics for Model 2 Figure 8. (Two-dimensional loading calculation) 376 Flutter of Flextble Hydrofotl Struts bare Ni aes ddl Sanne es PNeT at l/ I ' - Speed in Knots Flutter © ie | Experiment Speed _— = @ Ref. 4 81 Knots Theory (3-Dim) -0.8} -—o—Mode 3 0 Knots —o— New Mode 147 Knots ete | i ee ce las Figure 9a. Damping ratior Frequency in Hz Speed in Knots Figure 9b. Frequency Figure 9. Hydroelastic mode characteristics for Model 2 (Three-dimensional loading calculation) Bet Besch and Ltu 100 80, Pe o 60 i wd e 40 => 20 3-Dim. Value 0 0.6 0.8 1.0 Aerodynamic Center in Noncirculatory Modification Percent Chord Aft of L.E. Factor p 140 120 100 " ) ° c Z = u 60 0 20 4o 60 80 100 Lift Slope in Percent of 27 Figure 10. Effect of loading modifications on calculated flutter speed Up for Model 2 378 Flutter of Flextble Hydrofotl Struts Flutter Experiment Speed @ Ref. 4 18.4 Knots NSRDC 0 Mode 1 None 18.1 Knots 14.1 Knots 18.8 Knots fem ee = a a [a ae en ee lies Fpaet fen < pperievehttst | | | esis. aes YY) 2 Do Pest ro bull -) ef ARE os \1 as in Knots Figure 1b. Prequency (o) Frequency in Hz Figure 11. Hydroelastic mode characteristics for Model 2T Besch and Ltu Aerodynamic Center in Noncirculatory Modification Percent Chord Aft of L.E. Factor p Ur in Knots 0 ; 20 MY) 60 80 100 Lift Slope in Percent of 2n Figure 12. Effect of loading modifications on calculated flutter speed Up for Model 2T 380 Figure 13. Flutter of Flexible Hydrofotl Struts foe it, ist co W ct AIM t=t + 4at At = 0.02 Sec Calculated transient response for Model 2T at flutter 381 Besch and Liu 60 = — 40 x a 20 x 2 i , ae ag aS 3 2 3 4 5 6 |- oO Mtorsion 2 2 i > Figure 14, Comparison of experimental and theoretical flutter speeds for torsion-type struts without foils 382 Flutter of Flextble Hydrofotl Struts Mode 3 36.2 Hz Figure 15a. Strut Mode 2 10.9 Hz Mode 3 15.4 Hz Figure 15b. Strut with pod Figure 15c. Strut with pod and foils Figure 15. Calculated nodal lines for strut of Reference 2 at 83 percent submergence in water 383 Besch and Ltu uw ~ {2} Cc Ba c wv o v a 2) Flutter Speed -0.21 Experiment 18.1 Knots ( 16.6 Knots @ Ref. 2 Theory Dim. 5.8 Knots —o— Two- Damping ratio § Figure l6éa. ZH ul! Aduanbes4 A H Speed in Knots Figure 16b. Frequency Figure 16. Hydrostatic mode characteristics for strut of Reference 2 with pod and foils (Two-dimensional loading ) calculation 384 Flutter of Flextble Hydrofotl Struts Ok 0. g 0. 0 ‘ Na | Speed in Knots -0. \ Flutter is Experiment Speed “hy. 18.1 Knots (-2°) 16.6 Knots (-4°) -0. im. 6.2 Knots -0. -1.0 N Se = ~ 1S} c ev =} oo 0 ‘= we Speed in Knots Picure lib. “Py requency Figure 17. Hydroelastic mode characteristics for strut of Reference 2 with pod and foils (Three-dimensional loading calculation 385 Besch and Ltu APPENDIX DESCRIPTION OF STRUT MODELS MODEL 2 Model 2 was a blunt-based strut [4] constructed of solid steel. Its dimensions are shown in Figure 18, and its structural characteris- tics are summarized in Table 1. Spanwise distributions of lift slope, aerodynamic center, and noncirculatory modification factor are shown in Figure 19: MODEL 2T Model 2 T was constructed to the same specifications as Model 2, except that a 2-inch diameter pod was welded to its tip [4] ‘ Pod dimensions are shown in Figure 20. Weights were placed in the body of the pod to produce different inertial configurations. STRUT-FOIL MODEL The strut [2] consisted of a solid bar of copper alloy covered with flexible silicone rubber. The leading edge of the strut was coated with a thin layer of plastic which was slit to reduce its stiffening effect The strut profile closely resembled that of Models 2 and 2T, except for a rounded leading edge. Strut dimensions are shown in Figure 21, Structural characteristics are given in Table 2, The three-dimension- al loading curves for this strut were essentially the same as those for Model 2, differing only in sweep angle. The pod and foils were machined from solid aluminium and were attached to the strut with bolts. Dimensions for these compo- nents are given in Figure 22. 386 Flutter of Flextble Hydrofotl Struts CIRCULAR ARC FROM LEADING EDGE TO MIDCHORD AND CONSTANT THICKNESS FROM MIDCHORD TO TRAILING EDGE 12.0 = aA SECT. A-A DIMENSIONS — INCHES Figure 18. Model 2 dimensions 387 Besch and Ltu 40 uw o 30 ~ ce eo =) oe 20 ee ee ge E x nee) Free Surface vo 9 oOo & oe <5 “are 0 is 2 fs >: Ris Nek ive a WEE SE REN Be BoP es bh bale vrkition? 2 Hid? Gonodee oe oe pea a Ee Gy s Presi J) Bing ered Ue F.42. loratwog edd titerest < Chavwitaey $0 dears 24r, Siro wel a Re i ree. tee’ « 2 ‘ t 7 y . j re at P » * ‘ = mf ; ry 2A “ msn iin of ie ; oan td % » * or 1% a ¢ i) NOWRUO = Ot 1 art 1 LORs * #4 a ag , ot TAOS COrT eh pont 283 vine Vig ee rie { i : “ © ee: f rgaevond odes’ eA i? Me ake eet) . g° 4 ri b — _ mverm Mac Gi» Pe te r] ‘ n } 8 Mei 4 a BT ay BPE ON Ferd aq PAY Re TRB P Ghd Fi Muro iel rE D ; Py weed HY Ss to leo tfhde SRR? Tos Rete Pact: HO pets = ere eet reed , a : x 5 4 “EE! Heey Bit 4a red Chey G: Sirs i FF Di LiAte | a rear) ek teil ' eats ‘ ; : + * 19 la bee 7 ? ee rethee Ht ni SLED" ile iogeev. daagt4! of ef 1h: tem) both a ra] r ' , a.) & ‘ a a we i } ta r : ‘ x 71e wie © c nirrty o CRC UY; % I YAR ’ \<¢ . Gens ? | ma teke Pleee onby wher 2 o caed li f iit 4 my y 7 , : : “oe ey Dat aR Tip od CaN ; Peat a Sk fg ein TP Cle onthe WO Rye es om ge ir Roce Pel eh ae ee rey . ae ; ie Come tude cae Vi on aN PARR Vek Che cried: ae eV AROS ah Mesign af hydrelallip, ge Ct con side raty ign ete tad ae ie ye tion of she Meydaodetberd:. Comolex io canee hn « . ue ite actGune.. Information on one such twas eatodl: - heh wae, p : -' , ey ‘ie Kt ee j ihe » ; HYDRODYNAMIC DEVELOPMENT OF A HIGH SPEED PLANING HULL FOR ROUGH WATER Daniel Savitsky Stevens Instttute of Technology Hoboken, New Jersey, U.S.A. John K. Roper Atlantte Hydrofotls, Ine. Hancock UN Hi Uo A. Lawrence Benen Naval Shtp Systems Command U.S. Navy, U.S.A. ABSTRACT The hydrodynamic development of a new planing craft intended for sustained high-speed operation in a seaway is discussed. The design philosophy is presented andthen implemented toachieve optimum hull form and loading for both smooth and rough water operation of the craft. The resultant hullform is a high length-beam ratio, highly loaded, double chine configuration which provides greatly improv- ed seakeeping, high speed and high maneuverabili- ty. Extensive model tests were conducted to predict the SHP ; EHP; seakeeping ; course-keeping sta- bility ; and turning characteristics of the design. The extensiveness of the model test program and data analysis are unique for a planing craft. These results are presented in a form which should be of general interest to the designers of a high-speed planing craft. INTRODUCTION Although world navies are traditionally considered to be high sea state fleets with ocean-spanning capabilities, the small, high- AYV9 Savttsky, Roper, and Benen speed craft is an essential complement to this fleet. These small craft are called into service to assist in ASW operations ; to patrol in coastal and riverine environments ; and to act in concert with larger naval units. Although ubiquitous in numbers, small craft are not in- dividually a high portion of total naval capability, cost, or personnel. In fact, on an equal cost basis, it appears that more units can be pro- cured which can cover more areas than can larger ships. These fea- tures have made the small, high-speed craft attractive as the princi- pal naval force for many small countries with large coastlines. There are basically four different types of small, high-speed craft, i.e., round-bottom boats, hard-chine planing craft, hydrofoil boats, and various forms of air-supported vehicles. The most nu- merous of these craft is by far the hard-chine planing hull-especially when considering speed-length ratios in excess of approximately 2.0 where dynamic lifting forces are significant. Because they are equip- ped with large power, lightweight engines, it is not uncommon for planing craft to operate at speed-length ratios in excess of 5.0. Further, it is also not uncommon for these craft to operate sufficient- ly removed from the coastline so that moderate to high sea states are their normal environment. Thus, the small boat designer is faced with the formidable task of producing craft whose high speed potential is not seriously compromised in rough seas, The purpose of the present paper is to describe the hydro- dynamic development of a new planing craft intended for sustained high-speed operation in a seaway. The design philosophy is presented and then implemented to achieve optimum hull form and loading for both smooth and rough water operation of the craft. The resultant hull form is a high length-beam ratio (6.5), highly loaded (beam loading = 0.75), moderate deadrise ( B = 20°), double chine configuration which provides good seakeeping, high speed, and large maneuverabi- lity. Extensive model tests were conducted to predict the SHP ; EHP ; seakeeping ; course-kKeeping stability ; and turning characteristics of the design. The extensiveness of the model test program and data ana- lysis are unique for a planing craft. The present paper presents these results in a form which should be of general interest to designers of high-speed planing hulls. POSTULATED PERFORMANCE OBJECTIVES The design of any marine craft is based upon specifications which have been prescribed to achieve desired performance objecti- ves, Among the more significant requirements which have a pronounc- ed influence on hull form are : operational speeds ; dimensions and 420 High-Speed Planing Hull for Rough Water weight of payload components ; sea state; tolerable ''g'' loadings in that sea state ; restrictions on draft, length, and beam, metacentric stability ; and maneuvering qualities. Certainly, special purpose craft have additional restrictions but these are not included in this study. For the present paper, the following set of performance spe- cifications are prescribed. 1. Full Load Displacement: 150,000 lbs. A designer's experience is invaluable in making the first engineering estimate of full load displacement. Nevertheless, design studies and possible model tests are carried out at additional displa- cements of approximately 80% and 120% of the initial estimate. 2. Maximum Speed: In excess of 40 knots, Interception and attack missions require high speeds. For the present study, a nominal speed of 45 knots is assumed. 3. Cruise Speed : Approximately 12 knots. A patrol mission requires long endurance at slow search speeds, A cruise speed of 12 knots is selected. PViaxihniuim nll irate 295.5) 2. 5. Operational Sea State: 3. It is desired that the craft operate at 45 knots ina state 3 head sea having a significant wave height of 4.6 ft. 6. Average Center-of-Gravity Impact Acceleration : 0. 4g “cglavg i It is specified that the average center-of-gravity impact ac- celeration should not exceed 0.4g while running ina state 3 head sea at 45 knots. (. ‘Metacentric Stability: GM =3.0 ft. AGM of 3.0 ft. was selected to provide metacentric sta- bility under conditions of high wind and severe super-structure icing. The above set of requirements do not pre-specify the length or beam of the craft. There may, of course, be operations where 421 Savttsky, Roper, and Benen either of these dimensions must be fixed in advance. The design pro- cedure developed in subsequent sections of this paper can be equally applied to those cases and will show the extent to which the pre-spe- cified length, for example, may inhibit attainment of the performance requirements, DESIGN PROCEDURES FOR BASIC HULL DIMENSIONS In this section of the paper a methodology is developed for rationally selecting the length, beam, longitudinal center-of-gravity, nominal deadrise, and effective horsepower of the basic hull form which will satisfy the performance requirements previously specified. No attempt is made to optimize the hull design to attain, say, mini- mum resistance while satisfying the seakeeping requirements. This can be developed as a subsequent study using the basic design proce- dures developed herein. The design procedure is primarily based upon a combina- tion of smooth water prediction techniques such as given in References 1 and 2, and rough water prediction techniques such as given in Reference 3 . While both studies are concerned with prismatic planing hulls (constant beam, constant deadrise, buttocks parallel to the keel) these techniques have been successfully applied to actual hull forms by proper selection of an effective: constant deadrise and beam, Separate considerations are first given to relating hull di- mensions to the following hydrodynamic characteristics 1, Hydrodynamic Impact in a Seaway 2. Hydrostatic Displacement 3. Smooth Water Planing (High Speed) 4, Smooth Water Operation (Low Speed) 5. Metacentric Stability The results of these elemental studies are then combined to ee ies a hull form, overall dimensions, center of gravity, and ef- fective horse-power to achieve the operational objectives. 422 Htgh-Speed Plantng Hull for Rough Water Hydrodynamic Impact in a Seaway In Reference 3 , Fridsma presents the results of a syste- matic study of the effects of deadrise, trim, loading, length-beam ratio, bow section shape, speed, and sea state on the performance of a series of prismatic planing models operating in irregular waves. A statistical analysis of the measurements of added resistance, heave motions, pitch motions and impact accelerations resulted in quanti- tative relations between these measured quantities and hull dimensions and operating conditions, The results of those parametric studies are summarized in design charts which enable full-scale performance predictions for planing craft. Using the procedures described by Fridsma, the average center of gravity acceleration is computed for a range of combinations of beam, length-beam ratio, deadrise and trim angle for a maximum speed of 45 knots ina state 3 head sea. To enter the design charts of Reference 3 , the following coefficients are evaluated : on = Beam loading = —“, wB Vay = Speed-length ratio L/By = Length-beam ratio Hy 7/8 = Significant wave height/beam = H) /3/B C. = Speed coefficient = V/WgB where A = displacement = 150,000 lbs. Wye = maximum speed = 45 knots V = maximum speed = 76 ft/sec. L = length between perpendiculars, ft. B = average beam over aft 80% of hull, ft. Ay = significant wave in state 3 sea = 4. 6 ft. w = weight density of water = 64 lbs/ft? 423 Savttsky, Roper, and Benen p = mass density of water = 2 ineaee vat and 6 = deadrise angle at station 5, degrees T = trim angle of mean buttock line, degrees — + 1 4 Woll Gretel average vertical acceleration at center of gravity ''g (16g) /1o 1/10 highest vertical acceleration at center of gravity "o'!| From the statistical analysis of Reference 3 , (toghijio = 3:3 (Glave The previous coefficients are evaluated for a range of initially assum- ed values of beam : The following relations between initially assumed beam, as- sumed length-beam ratio, and speed-length ratio will also prove use- ful : 56 ft, Referring to Figures 16, 17, and 18 of Reference 3 , which are reproduced as Figures 1, 2, and 3 of the present report, itis seen that the average value of center-of-gravity acceleration is obtained 424 Htgh-Speed Plantng Hull for Rough Water directly from these dopiee charts for arbitrary combinations of L/B, ie. , V,/VL, and H ‘3 /B such as listed above. It is to be not- ed that thane results are for a trim angle of 4° anda deadrise angle of 20°. Corrections for other combinations of trim and deadrise angle will be described subsequently. For each assumed value of beam, the average CG accele- rations for 45 knots ina state 3 head sea is obtained from the de- sign charts of Figures 1-3 and plotted on the right half of Figure 4 as a function of length-beam ratio, These results are obtained by extrapolations of the design charts as suggested in Reference 3 . The ordinate of the plotin Figure 4 is the quantity : KOE (Average) | x | + ; | which defines the dependence of acceleration upon trim and deadrise as developed in Reference 3. For 7+ = 4° and 6 = 20°, the quan- tity in the square brackets is unity so that 1 cc(Average) as given in Figure 4, corresponds to the design charts of Reference 3 . Super- posed on Figure 4 are curves of constant boat length for various combinations of beam and length-beam ratio. The quantity 17°/4 (5/3 - B8°/30) is plotted on the left half of Figure 4 for ease of applying these results to arbitrary combi- nations of 7 and 8B Some interesting observations can be made by an examina- tion of the results in Figure 4 Effect of Beam on Hydrodynamic Impact : All other conditions being equal, a reduction of hull beam leads to significant reductions in impact load. For example, a 28% reduction in beam (from 18 ft. to 13 ft.) resultsina 69% reduc- tion in impact accelerations (from 1.10g to 0.35g). Evena 1 ft. reduction in beam, from 15 ft. to 14 ft., decreases the impact ac- celeration by approximately 29%. This powerful effect of beam results from the large dependence of impact upon the inverse of beam loading coefficient Ca, = 4/wB_ . The effect of Ca has long been familiar to the designer of water-based aircraft (References 4 and 5 ) and has just recently been quantitatively identified by Fridsma? for the case of the planing hull. 425 Savttsky, Roper, and Benen Effect of Trim Angle on Hydrodynamic Impact : The hydrodynamic impact load is linearly dependent upon trim angle so that, within the range of data acquisition, a 50% reduction in trim angle (from 4° to 2°) results ina 50% reduction in hydro- dynamic impact load. The reference trim angle is the smooth water running trim of the craft for the considered hull dimensions, loadings, and speed. Effect of Deadrise Angle on Hydrodynamic Impact : The accelerations decrease linearly with increasing deadrise so that a 50% reduction is achieved by increasing the deadrise from LO ‘to * 50". In order of importance, then, impact loads in a seaway can be reduced by providing a hull having a narrow beam, a low running trim angle, and a high deadrise. As will be subsequently developed, this results in as long and narrow a hull as can be accepted without seriously compromising other essential operational conditions. Relation Between Beam and Trim Angle to Achieve ("cG)aveg = 0.4g: Considering an initial deadrise angle of 20°, the relations between beam and trim angle to achieve ( 7 cc) eo 0.4g can be established. The following tabulation follows from erigure 4. (1) Average value for range of L/B as given in Figure 4. 426 High-Speed Planing Hull for Rough Water . 40 ree ¢ a ; Peay hs tae oe ale , = Py) (Gq) avg ( : oe) (3) 7 = equilibrium trim angle to achieve (ree? ae = 0.40 g¢g for Be 20° Vie = 4ae PAL 150,000 lb. Static Displacement Considerations ee EE Een SICeEALlon's )) The block coefficient for planing hulls can vary between 0.40 and 0.50. Having fixed a draft of 3.5 ft. for the present design, the relations between length and beam will take on a more limited set of values than those previously developed from considerations of only hull impact. A relaiond 7 BG Ww where A =" 505,000" Ths: 14 = length between perpendiculars, ft. d = maximum dratt, 3.5 ft. w = weight density of water, 64 ieee 427 Savittsky, Roper and Benen These combinations of length and beam are superposed on the plots of Figure 4. This results in a substantial reduction in length- beam combinations for further study. Smooth Watcr Planing (45 knots) A computational procedure for predicting the smooth water equilibrium conditions of a planing hull is given by Savitsky! . this procedure has been programmed for high-speed computers and is ge- nerally available to the small boat naval architect. In Reference 2 , Hadler extends this work to include the effects of propellers and ap- pendages. Unfortunately, a computer program for this extended com- putation is not yet generally available to the small boat naval architect. For the present study, which is intended to define the prin- cipal hull dimensions for preliminary design, the simplified computa- tional procedure developed by Savitsky is considered adequate. A 20° deadrise hull is initially assumed and the equilibrium trim and wetted keel length are computed for values of beam between 13 ft. and 18 ft; for longitudinal center-of-gravity positions between 22 ft. and 44 ft, forward of the transom, for a planing speed of 45 knots anda dis- placement of 150,000 lbs. The results of this computation are plot- ted in Figure 5. For each value of beam, the trim angle required to achieve (cc) awe = 0.4 is also indicated. Using the results of Figure 5, the following relation between beam, LCG, wetted keel length and suggested L (load waterline length) is obtained. These values are also plotted in Figure 6. 428 High-Speed Plantng Hull for Rough Water 106 7A The length L is load waterline length and is taken to be equal to 1.10 Ly. It has been found that this relation between L and I, is most satisfactory. If L is less than 10% greater than Ly, , there is substantial bow immersion at high speed resulting in a significant increase in smooth water resistance. When L is much larger than 1.10 lL, , the excessive hull length forward of Ly pro- vides additional impact area when running in a seaway with resultant increasing impact loads. Also included in the previous tabulation are the correspond- ing values of block coefficient C, and slenderness ratio L/vi3 These will be subsequently discussed. Smooth Water Operation (Low Speeds) Since one of the operational objectives for the craft is to cruise at 12 knots for extended periods, the hydrodynamic resist- ance should be minimized. Fora displacement of 150,000 lbs., the volume Froude number V 20.4 g Bin Nee / S ji 1y/ eV / 32.2 =(2350).~ From a basic study of the resistance of planing hulls in the prehump speed region (Reference 6 ), it has been found that low speed resistance is primarily dependent upon slenderness ratio ay ale Figure 7 of this report shows the variation of resistance- weight ratio of the Series 62 hull form (Reference 7 ) asa function of L/yI3 for F =1.0. It is seen that, although the data points represent a wide range of hull loadings and length-beam ratios, they 429 Savttsky, Roper, and Benen are closely represented by a single curve relating the data to L/v'B. Similar plots for other hull series are given in Reference 6 and also indicate the pronounced dependence of low speed resistance on slen- derness ratio. Using these results as a guide, it is concluded that a value of L/vs_ equal to at least 7.0 is essential for low resist- ance at pre-hump speeds, For a displacement of 150,000 1bs., this results in a minimum hull length of 92 ft. Comparing this criteria with possible combinations of hull dimensions of page 429, it is seen that a hull beam should be at least equal to or greater than 14 ft. This limitation is also plotted on Figure 6. Metacentric Stability The specified requirement for metacentric stability is that the GM be equal to at least 3.0 ft, An engineering estimate of GM for a planing craft with deadrise is made using the following assump- tions. KB =a = — (3.5) = 289 Ft. I = water plane moment of inertia = .55(—| LB ) Preliminary weight estimates for the present design showed that the vertical center of gravity of the craft, KG, is 6.2 ft. above the Keel. Considering a minimum boat lenght of 92 ft., as determined from the low speed requirement, the following values of GM are comput- ed for each assumed beam, 430 High-Speed Planing Hull for Rough Water Oe Oe. 11,400 14, 000 16,410 24,200 where KG misty hele Cara ae BM = 1/V 3 V = 1502 000/04 — 2,340) tt KM = KB + BM GM = KM - KG A minimum beam of 15.7 ft. is required to attain the spe- cified metacentric height of 3.0 ft. Selection of Basic Hull Dimensions The previous elemental hull studies developed the following restrictions on hull dimensions to satisfy the postulated operating con- ditions. Low Speed Operation (V,_ = 12 knots) For low resistance in the cruise condition, the hull length should be equal to or greater than 92 ft. Metacentric Stability (GM = 3.0 ft.) For a hull length of 92 ft., the required hull beam is WSy Cees 431 Savttsky, Roper, and Benen Hydrodynamic Impact ( 1G) ae = 0.4 g in state 3 head sea) Two possible options are available for a 20° deadrise hull. = ea Tear ROFL EG he: (1) = Vas), 5 ae = "h.36 B =e ae L SU es (2) IkOGi ss = SB ft C = eee Option (1) satisfies all the hydrodynamic and operational requirements while Option (2) does not satisfy the basic metacentric height requirement. The 116 ft. boat lenght of Option (1) was con- sidered excessive when compared with space requirements for internal arrangements, This excessive length would increase the cost of the boat without materially improving its performance. As a compromise design, a double chine hull having a length of 92 ft. was selected. A mid-length section through this hull is shown in Figure 8. The inner chine beamis 14 ft. and the outer chine beamis 15.7 ft. In the static or low speed condition, the wetted beamis 15.7 ft. so that the metacentric stability requirement is sa- tisfied by the combination of L = 92 ft, and b= 15.7 ft, Atthe 45- knot condition, the flow separates from the lower chine so that the 92 ft. long hull has an effective beam of 14 ft. This satisfies the high-speed impact requirements, The metacentric stability of the craft is now satisfied by considering the additional roll stabilizing mo- ments developed by the dynamic loads generated at high speeds. In summary then, the following basic hull dimensions and loadings are selected for preliminary design L = 02) ts 432 Htgh-Speed:Plantng Hull for Rough Water me aes outer : = 14 ft. inner LCG SS ae B = aie (Sta. 5) A = eel EUS OOO elllosy, At a smooth water planing speed of 45 knots, this craftis expected to run ata trim angleof 3.3° and develop (1Gq) ave 0.4 ¢g ina State 3 head sea. It will be noted that a nominal 20° deadrise hull has been selected. Considerations were also given to deadrise angles of 10° and 30°, The details of these calculations (which follow the previous procedures) will not be presented, but the results will be summariz- ed, Figure 9 presents a plot of resistance-weight ratio (R/A ) versus trim angle for deadrise angles of 10° , 20°, and 30°. These results follow from Reference 1 . Superposed on Figure 9 are the maximum trim angles for combinations of deadrise and beam which will result in (1cGG) ayg = 0-4 g When the 150,000 1b. craft runs at 45 knots ina State 3 head sea. Considering the 30° deadrise case, itis seen thata 14 ft. beam requires a trim angle equal to or less than 5° to satisfy the impact requirements. Further, the minimum drag-lift ratio for a 30° deadrise craft occurs at a trim angle of approximately 5°. This high trim angle could impair visibility during high-speed operation and was thus not considered acceptable. A reduction in trim angle to improve visibility would reduce the impact loads below the maximum acceptable value but, according to Figure 9, would significantly in- crease the drag-lift ratio to values substantially larger than for the 20° deadrise surface. Thus, the 30° deadrise case was not further considered, A 10° deadrise surface having a beam of 14 ft. would be required to run ata trim angle of 2.4° soas not to exceed the de- sign impact acceleration. For this case, the drag-lift ratio would be slightly less than that for the 20° deadrise surface. The boat length and 5CGG, jpositionyfor 8 = NOC ) Te=w25 4 ee bel l4ecft, pA 150,000 lbs. and V, = 45 knots are computed from the monograph given in Figure 9 of Reterence.; 1): 433 Savttsky, Roper, and Benen L = 69 ft. LCG =) Sa CBE: L = 1.10xL =; Gy Et. The boat length of 76 ft. is 20% less thanthe 92 ft. length required for low resistance at 12 knots and is thus not acceptable. In summary then, the 20° deadrise hull was accepted as best meeting the design requirements of the craft. Description of Final Hull Form The lines are shown in Figure 10 and show that the princip- al dimensions of the craft are in substantial agreement with those de- veloped by the design procedures presented herein. Length, design waterline 92'-0"' Beam, lower chine (nominal) 14'-0"' Beam, upper chine (nominal) 15'=7" Deadrise, station 5 20° Deadrise, station 10 10° Draft (full load) 3.5 It is seen that the craft is a hard, double-chine hull whose high-length beam ratio is favorable for low resistance and good sea- keeping. Several detail design features are of interest and will be se- parately described. Koelbel (Reference 8 ) provides excellent design guidance in this regard. Chine Configuration It has been found that, for planing craft operating at speed- length ratios greater than approximately 2.0 - 2.5, a hard chine is required to assure complete separation of the flow from the bottom. At these speed-length ratios, a round bilge hull will prevent flow se- paration and result in significant side wetting and thus increase the hydrodynamic drag. The present 45-knot design condition corres- ponds to a speed-length ratio of 4.6 and clearly requires a hard chine 434 tgh-Speed Planting Hull for Rough Water configuration, In designing a double chine configuration, it is important that the upper chine location be within the cavity found by the boundary of the flow separation from the lower chine. As shown by Korvin- Kroukovsky (Reference 9 ), the trajectory of the free streamline re- presenting the cavity of the boundary is a function of deadrise angle. Figure 11 of the present report, which is taken from Reference 9, plots the separation trajectory for various deadrise angles, It is seen that the width of the separation cavity increases with decreasing dead- rise angle. If the upper chine is located outboard of this cavity bound- ary, the originally separated flow from the lower chine will reattach to the bottom somewhere between both chines and thus preclude com- plete flow separation from the lower chine. In the present design, the outer deadrise angle is 45° and the outer chine is approximately 0. 80 ft. outboard of the inner chine. This is sufficient to clear the lower chine trajectory at station 5 and result in complete separation from the lower chine, Observations of the wetted bottom areas during model tests confirmed this prediction. Section Shapes The section shapes are slightly convex. This section pounds less than others of equal deadrise because there is less likelihood of instantaneous water contact over large bottom areas, Planform Shape At high planing speeds, when dynamic lift predominates, itis usual to narrow the beam towards the stern. This reduces bottom friction without a noticeable loss in lift. The narrow transom also avoids the possibility of reattachment of the separation cavity formed in the region of maximum beam, For the present design, the transom width was determined by considerations of space requirements for the auxiliary machinery in the stern area. This resulted ina slight reduction of beam towards the stern which, in the model tests, was found to be sufficient to avoid flow reattachment. Bottom Warp The increase in deadrise with length forward of the transom is referred to as bottom warp and is required to provide a relatively high deadrise in the bow regions. Brown (Reference 10) has shown that there is a slight reduction in planing efficiency for moderate va- lues of warp. The slightly convex bottom sections, from keel to lower chine, used in the planing area aft of the high-speed stagnation line 435 Savttsky, Roper, and Benen are easily warped to result in increased deadrise and curvature in the bow sections, This combination reduces pounding and impact pres- sures ina seaway. The transverse section shape above the lower chine is increasingly more concave as the bow is approached, This upper ''flare'' is desirable to deflect the bow spray outboard of the deck and to provide additional buoyancy to reduce low-speed pitching in a seaway. Spray Rails Spray rails are provided along both chines to assure flow separation from the chines, The spray rail for the upper chine must not extend into the separated flow cavity formed by the lower chine. Otherwise flow reattachment will occur at high speeds. Separation from the upper chine occurs at a speed-length ratio between 2.0 to 2.5 while separation from the lower chine is expected to occur ata speed-length ratio of approximately 3.0. Final Design An artist's conception of the final design is given in Figures la and 13, MODEL TESTS Model tests were conducted at the Davidson Laboratory, Stevens Institute of Technology to evaluate the performance of the craft in smooth water and waves. A 1/11-scale model was used to determine EHP and SHP. A 1/16-scale model was used to inves- tigate the seakeeping, maneuvering, and turning ability of the craft. Some of the principal results and test procedures are presented here- in, Resistance and Propulsion Smooth Water Resistance A 1/11-scale model was constructed according to the lines of Figure 10. To assure flow separation from the bottom, the upper and lower chines of the model were sharpened by the addition of mylar plastic strips which projected vertically a distance of 1/32 of an inch below each chine. Tests were made for a range of loadings and speeds. The test procedure simulated towing the model through the shaft axis. Measurements were made of the heave, trim, drag and wetted areas. For the purpose of the present paper, Figure 14 presents a comparison between values of trim and drag computed by 436 High-Speed Plantng Hull for Rough Water the procedures of Reference 1 and the results of model tests, The comparison is for a displacement of 150,000 lbs. with an LCG of 38 ft. In the computational procedure, the upper chine beam (15.7 ft. ) is used for speeds up to 40 knots and the lower chine beam (14 ft. ) is used for higher speeds. This was consistent with test results where complete flow separation from the lower chine was observed at speeds greater than approximately 40 knots (full-scale equivalent). An ef- fective deadrise angle of 20° was used in the computations, It is seen that the computed and measured results agree well enough to justify use of Reference 1 for engineering estimates of planing boat performance. At speeds below 20 knots, extensive bow immersion precluded application of the methods of Reference 1 which are restricted to prismatic-like planing hulls. Reference 6 will provide procedures for performance estimates at low speeds where bow immersion is significant. It is interesting to note the complete absence of a "hump'' trim in Figure 14, This is attributed to the high-length beam ratio hull which, for normal LCG positions, is constrained to run at low trim angles. The low trim is, of course, most beneficial to improv- ed seakeeping. Self-Propelled Tests Self-propelled tests of the 1/11-scale model were carried out to determine propulsion characteristics, e.g. wake fraction, thrust deduction coefficient, relative rotative efficiency and, sub- sequently, predictions of delivered horsepower. The test program included resistance tests of the partly appended model, open water tests of the stock propellers used in pro- pulsion tests and self-propelled tests of the 1/11-scale model for overload and underload conditions (so-called 'British'' method) ata number of speeds and displacement conditions. The open-water tests were carried out with the shaft horizontal and with a shaft inclination of 12°. Self-propelled tests were made with all three propellers driving and instrumented. The rudders were not fitted for these tests since they are located approximately 4 propeller diameters aft of the propellers, out-of-line with any of the propeller races and, consequently, could have little influence on propeller-hull interaction. Three propeller dynamometers were installed in the model 437 Savittsky, Roper, and Benen for measuring thrust, torque and RPM. These were "reaction" type dynamometers having capacities of 101b. thrust, 5 in-lb. torque, RPM upto 10,000 and 0.50 HP. The averaging of the force and motion signals, as well as additional data processing, was accomplish- ed using a PDP-8E computer on line. The computer has a built-in analog-to-digital converter and is programmed to carry out operations such as signal averaging, correcting for zero levels, and multiplying by calibration factors to obtain results in engineering units. Davidson Laboratory uses the overload and underload testing procedure where a group of test runs are carried out at fixed speed with various rates of propeller rotation. This type of test provides information which may be applied for any desired assumptions concern- ing appendage drag, roughness allowance, scale ratio, air drag or rough water-drag increment. Some typical propeller-hull interaction factors, derived from the test data for the mcdel self-propulsion point (towing force = 0), are given for a speed corresponding to 45 knots and a displacement of 150,000 lbs. I = we = 0.99 Wake fractions = l-W a. Pine Q Relative rotative efficiency = etinws Q/2, = 0. 92 Thrust deduction os ed GE) eee NS ee These values of wake fraction, relative rotative efficiency and thrust deduction are used to select the particular propeller design which absorbs the installed power at the proper RPM and speed and has good efficiency even while operating under cavitating conditions. It is interesting to note that the thrust wake fractionis 0. 99 indicating an essentially undisturbed flow to the propeller. The thrust deduction is small, 1-t = 0.94, indicating a small effectof propeller- induced flow on the hull resistance, Rough Water Tests The rough water performance was measured for several loads and LCG positions in a variety of sea states. Measured quan- tities included heave and pitch motions, vertical accelerations at the bow and CG, and mean resistance in waves. During each test run, the 438 Htgh-Speed Plantng Hull for Rough Water data were processed by a PDP-8E computer on line, Each channel of data was analyzed at the rate of 200 scans per second and, at the conclusion of each test run, an ordered listing of the peaks and troughs of the pitch and heave motions and the accelerations at the bow and CG were printed out in addition to statistics suchas 1/10, 1/3, and average values. This instantaneous output of processed data was extremely useful in interpreting the results, A comparison between the computed average CG accelerat- ion and the results of model tests is tabulated below for a displace- ment of 150,000 lbs., a speed of 45 knots anda range of LCG ina head State 3 sea. Computed Measured 0.40 g 0.35 ¢ 0.50 g 0.45 ¢ 0.60 g 0. 559 It is seen that the computed values are approximately 0. 05g larger than the measured values. The average values are used in this comparison since, in random sea tests, the average statistics include considerably more impact peaks than do the 1/10 highest statistics. Thus, the comparison between measured and computed results are expected to be more reliable. It is interesting to note that a forward movement of LCG from 34 ft. to 38 ft. reduces the impact acce- lerations by nearly 35 %. The measured pitch and heave motions and added resistance in waves are not presented in this paper, but are in substantial agree- ment with results computed by the methods of Reference 3 Coursekeeping Stability and Turning Performance The calm water stability and maneuvering characteristics of the 1/16-scale model with appendages were investigated by means of straight course tests and by rotating arm tests. In both tests, the model was free to heave and pitch, but was restrained in yaw, roll, surge and sway. The restraining forces and moments were measured tao Savitsky, Roper, and Benen in a body axis system having its origin at the center-of-gravity. The straight course tests were made at port yaw angles up to 12° and at port roll angles up to 20°. Also, at zero yaw and roll the effects due to rudder deflection up to 35° were measured. The rotating arm tests were made at port yaw angles up to 12°, port roll angles up to 20° with the boat making port turns at radii corresponding to 2.5 and 5.0 boat lengths. The model was tested at a displacement of 120,000 lbs. at an LCG = 34 ft. and at speeds corresponding to 14 and 45 knots. The data obtained during the tests were processed on a digital com- puter and tables of drag, side force, yaw moment, roll moment, trim and heave were produced as a function of yaw angle and roll angle for each of the radii and speeds investigated. The reduced data were plott- ed and cross-faired as a function of yaw angle and radius for each of the speeds and roll angles. From these plots, which are not reproduc- ed here, the coefficients needed for stability analysis were determined and are tabulated below, These include the rates of change of side force and yaw moment with yaw angle and radius at zero roll angle. Hydrodynamic and Inertia Coefficients where 0s ech cilble ee 2buheiee m! Son! = /20wy/ pee Z Z Z y 3 ~ 2 N' = N/gB Yi te Ae 2 Gassatel) f2 B = beam = 15,7 ft. N = yaw moment, ft-lbs. Y = side force, lbs. W = 120,000 lbs. 440 Htgh-Speed Plantng Hull for Rough Water Dynamic Course Stability Dynamic stability relates to the track of a vessel following a small disturbance in, for example, heading angle when no corrective action is taken (i.e., controls fixed), A ship is said to be dynamically stable when, having suffered a disturbance from an initial straight path, it tends to take up a new straight path. The vessel may perform diminishing oscillations about the new track. The degree of stability is measured by the magnitude of a stability index which is negative if the vessel is stable and vice versa. The course stability may be found from linear differential equations governing the craft's motion, The coefficients in these equations may be found from the forces and moments measured dur- ing steady state turns, as described above. At both 14 and 45 knots, the craft is statically stable, that is to say that when run at a constant yaw angle the yaw moment tends to reduce the yaw angle. Static sta- bility is measured by the coefficient NY and is positive for static stability. However, the degree of static stability is not enough to im- pair maneuverability. Since itis statically stable, it follows that the craft is also dynamically stable, though oscillatory. These oscilla- tions decay very rapidly, however, being damped to 40 % of the initial disturbance by the time the craft has traveled one boat length. The stability index has been calculated to be Speed Stability Index 14 knots Gr= f= 0.219 45 knots es oc 0), 42 Turning Performance The straight course tests with the rudders deflected showed that the longitudinal position of the center of pressure coincided with the quarter chord point of the mean rudder chord and the vertical lo- cation coincided with the depth of the mean rudder chord. Thus, the effect of the rudders can be represented by a force acting at the aero- dynamic center of the rudder. The magnitude of the rudder "lift force'' was calculated from aerodynamic theory and confirmed by experiment to be represented by a lift curve slope of 0.0373 per degree. The forces acting on the boat when making a steady turn to port are shown in the following sketch. 44] Savttsky, Roper, and Benen The equilibrium equations in side force and yaw moment are Y=F+H+Y and N = f/_yY The component of centrifugal force in the x, y plane when the craft has yaw and rollangle of 8 and @¢ is Fi) =)\o(w/g) (v7/R) cos B cos ¥ For each speed and radius, the quantity Y - N/LR is plottedasa function of yaw angle for each roll angle and the yaw angles necessary to satisfy Equation (1) are found. At these intersections, the roll moment due to the rudder is found from K, = (Y-F)d, 442 High-Speed Planting Hull for Rough Water where dp is the distance of the rudder mean chord below the craft VCG. This roll moment is superimposed on a plot of roll moment versus roll angle to give the roll angles at equilibrium. The results of this calculation show that the craft turning diameter is less than 15 boat lengths and that it will roll inboard during turns. CONCLUSIONS A quantitative design procedure is described to determine the principal hull dimensions for planing craft intended to satisfy prescribed operational conditions. The method.is applied to establish a hull form required to operate at high speeds in moderate sea states. Principal design features of this craft are described, Extensive model tests were conducted to predict the SHP, EHP, seakeeping, course- keeping stability and turning characteristics of the design. Some of these model test results are presented. ACKNOW LEDGEMENTS The authors would like to express their appreciation to Mr. Joseph G. Koelbel, Jr. and to Mr. G. Gordon Sammis for their in- valuable assistance in all phases of the development of this new plan- ing craft. REFERENCES ] SAVITSKY, Daniel, 'Hydrodynamic Design of Planing Hulls" Marine Technology, SNAME, Vol. 1, No. 1, October 1964. 2 HADLER, J.B., ''The Prediction of Power Performance of Planing Craft'', SNAME Transactions, Vol. 74, 1966. 3 FRIDSMA, Gerard, ''A Systematic Study of the Rough-Water Performance of Planing Boats, Irregular Waves - Part II" Davidson Laboratory, Stevens Institute of Technology Report No. 1495, March 1971. 4 MILWITZSKY, Benjamin, ''Generalized Theory for Seaplane Impact'' NACA Report 1103, 1952. 5 MIXSON, John S., "The Effect of Beam Loading on Water 443 10 Savttsky, Roper, and Benen Impact Loads and Motions", NASA Memo 1-5-596, 1959. SAVITSKY, Daniel and MERCIER, J., ''Resistance of Transom-Stern Craft in the Pre-Planing Regime", Davidson Laboratory, Stevens Institute of Technology. Report 5 IT- DL-73-1667. July 1973. CLEMENT, E.P. and BLOUNT, D.L., ''Resistance Tests of a Systematic Series of Planing Hull Forms", Transactions SNAME, Vol. 71, 1963. KOELBEL, JosephG., Jr., ''The Detail Design of Planing Hull Forms'', SNAME South East Section on Smallcraft Hydrodynamics, Miami, Florida, May 1966. KORVIN-KROUKOVSKY, B.V. and CHABROW, F.R., "The Discontinuous Fluid Flow Past an Immersed Wedge'' Davidson Laboratory, Stevens Institude of Technology Report 334, October 1948. BROWN, P.W., ''An Experimental Study of Planing Surfaces with Warp'' Davidson Laboratory, Stevens Institute of Tech- nology (Report to be published). 444 High-Speed Planing Hull for Rough Water L=2 Figure 1. Average CG acceleration at V/ B = 20° ) eee ( Savttsky, Roper, and Benen ( 02 =9 ai P= TIA/A 38 UvoTJeteTeD0e 59 eselsay ‘2 ean3staz 20 ay ely £0 vO sO b=8/7 ct 446 Htgh-Speed Plantng Hull for Rough Water ( 002 =9 ai” SS St 9="IA /A 3 uoTJerTETID0e ) DD) oser9ay “°¢ oansTy ay lH ie) Z'0 £°0 v0 me) 447 Savttsky, Roper, and Benen uleoq JO SANTA SNOTALA IOF UOTJeETOTADIVe HOO ssert9AVy 8/1 Zz 9 G > € 0 oro cp=Xa #000'0S!1=V es o0Z= g fo} 340) 8 el=9 os'O="9 ovo oso Sv'0 —4 090 —{} 020 Ovo os’0 060 —j 001 orl (ey SAV 99 g-£/S)¥/o2] omy ys [102 7, x o0f / 002 ‘Pp oanstz g ec= o9= 1 oO 00 020 Ovo og90 o8'0 Ov! 09"! 08’! 00'e 448 Htgh-Speed Planting Hull for Rough Water V_ 245 A =150,000 Se Bi 20" eee eee 120 80 ree Lal 40 LCG, FT Figure 5. Equilibrium conditions in smooth water for various values of beam and LCG 449 Savitsky, Roper, and Benen 130 120 100 L,FT ELIMINAT SINCE L/V"3K7.0 LCG LCG, FT B,FT Figure 6, Relation between beam and length to achieve = 0.4 VL. = 45 kn 4= 150,000 1b. ( "cc ave Sree K hig Ses 450 High-Speed Planing Hull for Rough Water sting z9 sotzes O°T = Sa sox V/A “2 Sansa GA) al 8 Z 9 S v € 2 ) SS Y, rs On en Q/27 40 3NTWA a iS © WAWINIW GaLS3a99NS 4B \ } y¥~G y D GIONLN3D JOLlsv S91 % 21-0 AY, 997 40 39NVY s Q = Vv ES G8zezA/tV SO 0 Ww) ©) ONIGVOT 40 SONVY O) SJ BES Cee Z S p tS 2 g/ 020 451 Savitsky, Roper, and Benen Figure 8, Double chine hull form selected for design 452 Htgh-Speed Planing Hull for Rough Water 20° MAXIMUM T FOR 0.20 B=15' (SEE NOTE!) MAXIMUM T FOR = ° ‘ B=10 B=14 Qs a2 JJ ~ ar 0.08 NOTE: (1) MAXIMUM VALUE OF T FOR COMBINATION OF 8 AND BEAM WHICH WILL RESULT IN aon (Moe) ave 049 Vu 245, A=150,000#, STATE 3 HEAD SEAS o° 2° 4° 6° , ge T, = TRIM ANGLE, DEG Figure 9. Variation of R/A for various7 and 8B 453 10. Line plan of final configuration Figure High-Speed Planing Hull for Rough Water Figure 11. Shape of free streamline for immersed V-bottom 455 Savttsky, Roper, and Benen uoTJeAINSTjUuoS Teul_y “7_ eanstz 456 High-Speed Planing Hull for Rough Water MOTA ueTd pue eTtjoad preoqyno "ET + ernst a 457 Savttsky, Roper, and Benen O TEST DRAG O TEST TRIM —— COMPUTED VALUES OF 20,000 |— DRAG AND T 2 A=150,000 B=20° LCG=38° DRAG 4 é a a ve) s 3 oO q 3 10,000 0 fe) re) C) Ww 2 a re t fe) fe) (e) 10 20 30 40 50 Vx» KNOTS Figure 14. Computed and measured values of resistance and trim in smooth water 458 High-Speed Planing Hull for Rough Water DISCUSSION Manley Saint-Denis Untversity of Hawat Honolulu, U.S.A. I am very happy that seakeeping has been treated in this paper as the pre-eminent factor in the design of planing craft, for to the present, seakeeping has been introduced in the design of small craft hardly at all. Indeed it has been ignored except for raising the chine at the bow and narrowing the beam, And then the designers have simp- ly put their trust in God, hoping that he would be kind to them and to the sea-beaten crews that would man in the open sea the craft they had designed. Therefore, I am grateful that something is being done be- cause, waves being independent of the size of the vehicle that ventures over the surface of the sea, I suppose it is not a profound revelation to state that the smaller the craft the more she suffers. For a large vessel, even a heavy sea can be only an inconvenience, but for a small craft even a modest sea can lead to a very miserable experience. Therefore, starting the design of planing craft by considering the sea behaviour as the very first step in the process is the correct way to proceed, and I am glad to see the authors have done just this. My second point relates to the authors' conclusion that if you narrow the beam, reduce the trim angle and up the dead-rise, things will be better rather than worse ; and while the exposition of the paper itself gives quite some insight into the sensivity of how impact, sea behaviour and other effects are related to the design features, the designers do not unfortunately go further into the matter. I should like to point out that if all the authors wanted to do was to show how to develop a design to fulfil some very rigid specifi- cations, such as the inflexible ones they have stated, the design pro- cess could be shortened considerably. In fact, one could develop a simple computer program that would yield an almost instantaneous answer, for the line of iogic is simple and unambiguous in such a case. However, the point I should like to raise is that the specifications are not always quite as rigid as the authors have listed them, that indeed one has to play with them somewhat, giving up somewhat little here to gain somewhat more elsewhere : for example, take the problem of the transverse metacentric height, the metacentric height is reduce by 459 Savttsky, Roper, and Benen decreasing the beam, but this step also lessens the impact force, and so one might be better off. Is it worthwhile ? How much is it worth- while ? The answer is not easy, of course, but such type of problems - so called trade-off problems - are not treated. The computer tech- nique is in hand for coping very nicely with such problems and it is feasible to set up a programme that would yield a design, by satisfy- ing in an optimum manner an imposed set of trade-off criteria. There- fore I humbly suggest that the authors, having been successful so far, should continue their quest for further success by applying themsel- ves to this step. DISCUSSION Reuven Leopold U.S. Navy. Naval Shtp Engtneertng Center Hyattsville, Maryland, U.S.A. The importance of high endurance, and hence low resistance, at low -about 12 knots- cruising speeds is emphasised, Certainly a broad transom will have an adverse effect on resistance at these low speeds. This point is not discussed in the paper. Was the possibility .of incorporating a method of trim control in a design to reduce or eliminate transom immersion at low cruising eperés considered in this design methodology ? REPLY TO DISCUSSION Daniel Savitsky Stevens Instttute of Technology Hoboken, New Jersey, U.S.A. Yes. Deliberate trim control was considered, but it is not presented here, 460 Htgh-Speed Planting Hull for Rough Water DISCUSSION Reuven Leopold U.S. Navy. Naval Shtp Engineering Center Hyattsville, Maryland, U.S.A. The requirement to achieve 45 knots in state 3 seas is empha- sized. However the effects of air drag and sea state on the power re- quired to propel the craft at 45 knots is not discussed. Assuming that a state 3 sea is generated by a 15 knot wind, how much is the calm water 0 kn. relative wind resistance of the craft at 45 knots increas- ed by the presence of a 60 knot relative wind and a state 3 sea ? REPLY TO DISCUSSION Daniel Savitsky Stevens Institute of Technology Hoboken, Wew Jersey, U.S.A. It is important to emphasize that the present paper presents a methodology for rational design of planing hulls. The method has been applied to a particular set of design parameters to demonstrate its validity. It now remains to use this technique to develop optimum de- signs as suggested by Dr. Saint-Denis and Dr. Leopold. 461 2ST ae ee eee 4 1% ." Lal oa | ~ Ww acy oe Ys y, aie Geardaning the dean, Kat this ‘steps Vio wens maiy ht be better oat, Ts Ht word , whihe * The pnewer ie ha eee Célled abun atin hibe erin “ ate MO © in TR compuker'l nique ig it hand for < Fea sinic fo eet-op a 4 ssiygons 2t apse & o2ate yt eid oniscnuasA Deeauoeibd piso eft a: torn wor wesoionr aicnsd eh is fieto {e093 € oisie Bs bast S UOif2U08IG OF YIIaA f r ry < ‘ : ‘ ,e.7 ‘ 7 TAed ia FAS % 7 r - u ‘ % » - eats ehal. ae r 4 i ales vé © vay O78 ‘ ra. . ; . f.. ke a ’ ” e a a o . , ac : thc EP ? . i . * ad : ‘ ri ; . 2 4, eras} a < : an) BS Siners 74 TSCLo Tse] SAF 1. S31 Bnaecie of Jans > ehee be i sf fe 6 a a a GSec 2aii HSoalac aij ati guinsia to aglesd\imrover vol 1301 Chama a ati sistiationrwhb of ess%oc%1s 6a tuiecb to ea lun kb aq: & og “ab ctivaiitac qolaval oF ow - ‘bieqoaw 7 ow i \ 6 ’ ¥, tic BH os otdne: ‘OF Insite stivps: odT “92 towog $d} no oj43e 498 See gaxh tis-40 aipuiis of? raveweoH 7 eck prodleass and 4 ‘phd a i pyar tir schon muscons ay. 2 pte BS : aa e : y ; : ee son nt: ere we te Nats os} leyoug. of bain gon Set +e beta:sasy Ui aoe B.¢ oe atk ray aie sf at Boba: iw svivelow cob. D0 vw ovitalo: tond OO # to comeaetg wad ined ehs at of 2ntasritot wont). emis att “a 4 _ ¢ * + baa ; i caret © Deltog gue es ; OF ‘AZ BST ‘ -_ ’ =e MOTION AND RESISTANCE OF A LOW-WATERPLANE CATAMARAN BoC. Pien and ©. MM.) ee Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. ABSTRACT The unusually large, useful deck area is the advan- tage normally associated with catamarans. In addi- tion, small-waterplane-area-twin-hull-ship (SWATHS) can exhibit good seaworthiness characteristics in rough seas but at the expense of high powering re- quirements due to the large wetted hull surface. A well-balanced design of suchacraft must be the re- sult of a compromise among motion, powering, and structural weight considerations. Inthis paper, how- ever, only the hydrodynamic aspect of the design is discussed. It consists of two parts. The first part deals with ship motion, while the second part deals with resistdnce. In the first part, a theoretical method of predicting the motion and hydrodynamic loads of catamarans in a seaway is given. Based on theoretical analysis, tolerable limits on hull characteristics are determin- ed to ensure the desired motion characteristics. In the second part, a set of lines is developed within these limits such that the powering requirement is an optimum. This is done theoretically, based on existing wavemaking-resistance theory. Finally, a design example is given to show how hydrodynamic theories are used in designing a SWATHS. 463 Pten and Lee INTRODUCTION There are many types of naval ships which are volume- limited. In such cases, the proposition of using a catamaran hull con- figuration becomes very attractive because of its large deck area. A conventional catamaran, however, was found to have some bad mo- tion characteristics and cannot offer a stable platform in heavy seas. Since the sea-excited ship motions can be reduced by submerging the hull, a small-waterplane-area-twin-hull-ship (SWATHS) has become a subject of great interest. To explore the potential advantages of a SWATHS, a few experimental models were developed and tested for resistance as well as for motions. The expected favorable motion characteristics were generally confirmed. However, the resistance level was found to be unusually high, and the power requirement was much higher than ex- pected. The percentage of structural weight to total weight of a SWATHS is also expected to be high. This, coupled with large machinery and fuel weights, greatly restricts its payload. Before building a SWATHS which can accommodate a rea- sonable payload, it is necessary to reduce the power requirement by controlling the hull resistance, and to reduce the structural weight by controlling the hydrodynamic loading on the hull structure. Since the hydrodynamic loading depends upon the relative motions between the hull and the surrounding water, it is essential to control the ship mo- tions in such a way that the hydrodynamic loading is minimized. In attempt to solve this problem, an investigation of cata- maran hydrodynamics was undertaken at the Naval Ship Research and Development Center. Research efforts in catamaran motions were made by the Ship Dynamics Division, while parallel efforts in cata- maran resistance were made by the Ship Powering Division. Since the development of a successful catamaran design is contingent upon solving the motion and resistance problems simultaneously, the re- sults of these efforts were incorporated, and are presented here as a single contribution. This paper consists of two essentially independent parts written by two different authors; the first part by Lee, and the second part by Pien. In dealing with the motion problem, it was necessary to carry the theoretical work beyond that of solving the motion pro- blem of a single hull. Since this additional theoretical work has not previously been published, it is discussed here in its entirety. 464 Motton and Reststance of a Low-Waterplane Catamaran Several examples of comparisons between the theoretical predictions and the experimentally obtained motions are given. In dealing with the resistance problem, it was found that, by using the concept of an effective hull form, the design problem of a catamaran and a conventional hull became the same. Hence, no additional theoretical development was necessary. Because of this, it was possible to devote this section exclusively to the catamaran design procedure and design examples. Based on the motion work, the principal dimensions and hull coefficients which control ship motions in a given seaway can be specified. These specifications constitute part of the hull-form design conditions. Within all the design contraints, an optimum catamaran hull form based on powering considerations can be developed by fol- lowing the design procedure. When this is done, a table of hull offsets is available which can be used to make a final check on ship motions. Since the hull characteristics required for ship motion considera- tions may conflict with those required for the optimum power requi- rement, a compromise between ship motion and ship powering is necessary. Based on the work given in this paper, a well-balanced design can be developed. I - MOTION OF CATAMARAN. I.1 - Background. One of the obvious advantages of a catamaran is the large available deck area. If this large deck area is to be efficiently utili- zed, it must behave as a stable platform. From a seaworthiness viewpoint there are some special features associated with twin-hull configurations. First, an increase of overall beam results ina decrease in natural period in roll. A smaller natural period in roll makes catamarans very jerky ships. Most conventional monohull ships have a greater natural period in roll than in pitch. In case of catamarans, the pitch period may be slightly larger than that of monohull ships of equivalent length and displacement. This fact together with the de- crease in the roll period for catamarans tends to bring the natural period for roll and pitch closer to each other. This could cause simultaneous excitation of large roll and pitch motions, which make very uncomfortable riding for the crews. Second, the existence of a cross-deck structure between 465 Pten and Lee the two hulls above the water can result in slamming of the bottom of the cross-deck either by a train of sharp-crested waves or by a large vertical motion of the ship. The slamming of the cross-deck can cause structural damages due to water impact and the hull vibra- tion initiated by the impact. Thus, designing catamaran hull forms which could avoid the afore-said disadvantages requires different experience and knowledge of seaworthiness characteristics from what is needed for monohull ships. Motion of a ship is mainly excited_by waves, and unless ships are deeply submerged, like submarines, the influence of waves cannot be avoided. Wave influence on the hull could be minimized on a ship whose main hull is submerged and connected to a deck by a vertical strut. This is the main idea behind the semisubmersible or low-waterplane-area catamarans as they are referred to in this paper. From a motion standpoint, the concept of the small-water - plane-area-twin-hull-ship (SWATHS) configuration may be traced to the so-called wave-excitationless forms which are extensively studied by Motora and Koyama! (1966). This configuration, which has small waterplane area but carries large volume beneath the waterplane, in- creases the natural period of heave, since the natural period is pro- portional to the square root of the ratio of virtual mass to waterplane area, This fact means that only long waves may excite a large motion, Moreover, as investigated by Motora and Koyama, depending on the scantling of the strut width and height and maximum breadth of the submerged hull, the wave damping? can be reduced to a small value in a certain frequency range. Smaller wave damping means a smaller wave excitation force and moment ; see Newman (1962). Caution is necessary, however, in reducing the damping factor of a dynamic system for the purpose of reducing the forcing function. If we let c be the damping coefficient of a harmonically excited mass-dash pot-spring system and F the forcing function, References are listed on page 539. 2 Here the term ''wave damping'"' means the damping associated with generation of progressive water waves which carry away ener- gy supplied by an oscillating body. 466 Motton and Reststance of a Low-Waterplane Catamaran then the motion x expressed by o of the mass at the natural frequency w, canbe In the case of ship motion, F corresponds to the wave exciting force and is proportional to c as shown by Newman (1962). Thus we have X9~c -»-. This means that a reduction in damping at the natu- ral frequency could result ina large motion. However, the natural period may be increased by proper design to such a magnitude that the corresponding wavelengths may not be frequently encountered by ships in the ocean. Furthermore, the concern for an expected high- peaked resonant motion resulting from a reduction of wave damping of the system may not be serious because of an augmentation of viscous damping due to an increased wetted surface on the LWP catamarans. Although reduction of motions of catamarans may be accom- plished through SWATHS configurations, such configurations present formidable structural problems. The decrease in waterplane area re- duces the restoring buoyancy, and this, in turn, makes the LWP cata- marans weight sensitive. The limited tolerance for additional weight requires a narrow margin for safety factors on structural weights. An additional complication to the structural problem is the lack of data for wave loading. The wave loading includes the contribution from impinging waves as well as from motion-generated inertial and hydro- dynamic forces. To obtain an accurate wave loading the effect of the wave diffraction by two hulls and of the motion of the body should be taken into account in the theoretical analysis. In this work an analytical method has been developed for predicting characteristics of motion and hydrodynamic loads of cata- marans, either conventional or SWATHS. The equations of motion for catamarans are derived in the frequency domain under an assumption of linear excitation-response relationship. The hydrodynamic coeffi- cients involved in the equations of motion are determined from strip theory, assuming slender geometry of each hull of the catamaran. The effect of forward speed on the hydrodynamic coefficients is treat- ed as if there were no perturbation on the fluid due to a translation of the ship. An apparent underestimation of damping by potential theory results in an unrealistically large motion amplitude at the resonant 467 Pten and Lee frequency. Thus, introduction of supplemental damping into the equa- tion of motion is needed to achieve a reasonable prediction of cata- maran motions. The supplemental damping is introduced into the equations of motion in a form linearly proportional to the oscillation velocity. This damping is found to depend on the ratio of the ship speed to the celerity of motion-generated waves. This fact implies that the interaction between the wave systems, created by oscillation and for- ward speed, is important and should be included in the evaluation of damping. Prediction of statistical averages of motion amplitudes for catamarans in irregular seas is made by using the frequency respon- se-amplitude operator in conjunction with the Pierson and Moskowitz (1964) sea spectrum. The probable frequency of a water contact with the cross-deck structure is computed for given conditions such as height of the cross-deck structure from the waterline, significant wave height, and forward speed. The formula used is based ona truncated Rayleigh's probability distribution for slamming and is similar to the formula developed by Ochi (1964) for bow slamming of monohull ships. The expressions are developed for various loadings contri- buted by inertial and hydrodynamic effects, such as bending and torsion moments and shear forces on both the cross-deck structure and the supporting strut (shear and bending only). Presentation of the work on motion of catamarans is given in the following order. In section 2, the subjects covered are: formulation of equations of motion, derivation of the hydrodynamic coefficients and derivation of an expression to estimate the number of slammings of the cross-deck structure per given period in regu- lar and irregular waves. In Section 3, the derivation of expressions for various hydrodynamic loads on catamarans is given. In Section 4, a presentation of comparisons of theoretical and experimental results is made, and concluding remarks are given. I..2 - Motion. Equations of Motion The assumptions or conditions made in this paper for stu- dying motions of catamarans are as follows. A catamaran which is made of two symmetrical hulls is cruising with a constant speed, while it is experiencing an undulatory motion due to sea waves. The sea waves are assumed to be made of a linear sum of unidirectional 468 Motton and Reststance of a Low-Waterplane Catamaran regular waves of different frequencies. The response of a catamaran to these waves is assumed linear in amplitudes and frequencies. The amplitudes of the waves and the motions are assumed to be small, and, consequently, the fluid disturbance generated by the motions of waves and ship is also assumed small. The depth of the ocean is assumed infinitely deep and the effects of wind and current on the mo- tion are not considered. Within a linear approximation of the motion and with the conditions prescribed in this work it is convenient to choose Oxyz, a coordinate system representing the mean position of the catamaran as the reference frame for which the equations of motion are to be formulated. When the catamaran has only steady translation, the Oxz plane coincides with the longitudinal plane of symmetry of the catamaran, the Oxy plane coincides with the calm water surface, the Oz axis is directed upward, the 0x axis is directed toward the bow, and the Oy axis is directed toward the port side. Since the wave excitation is assumed to be of harmonic nature in time, the equations can be formulated in a frequency domain. With the conditions stated in the foregoing paragraph, the linearly coupled motion of a catamaran in six degrees of freedom can be written with the motion-generated displacements from the mean position denoted by gs ; where the values of i represent | for surge, 2 for sway, 3 for heave, 4 for roll, 5 for pitch, and 6 for yaw, as 6 ae oer pena OME ore = k 1 k=1 ik °k* “Gk for i=1.2. ..., 6. The equation shown above is a degenerate case of the equations of motion of floating bodies in waves, formulated in the time domain which has the form of integro-differential equations as shown by Cummins (1962) and Ogilvie (1964). In Equation (1) Mj, is the mass or moment of inertia of the catamaran, Aix , the added iner- tia, Bi, , the damping, Cj, , the restoring constants, Fj;\©/, the wave excitation in the form of complex amplitude, and j is the imaginary unit associated only with a harmonic-time function. The expression ''added mass (or inertia)'' which will be frequently referred to in this paper is used for mathematical conve- nience. Thus, it does not have the same meaning as the classical added mass which is an intrinsic property of the geometry of the body only and is independent of motion, frequency, and forward speed. The mathematical relation between the added mass of the classical defini- tion given in Lamb's Hydrodynamics and the one referred to in this 469 Pien and Lee paper is derived in, e.g., Wehausen (1971, pp. 243-245). To be compatible with the complex expression on the right side of Equation (1), the motion displacements oe are assumed to be complex functions in the form of g,(=Re, [te Set Jere, [C 2, +58.) 6 Mt] @) where Re; means that the real part of a complex function in terms of the imaginary unit of j should be taken, and a and ee are real functions. Each hull of the catamaran is assumed to be slender so that the change of the surface normal in the length direction is small compared to the change in the transverse directions. This slender- ness assumption together with the symmetry of the two hulls lead to decoupling motions into three independent groups of motion : (1) sur- ge, (2) heave and pitch, and (3) sway, roll and yaw. In this work the surge motion will not be considered. The explicit forms of the remai- ning two groups of motion are given as follows. Heave and pitch equations : “ (e) -jwt UT Pioe! stat 3¢ Gatt Ae Eat Baie SG pi ee (3) Em -jwt (I, + A,,) BAR f+ C244, €, +B, PULSE: e (4) Sway, roll, and yaw equations : (M + A,,) emit Bs fot: Aoges Mz) Egct By, 62 +A,,5,+B,. ee* Fle). - jot (5) (I, “. AY, ty + Bie Et Gn goby. ilAracie s Mz.) t+ By aa + Age Eg + Bag fe - tai e I°" (6) (I, +A.) & + Bee E+ Ago t,t Beata + Ag, ty + Beg fy = rie) aes jot (7) 470 Motton and Reststance of a Low-Waterplane Catamaran In the previously given equations, M is the mass of the catamaran; I, ,1I, and Ig are mass moments of inertia about the 0x, Oy and Oz axes, respectively; Zp is the z coordinate of the center of mass; and the restoring constants C33 , C35 ,Cs55 ,C53 and C4, are given by BS w aoe cen Bis 5 E 7 M (GM), Cares) VEN se™ Here A. is the waterplane area at the mean position of the catama- ran, M,, the area moment of the waterplane about the Oy axis, (GM), and (GM), the restoring moment arm in pitch and roll respec- tively. The major task in solving the equations of motion shown previously lies in determining the hydrodynamic coefficients, A; Eye and E©) ijk = 2, 3; ..., 6. They arelfunctions of hull geometry, wave frequency, and forward speed. The method of deter- mining these coefficients is described in Appendix A, and the results are presented in Table 1. The lowercase letters aj, and bj, shown in Table 1 are sectional added mass and damping, and U is the for- ward velocity. These are obtained by solving two-dimensional boun- dary-value problems for velocity potentials representing the fluid motion generated by an oscillation of infinitely long twin cylinders. The cylinders are semisubmerged horizontally, have a certain sepa- ration distance between, and are rigidly connected together from above. The twin cylinders have a uniform cross section which is identical to the cross section at any given location along the length of the catamaran. The method of distribution of pulsating sources along the sumberged contours of the cylinders is employed in solving the velo- city potential; see Lee, Jones, and Bedel (1971). The method used is similar to the one developed by Frank (1967) for single cylinders. 471 Pten and Lee Table 1 - Strip approximation of hydrodynamic coefficients * 4 ny) ime) ine) Q * at alc a ia?) ia) = 1S a LO | iT] * ine) oO W W Qa * ob a E W WwW * o ine) MN eV w i] (e! > ime) ine) > 1S! W i! SS, * © W W oF rv] + e NM ios) W W wo ies) ws iH} | wr OQ. * > 1 Qn i oes * ia?) ny) W Oo Qu * s lq, > W W > WN WN \ jel) Ww Ww [oh * wo onl wn \ SS * on WN WN Q * 1 Cc > W W bw WN WN | < WN Q ma > lon lon Sa * ine) ie) ia) ine) jo * + 4 ime) _ ine) ie?) o Q s ob ‘a > > iH rs we Wm \ 1 ! re om » Ww W oh * 1 a C 0 Ww Ww wo Oo lon \ 4 ine) lop lee) lee) Q Pe sb | ae) x ip?) Mm WW W WW W oO N os) | a Qu » > iT] > i" ll —, — a od ry is) oO ie) iH} oe oP. ae a Nh Q ra + (eS > lye) lye) crea gl Bogs fx Pag de + UA, * Lowercase letters mean two-dimensional hydrodynamic coefficients; integrals are from the aft to the fore perpendicular. 472 Motton and Reststance of a Low-Waterplane Catamaran tr Da —~ ct ~~ iH] =i gi Re, es | as defined in Equation (2), we have -jowt Okc “JOG r E = = we Ainjgont k See in which we have dropped Re; for the sake of brevity; however, it will be understood in the sequal that the real part is meant whenever a term is involved with e~/®'. Substitution of the foregoing results into Equations (3) through (7) yields two sets of complex algebraic equations of the form Ls dl 1 (8) A, x5 = B, where E E59 x) = 30 x5 = E40 E50 S50 F_(e) te = = (e) _ F,.(2) *2 ue 5 Fs = am (M+ A |) 4 C -jwB aa: + C¢ -jsB 33 33 33) 35 35 35 Ay = : = ec) = (8) (8 an ail ecu ona (IztA pe) + Coe - jas and 473 Pten and Lee 2 5 2 2 : a (M+A,,)-jwBoo -@ (A, ,-Mzg)-jaBo, -wf Az, -j@B5¢ 2 : : Zz P A= —w (Ag- Mz, )—j Baa -F CO, +4, 4,508, —W Aye -jOBee 2 : 2 : 2 3 pe Aas —yer tes =-OUMGr - I es ~0 Ig +.A¢¢)-JoB gg After an inversion of the matrix in Equation (8), we can obtain the absolute motion amplitudes by atid 2 oi d( on ia fof [o\act (9) 8 Ja ae 6 AS and the phase angles with respect to the wave motion at the origin 0 by fle tan ; a. be / to} (10) lee ae a © Once the motion amplitudes be are obtained as previously described as a function of wave frequency, we can obtain various statistical averages of the displacement and velocity of the ship, using the method introduced by St. Denis and Pierson (1953). We can show that the statistical averages of the short-term response of ships to sea waves can be given by ( ae, stat ave rite. Ed ( eee ave = TN By k , at 1 1 (S) 6 [ : ; i ue ou ‘ Etro ds (28) c = me k 6 he ee at = = * ko [eo2 : c(s) ior tS 2, 3 BOR an where @y, , for instance, means to the last term of Equation (68) , and the last term corresponds If we let x = fie [ >, 0. dea Ay (x) + jw Bx (x) (29) C(s) for i= 2,3, and substitute this expression into Equation (28), we get 481 Pten and Lee , (s) PU 2 cr >, o,, a + 2, (w A, (x) + jo B.. (x)) ee + UD Lilie ea tage) (30) = where a, (x) and b, (x) are the sectional added mass and dam- ping of the cross section at x. The expressions of A, (x) and B,, (x) in terms of sectional added inertia and damping are almost identical to those for Aix and Bix shown in Table 1, except that the integrals in the table should be replaced by fi , and Aj, and B, should be replaced by A, (x) and By (x) . For instan- ce, xX U LP oe a 5a,, te) de tir Pay where 1 x B,, (se), = a Bas (s) ds Returning to Equation (24), we can now write I ao) S ©; Oo Q oS ' £ > « +b eS to + (S str ae [Ss ey) nN na ‘2 ' o tN x # ey — in which 482 Motton and Reststance of a Low-Waterplane Catamaran Xx -ofoas | (jo n, + 9) ds LN ols) 3 on I (32) jeu : + LEU | 6,0, of - (0%, (0) +1% By) £4, 5 c(S) =( @ A, (x) +joB,.(x))&,)+U (jw a, 4(*) b,,(x))E 35 in which The horizontal bending moment Mp5 and the longitudinal bending moment M, can be obtained by X M, (G2) = i i (s) ds (33) e , X M, Go)in= i We (s) ds (34) The previously described loadings V, , V, » M; and M, are obtained in exactly the same way as for monohull ships. However, to obtain the remaining loadings, the loadings on each hull should be separately studied. We would like to obtain the shear forces and bending moment at the various locations over a transverse cross section. The loadings V, , Vs and M, fall in this category. Let us consider the right half of a cross section, located x from the origin. The structural members of this section will be divi- ded into three parts : the cross-deck, the strut, and the main hull - 483 Pten and Lee which will be denoted by A, B and C_ respectively, as shown in figure 2. First we will consider V, , V, and M, at the left edge of the member A, which is the longitudinal centerline section of the catamaran. M, TLL Cy, YN Figure 2 -A simplified cross section for loading analysis The expressions for these loadings can be given in the form Vio (0) = — 2 i pn. ak -R! (35) CR Lore d= 925.3) my Or=mj- fpf ym, +0, -2) mp} al - Ri (36) CR Here ft; are the inertial forces contributed by the mass of one-half of the cross section, C, means the line integral in counterclock- wise direction along the immersed contour of the right half of the cross section at its mean position, the subscripted R's are the restoring forces and moment, M, is the mass inertial moment, and hy, is as shown in figure 2. We can show that of eae om! {Eg t S60 7° 2 oie en 484 Motton and Reststance of a Low-Waterplane Catamaran carta ane (a Aap I) pO a IN (38) Me wi m'{ Yessir Eoiiete. Eo} Ly wf I! + m! { (oin- hn) + (v7 }] (39) where m!' is the mass of the half section, ly is the mass moment of inertia of roll about the center of mass of the half cross section, and (y', z') is the coordinate of the mass center of the half section. The restoring forces and moment are given by R, = -2egab( & mess ue 52 GNA 30 og) 2S SGM £ 40 (40) where b and a are as shown in figure 2, S'y is one-half of the immersed total sectional area, and GM' is the transverse metacen- tric height from the mass center. Rj = 10 (41) ! = é 2pgean() Sse fe +b & (42) The pressure p at any point on the immersed contour of the right half section is given by 6 p= p(jvtuF)(o +>, + Re (43) Then for i= 2,3,4, where nz andnz are respectively the y and z compo- nents of the unit normal vector on the body surface and ng = yn3-znp. Note that we can no longer apply Haskind's relation (1957) to express the diffracted wave force for a half cross section in terms of Ot and Ok as shown in Appendix A. This means that we 4 To be strictly true, the application of Haskind's relation in strip approximation is incorrect. Faltinsen (1971) has shown a three-dimen- sional diffraction effect. 485 Pten and Lee have to obtain the solution of > before we can evaluate complete hydrodynamic loadings. In Appendix B, the method of solving Dp by source distribution similar to that used in solving the motion potentials 0, is given. We shall not attempt to reduce the last inte- gral of the right side of Equation (44) to added inertias and dampings. However, attention should be paid to the fact that decoupling between the motions of the vertical and horizontal planes can no longer be made in this integral. This is because we are dealing only with one hull for which there exists asymmetric pressure distribution even for a symmetric demihull, due to the blockage effect of the other hull. Thus, every term of the summation in Equation (44) would sur- vive. Applying the foregoing results into Equations (35) and (36), we obtain the following expressions V,(0)=- w'm'( E50 +e Eo Z E44) (45) + Dy bxo | Sond # -ief (0+ ¥ Op) mga k=2 Ce Cp 2 ¥, 0} eee mine WX By FT ag) sof ( we f- acl )n ad - I p’ 3 (46) te Eko 93 at ki2 CA R 6 rr pyar 04, 0, a k=2 Cp +2 pegab(é.)-x 5) +e gS, GM' ys (47) 486 Motion and Resistance of a Low-Waterplane Catamaran Once the loadings at the centerline section of the cross-deck member are known, we can find the loadings at any section of the structural members in the following way. For -hi< Z

w~ exp ape z se (266) | 2 § -= 4 Ae Bp? where S_ | (f{) is the spectral density of s (t) and fg = —— aoe 217 = We note that Carrier's theorem is obtained by setting P=0. When s (t) is not a gaussian white noise this asymptotic expression does not hold any longer . Nevertheless , Carrier has shown that this formula is a good approximation in the following cases a/ s(t)=s, cos (47Tfp t+ € ) where€ is a random variable distributed uniformly over the interval (0, 2/7) by esi (Eyicea gaussian process with an auto-correlation function given by -k |T| Ra )) a Pa eae (k >0) 1065 Dern For lack of results concerning the case when s(t) is a. gaussian process with any autocorrelation function we use the asymp- totic expression given by the assertion . This expression permit us to give the stability criterion of equation (III-3. 1) Bo 2p (b aa 5 (Teac) ae (2£0) < 2 For all types Fig << % , hence (111-3. 4) _ (2464 << hae 0 This stability criterion may be written in another form by the following transformations : Sep (f) ' SS . S pp = H = indep Z of H H. , (2 7rif ) = transfer function of relative motion S (f) =r Gay 9 (aT) Sp p (£) ss af S (f) 2 Jt A) its indep. of He (III-3.4) becomes Hp K (Tv, fo ) (i11='3- 5) (2) mg Vs? ss-(2 fo) K (TL , £¢ ) depends on TY , £ and on the heave characteristics of the buoy . 1066 Unstable Motton of Free Spar Buoys tn Waves In the plane (HY ale Ls) the instability region is located above the curve defined by ¢* IlI-3. 5 ) .As an example , figure n° 31 gives the range of He and ie in the North Atlantic . Figure n° 32 gives the region of rolling for type 11 buoy . In practice , relation (III-3.5) does not allow us to study by means of calculus only the effect of the Boy characteristics on the presence of a rolling motion because Nop Ys not known theoretically but only experimentally . IV - EXPERIMENTAL STUDY - 4.1 - Experimental apparatus and procedure - In order to verify the validity of the rolling criterion (III-3.5) we have performed experiments in irregular waves with a spectral density given by the modified Pierson-Moskowitz formula . Only type 1, 11, 14 and 15 buoys have been tested in tank n° 2 of Bassin d'Es- sais des Carénes . This tank is equipped for generating irregular waves [ 24 ] : any spectral density may be simulated by using a dri - ving voltage . This driving voltage is obtained by running a pseudo - random white noise through a linear filter so designed that the square of its frequency response has the desired shape . Figures 33 and 34 show an example of a measured spectral density of waves and of re- lative motion _ The experiments were carried out in the manner discussed below . A driving voltage was selected , corresponding to a given value of a . Then the value of He. was set by adjusting the gain of an amplifier located at the input of the generator of the Ward Leonard group . Unfortunately , during these experiments , the gain could be adjusted only by step . Consequently , the critical value of H_ (i.e. the value above which the model rolls) was not precisely determined . 4.2 - Results - Figures 32 and 35 to 37 show the results for type 1 ee, yi eee and 15 buoys . In these figures we can see the ''theoretical curves" which give the critical value of Ae versus Tas given by equation (III-3.5) . As was said before , the number of experimental points are too small for the critical values of H to be well-defined . Yet it seems that there is no fundamental discordance between the theoreti- cal curves and the experimental results . 1067 Dern 10 Hy ( meters ) Fully developped sea (Pierson - Moskowitz ) [26] ABCD: total massz0,7 70 15 FIGURE 31_ MOST PROBABLE VALUES OF(H,,T,) IN NORTH ATLANTIC (station am of rét{25] ) 10. Hy (meters) Experiments () no rolling ROLLING Theoretical curve NO ROLLING \ Tt ( sec.) Qo Ss 10 15 FIGURE 32 ~ REGION OF ROLLING OF TYPE 11 BUOY 1068 4 Unstable \ Measured Predicted Na FIGURE 33_ SPECTRAL DENSITY OF WAVES Sss (f) m?/H, for réal scale Measured Predicted 04 FIGURE 34_SPECTRAL DENSITY OF RELATIVE MOTION 1069 Motton of Free Spar Buoys tn Waves TYPE 11 BUOY Hy = 0.519m Real scale Frequency for real scale IN TANK N 2 TYPE 11 BUOY Hy = 0.519 m Real scale Frequency for real scale ——__—_ a2 rz) f (Hz) IN TANK N 2 Dern Hy (meters) 2 See key below figure 37 Theoretical curve 1 8 e l l l l Ty (sec.) 0 2 3 4 5 6 FIGURE 35_ REGION OF ROLLING OF TYPE 1 BUOY 2 Hy (meters) See key below figure 37 1 Theoretical cine Ty (sec.) 0 erie (AI. ARE NY | SEEN SN SARE | PPL Rar YE aT RT OTA ee 2 3 4 5 6 FIGURE 36_ REGION OF ROLLING OF TYPE 14 BUOY 1070 Unstable Motion of Free Spar Buoys tn Wr 28 Hy (meters) See key below this figure b Theoretical curve FIGURE 37_ REGION OF ROLLING OF TYPE 15 BUOY KEY FOR FIGURES 35 TO 37 Experiments =: frequent-and important rolling ( “OR > 10°) fal frequent but not important rolling ( a <10°) b no frequent but important rolling ( 5 ee, 10°) @ no rolling L071 Dern IV - CONCLUSION OF SECTION II - Not enough experimental data has been collected for the validity of rolling criterion to be verified without doubt . Neverthe- less this criterion seems reasonable . We must note that it was necessary to introduce in the rolling equation a damping term expe- rimentally determined for J.N.Newman's theory gives too small a value for this term , and , hence , for the critical values of Ho 2 GENERAL CONCLUSIONS - We have shown in sections I and II that J.N.Newman's theory is not valid except if the following conditions are fulfilled : a/ - the upper part of the buoy which is normally out of water must be of constant sectional area and must be high enough so as to avoid a double regime in the heaving motion . b/ - It is more difficult to state the conditions for avoi - ding rolling motions and unstable pitching motions but lowering the center of gravity is an effective method . When these conditions are fulfilled , J.N.Newman's the- ory is well verified except for the values of the maxima of the fre - quency response functions in heaving and pitching . Therefore it seems necessary to determine experimentally the damping terms in heaving and pitching . In section III we have proposed a criterion for roll in irregular waves . It seems that the criterion is not in disagreement with the few experimental results which are available , provided that one uses an experimentally determined damping term in the rolling equation . 1072 Unstable Motton of Free Spar Buoys tn Waves REFERENCES J.N.NEWMAN - '' The motions of a spar buoy in regular waves '' - Report 1499 , David Taylor Model Basin, 1963. Philip RUDNICK - ''Motion of a large spar buoy in sea waves ''- Journal of Ship Research , vol. 11, n° 4, 1967. C.BRATU - '' Comportement dynamique des bouées oceanographiques: "= A. TM. A. 1970: Bruce H. ADEE , KWANG JUNE BAI - " Experimental studies of the behaviour of spar type stable platforms in waves '' - Report n° NA-70-4 , Collége of Engineering University of California . H.LAMB - ''Hydrodynamics' - Cambridge University Press , 6th edition , 1932 . AZ PAAPE, 3 oN. C. BREUSERSs, J.D. van den BUNT ' L'estimation des forces hydrodynamiques sur les pieux'' - Colloques sur la connaissance de la houle , du vent , du courant pour le calcul des ouvrages pétroliers - -Editions Technip , 1970- E.G.BARILLON - ' On the theory of double systems of rolling of ships among waves '"' - Institution of Naval Architects , 1934. R.BRARD - "' Contribution a l'étude du roulis . Régimes multiples d'oscillations forcées d'un oscillateur non linéaire '' - Bassin d'essais des Carénes , 1944. BAUMANN - " Rollzustande grosser amplitude in seitli - cherdtinung '' - Schiffstechnik , 2, 1955. “ EORS Dern BLAGOVESHCHENSKY - " Theory of ship motions "' - Dover Publications , 1962. Austin BLAQUIERE - '' Nonlinear system analysis '' - - Academic Press - 1966. Otto GRIM - '' Rollschwingungen , Stabilitat und Sicher - heit im Seegang '' - Forschungshefte fir Schiffstechnik - - Heft 1, 1952 .- J.E.KERWIN - '' Notes on rolling in longitudinal waves "' - International Shipbuilding Progress , vol. 2, n° 16, 1955": J.R.PAULLING , R.M.ROSENBERG - '' On unstable ship motions resulting from nonlinear coupling '"' - -Journal of Ship Research , vol. n° 3,n° 1, June 1959. W.D.KINNEY - ''On the unstable rolling motions of ships resulting from nonlinear coupling with pitch including the effect of damping in roll '''. - Institute of Engineering Research , University of California , Berkeley , October 196!" : W.D.KINNEY - ''On the normal modes of nonlinear cou- pled ship motions and their stability '' - Institute of Engineering Research , University of California , Berke- ley , August 1962. M.R.HADDARA - '' On the stability of ship motions in regular oblique waves "' - International Shipbuilding Progress , vol. 18 , November 1971 , n° 207 . Robert CAMPBELL - "' Théorie générale de l'équation de Mathieu '' - Masson & Cie , Paris , 1955. MINORSKY - '' On the stability of nonlinear non autono - mous system '' - Proceedings of the US National Congress of Applied Mathematics '"'- 1954 - 1074 Unstable Motton of Free Spar Buoys tn Waves S. KASTNER - "' Hebelkurven in unregelmabigem seegang'! = Sehitistechoik , vol, be, nu” 17 .,.1970°. G.F.CARRIER - '' Stochastically driven dynamical sys - tems '' - Journal of Fluid Mechanics , vol. 44 , part2, 11 Novembre 1970. Proceedings of the 2nd International Ship Structures Congress 1964 - Delft - Netherlands - 20-24 July 1964 . J.B.KELLER - '' Wave propagation in random media "' - Proceedings of Symposia in Applied Mathematics , XIII , 227 - 1960. J.L.GIOVACHINI , P.WANTZ -''' Generation of Irregu- lar waves of Bassin d'Essais des Carénes , Paris "' - 10th ITTC - Teddington - September 1963 . H.U.ROLL - ' Height , Length and Steepness of Seawa- ves in the North Atlantic '' - '' Dimensions of Seawaves as functions of Wind Force ''SNAME T.R. - Bulletin n° 1-19 - 1958. W.J. PIERSON and L.MOSKOWITZ - ''A proposed spec- tral form of fully developped wind seas based on the similarity theory of S.A. Kitaigorodski '' - Journal of Geophysical Research , vol . 69 , November 1964 . J.R.MORISON , M.P.O'BRIEN , J.W.JOHNSON and S.A.SCHAAF -'" The force exerted by surface waves on piles '' - Petroleum Technology , vol. 2, n° 5 - May 1950. 1075 0 3 & Dern NOMENCLATURE Coefficient of drag appearing in Morison's formula Coefficient of added mass appearing in Morison's formula Vertical prismatic coefficient pH S ©) Buoy draft Describing function Transfer function of relative motion Significant wave height Transfer function of pitching motion Moment of inertia of buoy in roll about the center of gravity Moment of inertia of buoy in pitch about the center of gravity Added moment of inertia of buoy in roll about the center of gravity Added moment of inertia of buoy in pitch about the center of gravity Radius of gyration in pitch Overall length Damping coefficient in heave Damping coefficient in pitch Damping coefficient in roll Fixed coordinate system Body coordinate system = he (2G - z)"S(z) dz fe) H -kz p e n oxi (zG - z) S(z) dz Buoy radius 1076 Unstable Motion of Free Spar Buoys tn Waves S (z) Section area of buoy De (f) Spectral density of relative motion Sop (f) Spectral density of waves a Period of pitching motion oe Period of rolling motion Ts Period of relative motion T. Period of irregular waves qT. Period of regular waves Ny Instantaneous immersed volume of buoy W Displaced volume of buoy in calm water x Surge displacement of buoy ¥ Sway displacement of buoy Z Heave displacement of buoy = Te atts ! aie 26 a. Acceleration of fluid particles due to orbital motion of the on-coming waves — — —> a (Ax. x) x — —_ —, —> a = (a Zz 0ZO fo) o Oo f Frequency f Natural frequency of heaving motion fy Natural frequency of pitching motion g Acceleration of gravity h 1 Characteristics of the upper part of buoy ~ i=V -1l k Wave number , 277 A m Buoy mass m Added mass of buoy in the x direction Ap Added mass of buoy in the y direction ms Added mass of buoy in the z direction 1077 Be a AG AG ! oat 9 ft Te! Dern Coupling coefficient between surge and pitch equations Coupling coefficient between sway and roll equations Metacentric radius Relative motion Time Cartesian coordinate system Fixed cartesian coordinate system Unit vectors of Ox yz Unit vectors of O x y_z o 6 oO. oO Position of center of buoyancy /Oxyz Position of center of gravity /Oxyz Position of center of buoyancy / baseline Position of center of gravity / baseline Wave elevation Half-wave amplitude (preferably to ''wave amplitude") Wave height Pitch angle of buoy Length of regular wave Density of fluid Circular frequency of oscillations Roll angle of buoy Righting moment in roll 1078 AUTO-OSCILLATIONS OF ANCHORED VESSELS UNDER THE ACTION OF WIND AND CURRENT A.V. Gerassimov, R. Y. Pershitz, N.N. Rakhmanin Kryloff Shtp Researeh Institute Leningrad, U.S.S.R. ABSTRACT Horizontal plane auto-oscillations of a vessel laid at one anchor are investigated and condi- tions under which auto-oscillations can be eli- minated are established. Ship structure cha- racteristics providing auto-oscillation elimi- nation are considered. INTRODUCTION It is known from practice that in the presence of wind an anchored vessel swings from side to side with respect to wind direc- tion line. It performs angular (yaw) and translational (drift) oscilla- tions in the horizontal plane. As is shown by full-scale observations, the intensity of oscillations depends on the wind force, and their am- plitude values may reach 90° to 100° for the yaw while for the drift they may be equal to the depth at anchorage or even more than that. The dependence of the oscillatory period on the wind force is weak. When lying at anchor is a working condition for a vessel, such oscil- lations may prove to be extremely undesirable. Yawing and drifting of anchored vessels are auto-oscillatory in nature as they may be caused even by the wind of constant direction and force. Such character of motion is due to nonlinear relationships inherent in an oscillatory system formed by the anchored vessel. The most significant of these manifest themselves in the nonlinear relation- ship between the horizontal component of the anchor chain tension and the shifting of the hawse-hole with respect to the sea bed, as wellas 1079 Gerasstmov, Pershitz and Rakhmantn in the nonlinear relationships between the aerodynamic and hydrody- namic forces and the kinematics of the ship's motion. For the purpose of making a detailed analysis of yawing and drifting of anchored vessels this paper deals with the discussion of forces acting on the vessel in the circumstances, and the derivation of relevant differential equations of motion. In the derivation of these equations great attention was given to determining the tension of the anchor chain as dependent on the shifting of hawse-hole. The aerodynamic and hydrodynamic forces are defined in accordance with the known results ay : rie | . ‘ee . The general equations of motion obtained for an anchored ship are used for finding her equilibrium positions and analyzing stability of the same. Itis shown that the main reason inducing the ship to yaw is instability of her equilibrium position due to wind. Consideration is given to condi- tions in which stability of equilibrium is ensured for anchored vessels while periodic yawing and drifting is ruled out. 1, Coordinate systems and Nomenclature To solve the problem under review, four coordinate systems are used. Two of them are applied for the description of ship's motion in the horizontal plane, viz., the fixed coordinate system XOY with OX-axis directed oppositely to the wind and the origin O which coin- cides with the center of gravity (CG) of a non-diverted vessel, and the body axis system £047 with 0, -axis directed forward and the origin in CG. The 0; -axis is directed to port side. Figure 1 shows the directions of coordinate axes and positive directions of angle reading for the two systems. The notation B denotes a point of the anchor chain breaking away from the ground, Ho = initial position of the hawse-hole, H, = current position of the same. Two more coordinate systems (Figure 2) are required for the description of the anchor chain positioning in space. One of these, the s' Ax' system is situated in the plane of the anchor chain sag- ging. The origin A is madecadncident with the anchor lying on the ground. The As'-axis is directed vertically, while the horizontal axis is coincident with the ground plane and directed to the hawse-hole Ho. The other system of coordinates hoé is characterized by the fact that the vertical axis oh always passes through the point B where the anchor chain breaks away from the ground, and that the origin O is at a distance of 1080 Auto-osetllations of Anchored Vessels AL ater a (1) below the ground level. Here T denotes the horizontal component of the anchor chain tension and W is the weight per unit length of the chain submerged in water. Besides, the following designations are also used in this paper: i and si mass density of water and air, = acceleration due to gravity, Mes = length between perpendiculars, A = lateral area of the underwater body, Q = sail area, m = own mass of ship, Ma] and 50 = longitudinal and transverse added mass, Ji = ship's mass moment of inertia for cen- tral vertical axis, rg = added mass moment of inertia for the same axis, V = wind velocity, Nis = flow velocity, ao = angle between wind and flow directions, CA = center of sail area, Ee = abseissa of CA, eh and a} = hawse-hole coordinates in the body hh hh ; coordinate system £07 , i 1081 Gerasstmov, Pershitz and Rakhmanin aerodynamic force and its projections on the body axis, hydrodynamic force and its projections on the body axis, projections of horizontal component of the anchor chain tension on the body axis,’ aerodynamic moment about the central vertical axis, moment of resistance to ship's rotating about the central vertical axis, moment of anchor chain tension about the central vertical axis, angle between wind direction and ship's longitudinal axis; the amplitude value of the same angle, same angle at static equilibrium, projections of CG velocity on the body axis, lateral (normal to the wind) displacement of CG, displacement of CG towards the wind, displacement of hawse-hole towards the wind, lateral displacement of hawse-hole, projection of absolute displacement of ~hawse-hole onto the anchor chain sag- ging plane, angle between wind direction and the anchor chain sagging plane, depth of sea at anchorage, hawse-hole elevation over the sea bed. 1082 Auto-Osetllattons of Anchored Vessels 2. Basic assumptions The discussion of yaw and drift problem for the anchored ships is based on the following assumptions : 1, Itis assumed that the coupled pitching and heaving motions do not affect the ship's movement in the horizontal plane. 2. The magnitude of hydrodynamic forces is taken as in- dependent of athwartship inclinations. 3. In the estimation of inertial forces the vessel is con- sidered to be symmetric not only about the centerplane but also about the athwartship plane, and the center of gravity to be located in the athwartship plane. 4. In predicting the noninertial forces and moments acting on the vessel use is made of steadiness hypothesis. It is also assumed that the ship's movement is so slow that the anchor chain inertia forces can be neglected when determining the tension of the chain. 3. Differential equations of motion According to fag the differential equations of the ship's horizontal motion in the body coordinate system 0,7 can be writ- ten as follows: (m+,,) €- (m+,,) Vn ae ar eee ea eek) Vi B = iar 4 a= NB a See Neeries = HVE. (2) The right-hand side of equations (2) could most convenient- ly be written as the sums | 3 Ir W t 1 1 '] | 1 J 9) a0) J + ug) | =| cd Z 1083 Gerasstmov, Pershttz and Rakhmanin The terms included in the expressions (3) are determined by the aerodynamic forces acting upon the above-water body in the presence of wind, the anchor chain tension, and the noninertial hydro- dynamic forces generated on the underwater body during its motion. The inertial forces considered in this problem are taken into account in the left-hand side of equations (2). When defining the signs of formulae (3) it was thought that the forces and moments were cal- culated for the positive shifts. In equations (2) provision is made for taking account of the constant current in the vicinity of anchorage. For this purpose you need only to represent the CG velocity projections with respect to water in the form of the following obvious expressions (Figure 1. ): Vv F ht Woe Ce ete p eee pee sa le sea at * (4) In the absence of current V, and a, are equal to zero. Thus three unknown values can be derived directly from equations (2): yaw angle 6 and projections — and 7 of the CG velocity. In the fixed coordinate systems these projections will have the form 2 EVGos 8 = 4 Sime 0 4 YA & — Sin B + % CosB (5) By integrating expressions (5) time functions X,(t) and Y(t) can be found which determine the position of CG in space. The position of the hawse-hole can be found from the following obvious relationships : (1) = Cosh lar © Sick poi Ep Sin B (6) hh pa hh Along with the relationship for 6 (t) the functions of X(t), Yo(t), Xpn(t) and Ypplt) give rather a full idea of the yawing and drifting of an anchored vessel under the action of wind and current. 4, Estimation of aerodynamic forces Projections of aerodynamic forces on the axis of the body system of coordinates & 0; ». are defined by expressions 1084 Auto-Osctllattons of Anchored Vessels Safe = Cre — O"Cos: §. . — J fe) ve P 2 ne 5 Ossie ; (7) and the moment about the central vertical axis by 2 M = C¢ Seowinrus 1 a ma 2 In the latter expression G = 3S ceilaiGn tet fgew ble Syawpis (9) ma us where non-dimensional parameters 6, and C, are dependent on the relative position of CA and defined by the generalized curve (Figure 3) plotted against the data obtained from [2] and [3] . Irrespective of the CA position coefficients Gs é and Gs n may be considered as constant, viz. , CaO, ganda C1,,.9 -ni0m95-1e05) 5. Estimation of hydrodynamic forces The hydrodynamic force components, the longitudinal one Pine and the normal ing ,» as wellas the moment Mp originating during rotation of the vessel may be estimated approximately from the formulae = v anillh ae a S Sean ae P| +¥,|%,|)—4 = = oO 5 5 = a SA Pip) Cr ( ‘| + Vn | ¥, | peat (11) M, z eae ele 1085 Gerasstmov, Pershttz and Rakhmanin Here coefficients Cp and Chen are chosen in conformity with recommendations of Ref. [2] » and C,),, is determined by the expression Cc Dred iagadiapaey gr (12) n established under the assumption that the centre of ship's rotation in the horizontal plane coincides with the athwartship plane. The second terms of formulae (11) allow for the presence of current V: = + Cos(a_ + Bx), V, = V , Sint Fisk B }. (13). 6. Estimation of tension of the anchor chain At an arbitrary moment of time the longitudinal axis 0, g forms an angle ( 6 + ¢) with the anchor chain sagging plane (Fig. 1) Projections of tension T for the latter on the body axis will be expressed by the relationships Pry =D, Cie ein dete be Pr, = (T Sino (28 ox). (14) The moment of force T about the central vertical axis appears to be equal to M. = T-¢ r ny Sin ( 8 +) - Tanh-Cos( 8 + @ ). (15) The horizontal component of the anchor chain tension is represented by the sum i eee dy Pa AL Te (16) Here To is taken as T = c?, —\ vi, (17) fe) aé i 1086 Auto-Osetllattons of Anchored Vessels which corresponds to the longitudinal component Pee of the aero- dynamic force for B = ¢ = 0 . Thetension increment AT is estimated by the hawse-hole shifts AxX' in the course of drifting or yawing of the vessel. The curve of T against AX' plotted with allowance for the chain line characteristics is presented in the dimen- sionless form in Figure 4 for the case when K, = 52—=0 - Ae other cases the relationship of AK = f(A X') is easily determined, using the same figure, by shifting the origin along the curve to the point where the latter is intersected by the straight line K=K,. 7. The final form of differential equations of motion Taking into account the results given above and conver- ting the equations (2) to the form where the coefficients for the second derivatives of variables § , » , and 86 are equal to unit, the set of differential equations of motion for an anchored vessel in the presence of wind and current can be presented in the following final form which will be convenient for further analysis : ee e . 3 2 eine Pee ElE|+P,, Ve Cost a +B)|Cos( a +6)|- - Pp AkCos( 6 +¢)=P V Cus 4) oP Mia Gos 6 £3 £4 £4 mg B a al i|-P, Vi, Sin( a +6)|Sin(« +8) |+ Mp Ak Sings +o) =. P ay sin 6 (18) Zais ma ree Sin( B+¢)+ Fs 6 Be Pa, Vp Wait Pe, 38 |é [+ Pay AkSin( 6 +¢) = = PV sing -P,, v°|¢|sin 8 -P,, V" Sin( 6+¢) + B4 B5 B 6 +P Wig anise eieere 87 AkCos( Bt+¢) B8 In equations (18) the values of CG velocity,projections V¢ and V , are determined from formulae (4) and the following designat- ions are used : 1087 Gerasstmov, Pershitz and Rakhmanin Oo m+ .,, eh A P att = > Eig = ? mtr), 2 (m + iy (19) Oo WH a ES as alle pias. mtr), 2 (m + 1) oO m+), ce sa Eggi Ba gat ; mth, 2(m+,,) (20) oO r@) - yn ERS anJo Py 7 ? Py an ’ | 5 « > a m+ ? 2 (nwt k..) AG 2 (m+ A.) ZZ 22 22, oO 3 tA + aR MP h Le) ey Fal -- : B2 32 Z (ok ) T+ roe 66 WED iby cles + 6)F al. P a3 = : Pea = A I+ dy, 2(J+ 66) (21) oO yi Lae aye C ee age a ht ” 8g ¢ , 27 2(J + 66) 2(5+r¢-) Oo E de tee | : WE tun B7 , B8 2( T+ r¢) T+ hee 8. Equations of equilibrium Equations of equilibrium for anchored vessels subjected to wind and current action can be derived from differential equations (18) providing £ =» = 0; £ =-y7 = 03 B=. 0 and BP -e2 1088 Auto-Oscetllattons of Anchored Vessels The set of equations thus obtained makes it possible not only to define the equilibrium position of an anchored ship, with the wind and current prescribed, but also to follow the dependence of this position on the ship's particulars and the coordinates of the hawse- hole. In the absence of current (V,=0) this set of equations is reduced to a single equation which determines the angle of equi- librium 6, : | 8| Gr ohh 6 6 fase Ss Yl fee lez. i [ a » eG Me | sin p + re) (22) G n ae 9 ih - 0 + = E Cos 6 It follows from equation (22) that the angle 6. is dependent on the coordinates of the hawse-hole §}, and 7j, and the lengthwise position of the centre of sailarea 6, . The ordinate 7}; has no appreciable effect on the equilibrium position of the vessel. Setting 1th = © we shall find that angle 6, is equal to zero if the hawse- hole abscissa satisfies the condition ie O gp tageS a88 Yee: 6)4. (23) an Otherwise angle 8, is defined from the formula = ia ahaa ear tes) epi Sau ae! o E | an hh re) fe) Ss G 4 ; (24) 9. Stability of equilibrium positions A vessel may stay in the positions of equilibrium as defined above only on condition that these positions are stable. Considering the stability of the vessel with respect to yawing in con- formity with A.M. Liapunov's general theory [4] , the following criterion of stability can be obtained : 1089 Gerasstmov, Pershttz and Rakhmanin oO oO oO . 3 Sc [cae : Crt Cos hie pss fe Sin B tg By “an | Ly Co Ax, : Co Hi L 25 » te 8. (25) sg SOP SE” Bese on 2a vs : where AX, is the non-dimensional shifting of the hawse-hole in relation to the anchorage depth Hy as the ship passes from the state of rest in the absence of wind to an equilibrium position with the wind having the velocity of V. oh It can practically be assumed that Z. = 0.5. In this case angle 6, is equal to zero, which can easily be verified by using formula (23), and the criterion (25) is simplified taking the form Shh 2 —s— >( 6 + 6) co," (26) Taking into account the curves of Figure 3 it can easily be shown that for the conventional arrangement of the forward hawse-hole condition (26) is not met, i.e. in the absence of current the anchor- ed vessel subjected to wind will not be stable to angular deflections from the course. Instability of equilibrium of a vessel held in place by the anchor is the main cause of drifting and yawing, which in the absence of current and with constant wind have the nature of auto-oscillations which are symmetrical with respect to the wind directions. Fig. 5 shows the curves obtained by computer simulation of the set of equations (18), which characterize the auto-oscillations of the an- chored vessel B. = fs hh = 0.5; 6. = 0.07) subjected Ce > - ) j to constant wind ( V = 12 m/sec) in the absence of current b Vig 90). Under the simultaneous action of the wind and current the yawing becomes asymmetric with respect to the wind provided that the direction of the wind differs from that of the current. The average angle 8 9 and average shifting of the hawse-hole Yy, increase 1090 Auto-Osetllattons of Anchored Vessels with the increase in the flow velocity V, and angle a ,. The am- plitude of steady cyclic yaw is but slightly dependent on the flow para- meters, On the contrary, the amplitude of lateral displacement of the hawse-hole is substantially decreased with the increase of the flow velocity. The increase in the flow velocity leads, all other things being equal, to increasing angle 8B .,. In consequence, as is seen from expression (25), the position of the vessel's equilibrium may change from being instable in respect of yawing to a stable one, which will involve complete ceasing of its oscillations due to yawing and drifting. In the example given the oscillations of the electronic model of an anchored vessel ceased at a flow velocity exceeding 0.8 m/sec. . It is obvious from equations (18) that period ic of the oscillations under consideration is mainly dependent on the depth H at anchorage (Figure 6). At the same time there is a clearly defined dependence of this period on wind velocity. The latter result, however, needs to be explained additionally. In consequence of the ship's motions and wave action the resistance to drift P73 and yaw P 8 > must increase much like the resistance of a ship moving in a seaway, which is not taken into account by the set of equations (18). Additional resistance to drift and yaw in a seaway brings about an appreciable reduction in drif- ting velocity and, consequently, an increase in the period of auto- oscillations of an anchored vessel, all other things being equal. Hence, seaways may be considered as the cause of significant weake- ning of the relationship between the period of yawing oscillations and the velocity of wind. According to full-scale data, the period of oscillations due to strong wind slightly differs from that when the wind force is 3-4 (on Beaufort scale). 10. Ways to eliminate the auto-oscillations of anchored vessels Solution of equations (18) indicates that the intensity of auto-oscillations for the given depth at anchorage and wind velocity may be in direct relation to the extent of instability of the ship's equilibrium position. This latter is defined by the difference bet- ween the right-hand sides of inequalities (25) and (26). In similar anchorage conditions the left-hand side of these inequalities is sub- stantially dependent upon the position of the hawse-hole along the ship's length. The right-hand side of the inequalities is eventually characterized by the initial (for By = 0) value of the positional 1091 Gerasstmov, Pershttz and Rakhmantn aerodynamic derivative coefficient (9) : C = 5 A ‘pele’ casos (27) i.e. by the lengthwise position of the centre of sailarea. Figure 7 gives an indication of the relationship between the intensity of yawing and the extent of instability of the ship's equilibrium position. ‘The intensity of yawing is characterized in this figure by the relative amplitude 6B = versus the derivative Cha. Here Buy = dimensional amplitude fo yaw, [2 Se = dimensional amplitude of yaw for the vessel with —-= 5.0, 6,= 0.068. The curve of Bingo against the anchorage depth is presented in Figure 8. Thus the elimination of the wind-induced auto-oscillations of an anchored vessel may be brought about if stability of its equi- librium position is ensured. This latter can be ensured, as evidenc- ed by the analysis of condition (25), by shifting aft both the centre of sail area and the hawse-hole. This same condition, along with (26), gives the quantitative value of the required shifting of the above points. — When the hawse-hole is located near the forward perpen- dicular, the auto-oscillations of the anchored vessel subjected to wind may be eliminated at the cost of shifting the centre of sail area well aft. As angle 8, = 0 corresponds in this case to the ship's equilibrium position, and the mas ratio is rather large, so the stability of equilibrium position, a’s follows from inequality (26), can practically be ensured if the right-hand side of this inequality is close to zero or negative. This will be the case if 65 << 0, 25 (28) So considerable a shifting of the centre of sail area, however, adversely affects the controllability of the vessel in wind. The shifting of the hawse-hole aft of the forward perpen- dicular must be greater than that where the ship's equilibrium is possible with the value of By, different from zero. As the angle 8, increases, the instability of equilibrium position decreases, and at a certain value of (~;~~) Cr_ the position becomes stable, viz. inequality (25) is satisfied. Thus, with 1092 Auto-Osctllattons of Anchored Vessels Ew Shh seas F poses (29) the auto-oscillations of the anchored vessel are eliminated. Even so, this conclusion based on the analysis of small perturbation stability quite satisfactorily characterizes motion in general. The test results shown in Figure 9 (a) and (b), for an electronic model of an anchored vessel (-6, = 0. 068, —_ 5/0) give an idea of the effect the longitudinal arrangement of the hawse- hole has on the intensity of yaw and drift. The dashed lines in the region of unstable equilibrium represent the curves of yaw ampli- tudes against abscissa hh . During the tests no auto-oscillat- ions were observed at (2 — ) values beyond the left ends of these curves. The solid lines indicate the B, and Y}, parameters for the ship's equilibrium position. These relationships were cal- culated from static equilibrium equations (see section 8) ; the results obtained from equations (18) are illustrated by points on’ the solid lines. It is obvious that the results presented are in good agreement with the boundary of auto-oscillations region determined by calculation from formula (29). Setting 7}, = O for the critic- al abscissa of the hawse-hole the following formula can be obtained ohh Sa. ce ( ae cr roo oe CaL ol (30) In contrast, for eliminating the auto-oscillations of the anchored vessel by shifting the hawse-hole aft it is desirable that the centre of aerodynamic pressure (CP) should be shifted forward. Really, in the position of ship's equilibrium the line of aerodynamic action coincides with the anchor chain horizontal projection and passes through the hawse-hole. Hence, no auto- oscillations are present if the following inequalities are met simul- taneously : (31) 1093 Gerasstmov, Pershttz and Rakhmantn where a = abcissa of CP. But c iat a 2 = Se [6 o| ) (32) prs LL, Sir B eal 20 It is evident from relationships (24), (30) and (32) that if CP is shifted forward, given the position of the hawse-hole along the ship's length, this will involve an increase of 6, anda reduction of 2hh , which will allow satisfaction of the second inequality (31). So, the farther forward CP is displaced, the less is the necessity of shifting the hawse-hole aft of the stem so as to eliminate the wind- induced auto-oscillations of the anchored vessel. The shifting of the anchor hole aft of the stem is equivalent to springing the vessel as is accepted in maritime practice. Let us consider the scheme (Figure 10) showing the spring- ing technique. The lengths of the forward Hs H, and after Hz H, portions of the spring must be chosen so that in the ship's equilibrium position they will be tensioned. As long as the spring remains ten- sioned during oscillations, its presence will be equivalent to the hawse-hole shifting to point H,, , and the tension line of the anchor chain will intersect the centre line plane at point K which is coin- cident with CP. It is evident that the position of equilibrium will not be disturbed if a single anchor rope is secured to the vessel at the point H,; As far as research and fishing vessels are concerned for which lying at anchor at various places of the water area is the basic condition of operation, it may prove to be convenient that a special anchor gear be designed so that the point where the anchor chain is secured to the vessel is shifted aft of the stem when at station. This point must satisfy the conditions (31). In the case of a fishing vessel it was found that you need only to locate such a hawse-hole in the 1094 Auto-Osctllattons of Anchored Vessels shaded region (Figure 10) covering the centre of sail area in order to eliminate yawing of the anchored ship. This region is likely to be equal for ships which do not differ much in respect of the deck-house architecture. It is expected that such an anchor gear, if properly designed, will create favourable conditions for the operation of the above-mentioned ships. REFERENCES 1 BASSIN, A. M., ''Khodkost' i upravliayemost' sudov"' (Performance and controllability of ships), Izd. Transport Leningrad, 1968 (in Russian). Z VOLEKUNSKY, Yo. PERSHITZ,-R.Y:, LLEOV, 1.A-.; "Spravochnik po teorii korablia'' (Reference book on ship theory), Sudpromgiz, Leningrad, 1960 (in Russian). 3 GOFMAN, A.D., ZAIKOV, V.I., SEMIONOVA-TIAN- SHANSKAYA, A.V., "K raschetu upravliayemosti sudna pri vetre'' (Analysis of controllability of vessels subjected to wind), Trudy LIIVT, vyp. 81, Rechizdat, Leningrad, 1965 (in Russian). 4 LIAPUNOV, A.M., ''Obshchaya zadacha ob ustoichivosti dvizheniya" (General problem of stability of motion), ONTI, Moskva, 1933(in Russian). 5 SAVELOV, A.A., ''Ploskiye kriviye'' (Plane curves), GIFML, Moskva, 1960. MA BIOLOGICAL LABORATORY LIBRARY errr $5 W. he Cy. RINE 1095 Gerasstmov, Pershttz and Rakhmantn > Us A | 7 | “A | 4 | | oe | | of | mace aaiea | M, / C Sy \ » oN \ Figure 1 System of coordinates for the description of the anchored ship's motion. 1096 Auto-oscetllattons of Anchored Vessels ‘aoeds ut sutuotjtsod uteyo z0oyoue ay} JO UOTIAIIOSOp 9Y} TOF SsOJeUTPIOOD Jo uTa3sAG 7 OTNST AT eet same oe 5 en : {RQ oy ASS [ 2 *= ye PA Mg le, ax / | ‘ ia Se eee > y | 7 | | / ! | A | | l/ | v4 4 rary °7 V0 / 4 pel | / 9 ds @ 1097 Gerasstmov, Pershttz and Rakhmantn 1098 and 6. versus the lengthwise ° position of the centre of sail area. Figure 3 Coefficients C, Auto-osctllattons of Anchored Vessels : : et foal eee ———__| Figure 4 Dimensionless tension of the anchor chain against displacement of the hawse-hole. 1099 Gerasstmov, Pershttz and Rakhmantn Figure 5 Development of auto-oscillations of the anchored vessel (sea depth H = 100 m, wind velocity = 12 m/sec.) 1100 Auto-osetllattons of Anchored Vessels 1101 7 m/sec). Figure 6 Sea depth effect on the period of oscillations (wind velocity Gerasstmov, Pershttz and Rakhmanin ‘yuouIOUI OTUTeUAPOTEe FO DATJEATIOPp oy} Snsio9A Sutmed jo Ajtsuojur oyL L oinst 7 o£0 O20 OID aa 1102 Auto-osctllattons of Anchored Vessels LG sO i go oe a rg ge sme Oma ic eal OD a ig ‘ee ial eee sce ee on el fea 0 ee ee RF I ik Ee Pe a Ca: a el io is aa a a a Ae Ts [See ee eee ee Wee eee oS Oe eee Se eit mens 1103 Figure 8 Sea depth effect on the amplitude of yaw. Gerasstmov, Pershttz and Rakhmantn Figure 9 The effect of longitudinal shifting of the hawse-hole on the intensity of yaw (a) and drift (b). 1104 Auto-osetllattons of Anchored Vessels Figure 10 Scheme showing the springing system. 1105 Gerasstmov, Pershttz and Rakhmanin DISCUSSION J.C. Dern Bassin d'Essats des Carénes Paris, France I have been very interested by the paper we just listened to, for it explains phenomena that I have personnaly encountered during model tests. I would like to make a few remarks. In the introduction of the paper, it is said that the mean reason for the oscillations in the position of an anchored ship is the instability of her equilibrium position due to the wind. However, it seems to me that under certain conditions, current and waves may also create instabilities. For example the instability due to current is related to the course instability of a towed ship. This instability ap- pears not only with ships but also with buoys. Tests ona 2-ton coas- tal buoy have shown the presence of transverse oscillations when the current velocityyis less than 4 knots. Beyond 4 knots, however, these oscillations disappear. These results are in agreement with the conclusions of pa- ragraph 9. The authors assume that the coupled motions of pitching and heaving have no influence on the motions in the horizontal plane. How- ever, in paragraph 9, it is stated that the increase in the period of the sway of an anchored ship is probably due to waves. It seems to me that waves may have a great influence on the behaviour of an an- chored ship. For example, for very long waves, the ship may, under certain conditions, go up the waves and her hawser or chain may slacken partially or even completely. In this case, the wave length is the main parameter though it is not taken into account in the equations of the paper. In any case, waves change the conditions of applications of Liapounov's stability theory since when waves are present, the solu- tion of the motion equations is an oscillatory solution whose stability may notably differ from that of the steady solution of the no-wave case. In some cases, waves may have a stabilizing effect such that the instability oscillations disappear completely. In the case of the 1106 Auto-oscetllations of Anchored Vessels coastal buoy, which I mentionned earlier, the sway oscillations to- tally disappear in I meter waves. The authors discussed fully the influence of the position of the centre of pressure of the superstructures on the stability of the equilibrium position, I would like to know the effect of prestretching the chain on the ship oscillations. REPLY TO DISCUSSION N.N. Rakhmanin Kryloff Shtp Research Instttute Lentngrad, U.S.S.R. First of all I must thank Dr. Dern for his comments. I should like to make two remarks about them and to answer them. When we compare the behaviour of a vessel with a buoy we must remember that the buoy has a small LB ratio, which is about I. As to the conventional fishing vessel, her length breadth ratio is about 5 or more. In the Paper there is an indication that this LB ratio has an influence on the oscillation. To my mind these auto- oscillations of a conventional vessel and a buoy have a different cha- racter. The auto-oscillations of a vessel, I described in detail in the paper, are connected with the quality of a vessel to turn across the flow, when being the free floating body. Such body has a steady equi- librium position when it is located across the flow. te, As to the oscillations which one can observe at a buoy in the current, to my mind the cause of these oscillations is connected with another phenomenon. This phenomenon is the vortex separation be- hind the cylindric body of a buoy. The eddies separate in turn from the left-hand and the right-hand side of a buoy. As it appears to me that such character of the vortex separation causes the transverse periodic forces to excite the buoy oscillations. Another point concerns the comparison of the results of the theoretical calculations and that of the experimental data. I must say that this investigation has been started not from a theoretical consi- deration but from the full-scale observation of the fishing vessel 1107 Gerasstmov, Pershitz and Rakhmantn behaviour in the sea. The fishermen reported that when they observed the behaviour of a vessel in windy weather (the yawing angle and the drifting of the vessel), they were very surprised that the vessel turn- ed not only beam to the wind direction but even more ; the stern of the vessel had turned almost towards to the wind. We checked this behaviour in details and then set the problem theoretically, I must say that the agreement is good between the theoretical results and the full-scale data. The main purpose of this paper was the representation of the theoretical considerations of the phenomenon. I certainly agree with Dr. Dern that the presentation of the results of the comparison between the theoretical and the experimen- tal data would have been of interest. I regret that the authors were not able to check the theory against model tests in order to determine how well their theory is capable of predicting quantitatively the amplitude of the ship os- cillations. yy U.S. GOVERNMENT PRINTING OFFICE : 1975 O—565-528 1108 wien in Peas nn, ‘Sa: vein oa ie ait, weeawl ST OCRS a cae RUE ieuiesie: of thes ‘oe nel}, Uaky awe 1 Reba Tite tail et 0% na) ‘ baseitiy ed thee ole ay eRe soup sown : ius Baced bine barnes | BALRO AS WINE ee i wat 3 ne eas f my We thaeckhed thi + Beene z bi Tie TSW: fheometicaliy, < runtt, oF. yet tae ene aia : a : * ” ogh titan ne - &, ‘ = theoretioss greeulis Ama tye fadlinan peve Spl paper wae the reprepenmtige of the Suan on Fathi Bie i - ni: sida St a he iH i i totoeet ey ata haeyee if thee Ci a tear et Leeibe bestia) i ihe Hi tnt (vit ee pat i le pan ptaed Noakaal 1h bbe witha +1 cat Mou cate i Mie deDel edeneyly iit ot rey att 4 madd cb (\ 44 ae iy Pete i atgtat vy ci Met fs lode ne Hated) Ht ti \ (eatsite teat Ht di dAdaedsd dal seh stat th Hiatt : hea) ited iy Het cana lat Dian rises het oun I . miheciyts na #1 64 : nt aaa Pitot is ba Pine tala jaetsta Choe +4 i set bau it it} Ht 1 ii it rev Wieler rite = Hi 4 4) » Hthe} ye nett Hehehe a Hepat i ie it Wicks herigady aiAavessesth sprntitie tit tiehy (Heit Metih ia Heth Hanbd ade pret ctr ati 1 Hees fi hi ie iit itt Na Hal Hcvitayt a) What mM aaa fi it ‘ iT 2 i ' tate nahh +4) ih tai HE DANE rg: by Peele hie aie if 4 iH ce v ae Haat iy ie +e Has Hatt Wp tey it ai Hy 4 } act Hist fenethons He oe bites hed i} a seed ee sat hah an tits ai Tine nit Wikiven hall sath 161) Ried | abate eh ath ah th a vB Hey ae pies Hatt it ait tee Nit Per lit inh ty et ha ha “ ibe yah 7 ui) ; bibl eb 40 | Athi - Hi Pees yu 4 Hf thn ait }Awhhab hat hy th vet iit * ce HAAN HI hihi bh Ninh ty} ret Hi POR bbe ge) Hate pebrranitatel att ayy Sit atin i ie rah ath wis hid aan ee sf i it Auster fh put ' Pe yeed faite phy typ ieys aan 4 aE rt " iy t bb ti 4 tbe 9 ay) Bi ‘ Hi hy | abana IiHRah ae | pe jot) ete sh] = A Pat ied att Wis eye “4c a i ne Hits nie hey Aah ideheey sts slit Ul Feu bares NaS RR i iy sii Heth ‘t! dant i} a i Abe + He 4 iiyit Hi nh tea 4 hi . sdb ; bide ebay a rOyeay ti fr ue Hits i pe tH yy htt bo) feore a aS Ms Havithy we it o H al Hal Hitt tia! anette is nH ‘ Waveeeeben 7 thie | haat tani neat mye " bee abana sites Nien rite tt "y My Muitaes itd hae sts tte i bade te hae i aii hee tote Hirpienee a i ia Otis Hb) ee 1 HR weayeate baer tte a ae pee y it Is Be baie bp i Nad bitte f dle Hf, dr) iingh sinbe Hitt Mi Bey tier) myer Hi ne Priecey it Rete ttn) 14 inp lti hee Pe PeBe bem Ube orbs ae LEM Rr heg Ueeebene Galaaiiel citi tt vd Hy Hi PR Orbs J Oe) 1 hes Weel H fi MANE bi { Hea ya CAs En Destvepbateth seh tboie few AEE Fie boy ft Hthd he . sae, te ait me Nea lye abhi tee hn Pe bh he mepen fie fet) (eleben 1H ua whet siebeute a ate ae iit i Hine ee { hades HeHely it tap ft pikes fle nity f! iy sui hit hei Heath if me | 14 aa bh ihe) Pity: H| te ti ut) +) iH Ht redid i vere ved Wvieb eds} A ad ate) Nit ehh ney Dat ebheod, Ly9) iI irre a 7 DEE ep bret Fare ae bh Peoeerlions POR ao it the eh iy iN Fe} Pe Ci by FYE bil | HAL e hig Sen Piitieiis any ii (ari iH MH Hn Wi " ] iad rye Ar bts nati) Ho! (PMbRtitete it HOPE Meiiiplath any (au it ‘ i | Ma itt ioe a tat} nie iN Pr ebisah start ; Ni ‘ att bayabate Ua Hea a , mere aR shesiiny i] et Sabet k Resvataeacat ty UU lath tet a t tvs ait if rite dy i Apa it Hite bis) , it} At Hit t hi if tbh i Witt tit ity H$o4 fib seye Heit | sagan f\ He Watt ih ct haat CF tone fi ead i att } ited fate itt ti bat + Panat eh cta nts PPOE We he HM | ‘ ahah Nh Ki ; Hilts wit) uy MMA 4 (it) he + us te ate hie i a iis f sistas iitnd syorpeuepeie : Tlateirey! hh Atheps bene Hy iitat 1h Hit a tHebene hh ainettetey Abit roe: +f) weeny sitet lte ith a te Ls ia ae Maske tae by ia Uhie Pras be fri ey Hatt itt 4 igets Min} tit 7 Qed npi tye 0 bh pitt ey ‘put Hint sii iT nit Hpehepe peepee tebe fii beieieba ete iNtsiteneit hit if heel ty tbs *hab \4 iene WH SEA, rile Mae a jaeeneas rm U] | Ril rit efeheeney 7 ith Heh, iit # Hh Hitt HPL ee BAS BeAr hp ate type ay aati i NAL ‘ y PAP RDN eA i! + ni At vilbi te uy) ttt HPtTETL / tl i Neng ba) siti pees iii Meth araeih H. aL iy Tnait Hines paes masa aay , tight hes ty Lites mnie i fii AH EDEL eae Pah, heat ya tipe 4, Fist beh gt aise th abivaet these wed | i Hansuce ’ iis sh Gi abit ' a {t eat arin Heli Peet at be iyel iM My en +f Wil feet ‘ ann hi Hie aint MN DEPTH EP ADF iy Bete Hineety diy Hah + Lee | VOT OPE? Las theta, pened ied eG ete tpt Ht gp Ven beg iey, where Wien pee tes i} the hee Beet ety Pore be tls vert Crt Beli itis HMeeprerd a Hehe hi ity it 4 by i AY ee He Piet Ah OUR: a unta Mfsheheieht ¢ Saedats hat ii frorka bebe ti te ih ;

* Ko [2 %. y=b a’ 2, aneliech ) act where my, _ is the mass of the member A, and mp (6) is the mass distribution of the member B. If z > 0, the expression below the dotted line should be discarded. 2 2 M,(z) = M,(0) - © °m! (b ee ~h? + 2h % - aby - 227) (@) wile D y=b-a y=b+a -iof (oo 9+ wbp)| +(e 6 O,+ © 5)| f (5-2) dg Z y=b-a y=b+ (49) where m' (7 ) is the mass distribution of the member A. If z > 0, the expression below the dotted line should be discarded. y V(y) = V_(0) + 2? Png ay | ge en ea PaCS: NTT.) 487 Pten and Lee y 0? fmg Cad r= 9) bag = E eget atl Eo) dn (51) fe) Let us proceed to derive the expressions for the torsion moments T, and T, We assume that the twist center is loca- ted at the center of mass of the catamaran. The torque T5 and T3 are induced by out-of-phase hydrodynamic moments of yaw and pitch acting on the two hulls of a catamaran. From the equilibrium law, each hull should exercise equal magnitude but opposite torque load on the cross-deck structure. Thus, we candetermine T, and T, by considering the yaw and pitch moment contributed by dynamic effects on one hull. Utilizing the oi derived earlier, we can write AF in x F(x ) dx fi —- pn, al (52) and A, YT, = i ) dx + i x dx pn, at -A, Cp = Pg , xb.(x) () Ego We biggie oO (53) where - i Penh are the locations of the aft and fore perpendicu- lars; f,(x) and f, (x) are given by Equations (25) and (26), respecti- vely; Cp is the contour integral of a half cross section; p the Tie aes pressure and the expressions for Sc p njd for = 2, 3 are given by Equation (44); and b (x) is the half beam af ane cross section at x. I.4 - Results and Discussion. Numerical results from the theory developed in the preceed- ing sections are compared with available experimental results. The presentation of the results is made in the following order. The first part deals with the two-dimensional hydrodynamic coefficients. The second part deals with the three-dimensional hydrodynamic coeffi- cients. The third part deals with the heave and pitch motions of three catamaran models. Two of the models are of conventional-catamaran configuration and the other one is of a SWATHS configuration, Thedes- cription of these three catamarans is given in Table 2 and figure 5 488 Motton and Reststance of a Low-Waterplane Catamaran The strip approximations employed in our analysis of mo- tions and hydrodynamic loadings are based on a two-dimensional approximation of the fluid motion at each transverse cross section. A solution for the velocity potential associated with heaving, swaying, or rolling twin cylindrical bodies of arbitrary but uniform cross sectional forms has been developed by the method of source distri- bution. Lee, Jones, and Bedel (1971) show good agreement between theoretical and experimental values of heave added mass and damp- ing of four different types of twin cylinders. Figure 3 compares heave added mass and damping in nondimensional forms a33/( e+) and b53/( (py¥w) where V is the eee volume at the mean posi- tion versus the frequency number ° for twin rectangular cylinders. The dotted lines in Figures 3a and b are the theoretical results for one cylinder only. If the separation distance between the two cylin- ders is large, one would expect that the added mass and damping coef- ficients for the twin cylinders approach those for the single cylinder. Thus the difference between the solid and dotted curves in these two figures can be regarded as the measure of a hydrodynamic interac- tion between the two hulls. The added mass reflects the local beha- vior of the fluid motion near the body, whereas the damping is sensi- tive only to the farfield behavior of the fluid motion. Here in figures 3a and b we can observe that both local and far-field behavior of fluid motions generated by a single cylinder is quite different from the behavior of twin cylinders. Two types of singular solutions may occur at certain frequen- cies in the problem of oscillating twin cylinders. One is associated with a mutual blockage effect between two cylinders, and the other is associated with the method of singularity distributions; see John (1950). The former is of both mathematical and physical origin; the latter is strictly of mathematical origin and applies to both single and twin cylinders. The former type of singular behavior is shown in figure 3c atthe frequency number of about unity. The experimen- tal results seem to confirm the singular behavior. The frequencies at which such singular behavior occurs can be determined by ow a hse = Mada aa Sa LOG nee lee eae & 54 g (b/a - 1) (64) where the definition of a and b is as shown in figure 3a. The second type of singular behavior is shown in figure 3d by the solid curve. This type of singular behavior results from the break down of the solution of the Fredholm-type integral equation at 489 Pten and Lee certain eigen frequencies. Existence of such singular behavior in the solution of sec- tional hydrodynamic coefficients can present troubles in applying strip approximations to three-dimensional hydrodynamic coefficients. Removal of the second type of singular behavior has been achieved by imposing a rigid wall condition, i.e., o, = 0 or a pressure relief condition, i.e., @= 0, onthe line z = 0 inside cylinders? A mathe- matical proof of legitimacy for employing this technique will be publi- shed in the future. Figure 3d shows heave added mass of a rectan- gular cylinder obtained with and without the additional interior boun- dary in the solution of the boundary-value problem. Removal of the other type of singularity may not be possible unless a full threee-dimensional solution of the problem is achieved. However, for catamarans having the inner hull separation distance on the order of the beam of one hull, the lowest frequency at which this singularity occur may lie out of the practical range of interest in ship motions. For example, if we take b/a = 2, the longest cri- tical wave length A which can be encountered in head waves can be obtained by 42 Row ke = (55) . | 2 g(l+af/1+4U V5t?2 We) For a=15', we have U (knots) - (ft) 10 17.8 20 Glee 30 124. 1 Comparison of theoretical results with experimental results of three-dimensional added mass and damping for NSRDC Model 5061, the description of which is given in Table 2 and figure 5 , is shown in figure 4 for Froude numbers F, = 0 and 0.253 (10 knots). The three-dimensional added mass and damping are obtained from the two- dimensional added mass and damping by the strip approximation as 5 Employment of this technique was first made by Paul Wood of the University of California at Berkeley. 490 Motton and Reststance of a Low-Waterplane Catamaran shown in Table 1. The results in figure 4 are extracted from Jones (1972) and are nondimensional values defined by Ba | eas ns | eae 3/3. . aii 33 7 MY ¢g PAU, aa Sail Ba TES cub a BB 5B oa a eel oi M L MY gL des Ags ba Phe sis 2 Phy ML ML gL sieges 8 o=WYy— g The experimental results are taken at several amplitudes of oscilla- tion. Agreement between the theoretical and the experimental results is good for the zero-speed case, whereas some discrepancies can be observed for the case of F, = 0.253. Comparison of theoretical and experimental values of non- dimensional heave amplitude ey A and pitch amplitude Ev (2 A) versus wavelength \/L for the catamarans shown in figure 5 are presented in figures 6 through 8 . A is the wave amplitude, A is the wavelength, and L is the ship length. Most of the results shown in these figures are from Jones (1972). Unrealistically high-spiked theoretical values of heave and pitch amplitudes for 30 knots shown in figure 8 imply that damping values obtained from theory have been underestimated. The deficien- cy of theory may be traced to several assumptions or approximations made in the present analysis : the ideal-fluid assumption, the strip approximation of three-dimensional hydrodynamic coefficients, and the assumption of neglecting the second-order effect of coupling bet- ween the steady and oscillatory perturbation potentials. None of these assumptions can be removed without undergoing major renovations in the analytical procedures. Nevertheless, an attempt to introduce supplemental damp- ing in the equations of motion has been made by using a trial-and- error approach. The first approach attempted was to express the 491 Pten and Lee sectional heave-damping force in the form of Be Bs te) ee where W(t) = - -x ae , and c and c are constants. An expres- sion similar to Equation (56) is given in Thwaites (1960) for a slen- der body for a moderate angle of incidence in an unbounded fluid. The second term of the right side of Equation (56) is called viscous lift and the third term is called cross-flow drag. From the model test results, it was found that the damping obtained from theory for the case of zero speed seemed adequate. Thus, addition of the cross- flow drag which is independent of speed was considered unnecessary. However, the viscous-lift term, which depends on forward speed, seemed proper to be retained. Use of the test results for Cc; > given in Thwaites (1960, pp. 415-416), and a modification of the sectional heave damping obtained by adding c, U to b,3 , has not been successful. From this trial-and-error approach, it was learned that additional damping seemed to depend on a parameter wU/g. This parameter is the ratio of ship speed to the celerity of motion-gene- rated waves. When the ratio is less than one-fourth, there can be generated a chain of ring waves propagating ahead of the ship; see Wehausen and Laitone (1960, p 494). Dependence of the supplemental damping on this parameter implies that the hitherto neglected inter- action effect between two wave systems, one produced by oscillation and the other by translation of ship, is important. ,UW +c, |wl w (56) The strip approximation may exaggerate the effect of the hydrodynamic interaction between the two hulls. When a catamaran sails with a forward speed so that wU/g> 1/4, the motion-generated waves will be swept back by the forward speed. Especially between the two hulls, the steady horizontal flow can be accelerated by a chan- nel effect which leaves less chance for the oscillation-generated waves by the two hulls to interact in this region. To examine whether the foregoing postulation is true, the heave damping and heave and pitch amplitudes of Model 5061 and the demihull of this model are compared in figure 9 together with experimental results for twin hulls. There seems little change in the motion results between the twin and the demi-hulls except at the resonant wavelengths. At the shorter wavelengths, the heave damping of the demihull shows a bet- ter agreement with the experimental values. However, a similar comparison to that previously described for the heave and pitch amplitudes of 5266 at 30 knots revealed that the demihull has higher motion amplitudes at the resonant wavelengths than the twin hulls do. This seems to imply that for LWP catamarans, the underestimation 492 Motton and Reststance of a Low-Waterplane Catamaran of damping does not necessarily arise from an over-estimation of the mutual hydrodynamic interaction effects of the two hulls by the strip approximation. The dotted curves in figure 8 are obtained with modified heave and pitch damping values. These are obtained by modifying the sectional heave damping by b,,*(x) = b5, (x) + apwSp(x) WU where b,, (x) is the old heave damping at a cross section at x, p_ is the density of water, S,(x) is the sectional area and a = 3.0 for Model 5266. The constant a@ is a function of ship geometry and is obtained at present from the comparison of the theoretical and the experimental results of motion. The hydrodynamic coefficients affect- ed by this change are B53 A As, p Ax; , and Bee (Table 1). A further investigation to remove the discrepancy of the motion pre- diction at the resonant wavelengths by better techniques seems to be definitely necessary. The absolute and relative vertical motions of Model 5266 at a speed of 30 knots, which are computed with the modified damping, are shown in figure 10 . If we assume the height of the cross-deck of Model 5266 from the designed waterline is 30 feet, the water con- tact could be made when a sinusoidal incoming wave having an ampli- tude of 27 feet is encountered with an interval period of 18 seconds as the ship runs at 30 knots. The irregular-sea computation showed that the chance of water contact of the cross-deck of Model 5266 is zero toa significant wave height of 20 feet at 30 knots. The results presented in this section cover only part of the analysis made in part 1. Numerical results for the sway, roll and yaw motions and the hydrodynamic loads were not available at the time of this writing. I.5 - Concluding Remarks on the Prediction of Motion of Catamarans. 1. The strip approximation seems to yield a satisfactory motion prediction, except at the resonant wavelengths. The short- coming of the strip approximation is considered to arise from inabi- lity to account for the correct forward speed effect on the hydrody- namic coefficients over the range of resonant frequencies. The area to be improved in the theory seems to be the evaluation of the dam- ping coefficients. Proper incorporation of the interaction effect bet- ween the waves generated by the forward speed and the body oscilla- tion is considered to be the most important factor to be investigated. 2. Although an improvement of the analytical prediction at 495 Pten and Lee the resonant wavelengths should be made, with proper supplemental damping the prediction method can be utilized for parametric study of catamaran hull geometry with respect to seaworthiness characteris - tics. II - HULL FORM RESISTANCE AND DESIGN PROCEDURE. II ..1-A General Discussion of Catamaran Hull Resistance. A SWATHS is a special category of catamarans, First it will be appropriate to discuss a general catamaran, The resistance pro- blem for a catamaran hull configuration is far more complex than that ofa monohull. This complexity arises from two interference effects between the demihulls : a surface wave, which is familiar phenome- non, and the flow curvature, induced by the displacement of each demihull. This displacement interference exists even in the absence of a free surface. The pressure distribution over a single hull towed alone is different from that when one hull is towed alongside another hull in that the hydrodynamic property is significantly changed. To predict the combined resistance of two ship models towed side-by-side would be rather difficult, even if we knew all the hydrodynamic properties of each model when towed alone. In the first place, the stagnation points would be altered, and the flow on both sides of each hull would no longer be the same. As a result, a crossflow and a side force would be produced. Such a crossflow would increase eddying, and the side force would have additional resistance similar to the induced drag of a lifting body or lifting surface. For these reasons, the surface wave of each model is also altered due to the presence of another hull. Indeed, for a catamaran hull configuration, we are not only confronted with the added comple- xity in wavemaking resistance and viscous resistance but also witha new problem of induced drag. Faced with this situation, it seems very tempting to obtain a catamaran model series for prediction of resis- tance. Such a series, however, would be very expensive to accomplish. Furthermore, its usefulness would be very limited because the inter- ference effect depends not only upon the hull distance but also upon the hull geometry. The interference effects obtained from a catamaran model series could not be applied to a catamaran having a hull geome- try. that was different from that of the series. A catamaran series would only be useful if it had good resistance performance and if it were possible to confine our catamaran hull designs within the series Then we would be faced with the same problem in designing a catama- ran hull forms with good resistance performance. This is the subject 494 Motton and Reststance of a Low-Waterplane Catamaran of this part of the paper. II .2 - Design Procedure. Designing a catamaran hull form with good resistance quali- ties is a much easier problem than predicting the resistance of a given hull form. For instance, for the purpose of reducing the resis- tance, the crossflow around each demihull is eliminated. Hence, in a design problem, we have to deal only with the hull form without crossflow, and we need not be concerned with the complications crossflow would create. Since the basic flow around a demihull is not straight but is curved due to the presence of another demihull, the geometry of a demihull cannot have symmetry with respect to both sides if the crossflow is to be avoided. The amount of asymme- try depends upon the geometry of, as well as the distance between, the demihulls. Since hull geometry and hull distance are the objecti- ves of the design problem for a catamaran hull form, it is very diffi- cult to solve the problem in one step. To simplify the situation, a concept of an effective hull form is introduced as follows. Concept of an Effective Hull Form. We define a monohull in a straight uniform flow to be an effective hull for a demihull in a curved flow, if the flow relative to the hull is the same in both cases. If a demihull were towed alone and then towed with another demihull, a different effective hull form would result, even though its physical geometry had been kept the same. To maintain the same effective hull form, changes must be introduced to the hull geometry. Whenever the distance between demihulls is varied, a corresponding variation in hull geometry is also needed. Since the geometrical difference between a demihull and its effective hull can be determined after the effective hull geometry and the hull distance are given, it is rather logical to divide the de- sign problem into two steps. In the first step an effective hull form is developed, and in the second step the geometry of the demihull is determined. Designing an Effective Hull Form for the Demihull of a Catamaran. The viscous drag and the free wave system of a demihull are considered to be the same as those of its effective hull form. Hence, the resistance of a catamaran can be optimized by developing an optimum effective hull form. Since effective and geometric hull oo Pten and Lee forms are the same ina straight uniform flow, the design of an effective hull form is equivalent to the design of a monohull. There- fore, the concept of an effective hull form enables us to link together the design problems of a single hull and a catamaran. This link is an important step in the future development of catamaran technology. With this link, it becomes possible to utilize all the knowledge and information of single hull form design to the design of catamaran hull forms. In designing a single hull form, there are two possible approaches. One is the empirical approach, based on model series work and successful ships built in the past. The other approach is theoretical, the essential foundation of which is the wavemaking- resistance theory. With the first approach, many good hull forms have been developed. However, since it is not possible to know what makes good hull forms, this approach is very difficult when design- ing unusual hull forms for which there is very little available infor- mation. On the other hand, due to the oversimplification of a theory, the second approach cannot always produce satisfactory results. Perhaps the most rewarding approach would be a combination of the two and a great deal of intuition. At this point, we shall presume that a good monohull can be developed one way or another, so we shall not discuss the design of a single hull form further, except to make a few remarks pertinent to demihull design. The attractive feature of a catamaran hull configuration, as far as wavemaking resistance is concerned, is the added freedom in the displacement-volume distribution. In the case of a monohull, the required transverse stability limits the freedom of the displacement- volume distribution in the vertical direction. The benefit of wave cancellation can mainly be achieved by the proper longitudinal distri- bution of the displacement-volume. In the case of a catamaran, the transverse stability does not depend solely on the waterplane area of each demihull. Hence, we can have more freedom in distributing the displacement-volume in the vertical direction. It then becomes pos- sible to have cancellation between waves produced by displacement volume at various depths. In the case of a SWATHS, for ins- tance, it is possible to have wave cancellation between a strut and a submerged body. Since the distance between two demihulls can be varied to a certain extent, another freedom of displacement-volume distribution is obtained. Asa result, a far greater degree of wave cancellation is possible for a catamaran hull configuration than for a monohull. In designing an effective hull form for a demihull, the advantage of greater wave cancellation should be achieved. 496 Motton and Reststance of a Low-Waterplane Catamaran If the effective hull form is to be chosen from a group of existing monohull forms with good resistance qualities, the one with the lowest resistance value at design condition is not necessarily the best choice for the effective demihull form. The one with the lowest resistance at the design speed may not give the best wave cancella- tion between the two demihulls. The wave cancellation at a given Froude number and a given hull distance depends upon the free wave amplitude-spectrum distribution of each demihull. The wavemaking resistance of a catamaran depends not only upon the level of the free wave amplitude curve, but also upon the shape of such a curve. Befo- re the best choice can be made, it is necessary to obtain the ampli- tude-spectrum curve of each monohull, either experimentally or theoretically. Experimental methods are available for obtaining wave ''cuts'' from which a wave amplitude-spectrum curve can be deduced. By using the Douglas program and the existing wavemaking resistance theory, the wave amplitude-spectrum curve of a monohull can also be computed. If the possibility of obtaining the wave ampli- tude-spectrum curve does not exist, hull forms with pronounced hollows and humps in their C, curves should be avoided, even though the C, values at the design speed are relatively low. If a theoretical approach is used in designing an effective hull form, two singularity distributions, one for each demihull, are placed the desired distance apart. On the basis of linearized wave- making resistance theory, the surface wave of a catamaran is a li- near superposition of the waves produced by the demihulls. Within the constraints of design conditions, the optimum singularity distri- bution is obtained by minimizing the wavemaking resistance ofa catamaran. The final design, of course, has to be chosen on the basis of total resistance rather than on the basis of wavemaking resistance alone. By tracing a number of streamlines generated by one of the singularity distributions in a uniform flow, the hull geome- try of an effective hull is obtained. Based on theoretical insight and whatever practical informa- tion is available, we assume that the design of an effective hull for a demihull can be carried out successfully. Our remaining task is to find a way of obtaining the geometry of a demihull from the geometry of its effective hull form. This will be discussed in the next section. Developing the Geometry of a Demihull from its Effective Hull Form. Before discussing the procedure for obtaining the geometry of a demihull from the geometry of its effective hull form, let us first consider a two-dimensional, thin, symmetrical section. Ina straight 497 Pten and Lee uniform flow, there is no circulation on this symmetrical section. However, if we place this section in a curved flow, we have to curve the section in such a way that its original plane of symmetry coinci- des with the flow in order to maintain the stagnation points and thus avoid the creation of circulation on this section. Similarly, we have to modify the single-hull geometry in the presence of another demi- hull so that there is no shift in the stagnation points. In this case, however the beam of each demihull is relatively large in comparison with the distance between the demihulls. The flow curvatures on two sides of a demihull are quite different, and the usual practice of ad- ding thickness distribution to a mean cambered surface may not al- ways be applied here. Instead, we have chosen the following proce- dure. The Douglas program, used to compute the source distribu- tion, has been extended to trace offbody streamlines. Our objective is to obtain the distortion in the effective hull form placed at the location of one demihull in the presence of another for the purpose of obtaining its final geometry. We shall measure the distortion of the effective hull form due to the flow curvature with respect to the mid- ship section. In other words, the midship section of the demihull is identical to that of its effective hull form. Let us consider an after- body plan of the effective hull form placed at the midship section. If a number of points along a given station of this body plan are chosen as starting points of streamline tracing, all streamlines are parallel in a uniform flow in the absence of the other demihull, and the sta- tion section defined by these points will not be distorted. Due to the presence of the other demihull, however, the streamlines so traced will no longer be parallel but will be distorted according to the flow curvatures due to the other demihull. The points obtained by inter- secting these distorbed streamlines by a cross plane at the corres- ponding longitudinal location will then define the required hull cross section of the demihull. In this manner the afterbody of the demihull is obtained. Similarly, by placing a forebody plan at the midship section and by reversing the direction of the uniform flow at infinity, we can obtain the forebody of the demihull. A computing program has been developed, using the exis- ting Douglas program as a starting point for computing the body plan of a demihull. Using the offsets of the effective hull form as input data, the required offset table of the demihull is generated by this computing program. By using this computer program, an existing single hull can easily be converted into a demihull of a catamaran without changing its hydrodynamic properties. 498 Motton and Resistance of a Low-Waterplane Catamaran II. 3. - Design Examples - SWATHS. Remarks Pertinent to SWATHS Hull Form Design, The design procedure previously described can easily be followed in designing a SWATHS. Inthis case, a theoretical approach is necessary since there is very little information available. Due to the relatively simple geometry of a demihull, its singularity repre- sentation can be much simpler in comparison with that of a usual mo- nohull. A thin wall-sided strut can be approximately represented by a surface singularity distribution on a vertical plane with density vary- ing longitudinally but not vertically. The depth of such a distribution is the same as the strut draft. In addition, vertical line source and line doublet can be included to represent a strut bulb. A submerged hull is represented by a line-source distribution which generates a body of revolution in an infinite fluid with uniform flow. If the main body is not exactly a body of revolution, it is approximated by a line- source distribution which would generate a body of revolution having the same sectional area curve as the main hull. For a surface and a line singularity distribution on a com- mon vertical plane, the body generated in a uniform flow with a free surface would be symmetrical with respect to the vertical plane. The strut horizontal section would vary with depth, and the submer- ged main hull would sag in the middle. To retain the wall sidedness of the strut and to prevent the main hull from sagging, an additional singularity distribution is necessary. Since the main objective of a theoretical analysis is developing a hull form with good resistance performance rather than achieving an accurate resistance prediction, such an additional singularity has been ignored. In other words, each demihull is represented by a surface singularity distribution ona central plane and a longitudinal line-source distribution. First Design Example - A 100,000-Ton, 30-Knot, SWATHS Model 5266. Singularity Distributions and the Theoretical Wavemaking Resitance. Within the limits of constraints imposed on the hull geo- metry, an optimum singularity distribution of an effective hull form for both demihulls was determined. These constraints were specified based on functional requirements on ship displacement and space as well as ship motion. The optimization was done on the wavemaking resistance first, and then the total resistance was computed. Within 299 Pten and Lee the design constraints, hull parameters which influenced the wetted surface most were varied: The optimization of wavemaking resistance and the conputation of total resistance were repeated. After a few such repetitions, a final singularity distribution was chosen. The theoretical analysis was performed with the aid of an existing wavemaking reistance computer program. Since the sub- merged main-hull length was different from the strut length, some difficulty was encountered in using this program. For this reason, discrete source points, rather than a continuous line source, were used for a main hull. The final singularity distribution of this design is given as follows. Submerged main body: x +y —Z m 13 37/5) . 295 aA -000470 1.1250 =295 esecdut -000470 e150 295 Si weal -000470 1.1000 295 aylléra i -000470 1.0800 - 295 1411 -000470 1.0550 -295 ital -000470 1.0250 295 plus bil -000470 -9900 295 1411 000470 -9500 295 aA 000470 -9050 .295 ATL .-000470 8550 - 295 KIA TVA .000470 . 8000 295 CL4LL -000470 - 7400 295 1411 -000470 .6750 .295 1411 -000470 -6050 .295 1411 -000470 - 5300 .295 ait -000470 -.3300 3295 ehh. -.000235 —. 3832 RYE. 1411 -.000235 -.4363 -295 .1411 -.000235 -.4895 .295 1411 -.000235 -.5300 295 1411 -.000235 -.5426 ~295: SEWAIUIL -.000235 -.5958 295 WA TAL -.000235 -.6050 295 1411 -.000235 -.6490 .295 1411 -.000235 -.6750 . 295 Sent -.000235 -.7021 3295 1411 -.000235 -.7400 .295 1411 -.000235 -.7553 -295 a Asda -.000235 -.8000 295 1411 -.000235 -.8084 295 1411 -.000235 -.8550 .295 1411 -.000235 -.8620 205 -1411 -.000235 -.9050 295 replete -.000235 -.9148 - 295 -1411 -.000235 -.9500 ~295 Se -.000235 -.9679 25 1411 -.000235 -.9900 295 1411 -.000235 500 Motton and Reststance of a Low-Waterplane Catamaran x ty -Z m -1.0211 205) 1411 -.000235 -1.0254 . 295 1411 -.000235 -1.0554 3295 oA ib -.000235 -1.0742 ~ 295 = ali -.000235 -1.0854 5 28)5) pelea -.000235 -1.1054 295 - 1411 -.000235 -1.1154 295 1411 -.000235 -1.1254 295 1411 -.000235 -1.1274 ~295 ~ 1411 -,.000235 -1.1274 3295 seep Te -.000235 where x, y, 2Z are coordinates of a source point, and m_-is the point-source strength. The origin is taken at the midship section in the center of a catamaran at the undisturbed free surface. Sitmute: 2 3 4 3) MEae CuO cSoss leno exc e4OLes Gl Jaa On 52 Osni te Oe Ode ds for forebody with 0 < x <1 ine ee ite eoscle lt conscocle| otee cosas ee 1710, ee for afterbody with -1< x < 0 In addition, there is a line source of 0.0l at x=1 anda line sink of the same strength at x = -1 . The singularity strength is normalized on the base of a unit of forward ship speed and the strut singularity distribution length as two units. The depth of the previously described surface and line singularity distribution is OMO99: The wavemaking resistance coefficient Cy is normalized by wetted surface rather than the length square of the strut. For the purpose of direct comparison with the experimental residual resis- tance coefficient Cr. , it is plotted against V /Yu ; where L is related to the main hull length. The theoretical wavemaking resis- tance curve is shown in figure ll. Model Experimental Results The experimental hull form 501 Pten and Lee of each demihull was obtained in two steps. In the first step, an effec- tive hull form was derived from its singularity distribution bv the method of double-model or zero-Froude-number technique. Further simplification was‘introduced here by obtaining strut and main-hull geometries separately. A load waterline for the strut was computed from the strut singularities alone, and ali the other waterlines were the same as the load waterline. The main-hull geometry was obtai- ned by tracing a streamline generated by the line source points in an infinite medium. Then the strut and main-hull were joined together with the proper fillet. In the second step, the demihull geometry was computed from its effective hull form, and the singularity distribu- tion of the other demihull properly located. Due to the small width of the strut as compared with the flow curvature induced by the other demihull, the load waterplane was cambered by computing the dis- tortion of its centerline. To maintain the wall sidedness of the strut, this camberline was used at all the depths. Furthermore, it was also used to camber the submerged main-hull. For the purpose of internal functional arrangement and twin- screw propulsion arrangement, the main-hull cross sections were changed gradually from circular to elliptical sections, and a beaver- tail was formed at the stern. To increase propeller submergence, the tail end was slightly bent downward. Figure 12 shows the lines of this model, and figure 13 gives the hull characteristics. A 16-ft model was built and tested for resistance and powe- ring. The’’C, values were also plotted in figure 11 for compari- son. Second Design Example - Model 5276. Singularity Distributions and Theoretical Wavemaking Resitance. This design represents a 4000-ton, 30-knot ship. The singularity distribution for each main-hull was chosen to be the same as that of Model 5266. However, the main-hull geometry was kept as a body of revolution. The operating Froude number was much higher than that of Model 5266. Struts were redesigned to give better performance at operating condition. At low Froude number, it is possible to achieve great beneficial wave interference between struts and main-hulls. The theoretical wavemaking resistance of a SWATHS way have lower wavemaking resistance with than without struts, despite the added strut-displacement volumes, as for Model 5266. At high Froude number beyond the last hump, it is difficult to achieve such great 502 Motion and Reststance of a Low-Waterplane Catamaran beneficial wave interference between struts and main-hulls. After a number of computations, the following singularity distributions for each demihull strut were chosen : Forebody Surface Source 2 3 4 5 m(x) = 9.4498x - 69.5632x + 189.4110x - 213.4796x + 83.5701x Surface Doublet 4 De SrSreA) Peel Mer seee) with .4 a CR 2 w7a/g w?a/g c. Added Mass d. Added Mass cf a Rectangular Cylinder Figure 3 Two-dimensional heave added mass and damping coefficients. 507 Pten and Lee F,=0 F, = 0.253 0.25 — COMPUTED o 860.125" o ~—-0.250"" =0:25 y 0.375" 0.25 — COMPUTED = 0.159° C 0.318° 0.477° —0.25 F_.= F. = 0.253 — COMPUTED C) 0.159° ° 0.318° — COMPUTED o 0.125” Figure 4 Added mass and damping coefficients of.Model 5061 508 Motton and Reststance of a Low-Waterplane Catamaran TABLE 2 — Particular Dimensions Beam (Each Hull) in Feet at the Waterline Draft in Feet (Station 10) Ee Length in Feet at the Waterline Displacement of Each Hull 14,000 1386 in Long Tons (S.W.)* (S.W.) Hull Spacing in Feet** ES Longitudinal Center of 285.3 105.6 Gravity Aft of F. P. in 0.24L 0.233L 0.26L Feet Gyration in Feet [Block Coefficient «| 02 | 088 | 0.864 | [Scale Ratio —=S=s=‘g;=C‘ilé TB | eso | 600 | cet ae Sen ee eee wl *Salt Water **Between the inner hulls FOREBODY AFTBODY U-@ Figure 5 Body plans 509 Pten and Lee c8TS [2pow fo suotjowr youtd pue sAeoyYy o08L— 96z'0 = “4 g einst7 g3aindWwoo — LNAWIYAdx3 © 510 Motton and Reststance of a Low-Waterplane Catamaran 15 KNOTS THEORY EXPERIMENT oO 15 KNOTS Figure 7 Heave and pitch motions of Model 5061 BEE Pten and Lee 30 KNOTS 0 fs A ene evelene. THE ORY 0 KNOT 30 KNOTS — _) sWITH) ADDED DAMPING EXPERIMENT Oo ) tsi 27A Figure 8 Heave and pitch motions of Model 5266 5FZ Motton and Reststance of a Low-Waterplane Catamaran HEAVE 15 KNOTS SPEED 15 KNOTS THEORY TWIN HULLS ——— DEMIHULL EXPERIMENT O eel] 33 0 1 2 3 4 X/L WL a. Damping b. Heave Amplitude PITCH 15 KNOTS WL c. Pitch Amplitude Figure 9 Comparison between twin hulls and demihull of Model 5061 513 Figure 10 Pten and Lee SPEED 30 KNOTS RELATIVE ABSOLUTE 4 6 8 10 12 14 16 18 20 ENCOUNTERING PERIOD (SEC) Absolute and relative vertical motion of the front edge of cross-deck of Model 5266 514 22 992ZS [®poyW TOF soaano dIHSGIW IV STINHINGG 4O YAINAD NAAMIAG) “3¥ 9TZ ONIOVdS TINH Motton and Resistance of a Low-Waterplane Catamaran I ) pue Mw >) uereurezen I T puo) ustsaq ct ° t=) c00°0 BL) Pten and Lee uereurezeos (MT 107 uetd Apog pue soul], poyetasriqqy 9925S [epow Aq pejuesoadar ZT ernst 516 Motton and Reststance of a Low-Waterplane Catamaran APPENDAGES: None DIMENS LONS* LOA COEFFICIENTS* SHIP MODEL ; Length (LOA) FT 850 770 C, 0.616 Coe 0.82 Length (LBP) FT — —— C, 0.776 ON 0.88 Beam (B,,) FT 70 1.4 Cy, 0.794 L,/L 0.50 Draft (H) FT 38 0.76 C, 0-846 L/L 0.0 Displ.in Tons 39790S.W. 0.310F.W. C,, 0.80 L,/L 0.50 Wetted Surf.Sq Ft 117097 46.84 Coa O72 L/B,, 12.14 Design V in Kts 30 4.24 Cop 0-80 Biy/H 1.84 LCB, 4, 0-491 Aft of F. End Cop 0-75 A/(OILY 64.79 LCF, 4 0-517 ° Aft of F. End Coy 9-73 S/V AL 20.13 d = 50.0 Te) BOG Coy, 0-68 ® = 2.995 ® = 0.872 beck (hot 78 LINES: NSRDC Shop Plans. Model 5266 *NOTE: The coefficients and dimensions presented above are for that portion of a single hull that is below the 38 ft waterline. This draft corresponds to the total height (minor axis) of the largest elliptical body section. The total full-scale displacement, wetted surface, and draft, for a single hull, are 50,500 tons S.W., 164,780 sq ft, and 69.5 ft, respectively. That portion of the full-scale hull above the 38 ft waterline (strut) has a waterline length (LWL) of 757.5 ft, a beam (B 2) of 30.4 ft, and a waterplane coefficient (¢,) of 0.515. ETRE TEE SGI BE ERE Ete EHLERS se ceeaaeeel taeda pda es nee HUGE BET EE GG EP ait CSRSH BE ee rN i ee ne + +? +htt Bee altel 4 odue g ’ ttt tber +tetheye tr settbr pt teteeee tr +tre ttPr t t tttttrrs ttt ' + 4 + Ht RRS fete H trtt thir Oe ne He 20 16 2 8 4 0 STATIONS Figure 13. Ship and model data for a LWP catamaran represented by Model 5266 SRT Pten and Lee 2 ay 2 ee | ~ 518 Profile and strut section for an LWP catamaran demihull represented by Model 5276 Figure 14 Motion and Reststance of a Low-Waterplane Catamaran PHOTOGRAPH LAY-UP SHEET NDW-NSRDC-10700/10 (6140) SHIP AND MODEL DATA A LWP eAMeAn RARER EAE BY MODEL 5276 APPENDAGES ; NONE DIMENSIONS * LOA COEFFICIENTS * | _MODEL__[Cx 0.602 Cwr 0.88 LENGTH (LOA) FT 287 14.07 p 0.758 Cwa 0.81 LENGTH (LBP) FT Saas | a en CBE Lg/L 0.36 SEAM (Oy) #7 Cy ORT py 0-2? ORAFT (Hl FT peO-80 g/t 0-42 DISPL. IN TONS | 1486 sw.| ‘| Cpao. 71 L/ Bx16.55 WETTED SURF SQ FT. Cpe 0.73 Bx/H 1.0 DESIGN V IN KTS. ERR ee Cpr 0.66 A/Lon)362.9 LCB, on = 0-479 art of F. End 1Cpy0.71 S/¥AL 19.12 LCFioa = 0.486 AFT OF F. End Cpvad. 72 f WL ENTRANCE HALF ANGLE = pve 0.73 t h# 20.4 Vi Se LBP COEFFICIENTS Qe 5.525 (®): 1.619 B L/Bx “LINES Cp Q4/(oiL)$§ *NOTE: 1, The coefficients and dimensions presented above are for that portion of a single hull that is below the 17.3 ft waterline. This draft corresponds to the total height at the top of the body of revolution. The total full-scale displacement, wetted surface, and draft, for a single hull, are 2026 tons S.W., 19197 sq. ft., and 32 ft., respectively. That portion of the full-scale hull above the 17.3 ft. waterline (strut) has a waterline length (LWL) of 226.5 ft., a beam (By rote Se Onis a andaral water plane coefficient (C_ ) o£ 0709): 2, Sectional area curve same as Model 5266 For detail see Figure 2-3 Figure 15 Ship and model data for a LWP catamaran represented by Model 5276 En. Pten and Lee } Hey + ti} + HEN i 2) esi Hdsgtacedt teas : pa peat saeee eanne T4444 SEs o Af ee : ce > em an ee ee ATT Eile Ih iL ESE) ESE ee EH 520 POU ULE ea py 1444 {| LE at tHt ttt rH FET Fen EE re HEE REE ro tt tt 3 . t Bak a it { By ee ee curves for Model 5276 tested asa ee and C “ single hull Cc Figure 16 Motton and Reststance of a Low-Waterplane Catamaran 9-9L72ZS TSPOW ‘Tiny eTsuts e AOF UOTIDES 4NI4s puke oT[IjJOIg " 4) £1 +! si LT canst ay Ai iu Hine NT EE | te HA TUTE HER te one ne HEHE a ESE EE EEL Pten and Lee 522 curves for Model 5276-C tested as and C le ae Ww, a sing Cc Figure 18 Motton and Reststance of a Low-Waterplane Catamaran A-9LZG [PPOW ‘T[NY eTSUTS e TOF UOTJOES 4naA4s puUe oTTjoIgG 61 ernst ga 523 IIny e[suIs e se poise} F-9172gG TePOW OF seaArno are) pue 95 OZ eanstyq °C (a4 0°¢ eae ee Wel Cap O°T Pten and Lee se = A SS + id: Suess ass sant sensi comes cessceveet tate rats 3 aEa0 GE pomee seees coat + } : Beaapeesabred: T00°0 os yy some = |_(Livad Li ze) JI : west (Lavud Li 8Z) oj; =fe at assess nest oan isstessss ts : Z00°0 ttt + eae $ - ssscaesess seueecegaaaesueuae (lava Li ZE) 9 a3: + POR sacascscss: : + | (Lavud La 82) 9 (5 €00°0 & oases soseveeaes feessess . i a +1 +. — Ze + eal ai ee: aes 36 paoaet =} ee anans SEBSs +4 E pas a fo) eres = ssesuesees sees: [e) > + te age 5 gas beat bans: a + a : oe Lo ++ ae i fe) eee 2 = A se: + Sages = reese +t = a 524 DEE E Dadate: oc U EU WEBS 2 Oe ge Colle mw ‘)-9275 TOPOW usemjoq seaArTnd Oo pue OO jo suostaeduroD Iz eansta Motton and Reststance of a Low-Waterplane Catamaran 27°C ce 1 0'C 8°T eal 7m GI rir (0-947S) 9 q-9/7S) “9 uoTI_puCD usTseq 1T00°0O c00°0 £00°0 525 Pten and Lee 526 0.002 6.001 ‘amid G curves between Model 5276-C, hull at 28-ft draft w 7 -E and a main Comparisons of C Model 5276 Figure 22 Motton and Reststance of a Low-Waterplane Catamaran APPENDIX A - DETERMINATION OF HYDRODYNAMIC COEFFI- CIENTS. If we assume that the fluid surrounding a catamaran has irrotational motion, we can introduce a velocity potential Gi) (x, Y,Z, t) in the fluid region. The velocity potential should satisfy, in addition to the Laplace equation, the following boundary conditions z (—9-- U2) Ox, y, z t)+ eG =0 on z = 0 (57) where g is the gravitational acceleration, : = 8 Lata ae lg on S (58) where n is the unit normal vector on the body surface pointing into the body, V, is the normal component of the velocity of the body surface, and S,_ is the mean position of the body, W) (xy, = oo", *t)' = *0 (59) =i and a physically appropriate far-field condition for (x? ota y? j2Z—7 2 to make the problem well posed. Assuming the flow disturbance to be a small perturbation from uniform flow, we can express “) in the form @ = - Ux +, (x, y> z) +O. (x, Ne aie RoE (60) where U is the forward velocity of the ship; ~, is the steady potential, representing the wavemaking disturbance due to the for- ward velocity; and 0, =e ty Or is the potential associated with the oscillatory fluid disturbance. We can further divide O, into three distinct origins of the oscillatory fluid disturbance as OF Beer et ol Pa (61) The incoming wave potential, o, is given by K (z+jxcosB-jy sinB) Or ieadhen 067-0 (62) where 8 is the wave-heading angle with respect to the positive 527 Pten and Lee X-axiS, Wo is the wave frequency, K, is the wave number given by wé / g, and A is the wave amplitude. The wave diffraction is represented by P>) and the fluid disturbance caused by the motion of the body in initially calm water is represented by Ou . Within a linear approximation to the solution of the velocity potential D., we can let 6 ORDO ai M k-2 ko k (63) where o, is another set of velocity potentials, and ie Ki 25a 6 are the complex amplitudes of the displacement of the body from its mean position in sway, heave, roll, pitch, and yaw modes, res- pectively. The pressure at any point of the hull is obtained from Bernoulli's equation by p= SU- oO +g x(t) +31 VG" ) (64) At this point, we will establish the following conditions : (1) the motion of the body is small, so that the pressure at a point on the body surface at any instant can be obtained via Taylor's expan- sion of the pressure at the mean position of the body; (2) the terms oR OG, Ol 3 Fo. EO. E.,) will be discarded in the evaluation of the pressure; (3) only those components of the pressure which have harmonic time dependence will be considered, and (4) the static-pressure component and the component which contributes to the static restoring force or moment of the body will not be inclu- ded in evaluation of the pressure® With the foregoing conditions, the complex amplitude of the pressure at a point on the body surface can be expressed by Ox fo) 2 (65) p= (jo + uS-) (+0) + 2, nae evaluated at the mean position of the body. 6 These conditions are also applied to monohull ships by Salvesen, Tuck and Faltinsen (1970). 528 Motton and Reststance of a Low-Waterplane Catamaran Integration of the pressure over the wetted surface of a catamaran should yield the hydrodynamic forces and moments. Thus, we have (H) F. = p n. ds So 6 = j ay a7 + le (jw JS ) (, + , a8 ko x) 28 So fornia 21 Sen Ebon Osmavihie he no and nz are respectively the y and z components of the unit normal vector and ny = yn3 - 2n2, Ns = -xn3 , ng = XN. We can further decompose the hydrodynamic force into two parts i.e. P(e) te, pen) i 1 1 where pie = wave-excited force - off, (jo + ug) (), + 9) ds (66) and o 22 (m) = motion-excited force 6 Dial fhe (jw + v2) o, ds (67) = So Applying the results of Ogilvie and Tuck (1969) to the simpli- fied case in this work, we can show for any differentiable scalar function ) with the usage of Kronecker's delta function that Ino dis [f . ee Say sce ff, n. Dal (68) S S) C(x) where S is the immersed hull surface forward of the cross section at x, and c(x) is the line integral along the contour of the cross section. Utilization of Equation (68) in Equation (66) yields 529 Pten and Lee r(e) -|f n, (jo + v9) @, ‘ o,) ds so = jn, (¢ + UK) cos 6B) O,ds S elf n,+Un, bis -Un, 5.) o, ds (69) So where the line integral at the stern section is neglected. Within the accuracy of the order of approximation in this work and together with a slenderness assumption of each hull of the catamaran, we can derive from Equation (58) and (61) that seer ye (5 +" Oeil). m= {x ey) on S| (70) where Pgh oe Fc dea, = (x, y> z) and @j is the unit vector inthe x direction. Substitution of Equa- tion (63) into Equation (70) yields 9,77 Je n +Un, 6,,-Un, 8 6 (71) Using Equation (71) and the relation ”+ UK _ cos Be ey oO Oo in Equation (69), we get 530 Motton and Reststance of a Low-Waterplane Catamaran So Since Ifo. 5 ges Ifo, 9. us So So by Green's theorem and o = =O on S In Dn re) from the kinematic boundary condition, we can show that (e) _ ; eu 2 5 ya fone (Oa ?s. fer ys Pre an ne So The above procedure for eliminating the diffraction potential D from the expression for the wave-exciting force and moment was first shown by Haskind (1957) for zero speed and later it was exten- ded by Newman (1965) for the case of forward speeds and is referred to as the Haskind-Newman relation. Similarly, we can derive (aa) 6 a p 2 Fo (j w n. + Un, b.5 = Un, 556) >, ds = ote NS (73) 6 2U 2U I (8 Tay ayn SCG eee big), ds (m) So i in the form | L} > nh Str a (e) If we express F 6 F (™) = pe: ( oA. + jwB. i kz2 ko joBy ) k where Aj, are the so-called added mass quantities, and Bx are the damping quantities, we find from Equation (73) that a p eu al ae Res { ff, bene oF 6 is =e Oe 56) ae ds} (4) 551 Pten and Lee Bie 13 SHI, mee co oe -= 6, 5:4)? as} (15) Jw Using Newton's second law, we can show that the equations of motion can be expressed in the form z = eee (m) (s) pas x Me Re am en ae oe where M 0 -Mz 0 0 g 0 M 0 0 0 M,, = Mz, 0 ly 0 0 0 0 0 0 7 0 0 0 0 is z is the z-coordinate of the center of mass, and ie = Hydrostatic restoring force 0 0 0 0 0 0) -C,, 0 -C,, 0 = 0 0 sis! 0 0 0 -C., 0 0 -C.. 0 0 0 0 0 Thus, the equations reduce to the algebraic equation : 6 2 hfe) 2a eae le ee a poe SZ (76) (77) Motton and Reststance of a Low-Waterplane Catamaran where A, , Bix and F; 5 are frequency and speed dependent. For slender catamarans we can assume that no coupling exists between the motions on the horizontal and vertical planes so that or ee =o for We = 25:94, 6 As has been seen previously, the hydrodynamic coefficients appearing in the equations of motion can be obtained, if the solution of the velocity potentials 0; (i= 2, ..., 6) are known. In the solu- tion of ; , the flow around each transverse section is assumed to be two-dimensional, and, thus, the variable x enters as a para- meter in the expressions for Qj . A correct mathematical deve- lopment to lead to such an approximation from the slender-body theory is given by Ogilvie and Tuck (1969), and a comprehensive review of this approximation is given in Newman (1970). Approximation of three-dimensional flow by two-dimensio- nal flow as described previously is often called the strip approxima- tion. The strip approximation of the hydrodynamic coefficients has been quite successful in the prediction of motions of monohull ships. Regardless of the possibility that the two-body interference of the flow between the hulls may weaken the two-dimensional approxima- tion, the strip approximation is applied in this work to check both the reliability of the approximation and the areas which might be improved should the approximation prove unreliable. The assumption of linear excitation-response relationship and the strip approximation lead finally to determination of the hydro- dynamic coefficients shown in Table 1. The solution of two-dimen- sional added inertia and damping is obtained by using the method of source distribution. The description of the method for heave added mass and damping is given in Lee, Jones, and Bedel (1971). For illustration, the derivation of As3 a B., and Ba. will be shown here. From Equation (74) and the strip approximation, we have 533 Pten and Lee p ZU A = = ———— +— 53 SE ii [= (9, ae an) 93 4 L C(x) where means an integral over the length of the body from the aft to the fore perpendicular. From Equation (7!) we have . a Ree ae Ua, y On =-jen, + Un, Tide (x Pr a )= -(x er ) o,. Thus, A jamie ela bk ae Jo. oe 53 we j $n. 3 C(x) Ags fh - i x dx { 3 Re >, o,, a C(x) Be ek Q + 2 \- Im dx o, >, d L cx) = |. a5 Gepax® + —s B,, (2 where siatvad £ a.5 (x) re Re. | o, o, d C(x) Similarly, we can show, using Equation (75), that m p 2U PyauS vorigiat ey [ «=| (Ps, jw D sex) Ds 1 L C(x) Ss paste ae. eb 4 im, f if (-x + )>, 9, 4 | L C(x) xiv i eh = po Pag (x) dx - U { =o Re, dx , o,. at} if rE C(x) = x b,, (x) dx -UA,, where b3 (x) = - —— oo [ ?, >, af Motton and Reststance of a Low-Waterplane Catamaran From Equation ( we have By, = - qos fate al C(x) Since by inspection we can deduce of Equation (71) that U OF = (x TBaio ) 0, we can show that Bog = [« {- Sim, >, , at jk F C(x) i uf Sees | ee at L C(x) xb,, (x) dx - UA,, L APPENDIX B - SOLUTION OF DIFFRACTION POTENTIAL In the following, the solution of the diffraction potential o will be obtained in the Oyz plane. The boundary-value problem for or is described as by Oz OF ae On On 0 (78) where Kiy(ec or fg Un mrey oh (79) Co Co where cp, is the contour of the immersed portion of a catamaran cross section located at x , and the incoming wave potential ®, is given by ogee Ag e) Ko x cos B eo (z -jy sin 6 ) (80) 5315 Pten and Lee where Since the variable x enters into this problem as a parameter, we take e 7*o*%°0SB as a constant term and let it be denoted by c o- pe K2tikoly| as |y|—2 (81) where B is a constant. Substituting Equation (80) into Equation (79), we obtain | =- AC (-jn, sin B tn, Jeno - ix eam Co fe) z 3 (82) where nN» and nj, are the y and z components of the unit normal vector on cy, . For brevity, we let y’ -saypem? n, = ci sin B A'=-=- w AC ° = ! Oo Z ! A'e (n, > 1 (83) K '- in K i cos K y'- nj) sin aan K avyaAtNe ? 2 (n! cosK y'tn 7 ! > sin Ky ) 3 Since we assume that catamarans are made of two symme- trical hulls, we know that nz is odd, and n3 is even, with respect to y . Then, it is clear that the real part of the right side of Equa- tion (83) is an even function, and the imaginary part is an odd func- tion. Now, we let 536 Motton and Reststance of a Low-Waterplane Catamaran where a) and on are even and odd functions in y , respectively, and satisfy the following body boundary conditions (n, cos Ky =, sin K | y') (84) leQ ta a2 ah ee ' oO ' ; ' j ' = i= JAN ie ( n, cos K y'+n, sin Koy ) (85) The potentials o. and 0, also independently satisfy the remaining conditions prescribed for ©Mp . With this arrangement, we can easi- ly relate De to the problem of heaving twin cylinders and relate , to the problem of swaying twin cylinders with the only trivial differen- ce being the magnitudes on the right side of Equations (84) and (85). The solution for oscillating twin cylinders can be obtained by using the method of source distribution along the contour Cog The expression for the source is given by Se 4 |e {y-n) #2 ob yt 2)? + (ee 2)" GE, per (Ga eet eh Wee ee 1 co k(zt ©) ce oe fooe k y- 7) + cos k (yt 1) + ak 0 (@) wee (ee Sa) { cos K, (y- 1) #08 K, (yt) f (86) where the plus sign corresponds to heave (G,) and the minus sign cor- responds to sway (Go) , and f£“ means Cauchy's principal-value integral. Let >. = f Q.G df (87) CR @, = - Ones af (88) where Cp is the integral along the immersed contour of the right- half of the cross section, and Qe and Qo are the unknown source strengths. Applying the boundary conditions given by Equations (84) and (85) to Equations (87) and (88), respectively, and solving the integral equations for Q, and Q, , we obtain the solutions of %. Sei Pten and Lee and Py and, hence, the solution of 05 ACKNOWLEDGMENTS. The authors would like to express their thanks to Mr. J.B. Hadler for his encouragement and support of this work. The authors are also indebted to those who have contributed their experimental results to guide the analytical developments described in this work. The authors would further like to acknowledge the funding support received from the following programs during the course of their work : The in-house Independent Research Program of the Naval Ship Research and Development Center, and the Research, Development, Test and Evaluation Program and the General Hydro- mechanics Research Program of the Naval Ship Systems Command. 538 Motion and Reststance of a Low-Waterplane Catamaran REFERENCES Cummins, W.E., ''The Impulse Response Function and Ship Motions", Schiffstechnik, Vol. 9 (1962) pp 101-109; reprint- ed as NSRDC Report 1661. ” Faltinsen, O., "A Rational Strip Theory of Ship Motions; part II 3 Report 113, College of Engineering, University of Michigan (1971). Frank, W., "Oscillation of Cylinders In or Below the Freee Sur - face of Deep Fluids'', NSRDC Report 2375 (1967). Haskind, M.D., ''The Exciting Forces and Wetting of Ships in Waves" Isvestia Akademic Nauk SSSR, Otdelenie Teknicheskikh Nauk, No. 7 (1957); NSRDC T-307 (1962). John, F., ''On the Motion of Floating Bodies : II]. Simple Harmonic Motions, '' Communications of Pure and Applied Mathema- tics, Vol. 3 (1950) pp. 45-101. Jones, H.D., ''Catamaran Motion Predictions in Regular Waves", NSRDC Report 3700 (1972). Lee, C.M., Jones, H., and Bedel, J.W., ''Added Mass and Damping Coefficients of Heaving Twin Cylinders in a Free Surface}' NSRDC Report 3695 (1971). Motora, S. and Koyama, T., 'Wave-excitationless Ship Forms }' The Sixth Naval Hydrodynamic Symposium, Washington, D.C. (1966) pp. 383-411. Newman, J.N., ''The Exciting Forces on Fixed Bodies in Waves", Journal of Ship Research, Vol. 6, No. 3 (1962) pp. 10-17. Newman, J.N., ''Applications of Slender-Body Theory in Ship Hydro- dynamic", Annual Review of Fluid Mechanics, Annual Reviews, Inc. Palo Alto, U.S.A., Vol. 2, (1970) pp. 67-94. Newman, J.N., ''The Exciting Forces on a Moving Body in Waves'; Journal of Ship Research, Vol. 9, No. 3 (1965), pp. 190- NS: Ochi, M.K., ''Prediction of Occurrence and Severity of Ship Slam- ming at Sea'; The fifth Naval Hydrodynamics Symposium, Bergen, Norway (1964) pp. 545-595. 53 Pten and Lee Ogilvie, T.F. and Tuck, E.O., "A Rational Strip Theory of Ship Motions; Part 1'' Report 013, College of Engineering, Uni- versity of Michigan (1969). Ogilvie, T.F., "Recent Progress Toward the Understanding and Prediction of Ship Motion", The Fifth Naval Hydrodynamics Symposium, Bergen, Norway (1964). Pierson, J. W. and Moskowitz, L. ''A Proposed Spectral Form for Fully Developed Wind Seas, Based on the Similarity Theory of S.A. Kitaigarodskii', Journal of Geophysical Research, Vol. 69, No. 24 (1964) pp. 5181-5190. Salvesen, N., Tuck, E.O. and Faltinsen, O.,"'Ship Motions and Sea Loads'! The Society of Naval Architects and Marine Engineers Transactions Vol. 78 (1970) pp. 250-287. St. Denis, M. and Pierson, W.J., "On the Motion of Ships in Confu- sed Seas", Transactions of the Society of Naval Architects and Marine Engineers, Vol 61 (1953). Thwaites, B., 'Incompressible Aereodynamics'; Oxford Press, England, (1960). Wehausen, J. V.,and Laitone, E. V.,"'Surface Waves'} Encyclopedia of Physics, Vol. 9; Springer-Verlag, Berlin, (1960) pp. 446-778. Wehausen, J. V.,''The Motion of Floating Bodies'; Annual Review of Fluid Mechanics, Annual Review, Inc., Palo Alto Us5. 4. , Vol. 3 (1971) pp. 232.268: 32 Eo ME 540 Motton and Reststance of a Low-Waterplane Catamaran DISCUSSION J.N. Newman Massachusetts Instttute of Technology Cambridge, Massachusetts, U.S.A. (Discussion read by Professor Beck) Drs. Lee and Pien have applied up-to-date hydrodynamic techniques to the prediction of catamaran motions and wave resistan- ce, and while they have each experienced isolated cases of poor agreement with experiments, the overall success in both problems is quite striking. My remarks will be focused on the principal discre- pancy experienced in the seakeeping portion of this paper; namely, the substantial overprediction of pitch and heave, with forward speed, at a critical wavelength. It seems clear that this is associated with the presence of near-zero heave damping at zero forward speed, and the use of the zero-speed hydrodynamic coefficients ina strip-theory manner. This problem does not show up for conventional hull forms, even when (zero-damping) bulbous sections are present locally such as in the bow. But for a body such as Model 5266 where bulbous sections are dominant, the total heave damping coefficient will be more seriously affected and, in view of the strong coupling due to forward speed, both the pitch and heave motions will be exaggerated as shown in figure 8. To be more specific, let us consider the case of a thin-body section, so that the explicit results of thin-ship theory are applica- ble. Then it is known that the two-dimensional damping coefficient for a section with offsets y =tf(z) is proportional to the square of the integral or to the Laplace transform of the hull slope in the vertical direction. Here T is the ship's draft, and K is the wavenumber. Clearly, for a bulbous form where df/dz changes sign, the above integral will vanish for suitable combinations of the wavenumber and hull shape. Now in three dimensions and with zero speed, the same conclusion 541 Pten and Lee will apply, for suitable shapes f (x, y), essentially because here the damping coefficient depends on an integral, with respect to the waveangle 9 , of the square of the surface integral. i ateee - iK 8 f ‘ Kz + iKx cos O f(x, z) Ae 0 -L/? dz and since the wavenumber K is independent of the parameter @ , the above integral will vanish for cylindrical vessels having the appro- priate two-dimensional shape. Based on this argument, we can con- clude that zero damping can occur not only in two dimensions but also in three, provided the forward speed is zero. Finally, let us consider the case of forward speed. Then, as shown, for example, in my paper in the June 1959 issue of the Jour- nal of Ship Research the threee-dimensional damping coefficient is again proportional to the square of a surface integral similar to that shown above, but now the wavenumber K is no longer a single cons- tant, but, depending on the value of the Brard parameter U / g takes on either two or four discrete values, each of which depends on ® . Thus, since the square of the surface integral is integrated over a continuous spectrum of K ; the probability of zero damping is greatly reduced. The three-dimensional theory which I presented in 1959 was eclipsed by the subsequent success of the simpler strip theories, which could be more easily refined to account for the effects of finite beam. But Dr. Lee has discovered a situation where the three-dimensional and forward-speed effects may be more important, and I hope to have the opportunity to pursue this matter further, by resurrecting my 1959 theory and applying it to the catamaran configu- ration. DISCUSSION Robert F. Beck Untverstty of Michtgan Ann Arbor, Michtgan, U.S.A. Personally, I am wondering if you have done any non-head- sea calculations. What you have shown in the paper is just for pitch 542 Motton c id Reststance of a Low-Waterplane Catamaran and heave in head seas. It would seem to me that the non-head-sea case may be more critical for certain combinations of ship speed and wave heading angles. If you have calculated these modes of motions, did you consider any viscous effects, particularly roll-damping, as Salvesen, et. al. had in their paper when they included an empiri- cal viscous damping term. REPLY TO DISCUSSION Choung M. Lee Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. I will answer Professor Newman's discussion first. As I discussed in the paper, the under-estimation of the damping is expect- ed to come from the deficiency of the’strip theory which cannot take care of a three-dimensional flow when a catamaran has forward speed. I agree with Professor Newman's viewpoint that the forward speed effect on the motion of LWP catamarans could well be repre- sented byathin-ship approach. The difficult part is to incorporate the three-dimensional wave interference between the two hulls. As to Professor Beck's question, I have pointed out that I formulated for five degrees of freedom of motion. The only numerical results we have obtained so far is up to coupled heave and pitch mo- tion in head seas. As to roll damping, I suspect that we may have to introduce a similar kind of supplemental damping to the one introduc- ed in the heave and pitch motion for LWP catamarans, However, at the present stage we are not sure if the supplemental damping should necessarily be governed by viscous effect alone as treated by Salve- sen, Tuck and Faltinsen in their SNAME paper of 1970. More signifi- cant factor governing the damping could be the forward speed effect. I hope that our future investigation on this subject would clarify the question. 543 Pten and Lee DISCUSSION T.K.S. Murthy Portsmouth Polyteechnte Portsmouth, U.K. I have not had occasion so far to read the paper which Dr. Pien and Dr. Lee have presented, but from the presentation and the discussion I can see that they have used strip theory with two-dimen- sional damping values and that they have also made the assumption that the waves are not disturbed by the catamaran. These are perhaps two shortcomings in the work but it is obvious from the talk I gave this morning that if we put the parameter beta equals zero - we have chosen four parameters - delta, beta, alpha and epsilon - if we put beta relating to the cushion pressure equal to zero we get all the results we require for a catamaran. In my treatment I did not start from the equations of motion, as is normally done. I have not specifically used added mass and damping values, but these are implicit in the equations. The strip theory is always suspect, particularly at forward speeds, and it is better not to use it. Although my theory is applicable only to the thin hulls of the conventional catamaran, it can also be applied to circu- lar cylinders well below the free surface, whatever the dimensions might be, because then the wavemaking could be considered fairly negligible. I would strongly recommend that Dr. Pien and Dr. Lee use the method talked about this morning to see how it compares with their results. We should be very glad to collaborate with them in this respect and do some work at the University if required for confirmation of their results. This is a different approach. It does not make any explicit assumptions and we get the same results by a different approach. I strongly recommend some sort of collaboration between NSRDC and the University. 544 Motton and Reststance of a Low-Waterplane Catamaran REPLY TO DISCUSSION Choung M. Lee Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. Dr. Murthy's suggestion of using three-dimensional singu- larity distribution is well understood. In principle we can always use the method of singularity distribution to obtain the solution of a linear boundary-value problem in the three-dimensional space. The only difficulty is in the evaluation of the hydrodynamic quantities, which takes much more time than the two-dimensional solution does. This is the main reason why we have relied on the strip theory. Owing to the short time at my disposal I did not explain too well, but when I said that the presence of body did not disturb the incoming wave this was applicable only to the case of obtaining relative motion. In obtai- ning ship motion the wave diffraction effect due to the presence of body is incorporated in the wave-exciting term by use of the Haskind relation. 545 “patie Laraseorsiinyg ts) Gite ia lnaiieitaribis: PMR? UTA his 3, dhin aan asd sw Rieko wag aks pode mgt, Ligne ss Pits Petar. ; t@evsl pale 00 ign ioe orl aintdg »), coed eiiaih. ud) it io boy teag viap ash ro Crey bee eafaoosas? mes Tap mip vicerseapl Sede (eile sagitisceic oleopnykethedade giek avisye ods chee Mind tO aesok wOstel re, Lnaodacvatuh- owt ost Reds ws posted Poin. “proud qivie ad po. boileieved om x dw TTA Prt ities a6) ew pot cisigze, 3 hee 1 Jaeog' akb oven Je att ET ese Hue gir moon: O07 Gastaes,cos dih, YR tO 292 go20' pets: | had iu MODE sc avg win’ wilwneacse tu) oA a2 oI 7 Ga? 4 md oid sotle to asrieewig «fh oF sub teats agiioerttiih avaew ed? aciior: ¢ + 2 ~ . ’ Inistéoh srt 408.00 ix sort ox 4savew,-act oe Pass SIRI ‘ ory a &: . a t ’ . , —-= 4 < ‘ . P y ae 7 = ’ 4 re AM 4 < * YY « ae ¥ , £ .; ft , ¢ y is ms a! “ “4 and, AT ‘ : 1% cf . 5 Me ~alv - = 2 ; ; : f ; © .DrLeC te eam f Semi-Submerged Ship. T.G. Lang (Naval Undersea Research and Development Center, U.S.A.). Propeller Excitation, and Response of 230 000 TDW Tankers, C.A. Johnsson (Statens Skeppsprovningsanstalt, Sweden). T. Sdntvedt (Det Norske Veritas, Norway). Motions of Moored Ships in Six Degrees of Freedom. I, Ming Yang (Tetra Tech. Inc., U.S.A.). Analysis of Ship-Side Wave Profiles, with Special Reference to Hull's Sheltering Effect, K. Mori, T. Inui, H. Kajitani (University of Tokyo, Japan). 547 Page 549 581 671 687 = eget iga Sve .S& tango ahaa : nokaeed prinvoM sqid2 begrerkdie- ina? "2 sn Ip agiael a ienagy Od ; . ae ws KHOVAstesd, oft" tet i fey Vt) BA t ey) Pe ; 0a 5 se. «tetas. In somoteved «) Wat O00 G&S b% ekoeceeH bat worse Jistansegubave sgeqqaxts anniale) poagadolt uh ry en 1 ( lyawsoU .4chiteV ete7t0oY InW) i SoviaRe e: mobeese tl to seoaxyedl iB at qid® be100M 26 & { ALG, 0.4\on8 ,doeT extoT)} gas gail 4 44 intveqe Siw solders oveW ebi¢-qit2 to | A909. goitetieds «lint of » 36 bo yiietevial)) inetiied .H ual .T .ftoM e Ben: eg el ole: a HYDRODYNAMIC DESIGN OF AN S3 SEMI-SUBMERGED SHIP Thomas G. Lang, PhD Naval Undersea Research and Development Center San Dtego, Caltfornta U.S.A. ABSTRACT The S3 semisubmerged ship concept is described, and hydrodynamic characteristicsare presented. Va- riations of the basic form are discussed and results of model tests and theory are presented on static and dynamic stability, drag and power, motion in waves and effectiveness of an automatic control system for motion reduction. The results show that an S? is in- herently stable at all speeds, well damped in all mo- des, and should provide a near-level ride in high sea states if equipped with an automatic control system. Furthermore, an S? should have less drag thana monohull at the higher design speeds. INTRODUCTION Military and commercial users of ships are continuously searching for new design concepts which would provide improved speed, range, payload ratio, seaworthiness, or reliability. Such improvements are preferably to be attained at reduced cost, although cost tradeoffs are the general rule. Since monohulls have long been the most widely used hull form, it is generally accepted that their lead position is not easily challenged. The large monohulls can carry a very large payload ratio, they have a long range at moderate-to-high ship speeds, and they offer good seaworthiness ata relatively low initial and operating cost per unit of payload. The small monohulls, on the other hand, have 549 Lang other advantages, suchas : low unit cost, more flexible utilization resulting from greater numbers for a given total cost, more frequent scheduling, less net cost when small payloads are required, and less target value in the case of military applications. Unusual ship designs such as hydrofoils and various types of air-supported vehicles have already taken over some of the missions performed earlier by monohulls. These types of craft are high per- formance vehicles, and tend to be used when higher speed is impor- tant, such as certain passenger craft and special military applica - tions. These craft require considerable power, are more complex in design, and are therefore more costly than monohulls. There is a need for a new type of small displacement ship which has low cost, has all the desirable features of small ships, and yet has many of the desirable features of large ships. One new type of displacement ship which has been receiving considerable attention lately, especially in the oil drilling industry, is called a semisubmersible. Typically, semisubmersibles are low-speed ships having two or more submerged cylindrical hulls with several vertical cylinders supporting a platform well above the water. These craft have been found to withstand very high sea states and winds, and exhibit small motion in waves relative to monohulls. The term S? refers toa certain class of related semisub- merged ship designs and their characteristics. The S? semisub- merged ship concept discussed in Reference 1, and shown in Figure l, belongs to the family of semisubmersibles ; however, it is designed to provide low drag at higher speeds, and to have good seaworthiness not only at rest, but underway. An S? tends to fill a gap in ship design since it can be small, having all of the advantages of small ships, and yet have the speed, deck space, and seaworthiness of large ships. The S° concept stemmed from designs of the writer dating back to the 1950's. This concept was introduced at the Naval Undersea Center (NUC) in 1968, where it has been under active investigation ever since. The S> is not the only higher-speed se- misubmerged ship concept, however. Several other types have been designed, as discussed in Reference 1, including a single-hull version conceived by Lundborg dating back to 1880, a multihulled version described by Blair in 1929, a twin-hulled version by Creed in 1945, the Trisec by Leopold at Litton Systems in 1969 (Ref 2), and more recent versions called Modcats designed by Pien at Naval Ship 550 Hydrodynamtce Destgn of an oe Semtsubmerged Shtp Research and Development Center (NSRDC) (Ref 3) DESCRIPTION The typical design of an s3 semisubmerged ship, shown in Figure 1, consists of two parallel torpedo-like hulls which support an. above-water platform by means of four well-spaced streamlined vertical struts. Stabilizing fins attached to the aft portions of the hulls provide pitch stability at higher speeds. The water plane area and spacing of the struts provide static stability in both roll and pitch. Small controllable fins called canards may be placed near the hull noses. These canard fins can be used in conjunction with control- lable stabilizing fins at the rear to provide motion control over heave, pitch and roll. If rudders are placed in each of the four struts, mo- tion control over yaw and sway can be obtained, especially when an S? travels obliquely to waves. It should be noted that an S72 design is inherently stable at all speeds, without the use of control surfaces. Some of the advantages of the S? hull type relative to a mono- hull are : greatly improved seaworthiness, both at rest and underway; reduced wave drag at higher speeds , greater deck area and internal volume ; certain advantages of the unusual hull shape for placement of a central well, mounting sonars, carrying small craft, placement of propulsors, and potential for modular construction ; improved propulsive efficiency and greater cavitation resistance ; greater top- side weight capacity; and the potential for a near-level ride in high sea states. These advantages are to be weighed against the disadvantages. The primary disadvantage is the increased structural weight due to its relatively dispersed design form. Other possible disadvantages include the large draft, and the need for ballast control over trim. Many variations of the typical design shown in Figure 1 are possible. The strut thickness and chord lengths can vary, the hull lengths and diameters can change, the hull cross-sectional shape can vary, the rudders can be located behind the propellers, the sizes and positions of the stabilizing and control fins can be varied, and the ship can be propelled by means other than propellers, such as pump- jets. Still other S? variations from the typical design form are presented in Figure 2; these include a two-strut and a six-strut, twin-hulled design, and several types of single-hulled designs. There is no "best" S? hull form, since the form will vary as a function of size, mission, and design constraints. 551 Lang The primary objective of this paper is to describe the basic characteristics of an S? so that it can be compared with other types of ships for various types of applications. Todo this, the drag and power, stability, motion in waves, and automatic control characteris- tics will be discussed. DRAG AND POWER In calm water, ship speed is a function of drag, and is there- fore limited by the installed power. The maximum speed may be less in the higher sea states, since speed may be limited either by ship motion or by increased drag due to waves. In the case of monohulls, speed limitations in the higher sea states can be severe. In order to compare the drag and power of a wide variety of ship forms, sizes, and speeds, the following equations are used : D . . = = 2 = drag coefficient Cy v3 P y2 Cooist an f E Dp), ¥ power = P= ——= C vA aye n D Z Vv Vv displacement :Proude number =. (Eiyg< = Sas _ me fe 1 Vevis sak phe }. iA. TR AS hull efficiency = E = aan Pep todt A 2 A dey witht ont 1 range = Resign oie 1+ spe =, © “gre where D = drag, V = displaced volume, p = mass den- sity of water, V = speed, 1 = propulsive efficiency, g = accele- ration of gravity, 4 = Veg = displaced weight, Af = weight of fuel, and SFC = specific fuel consumption = weight of fuel consum- ed per unit power per unit time. The units used may be any consis- tent set. The term a, is the frictional drag coefficient, and is assumed to be purely a’ function of Reynolds number ; the term Cp, reduces as the size or speed increases. The term C is the residual drag coefficient ; it includes the wave drag and all other sources of drag except frictional drag, and is assumed to be purely 552 Hydrodynamte Destgn of an ee Semtsubmerged Shtp a function of FG : Figure 3 is reproduced from Reference 1, and shows the approximate hull efficiency E at maximum speed for a variety of ship types as a function of displacement Froude number Fy in calm water. Hull efficiency is an important parameter since the equation shows that it is directly proportional to range. Note that the hull efficiency of an S? is somewhat less than that of a monohull at low Fy , but somewhat greater than that of a monohull at high Fy where monohull wave drag becomes large. The reason for this re- sult is that an S? has a greater frictional drag than a monohull due to its increased wetted surface area, but has less wave drag at higher speed due to its unusual hull form. A Cy of 0.05 and an n of 0.80 have been used for the S? curve in Figure 3 at ta = 2.0, with Cy /n reducing slightly at lower FY , and increasing slightly at higher Po to reflect reduced propulsive efficiency. The propulsive efficiency 1 is somewhat greater for an S$? than for monohulls since the boundary layer inflow to the propulsors will be more axially symmetric ; therefore, the S? propulsors can be more completely wake adapted, as in the case of torpedoes where propul- sive efficiencies of 85% to 90% are not uncommon. The line shown in Figure 3 for monohulls is the locus of the highest measured values of E. In rough water, the value of E for monohulls will reduce considerably, as shown later, while E for an s3 ship will not change appreciably. The dashed lines in Figure 4 show the measured C,) from model tests. The model data relate to a small-craft S? design. The solid lines are the estimated drag coefficients for several 3000-ton ships, including an improved low-wave-drag four-strut S? , and the estimated Cp of an improved two-strut design taken from Ref 3. Notice that the values of Cp for the 3000-ton ships are significant- ly lower than those of the small models, primarily due to the Reynold's number effect on frictional drag and the use of thinner struts. The wave drag portion of the estimated value for the $3 ship was calculated by Dr. R.B. Chapman of NUC using linearized thin ship theory in which all strut-strut, strut-hull, and hull-hull interactions were included, This same theory has provided excellent agreement with a large number of tests conducted on various struts, strut-hull combinations, and complete S? models. Reference 4 by Dr. Chapman contains data for estimating the spray drag of surface- piercing struts at high speeds, Figure 5 shows the ratio of the drag in waves to the drag in calm water for tests ona 5-foot model of a DE-1006 destroyer 553 Lang (Reference 5), and for tests on a 5-foot model of an S? . The drag of the destroyer model increases by factors of five or more in waves, while waves are shown to have no significant effect on the drag of the S? model. Figure 6 shows the power required for a 3000-ton, four- strut S? compared with the estimated power requirements for a hydrofoil, a high-speed surface effect ship and a destroyer. The results show that the S? requires significantly less power than either a hydrofoil or SES at speeds up to around 50 knots, A photograph of a model of a 3000-ton S? is shown in Figure 7, together with a list of some of its estimated characteris- 11S; STABILITY A wide variety of model tests have shown that the S? is inherently both statically and dynamically stable. In regard to static stability, the metacentric height in roll can be calculated from the equation DiiGarece sak V be where I su a A = moment of inertia of the total waterplane area A, b = strut center-line spacing, displaced volume BG = is the distance upward from the center of buoyancy to the center of gravity. q 1 Large topside loads can be carried,even with a small waterplane area,due to the substantial transverse and longitudinal strut spacing. Tests in large waves and high simulated winds have shown that GM in roll should be around 3/4 of the hull diameter (alternat- ively, approximately 8% of the beam), although values as little as 1/4 of the hull diameter are acceptable. Tests indicate that motion in beam waves reduces as the roll GM increases, contrary to some monohull results. However, since both wave drag and structural weight increase as the strut waterplane area and spacing increase, the roll GM_ should be made no larger than necessary. 554 Hudrodynamtce Destgn of an s° Semtsubmerged Ship The metacentric height in pitch is calculated from the same equation as for roll, except I now refers to the longitudinal area moment of inertia. Tests to date on S? models have shown that motion in waves reduces as the pitch GM increases. In other words, the struts should be well-spaced in the longitudinal direction, This is one of several reasons why the four-strut configuration was selected as a typical (but not the only) design form for an S$? Figure 8 shows typical waterplane areas for a monohull, a catamaran ship, a two-strut low waterplane ship, and a four-strut S3 . Note that the S? has the greatest static stability in both roll and pitch per unit waterplane area because the waterplane area is concentrated in the four corners of the ship where it is most effec- tive. Another advantage of the four-strut configuration is that it has less virtual mass in the transverse direction than the two strut design, and therefore will have less motion and hydrodynamic load- ing in beam seas. One of the first questions explored ina series of S3 model tests conducted in 1969 concerned the dynamic stability of an S? Figure 9 shows pitch data obtained on several 5-foot model confi- gurations tested in calm water in the General Dynamics Aeromarine Test Facility model towing basin in San Diego, California. The hull diameters were 4 inches, Figure 9 shows that all models were stable at all test speeds except the non-S? model designated C+N which had no stabilizing fins. Thus, these tests showed that the S? sta- bilizing fins were necessary for dynamic stability at Fy greater than about 0.9. This result was in good agreement with S? design theory which shows that the dynamic instability of bare hulls will overcome the static stability provided by the struts above some critical speed unless stabilizing fins are incorporated. A very useful device to further investigate the dynamic stability of an S3 is the 5-foot radio-controlled model shown in Figure 10, which was tested in 1970. This model was stable under all test conditions and controlled well. All motions were well damped at rest and highly damped when underway. It operated well in waves and wind at all angles, although the greatest motion occurred in large following waves. Figure 11 shows an 11-foot model built and tested at the Naval Ship Research and Development Center in 1971. This model performed similar to the 5-foot model suggesting that model tests and the known scaling relationships are valid. 555 Lang MOTION IN WAVES During the 1969 towing tests, various S? model configurat- ions were tested in 4-inch X 80-inch waves in head and following seas, The non-dimensional pitch and heave amplitudes for two S$? models in head seas are shown in Figure 12 together with the pitch and heave amplitudes of a 5-foot model of a C-4 monohull ship. Note that the motion of the S? models is significantly less than that of the monohull model. The S? models were also tested ina variety of wave lengths, and no resonance was found in head seas. The test results in following waves showed significantly more motion, as seen in Figure 13. The monohull was not tested in follow- ing waves. The wave height was equal to the hull diameters, so the waves were relatively high. Testsin 2-inch waves showed consi- derably less motion, Data taken on the lift force and pitching moment indicated that small control surfaces and an automatic control sys- tem would significantly reduce motion in following seas. Tests at rest in beam seas showed that the roll of the S? models was significantly less than that of the monohull model, and no resonance occurred at any of the wavelengths tested. AUTOMATIC CONTROL SYSTEM The combined use of horizontal canard control fins near the noses of the hulls, and controllable stabilizing fins near the aft end of the hulls, provides motion control over heave, pitch and roll in high sea states. Figure 14 presents computer results obtained by Dr D.T. Higdon of NUC showing the reduction of heave and pitch in ae waves which is achievable by automatic stabilization of an s3 ship similar in shape to the radio-controlled model of Figure 10. The already small motions are reduced by a factor of four or more. Figure 15 shows the computer results for motion reduction in following waves. In this case, the result is much more dramatic. Heave is reduced by factors of twenty or more, and pitch is reduced by factors of five to ten. 556 Hydrodynamte Destgn of an ar Semtsubmerged Shtp SUMMARY A considerable number of model tests, theoretical studies, and design studies have been conducted on the S? concept. The results show that the S? is highly stable and seaworthy (both at rest and underway), more efficient at higher speeds than conven- tional ships, and will provide a near-level ride if automatically controlled in high sea states. Also, many advantages result from its unusual hull form for various kinds of military and non-military applications. REFERENCES 1 LANG, T.G., "S? New Type of High-Performance Semi- submerged Ship", American Society of Mechanical Engineers, Paper No, 71-WA/UnT-1, Winter Annual Meeting, Nov 28 - Dec 2, 1971. 2 LEOPOLD, R., "A New Hull Form for High-Speed Volume- Limited Displacement-Type Ships'', Society of Naval Arch- tects and Marine Engineers, Paper No. 8, Spring Meeting, May 21 - 24, 1969. 3 STEVENS, R.M., ''New Dimensions in Naval Catamarans", American Society of Naval Engineers, ASNE Day Meeting, May 4 = 5, 1972. 4 CHAPMAN, R.B., "Spray Drag of Surface-Piercing Struts", Naval Undersea Research and Development Center, TP-251, Sep 1971. 5 SIBUL, O.J., "Ship Resistance in Uniform Waves", Insti- tute of Engineering Research, University of California, Berkeley, California, Report No. NA-64-1, Jan 1964. AD # 606272. 55a Lang Figure 1, Basic s3 semisubmerged ship concept 558 Hydrodynamte Destgn of an S® Semtsubmerged Ship SINGLE HULL \ TWIN HULL Figure 2, Alternative designs of the S? concept 559 Lang super tankers tankers streamlined submarines ocean liners BS a) 100 = 5 eae la e\=£ aircraft carriers aC e,= Ss\o a cruisers > 4 e 2 \ = ° destroyers ———> UL 10 wi “i < 5 5 = ° 2 GEMS = ° a 2 $ planing boats 1 0.1 1.0 10.0 164 i: FROUDE NUMBER, F,, = Se eS vavi/s Figure 3. Hull efficiency of various ship types 560 Hydrodynamte Destgn of an 8 Semtsubmerged Shtp ir 4 sustsop diys u0}-O000E pue STepour -G SNOTIeA Jo SJUSTOTZZI09 Beaqy “p 9AinstqT Ay ‘saaquny epnosy st ot s ross Ud! AA ‘383) ¢S YL OOOE “yn4Is-7 (Sula Of ‘3S) LYIGOW 4OL-OOOE ‘ 3135-2 usiseag eS JZE1D [JEWS PIsipow & 3o Japom "34 TE V7 uZisaqg ge WEAD-HEWS & 50 [9POW “34-S O or < “ sv Gy Syc9919193209 Ze2qg 561 Lang ~ DE 1006 (Wave Lengths-=1.5 Hull Length) H=Wave Height L= Model Length H/L=0.041 H/L=0.028 s? Model A for H/L=0.035, 0.070,0.105 Ratio of Drag(Waves)/ Drag (Calm Water) Ci) 1.0 2.0 Froude Number, FL Figure 5. Effect of waves on the drag of five-foot destroyer and S? models 562 Hydrodynamte Destgn of an S? Semisubmerged Ship sustsop drys U0}-Q000€ JO SPUTY SNnotreaA TOF sJUSUTOAINbeAT TOMOg ‘9 2aANnsT SLON™ ‘G33dS QoL 06 08 yd 09 0s (0) 0€ 02 ‘OL 0 (,S) pabuawiqnsiwas (4q) 42A02}s9q (S3S) diys 399433 a9e41NS 1!0304pAH a 08 Sas oz / 1O4OYGAH 002 563 SGNVSNOHL ‘Y3MOd3SYHOH LAVHS Lang dtys “Ld VT = YALAWVIC TINH “Ld ZET = WV3d ‘Ld STE = HLONI1 cS U0I-000€ e Jo TEPOW *2 dans 564 Hydrodynamic Destgn of an §% Semtsubmerged Ship soedAj diys snotzea jo seare euetdiojem °g candy aD NVYVNV LVI NOL-OO0E a a eres See gs NOL-O00E Se IVIGOW NOL-000€ a LOL IMA LLL Oa 565 Lang ie 4 zrequinu epnoi.7 yuoursoe{ds tp yo uoTjOUNny e Se STOpour es JO 193M UWITeO UT YI A, ‘saquinyy epnojiy oe st oT s*0 4+N+@ 4t+n+y V Vv °6 eansty "Sap ‘yd3!d 566 Hydrodynamte Design of an S® Semtsubmerged Shtp Topol -S§ pe[[O1}UOd-oTped JOOJ-9AT T ‘OT Sanst gy 567 Lang [epout OCUSN peTtedosd-j;Tes jooz-usASTA "TT eansty 568 Hydrodynamte Destgn of an 5° Semtsubmerged Ship LX Model A+N+F 0 Model B+N+F V7 ship (C-4) @® w wld aja bm ee) cS al< =\|= 1.0 5/< o|= ra) ui > < WwW xr te) 0.5 1.0 1.5 2.0 2.5 3.0 FROUDE NUMBER, Fo Figure 12. Pitch and heave of S? models in head seas 569 PITCH AMPLITUDE MAXIMUM WAVE SLOPE HEAVE, DOUBLE AMPLITUDE Lang 2.0 LX Model A+N+F O Model B+N+F 9 1.0 Ld 0 0.5 1.0 1.5 2.0 2.5 3.0 FROUDE NUMBER, Fy 2.0 Z\ Model A+N+F OO Model B+N+F WAVE HEIGHT ° ) 0.5 1.0 1.5 2.0 2.5 3.0 FROUDE NUMBER, Fy Figure 13. Pitch and heave of S? models in following seas 570 Hydrodynamte Destgn of an 58 Semtsubmerged Ship be) ro) [o) NONDIMENSIONAL HEAVE AMPLITUDE, Z/v!/3 (o) (>) Oo = on De) [o) Nh (63) W (=>) 2 I) controlled uncontrolled PITCH AMPLITUDE, rad Oo 2 ron) = a9 2 N) fo) 5 10 15 20 25 30 NONDIMENSIONAL WAVELENGTH, d/v2/3 CONTROL DEFLECTION AMPLITUDE, rad fo) Figure 14, Effectiveness of automatic control in head seas at Fy = ie 65 5a! Lang following waves NONDIMENSIONAL HEAVE AMPLITUDE, Z/v!/3 controlled uncontrolled = See me ye PITCH AMPLITUDE, rad 0 5 10 15 20 25 30 NONDIMENSIONAL WAVELENGTH, X/v!/3 CONTROL DEFLECTION “AMPLITUDE, rad (=) nN Figure 15, Effectiveness of automatic control in following seas at Fo (= 1/05 Hydrodynamte Destgn of an oe Semtsubmerged Shtp DISCUSSION Gerald E. Bellows Universttat Hamburg, Institut fur Sehr ffbau Hamburg, Germany I would like to thank Dr. Lang for an excellent presentation. It is evident that this type of ship has extremely good seakeeping pro- perties. This is perhaps its major advantage over the monohull, I have studied ships of this type and have found that, with the cargo loaded on the deck, it is difficult to obtain sufficient GM in both the longitudinal and transverse directions. I wonder if Dr. Lang could give us a weight breakdown, including the structure, fuel, and cargo (or payload). I also would like to know what type of power plant would be used and if it would be located in the underwater hulls. I would like to know if it is possible to alter the draft by ballasting. Dr. Lang has mentioned in his report that the draft could be a problem, but this could be overcome by using ballast to alter the draft when entering port. REPLY TO DISCUSSION Thomas G, Lang Naval Undersea Research and Development Center San Dtego, Caltfornta, U.S.A. To answer the first question, the metacentric height (GM) increases a little faster than the square of the distance between the waterplane areas, Thus by concentrating the waterplane area near the four corners of the platform through using four struts we obtain a maximum GM for a given waterplane area in both roll and pitch. As a result, itis possible to have a GM that is on the order of three or four times that of the conventional monohull in roll, anda GM which is much less than a monohull in pitch but yet one that gives very good pitch response. The result, as seen in the film, is an acceptable me- tacentric height in both roll and pitch. 573 Lang As far as the ratios of the various weights are concerned, our preliminary calculations indicate a structural weight to total weight ratio on the order of 40 per cent for a moderately sized ship, if the ship is made out of aluminium. Thus, we are proposing that a ship of this type be made of aluminium to help to solve the weight problem. In regard to the power plant, we would propose that the po- wer plant on small craft be located in the top cross structure, with some type of a drive mechanism to transmit the power down to the propeller. In ships of a few thousand tons or greater, we would pro- pose that the power plants be located in the tailcones of submerged hulls, with direct drive to the propeller through a gearbox. Alterna- tively, a cryogenically cooled electric power system We are proposing gas turbines for power because of their light weight and efficiency, especially in the larger size range. In answer to the last question, on ballasting, we do have pro- vision for ballasting in all the proposed designs. In very shallow water ports, the loaded ship minus fuel provides a draft about equivalent to that of a monohull ; therefore such a ship could be fully loaded at the dock and the fuel topped off in the deeper region of the harbor. DISCUSSION Nils Salvesen Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. I have recently made an investigation of the seakeeping cha- racteristics of low-water-plane catamarans and I found it a very inte- resting subject, Reference 1. What makes it so much more interes- ting than for conventional ships is that the seakeeping characteristics of catamarans and in particular LWP catamarans are extremely sensitive to changes in the hull geometry. For conventional monohulls, on the other hand, large changes in the hull parameters are required in order to produce any substantial effects on the seakeeping charac- teristics. It is recognized that as far as the seakeeping is concerned, the major advantage of the LWP catamaran over conventional hulls is that they have a very low natural frequency due to their small water- 574 Hydrodynamte Destgn of an e Semtsubmerged Shtp plane area. Therefore, in head seas, the maximum heave and pitch motions are usually at wave lengths approximately five times the length of the hull. This means that a long LWP catamaran (say longer than 300 feet) will only experience large pitch and heave motions in extremely severe sea conditions. On the other hand, a 100-foot cata- maran will experience in ocean operation maximum pitch and heave motions a large percentage of the time. This aspect of the motion responses of LWP catamarans is well-known ; however, itis less recognized that the added mass and damping coefficients as well as the exciting forces are all much smaller than for conventional hulls and that the maximum pitch and heave motions are extremely sensitive to small changes in these quantities. In particular, it is important to recognize that the damping coefficient for certain catamaran configurations can be so small that it results in pitch and heave motions several times larger than for conventional hulls as shown in Reference 1. Another seakeeping aspect which deserves attention is the pitch and heave motions in following seas. Some of the LWP catama- rans have about twice as much pitch motions in following seas than conventional monohulls and the maximum pitch motions occur at wave lengths of the order of magnitude of ship length (see Reference 1). Dr. Lang has demonstrated that these vertical mations in following seas can be considerably reduced by use of automatic control surfaces. I would like to ask Dr. Lang if he is of the opinion that LWP catama- rans in general will need automatic control surfaces in order to have acceptable motions in following seas, 1 SALVESEN, N., ''Seakeeping Characteristics of Small-Water- Area-Twin-Hull Ships", presented at AISS, SNAME, USN Ad- vanced Marine Vehicles Meeting, Annapolis, Maryland, 17-19 July 1972, and published in the Journal of Hydronautics, Vol. 7 No, fy Jan h973: 55 Lang REPLY TO DISCUSSION Thomas G. Lang Naval Undersea Research and Development Center San Diego, Caltfornta, U.S.A. In answer to the first comment, we have not seen in our ex- periments any of the head sea resonance problems that Dr. Salvesen mentioned. As you saw in the motion picture, the S? model has extremely high damping when under way, in both roll and pitch. This large damping is provided largely by the canard fins at the front, and the stabilizing fins at the back. It is probable that the differences from Dr. Salvesen's results are due to differences in design form from the S? model. We have found relatively good comparisons bet- ween experiment and theory ; in general, the experimental results tended to show less motion than theory would predict, especially in the following sea case. As far as automatic control is concerned, all of the tests you saw in the motion picture were without automatic control, so itis seen that the craft can operate effectively without automatic control. In the case of following seas there is no motion problem until the waves reach a certain height. When not using automatic con- trol, large craft motion in the very highest waves in a following seas can be alleviated by slowing down approximately the wave speed, or less, 576 Hydrodynamte Destgn of an g Semtsubmerged Shtp DISCUSSION Edmund P, Lover Admtralty Experiment Works Haslar, Gosport, Hants, U.K. I have a point to make concerning this most interesting paper, and a question to ask. Firstly, in Figure 3 of the paper a comparison is made bet- ween the S* form and conventional ships. I would like to make the point that the "hull efficiency'' WV/P 7 conventional monohulls can be improved significantly above that shown by increasing the length to displacement ratio. These longer vessels would also have improved seakeeping characteristics as well as the Froude number of cross- over above which the S? shows to advantage. My question is this. Was any simulation made during the model experiments of an emergency crash situation and, if so, did this exhibit any problems with maintenance of trim ? REPLY TO DISCUSSION Thomas G. Lang Naval Undersea Research and Development Center San Dtego, Caltfornta, U.S.A. I agree with your comment of reduced drag for the larger length-beam ratio monohulls designed for higher speeds. In the non- dimensional graph of hull efficiency, this effect is already included to a certain extent since the line for monohulls represents the maximum value of existing monohulls wherein the higher-speed monohulls al- ready have a larger length-beam ratio. Thus, the monohull line re- presents the best of the known data, so the majority of monohulls will lie below that line as far as efficiency is concerned. Regarding the question of a crash situation, the radio-con - Bin 7 Lang trolled model was used to test light crack avoidance and the problem of hull flooding in case a crack cannot be avoided. The model was operated at top speed and suddenly given full reverse thrust. In no case did the bow submerge or was there water over the deck. The main reason for this good behavior is that the relatively low water- plane area has been concentrated near the front and the back of the craft by making use of four struts. Thus, the metacentric height in pitch is a maximum for a given waterplane area, This metacentric height is adequate to prevent excessive pitch in the case of full re- verse thrust from a condition of top forward speed. In the case of hull flooding, the control surfaces were adequate to raise the platform to trim conditions as speed increased, Trim at rest could be adjusted by blowing water ballast or fuel on the damaged side and/or counter - flooding on the opposite side. DISCUSSION Hans Edstrand Statens Skeppsprovntngsanstalt Goteborg, Sweden I have not read Dr, Lang's paper, but when I looked at the beautiful film I wanted to ask a question. The model size seems to me to be enormous compared with the tank dimensions, and I should like to know if Dr. Lang has checked his results in a larger tank and if he has used the measurement for quantitative development, REPLY TO DISCUSSION Thomas G. Lang Naval Undersea Research and Development Center San Diego, Caltfornta, U.S.A. The model size is slightly large for the tank it was tested in. The model hull diameter was 4 inches and the model was 5 feet long; the tank was 6 feet deep, 12 feet wide, and 300 feet long. The ratio 578 Hydrodynamte Destgn of an sé Semtsubmerged Shtp of hull diameter to tank depth was 1/16, Dr. Chapman's analysis of drag indicates that the model drag would increase at most a few per- cent, depending on speed ; corrections for tank size have been made in the model tests. There is no evidence that tank depth has an effect on motion. One reason why tests were conducted in the larger model basin at Escondido, California, and in San Diego Bay was to verify that there were no significant effects of tank size on performance. The results on the 11-foot model at NSRDC showed essentially equi- valent results. Thus to date, we have seen no significant effect of tank size on the model characteristics DISCUSSION Christopher Hook Hydroftin Bosham, Sussex, U.K. This extremely interesting paper by Dr. Lang I must disagree with on the matter of his comparisons as given in Figure 6. If we refer to Figure 14 of the Silverleaf 4294 Thomas Gray Lecture (paper) given to the Institution of Mechanical Engineers and entitled "Developments in High Speed Marine Craft'' we see that whereas the SES or amphibious Hovercraft shows higher speeds and better 7(L/D) values in calm water than the Hydrofoil, this situation is reversed in even moderate sea conditions and we get : Hovercraft 7 (6/ Die = 2 to 3 against submerged Hydrofoils approximately 6, and surface piercing hydrofoils 3 to 6. Now since the whole point of the raised platform is to elimi- nate wave effects as much as possible, it follows that it is unfair to present SES curves based on performances restricted to calm water and progressing right off the graph to the right, i.e. to a speed range which is far from having ever been demonstrated. I am informed on the best authority that until new skirt techniques have been developed, 100 knot speeds remain out of the question. To be specific, in Figure 6 Dr. Lang appears to claim some 10 knot more speed for a given power than for a hydrofoil but that is not what he claimed verbally in this presentation. Surely there is a mistake here. 579 Lang REPLY TO DISCUSSION Thomas G. Lang Naval Undersea Research and Development Center San Dtego, Caltfornta U.S.A. As pointed out by Mr. Hook, vehicle efficiency sould be com- pared in both calm water and rough water. The objective of the paper, however, was the semisubmerged ship concept ; consequently, full comparisons were not made between hydrofoils and SES. In regard to Mr. Hook's comment onFigure6, it should be recalled that all curves in this figure pertain to 3,000-ton vehicles. Conventional hydrofoils of this size are generally considered im- practical ; hydrofoil weight tends to be high, and the design Froude number tends to be too low. Design tradeoffs for a conventional 3,000-ton hydrofoil between structures, cavitation, and hydro- dynamic drag would tend ro result in reduced hydrodynamic efficiency of the order shown in Figure 6. On the other hand, the value of n L/D for semisubmerged ships, and displacement hulls in general, increase with displacement, for a given speed. This result should in no way detract from the good efficiency exhibited by small, high- speed hydrofoils and their excellent performance in rough water. 580 PROPELLER EXCITATION AND RESPONSE OF 230000 TDW TANKERS C.A. Johnsson The Swedtsh State Shtpbutlding Experimental Tank, Gdteborg, Sweden Al Bae Séntvedt Det norske Verttas, Oslo, Norway ABSTRACT In cooperation between Uddevallavarvet AB(UV), the Swedish State Shipbuilding Experimental Tank (SSPA) and Detnorske Veritas (DnV) a comprehensive inves- tigation concerning propulsion, cavitation and vibra- tion has been carried out on two 230 000 TDW tanker ships. The results with reference to propulsion, pro- peller cavitation and erosion have been reported at the 1972 Spring Meeting of the Royal Institution of Naval Architects [4] : } In the present report the results concerning vibra- tion are reported. Full scale measurements of pro- peller induced pressure fluctuations at different po- sitions on the stern, static and dynamic thrust and torque in the shaft, as well as vibratory response in different parts of the structure have been carried out. In model scale the experiments of primary interest in this connectionare the measurements of propeller induced pressure fluctuations in the stern, carried out under cavitating and non-cavitating conditions, the transducers being placed in the same positions as in full scale. These tests were carried out in the new cavitation tunnel at SSPA, allowing the use of the same ship model for the cavitation tests as for 581 Johnsson and Séntvedt the tests in the towing tank. The results of the full scale and model experiments for different ship-propeller-configurations (five-and six-bladed propellers, hull with and without stern fins) have been analysed and compared. Included inthe comparison are results of theoretical calculations of the extent of cavitation over the pro- peller blades, different kinds of shaft vibrations and propeller induced hull pressure fluctuations from ca- vitating propellers. I, INTRODUCTION The ships included in the present investigation are the first two ina series of tankers of about 230 000 tons deadweight, which are to be delivered from Uddevallavarvet AB(UV) within the next few years. The first ship, T/T ''Thorshammer'', was delivered in December 1969, being equipped with a five-bladed propeller. The model tests in the towing tank indicated very good propul- sive performance, which was confirmed by the results of the speed trials with the first ship. At the trials with this ship extensive measurements of vibra- tory response and pressure fluctuations on the hull near the stern- post were carried out by Det norske Veritas (DnV) together with re- cordings of different kinds of shaft vibrations [1] . The results of these measurements indicated that the levels of the pressure fluctua- tions registered by some of the transducers were very high ; in addition a loud pattering noise could be heard in the aft part of the ship. The vibration levels recorded were, however, not annoying. At the docking of the ship, shortly after the trials, erosion was detected on four of five blades. The blades were modified but after one voyage to the Persian Gulf and back, erosion was again observed, The high level of the pressure fluctuations was attributed to significant development of various forms of unstable cavitation and it was felt that there was a risk that the structure of the stern might be damaged. A proposal was made to fit the ship with streamline fins on the afterbody above the propeller and after extensive model testing of different alternatives in the towing tank the ship was fitted with fins, 582 Propeller Exettatton and Response of 230000 TDW Tankers as shown in Figure 1, and new full scale measurements of vibrations and pressure fluctuations were carried out [2 ] . The results of these measurements were promising. The pressure fluctuations, as well as the noise and vibration levels, were lower. Further the service re- ports confirmed the results of the model tests, indicating no increase of the power consumption, The problem of erosion of the propeller blades was, however, not solved, although some improvement could be noticed. The main modification applied to the second ship was that fins (somewhat different from those of the first ship, see Figure 1) were fitted during construction, In addition, a six-bladed propeller was fitted, having a radial load distribution different from the original five-bladed design. Successful trials were carried out with this ship aks Further, the new propeller has so far worked without any trouble. i SCOPE OF THE, PRESENT INVESTIGATION When the project work was completed it was felt that, if com- plementary model tests and full scale measurements could be carried out with the second ship, an opportunity had arisen to obtain unique material showing the influence of different modifications on the model as well as on the full scale, whereby investigations concerning the correlation factors for propulsion, vibratory response and cavitation could be made, Accordingly a research program was established, which was carried out in close cooperation between Uddevallavarvet, SSPA and DnV. Together with the investigations carried out on the first ship the full scale program included : 1, Speed trials with measurements of propeller thrust and thrust variations (second ship only) 2. Photographing of cavitation patterns at different propeller loadings and angular blade positions (second ship only) 3. Inspection of the propellers with regard to cavitation erosion 4. Measurement of propeller induced hull pressure variations 583 Johnsson and Séntvedt 5. Measurement of hull vibrations Corresponding model experiments were carried out, including studies of propeller cavitation behind complete ship models, In addi- tion the model investigations included measurements of wake patterns and static pressure with and without propeller in the towing tank as well as in the cavitation tunnel, special streamline tests and tests with a ducted propeller. The present report will deal primarily with items 4 and 5 above. The results of the measurements under items 1-3 above were summarized in [ 4] : Ill. SHIP AND MODEL CONFIGURATIONS TESTED The main dimensions of the ships are: Length between perpendiculars Se = 1020'-0" = 310.89 m Breadth Buy = 157'-10" =: 469710 Draught, fully loaded condition T = ),67'-7/8'o=) + 20¢44ame Draught, ballast condition, fore Th = LoVe Draught, ballast condition, aft Ta = 13.0: ¢m Displacement 260 850 long tons Deadweight 228 250 long tons Block coefficient 6 = 0,844 PP Capacity of cargo oil tanks 10 124 300 cft Clean water ballast CT No 3 836 900 cft Class Det norske Veritas + 1.A.1. '"'Tankskip for oljelast, F, EO." Main engine, General Electric steam turbine, 32 420 SHP (metric) at 85 rpm Model scale 1:43.5 The different ship and propeller configurations tested are sum- marized in Table l. Propeller Exettatton and Response of 230000 TDW Tankers Table 1 Ship Propeller Model Model > Polo P/D Name Fins a, 0.7 se 0 Thorshammer | 1600-A Without|] P1378 5) litte, 0) |] Os Bree} 10) 738° 1600-B!)| with Norse King - With P1493 618.50] 0. 64 |0. 795 1) Model No 1600-B (Thorshammer) has a slightly different fin (smaller) than 1600-C (Norse King). Model 1600-B was not tested. 2) Refers to the full scale propeller. The corresponding value for the model propeller was 2% lower, i.e. e/a = Oa Ar 3) Nozzle propeller. Dimensions of nozzle, Drax aOR Oana, t= Alona. The two conventional propellers were designed using the vortex theory in accordance with the standard procedure used for merchant ship propellers at SSPA. Different radial circulation distributions were used for the two propellers, resulting in different radial camber and pitch distributions, ID/g PUNE Ep SX Qa, NDS ES IV.1. Test Equipment:and Test Conditions On the first ship tests were carried out at two different occa- sions ; at the delivery trials and at trials arranged after the ship had been fitted with stern fins. On the second ship tests were carried out at the delivery trials and immediately after, the test on this ship in- cluding photographing of cavitation patterns on the propeller blades. The equipment for taking these photographs is indicated in Figure 1. It is further described in[ 4]. 585 Johnsson and Séntvedt Speed, Number of Revs and Sea State Most of the tests were carried out in connection with the speed trials but not at the runs on the measured mile. The speeds for the tests reported here were recorded with the log of the ship, whichis not a very accurate method, Accurate registration of the number of revs was, however, carried out in connection with all measurements by using a photoelectric cell, mounted on the shaft. Most of the re- sults in this paper are therefore related to number of revs instead of speed, In the table below the sea state during various tests is present- ed. Sea State during Trials Wind Loading Conditi biel 0 a Beaufort Loaded Ballast Loaded Ballast Norse King Loaded - Ballast Vibratory Response in the Structure of the Ship Vibratory response was registered at about 20 points in the structure (wing tank), engine room and deck house with the use of velocity transducers manufactured by CEC. The positions of these transducers, which were fitted at least at two of the trials, are shown in Figure 1. The fins, when installed were fitted with accele- rometers, positioned in the vicinity of the pressure transducers, see below. 586 Propeller Exettatton and Response of 230000 TDW Tankers Table II Positions of Pressure Transducers in Full Scale ducer ae eee No 587 Johnsson and Séntvedt 1) From aft end of stern post at the height considered 2) Not working at fully loaded condition 3) Fin, bottom 4) Fin, top Pressure Fluctuations in the Afterbody Measurements of the pressure fluctuations were carried out at different points in the afterbody using semi-conductor strain gauge transducers (ENDEVCO), see Figure 2 and Table II, having a mem- brane diameter of 8 mm, The circuits included low pass filters (32 Hz). Accelerometers were fitted to the hull close to the transducers, in order to record the influence of the hull vibrations on the propeller induced pressure field, [28]. Propeller Cavitation The equipment used for observation of propeller cavitation is described in [4 ] : Noise Level in the Afterbody A tape-recorder Tandberg TB 11P was positioned in the aft of the main engine room and in locations adjacent to the propeller (the emergency exit channel), Thrust, Thrust Variations and Shaft Response were measured by strain gauges (Hottinger) on the thrust bearing fundament and shaft and the signals telemetered to a digital voltmeter (thrust) and a UV- recorder (thrust variations), Aslo this circuit included a low pass filter, Further, axial shaft resonances were recorded by a spring transducer, working against the flange coupling between propeller and intermediate shaft. * IV..2. Test Results Vibratory Response in the Structure of the Ship The results of the measurements of the vibration levels at 588 Propeller Exettatton and Response of 230000 TDW Tankers different points in the deck house and in wing tank No 5 SB are shown in Figures 3 and 4. The levels are given as amplitudes, In Figure 3 some levels, indicating the degree of unpleasantness, are marked for comparison. They were calculated from the ISSC criteria, based on acceleration, by using the formula y = af(2nf)” where y = vibration amplitude a = acceleration, vertical f = frequency of vibration The results from the measurements in the wing tank indicate weak resonance at certain shaft speeds. The results of Figure 4 are the largest values measured over the speed range. The diagrams of Figures 3 and 4 show that the vibration am- plitudes in the accomodation spaces were of reasonable magnitude. On the first ship without fins, the amplitudes measured in the wing tank were, however, considerable due to resonance at certain shaft speeds (in general not the normal service speed). Unfortunately no measure- ments were carried out in the wing tank on this ship, when fitted with fins. From the results of the measurements in the deckhouse at the different occasions it can, however, be concluded that the introduction of fins reduced the vibration level considerably, but that the replace- ment of the five-blades propeller with a six-bladed one caused no fur- ther reduction of the vibration level. (In this connection it should be remarked that the structures of the hull and deckhouse are identical for the two ships apart from the fact that the second ship was fitted with bilge keels). Pressure Fluctuations on the Hull The results of the measurements of the pressure fluctuations at different points in the stern are given in Figures 5-6, In Figure 5 the two versions of the first ship are compared at fully loaded and ballast conditions. The results shown were obtained with transducers Nos 3 and 5 respectively, approximately corresponding to measuring point A in model tests. The diagram shows faired mean values of the peak to peak values 2p , obtained in the way shown in Figure 7, nor- malised as 589 Johnsson and Sgntvedt These results show the beneficial influence of the fins and it is further evident that the amplitudes in ballast condition are larger than in fully loaded condition, In Figure 6 the two ship configurations having five-bladed (first ship) and six-bladed (second ship) propellers are compared, both being fitted with fins. The results were obtained for the trans- ducers 4 and 2 respectively, which were placed at the bottom of the fin, on the starboard side, close to the centerline of the ship. The positions were slightly different for the two ships, as is shown in Table II and Figure 2. The difference is, however, small enough to allow direct comparison of the results. The positions correspond approximately to measuring point D in model scale. The diagram shows that the pressure fluctuations measured on the two ships are very similar, in spite of the fact that the first ship was equipped with a five-bladed and the second with a six-bladed propeller. In fully loaded condition the six-bladed propel- ler caused somewhat larger amplitudes than the five-bladed. (The fact that only the second ship was fitted with bilge keels is not consi- dered when making this comparison). A sample record, obtained at the full scale measurements, is shown in Figure 8. Propeller cavitation Some of the photographs taken during the trials with ''Norse King'' were reproduced and discussed in [4] . In the present report some sketches based on these photographs are given in Figures 14 and 17 together with the results of the measurements of the pressure fluctuations and the corresponding cavitation patterns, obtained at the model tests. Noise Level in the Afterbody The noise level in the afterbody of ''Thorshammer'" is ofa transient type, varying directly as the vibratory response in the struc- ture of the ship and the recorded pressure fluctuations. This observa- tion is valid for the ship with and without fins installed. The noise level onboard ''Norse King" is stable (mainly pulses of short duration, occurence frequency equal to blade frequency) in the higher speed range (70-85 RPM). 590 Propeller Excttatton and Response of 230000 TDW Tankers Mean Thrust, Thrust Fluctuations and Shaft Response Recordings of mean thrust are presented in [2] j [3] and [4] : In Figure 9 results of calculations of the natural frequency in the axial mode for the two ships are shown, assuming different values of the thrust bearing stiffness. As shown in the Figure, the measured natural frequencies for the two cases correspond to almost the same value of the thrust bearing stiffness, indicating that recorded reso- nances of shaft vibration in the axial mode are predicted with fairly good accuracy by a method now in use at DnV [5 | . The vibratory output from the shaft through the thrust bearings is of moderate ma- gnitude for all ship - propeller configurations. Propeller Blade Erosion The results of erosion studies in full scale and model scale were given in [4 ] and will not be discussed in the present report, It should, however, be noted that the eroded areas were similar in model and full scale. Regarding the relative merits of the different configu- rations, the full scale, as well as the model tests, showed that the area of erosion was not reduced essentially by the introduction of fins on the first ship, but was eliminated by fitting the second ship with a six-bladed propeller of new design. On the propeller of the first ship the erosion was, however, less rapid after the introduction of fins, V. MODEL TESTS V.1. Test Arrangements and Facilities Cavitation Tunnel The cavitation tests were carried out in the new, large cavita- tion tunnel of SSPA, This tunnel, see [6 | and Figure 10, is powered by a 1000 Hp motor and has two interchangeable test sections, one being circular, as the remaining part of the circuit. The other test section, which is of interest in this connection, is of rectangular shape with a breadth of 2.6m, a height of 1.5m anda length of about 10 m. The section is covered by a recess in which the ship model is placed. This model is the one used in the towing tank for the self pro- pulsion tests and it is normally made of paraffin wax. The vertical position of the model is adjusted in such a way that the waterline, corresponding to the level of the free water surface in the towing tank, is flush with the top of the test section, Individual- ly cut wooden plates are then fitted to simulate the free surface, and 59 Johnsson and Séntvedt the test section and the recess are filled completely with water. Up to now flat plates have been used and no attempts have been made to si- mulate the wave system around the hull, The maximum water speed is 6.8 m/sec. An electric motor and a strain gauge dynamometer for measuring thrust and torque for the propeller are placed in a water- tight cylinder in the model, Measurement of Pressure Fluctuations on the Hull For the measurements of the pressure fluctuations differential transducers were used, being of the strain gauge type, manufactured by Statham, The maximum range for the transducers, used at the tests in the cavitation tunnel is + 25 psi, the natural frequency being about 9 kHz. The diameter of the membrane is 1/4" (6.35 mm), One end of the transducer was connected to the atmospheric pressure, The signals were amplified and registered on an oscillographic recorder, two channels being used for each transducer. One channel was used for the original signals, on the other a filtered signal was registered, the filter being tuned on the blade frequency. The natural frequency of the galvanometers was 1 650 Hz (original signals) and 400 Hz (filtered signals), For obtaining higher harmonics a frequency analyser (manufacturer Brttel and Kjaer, type 2107) was used. The range 63-2000 Hz was used, the total sweeping time being 6 min for this range. The band width is about 6% of the frequency registered. No accelerometer was fitted to the model during the tests as, at earlier measurements of a similar kind, carried out at SSPA, only low levels of the accelerometer signals were obtained. V.2. Test Conditions in the Cavitation Tunnel The first condition to be fulfilled in order to obtain reasonable results with regard to cavitation patterns, erosion patterns and pres- sure fluctuations is to accomplish a realistic wake distribution behind the ship model in the cavitation tunnel. This problem was thoroughly discussed in [4 ]and it will only be stated here that, if the wake dis- tribution in the towing tank is used as that to be aimed at, the agree- ment obtained in this case was very good, in particular in fully load- ed condition, as-is evident from Figure 11. Thus it has been demons- trated at these tests that a representative wake distribution can be realised without incorporating a free water surface, The comparison was made by using ordinary Prandtl-tubes. Recently, when using five-hole spherical pitot tubes on another, similar project, the same degree of agreement was obtained also for the tangential velocities and flow angles. 592 Propeller Exettatton and Response of 230000 TDW Tankers For these kinds of tests it is further required to define the loading cases for the propeller. In the present investigation two com- binations of advance ratio J and cavitation number o were tested, one corresponding to the values of J and effective wake wy , obtained from the propulsion tests in the towing tank, the other corresponding to the predicted full scale values of J and wyc. The prediction method used for obtaining the latter values was discussed in [4 ]. The desired combination of J and o was realised in the tun- nel by using thrust identity with the open water tests. Most of the tests in the tunnel were carried out at a water speed of V=4m/s and an air content ratio of a/a; % 0.4, but in some cases these parameters were varied. Non-Cavitating Flow : V.3. Test Results, Pressure Fluctuations in Non-Cavitating Flow, Comparison with Theoretical Calculations In Figure 12 the first harmonics of the pressure fluctuations obtained for the model with fins, fitted with the six-bladed propeller, are shown in fully loaded, as well as ballast condition. Amplitudes, obtained in the following manner, are included in the diagram : A) Measurements in the towing tank, model speed 1.2 - 1.5 m/sec. Amplitudes registered on oscillographic recorder, original signals. These curves give the mean values over the speed range 11-18 knots. B) Measurements in the cavitation tunnel, water speed 4 m/sec. J-value = Jtowing tank - “mplitudes registered on oscillographic re- corder, filtered signals. Measuring accuracy about the same as in towing tank, C) Theoretical calculations, carried out by SSPA. Wake influence considered. Method of calculation described in|[7]. D) Theoretical calculations, carried out by DnV. Wake influence considered, Method of calculation described in [8 -9 | : From the diagram in Figure 12 it can be concluded that the agreement between the measurements in cavitation tunnel and towing tank is reasonably good. Also the agreement between calculation and experiment is satisfactory. The influence of number of blades and fins on the pressure 595 Johnsson and Sgntvedt fluctuations in non-cavitating flow is demonstrated in connection with the results in cavitating flow, see section 5, 6. Cavitating Flow : V.4. Measurement of Pressure Fluctuations in Cavitation Tunnel. Variation of Test Parameters, Wall Effects Apart from the wake distribution the two most important test parameters in the cavitation tunnel are water speed and air content. Both parameters are known to affect the extent of the cavitation, With regard to the air content a /ag it is plausible that it might affect the damping of the water and thereby the level of the pressure fluctuations registered by the transducers, In order to investigate this influence, four different combina - tions of water speed and air content (4 and 5 m/sec, a/a, = 0.13 and 0.40) were investigated for one propeller - hull configuration (six- bladed propeller, model without fins), From the results, some of which are given in Figure 13, it can be concluded that the influence of these parameters on the amplitudes was considerable for the filtered signals but rather small for the maximum values of the non-filtered signals, A problem, which should not be neglected in connection with this kind of measurements, is that of wall effect. This has been dis- cussed by Huse in [10 | » from whom we quote : "The pressure signal recorded by the pressure transducers may be split into two parts, 1) the "direct pressure wave'' induced by propeller and cavity, propagating directly to the field point (pressure transducer), and 2) the reflected pressure wave induced by propeller and ca- vity, propagating to the field point by one or more reflections at the tunnel walls, "' On the basis of some approximate calculations Huse concludes that for the combination of test section and propeller size used at his experiments "wall effects are of minor importance in the case of a non- cavitating propeller and also in the case of cavities of constant 594 Propeller Exettatton and Response of 230000 TDW Tankers volume during the propeller revolution, The pressure field due to volume variation, however, is reflected from the tunnel walls in such a way that the reflected amplitude may possibly be of the same magnitude or even higher than the direct wave amplitude,"' In fact, for one of the field points investigated, the calculations gave reflected amplitudes 4 times larger than those of the direct pres- sure wave. It could be expected that, in the much larger test section used for the experiments reported here, the wall effects are of less impor- tance, V.5. Test Results, Comparison with Full Scale In this paragraph some test results will be given, which illus- trate the correlation obtained between the measurements of pressure fluctuations in the tunnel under cavitating conditions and the corres- ponding full scale results. Analogous problems in connection with ca- vitation patterns and erosion patterns were discussed in| 4]. In [4] it was shown that the erosion patterns as well as the ca- vitation patterns in fully loaded condition agreed very well in model and full scale. In ballast condition, however, the extension of the ca- vitation, as observed in the cavitation tunnel, was somewhat smaller than in full scale. Some of these comparisons are shown in Figures 14 and 17 in connection with the comparisons of the pressure fluctuations. Pressure fluctuations, measured in model scale and full scale, are compared in Figures 14-17. In all the diagrams the following kinds of results are given in non-dimensional form : A) Model tests, filtered signals, mean values over about 10-20 revs. B) Model tests, signals obtained without filters, max values during 10-20 revs. C) Full scale results, max values obtained as shown in Figure 7. This kind of curve should correspond to results according to B for the model, (For the case shown in Figure 16 this kind of value was not available. ) D) Full scale results, mean values obtained as shown in Figure 7. 595 Johnsson and Séntvedt In Figures 14, 15 and 17 some values of the following kind are also shown : E) Full scale results, mean values of the first harmonic of blade frequency, obtained from the energy spectrum (with the use of UV recorders, D-Mac curve follower, paper tape and a computer pro- gram, estimating the energy spectrum of stationary stochastic pro- cesses, see [29 ‘ These results correspond approximately to results of type A from the model tests. In Figures 14 and 17 also the cavitation patterns obtained in full scale and model scale are shown, The full scale patterns were obtained from photographs taken in connection with the speed trials, the model patterns were sketched directly when observing the cavi- tation in the tunnel, the position angle for the blade being 20° from upright position for the sketches in Figure 14 and 25° for Figure 17. From Figures 14-16 it is evident that, in fully loaded condi- tion, the agreement between the pressure fluctuations in model and full scale is reasonably good. This applies to the non-filtered signals as well as to the few cases where a comparison was made for the fil- tered signals. In ballast condition, however, the amplitudes were lower during the model tests, which seems to be due to the fact that the ex- tension of the cavitation was smaller on the model propeller in this case, When the propeller was run ata lower J (J and o correspond- ing to self propulsion tests in the towing tank) the level of the unfil- tered signals increased, however, to values reasonably close to those measured in full scale, V.6. Test Results. Comparison between Different Propeller -Hull Configurations In Figures 18-20 the four different configurations are com- pared in the fully loaded condition, In the diagrams the amplitudes for the measuring point B are shown for three combinations of ad- vance ratio J and cavitation number @ which correspond to three different speeds for the full scale ship. In Figure 20 the cavitation patterns in the blade position 20° from upright are also shown. More detailed cavitation patterns for the propellers in different blade posi- tions are shown in Figures 22-23 for the speed 16 knots. In Figure 18 the mean values of the filtered signals are given, in Figure 19 the corresponding maximum values, Further the diagram in Figure 20 shows the maximum values of the non-filtered signals. 596 Propeller Exettatiton and Response of 230000 TDW Tankers Examples of different kinds of signals are shown in Figure 21 for some of the runs (results from the frequency analyser are also included), In Figure 18 the results obtained at atmospheric pressure cor- respond closely to non-cavitating conditions, although limited cavita- tion was present at some configurations. The diagram shows that, under non-cavitating conditions, the difference between the amplitudes for the different hull-propeller configurations is small, The diagram in Figure 18 further shows that, if the filtered signals are taken as representative (they should correspond approxi- mately to the first harmonic), the influence of cavitation on the ampli- tudes of the signals of the five-bladed propeller is very small and in- dependent of the model being fitted with fins or not. Especially in the case of the model without fins the difference between the amplitudes of the five-and six-bladed propeller in cavitating condition is ap- preciable, in spite of the fact that the cavitation patterns are very si- milar, see Figures 22-23. This situation is not changed very much if, instead, the maximum values of the filtered signals are compared. If, however, the maximum amplitudes, obtained without filter, are compared, a more reasonable relation is obtained between the different configurations, see Figure 20. This diagram shows the bene- ficial influence of the fins for the five-bladed as well as the six-bladed propeller, which has been confirmed in full scale, both by vibration measurements and measurements of pressure fluctuations. Both with and without fins the five-blades propeller is better that the six-bladed, in agreement with the tendency of the corresponding full scale mea- surements. According to the vibration measurements in full scale the two propellers should, however, be roughly equal. The amplification of the signals, caused by cavitation, is appre- ciably higher for the six-bladed than for the five-bladed propeller. The same tendency, although less pronounced, was found at systema- tic tests carried out at Wageningen, using a dummy model [1 1 | er he maximum amplification factors found at the present tests were 4 (filtered signals) and 17 (max non-filtered/filtered without cavitation). Apart from being observed from the non-filtered signals, the difference between the five- and the six-bladed propeller can be seen in the diagrams obtained from the frequency analyser, see Figure 21. It is evident that the content of higher harmonics is larger for the five-bladed than for the six-bladed propeller. Results of comparisons of higher harmonics have not been included here as, in several cases, it was difficult to determine the amplitudes with reasonable accuracy. BOW Johnsson and Sgntvedt From the diagrams in Figures 22 and 23 it is evident that the maximum extent of cavitation in different blade positions is rather similar for the different hull-propeller configurations, The main difference between the cavitation patterns for the five- and six-bladed propellers seems to be that, for the five-bladed propeller, the extension of the cavitation was more fluctuating with time than for the six-bladed. This may be one explanation of the fact that the high pressure pulses from the five-bladed propeller were of such a short duration that they were not manifested on the registra- tions of the filtered signals. It should also be mentioned that rather small band widths were used when filtering the signals. The extension of the cavitation was rather similar, whether the model was fitted with fins or not. In spite of this a beneficial in- fluence of the fins could be noted on the amplitudes of the pressure fluctuations, A type of cavitation, which is regarded as important in connec- tion with fluctuating pressures on the hull, is the so called propeller - hull - vortex cavitation [1 0]. This type of cavitation was observed fre- quently during the tests, but to about the same extent for the two pro- pellers. This kind of cavitation was probably present during the full scale trials with the first ship, see [12] , but could not be observed on the second ship, when making visual observations in connection with the photographing of cavitation. VI. CAVITATION PATTERNS AND PRESSURE FLUCTUATIONS, THEORETICAL CALCULATIONS AND COMPARISON WITH EXPERIMENTS VI. 1. Calculation of Circulation Distributions Proper results for hydrodynamic loading particulars require the solution of a complicated lifting surface problem. With boundary layer aspects included, the downwash surface integral equation for a twisted wing of finite span should ideally be completely solved. The importance of obtaining a reliable method for prediction of external loading was strongly emphasized by various authorities some years ago. Research was then initiated in Scandinavia to meet this demand along the following lines of approach : A method of cal- culation was desired, which should be able to reproduce open water diagrams within experimental accuracy for all relevant values of J. By combining unsteady effects, effects of curved flow and interaction with the inlet wake field one should then be able to simulate experi- 598 Propeller Excttatton and Response of 230000 TDW Tankers mental results from tests in grid wakes, as well as in the actual be- hind conditions. Two parallel methods of approach have been out- lined : 1. Unsteady lifting line technique 2, Unsteady lifting surface technique The steady part of 1,i.e study of open water performance has been completed [13, 27 | . For 14 propellers tested the method des- cribed in |13] reproduced recorded open water characteristics within experimental accuracy for all values of advance ratio. The method employs results from an experimental study of pressure distribution across a propeller surface [14 Jas the basic empirical ''tool' for obtaining a realistic lift distribution along a lifting line. Preliminary results from use of 2(unsteady lifting surface theory) for open water work indicate that the method works poorly for off design cases (low advance ratios) [15]. Combined with effects indicated above, the lifting line tech- nique in use at DnV has been advanced to a stage where it has been possible to reproduce the experimental results in behind condition for the relatively few experimental results available. In Figure 24 is shown how both the lifting line and lifting surface technique may work poorly, when interaction between propeller and hull wake field is not considered. The propeller model in question works behind a 220 000 TDW tanker. Clearly, our method of approach, which includes in- teraction corrections, based on simple continuity of flow, reproduces the experiments ''within experimental accuracy" [16 : Local advance ratios, as may be experienced in the tip region of the blade when passing a wake peak, will lead to a significantly non- linear C, - a relationship. This effect has been approximated by use of results of experiments for low aspect ratio wings described in [17] and [18]. To decrease the risk for a "happy coincidence to occur'' we have performed other comparisons with experiments. Thus in Fig. 25 results from a one-blade dynamometer test are compared with cal- culated values, obtained by using our approach ; the calculations being based on the nominal wake field and the detailed propeller geo- metry. As far as we understand, the practical implications of the ob- servations given above, are the following : 1, The wake survey should be performed in the propeller plane of the eg Johnsson and Sgntvedt towed model - from the shaft CLto at least 1.3 x R (R = propeller radius). 2. It is important to include both the axial and tangential wake field in the analysis. 3. Itis possibly necessary to extend the lifting surface theory to in- clude non-linearity and effects of interaction with nearby boundaries. The research now initiated will continue in the 1972-1974 pe- riod. VI.2. Calculation of Radial and Chordwise Pressure Distributions The corresponding detailed pressure distributions are then found, applying a method presented in [14 ] and [19] and briefly outlined in Appendix A, In Figures 26 and 27 detailed pressure distributions calculated in accordance with the said appendix are shown to correlate well with Hgiby's experiments (see [14] , Figure 21, J = 0.1068). VI.3. Calculation of Cavity Formation For the ships considered in this report, the pressure distri- butions for the propeller blades in upwards vertical position (% = 0) and corresponding extent of cavitation are given as follows : Figure 28 illustrates the calculated extent of cavitation on the full scale propeller mounted onboard T/T ''Thorshammer' with ob- served erosion on the blades included. In [4] it is concluded that mo- del and full scale erosion patterns are similar (Figures 31-32- cor- responding pressure distribution - calculated), Figure 29 gives the observed versus calculated amount of cavitation in loaded condition onboard T/T "Norse King". Figure 30 illustrates a similar compa- rison in the ballasted condition for RPM = 66, Vos = 12.5 knots. More interesting are the theoretical/full scale correlation and the tehoreti- cal/model correlation presented in Figures 33 and 34 respectively. (Figures 35-36- corresponding pressure distribution). Assuming no scale effect on the cavitation tunnel wake field, we observe that the calculated difference in radial variation of the dynamic pressure re- lative to the static pressure is actually experienced by visual cavita- tion observations. Some difficulties reported with exact simulation of velocity, number of revs and tunnel pressure may also explain some of the discrepancies between model and full scale observations, Details connected with determination of type and extent of ca- vitation are described in [1 9] and briefly outlined in Appendix B. 600 Propeller Exettatton and Response of 230000 TDW Tankers Applying symmetrical hysteresis effect, (although non-symmetrical in the tip region) and assuming that time dependent factors, such as inertia and duration of transient pressure, do not influence the onset of cavitation, a simple 'maximum bubble radius"! concept has been used to establish the extent of cavitation. Further, it has been assum- ed that the degree of turbulence of the inlet flow is so large that no laminar separation occurs, Also when ignoring effects of sudden charges of angle of incidence and several other effects, we find that, for several cases considered, the quality of the results obtained, when operating in behind condition, is satisfactory for engineering purposes fi9] : The thickness of the sheet of cavities at 0.95 r/R is found by estimating the height of the tip vortex, as described in [1 8 | Per he radial thickness distribution is then found by linear interpolation, as the radial inception point is already determined. The method is briefly outlined in Appendix B. VI.4. Calculation of Pressure Fluctuations on the Hull Finally, in this section we will illustrate how simple mathe- matical models may be used in this case to approximate the compli- cated transfer function, giving rise to a fluctuating hull pressure field, during the formation of unstable cavities. The acceleration potential caused by the cavity formation may be found by solution of the Volterra integral equation, if the forma- tion be accurately represented at any time during growth and collapse. The vapour/liquid mixture representing a pulsating volume cannot be said to constitute a surface of known shape. Consequently, an ideal mathematical model of moderate complexity should be employed, together with empirical corrections found by experiments. The mathematical model now in use at DnV [20 | is described in some detail in Appendix C, The cavitation patterns observed on- board T/T 'Norse King", simulated as shown in Figures 37 and 38, have been used to obtain the results, presented in Figure 39. Clearly, the calculated values for blade frequency pressures on the hull clo- sely resemble full scale values recorded. Also twice blade frequency components, as calculated, correspond approximately with values recorded, see Figure 40. VII. SUMMARY AND CONCLUSIONS The most important conclusions to be drawn from the results of the present investigation can be summarised as follows : 601 Johnsson and Sgntvedt 1, From the vibration measurements in full scale it can be concluded that the introduction of afterbody fins lowered the level of the blade frequency hull vibrations. No further improvement seems to have been achieved by replacing the original five-bladed propeller on the first ship by a six-bladed propeller of different design on the second ship. For all configurations the amplitudes were somewhat larger in ballast than in fully loaded condition, 2. The conclusions drawn above were confirmed in general by the results of the measurements of hull pressure fluctuations in different points of the stern, made simultaneously with the vibration measure- ments. In fully loaded condition the amplitudes of the pressure fluc- tuations were, however, larger for the six-bladed than for the five- bladed propeller over most of the speed range, the vibration levels being about the same, as mentioned above. 3. Under non-cavitating conditions reasonable agreement was obtain- ed between cavitation tunnel, towing tank and theoretical calculations for the amplitudes of the blade frequency harmonic of the pressure fluctuations. The pressure fluctuations obtained were similar for the different hull-propeller configurations, 4, The amplitudes ofthepressure fluctuations in full scale and those obtained in the cavitation tunnel under cavitating conditions showed reasonably good agreement, This applies to the maximum peak to peak values, as well as the first harmonic for the fully loaded condi- tion. In the ballast condition the amplitudes were smaller in model scale, When judging the relative merits of the five- and six-bladed propellers from the vibration point of view on the basis of pressure fluctuations, measured in the cavitation tunnel under cavitating con- ditions, the results have to be analysed very carefully. Thus, in the present case, a relation between the different hull-propeller configu- rations, agreeing with the tendency of the full scale experiences, was obtained only when the maximum peak to peak values were used for comparison, Apparently, the pressure signals, recorded for the five- bladed propeller, were of pulse nature with rather unstable phase shifting. Consequently they did not affect the levels of the filtered signals, obtained in non-cavitating flow. In the cases considered the amplification of the amplitudes of the pressure fluctuations, caused by cavitation, was larger for the six-bladed than the five-bladed propeller. Results reported in [1 1] show the same tendency. 602 Propeller Excttatton and Response of 230000 TDW Tankers 5. The type and extension of the cavitation, as observed on the full scale ship (six-bladed propeller) and in the cavitation tunnel, are rea- sonably similar for the fully loaded condition. In the ballast condition, however, cavitation is more extensive on the ship than on the model propeller. 6. The observed type and extent of cavitation were confirmed with reasonable accuracy by the calculations illustrated in the present paper. 7. The pressure impulses of both blade and twice blade frequency, recorded onboard the ship, correlated well with those calculated with the use of the method developed for the calculation of pressure fluc- tuations from cavitating propellers. VIII ACKNOWLEDGEMENT The investigation was carried out in close cooperation between the initiator, Uddevallavarvet AB, the Swedish State Shipbuilding Ex- perimental Tank and Det norske Veritas. The work has been sponsor- ed partly hy the Swedish Board for Technical Development and partly by the shipyard and the shipowner companies involved, i.e. A/S Thor Dahl, Sandefjord and Odd Godager and Co, Oslo. The authors would like to express their sincere thanks to all those members of the staffs of the above-mentioned establishments and companies, who have taken part in the investigations and contri- buted to the analysis of the material. LIST OF SYMBOLS Ayo = developed blade area ratio 1 2 Cy = dL/—pcdbV = lift coefficient 1 2 i € = (p-p )/— pV = pressure coefficient p of 2 Cc = blade chord D = propeller diameter dL = lift of profile 603 Johnsson and Sgntvedt Vv ,/nD = advance ratio for propeller 2 2p/p ioe = non-dimensional coefficient for blade frequency amplitude 42 pie T/pD n = thrust coefficient correction factor for ideal angle of incidence, due to lifting surface effect correction factor for angle of incidence, due to thick- ness effect number of revolutions propeller pitch static pressure static pressure in undisturbed flow cavitation pressure vapor pressure radius of propeller blade section thrust maximum thickness of profile advance velocity of propeller ship speed local wake effective wake from thrust identity non-dimensional chordwise coordinate, measured from leading edge number of blades air content 604 Propeller Exettatton and Response of 230000 TDW Tankers p = density of water Q TT 1 (p - Bi)i=5 an = cavitation number for propeller REFERENCES [1] NESS, L.M., 'T/T Thorshammer, Uddevallavarvets b nr 226, Vibrasjonsmalinger'', DnV Report No 70-31-00, Oslo, May 1970 (in Norwegian). [aa J@RGENSEN, @., ''T/T Thorshammer, Uddevallavarvets b nr 226!' DnV Report No 70-46-0, Oslo, April 1970 (in Nor- wegian),. [3 | J@RGENSEN, G., ''T/T Norse King, Uddevallavarvets b nr 234'', DnV Report No 71-20-0, Oslo, Febr 1971 (in Norwe- gian). [4 | LINDGREN, H., JOHNSSON, C-A., SIMONSSON, E., "Propulsion and Cavitation Investigation on 230 000-tons dwt Tankers - Full Scale and Model Experiments", The Royal Institution of Naval Architects, Spring Meetings 1972, Paper No8. [5 ] "Computer program NV517 : Holzer Tabulation of Axial Vi- brations in Straight Marine Shaft Systems"! [6 | EDSTRAND, H., ''Kavitationslaboratoriet vid Statens Skepps- provningsanstalt (The Cavitation Laboratory at SSPA)", Swedish State Shipbuilding Experimental Tank Circular No 26, GSteborg 1970 (in Swedish). [7] JOHNSSON, C-A., ''Pressure Fluctuations Around a Marine Propeller. Results of Calculations and Comparison with Ex- periment'', Swedish State Shipbuilding Experimental Tank Publ. No 69, Gdteborg 1971. [3] HUSE, E., ''The Magnitude and Distribution of Propeller- Induced Surface Forces on a Single-Screw Ship Model", Norwegian Ship Model Experiment Tank Publ. No 100, Trondheim 1968. 605 [9] Lio] [11] [12] [13] [14] [15] [16] [17] [is] Li9] Johnsson and Sgntvedt RAESTAD, A.E,, ''Computer Program NV538 : The Free Stream Pressure Field Induced by the Propeller", DnV Report No 70-22-M, Oslo 1970. HUSE, E., ''Pressure Fluctuations on the Hull Induced by Cavitating Propellers", Norwegian Ship Model Experiment Tank Publ. ‘No 111, Trondheim, March 1972. VAN OOSSANEN, P., VAN DER KOOY, J.;, ''Vibratory Hull Forces Induced by Cavitating Propellers", The Royal Insti- tution of Naval Architects, Spring Meetings 1972, Paper No 9. SO@NTVEDT, T., ''Propeller Induced Excitation Forces", DnV Publ. No 74, Oslo, Jan, 1971. SONTVEDT, T., ''Theoretical Calculations of Hydrodynamic Loading on the Marine Propeller. PartI, Open Water Per- formance, Progress Report No.2", DnV Report No 71-64-M, Oslo 1971. H@OIBY, O.W., ''Three-Dimensional Effects in Propeller Theory", Norwegian Ship Model Experiment Tank, Publ. No 105, Trondheim, May 1970. KUIPER, C., "Unsteady Lifting Surface Theory", Design and Economical Considerations on Shipbuilding and Shipping, Rep of the Post Graduate Course, May 1969, pp 125-149, (Royal Institution of Engineers), Wageningen, Holland 1970. RAESTAD, A.E., ''Estimation of a Marine Propeller's Induc- ed Effects on the Hull Wake Field", DnV Report No 72-3-M, Oslo 1972. KUCHEMANN, D., "'A Simple Method for Calculating the Span and Chordwise Loading on Straight and Swept Wings of any Given Aspect Ratio at Subsonic Speed", Aeronautical Research Council, R and M, no 2935, London 1956. KUCHEMANN, D., KETTLE, D.F., ''The Effect of End- plates on Swept Wings"', Royal Aircraft Establishment, Farnborough, Report No Aero 2429, June 1951, HOLDEN, K., ''Type and Extent of Cavitation on Hydrofoils and Marine Propeller Blades"', DnV Report No 72-2-M, Oslo 1972. 606 Propeller Exettatton and Response of 230000 TDW Tankers [20 | HOLDEN, K., SONTVEDT, T., ''Propeller Cavitation asa Source to Vibration'', DnV Report No 72-5-M, Oslo 1972. [2a] SCHOLTZ, N., ''Strémungsuntersuchungen an Schaufelgit- tern", VDI-Forschungsheft 442, Dusseldorf 1954. [22 ] SCHLICHTING, H., ''Berechnung der Reibungslosen in- kompressiblen Stré6mung ftir ein vorgegebenes Schaufelgitter", VDI-Forschungsheft 447, Dusseldorf 1955, [23] POLLARD, D., WORDSWORTH, J., 'A Comparison of Two Methods for Predicting the Potential Flow around Arbitrary Airfoils in Cascade'', Aeronautical Research Council, C P No 618, London 1963. [24 | MORGAN, W. 5. SLLOVIGA i. DENNY, ©.; | Propeller Lifting Surface Corrections", Trans Society of Naval Archi - tects and Marine Engineers, Vol. 76 (1968). [25] STRASBERG, M., ''The Influence of Air-Filled Nuclei on Cavitation Inception'', David Taylor Model Basin Rep 1078, Washington DC, 1957. [26] | JOHNSSON, C-A., "On Theoretical Predictions of Charac- teristics and Cavitation Properties of Propellers'', SSPA Publ. No 64, Gdteborg 1968. bem HUSE, E., "Hull Vibration and Measurements of Propeller- Induced Pressure Fluctuations", Proc. 12th Int Towing Tank Conference, Rome, Sept. 1969. [234 DENOOTJ, F.E., 'Auto-Correlation Functions and Energy Spectrum of Stationary Stochastic Processes'', DnV Report No 69-37-S, Oslo 1969. [29] WEBER, J., "The Calculation of the Pressure Distribution over the Surface of Two-Dimensional and Swept Wings with Symmetrical Aerofoil Sections', Aeronautical Research Council, R and M, No 2918, London 1956. 607 Johnsson and Sgntvedt APPENDIX A CHORDWISE PRESSURE DISTRIBUTION, METHOD OF CALCULA- TION From [14] and [19 | we have for the contributions to the induced velocities : 1, Velocity distribution due to blade thickness distribution, (2) a2 all = cera Au, (x, z) a : A) 7 ahs 5 == (1) NO sage at si@higgor't Z = profile thickness ordinates Zz . s = camber ordinates Uno = Uy . cosa, cos 2 Q = effective skew angle « = correction factor due to blade thickness taper Mn and f{(Q) = factors taking account of the effect of the centre section and the tip region 608 Propeller Exettatton and Response of 230000 TDW Tankers : bal i 2 2 2 Uy = resultant inflow velocity = (wr+V,) Nats ee 2. Angle of incidence Au, (x, z) ‘ 1 ns: arts ——— = .(—S). (. +—“*— ) (2) Oo 57) ag cos(1-y_,)2 (x) rT veo = Uo. sina. cos y 1 l ! 2} 1 ! 5!) (x) -= (2). z(x') : dx ue 0 1 -(1-2x') x-x! y = rake angle Qu p eae A a n/2 = - ——_1_______.. ag: cosQ up 1/4 2 {1 +_——————) } TA 3. Mean Line Camber Q Au, (x, z) tye) i Ho. tan dz | (x) : ; ; eae Race ee ee CG 2U No ise fi pba es) ce Losey é Atanro T 1 Qe. 1 -x 3) ( (3) Se 1 1 oan (x!) x! 9 dx' all ) 1 0 dx! ] - x! x=-x! Bore - 1 6 = ‘ey eee arccos “T 2 4. Thickness of the Other Blades The cascade effects are obtained as shown by 0. H¢ib 14] » Scholz [21], Schlichting [22]and Pollard and Wordsworth [23 609 Johnsson and Séntvedt = ' ! HS Ss dx! BS Pa Ra (4) uw. = chordwise induced velocity a 4 an x Z) 4 Chord <" = Os25r 8 2005 dc k = /2n da ast ~ hf 2 C*)i4n“”..;,coa 2 p 2 x-X B oa s z a= 1 x-x x-x' ( ) +n +2n° .cos2 ¢ ( ) s cS é =, r/2-¥ v = pitch angle 5. Bound Vortices on Other Blades. 1 ie Nae Sa + = Th Fy rs 2 u' ° (a2) dp (5) NO NO c A b a = chordwise induced velocity P{p,2) = circulation r. sin y(r). sin(g-O+ 5,,)-(¥(2) 0-¥()9)cos(s-0 - 6.) cosy(r) u! = ieee eT es a ee Seo ll” 2 2) 2 3/2 P 4m { (y(r)O-¥(p)d) +r tp - 2rp cos(g-O+ 5 Of 6k = —— ; RS 2y tes a = chordwise induced velocity due to a bound vortex of unit strength at (5.5 P,; g) The angle of incidence is found from the following expres- sions : 610 where ey Propeller Exettatton and Response of 230000 TDW Tankers ‘ie L da ata er 1 = “2 O(a DCG 2B). °C jibod iby foe dx (6) a i ny tae” wel U Oo fe) NO Ch» B. and £6 are obtained from lifting line calculations nf) 24 COS Mp Q Zon a Sow mcinanet * 7 = wn(cotmn - cot7n ) fe) fe) 1 Q = ell + Map == 1/2 t = 10. 941 + 0.8—) as 1 Z os { PaaS ai(al atiieo Zi .48 NO r k=2 = induced velocity normal to the chord line -r cosy(r). sin(g-6+6, ) + (¥(r) O- ¥(p)¢) cos(g-0+6, ). siny(r) i Semmes CCR RIDA DENT (aD Gacy. vampypecuts svat SL De The) muh 4r{(¥(r)0-¥(p)p)> + x? +p” - 2rpcos (g-046,) }°/" = induced velocity normal to the chord line, due to a vortex of unit strength 1 Ve (x) = B - B+ dx rL f Uno fo) k Vedas") == ii ——. 1(¢, 5, x") ax! = induced velocity normal to the chord line due to thickness of the other blades Za/ 2. 3 12 case y Tinie Srae — ie Shaki (ay . + ce + Deae . cos apy es ) ig = Il s Ss Johnsson and Sgntvedt dc, dC a = (Va bs el) ay e a= effective angle of incidence, corrected for lifting surface effects i3 i2 a t*? at where kK. and k are found from lifting surface theory [24 | : From [14 ] we quote : '" For a swept wing Weber [30 | has shown that a flow parallel to the span must be added, so that the total pressure distribution is given as: poets —~icos @ ‘sin < is 3 i Us + Au, (x) + Au, (x) + Au_(x) + u(x) - se APPENDIX B CAVITY FORMATION, METHOD OF CALCULATION 1. Extent of Cavitation When the detailed blade pressure distribution is known, the amount of cavitation may be determined, see [2 6, 27 | 2 ae op dp = ik f af 4 (1) = ' 3 x = Pp 1 2 where x) = the point at which the local pressure falls below the cavi- tation pressure 18 Wea Papor x = the point at which the local pressure has increased and reached the cavitation pressure 612 WY Propeller Exettatton and Response of 230000 TDW Tankers T, = the time when SH ay Se p = local pressure eo may © T, and x, are found by an iteration process x) +P x, = extent of cavitation For symmetrical hysteresis k is equal to unity. In the tip region the hysteresis will be non-symmetrical and the factor k is below unity. The radial distribution of k is found from propeller ca- vitation tests in homogeneous flow, see [26] . 2. Thickness The diameter of the tip vortex obtained when separation takes place at the tip may be determined by the expression = = nitmepiaar fc = Sine (2) where h = diameter of the tip vortex b = blade length a = = (8 d = pitch B = advance angle Co = blade section length at 0.95 R G = mean blade section length A = aspect ratio a = angle of incidence for separation at tip (relative tof). Based on experiments in [17 @= 8° inthis report The pressure in the centre of the vortex : 613 Johnsson and Séntvedt fe p (T_) i (3) e fe) 8 z ay ae Py = pressure at the radial section close to the boundary of the vortex ar r = ae .dr free vortex strength F Or Ds = bound circulation Two cases A and B must be considered : aD Pp. <2 oO cav The maximum cavitation thickness will be equal to the tip vortex diameter, The radial thickness distribution is found from ob- served shapes of thickness along the radius terminating at the radial inception point (calculated or observed). B. p> 'p > p The diameter of the cavitating tip vortex is found from the pressure distribution across the tip vortex radius assuming symmetri- cal hysteresis as outlined in point 1 above, APPENDIX C PRESSURE FLUCTUATIONS ON THE HULL, METHOD OF CALCULA- TION The velocity potential caused by a pulsating cavity may be found by solution of the Volterra equation if the cavity formation be ac- curately represented at any time during growth and collapse. The vapour/liquid mixture representing a pulsating volume cannot be said to constitute a surface of known shape. Consequently, an ideal mathe- matical model of moderate complexity should be employed together with empirical functions found by experiments. The net velocity potential g at any field point (x, ve z.-) caused by the small volume source Ar Ax Ar: 614 Propeller Excttatton and Response of 230000 TDW Tankers TOs Or ee OT a pS esha oe A —— —- A ray 4n¢g ae Ar Ax + ar Ax r + cor r 1) OT: le x ‘eats bg ( mit Ph tee: A el pd eae eo ae Ar Tes aon aie Ax AT where d = distance between volume source and field point At = thickness of volume Ax = length of volume along heiix Ar = length of volume along radius t = time U = velocity of advance of volume source eye aes -The first part of the right hand side of the equation is due to the motion of cavities and derivatives of +, x and r with respect to time are zero. The last part is due to volume variation of cavities. For unsteady, irrotational flow the fluctuating pressure p field at a field point : E = _99 = gt ue ~ Od Sp ic 2 ot where p = density of fluid s = solid boundary factor [10 | U = velocity at the field point, negligible at the hull surface Assuming Whe = U_ = 0 and combining eqs. (1) and (2) : 47 p 0 1,0 ptmadeaae mi ® ar T ea las ax Ux tara art a at rae v 2 (angry, a i 0x On plkwor lide x [at i) gp ae Be Jo oa Ee ae dey 2 97 0x 2 dror 2 ox Or + — >——-——Ar + ——— — doce dapior o> ae (3) 615 Johnsson and Séntvedt 2 2 2 Z d = (x, -%,)° + (yp-¥,)° + (2-2) (4) where 1 9 2 = coordinate of volume source 5.0 <6 0 ,l 2arn : ae @ moa [ ,-x,) sing - (y,-y,) cos¢ |= e (5) where n = RPS dé = angular position of volume source U_ ®& 22rrn/cos B x where 6B = geometric pitch angle We assume that the cavity formation at time t displaces a volume of liquid bound by the helicoidal blade surface and the plane P parallel to the shaft (Z-axis) (Figure 41). In accordance with Figure 42 the thickness distribution in the x-direction ee, Sy PS ee ee (6) Admittingly, the proposed distribution has been designed to suit mathematical interpretation of the extremely complex ''shape'! of the vapour/liquid formation. The chordwise extent of the volume at any time when perio- dicity may be assumed a valid assumption : co 2 4 nm mt = 8 a Dale 2 en sin pee 1 ms (7) m= oO = fo) where od = phase angle n = RPS 616 Propeller Exettatton and Response of 230000 TDW Tankers m = harmonic order iN II angular position for collapse of cavitation Ww i] angular position for inception of cavitation . Diese Dividing equation (3) by n D and making use of derived quantities in given eqs. x oe Ge a )ydac + ly (8) Oo = max. length of cavity along radius | I max, thickness of cavity along radius Z = number of propeller blades If we assume that f= 1 in equation (6) 1 = ! 7 = Reeve V/s at Fae Tot a (9) 1,/K = esr (10) Se Ogee 1,/K = CSET shes (11) = 1,/K =) (aeebe2) sc) = (12) uf Tr 2 1,/K oe ee (13) TN par eae (14) 6 d The expressions for some of the variables in the above equa- tions are finally listed : 617 Johnsson and Séntvedt ee 4 Je sg 00 Te ee Svineg 7 sina (15) RLS m= 1 we 4 2 ar baa ope! yy» m x_ sina (16) | ae © m=1 is where 2n(g - do) Q@ SS eG) b ‘ in ”~ = aspect ratio of the cavitating body k = reduced frequency S K = ——~——_ 2 4wn p* 618 Propeller Exettatton and Response of 230000 TDW Tankers Navigation deck Upper Deck and 2nd Poopdeck Wing tank No5 SB Pressure transducers Figure 1 General arrangement and positions of vibration pickups 619 Johnsson and Séntvedt Model tests O Full Scole;Thorshammer” 5 bl without fins Zs oon n « ~=with " [J « «= “Norse King” 6bl with fins Fro Fr 1/4 Figure 2 Positions of transducers for measuring pressure fluctua- tions on the model and full scale ship. For exact positions of transducers in full scale, see Table II. In model tests all transducers in centerline of model 620 Propeller Excttatton and Response of 230000 TDW Tankers —--o--— First ship; Thorshommer" 6 bi. without fins --0--- 7) o 7) o with Fy Ampl. ——— Second ship‘"Norse King"6bl. » ; mm slightly _uncomf. i) \ { \ Ballast | \ noticeoble H Meas. point Direct. of vibr. Di D2 D3 V eLMT Figure 3 Vibration levels in deck house. For positions of pickups, see Figure 1. Numbers indicate approximate number of revs per min. 621 Johnsson and Sgéntvedt ——©——First Ship; Thorshammer'Sbl. without fins ——s— Second Ship; Norse King’’6 bl. with fins Ampl mn BALLAST FULLY LOADED \/ 616 \ c \ {062 / \. 5907 4 60 G gi 733 75 0 Meas.point V2 V2 V4 V4 V5 V5 V2 V4 V5 V1 V3 V6 V6 Figure 4 Vibration levels in wing tank No 5, SB. For positions of pickups, see Figure 1. Numbers indicate approximate number of revs per min, 622 Propeller Exettatton and Response of 230000 TDW Tankers 5-bladed propeller with fins ——— without fins ase ea 30 01 Number of revs/min Figure 5 Pressure fluctuations in full scale. ''Thorshammer'"' (first ship) with and without fins. Transducers 3 and 5 resp (measuring point A, see Table II and Figure 2) 623 Johnsson and Séntvedt — 6-bladed propeller Kp —-—-— §-bladed propeller Number of revs/min Figure 6 Pressure fluctuations in full scale, First and second ship (5- and 6-bladed propeller) with fins. Transducers 4 and 2 resp (measuring point D, see Table II and Figure 2), 5 waluge based on diameter of 6-bladed propeller 624 Propeller Exettatton and Response of 230000 TDW Tankers 4 a =) rT) plitude 2p kp/cny Pressure am ° ~ Q3 70 90 Number of revs/min Figure 7 Pressure fluctuations in full scale. ''Norse King"! (with fins, 6-bladed propeller), Measuring pointD. Ballast condition 625 Johnsson and Sgntvedt T/T "“NORSE KING” BALLAST RPM = 85.4 VS = 17.8 knots RECORD FROM SHIP MEASUREMENTS, CELL 2 {D) ENERGY SPECTRUM FS FPS TEE | S(w) Amp = V2-5(wlmax: 4 6-10" 4 Aw =i1 3! fg! 2 i 5 E w 8.11 s7! Ag lord. 2ord. Figure 8 Pressure fluctuations in full scale. Sample of energy spectrum 626 Propeller Exettatton and Response of 230000 TDW Tankers NATURAL FREQUENCIES, AXIAL VIBRATIONS 4+ — T/T “THORSHAMMER,” CALCULATED THRUST VARIATION OF 10th AND 15th ORDER: + 22 MET. TONS -—-O== o—tT/t “NORSE KING,” CALCULATED THRUST VARIATION OF 6th ORDER: + 20 MET. TONS “SNORSE KING” (6 BL.) 12. ORDER, 73 RPM “THORSHAMMER” (5 dead 10. ORDER, 62.5 RPM ; NATURAL FREQUENCY (vib/min) PROP. RPM = 85 ORDER EXCITATION FREQUENCY 0.5 1 ; 2 3 4 THRUST BEARING STIFFNESS 1076 Kp (kp/cm) Figure 9 Axial mode of shaft vibration. Natural frequency of actual vibrations 627 Johnsson and Séntvedt 20 HUTT 12 s cin @ ‘wa JS ily Figure 10 SSPA Cavitation Tunnel No 2, Large test section in place 628 Propeller Exettatton and Response of 230000 TDW Tankers [—s—sd]sSCFACILITY [WATER SPEEO TOWING TANK ie M/S CANS TUNNEL [ee FULLY LOADED WITH FINS 100 Mie vn en : iN : © e a © 75 50 = xX NN ITHOUT PROP re) = 50 —_ S y 25 = reine 25 bass 5 0 A eee er 0 45 90 135 180 BALLAST WITHOUT FINS 100 "le We 100 75 n ~ ; 9 4 a 75 so © = > a 50 25 = > 5 0 Figure 11 Wake patterns in towing tank and cavitation tunnel 629 Johnsson and Séntvedt Kp 08 a (towing tonk) —o——. Covtunnel V=4 m/s ---O-- Towing tonk V=12-14 m/s —--&--— Colculotions , SSPA Bee “ , DNV 0.07 : Pants ameain 4 0.05 Ld J 004 me GTS Tee ee Fe section A-A A pressure wave transmitter / Figure 1 Afterbody model in tunnel test section iron tube rubber membrane coil winding \iron piece glued to rubber _membrane dS oe , 4 CLLLALLALL ALAM closed volume of air pressure transducer for Measuring volume oscillation PAMCOSHM NS VOI wos eee Figure 2 Pressure wave transmitter 668 Propeller Exettatton and Response of 230000 TDW Tankers REPLY TO DISCUSSION Carl-Anders Johnsson Statens Skeppsprovntngsanstalt Goteborg, Sweden Thank you, Mr Huse, for your comments. We read your con- tribution with interest and also with some relief, to get, without pay- ing for it, a good investigation of the wall effect of a tunnel of similar dimensions as that of SSPA. On trials we have several times observed that pressure fluc- tuations certain up to 5-6 times blade frequency. Preliminary, it was required to study the mechanism by which such components could be included. The m% term in Equations (15) and (16) of Appendix C however, leads to large magnitudes of the component of that frequency, as the corresponding inaccuracies in the Fourier term become significant. Consequently, empirical damping of the signal was required ; for con- venience the reduced frequency/oscillating blade relationship was chosen as the "building brick''. Presently, we terminate computations at twice blade frequency and can neglect the method originally includ- ed in the paper. 669 AP oy ie Me ed st Ps YA CUS Leo Nal ‘al) be aus Ar i% y Lie ey a La es Ae ae on 9 ee aie ann A SPSS a ie EST Rip natene Se Pian PRN Geb ES ey eae iene é ) PLOT el lege statin? WIT COOMSS/Ro Re ‘ ik Sieh lage Seed Cor df vi ree eee Tix: CG Ms hy * MS % J a we. if, sf as ' rae * , Voizeuozid OF VAR nosendot aiolmAatiad : 2! of wma gee R.. Fposens : ‘a i ae yh M Tet ieee + 2 Seta te ES. of ; \ A Sebiiatvabties . ae Nan mins ‘ Pi 1 5 ‘ : 4 ‘ } Josqn DW leidpracnion shoy 40) \eheH tM wog-ataed Tt ¥iGy ELA ‘fiw tad iy rt: 9 iy dfiw oata bone tos tos dsdux, £8 ud “weal breed s¥ensdTR to inpite Lh erty To Tretia aoe yi a body. & JF ape ASSES to 1sHT ee BAOE ~ ‘ ' ; * pane vr “panei bet a - ‘f + - — << ¢ © ih i ae j Ce , a +A cs 8 ¥ * 4, > > et , 2 I} Y 8192 ow . | oe wits m fay) at) siuod ai orvet 2a. OTE ble tne ) ai | aobsclingan sutal Ol ee tj Ts got. odd el [se1g03 eal yibdoses ae : rris va £5 LI aan Gite AMOL BI UGE Qo “DINO: Lian satwewr ies ant 1 5 bs : 17S8QG AQ ad? 4 * n ° : ¥ ‘ Fy rn he f o : ; , » ree gy ; aT Ae i P a MOTIONS OF MOORED SHIPS IN SIX DEGREES OF FREEDOM I -Min Yang Tetra Tech, Inc. Pasadena, Caltfornta U.S.A. ABSTRACT The equations of motions of a moored ship hav- ing six degrees of freedom were formulated. The mooring force is nonlinear and asymmetrical. A new approach is developed to solve the resulting nonlinear and asymmetrical problem. INTRODUCTION Recently the development of large ships has attracted many investigators to study motions of moored ships. The introduction of container ships and the increase of oil exploration in deeper water depths make it necessary to have a through understanding of motions of moored structures. For container ships, the operation of loading and unloading containers are controlled by land based huge cranes, extensive ship motion may greatly reduce container loading and un- loading rate. Oil exploration in deep sea needs to drill through the ocean floor from a moored ship (or other moored structures), large motions of the ship may hinder drilling operation. Another related problem is that of a moored buoy system. The effective design and development of such system also require an ability to predict its os- cillatory motions. The study of motions of a moored ship is usually limited to the 671 I-Min Yang surge motion flr , that is, the moored ship is considered as having only one degree of freedom, and it is known that the force-elongation relationship of a mooring line is highly nonlinear [2] . Kaplan and Putz [3] , and Muga [4] have investigated moored structures in six degrees of freedom. In their study, the force-elongation relationship of mooring lines is assumed to be linear, thus the problem is linear and the solution can be readily obtained. In this paper, we consider a more general six degree of freedom problem. The mooring force is a nonlinear function of elongation ; since a ship can not be symmetric- ally moored, and fenders are only at one side of the ship, motions of the ship are asymmetric. An approach has been developed to generate an approximate steady-state solution to this nonlinear asymmetric problem. FORMULATION The motion of a moored ship in waves is an oscillating system with six degrees of freedom corresponding to surge, heave, sway, roll, pitch, and yaw. The ship is considered as a rigid body and its deformation is neglected. Usually the length of a ship is much longer than its beam, and the slender body theory can be applied to find the hydrodynamic properties of a moored ship. According to this theory, for an elongated body where lateral dimensions are small compared to its length, the flow field at any cross-section is independent of that at any other sections. Hence, the flow field of an elongated body, like a ship, is reduced to a two dimensional problem of its cross-sections. The total hydrodynamical properties is found by integrating over the length of the body. The six dynamic variables surge, sway, heave, roll, pitch, and yaw of a ship are expressed in terms of two right-hand cartesian coordinate systems. A moving systems which is fixed in the ship, and a fixed system which is fixed in space. The moving system (y ,y_, 2 has its origin at the center of gravity of the ship and its three axes (y, ibs OY, ) coincident with the three principal axes of the ship. The y -axis is positive toward the bow, the y_,-axis is positive to port and the Y3 -axis is positive upward. This system will move with the ship and the angular displacements about the three axes are respecti- vely, the roll, the pitch and the yaw of the ship. They are positive for rotations about the positive directions of y , Y> and V5 in a counter- clockwise direction. The fixed systems(x,, x, x,) is chosen suchthat the two coordinate systems are coincident when the shipisatrest. Thenthe * Numbers in brackets designate References at the end of the paper. 672 Mottons of Moored Shtps tn Stx Degrees of Freedom three components of translational motion of the origin of the moving system relative to the fixed system are defined as the surge, the sway, and the heave of the ship. The positive directions of forces and moments are defined in the same way as their corresponding displa- cements, The forces and moments on a moored ship can be divided into four categories : inertia, damping, restoring and exciting forces and moments. The details of determining these forces and moments are discussed in [5] , and only a brief discussion will be given be- low. The inertia forces and moments arise from change of velo- cities of the ship and water particles around it. Although the change of velocity of water particles depends on their position in relation to the ship, it can be assumed that a certain amount of water behaves as if integral with the ship and moves with it. The amount of entrain- ed water is different for different components of motion and the mass or the moment of inertia of such entrained water is called the added mass or added moment of inertia. By applying the slender body theo- ry, the inertia force or moment on a section of the ship is then equal to the product of the virtual mass (the sum of natural mass and added mass) or the virtual moment of inertia (the sum of natural moment of inertia and added moment of inertia) and acceleration. The added mass and added moment of inertia depend on wave frequency, water depth, shape of ship sections and the clearance between ship sections and side walls. Damping forces and moments arise from wave generation and are proportional to the relative velocity between the ship and wa- ter particles. The proportional constants are called the damping coef ficients and depend on wave frequency, water depth, shape of ship sections and the clearance between ship sections and side walls. Restoring forces and moments come from three different origins and will be discussed separately in the following. Hydrostatic restoring forces and moments are due to the buoyancy effect resulting from ship displacement. The total hydro- static restoring force has only one component in the vertical direc- tion and the hydrostatic restoring moments has components in the roll and pitch direction. The second restoring forces and moments come from moo- ring lines. The behavior of a mooring line under tension has been investigated by Wilson [2] . The relationship between force and elongation is highly nonlinear and in general can be represented in 673 I-Mitn Yang the following form : = 0 220 T ATs C0 where L is the moored length of a mooring line, the change in length due toa tensile force T is AL, and c and n are two constants depending on the type of mooring lines. The total restoring force and moment due to all mooring lines have components in all three directions of translation and rotation, Another restoring forces and moments come from fenders which will be in action only when they are in contact with the ship. The total restoring force has only one component in the sway direction and the total restoring moment has components in the roll and yaw direc- tions. Finally there are the exciting forces and moments due to water waves. The waves are assumed to be sinusoidal standing or progressive waves and have a unique frequency, then forces and moments on ship sections can be obtained from certain wave potential. The six equations of motion for a moored ship are obtained by balancing the various forces and moments discussed above and may be represented in the following matrix form : Mx + Cx + K x + #(x) = g(t) (1) where M = virtual mass and moment of inertia matrix C = damping matrix Ko = stiffness matrix due to linear restoring forces and moments f(x) = force vector due to nonlinear restoring forces and moments g(t) = force vector related to water waves = q.cos (Wt +W,), i=l, ..., 6 % 5 (ys 51 % 32 Oy» Op» Og) 674 Mottons of Moored Ships in Stx Degrees of Freedom The first three elements x, , x. , x. in the displacement vector X represent the surge, the sway and the heave of the moored ship, and the next three elements @, 0, and 0, represent respectively the roll, the pitch and the yaw of the ship. The force vector f(x) is nonlinear. If its argument x is replaced by -x, f(-x) will in general differ from f(x) in magnitude as well as in sign. Hence f(x) is asymmetric. The quantitiesW,q. andy. represent wave frequency, force or moment amplitude and phase angle for the ith element of the vector g(t). METHOD OF SOLUTION Since the excitation vector is harmonic, we assumed that an approximate steady-state solution for the response of the system (1) may take the following form : =) 2. iy COs (Wt ida) ie eo (3) z. is a constant introduced to account for the asymmetry of f(x) in (1). If f(x) is symmetric, then Z will vanish. Y; is a harmonic function whose amplitude and phase are Ys and Pa: Consider a linear system defined by My + Cy + (K)+ K)y = g(t) (4) where K is an unknown matrix. If K is known, this linear system can readily be solved to give : (5) where w,'s are given by 675 I-Min Yang -I wi vy - Vv, q) cos Wi 26 cos V6 = q, sin y1 (7) Wi ve - Vv, qe sin "V6 in which V =i! Garde 5% - Muw~ (8) 1 fe) Vv, = CAG (9) and the superscript -1 for a matrix denotes its inverse. If the exact solution for the linear system is used as an approximate solution for the nonlinear system, direct substitution gives E (z,, k,) = £ (x) - Ky +K 2 (10) where kj; are the (i, j) element of K and € denotes the error vector. The unknowns kij are chosen in such a way that the average mean- square error over one cycle defined by the integral 6 271 EN ae anf dy Ste 6= wt (12) a 27 0 i i! Bee I is a minimum. This leads to JEa wh aKay. eee Sie eee (12) 676 Mottons of Moored Ships in Six Degrees of Freedom However, it can be shown that not all kij's are independent. In order to avoid this difficulty, we choose in this case, eae (0 Si ie (13) then, k.. can be uniquely determined as follows : il 2M 1 lee a Ny, of f. cos (0 ate) d@ (14) Six more equations are furnished to determine z; by averaging the nonlinear equations over one cycle which leads to Ona 27K z +f f(x)d@ = 0 dig eta Nae) Guay 6 (15) 0 Thus the solution of the nonlinear system is reduced to the solution of Equations (5) - (7), (14) and (15). They are nonlinear algebraic equations but can be solved numerically by the following iteration approach, First, set zj = 0 and assume a set of values k;j;. Then Equation (7) can be solved by simple matrix inversionand yj, and 9; are determined from Equations (5) and (6). Now a new set of values of kjj and z; are calculated from (14) and (15). This proce- dure can be repeated until required accuracy is reached. SUMMARY AND DISCUSSION An approach to the determination of an approximate solut- ion for the steady-state response of moored ships in six degrees of freedom has been formulated. This approach can be applied to moor- ed structures in open sea as well as moored ships in harbors. In this paper, the degrees of freedom of the system is specified as six. However, this approach is still valid for degrees of freedom other _than six. The accuracy of an approximate analysis is difficult to pre- dict in general. A different version of this approach where f(x) is symmetric has been employed to problems which possess known exact solution [6] , it shows that the accuracy of this approach is well within the limits of practical engineering usefulness. 677 [2] [3] [4] [5] [6] I-Min Yang REFERENCES WILSON, B.W., ''The Energy Problem in the Mooring of Ships Exposed to Waves'', Proceedings of Princeton Confer- ence on Berthing and Cargo Handling in Exposed Locations, October, 1958. WILSON, B.W., "Elastic Characteristics of Moorings", Journal of the Waterways and Harbors Division, ASCE, WW4, November, 1967. KAPLAN, P., and PUTZ, R.R., ''The Motions of a Moored Construction-Type Barge in Irregular Waves and Their Influence on Construction Operation", Contract NBy-32206, an investigation conducted by Marine Advisers, Inc., La Jolla, California for U.S. Naval Civil Engineering Laboratory, Port Hueneme, California, 1962. MUGA, B.J., ''Hydrodynamic Analysis of a Spread-Moored Platform in the Open Sea", U.S. Naval Civil Engineering Laboratory, Port Hueneme, California, August, 1966. HWANG, L.S., YANG, I., DIVOKY, D. and YUEN, A., "A Study of Wave and Ship Behavior at Long Beach Harbor with Application to a Modern Container Ship", an investiga- tion conducted by Tetra Tech. Inc., Pasadena, California for the Port of Long Beach, Long Beach, California, 1972. YANG, I., "Stationary Random Response of Multidegree-of- freedom Systems", California Institute of Technology, Dynamic Laboratory, Report No. DYNL-100, June, 1970. et + & 678 Mottons of Moored Ships in Stx Degrees of Freedom DISCUSSION Manley Saint-Denis Untverstty of Hawat Honolulu, Hawat, U.S.A. I am afraid I must begin by begging the author's forgiveness and the indulgence of the audience for the critical remarks I am about to make on this paper. Perhaps this is the wrong way to start a dis- cussion, but if it is not the right way, it is at least a diplomatic one. However I must confess with some alacrity that my remarks are go- ing to be rather suggestive and tentative and not at all forceful or ca- tegorical. This is due in part, perhaps, to the manifest voids in the paper which have led me to infer, perhaps mistakenly, what might be the full development. Having provided sufficient cushioning for my cri- ticism, it is time that I voice it. I have five specific comments and one general recommendation. The first is this : I have been unable to discover anything new in the paper. Lack of originality is not in itself condemnable, of course, if the paper contains other rewards, such as elegance of development or an efficient computer programme, etc... But these I find not to be present. My second comment is that the paper appears to consist of two parts : an adequate introduction and a short conclusion ; but of the essential development that should be the core of the presentation there is only a hint. It is this parsimony of the essential that I have found to be rather distressing. To rest the paper on reports that, if not pro- prietary, are not generally available militates against an appreciation Gilit: My third comment relates to the author's statement that the solution is valid for a ship moored alongside a dock, a condition that introduces an asymmetric non-linearity in the restoration but this is, in principle at least, a grave insufficiency, for the proximity of the rigid boundary, which is the dock, affects also the hydrodynamic mass, the damping reaction and the acceleration and velocity terms of the excitation, so that all these are non-linear, and the non-linearities are not readily written off as negligible. Indeed, they are quite power- ful. 679 I-Min Yang My fourth comment is that the equation of motion reveals an excitation that consists of a single component related to wave displa- cement. This is correct only so long as the excitation is linear (in which case its amplitude is frequency dependent), but not when the excitation is non-linear (in which case its amplitude is a function of both frequency and amplitude of motion). My fifth comment relates to the conclusion that the accuracy of the approximate approach is well within the limits of practical en- gineering usefulness. But comparison is against another computa- tional method and not against measured reality, and such a compa- rison leads to an appreciation of the validity of the approximation, not of the basic method. The paper raises a large number of stimulating questions but it provides a paucity of answers to them, and my recommendation is quite simple. I suggest that the author complete his paper and add thereto whatever experimental or trial data he can adduce in support of the technique described. Only then will it be possible to appreciate the paper and to comment constructively on its intrinsic merits, which at present stand unrevealed. REPLY TO DISCUSSION I-Min Yang Tetra Tech, Inc. Pasadena, Caltfornta, U.S.A. From the comments made by the discusser, it seems to me that he has some misunderstanding about the definition of nonlinear differential equations. Hence he can not find any originality in this paper. In equation (1), the independent variable is the time t , and the dependent variables are the components of the displacement vec- tor x. Since it is assumed that £(x) is a nonlinear vector function of x, the differential equation (1) is therefore nonlinear. The fre- quency w in this equation is just a parameter, that is, for a particu- lar case, it is a constant. The hydrodynamic coefficients (virtual masses and damping coefficients), and amplitudes and phase angles of wave forces depend on w and the dock in a very complicated man- ner, but for a fixed w , they are just constants. Thus they have no- thing to do with the nonlinearity of differential equations. (For 680 Mottons of Moored Ships tn Stx Degrees of Freedom example, in equation (1), if f(x) is a linear function of x, then equa- tion (1) is always linear. ) Many papers have been published in deter - mining hydrodynamic coefficients and wave forces and since the main purpose of this paper is to present an approach to solve the nonlinear, asymmetrical problem due to the presence of mooring lines, I did not even try to explain how to find these coefficients in this paper. The explanation of the approach is quite complete. If one reads this paper carefully, he will find that, although it is a quite short paper, it contains all information about the approach. It is true as the discusser pointed out I failed to compare the results with measured data. However, the comparison with an exact solution does indicate how good the approach is. If the hydrodynamic coefficients and the representation of mooring lines are adequate, then this approach will give very reasonable and practical results. DISCUSSION Paul Kaplan Oceantes Inc. Wew-York, U.S.A. My comments are somewhat similar to those of Dr Saint-Denis but perhaps they are motivated by different reasons. In some ways, when I looked at the paper and considered the content of it I was re- minded of a modern song that was popular a couple of years ago by Miss Peggy Lee. It is called "Is that all there is''. If you think about the song, it starts out on a sad tone and that is the way I felt initially, but the song proceeds by careful consideration to some sort of posi- tive outlook. Perhaps if you bear with me and my discussion and some interchange between the author and myself, the same result may be achieved. My interest in this particular subject is indicated by the fact that there is a reference to the paper I wrote ten years ago dealing with the motions of a moored ship with six degrees of freedom, and it was linear. Two years ago at thelast Naval Hydrodynamics Symposium I presented a similar subject which was just concerned with a few pro- blems in the whole general area of mooring and positioning ships at 681 I-Min Yang sea. Therefore I am concerned about some aspects of that is in this paper and how they modify what may be considered as some results of my own some time ago, and therefore raise questions. For one thing, this being a naval hydrodynamic symposium, there is no indication as to what is unique in the paper from a hydrodynamic point of view. There are walls and there is a mention by the author of the fact that the added mass and damping are affected by the presence of the walls. However, how did you take account of it ? What mathematical proce- dures, what techniques, what approaches were used ? This is the sort of information that we professionals could make some use of and would like to hear about so that if we have an apposing point of view or some- thing interesting to say there can be some interchange in regard to it. Another point is that this was applied, we are told, to a con- tainer ship. Was there anything unique about the container ship ? Was it a standard ship ? Was it one that also had barges perhaps ? Was it an unusual form, like a Lash or a Seabee ? This makes a difference in the type of ship computation per se. On the question of non-linearity, what kind of non-linearities do we have ? I can imagine something with fenders. A fender will only provide a force, and it is one-sided, when you press against it ; when you go off, there is nothing. I would make an analogy with the so-called linear detector in electronics, which is one-sided in its va- riation. If you do that, you have a mathematical structure of whata non linearity looks like. Similarly with regard to other aspects of the non-linearity that comes from the mooring cable. For example, you made a particular assumption in your mathematical procedure which raises a very significant question in my mind. Equation 13 in the paper, which you alluded to also on your slide, said that you made the assumption that the Kij terms are zero. This means you are only using diagonal terms in the K matrix. The implication of that is very interesting. It means that you just take account of what I consider to be self-mode non-linearities - that is, non-linearities in each mode separately, like x”, x, UG y? . Moored ships certainly have inter - actions of a non-linear nature because the bow and the stern when you yaw together with sway have some xx y+x sine and y-x sine at either end. Non-linearity reflects itself differently in different pro- ducts. In fact, if you have similar mooring cables you can use the fact that the cross product term is opposite sines and this will allow you to get some information, something useful, and not as restricted as itis here. It is a suggestion. I do not know what you have perhaps done in considering that which led you to the-conclusion of a reduced number of terms. 682 Mottons of Moored Shtps tn Stx Degrees of Freedom Similarly, you are dealing with a case of a sine wave here which I do not think is quite apropos, but nevertheless it is an analy- tical exercise that has its utility. How different are the results for the case you have obtained here with the non-linearity included as compared to the linear theory ? There must have been some results. Is there any significant difference ? Does it occur when in a certain region we have certain wave amplitudes ? This will certainly be use- ful to us. Similarly, if you have an application to a real condition, even though it is a harbour and has a very narrow tune system, there are swells that come in and you have to treat the problem as one for a random system. There are techniques known as describing functions which you can use, and I am sure you can use them. Your reference says your thesis dealt with this problem area. It would be a natural and wonderful extension of this procedure. The question then is what are the theoretical results and how would they compare with the case Of linearity ? In concluding, I want to say something that is similar to the statement Dr Saint-Denis made. There is a statement which I used to hear years ago and perhaps it is applicable to my own delivery some- times, but it is of oriental origin. It is essentially that one picture is worth a thousand words. I think we ought to make a little inversion here and put some more words together with some pictures, and the end product comes cut really two years from now. This is a tremen- dous opportunity to make a substantial contribution, and I should greatly like to see it. REPLY TO DISCUSSION I-Min Yang Tetra Tech, Inc. Pasadena, Caltfornta, U.S.A. (1) The discusser said, ''...you just take account of what I consi- der to be self mode nonlinearities,...''. This is not true. It is true that the matrix K is assumed to be diagonal, but if you read this paper carefully, you will find that Kea, are unknown quantities and their determination depends on all six modes of motion. 683 I-Min Yang (2) Since the main purpose of this paper is to present an approach to find the nonlinear, asymmetrical motion of a moored ship, I have omitted the discussion of determining the hydrodynamic coefficients. In fact, I just used the data discussed in published papers. (3) The effect of nonlinearity depends heavily on the nonlinear sys- tem. If the nonlinearity in the force-displacement relationship is very large, like the case of mooring lines, the resonant period determined by linear analysis is quite different from that determined by nonlinear analysis. For example, in one case I considered, it may shift from 20 seconds to 50 seconds. (4) This paper discussed an analytical approach to investigate the motion of a moored ship. It seems to me that the lack of figures does not affect the understanding of this approach. DISCUSSION Ernest O. Tuck Untverstty of Adelatde Adelatde, Australta I do not want to let one of Dr Saint-Denis's remarks pass. I forget which point number it was but it was the business about the ex- citing force being of only one type when he thought it ought perhaps to be of several types. I think this is quite erroneous, There is no rea- son to believe that the excitation is anything but the most general li- near excitation in this particular problem. From my reading of the paper there is an input frequency-dependent phase as well as input amplitude, so I think that that particular point put by Dr Saint-Denis should not be allowed to stand. Another point of his was that there is nothing new in this paper and I think that is quite wrong. I think myself that it is quite a good paper. I think Dr Yang explained about the matrix Kij . The fact that it is diagonal does not necessarily mean that one is considering only self-linearities. I think that some of this criticism is simply misplac- ed and that the people making the criticism should read the paper a bit more carefully. 684 Mottons of Moored Shtps in Stx Degrees of Freedom DISCUSSION Grant Lewison Nattonal Phystcal Laboratory Feltham, Middlesex, U.K. I should like to make one small point about the paper. The author has assumed that the response of the system is entirely funda- mental mode -in other words, that there is only one harmonic present. This completely ignores the possibility of higher and lower mode har- monics which for many moored ships are well known to provide much bigger responses than the fundamental. REPLY TO DISCUSSION I-Min Yang Tetra Tech, Inc. Pasadena, Caltfornta, U.S.A. Thank you for the comments. Usually for a non-linear system there are some higher harmonical terms. But I compared the harmo- nic solution with the exact solution for some symmetric cases. The higher harmonical terms are not important. DISCUSSION Grant Lewison Nattonal Phystecal Laboratory Feltham, Middlesex, U.K. What about the lower one - i.e. sub harmonic ? 685 I-Min Yang REPLY TO DISCUSSION I-Min Yang Tetra Tech, Ine. Pasadena, Caltfornta, U.S.A. If you want to have high harmonic or sub harmonic terms, of course you can do it by purely numerical integration. But note that near resonance, the solution is not unique and depends on initial con- ditions. The region of convergence for the peak solution is very nar- row and therefore it is very difficult to obtain the most important re- sults by numerical method. That is the main reason why I developed the approximate method for the harmonic case. 686 ANALYSIS OF SHIP-SIDE WAVE PROFILES, WITH SPECIAL REFERENCE TO HULL’S SHELTERING EFFECT Kazuhiro Mori, Takao Inui, and Hisashi Kajitani Untverstty of Tokyo Tokyo, Japan ABSTRACT Attempt is made to find out the effective wave- making source ofa ship from the measurement of the hull-side wave profiles. The integral equation of the source distribution function is simplifiedand solved numerically under the specific limitation, (a) rectangular, vertical central plane, and (b) draughtwise uniform. Two Inuid models M 20 (B/L = 0.0746) and M 21 (B/L = 0. 1184), whose hull-generating sourcesare optimized to give the minimum wave resistance at Fn = 0.2887(K,L =12), are tank-tested and wave- analyzed. The obtained source distribution m(£) shows a clear discrepancy from the hull-generating source m( & )in a similar way to the so-called u-correc- ti0n, OF a(&)= m(£ )/m(£)= 1-H (1- [é]), (#= 0.4) Wave profiles, wave patterns and wave - making resistance are calculated in two ways, (a) from hull-generating source m(& ),and (b) from wave- analyzed source m( &), The gap between experiment and calculation (a) is satisfactorily filled up by calculation (b). From experimental results it is proved that the 687 Mort, Inut and Kajttant design procedure where a (£)is taken into ac- count is very significant. The second order calculations for hull - surface conditionas wellas for free-surface condition are found not enough to give the theoretical basis for the correction function a( &), which suggests the importance of the hull's sheltering effect. I, INTRODUCTION The wave analysis has two objectives, i.e. (a) to deter- mine wave-pattern resistance directly, and (b) to find out the actual wave-making mechanism of a ship-like floating body. This paper deals with the problem (b) by means of the measurement of ship-side wave profiles rather than by free wave pat- terns in the rear of ships. II METHOD OF ANALYSIS The co-ordinate system as shown in Figure 1 is adopted throughout the papers. All quantities in the following equations are dimensionless, where & (= 4/2), half length of the ship, and U, the velocity of the uniform flow are taken as the units of length and speed, respectively. Let us assume that the hydrodynamic singularity (source) is distributed on the surface i = IE Sa (1) Then the perturbation velocity potential at an arbitrary con- trol point P(x, y, z) is given by b (x, ys 2) = = ff mE, £)Gos ys 258.7, 5) 6 s | (2) 688 Analysts of Shtp-Stde Wave Profiles where U.m(&,5) denotes the source density at the point Q(&,7, 5) on the distribution surface S. The Green function G(x, y, z;&,1,{)is PIG Gs hl ae eines (3) where sag " ee) A GsPaNe n (ee 2 2 9 ea lec Spats (iy -n)° + (n+t)° (4) T “5 le ea [oof exp [k(z+5) + ee -Ecos 6+ y-nsind)] dic k - K Lsec 6 - iusecé By making use of the well-known free surface condition 1 (x,y) = ae x Oo the integral equation (2) can be converted to Soy) = a ff bs ceonatee| Se where {(x,y) denotes the surface elevation in general, For simplicity, let us confine ourselves to the specific limi- tations, (a) the distribution surface is the rectangular, vertical central plane (-1S&<1, -t<‘<0) (b) the distribution function is draughtwise uniform. Further, the ship-side wave profiles Sales y) are selected as the given information of the wave elevation Sal Gaines Thus we have the fundamental integral equation v0 y) Seams =a. arf ime f) 2 Gla ashe, ¢) ae oe (7) For numerical solution of Equation (7), the modified Fourier expans- ions are introduced as follows 689 Mort, Inut and Kajttant N N mae) = » a cos nré ‘+ ba bo sin n7&’ (one 5 (8) n= 1 n= 1 where 1 es Sebi aes ee (9) Using above expression for m(&), we divide the distribution plane into M number small meshes, and assume that within the mesh- es the source strength is constant, then the wave profiles are given by N o (x, y) = Df acab y) +b S (x, | (10) : M with C(x, a ME yas cos nrk’. G. (x,y 5&,, to) i M Sy) = au sin nif. G. (x,y 3, to) (11) where it] 9° G, (x, £., to) = ae af" 2 G(x, y,z3§,0,$) 2 = 08 (12) and erie Si y 2 (5; + S44) By preliminary studies of Equations (8) and (10) , five term truncation N= 5 are found suitable. Then 5x2= 10 numerical coefficients ja} , {b,} (n=1,2,... , 5) are determined by the least square method. Table 1 shows an example of such preliminary studies. Start- ing with the wave profiles which are calculated from the hull-generat- ing source m(£) of the model M21 at the speed of Fn = 0. 2887 (KL = 12), the wave analyzed sources m(£) are obtained for the cases N=4, 5, 6and8. Figures 2~4 also show the general features of the contribu- tion function G;(x,y;§;,to) as expressed in Equation (12). 690 Analysts of Shtp-Stde Wave Proftles III MODELS AND WAVE PROFILE MEASUREMENT Among a variety of hull form characteristics, the beam- length ratio (B/L) is supposed as the leading parameter for the shel- tering effect. Therefore a set of two Inuid models M 20(B/L = 0. 0746) and M 21(B/L = 0.1184) are prepared as shown in Table 2 and Fi- gures 5~6, The hull generating sources m(é) of M20 and M21 are optimized to give the minimum wave resistance at the speed of KL = 12(Fn = 0.2887) under the following restraints: 0. 018(M 20) fo) 1 War Riee | df i Ex ooma(hs) dt. = (13) -ty fe) 0. 036(M 21) and Three kinds of tank experiments, i.e. (a) towing test, (b) wave-profile measurement, and (c) wave pattern measurement, are carried out with M20 and M21 for the speed of K,L =7~20 (Fn = 0.3780 ~ 0. 2236). As the typical examples, the results of the wave-profile measurement at the speed of K,L = 8, 12 and 16 are reproduced here in Figures 7~9, where the two kinds of calculation, (a) from the hull-generating source m(£), and (b) from the wave-analyzed source m(é), are also presented by dotted lines and by plots, res- pectively. IV. ANALYSIS OF MEASURED WAVE PROFILES The proposed method of analysis is applied to the ’measured 691 Mort, Inut and Kajttant wave profiles of the two tested models M20 and M21 for the twelve speeds K,L=7 through K,L = 20. In this procedure, the measured wave profiles at twenty po- sitions x= -0.95, - 0.85, ..., 0.95 are adopted as the principal input data. In addition, some selected readings of the wave recorder on the longitudinal cut line y = 0.25 (x = 1.0~2.0) are adopted as the supplemental input data, which are useful for the definite deter- mination of the source around the stern. Figures 10~12 show the wave-analyzed sources m/(& ) of M20 and M21 for the three selected speeds K,L = 8, 12 and 16, where the hull generating source m(£é) is also given for comparison, Because of a very low level of wave elevation, the accuracy of wave analysis is rather poor with the thin model M 20, particular- ly at the lower Froude number Fn < 0,2887 (K,L > 12). In Figures 13 and 14, the similar results at K,L = 10, 11 and 12 are summarized with M20 and M 21, respectively. From Figures 10 through 14, a clear discrepancy, which is roughly proportional to the beam-length ratio of the models, is ob- served between the wave-analyzed source m/(£&) and the hull-gene- rating source m(€ ). For the convenience of further studies including the effect of Froude number, the ratio of the two kinds of sources, or the cor- rection function a (&)=m/(é)/m(£&) is calculated with the wide model M 2l. The results are reproduced in Figures 15~17., Here it must be remembered that the relative accuracy of a(&) is poor around midship, because of m(£é) being null for €= 0, With respect to the sheltering effect, the present authors (ia : [2] s [3] suggested a simple, empirical correction, like at Eh Serer: |e) (14) 692 Analysts of Shtp-Stde Wave Profiles with .4 for © By iy = One ~ 0205 1S i} i=) In Figures 15 ~17, Equation (14) which we call yw- correc- tion, is also given for comparison with the wave-analyzed result of M 21(B/L = 0.1184). It is noticeable that the general tendencies of the wave- analyzed correction function are of quite similar tendency to the simple, empirical relation (14), except the higher Froude number Fn> 0.30 (Keo bolt): At the. higher speed range, the inclination of a ( £) is gett- ing steeper with increasing Froude number. Therefore Equation (14) is modified here to a more general expression, like CuK igi= Ste tab |e | (15) The results of the generalized straight line approximation, which is applied to the wave analyzed correction function of M 21 , are given in Figure 18 together with the original proposal (14). V. COMPARISON WITH TANK EXPERIMENT Before entering the discussions on the theoretical basis for the obtained correction function a@ (£), its justification and useful- ness are examined by comparison with tank experiments on four items, i.e. (i) wave profiles, (ii) wave patterns, (iii) wave-making resist - ance, and (iv) amplitude functions, (i) Wave profiles By direct integration, the model's side-wave profile is cal- culated from the wave-analyzed source distribution m(£), which is reproduced by the plots in Figures 7~9, Its satisfactory agreement with the measured wave-profile (full-lines) makes a striking contrast with the rather poor result of the existing theory (dotted lines). 693 Mort, Inut and Kajttant A slight phase shift, however, is observed with the first wave crest of the wide model M 21, which appears to be attributable to the non-linear flow effects in close vicinity of the stem. To find out the accuracy of approximation, similar calcu- lation is also carried out with M21 by adopting w- correction (# = 0.4), whose result is also presented in Figures 7~9, (ii) Wave Patterns The measured wave contours of the tested two Inuid models M 20 and M21 are obtained at a single speed Fn = 0.2887 (KoL = 12) by cross fairing of the longitudinal cut wave recordings of every 5 cm intervals from y = 0.25 (close to the model's side) through y = 1.75 (tank side wall). The final results are reproduced in Figures 19 (M20) and 20(M21) respectively. The corresponding calculations are carried out in three different ways. (a) by existing theory, or from the hull-generating source m(é), (b) from the wave-analyzed source m(é), (c) by “- correction (# = 0.4) These calculated wave patterns are presented in Figures 21~ 24, The existing theory (a) (Figure 21) shows the poorest agreement with experiment. Particularly, the transverse waves are tremendously exaggerated in this calculation in accordance to the author's previous suggestions [1] ‘ The clear disagreement between existing theory (a) and experiment is again improved successfully by the present approach (b). In fact, not only the general features but also the details of the measured wave patterns of the tested models M 20 (Figure 19) and M21 (Figure 20) are beautifully reproduced in Figure 22 (M 20) and Figure 23 (M21), respectively. 694 Analysts of Shtp-Stde Wave Profiles With respect to the effectiveness of the simple correction (c), Figure 24 shows its relative merits and demerits in compari- son with (a) and (b). (iii) Wave-Making Resistance Ordinary towing tests are carried out with M 20 and M 21 by fitting the plate-stud stimulator at 0.05 L behind the stem. The wave-making resistance coefficients CGyee=- Rw =e oe (16) which are obtained by adopting Schoenherr-line with form factor K = 0.07 (M20) and K=0.15 (M21), are presented in Figures 25 and 26. In these Figures, the three kinds of calculations are also given for comparison with the experiment, (a) Cw calculated by existing theory, or from the hull- generating source m(£é), (b) Cw calculated from the wave-analyzed source m/(é), (c) Cw calculated by » — correction (» = 0.4), Calculation (b) again shows the best and the most satis- factory agreement with experiment in contrast to calculation (a) or (c), (iv) Amplitude Function Comparison is also made on the amplitude function of the total free waves for the speed of Fn = 0.2887 (K,L = 12). Figures 27 (M20) and 28 (M21) show four kinds of amplitude function, 5 Be ; (a) Amplitude function calculated by existing theory, (b) Amplitude function calculated from wave-analyzed sour- ce distribution, 695 Mort, Inut and Kajttant (c) Amplitude function obtained from the measured free waves by longitudinal cut method, (d) Amplitude function obtained from the measured free waves by transverse cut method. As easily observed, the amplitude function (b) shows the highest average level of free wave amplitude, particularly in the trans- verse wave range. It appears that the difference between (b) and (c) or (d) may be partially explained by the wave-breaking resistance. In Figures 25 and 26, Cw atthe specific Froude number Fn = 0.2887, which is obtained from the longitudinal cut method (c), is also presented. VI. SEARCH FOR THEORETICAL BASIS OF CORRECTION FUNCTION eee) The practical usefulness as well as the experimental justi- fication of the obtained correction function a (&) are clearly demons- trated in the preceding sections, From a theoretical point of view, however, its hydrodyna- mical mechanism still remains open for further investigations. From the standpoint of the boundary condition which is used to obtain velocity potential in.Equation 2 , two possible causes for the discrepancies between theoretical and wave-analyzed results can be mentioned. Namely, (a) finite Froude number effect for the hull- surface condition, (b) non-linear effect for the free-surface condi- tion, For the time being, preliminary calculations of these two higher order terms are carried out with respect to the amplitude func- tions, VI. 1. Hull surface condition Throughout the present paper, the hull-generating source m(&) is derived from the so-called double-model approximation m which is correct only for limiting case Fn-~0 or K,)L—> @ 696 Analysts of Shtp-Stde Wave Proftles To find out the effect of finite Froude number, the Green function G(x,y,z;&,%7, $) expressed in Equation (3) is rewritten as Gx,y,256,%,6) = G) +G,+G, (17) where 1 Senn eagtepeinaivar niga (18) : vel V2 ie. 2 Ko (45sec 8 2 G, = = 4K L aoe” de sin(K f psec 6)d@ avi 19 ; +(@) (19) TT if z+ @ ia) et 2 eee pear ie ° da] sec dcos(k Ko £2 + £)-ksin(k Kf z +f) 3 us oO ee a Ps Se ee ee) eee eee ee Pee ee 2 4- oe k + ) -5+@® sec 20 = KKoflp dk ie) (x -&) cos 06 +(y-7) sin 6 tan@= (y-7)/(x-&) (21) In Equations (17) through (21), the second term Gp, denotes the free wave component which propagates oscillatorily to the rear of a ship, while (G, + G3) represents the local disturbance. 697 Mort, Inut and Kajttant Differentiating Equation (17) asto x, y, z and integrating all over the distribution plane, the velocity components u, v, w are also written as u = i si a7 + u, A cay eR MRE 2 (22) Wy Samy + w, + Ww, In finite Froude number problem, the free wave terms Uys Vo > and Wo and a part of local disturbance uz, Vz , and w3 play important role which is shown in Figure 29 in the case of M21 and K,L = 12. Based upon the ''exact'' hull-surface condition, ny 0 f(x, z) ae O f(x, z) URE Of(x, z) (23) Ox Oz A Ox where y = f(x,z) denotes the half-breadth of the hull, the "exact" hull-generating sources for finite Froude number K,L = 12 are ob- tained with M 21. This result is presented in Figure 30 together with the double-modei approximation m(&) and the wave-analyzed source ime The corresponding amplitude function is also given in Figure 31. From Figures 30 and 31, it may be concluded that the effect of finite Froude number for the hull-surface condition can ex- plain the cause of the correction function a (é) partially, but not completely. V1.2. Free-Surface Condition Before discussing the problem (b), we assume that the se- cond order term for the free-surface condition is independent of the higher contribution of the hull-surface condition (VI.1). The velocity 698 Analysts of Shtp-Stde Wave Profiles potential in Equation (2) is obtained from the linearized free-surface condition U 2 O¢ on @ = eye ae ee Assuming that the second order term for the free-surface condition is independent of the hull-surface condition, the approxi- mate calculations are carried out as follows. Using small parameter e (we can choose € = B/L), we can get the well-known expansions Leip yo, (25) 3 2 3 = G ¢ Shh dot pe tae. Then the kinematical condition of free surface Q/ Suie BH plied take # lam oy Ue ) Ox 2 dy oy Oz B= i (26) is written as 699 Mort, Inut and Kajttant Similarly, the pressure condition 2 2 2 oes ee is (22) | é U Be a ee + a Se + gz =0 4 (28) z=f is expressed by O¢ 2 ry) 1 ju sotet, fe E 5 +g 2 he 5 Fi (29) Paying attention to order of « , we have following two equations as to e and e 2 respectively. (30) Equation (31), which is the second order term of the free- surface condition, can be written as + = WI Ge, 32 fas < (x,y) (32) 700 Analysts of Shtp-Stde Wave Profiles where 3¢1\2 3 ¢)\2 1\% Sere eh le aac (4) The second order term of the velocity potential ¢ is given in accordance with well-known Fourier double integral 2 00 Be uth o-4 i ik. Pa ® Ag / f Yi (xt y') G (x.y, 25 x's y', 0) dx dy (35) - 00 to) where G(x,y,z;x!',y',o) is given in Equation (3) and (4.) Once $, is known, 7, (x,y) can be calculated by Equa- tion (33) all over the z=0 plane, then ¢, is obtained by Equation (35. Equation (35) means that the velocity eS of second order term is just that of source distribution ¥, (x,y) which is distributed over the free surface. The second order amplitude function corresponding to v, is given as follows after some approximations. Ce l= 2 eee [fren cos(K L pS saya) -Q(6,y') sin(K £ Beae 6 sin 6 y') > contd. 701 Mort, Inut and Kajttant co S,(6)/L Zecca | fr(0.y" sin(k £ seene sin€ y') oO 2 ha + Q(6,y') cos(K L sec 0 sinoy')} dy' (36) where eo 2 P(@,y') = ; K £ seco] & sin(K £ sec 0x") dx' fo) fo) e ate; ytys= > K J secof 8? cos(K L sec@x') dx' (37) Figure 32 shows the first order amplitude functions, C, (oy/i % S, (6)/L of the model M21 at the speed of K,L = 12 which are calculated in three different ways. (a) by existing theory, or from m(£) (b) by wave-analyzed source m(£) (c) by # -correction (#= 0. 4) The difference between the result (a) and (b) is present- ed in Figure 33 together with the second order amplitude func- tions C2(6)/L, S2(0@)/L which are calculated Equation 36. Figure 33 suggests that the second order correction for the free-surface condition is important only for the diverging wave range, where the wave-slope is predominant. Consequently, it appears that the remarkable discrepancy which is observed in the transverse wave range cannot be explained by this kind of non-linear effect. 702 Analysts of Shtp-Stde Wave Profiles By summarizing the preceding discussions (VI. 1) and (VI.2) it may be safely concluded that the real mechanism of the sheltering effect a (&) should be investigated not only by the second order con- siderations but also from some different kind of approaches such as suggested by Brard [4] or Pien and Chang [5] “ VII. APPLICATION TO HULL FOR M DESIGN Although the theoretical basis of the sheltering effect still remains unsolved, the presently obtained results will be of practical use for the hull form design with least wave resistance, As an example, an asymmetry Inuid model M 21-Modified is designed under the same geometrical restraints as M2l except that (a) The approximate correction function a(£) whichis given in Figure 18 is applied in the process of minimization of wave resistance, For example, at the designed Froude number Fn = 0.2887 (KoL = 12), we have 0.3 + 0.8 |&| , fore body aes) (38) O77 0.3 | el | att body (b) The correction function a (&) being asymmetry fore and aft, the optimized hull form is also asymmetry under the follow- ing restraint, Total source = W = / ime ides) = 0 (39) (c) The modified Fourier expansions are used for m(é ). It is expected that the modified asymmetry model M 21-M will be superior to the original model M 21 as far as wave resist- ance at the designed Froude number is concerned. 703 Mort, Inut and Kajttant Figures 34 through 36 show the calculated results. i.e, (a) hull-generating source, (b) load water line, (c) amplitude func- tion, respectively. The towing tests of M 21-M were run on May 9th through 12th, 1972 and Figure 37 shows its result. As far as wave resist- ance is concerned, the agreement between experiment and the pre- sent theory is noticeable. Moreover, M21-M is much less, as ex- pected, than M 21 in wave-making resistance at the designed Froude number. These results may suggest that the principal leading fac- tors of the sheltering effect are B/L and the shape of water plane as discussed in the preceding sections. In this connection, it must also be remembered that the simple wu -correction of Equation (14) was originally proposed not with Inuid models but with Pienoid models. Therefore, the correction function a (&) which is obtain- ed with M 21(B/L = 0.1184) will be applicable not only to Inuids but also to more general hull forms with flat bottoms. VIII. CONCLUSIONS Analytical method for obtaining the effective wave-making source of a ship is developed, where the measurement of the hull- side wave profiles is adopted as input data, instead of the free wave patterns in the rear ofa ship. Two Inuid models with different beam-length ratio (B/L = 0.0746, 0.1184) are wave-analyzed . Clear discrepancy is observed between the two kinds of sources, i.e. (a) hull-generating source m(&é) based upon double- model assumption, and (b) effective wave-making source m{( &) analyzed from measured wave profiles by means of the proposed me- thod. The correction function a (&) = m(&)/m(&) which is ob- tained from the wide model (B/L = 0.1184) is almost identical with the authors' proposal for the sheltering effect (1968), a(—)= 1- w(l-[e]), (= 0.4) 704 Analysts of Shtp-Stde Wave Profiles The second order calculations for hull-surface condition as well as for free-surface condition are found insufficient to give the theoretical bases of the obtained correction function a (£ ), which suggests the importance of the hull's sheltering effect. Considering a ( €) asa correction function, M21-M is designed under the same restraint condition as that of M21. From towing test results, itis proved that M 21-M is much less that M 21 in wave-making resistance and such a design procedure is si- gnificant. REFERENCES eal INGE 2.5 KAJITAND EH. PUKUTANI No and YAMAGUCHI, M., ''On Wave Making Mechanism of Ship Hull Forms, Generated from Undulatory Source Distributions", Journ. Soc. Nav. Arch. Japan, Vol. 124, Dec. 1968. [2] Do. : Selected Papers from Journ. Soc. Nav. Arch. Japan, Wools: 45) 970; [3 | INUI, T. and KAJITANI, H., "Sheltering Effect of Compli- cated Hull Forms", Proc. 12th Int. Towing Tank Confe- rence, Rome, 1969. [4] BRARD, R., ''The Neumann-Kelvin Problem for Surface Ships'', Report 11CST, 1971. [5] PIEN, P.C. and CHANG, M.S., ''Potential Flow about a General Three-Dimensional Body'', NSRDC Report 3608, OW 34. IE VE 705 Mort, Inut and Kajttant Table 1. Analyzed source strength of M 21, based on calculated wave profiles Remarks : m{(& ) wave-analyzed source m(é) = hull-generating source m(§) Saaaekis Up nayses ea ares eee! -1.0] 0.3207 0.3172 0.2546 | -0.0791 -0.9] 0.2927 0.3015 0.3112 0.2772 -0.8] 0.3740 0.3796 0.3712 0.3891 -0.7| 0.4911 0.4980 0.4979 0.4865 -0.6] 0.5920 0.6042 0.6036 0.5872 -0.5] 0.6437 0.6595 0.6442 0.6649 -0.4| 0.6297 0.6444 0.6291 0.5992 -0.3] 0.5468 0.5577 0.5611 0.5327 -0.2| 0.4027 0.4108 ‘ 0.4219 0.4328 -0.1]| 0.2134 0.2218 0.2137 | 9 @ea123 0.1909 0.0] 0.0000 0.0110 0.0001 | -0.0231 | -0.0434 0.1] -0.2134 | -0.2007 | -0.2136 | -0.2281 | -0.2153 0.2} -0.4027 | -0.3911 | -0.4029 | -0.3863 | -0.4881 0.3] -0.5468 | -0.5376 | -0.5466 | -0.5198 | -0.6691 0.4| -0.6297 | -0.6197 | -0.6294 | -0.6324 | -0.5650 0.5 | -0.6437 | -0.6254 | -0.6438 | -0.6717 | -0.7467 0.6 | -0.5920 | -0.5578 | -0.5922 | -0.5853 | -1.1521 0.7 | -0.4911 | -0.4410 | -0.4910 | -0.4251 | -0.2983 0.8 | -0.3740 | -0.3198 | -0.3740 | -0.3400 | -0.5466 0.9 | -0.2927 | -0.2514 | -0.2936 | -0.3322 | -4.4894 1.0 | -0.3207 | -0.2909 | -0.3138 | 0.1284 | 36.416 706 Analysts of Shtp-Stde Wave Profiles Table 2. Principal particulars of M20 & M21 2.001 2.001 0.1492 0.2368 0.2424 0.2055 0.1405 0.1724 0.0178 0.0347 0.5390 0.6686 0.0746 0.1184 0.0702 0.0862 Condition of Restraint Distribution Plane Source Distribution Vv, =0.018 V,'=0.036 T/L=0.04 (Inuid) 2.17444 a, -2.05643 | -4.11286 eer a 1.12955 2.25910 1.08722 707 Mort, Inut and Kajttant Figure 1. Co-ordinate system 708 Analysts of Ship-Stde Wave Profiles (I) 3924s OqTUTZ Fo JYSTOY OAeM OATIOETZO OUT, “7 oaNn3IW A ae | (dv) O01 oo 1c W 40 od -- OZ W {9 3UI0"”d Pjs!i4 —-o— / (dv) Ol 709 Mort, Inut and Kajttant (Z) Jos eTUTF FO FYSTOY SAM SATIOOTO OUI, 12 W 30 OZ W 40 qUl0d Pjel4 --7—- (dV) Ol ‘¢ eanstg 710 Analysts of Ship-Side Wave Profiles 00 LOO (€) 32Ys eyTUTT Jo yYSTEYy oAeM VATJOOTJO ay, —— cl= Py ——— 00 ‘h oansta (Alla Mort, Inut and Kajttant ue 0.02 0.04 006 008 0.10 y/l 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 z/\ Figure 5. The body plan of M 20 Few 0.0 0.0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 z/l Analysts of Shtp-Stde Wave Profiles 0.02 004 006 008 0.10 Figure 6. The body plan of M 21 0.12 y/\ Mort, Inut and Kajttant @) (8 =I M) IZW 8 OZW fo Seqtyord eaeM (*L 8ansTy ie 0'0 07 02 0z ‘G'S pezAjeuy WO1s5q so ('7°0= 1) UO1} D940} -7/ YM OQ —-— payeynzjey -——- poinseow) —— O0Lx /s 714 Analysts of Ship-Stde Wave Profiles e) (21 =I M) IZW ® 0ZW Jo settyoad ove "8 ernst 7 Ol- 00 OL QS pezhjeuywoysq Ot (7°0=W)U0!204409-y YUMoq —-— OOL x Ve Payejnjyej -—~- Psunseayy —_. CLS Mort, Inut and Kajttant Oo (91=T M) IZW 8 OZW jo seqtyord caem =6"6 0OANSTA 9L=1°M Ol-- 0 ‘I- 0'0 Ol * Ol Ol Ol- 00 00 Ol Ol ‘Q'S pezAjeuy walyoq =o OOLX 1/¢ (7O=1)U0!}224409- UIM6Q —-— Dales) —==— polnseen == 716 Analysts of Shtp-Stde Wave Profiles (Qi Ty) UOT]NGIIJSTp 9dINOS pozATeue-9AeM “OT 2ANndT ZT a3 \ me O° SS. (dv) 3 aaals Os z SO 00 SO ICW 6g == (z)W ZX OZW PpezAjeuy —-— ‘xouddy japow e}qnog ——— —— 8-1 (44) 00 : Je ore) : (2)Uu fi coth Mort, Inut and Kajttant (2q*= TH) uUOoT}NqTI4STp 9O1nos pozATeue-SAeCM “TI ANdT ZT \\ cl= 1M ; x (44) an SO \\oo SO0- O1- (2) ZXOZW PpezAjeuy —-— . (¢) WW ‘xoiddy japow a}qnoq ——— 718 Analysts of Shtp-Stde Wave Profiles (Sites 3) UOTINGTIISTp Sdtnos poezATeue-oAeM “ZI 2aNnsTW SO- Boa ef Z aaa hae SI=1 Fi IN | i (44) gy) = go . \oo S0- Ol 00 00 7 SO ZW oa —— : SO (QU zx0ZW pazhjeuy —-— oe (¢)us ‘xouddy japow a}qnoqg 719 Mort, Inut and Kajttant (0Z W) Uotjnqt4j3stp eoanos pezATeue-oAeM ‘ET OaN3TAZ exe) we Maas Z1= 1° oe Li=Ty Ol= 1° \ (dv) 01 3 GO 00 S0 rA | (3)W At OL=1°H ‘xosddy jepow 21qnoq CXOCW S'0- Ol- 720 Analysts of Shtp-Stde Wave Profiles (IZ W) Uotynqtaystp e5.rnos pezATeue-oAeM “PT aANBIW S'0- 00 \ 00 SS — S0 gt a : \ s‘0 Woe \ W (g) OL=T1°¥ ee ‘xouddy japoyy aj]qnceq ——— (ced Mort, Inut and Kajttant (¥) 10 ° . ) KoL=7 a 5 eas 0-5 F ° 2 Wave Analysis ° U- Correction 0-0 £ (He 0.4) 1-0 ° ° “ ° \ ° eee ° 0-5 KL=8 ° ; M21 0-0 ° ° 1.0 ay Za ° ° ° ° 0:5 KeL=9 © ew tee te 0:0 : d 1. ° ° 2 0 10 ° r-) 0.5 ° ° \ ° SSE a SSP ten) ER er eS en (alae, Cee Tey coy | Bh eee -1.0 (i 10 os (ER) 05 00 4 05 (AP) Figure 15, The ratio of m(é)/m(é) (K OL = 7~ 19) 122 Analysts of Ship-Side Wave Profiles & (3) : 1-0 05 ° Wave Analysis 00 ard a tL (AP) Figure 16, The ratio of m( &)/m(£) (KUL = 11~14) Mort, Inut and Kajttani ° ° ° ° ° Wave Analysis M-Correction eS (u=0.4 0! 0.5 ws KeL=1 ° M21 ea ate. 0- : ° t+) ° Figure 17, The ratio of m/( é )/m( E) (K OL =1'5'S 20) Analysts of Shtp-Stde Wave Profiles & (3) =———-—) /U-Correction (K=0.4) 1kO Fore Part oy KoL =14,15 16,18 0.0 -1.0 -05 z 0.0 (FR) (D) Figure 18. The correction function for M21 type ships t2D Mort, Inut and Kajttant (7 = ty) OZ W jo uzoqed ovem porns eoy[ ‘61 eansty (@) 00 ( Ww ul ) PainsDe;W fL= 1° O2W Met roa 0 ALLIS 4 ‘4 YH, b = aay 0 726 KoL=12 M 21 Measured ( in mm ) Analysts of Shtp-Stde Wave Profiles 2.0 1.5 1.0 0.5 -1.0 (F.P.) tant (Ree Oo Measured wave pattern of M 21 20.. Figure KoL =12 & M20x2 M 21 (in mm ) Calculated Mort, Inut and Kajttant 728 re) Calculated wave pattern of M20 and M21 (K L = 12) Figure 21. Analysts of Shtp-Stde Wave Profiles Oo (ZT =I M) OZW fo (3? )w uot nqT414sSTp adINOS SUTHeUI-SACM POUTe}qO WOT uTEWed SAeM PoJeTNITeD ‘77 ans ay) (O) (44) CS= y)) ct (ww u}) ZL=1°X OCW 729 KoL=12 21 M Mort, Inut and Kajttant 730 Calculated wave pattern from obtained wave-making source distribution m(é) of M21 wag) (KL (e) Figure 23. M20x2 & M2! Analysts of Shtp-Stde Wave Profiles Calculated with H-Correction (in mm) -2 eor----, 731 Calculated wave pattern of M20 and M21 with 24, Figure (#=0.4, KL = 12) mM -correction Mort, Inut and Kajttant Cwxi0" 1.0 —— Calculated —— Calcuated( 4=0.4) ———_ TowingTest ° Present Md A Longl.Cut Md 0.5 0.0 025 030 035 = Fn. Figure 25, Wave-making resistance of M 20 732 Analysts of Shtp-Stde Wave Profiles Cwxi0" 1.0 Calculated —— Calculated(U=0.4) Towing Test ° Present Md a _ Longl. Cut Md 05 0.0 0.25 030 *‘ Fm 0.35 Figure 26. Wave-making resistance of M 21 Mort, Inut and Kajttant (oz W) poyjeur snotizea Aq uotjouns opnjzttdure pezAjTeue-oaemM “LZ Sansty saiBap 08 OZ 09 0S Ov Oe 0c OL Cl Ie OCW Poyre|] IND “sues, ——-—— poyjayy ing ‘|Guo7 ———— G0 poyrayy) }Uaselg — —— ooLx 1/(8) v payeynaje9 734 Analysts of Ship-Stde Wave Profiles (12 W) poyjyeut snotzea Aq uotjouNy opnyttdure pozAjTeue-ovem ‘97 eandtaq saiBap 08 OL 09 0S OV O€ Oc OL | | | ! \/ : i" I \ \\ =| oe \ \ tl iW cee Pee li =19M IZW fi le \S if v0 SH 4 Ppoyysa|| 1ND ‘suesy ——-—— 001x Vie) poyia] 31ND “"}buo7 ———— pouja|\] }uese1y ——-—— payejnajeg ——-—— \" ay €'0 Tig le) Mort, Inut and Kajttant (12 W) e9eFins-T[nYy Fo JUoUOdWIOD AyTOOTOA OY, “67 9ANSIWT 0'0=Z 1k a (dV) oe - oF I oo yuauodwo7j-m i 02 O1.% 1}/M Qe= (44) Ol- | 00 ot yUsUOdWO3-A } A OLxf/A | = oz \ yUsUOdWO)-n (dv) Ol *sres— 0:0 Ol- SUWa] BARN 9814 *|OXQ ——— cll ICW oLx n/n SWI] AACA B214 *|9U] 736 Analysts of Shtp-Stde Wave Profiles (IZ W) UoTtpuos 2deTANS-T[NY JOeXS ULOAJT pouTe}qoO UOTJNQI4A}STp 9oA1NnOS SUT “OE ernst G0- CL=1°M LOW (dv) : (d 4) Ot ‘ : l- 0:0 00 (3) U0 !}!PU0D QdPy4NS }]NHyexy --——— S-0 siskjeuy sAeKy ———— 5-0 ‘xoliddy jepo-= e}qnog USM Mort, Inut and Kajttant (IZ W) UoTIIpuod sdeFINS-T[NY JOexe ULOAT peUTe}qo UOTINGTA}STIp 9D1NOs oY} Fo uoTJOUNZ opnyTidure eyL “TE Inst 7 sei6ap og te) z'0 ZI= TM IZW & 70 U0!}!|Pudy S2PYINS J}NH yexQ ———-- s‘0 sishjeuy aaey ——— 001x1/(8) ¥ ‘xosddy jepoy-, 9}qnog 738 Analysts of Shtp-Stde Wave Profiles Calculated Sin Comp. —-— DQ with A-Correction CoyvL srey/L*100 (u=0.4) 1.0 ——-—- Analyzed Cos Comp. To OS Sin Comp. M21 KeL=12 Figure 32, Wave-analyzed amplitude function of M 21 (KUL =?) (39 Mort, Inut and Kajttant AC@)/L —w— Secondary Cos Comp. asie)/L* 10° ee Sin Comp. 1.0 —-— Analyzed Additive Cos Comp. emer (DE Sin Comp. M21 KeL=12 Figure 33, Comparison between wave-analyzed amplitude function and secondary contribution of free surface condition of M 21 (KOU = be) 740 Analysts of Shtp-Stde Wave Profiles -109 Sutsn) petjIpoN-1TZ W FO UOT{NQIA}SIp 9d4no0s oY J, SO Chi W-icW (uoTOUNF UOTE 0'890°0- OLT6S °0- vOzEO"O €Sses*o 00€00°0 E9LET*O- TZveoo OLESE*O TS200°0 ETLEE*O "pe ernsty (d3) Ol- 00 S‘0 ($)us 741 Mort, Inut and Kajttant P°TIPON-12Z W jo euetd tazem OUT, “GE eansTT dv l Zz € ¥ S 9 L 8 6 d3 00 00 Z200 \ 700 iZW —— Cn 0s’0 900 80'0 oro : 010 es YA fA 742 Analysts of Shtp-Stde Wave Profiles PeTJIPONW-1Z2 W fo uotjoung oprgartdure oy y, eoiBep 08 cL= 1 pezAjeuy-2aey 69 —— peyeindjeD LZW = — —— W-LZ2W OL 09 —— y ‘"9¢€ OInBT ZT iw iN 4 0 he = 0) N 9°0 001x1/(6),v 743 Mort, Inut and Kajttant 3 Cy x 10 1.0 M21—M Measured* ----- D2 Calculated ———— M21 Measured*t | —-— D2 Calculated (* Using Schoeherr’s Mean Line, K=0.15) 0.5 ‘ __ Designed Speed 0.0 0.25 0.30 En 0.35 Figure 37. Wave-making resistance of M 21-Modified 744 Analysts of Shtp-Stde Wave Proftles DISCUSSION Klaus W. Eggers Institut fur Sehtffbau der Untversttdt Hamburg Hamburg, Germany I feel that this paper deserves a very careful evaluation as well as comparison with concurrent approaches, I shall restrict my discussion to one remark only and two minor questions, The authors deliberately do not display any physical assumpt- ions regarding the effect covered by their correction, it may result from higher order effects of wave making theory for an ideal fluid or from viscousity or even from wave breaking. As far as the first method, called mw correction, is concerned, it retains the anti- sym- metry of a source distribution for a symmetric ship and thus will not take account of viscous effects. Now my questions : on Figure 12, I observe that the actual source density found from wave pattern survey deviates from this anti-symmetry, as could have been expected, but the deviation is smaller with the larger beam/length ratio. How can this be explained ? Second, on Figure 25 there are represented different curves of wave resistance : from total drag, from wave pattern and from calculation using the method proposed. I would like to know from which Froude number measurement the underlying source distribution for the latter curve was taken and if there was any sensitive dependence found of wave resistance upon the Froude number selected for deter- mining the source strength. 745 Mort, Inut and Kajttant REPLY TO DISCUSSION Kazuhiro Mori Untverstty of Tokyo Tokyo, Japan Thank you, Professor Eggers, for your kind comments and questions, especially as to the uw -correction. The comments of Professor Eggers about the results present- ed in Figure 12, the analysis of source distribution, are quite sound, There results concern a thiner model at a speed of KoL = 16, which is rather slow for which the amplitude of the absolute wave height is less than one centimetre. We measured the wave height by photo methods using scales drawn on the ship's surface, with a one-centi- metre mesh, I am afraid the accuracy of the measured wave height is not enough, which explains the result of the analysis distribution in Figure 12, Secondly, the wave making resistance coefficient is calculat- ed from the analysied source distributions for each speed. The source strength is therefore not constant with speed. DISCUSSION Roger Brard Basstn d'Essats des Carénes de la Marine Parts, France I should like first to congratulate warmly the authors for the valuable contribution to the ship wave theory that they have presented to this Symposium, The idea of deriving the wave resistance from the surface elevation along the hull and not from the free waves is probably not quite new, since Guilloton, for instance, has attempted, some years ago, to take into account the surface elevation along the hull in the 746 Analysts of Shtp-Stde Wave Profiles computation of the wave resistance. Nevertheless, this idea is develop- ed by the authors in a very original manner and they have done a con- siderable amount of theoretical and experimental work. To obtain a ship form and a singularity distribution adapted to each other, one can start - either from the ship hull and attempt to determine the sin- gularity distribution to be associated with it, - or from a given singularity distribution and determine the ship hull generated by that distribution. The first path was first followed by Havelock. The sources are distributed in the longitudinal plane of symmetry and calculated by means of an approximate formula which implies the assumption that the length/beam ratio is considerably large and that the free sur- face behaves as a mirror. This first approach gives a solution which can be termed as Zero Froude number approximation for thin ship. At this stage, the theory is interesting from a scientific point of view only. The agreement between the measured wave resistance and the "residuary'' resistance is too poor to be acceptable by naval archi- tects. The Zero Froude number approximation transforms the real problem into a Neumann exterior problem. For solving that problem, in a rigorous manner, one can use a source distribution on the hull surface itself. Methods have been proposed by several authors - Hess and Smith [1], Landweber and Macagno [2]et al - inorder to deter- mine the source distribution, To my knowledge, it does not seem that the discrepancy between the measured resistance and the residuary resistance sensibly decreases ; it may even increase in some cases. It appears therefore that the Zero Froude condition may be one among the most important causes for discrepancy between theory and exper- iment, The second path is that of Professor Inui and his School. The given source distribution is located in the plane of symmetry and com- pleted by its image with respect to the plane of the free surface at rest. The ship form is obtained by integrating the system of differ- ential equations for the streamlines of the relative motion. For the ship form so obtained the given source distribution can be considered as the rigorous solution of the Neumann problem related to the double model derived from that form. But this method does not permit to get over the drawback of the Zero-Froude number condition. In their paper, the authors present a ''wave-analyzed source distribution" such that the ship-side wave profile derived from this 747 Mort, Inyt and Kajttant new distribution be close to that yielded by the experiment. They show that the agreement between the wave amplitude functions calculated from the new distribution and those derived from a longitudinal cut remains unsufficiently good. They conclude that researches are to be pursued in one or two other directions. I just heard from Professor Inui that one obtains a very good agreement between the two kinds of wave amplitude functions if one adds to the effect of the sources that of a line integral which I had considered in a paper [3] submitted, for discussion, in June 1971, to my Colleagues of the I. T. T.C. Resist- ance Committee. I consider as an interesting attempt the introduction of the wave profile on the hull into the calculation of the singularity distri- bution. One of the reasons why this does not provide a sufficiently good agreement with experiment may be the effect of viscosity and turbulence. The wave theory for inviscid fluid does not apply inside the boundary layer and wake. The longitudinal cut method does not give rise to this objection. What is to be concluded from the effect of the line integral ? In the last months, Professor Landweber [4] has gone further that I did a year and half ago in the afore-mentioned paper on the Neumann- Kelvin problem. He has shown that, if the hull is represented by a source distribution the line integral canbe transformed into a source dis- tribution over the hull. Therefore one can solve the Neumann-Kelvin pro- blem by using only a source distribution over the hull. But this requires the treatment of an integral equation with a kernel depending on the speedofthe ship. However, if the first approximationis that obtained from the Zero Froude number condition, it can be shown that this ap- proximation is certainly inaccurate when the Froude number decreases, and that the line integral expressed in term of that distribution should greatly improve the calculated wave resistance [5]. Hence, more de- tailed information on the procedure used by the authors for determin- ing both ''the wave-analyzed source distribution" and the line integral associated with it would be very much appreciated. I would like to add that I am not convinced that wave breaking is responsible for the discrepancy between calculation and experiment, for the M-21 form is thin. Much more important is undoubtedly the effect of the boundary layer and wake, not only because it entails a sensible loss of total head, but also because the boundary condition on the hull is altered. For this reason, I suggest that (it would be desirable that) the exper - iments of the authors be completed by a wake survey. 748 Analysts of Shtp-Stde Wave Profiles ey SMITH, A.M.O., "Recent Progress in the Calculation of Potential Flows'', Proceedings of the 7th ONR-Symposium on Naval Hydrodynamics, Roma, 1969. [2] LANDWEBER, L., and MACAGNO, M., "Irrotational Flow About Ship Forms'"', Iowa Institute of Hydraulic Research, The University of lowa, IIHR Report n° 123, december 1969. [3] BRARD, R., ''The Neumann-Kelvin Problem for Surface Ships'', Bassin d'Essais des Carenes, Report 11 CST, January 1971 (unpublished). [4] LANDWEBER, L., ''Contributions on some Current Problems of Ship Resistance", May 1972, Paper prepared for the NSMB 40th Anniversary Symposium (August 1972). [5] BRARD, R., ''The Representation of a Given Ship Form by Singularity Distributions when the Boundary Condition on the Free Surface is Linearized'', see Weinglum's issue of the Journal of Ship Research (March 1972). REPLY TO DISCUSSION Kazuhiro Mori Untverstty of Tokyo Tokyo, Japan 1. As to the empirical method The so-called u-correction has been proposed after quite same considerations as you have mentioned. If we were to use the kernel which depends on the speed of the ship, the calculations would get very complicated , such a method is quite useful,as shown in our paper, for hull form design. Zz. As to the discrepancy in the amplitude functions It is true that the B/L is not that large and wave-breaking may not occur. You pointed out as a main reason for the discrepancy, the effect of boundary layer, however we must remenber that the wave-making resistance obtained from analyzed source agrees with (49 Mort, Inut and Kajttant that obtained from towing test (Figure 25 and Figure 26). So the dis- crepancy in amplitude functions between the present method and wave pattern analysis corresponds to the discrepancy between C,, obtain- ed from towing test (CZ) and that given by wave pattern analysis (CQ). In Figures 25 and 26, A denote oy : As to the causes for this discrepancy between ke and Gy " we can mention first the failure of the wave pattern analysis method (longitudinal cut method) itself. The value of Co itself varies with y (the distance of the parallel cut line from the center line). Second, the magnitude of the discrepancy between CY and ct is proportion- al to Fre, not to Fr » in medium speed range, this means that this discrepancy is related to some characteristic of the wave height. We thus mentioned the wave-breaking as one possible reason, together with the non-linear wave height reduction with distance in this term. Of course the wake or boundary layer effect cannot be neglect- ed, and should be considered in further studies. 3. As to line integral contribution Our investigations made clear that the second order contri- bution of the boundary condition is not so large and we reach the con- clusion that the line integral contribution must be taken into account. From Green's theorem and free surface condition, the velo- city potentiel @ at P can be given as follows ®(P) = - I 00) ase (P,Q) - G(P,Q) 2 (Q) dS, SM ii Ee jee & G(P,Q) - G(P,Q)-9- va) | dy (A) In the existing theory we take into account only the first term and neglect the second term which is the so-called line integra- tion. Figures 1 ~ 4 show the contribution of the second term Q) the existing theory, namely the contribution of first term only ; @) the contribution of line integration, namely the second term ; GB) a) + (2) , namely the left hand side of Equation (A) ; 750 Analysts of Shtp-Stde Wave Proftles (4) wave analyzed result. As to (2) , Figures 1 and 2 present calculated results from theoretical wave height and Figures 3 and 4 are obtained from measur- ed wave height. The calculation is only a preliminary one and concerns only amplitude functions, but we can see striking agreements between and (4) . This suggests that the neglect of the contribution of the line integral was the most predominant cause of error. q51 Mort, Inut and Kajttant S(6)/. x100 fo degree @ Hull Surface Cond. @® Ll. (from D.H.M. ) @ O+©@ @ Wave Analyzed Fig.4 Contributions of Line Integration (Kel x40) _ 152 ECANP 73.42 Analysts of Shtp-Stde Wave Profiles S*(0)/L x 400 degree @) Hull Surface Cond. / “ @ £.1. (from 0.H.M) ; \ ® O® © Wave Analyzed -08 Ace -Fig.2 Contributions of Line Indregation (Wol = 12) _ 1538 Mort, Inut and Kajttant so) /L x1e0 degree @ Hull Surface Cond. @ Line Integration ( from Measured W.p.) ®@ O+® Kel. =0 ®© Wave Analyzed a iT ES): = Fig.3 Contributions of bine Integration (kel =e) _ 754 Analysts of Shtp-Stde Wave Profiles seh x Joo @ Wull Surface (ond. 4.0 @ Line Integration (from Measured wp) ®@+® ® Wave Analyzed -Fig.4. Contributions ef Line Intregration (Kel =12)_ 155 Mort, Inut and Kajttant DISCUSSION Louis Landweber Untverstty of Iowa lowa City, fowa, U.S.A. The intensity of a ship's centerplane source distribution is related to the ship-side wave profile by an integral equation of the first kind. Since the strength of the source distribution is assumed to be independent of the depth coordinate, which is not true in general, the solution of the integral equation can yield only an approximation which matches well the ship-side wave profile, but not necessarily the far-field wave pattern, and hence the amplitude-distribution funct- ion and the wave resistance. The assumption of thin-ship theory in the derivation of the integral equation seems unnecessary. It would be more accurate to apply the expression for the ship-side wave profile at the actual later- al coordinate of the hull than at the centerplane, as can equally well be done. In their reply to Brard's discussion, the authors have shown that the agreement between the wave amplitude function from a lon- gitudinal cut and from the ship-side wave profile is greatly improved by including the contribution from a line integral of singularities around the contour of the intersection of the hull with the free surface. Since the discussor has shown in his paper for the 40th Anniversary NSMB Symposium that this line integral can be transformed into an integral over the surface of the hull in such a way that the resulting system of singularities consists only of a source distribution on the hull surface, it would be interesting to know more precisely what sin- gularities were assumed along the contour and how their strengths were obtained. 756 Analysts of Shtp-Stde Wave Profiles REPLY TO DISCUSSION Kazuhiro Mori Untverstty of Tokyo Tokyo, Japan It is true that the solution of the integral equation can yield only an approximation because the source density is assumed to be independent of the depth coordinate, but the wave pattern calculated from the obtained source (Figure 23) shows the rather well agreement with the measured one (Figure 20). Moreover the wavemaking resist - ance which is calculated in the same way agrees well with the results of the towing tank test (Figures 25, 26). This is the reason why the hull-side wave analysis is adopted here to find out the effective wave- making source of a ship rather than far-field analysis. Besides itis conformed that the variation of the source to the depth direction is small. As for the second comment, the ship-side wave profile, in this paper, is calculated at the actual lateral coordinate of the hull, as Professor Landweber has mentioned. From the Green's theorem and the free surface condition, the velocity potential can be written, &(P) -|f or Row,a - o 38 bas, al Ss ; fe) C aG PB) pe -cZo| wy here, S stands for the hull surface and C the real hull water line respectively. The second term is the line integration, which is com- posed by sources and doublets on Z=0. At the first step, assuming that the contribution of the first term is equivalent to the center plane source, the velocity potential of the first term is calculated. Next, using the first step values, the contribution of the second term can be calculated. This may be the iterative method to solve the integral equation. Figure A shows the results of this calculation. Though the iteration has done only once, the sum of the both terms shows the fairly well agreement compared with the results of the wave-analysis. ie Mort, Inut and Kajttant ----Double Model Approx. —-—Line Integration ® @ @--O-®@ ) —Wave Analyzed S(O)/L x100 05 ae. 0.0 -05 M21 KeL=12 wi \ C(e)/Lx100 0.2 Fig. A The Amplitude Function of M21 (KoL=12) 758 OCEAN ENGINEERING Tuesday, August 22, 1972 Afternoon Session Clnguirerccein 9 125°, Jo IDG Walia, IMienarsin Netherlands Ship Model Basin, Wageningen, Netherlands Wave-Induced Eddies and Lift Forces on Circular Cylinders. R.L. Wiegel, R.C. Delmonte (University Ot Caktornian Uns) Ae). Analyses of Multiple-Float-Supported Platforms in Waves. C.H. Kim, J.A. Mercier (Stevens Institute of Technology, U.S.A.). Some Aspects of Very Large Offshore Structures. G. Van Oortmerssen (Netherlands Ship Model Basin). Unstable Motion of Free Spar Buoys in Waves. J.C. Dern (Bassin d'Essais des Carénes, France). Auto-Oscillations of Anchored Vessels under the Action of Wind and Current. A.V. Gerassimov, R.I. Persnitz, N.N. Rakhmanin (Kryloff Research Institute, Leningrad, U.S.S.R.). Ges) Page 761 (815: 95 1003 1079 ‘ . Ar o “ - ; 4 OMIAZ MOSS: A « i ks : at ep ation ay ‘@ i Leal» ¢ ster ,ss PRES). veal ruby: : ¢ i i S(G)/i.« 100 noisesS envogmpestA Fe _ ed % i 4) ot . ; y Poy wi ai arse <1. ofS + cameo i Yet tee leboM qide ebneltediel! ; ‘ absalredio a Agoinegs A ifm \ my gard , * ‘ff ” ty v4 1 WABI. 30. aati ten cba ht ae . ' , i 5 - eg avi saomploed aa a yy a ; wo tte wi) + \ : Fy e - biG RELI TBO i ree . tei ocr C rie =o i vig hia as ajuittgal ag Mi ie ees w ‘VRO had set te, 7 vet . isuxic stoacaliO eR_taea / iQ aoe ge: ae qidé ebaus! ssi “ean xestizoO may 09 é nies lei Bc meme 001; EW at ayouTin: sqy2 $930 lo sonoM-@ } eran Laos + A vf ALF Ws wi aah, Reo a ie: 4 | es ‘ f i? pe A] at i Vee ey ‘ ‘ ACO Gee en eee aif besadoad 3 tellioeGe } ‘ ean) 79.) 219862 / baesarsadke PO ef & an oro. Ino11v2 base bot 26 wjiinaseS .. A .vomdeastey) .¥ A Pe wtytitend dovanaek RolveA) gisecwtasn Vi A, ' ‘ - ¥ * ce ' tne titan we wh Aytuie. 0 -batgae ' , | sages WAVE-INDUCED EDDIES AND “LIFT” FORCES ON CIRCULAR CYLINDERS R. L. Wiegel and R. C. Delmonte Untverstty of Caltfornta Berkeley, Caltfornta, U.S.A. ABSA Cr The frequency of eddies formed by and shed in the lee of bluff bodies in steady flow is well known, and toa lesser extent the associated "'lift'' forces have been studiedandare reasonably well understood. The problem is more complicated in oscillatory flows such as exists in water waves. Results of studiesare presented for the case of a vertical circular cylinder which pierces the water surface. The value of the Keulegan-Carpenter number (NC,) incorrelating the "lift'' forces with the flow and cylinder parameters is shown, For higher values of Nc, it is found that the oscillatory flow tends to some extent to the stea- dy state flow condition insofar as the ''lift'' forces are concerned. However, owing tothe fact that in os- cillatory flow the''wake'' becomes the upstream flow, with eddies the same sizeasthe cylinder, itis always more complicated. The "lift'' forces are irregular for higher values of Ncyx and should be described by a distribution function ; examples of such func- tions are given. INTRODUCTION The formation of eddies in the lee of a circular cylinder in uniform steady flow normal to the axis of the cylinder has been studi- 761 Wiegel and Delmonte ed by a number of persons (see, for example, Laird, 1971). It has been found that the relationship among the frequency (cycles per se- cond) of the eddies, f, , the diameter of the cylinder D and the flow velocity V is given by the Strouhal number N,, s N = V ey Ne (1) where Np is the Reynolds number, VD/»v ,in which vis the kinema- tic viscosity. Except in the range of laminar flow, the Reynolds num- ber effect in this equation can be neglected. For flow in the sub-criti- cal range (Np less than about 2.0 x 10 ), there is a considerable variation of N, ; in fact, it is most likely that a spectrum of eddy frequencies exists (see Wiegel, 1964, p. 268 for a discussion of this). Extensive data on N, at very high Reynelds numbers, as wellas data on Cp (Figure 1) and the pressure distribution around a circular cylinder with its axis oriented normal toa steady flow, has been given by Roshko (1961) for steady flow. Few data are available on the resulting oscillating transverse forces, Cp is the coefficient of drag in the equation D 2 D (2) where F,, is the drag force, p is the mass density of the fluid, A is the projected area of the cylinder and V is the speed of flow of fluid relative to the body. What is the significance of N, for the type of oscillating flow that exists in wave motion ? The horizontal component of water par- ticle velocity is now cosh 27 (y +d) /L 27rt ven sink 2imid/Lnion vol YP (3) 762 Wave-Induced Eddtes and "Lift" Forces on Ctrcular Cyltnders — Kovosznoy | FIG, | R DRAG COEFFICIENT AND RECIPROCAL OF STROUHAL NUMBER VERSUS LOPta2 , | LAR NG rom 20 REYNOLDS NUMBER Cm (From Roshko, 1961 ) FIG. 2 CORRELATION OF DRAG AND INERTIA COEFFICIENTS (From Sarpkaya and Garrison, 1963) Cnet l+Cg (b) CIRCULAR CYLINDERS Ca¥ Co (a) PLATES FIG.3 INTER-RELATIONSHIP BETWEEN COEFFICIENTS OF COEFFICIENTS OF DRAG AND OF VIRTUAL MASS FOR (a) FLAT PLATES AND (b) CIRCULAR CYLINDERS (From Mc Nown and Keulegan , 1959 ) 763 Wiegel and Delmonte For deep water the horizontal component of water particle velocity is approximately u=(m7H/T) cos2mrt/T at y=0. Anaverage of u can be used to represent V; i.e., VF ui, = «H/2T , where Usyo is the average" horizontal component of water particle velocity due to a train of waves of height H and period T. For at least one pair of eddies to have time to form it can be argued that it is neces- sary for’ T > I/f, sS2eDT 7 x H Newmat N_ 0,2, H > 10 Dae Keulegan and Carpenter (1958) studied both experimentally and theoretically the problem of the forces exerted on a horizontal circu- lar cylinder by an oscillating flow. In their experimental work the os- cillations were of the standing water wave type, created by oscillating a tank of water. The cylinder was placed with its center in thenode of the standing wave so that the water motion was simply back and forth in a horizontal plane. The axis of the cylinder was normal to the direction of flow (i.e., parallel to the wave front), and about half way between the water surface and the bottom, They found that Ch (and Cy) depended upon u,,3x T/D, (the Keulegan-Carpenter number Nic)» where u = uy a, cos 2mt/T. They observed that when Nec was relatively small no eddy formed, that a single eddy formed when NKco was about 15, and that numerous eddies formed for large values of the parameter. It is useful to note that this leads to a conclusion similar to the one above. For example, if one used the deep water wave equation for uma, = TH/T, then uyay T/D >7H/D > 15, and H > 15D/7_ for one eddy to form. It appears from the work described above that a high Reynolds number oscillating flow can exist which is quite different from that which occurs in high Reynolds number steady rectilinear flow, unless the wave heights are larger than the diameter of the circular cylinder. Even then, owing to the reversing nature of the flow, the ''wake"' dur- ing one portion of the cycle becomes the approaching flow during an- other portion of the cycle. It is likely that Nyc is of greater signi- ficance in correlating Cp and Cy with flow conditions than is Np (Wiegel, 1964, p. 259), and that the ratio H/D should be held cons- tant to correlate model and prototype results, or at least should be the appropriate value to indicate the prototype and model flows are in the same ''eddy regime'' (see Paape and Breusers, 1967, for similar results for a circular cylinder and for a flat plate oscillating in water). In studying forces exerted by waves on circular cylinders one usually uses the equation developed by Morison, O'Brien, Johnson and Schaaf (1950). For a cylinder with its axial normal to the direc- tion of wave advance the horizontal component of force per unit length of cylinder is given by 764 Wave-Induced Eddies and "Lift" Forces on Circular Cyltnders F. = Fon + Fin (4) with F ss pig D | | (5) Dhy a, 2 D bag and 7 rD Ou Sinela aBCIant Hout (6) where | u | u is used rather than u“ to account for direction of flow. Fp) is the horizontal drag force per unit length of cylinder, Fy, is the horizontal inertia force per unit length of cylinder, C), is the coefficient of mass, C, is the coefficient of drag, D is the cylinder diameter. du/ dt is used in place of du/dt when the dia- meter of the cylinder is small compared with the wave length. When the Keulegan-Carpenter number is sufficiently large that eddies form, an oscillating "lift'' force will occur. For a verti- cal pile the '"'lift'' (transverse) force will be in the horizontal plane normal to the direction of the drag force. Few data have been publish- ed on the coefficient of lift, Cy; , for water wave type of flow (Chang, 1964 ; Bidde, 1970 ; 1971). In uniform rectilinear flows it can be as large as Cp, although there are few results available (Laird, 1961), The horizontal "lift'' force per unit length of cylinder is given by 1 PK realy Se al @ where €@y is the coefficient of "lift". Photographs taken of flow starting from rest, in the vicinity of a circular cylinder for the simpler case of a non-reversing flow, show that it takes time (the fluid particles must have time to travel a sufficient distance) for separation to occur and eddies to form (Rouse, 1946, p. 240). The effect of time on the flow, and hence on Cp and Cy has been studied by Sarpkaya and Garrison (1963 ; see also 765 Wiregel and Delmonte Sarpkaya, 1963). A theory was developed which was used as a guide in analyzing laboratory data taken of the uniform acceleration ofa circular cylinder in one direction. Figure 2 shows the relationship they found between Cp and Cy, which was dependent upon L /d, where £ is the distance travelled by the cylinder from its rest po- sition and D is the cylinder diameter. They indicated ''steady state" (i.e., for large value of £/D) values of Cp=1.2 and Cy, = 1.3. The results shown in Figure 2 are different than those found by McNown and Keulegan (1959) for the relationship between Cp and Cm in oscillatory flow. They measured the horizontal force exerted on a horizontal circular cylinder placed in a standing water wave, with the cylinder being parallel to the bottom, far from both the free surface and the bottom, and with the axis of the cylinder normal to the direction of motion of the water particles. The axis of the cylin- der was placed at the node of the standing wave so that the water par- ticle motion was only horizontal (in the absence of the cylinder). Their results are shown in Figure 3. Here, T is the wave period and T, is the period of a pair of eddies shedding in steady flow at a velocity characteristic of the unsteady flow. The characteristic velocity was taken as the maximum velocity. They found that if T/T, was 0.1 or less, separation and eddy formation were relati- vely unimportant, with the inertial effects being approximately those for the classical unseparated flow, and if Este was greater than 10, the motion was quasi-steady. "LIFT'' FORCES EXERTED ON A VERTICAL PILE BY PROGRES- SIVE WATER WAVES Water Particle Motion and Eddies Studies in the Hydraulic Laboratory of the University of Cali- fornia have been made by Bidde (1970, 1971) for the case of "deep water'' and ''transitional water'' waves acting on a vertical ''rigid'™ * The problems associated with a flexible pile are more compli- cated, owing to interaction of the pile motion and the formation of eddies. The reader is referred to the work of Price (1952) and Laird (1962, 1965) for details. The problem of an array, with the fluid flow - eddy interactions is also more complicated, and the reader is referred to papers by Laird and his colleagues for details on this subject (1960, 1963). 766 Wave-Induced Eddtes and "Lift" Forces on Circular Cylinders Direction of Wave Propagation ———____» | SARA gare y Length, L = Crest Ja y Still Water Level, Surface Trough Water Direction of Wave Height, SWL Water Particle Particle Water Particle H Orbit Motion Water Depth, 4 Note: Vertical scale and particle orbits 4/L = relative depth exaggerated proportional depth Bottom Wave Crest, at time + = 0.722 ! Front of wave passing through | i SWL, +=0.528 5 SS ee ——== | SWL Water particle at + = 0.722 Water particle f % ? Es? =X parte Motion e uy) | e . at + = 0.528 | e tim - ia) -+=0, 27 Sis —T feet S “ as Mean position of water particle Water Particle Seite ECD i mY Orbit for: 6 feet = 10 seconds 90 feet -7 feet SFR GRAPHICAL DESCRIPTION OF TERMS a z, ¢ u, ae Coordinate System HYD-7705 Figure 4, Water particle orbit due to waves (From Wiegel, 1964) 767 Wiegel and Delmonte circular cylinder which extended from near the bottom through the water surface. For this case the undisturbed water particle motion was not simply a rectilinear back and forth motion, but the water par- ticles moved in an elliptical orbit in a vertical plane, so that they were never at rest. Furthermore, any eddies that formed were affected by the free surface at the interface between the air and water. The hori- zontal component of water particle velocity is given by Equation (3), and the vertical component by mH sinh27 (y +d) /L WESC 5 sinh 27 d/L ea (8) Z 2 In deep water the water particle speed, oy =u" +v , 1s given by q = «H/T (9) at the surface. Thus, the speed remains constant in deep water while the particle continuously changes direction. It is more complicated in transitional and shallow water. An example of the water particle path is shown in Figure 4. There is little reason to expect that eddies formed in such a flow would have the same characteristics as those formed in simple oscillating flow. © One of the most crucial factors in oscillating flow of this type is the fact that the wake formed during one portion of the cycle be- comes the upstream flow in another portion of the cycle (the paper by Laird, Johnson and Walker, 1960, is useful in gaining some under- standing of this problem), and little is known of the water particle mo- tions under these conditions. When eddies, form, they appear to be of about the same size as the pile. In this regard it is interesting to refer to an observation made by Bacon and Reid (1923) in some studies of fluid forces on spheres. They found that if the scale of the turbu- lence was small compared with the diameter of the sphere, Reynolds number was a good criterion, but if the grain were coarse, then Reynolds number no longer served even as an indicator. During the first stages of the study by Bidde, immiscible fluid particles with the same specific gravity as the water were made ofa mixture of carbon tetrachloride and xylene, with some zinc oxide paste added to make the particles easily visible. The fluid was inject- ed into the water by means of a long glass tube which had a rubber 768 Wave-Induced Eddies and "Lift" Forces on Circular Cylinders bulb mounted at one end. The other end of the tube was heated and drawn to make the tip opening the desired size. Stereophotographic sets were taken of the trajectories of these tracer particles, anda computer program (Glaser, 1966) was used to calculate the space position of them. However, it was found to be too difficult and lengthy a job to pursue. Owing to the difficulty described above, a description of the wake regime was developed by Bidde which was based upon his ob- servations of the water surface characteristics, using magnesium powder sprinkled on the surface in the vicinity of the pile. An example of the relationship between the wake characteristic and the wave height, with the wave period being held constant is given in Table 1 together with the values of Np and Nyc. Similar tables were con- structed for a number of wave periods. The generalized results are shown in Figure 5. It was found that Nyc correlated reasonably well with the different regimes of the surface wake characteristics. When Nec was about 3, one or two eddies formed, when its value was about 4 several eddies formed and shed, having the appearance of a von Karman vortex street, when it was 5-7 the wake started to become turbulent, and when it was larger than 7, the wake became quite turbulent, and the turbulent mass of water swept back and forth past the pile. Using the concept described previously, H > about 3D/7 , that is, the wave height should be about equal to one pile diameter. As will be shown later, this was found to be the case for two piles, one about four times the diameter of the other. The Reynolds number was between 4,000 and 7,000 for the values of the Nxc when the wake became quite turbulent with no de- tectable von Karman vortex street. A similar phenomenon occurs in steady flow for Np greater than 2,500, according to Rouse (1963). He states that for Np greater than this value a trail as such can no longer be detected. Rouse fur- ther states that the body continues to be subjected to alternating "lift" forces, but that each vortex becomes progressively more unstable during its formation with a resulting wake that consists of a hetero- geneous series of eddies. "Lift!'' Forces (Bidde) When eddies form, in addition to their effect on the longitudi- nal drag and inertial forces, "lift'’ forces are also exerted on the cylinder. For a vertical cylinder these lift forces act horizontally, but normal to the longitudinal forces (longitudinal being in the direc- tion of wave motion), and should more properly be referred to as 769 Wiegel and Delmonte Table 1. Observation of surface characteristics of eddies (From Bidde, 1970, 1971) Water depth = 2.0 ft, cylinder diameter = 1-5/8", Wave period = 2.0 seconds Surface Surface | Keulegan- Reynolds | Carpenter Number Number Observations No separation, no eddies (Amplitude of motion does not reach cylinder diameter) Small separation Very weak von Karman street Clear von Karman street Wake of prior semicycle, when swept back gives rise to addi- tional eddies o vs) at . . WNUOrRrNHNAOUN~W OW Eddies swept back by the time they are formed Becoming highly turbulent WOMONDODUNLSLWWNNH ooooocoooococooocooco0cdcsd Extremely turbulent, no more eddies visible Fwy KEULEGAN-CARPENTER (a REULEGAN NUMBER <2 CanPEn = NO SEPARATION >) = Museen 34 AMPLITUDE OF MOTION MORE THAN 2 1S LESS THAN CYLINDER “SD ——EBDIES sued Fe ae DIAMETER YJ tans WALF a —e you VON KARwAR ‘STREET (bo) EGAN-CARPENTER KEULEGAN-CARPENTER e€e SS mEUL mp SNYv.. NUMBER=2-3 CET ~~ 8 NUMBER S—7 SMALL SEPARATION, NO ta Rene eco EDDY DEVELOPED YET 2 2 TUAMULENT. ces = Mee ee ed * EULESAN-CARPENTER e KEULEGAN-CARPENTER NUMBER 3 (4 “=> 2 MUMBER >7 FIRST EDOY SHED co” EXTREMELY ASSYMETRY STARTS 2 3 TURBULENT LIFT FORCE BEGINS TO WA BE NON-ZERO a= vg 9 J 66. —— Figure 5. Wake characteristics as a function of the Keulegan- Carpenter number (Bidde, 1970) 770 Wave-Induced Eddtes and "Lift" Forces on Circular Cyltnders transverse forces. Examples of waves, "lift'' forces and longitudinal forces are shown in Figure 6 for three different values of Nyc (3.2, 6.2 and 10.2). The terms ''top'' and "bottom" associated with the lift and longitudinal forces refer to the forces measured by the top and bottom strain gages on the transducer ; the total "'lift'' and total longitudinal forces are the sums of the outputs of the top and bottom gages. There is agreement between the visual observations described previously and the force measurements, Figure 6a shows a set of records fora Nxc of 3.2. The "lift" force has just begun to be non- zero, For this value of Nyc the first eddies develop and shed. The eddy strength is probably very small so that the "'lift'' force recorded is negligible. The ''lift'' forces for this case have a frequency which is about the same as the wave frequency. This might be due to the fact that the flow is not perfectly symmetrical. The horizontal com- ponent of velocity in one direction (wave crest) are slightly larger than those in the opposite direction (wave trough), and for the thres- hold condition the eddies only shed for one direction of the flow. The Keulegan-Carpenter number is 6.2 for the run shown in Figure 6b. The eddy is distinct, and the frequency of "lift'' forces is approxi- mately twice the frequency of waves. This shows that there is time only for two eddies to shed in each direction. The ''lift'' forces are about 25% of the longitudinal force. The wake is not yet completely turbulent, and the lift force records show a more or less regular pat- tern. The Keulegan-Carpenter number for the run shown in Figure 6c is 10,2. The wake is fully turbulent. The transvers (''lift'') force record appears to be random, The ratio of maximum ''lift'' to maxi- mum longitudinal force is about 40%. An equation for "lift'' forces is given by Equation (7). Use of this equation leads to difficulties as the time history of the force does not necessarily vanish when u goes through zero owing in part to the inertial force. Thus, very large values of Cy; can be calculated from the laboratory measurements, This difficulty can be overcome par- tially by defining the relationship only for maximum values of the force as Chang (1964) found values of Cla, between 1.0 and 1.5 for value Nec greater than about 10. ee Wregel and Delmonte otpotzed urzojstun O12 GNV 602 SNNY 4O SON093¥ 29 914 LAA CUE 10 Wd 1 ae et VAAN Se LE He ei Ofege tate fag) atk 0 ( ‘sp1ooea 20107 pue 9ACM gi2 ONY Liz SNNY 40 SONOI3Y G9 914 PLATT ACA Meee i serra ZOI-WaLaWVYYd NVDIINIX ¢OS'01-438WNN SOIONAZY €2°9-U3L3IWVWd NV9IINIx ¢0IXS'9=43SNNN SGIONAaY &l=0 4le0=H puosasi— Jes = Leia %iza wezo=H puorasi—4 2S BL1= 1 ‘opptd) etdueg ‘9 oansty tz2 ONY €22 SNNY 4O SONOI3N 09-914 a a et uci TE f= NRINVENd NV9D391NIY 01* € €=4Y3BWNN STIONAIY WAI =O MZIO=H o08 B2I=1 Tté Wave-Induced Eddies and "Lift" Forces on Ctrcular Cylinders In the study by Bidde the ratio of ''lift'' to longitudinal force was used as a basic parameter rather than Cy, as this parameter is comparatively less sensitive to any systematic errors in the instru- mentation used to measure the forces, as similar errors would be present in both ''lift'' and longitudinal force measurements, and these errors would have a certain tendency to cancel out. Some of the data are shown in Figure 7 of the relationship between the wave height and the ratio of "lift'' force to longitudinal force. a (2) PILE DIAMETER «1 Ye —-—@7 O085sec WATER DEPTH = 2° = —0T?! 06 sec ——x T5115 sec ——~ 0 T=1 25 sec > -@ 7:1 % sec ——9 7-155 sec ‘ * 40 x 100, % y {e) 100 % D oO 9) NIG, 2) —— T:'78 sec —o T=208 sec 9 PILE DIAMETER O5 WATER DEPTH CW) 30) LIFT FORCE LONGITUDINAL FORCE LIFT FORCE LONGITUDINAL FORCE 1 fe Je —t- = © om 2 © ca OS GS 7 om ca i) 2 ee as ae Fe ee ee ae WAVE HEIGHT (tr) Relationship between ratio of lift to longitudinal force and Bisure: 7. wave height (Bidde, 1970, 1971) The relationships between Nyc and Np and the ratio of "lift'' force to longitudinal force are shown in Figure 8, This graph indicates that the "'lift'' forces startat Nxc of about 15 the ratio of "lift'' to longitudinal force shows a slight tendency to stop increas- ing. As can be seen in Figure 6c the amplitude and frequency of the "lift'' forces become irregular for larger values of Nyc, anditis necessary to specify what is measured. Bidde presented the ratios of average maxiznum '"'lift'’ forces to average maximum longitudinal forces, The longitudinal forces were uniform so that no problem exist- ed in measuring and reporting them. Bidde drew a line by eye through the crests of the larger "lift'' forces and another line through the throughs, and reported the ''lift'' force as the distance between the two lines. Furthermore, Bidde reported the ratio of the ''lift'' to lon- gitudinal force as measured only by the bottom strain gages. An ana- lysis of a few records showed that this ratio was the same as the ratio of the total forces. ees KEULEGAN - CARPENTER NUMBER -4 REYNOLDS NUMBER x IO 10 20 30 Wiegel and Delmonte 405 50~. GOs 370 “80290 100" "110° 120 UCB DATA (1972), 0.14 ft. PILE O - MAXIMUM VALUE O- MEAN VALUE BEB DATA,(I972 ANALYSIS © MAXIMUM VALUE Qu MEAN VALUE NUMBER IN © OR [] REFERS TO RUN NUMBER. |.06ft. PILE PILE DIAMETERS: ee 3 in. = 0.14 ft. Sa 05 Ff WATER DEPTHS: 2 ft, 4.5ff. and 5fft. (Bidde, 1970; 1971) oO AVERAGE VALUES IN STEADY FLOW OBTAINED BY BISHOP AND HASSAN (1964) HUMPHREY (1960) un STEADY FLOW eases) & - MAXIMUM VALUE } OPEN END &- AVERAGE VALUE J GAP 9 - MAXIMUM VALUE } INSIDE A- AVERAGE yve} SEAL 40 50 60 70 80 90 100 II0 120 "LIFT" FORCE a a a ee So, ConcitupinaL Force * !00, % 774 Ratio of "lift'' force to longitudinal force (%) versus Keulegan-Carpenter number and Reynolds number Figure 8, Wave-Induced Eddtes and "Ltft" Forces on Ctrcular Cylinders Bidde reported he was not able to compare his results direct- ly with those of Chang (1964), as Chang gave his results graphically as (Cy) max v8 Nxc and (Cy) max v8 NR with no values of (CD) max: However, Chang's graph of (Cy) max vs Nac showed (Cr) max to approach zero for Nxc of about 5, and approach a value of 1.0 at about Nec = 20. "Lift'' Forces (Present Study) The work of Bidde described above was continued by the wri- ters, to determine the effect of large values of NxKoc on the ratio of "lift" forces to longitudinal forces. The results for the 1-5/8 inch (0. 14 ft.) pile are reported herein. Tests were made for conditions that approximately modeled large scale tests which will be described subsequently. The pile was carefully checked to determine its sen- sitivity to direction and location of resultant force through a series of calibrations, Aluminium powder and flour were sprinkled on the water surface to help in visualizing eddies. However, owing to the high values of Nec (15 to 90) and high Nr , the wake was very tur- bulent and no distinct eddies were observed. The crest to through distances for thirty consecutive '"'lift'' forces were measured, while the crest to trough distances for only four consecutive longitudinal forces were measured as the longitu- dinal forces were quite uniform, The ratios of the averages of these two sets of data were plotted on Figure 8 (labeled UCB 1972), and tabulated in Table 2. The Coastal Engineering Research Center ( CERC ; formerly the Beach Erosion Board, BEB ) , U.S. Army Corps of Engineers, kindly lent to the writers the original records of forces exerted by waves on a vertical pile that were obtained by Ross* (1959) in their large wave tank (635 feet long by 15 feet wide by 20 feet deep). The active section of the test pile was 3.0 feet, with a dummy pile below and above the active section (Figure 9). The outside diameter of the pile was 1.06 feet. Data were not plotted in the form shown in Fi- gure 7, as the minimum wave height used in these tests was greater *Run numbers on original data rolls and those reported in T.M. 111 are not the same. See letter of 16 September 1971 from John C, Fairchild of CERC to Robert L. Wiegel of UCB for code. 5 Wiegel and Delmonte “H,09 7284 sutsn payepnopteo Ww bat a *(3S9a0 anaem ay} Ye A FO anTeA ayy ‘ST 7BY}) OR = ok Sutsn payeprnopteo sem e nh) pue ° xeu xeul ‘9 = A Butsn payepnoptes sem ( 1) le=" b/adige SoD ox Ss. “ba) Nn a}yepTnNopTeo 0} pesn ATOSsYy} BACM ABAUTT +:S9}ON ‘aot0F FFTT,, 242 0} predar ut ‘y uny ut Atasedord yrom you ptp yuowdtnby” sues aor 0g Teutpnzytsuoy] FFTT,, TROL wNnWT xe eT tpuadI1adq asear9Ay oder3ay os = | aes, aa se | se ape Z g awe Te YNVL AJAVM ADUVI NI aTId “G’°O (14a F1°O) HONI 8/S-1 HOA VLVd 2Z6T don “Se ATAVL spuooes yzdeq| potsed| yyusT SH ABM * “ON uny 776 Wave-Induced Eddtes and "Itft" Forces on Ctreular Cylinders Neided ring ¢ Moving bulkhead A 2 4 -Rubber sieve 12-inch pipe line ~—_ [EE * Rignpes ah eee a See ah Mele cise onipinornp ss) Yhea soe saan erste ae vo Tee Wave gage on side q SS _--Stop Raere ee Ne Sand beach, 1.15 slope —~ Arm to support sensitive section of 12” pipe i ~~ Hole in 10” pipe for arm i=) --—Bending arm pn SR-4 Stigin gage eS — -- Diaphragm ce) ~--—-10” pipe 0.25” AX A —=— 12% pipe - 0.25” — Still water level Wave gage on side of tank ~— Wave gage on platform -— 3' sensitive section = Dummy piles used in some tests 15’ Section A-A (a) DIAGRAM OF WAVE TANK (b) DYNAMOMETER FOR MEASURING WAVE FORCE Figure 9. BEB arrangements (Ross, 1959) than twice the pile diameter so that the "'lift'' forces were always substantial compared with the longitudinal forces. Longitudinal and "lift'' forces were measured for waves with periods from 3.75 to 16.0 seconds and heights from 2.1 to 7.7 feet. The two sets of data labeled ''BEB data, 1972 analysis'’ in Figure 8 are ratios of forces per unit length of pile, using the average of four consecutive waves for one set, and the maximum of the four waves for the second set. Representative examples of the BEB data are given in Table 3. Owing to the manner in which Bidde analyzed and reported his data, it would be expected that they would lie between the maximum and mean values obtained by the writers for the same Nyc. If this is the case, it appears that Bidde reached the value of Nec that divides the regime from a strong dependence of the ratio of lift to longitudinal force upon Nyc to one in which it is essentially inde- pendent of Nec. Bidde found for the smaller diameter (1-2 inch) pile that Np CCT Wiegel and Delmonte PeB[NOTBO eTeM | YN PUB N °(389810 anem ay} ye A JO onTeVA 9yz ‘ST 3BY}) pue ‘attd fo yooy azad aor10sy ute}GoO 034 (Suo, 4aeF ¢ Sem UOTJOaS 4Sa4 9y}) € Aq pepratp painsvou saor10y oy} ore (2) pue (9) ua uy ° xew = A Sutsn pazertnopeo sem ( n) “T= £/42¢ soo zoz (¢ *ba) wtFF¥I,, uNnuUTxXeW *(g) suun{oo ut peyroder saoroy ayL "d,09 384 Sutsn x eu Ps) 9 gre ig sutsn pezernoteos sem ghia N 8}e[NOTVO 0. pesn ArOdYy}Z 2ACM IBdUTT :S930N TIt ‘W’L Ut tequnu uny spunod aoI10g TetyOL aser30Ay GTId ‘d°O Id 90°T YOd (SISATWNV 2261) VIVd daa ‘¢€ FIEVL * 778 Wave-Induced Eddtes and "Lift" Forces on Ctreular Cyltnders appears to be as good a parameter for correlation purposes as Nyc. However, Np fails to correlate well with the ratio of ''lift'' to longi- tudinal force when the values of the larger pile are compared with those of the smaller pile. For the smaller pile the minimum value of Np at which eddies form is about 0.5x 104 , whereasitis 2.5 x 104 for the larger pile. For the same conditions the value of N is 3 to 5 for both the piles. The UCB (1972) and BEB (1972 analysis) data reported herein seem to show an effect of Np. When the oscillating flow is such that the NKc becomes large, one might expect the ratio fo "'lift'' to longitudinal forces would ap- proach the values obtained for steady rectilinear flows. The writers were not able to find many data in the technical literature, however. The results they found have been plotted in Figure 8 (Bishop and Hassan, 1964 ; Humphrey, 1960 ; the two sets of curves labeled Fung and Macovsky were drawn from data attributed to them as they ap- peared in Humphrey's paper). These points were obtained by calculat- ing the ratios of Cj, to Cp from the values of the two coefficients given by the investigators. Although it is evident that much work remains to be done before the problem is solved, it is clear that an engineer must consider a rather large "lift'' force as well as a longitudinal force in his design. If tests are made with piles of two different sizes, and com- plete geometric similarity conditions are met, and the Froude model lawis adhered to, it can be shown that the "prototype"! and ''model'' Keulegan-Carpenter numbers will be equal, but that the Reynolds numbers will not be equal, All geometric scale ratios must be the same, however. For example, in Figure 10 records obtained by Bidde of waves and forces for the 1.63 inch (0.14 ft.) and the 0.5 ft. diameter piles are reproduced, The wave height ratio is the same as the pile diameter ratio, and the wave period ratio is the square root of the pile diameter ratio. However, the water depth was the same in both cases, so that complete geometric similarity was not obtained. Note how different are the two sets of waves and forces. The records in this figure, and in Figure 11 show another interesting feature. There is not a ''one to one"! correlation of the wave to the "lift'' force time series. Rather, the "lift'' forces occur in bursts, with intervals of very small forces in between. Comparative records are given in Figures 12-14 for the UCB 1972 and the BEB experiments, Froude modeling was used and com- plete geometric similarity was maintamed. Owing to difficulties with the equipment, the ratio of wave heights was not quite correct for the Ui) Wiregel and Delmonte soqtd ToyouIeIp “UT G79 ‘T pue ul 9 ‘sooo; IIT pue [eurpnytsuoT jo uostaeduroH “OT eInst 7 4ov5 WwoL108 | 39v9 dOl Sbe NNY 39v9 WO1108 fit 39v9 dO 9b¢ NNY OWN oo6'sy=4n ‘o6 = %2p=OlLVY 39YOI LIDNOT-1417 44 0'¢ « 39N39N3WENS 4480 = LHOIZH JAVM 43Gb = Hid30 YILVM 298 6 p= GOINId JAVM uig= vid 31d 3AVM 39Y404 4317 39404 TNIGALISNO) 3AVM 462 NNY oosc UN 82h ey ) ef % O02 = Ollv¥ 39404 LIDNOT-1417 ‘ie 4452 0» 39NF9NINBNS H1id30 H3LVM 44810 = LHOIZH JAVM 4900'S «© H1d30 UI1VM 288 Gp 2 = GOIN3d JAVM vi €91« vid 31d it 39N39NIWENS wiogepe 780 Wave-Induced Eddies and "Lift" Forces on Ctreular Cylinders SeTid To},0UTeTp “UT G7Z9g *{ pue “ut 9 ‘SSOLOF JIT pue [euIpnytsuoy jo uostaeduioy ‘{[] oeanstg7 SWiL — 3AVM _ J9v9 WOlL08 = 39yY04 14 Le 5) Ni ' ad = 39v9 WOL1L08 $0 : : i A NTT ANA TAT 3oyN04 WA IW Te te ei er ore TTT Ly dO 6 NN et inelataida econ ted Lee wha 3AVM osi'6g= YN ton = Wy o6ze = YN tag 2 4N % EE «Ollys 39404 119N01-1417 % 02 = OllvY 39404 “LI9NOT- 1417 4) 0€ = JONI9NINENS les 4 G2'0* 39N39N3NENS 1)60 * LH9IZH JAY H1d30 Y31LUM 44.220 « LHOISH JAVA 4)S% = Hid30 Y31UM 4400'S = Hid30 Y31VM 2356): GOIN3d 3AUM Vid ad 28% Sw'z « GOIN3d 3AM wig = ¥IO 31d 39N39NIWENS WEST: VIO 210 781 Wiegel and Delmonte DATA SOURCE TEST NUMBER PILE DIAMETER , ft. WATER DEPTH , it, WAVE PERIOD , sec. WAVE HEIGHT , Ft: MAX. LIFT-LONG. FORCE RATIO U.C.B. RECORDS LONGITUDINAL FORCE + sae CREST : —— TROUGH BOTTOM GAGE TOP GAGE ——_— i LIFT FORCE -- | ee Ny fp mereREST i NIAAA rou BOTTOM GAGE TOP GAGE LIFT FORCE LONGITUDINAL Jf FORCE Figure 12. Comparison of UCB (1972) records and BEB records for NcK of about 25 782 Wave-Induced Eddies and "Lift" Forces on Circular Cylinders DATA SOURCE U.C.B. TEST NUMBER 5 PILE DIAMETER , ft. 0.14 WATER DEPTH , ft. 1.79 WAVE PERIOD , sec. 4.05 WAVE HEIGHT , ft. 0.64 MAX. LIFT-LONG. FORCE RATIO Sue U.C.B. RECORDS LONGITUDINAL FORCE \ ) = CREST WAVE —=—TROUGH BOTTOM GAGE - TOP GAGE | “LiFT FORCE oe —— CREST ti od Tedahe TROUGH BOTTOM GAGE ‘ eri COAL Gememia Fb! seas ryeveie LONGITUDINAL ma Eps a HHA rite LED NE TPIS arene Figure 13. Comparison of UCB (1972) records and BEB records for Nxc of about 42 for the UCB and 35 for the BEB runs 783 Wiegel and Delmonte DATA SOURCE TEST NUMBER PILE DIAMETER , ft. WATER DEPTH , ft. WAVE PERIOD , sec. WAVE HEIGHT Ft: ? MAX. LIFT-LONG. FORCE RATIO U.C.B. RECORDS LONGITUDINAL gee WAVE HEA —— CREST . s Set” "TROUGH TOP GAGE silat nh aah nf Geieen A LIFT FORCE —=— CREST BOTTOM GAGE TOP GAGE edness B.E.B. RECORDS LIFT FORCE LONGITUDINAL Figure 14. Comparison of UCB (1972) records and BEB records for Nec of about 80 784 Wave-Induced Eddies and "Lift" Forces on Ctrcular Cyltnders runs shownin Fig. 13, however. Thelengths of the waves used in the runs shown in these figures wereall rather long compared with the water depth, and were quitenon-linear, The run shownin Figure 14 was chosen to show the worst comparison that was obtained, which was for the case of the mostnonlinear wave tested. "Tift'' Force Distribution Function Referring again to Figure 6c, it appeared to the writers that a "lift'' force distribution function would be useful. Owing to this, some "'lift'' force records were analyzed in detail. The method by which this was done is shown in Figure 15. The ''lift'' forces for 30 consecutive force oscillations were measured for each of 10 runs. Total "lift'' forces (the sum of the outputs of the top and bottom strain gages) were measured. The characteristics of the waves in these runs are given in Table 2. The characteristics of representative examples of the BEB data are given in Table 3. The distribution functions are shown in Figure 16. Figure 15. Method of analysis of '"lift'' force records, UCB 1972 force measured from crest to following trough "Lift'' Frequencies Chang (1964) stated the frequency of the '"'lift'' forces was twice the wave frequency. However, an examination of the sample records reproduced in his report showed that for a pile diameter of 0.083 foot, and a wave period of about two seconds, the "'lift'' force became irregular as the wave height was increased. One record showed the average ''lift'' frequency to be about three times the wave frequency. The writer's data showed that ''lift'' frequency to become quite irregu- lar, as was the case of Bidde's data for a number of runs (see Figure 6c, for example). The BEB data showed the "lift'' force frequencies varied from about 1.3 to 6 times the wave frequency. Irregular Waves Waves in the ocean are irregular. Irregular waves can be ge- nerated in the laboratory using a wave piston type generator. An example of such waves, together with the induced longitudinal and lift 785 (Ratio to force at 50%) NORMALIZED TOTAL LIFT FORCE Wregel and Delmonte KEY FOR DATA (1972) KEY FOR BEB DATA 0.14 ff PILE UCB (1972 ANALYSIS) £06 TT. PILE a er? RUN 35 ar t+ (RUN 58 Sa BoStro RUN 22 RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN x*OoO0odC pd ot xe Fa =HGLavounFsun— -_— eS 0.1 va 5 fe) 20 40 60 80 905 295 % LESS THAN OR EQUAL TO NORMALIZED FORCE Figure 16. Total "lift''’ force distribution functions 786 es | 98 Wave-Induced Eddies and "Lift" Forces on Ctrcular Cylinders RANDOM WAVE CENTRAL FREQUENCY= lO cps Bt aR in det dd fe—Isec 4b -l- bed bd -0, NEW YI a a a ar Pe A tay 7 Ht I] ke TE am ell om = (a) RECORDS OF RUN 366 RANDOM WAVE CENTRAL FREQUENCY * |.Ocps <4 wave 1 aa Hiieeseetitts YY XI Ni V/A = QRATION BUFFALO, NEW YORK SSS SS SSS SS SSS SS SS SS SS SS SS SS SS SS SSS SS aS SS SS SS SS SS SS SS SS (b) RECORDS OF RUN 365 Figure 17. Sample wave and force records, irregular waves (Bidde, 1970) 787 Wiegel and Delmonte forces is given in Figure 17 (Bidde 1970). It is suggested that as an approximation the case for regular waves can be extended to the case of two dimensional random waves, For example, the highest wave in Figure 17b has a height of 0.3 ft. Using linear theory, Nyc for this wave (the wave ''period'' is about 1.1 second) can be calculated, and is approximately 7.0. One would then estimate the ratio of ''lift'' to longitudinal force to be about 15 to 20%. Waves in the ocean are three dimensional, having a direction- al wave spectrum, This presents a much more difficult problem than the case of an irregular system of two dimensional waves. Some work has been done in trying to analyse field studies made in the Gulf of Mexico by consortium of oil companies (Schoettle, 1962 ; Blank, 1969). The results of this attempt to analyze the field data to obtain informa- tion on "lift'' forces was unsuccessful (Abdel-Aal and Wiegel, 1971). CONCLUSIONS The Keulegan-Carpenter Number appears to te a useful para- meter to predict the ratio of "lift'' to longitudinal wave induced forces in a regular system of two dimensional waves. For the case of a ver- tical pile piercing the water surface, ''lift'’ forces start to occur for values of Nxc between 3 and 5, with the ratio of ''lift'' to longitu- dinal force increasing rapidly with increasing NxKc toa value of NKc of about 15 to 20. The ratio then remains about constant, or perhaps decreases to some extent with increasing Nxc. The value of the ratio of ''lift'' to longitudinal force, for Neco greater than about 20 appears to be similar to the ratio of Cy /Cp for one directional steady flow. For the higher values of Nec the "lift'' force becomes irre- gular and should be described by a distribution function, ACKNOW LEDGEMENTS The work presented herein was performed under Contract DACW-72-69-C-0001 between the Coastal Engineering Research Center, Corps of Engineers, U.S. Army, and the University of Cali- fornia. 788 Wave-Induced Eddtes and "Lift" Forces on Ctreular Cylinders REFERENCES ABDEL-AAL, Farouk, and WIEGEL, R.L., "Preliminary Report on an Analysis of Project II Data (Wave Forces ona Pile), Hurricane Carla, Gulf of Mexico", University of Cali- fornia, Berkeley, California, Hydraulic Engineering Labora- tory, Tech. Rept. HEL 9-18, June 1971, 40 pp. BACON, David L., and REID, Elliott, G., '!' The Resistance of Spheres in Wind Tunnels and in Air, U.S. NACA, Ninth Annual Report, Rept. No. 479, 1923, pp. 471-487. BIDDE, Devidas, ''Wave Forces ona Circular Pile Due to Eddy Shedding, Ph. D. thesis, Department of Civil Engineer- ing, University of California, Berkeley, California ; also, Tech. Rept. No, HEL 9-16, Hydraulic Engineering Laboratory June 1970, 141 pp. BIDDE, Devidas, Laboratory Study Lift Forces on Circular Piles, Jour, Waterways, Harbors and Coastal Engineering Division, Proc. ASCE, Vol. 97, No. WW4, November 1971, pp. 595-614, BISHOP, R.E.D,, and HASSAN, A.Y., ''The Lift and Drag Forces on a Circular Cylinder in a Flowing Fluid, Proc. Roy. soc. (Lendon), Ser. A,; vol. 277, 1964, pp. 32-50. BLANK, L.S., ''Wave Project II, Users Guide, '' Chevron Oil Field Research Company, La Habra, California, 23 May 1969. CHANG, K.S., ''Transverse Forces on Cylinders Due to Vor- tex Sheding in Waves", M.S. Thesis, Mass. Inst. Tech. ; January 1964, 94 pp. GLASER, G.H., ''Determination of Source Coordinates of Particles in Water by Stereophotogrammetry"', M.S. Thesis, Dept. of Civil Engineering, University of California, Berkeley, California, 1966, HUMPHREYS, John S., "Ona Circular Cylinder ina Steady Wind at Transition Reynolds Number", Journ. Fluid Mecha- nics, Voll 9, Part 4, °1960,) pp. 603-62. 789 10 11 1Z L3 14 15 16 17 18 1g Wiegel and Delmonte KEULEGAN, Garbis H., and CARPENTER, Lloyd H., "Forces on Cylinders and Plates in an Oscillating Fluid," Journal of Research of the National Bureau of Standards, Vol. 60, No. 5, May 1958, pp. 423-440. LAIRD, A.D.K., ''Eddy Forces on Rigid Cylinders", Jour. Waterways and Harbors Div., Proc. ASCE, vol. 87, No. WW4, November 1961, pp. 53-68. LAIRD, A.D.K., ''Water Forces on Flexible Oscillating Cy- linders", Jour. Waterways and Harbors Div., Proc. ASCE, Vol. 88, No. WW3, August 1962, pp. 125-137. LAIRD, A.D.K., ''Forces ona Flexible Pile''’, Proceedings, Speciality Conference on Coastal Engineering, Santa Barbara, California, October 1965, ASCE, 1965, pp. 249-268. LAIRD, A.D.K., ''Eddy Formation Behind Circular Cylinders, "' Jour. Hydraulics Div., Proc. ASCE, Vol. 97, No. HY 6, June 1971, pp. 763-775. LAIRD, A.D.K., JOHNSON, C.A., and WALKER, R.W., "Water Eddy Forces on Oscillating Cylinders, Jour. Hydrau- lics Div., Proc. ASCE, vol. 86, No. HY 9, Proc. Paper 2652, November 1960, pp. 43-54. LAIRD, A.D. K., and WARREN, R.P., "Groups of Vertical Cylinders Oscillating in Water", Jour. Eng. Mechanics Div., Proc. ASCE, vol. 89, No. EM 1, Proc. Paper 3422, Februa- ry 1963, pp. 25-35. McNOWN, J.S., and KEULEGAN, G.H., ''Vortex Formation and Resistance in Periodic Motion", Jour. Engineering Mecha- nics Division, Proc. ASCE, vol. 85, No. EM 1, January 1959, pp. 1-6. MORISON, J. R.,.O'BRIEN,.M. P.., JOHNSON, J. W...ame Schaaf, S,A., ''The Force Exerted by Surface Waves on Piles," Petroleum Transactions, Vol. 189, TP 2846, 1950, pp. 149- 154, PAAPE, A., and BREUSERS, H.N.C., ''The Influence of Pile Dimensions on Forces Exerted by Waves", Proc. Tenth Conf, on Coastal Engineering, Vol. II, ASCE, 1967, pp. 840- 847. 790 20 21 22 23 24 25 26 27 28 29 Wave-Indueced Eddies and "Lift" Forces on Circular Cylinders PRICE, Peter, ''Suppression of the Fluid-Induced Vibration of ‘Circular Cylinders", Jour, Ene, Mech, Div., Proc. ASCE, vol, o2, No, &M 3, Proc, Paper 1030, July, 1956, 22 pp. ROSHKO, A., ''On the Development of Turbulent Wakes From Vortex Streets", U.S, National Advisory Committee for Aeronautics, Tech. Rept. 1191, 1954, pp. 801-825, ROSHKO, A., ''Experiments on the Flow Pasta Circular Cylinder at Very High Reynolds Numbers", Jour, Fluid Me- chanics, vol. 10, Part 3, may 1961, pp. 345-356. ROSS, Culbertson, W., "Large-Scale Tests of Wave Forces on Piling (Preliminary Report), U.S. Army Corps of Engineers, Beach Erosion Board, Tech. Memo. No, 111, May 1959. ROUSE, HUNTER, ''Elementary Fluid Mechanics, John Wiley & Sons, Inc., 1946, 376 pp. ROUSE, HUNTER, ' On the Role of Eddies in Fluid Motion", American Scientist, Vol. 51, No. 3, September 1963, pp. 285-314, SARPKAYA,TURGET, ''Lift, Drag and Added-Mass Coeffi- cients for a Circular Cylinder Immersed in a Time-Dependent Flow", Jour, Applied Mechanics, vol. 30, Series E, No. -1, March 1963, pp. 13-15. SARPKAYA, TURGET, and GARRISON, C.J., ''Vortex Forma- tion and Resistance in Unsteady Flow", Jour. Applied Mecha- nics, Vol. 30, Series E, No. 1, March 1963, pp. 16-24. SCHOETTLE, V., ''Design, Construction and Installation of Instrumentation for Wave Project II'', Research Report 744, California Research Corporation, 1962. WIEGEL, Robert L., ''Oceanographical Engineering}' Prentice- Haliiy ine: 196457 532 pp, 36 OG ry Ie 791 Wiegel and Delmonte DISCUSSION Choung M. Lee Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. Perhaps I should direct this question to Keulegan anc Carpenter themselves, but since Professor Wiegel seems to have much expe- rience in Keulegan number, I would like to ask one question. Keulegan number is formed by the maximum fluid velocity which is multiplied by the period of the flow and divided by the diameter of the circular cy- linder, For a periodic fluid motion the maximum velocity can be simp- ly converted into a product of the amplitude and the frequency which is equal to 2m times the amplitude over the period. Then the Keulegan number becomes the ratio of the amplitude of the oscillation of fluid to the diameter of the cylinder. This means that the Keulegan number is independent of frequency. It is a puzzle to me that all the forces act- ing on the circular cylinder only depend on the ratio of the amplitude of oscillation of fluid to the diameter of the circular cylinder. I believe that there must be a physical reason for such a phenomenon, I would appreciate Professor Wiegel's comment on this remark, REPLY TO DISCUSSION _ Robert L. Wiegel Untverstty Of Caltfornta Berkeley, Caltfornta, U.S.A. One can make use of the wave equation, calculate the actual excursion of the water particle motion, and what one comes up with is a ratio of the distance that the water particle moves to the diameter of the pile. If you have water waves it is very simple to show that the controlling number there is probably only the ratio of wave height to the diameter, You are correct. I hope I put this in my Paper ; if not, I should have, because it is not really the time, it is the distance along the boundary that the fluid must move in order for the boundary layer to form and for the vortices to form, and so forth. So the time is really only used to calculate how far it has moved. I hope that has answered your question and I see that you indicate that it has. te ANALYSES OF MULTIPLE-FLOAT-SUPPORTED PLATFORMS IN WAVES C.H. Kim and J.A. Mercier Stevens Instttute of Technology Davidson Laboratory Castle Potnt Statton Hoboken, New Jersey, U.S.A. ABSTRACT Results of several studies of the behavior of floating platforms in waves are presented. The contents of these studies may be briefly summarized : MOTIONS RESPONSE OF A RELATED SERIES OF THREE AND FOUR FLOAT PLATFORMS Transfer functions relating heave and pitch motions to incident waves have been used together with assum- ed (Pierson- Moskowitz) wave spectra to derive di- mensionless spectral response information for these motions in waves as a function of jacob iA Oa height divided by (displaced volume) By The depend- ence on float slenderness, damping plate (or footing) size and metacentric height are presented in Figures and Tables. SOME PROBLEMS OF THE HYDRODYNAMIC INTER- ACTION BETWEEN TWO FLOATING BODIES IN BEAM SEAS An analysis is given of both wave- and motion-induc- ed forces and moments on the individual bodies ofa rigidly - connected twin-cylindrical body floatingin beam seas. The influence of the hydrodynamic inter- action effect on the sway-exciting force was found to be quite remarkable. The diffracted and radiated waves were evaluated and relatedto wave and damping forces and moments. Numerical calculations were carried out of some hydrodynamic characteristics of interaction between two different cylindrical bodies floating in proximity in beam seas. Kim and Mercter SLENDER VERTICAL FLOATS ISOLATED PERFOR- MANCE AND INTERACTION EFFECTS FOR LARGE ARRAYS Relatively simple analytical procedures provide sa- tisfactory description of the heave response of slender vertical floats to waves. It is found that viscous ef- fects must account for most of the heave damping for such slender floats and that it is advisable to introduce specially-engineered damping devices to control re- sonant heaving motions. Force and motion studies of alarge array (210 ele- ments) of slender floats reveal important interaction effects on wave-induced forces and,especially, heave motions. A satisfactory theoretical explanation of these effects is not yet available but a continuing em- pirical investigation is planned. INTRODUCTION This paper presents results for a somewhat ''mixed-bag" of studies of the hydrodynamic performance of floating platforms in waves. The nature of these investigations range from systematic ex- perimental determinations of the performance of multiple float sup- ported platforms in irregular seas to rather sophisticated theoretical evaluations of the behavior of two cylindrical bodies in beam seas and to a combined theoretical and experimental study of an unusual con- cept for a large expandable floating platform. The several studies are related because they all deal with floating configurations with zero nominal speed through the water. Two of the studies deal with the important problems of hydrodynamic inter- actions, although the nature and extent, as well as the modes of in- vestigations of the interactions, differ significantly. These studies represent a significant portion of the extensive research and development work which has been done on ocean plat- form behavior at Davidson Laboratory in recent years. They do not relate specifically to ad hoc testing and evaluation of drilling plat- forms and similar craft, but it is hoped (and anticipated) that the find- ings reported will prove to be useful for practical design purposes. 794 Analyses of Multtple-Float-Supported Platforms in Waves Each of the studies is prefaced by a section giving the back- ground of the work done. A Brief summary of the investigations has been given in the Abstract of the paper. Nomenclature for the three parts of the paper is, unavoidable, not completely uniform. The listings of symbols and their meanings, in the Nomenclature section, are separated into three parts, one for each of the studies. The original model test program of the systematically-relat- ed float supported platforms was supported by the U.S. Naval Air Systems Command. Further analyses of the original data have been carried out by Davidson Laboratory. The work on the theoretical evaluations of hydrodynamic be- havior of two bodies in proximity was sponsored by the National Oceanic and Atmospheric Agencies Sea Grant Program Office as part of a continuing effort to improve prediction procedures for platform motions. The study of large arrays of slender vertical floats was under- taken as a sub-contract to Goodyear Aerospace Corporation on a Pro- ject which is being technically monitored by the Office of Naval Re- search Ocean Technology Branch. 195 Ktm and Mercter MOTIONS RESPONSE OF A RELATED SERIES OF THREE AND FOUR FLOAT PLATFORMS BACKGROUND In the early 1960's a substantial program of research and de- sign analyses was undertaken on the concept of using slender vertical floats to support water-based aircraft, thus providing a comfortable and stable environment for crew and equipment. This work was sup- ported by the United States Naval Air Systems Command under the cognizance of E.H.Handler, whose review article on ''Tilt and Vertic- al Float Aircraft for Open Ocean Operations'' describes the progress of this concept up to 1966. Two early studies of particular significance established the validity of the concept. In one of these studies two PBM seaplanes [2] were tested in five-to-eight-foot high waves. One of these craft was supported avobe the water surface by vertical floats and was found to be quite comfortable while the crew of the convention- al PBM soon became uncomfortable. The other investigation involved a one-man helicopter which was equipped with tiltable floats [3] and tested in heavy, choppy waves, reacting with slow and gentle motions. Earlier model tests of this vehicle at Davidson Laboratory [4] led to the introduction of horizontal plates at the lower end of the vertical floats, which are found to be a vital feature in the effective employ- ment of this concept. In order to provide systematic design information for the dy- namic performance in waves of these types of craft, results of a series of vertical float configurations which have been model tested in ir- regular waves to determine their response characteristics have been described by Mercier [5] .» Variations in the series include number of floats, float spacing and slenderness, size of damping plate and meta- centric height, and heading of the vehicle relative to the waves. The information developed may also be useful for other types of craft, such as work platforms, research and range-tracking platforms, oil- drilling rigs and possibly buoys, which can be configured with vertic- al floats. Float supported platforms may be envisioned with any number * Superior numbers in text matter refer to similarly numbered references listed at the end of this paper. 796 Analyses of Multtple-Float-Supported Platforms in Waves of floats from one up. Single float platforms, such as the SPAR and FLIP research vehicles will not, however, be considered here. Only three and four-float supported platforms, applicable to air-sea craft, are treated in this report. The four-float configurations are arranged so that two of the floats support the main hull, and carry most of the weight while the other two floats are attached to the wings, producing a cruciform array. For the three-float configurations, all floats are of equal size in an equilateral triangular array. The total displaced volume of the floats must equal the vehicle’ Ss weight and static pitch and roll stability must be provided. Static stability criteria usually follow naval architectural practice, measur- ing stability by metacentric height. From this consideration then, the float slenderness (waterplane area) and spacing are related to the ver- tical center of gravity of the craft. Descriptions of the elementary fundamentals of the dynamic motions response of vertical float and ''semi-submersible'' platforms have been outlined by Barr [6 | , Julien and Carrive [7] , and most recently and thoroughly by Hooft [8]. Considerations both of natural frequencies and of minimization of wave excitation forces and/or moments are of key importance in this regard. A more complete quan- titative description of these considerations for heaving motions of isolated floats is given in the present paper as part of the discussion of the performance of closely-packed arrays of floats. DESCRIPTIONS OF MODELS Tests in irregular seas were conducted with variations of two basic model configurations. One represents a cruciform array of floats where two equal floats would attach to the aircraft fuselage and two to the wings. This type of system was investigated by Ling-Temco- Vought [9] for application to the XC-124A VTOL aircraft and by General Dynamics/ Convair [10] for application to the P5A seaplane equipped with hydro-skis. The second configuration uses three equal floats in a triangular pattern. Such an arrangement has been planned by Boeing- Vertol (4 1] for application to the CH-46A helicopter. Sketches illustrating the two basic configurations tested are shown in Figures 1 and 2. The basic frames, to which the floats are attached, are made of aluminum in a lightweight but rigid construc- tion. The floats, which are interchangeable and have adjustable spac-~ ings, are made of low density (2 lbs per cu ft) closed-cell sytrofoam with 1/8-in thick aliminum plates on top to facilitate securing to the frame. All floats used have a circular cylindrical body shape with a UCM Ktm and Mereter hemispherical end. Damping plates are attached, interchangeably, to the lower end of the float in way of the junction of the cylindrical float body and the hemispherical cap. These circular plates were made of 1/16-in plexiglass with a 60-deg included angle bevel on the periphery. Model weight was selected to be 20-lb, resulting in model size suitable for the available apparatus. The location of the gimbal-box is such that the center-of-rotat- ion corresponds to the center-of-gravity of the model for a particular "average' set of floats and damping plates. It was felt that the influ- ence on vertical center-of-gravity due to interchanging floats and damp- ing collars would be negligible, being less than about 1/8 inch. Floats are designed so that the height of the gimbal-center is a constant dist- ance above the stillwater level for all the float variants tested. The model moments of inertia, about the pitch and roll axes, were fixed using the ''average'' sets of floats and damping plates. While changes in floats and float spacings may alter the model's pitch and roll inertias by amounts up to about 3 per cent, the small changes may be expected to approximate those which would be associated with corres- ponding design modifications for full-scale vehicles. The geometric characteristics of the models are given in Table 1 for four-float (cruciform) configurations, and in Table 2 for three-float (triangular) configurations. Also shown in these tables are metacentric heights as computed according to the model design draw- ings and as measured in separate tests. The computed and measured values do not agree especially well, which is probably attributable to small discrepancies in the manufacture of the styrofoam floats, which would have an important effect on the waterplane inertias and, conse- quently, on GM (especially for the low values of GM used here). For a few cases where the GM was not measured, estimated values are enter- ed in the table, and these are distinguished by being enclosed in par- entheses. The experiments were carried out in Davidson Laboratory Tank No. 2 which is 75-ft square with a water depth of 4.5 ft. The motions apparatus used to follow the model's motions and provide a means of recording the six components of motion with a minimum of interference has been described by Numata [12] in an appendix to his paper on "Hydrodynamic Model Tests of Offshore Drilling Structures.'' Tests were carried out in irregular waves with a significant height of 2 inches. RESULTS Figure 3 shows a sample oscillogram taken during a test run. 798 Analyses of Multtple-Float-Supported Platforms tn Waves The information obtained from the tests of twenty-five variations in the configuration indicated in Tables 1 and 2 have been analyzed and are presented in two different ways. The transfer functions, relating waves and motion, are present- ed in Reference 5 in a series of Tables, such as Table 3 given here, as a function of the non-dimensional frequency parameter, d=Nv1/3/¢ where w = frequency, V = average displaced volume of the vehicle, and g is the acceleration of gravity. The displaced volume has been arbitrarily selected as a useful measure of size for non-dimensionaliz - ing. These were derived by spectral analysis of the irregular sea mod- el data using methods similar to those described in detail by Dalzell and Yamanouchi (1 3] . The sampling interval used for data analysis, 0.25 sec, was sufficiently low to assure unambiguous spectra, that is, no "aliasing." Only those spectral results for which the answers appear reas- onable are included in the tabulations. Generally, the measure of reas- onableness is taken to be the coherency and it is preferred that this quantity should have a value of 0.8, or greater. In a few cases, and in some frequency ranges, the test results do not give such satisfactory values of coherency and a judgment concerning the adequacy of a sample of data has been made by a subjective interpretation of graphs of am- plitude and phase versus frequency parameters. Examples of this type of graph are shown in Figure 4 for the results of Test Run 002 (the oscillogram for this run is shown in Figure 3). In this case the range of frequencies for which the coherencies may be considered good is rather wide, especially for the wave-heave correlation. In Figure 4 and in the tables, the phase angles are positive when the motion lags the passage of the wave trough by the craft's center- of-gravity. Positive vehicle motions are taken as : surge forward, heave downwards, sway to starboard, pitch bow-up, roll starboard down and yaw bow to starboard, while the maximum positive wave el- evation corresponds to a wave trough. Beam seas tests were conduct- ed with waves approaching the craft from the starboard side while head seas tests, of course, have waves approaching from the bow. The non-dimensional angular motion is presented as the ratio of motion (pitch or roll) to the maximum wave slope, as expressed by linear wave theory, ae This is felt to offer some advantage in interpolating the transfer function data for low frequencies - it being realized that for zero frequency (static conditions) the craft must as- sume the slope of the wave, and the amplitude ratio must then be unity. The phase of this motion is, however, reckoned relative to the wave amplitude measurement, and must correspond to +90° (i.e., phase 199 Kim and Mereter lag) for | =)0, Impulse response functions were obtained as the Fourier trans- forms of the transfer functions in accordance with the theory first in- troduced in ship motions by Fuchs and MacCamy [14] , and more fully developed by Cummins [15] . Examples, again for Test Run 002, are shown in Figure 5. Tables, such as Table 4 given here, are given in Reference 5 for the twenty-five configurations tested. A comparison of measured and calculated heave and pitch motions for a cruciform model in head seas is given in Figure 6. The computations were carri- ed out by the convolution integrals z(t) = sft k (FSET) a and where Kk and ky are from Table 4. DISCUSSION Transfer Functions A comparison of the results of several test runs indicates the influence of variations in the significant parameters. The transfer functions, particularly the amplitude ratio, afford a convenient graph- ical indication of the performance. Heave responses for several cruciform models of Group A (Table 1) are shown in Figure 7. The amplitude ratios are plotted against the non-dimensional frequency parameter, @ = AGNaTS .. AD auxiliary chart is arranged beneath the frequency parameter, in which the wave length corresponding to a particular craft displacement (long tons of sea water) and frequency parameter can be read. The displac- ement range covered extends both above and below the usual aircraft range, in case other types of platforms are to be considered. The heave-wave relations shown in Figure 7 all have similar shapes, having virtually no magnification or motion greater than wave elevation, and a sharp fall-off in response above a certain frequency. The damping plate size (which can be considered to be similar to the ''footings'' fitt- ed to some drill rigs) appears to have the most influence on the heave response for these configurations, but it must be noted that greater variations in geometrical parameters could be adopted. 800 Analyses of Multtple-Float-Supported Platforms in Waves Pitch response for the models of Group A is given in Figure 8, The auxiliary wave-length-displacement chart is also shown beneath this figure. An additional scale of wave length/V¥/3=»/y 1/3 is added to this Figure, which related to angular motions of the vehicle. This is incorporated for these cases because the wave exciting moment must be expected to be crucially dependent on the ratio of wave length to float spacing. If the wave forces which act on the floats were purely vertical, for instance, as is considered to be the case in the strip theory of ship motions, it would be found that when the wave length equalled the hull float spacing, the exciting moment would vanish since wave crests would occur at each of the ghia simultaneously, etc. Table 1 shows the hull float spacing/V to be around 4, while the response curves of Figure 8 indicate practically null motion for \/V 1/3 in the range of 5.5 to 7. The apparent discrepancy is, of course, ex- plained by noting that horizontal wave exciting forces on the floats contribute greatly to the moment about the center-of-gravity, which is quite high above the still waterline. Short-term Statistics While the transfer functions are interesting for making qualitat- ive comparisons, especially because they are familiar, quantitative judgments of performance in realistic irregular waves require consi- deration of the spectra of the irregular response. For purposes of design and analysis, particular waves must be considered. The 12th International Towing Tank Conference in Rome [1 6] adopted the following recommendation relative to wave spectra. "The Conference recommends that wherever possible, use shall be made of information on wave conditions for the ship's service in presenting predicted ship behavior in waves. When information on typical sea spectra is not available, it is recommended that the follow- ing sea spectral formulation shall be used as an interim standard : 4 Nee Soy ese Ee g we 2 ey = spectral density, ft -sec w = circular frequency, rad/sec (2) "If the only information available is significant wave height, H ft ue -3 2 2 -4 AOE Se oaiOlixe LOM Morse" \G) 3 Crit sec 33.56/04. 2) w M 1/3 801 Ktm and Mercter If statistical information is available on characteristic wave period and significant wave height bs Eo 4 A = Ep) /T, 4 B = 691/T, where T, = 2m m,/m, cos variance or area under spectrum, ft 2 mj, = first moment of spectrum about w = 0 axis, ft /sec Data suggest that T, can be taken as the observed characteristic period. The Seakeeping Committee should study further the use of the proposed two parameter spectral formulation." This formulation is based on work of Pierson and Moskowitz.[17] The statistics of the response are related to the area under the response spectrum, where, assuming that the amplitudes of response records are narrow-banded and follow Rayleigh distributions, the average apparent response (crest-to-trough) is ade VE (3) the average of the one-third highest responses, or ''significant'' res- ponse, is (x) = 4.0 VE (4) 1/3 x and the average of the one-tenth highest responses is 5.1VE, (5) 740 where = : 2 E.. = fla, (e) S,(w)dw (6) is the variance, or area under the spectral density curve, a, (w) is the frequency response function (transfer function) describing response inthe xth mode of motion to a sinusoidal wave excitation, and S,(w) is the input (wave) spectral density. It is possible to express responses to the ITTC (or Pierson- Moskowitz) sea spectra in dimensionless form, applicable to any size 802 Analyses of Multtple-Float-Supported Platforms in Waves of configuration. Thus, if @_(@) is a non-dimensional ratio of res- ponse, x, to wave elevation, ¢, a non-dimensional variance may be obtained as i ee) ~ By ee et Oe T ae) E = = fie (3)| = 6 da x y2/3 % 5 fe) @ where the spectral formulation with only significant wave height being available was chosen so that only one parameter ( vl */Hy/3) would describe the relative intensity of the sea state. Since wave elevation has dimensions of length, the response it- self must have the same dimensions. This is, of course, the typical case for heave or surge but other examples of non-dimensional res- ponse are : avil§ Syl Gs Eten Ib for pitch angle (8a) ‘h vi/3 Grete ce for acceleration (8b) A g ¢ a al for relative motion (8c) =A 1/3 Av a. = 2 for relative velocity (8d) M GLA = for bending moment 8e) M peVve 3 pet gece for heave (8f) Z ¢ and so forth. The dimensionless presentation of spectral response was, to the authors knowledge, first introduced by Bennet [18] in 1966, who retained the two parameter (significant height and period) description of the irregular seas. This very useful form of presentation has been adopted by Lindgren and Williams [19] » among others, and deserves wider application. Fridsma [20] has exploited the useful scaling char- acteristics of the ITTC (Pierson-Moskowitz) spectra to present dimen- sionless results of motion and acceleration measurements carried out with planing model hulls in waves whose spectra corresponds with the ITTC recommendation. Thus, information for these highly non-linear responses can be given for craft of various sizes in realistic irregul- ar seas. 803 Kim and Mereter The transfer functions previously presented by Mercier [5] have now been used to obtain spectral information on performance in irregular waves in accordance with Equation (7). Carpet plots showing significant (average of the one-third highest) heave and pitch respons - es as a function of significant wave height divided by (displaced vol- ume)¥/3 and other parameters are shown in Figures 9 to 13. The de- vice of carpet-plotting, which is common in aeronautical research and testing, permits cross plots to be presented on a single sheet of paper, and greatly facilitates performance comparisons and exhibit- ing dependencies of results on various parameters. The effect of damping plate size on heave and pitch for a cru- ciform array of floats (test runs 002,012 and 018, Table 1) are shown in Figure 9. The same information is given in Figure 10 for platforms with a different distribution of displacement among the ''hull'' and "wing'' floats ; hull float diam/wing float diam = 1.25 (rather than 1.5 for the results shown in Figure 9) (test runs 060,058 and 062, Table 1). In both cases, larger damping plates reduce heave response but increase pitch response. For three float configurations, having some- what more slender floats, the corresponding results are given in Figure 11 (test runs 074,091, and 087) ; the influence on heave mo- tions is similar but for pitching motions with low relative sea condi- tions, Hy 3/ vi 3< 0.7, a suitable selection of damping plate size ap- pears to lead to minimum pitch motions. The effect of float slenderness on heave motions is exhibited in Fig. 12 for the cruciform (test runs 002,047 and 032) and triangular (test runs 074,096 and 083) float array models, both with damping plate diam/float diam = 1.6. Since the pitch response (but not the heave response) is strongly dependent on metacentric height, and since the tests with variations of float slenderness were not executed with uni- form values of GM, the influence of float slenderness alone on pitch motions is not presented. It is interesting to note the distinctly differ- ent trends of heave response as a-function of float draft-to-diameter ratio ; for the four-float configuration it appears that a maximum oc- curs for T/Dp between 2 and 2.5, while for the three-float array a minimum exists around T/Dr = 5. These different behaviors may be due to the different arrangements of floatation (three floats of four) as well as the different ranges of slendernesses investigated. The influence of metacentric height on heave and pitch response is shown in Fig.13 for a four-float configuration (test runs 002,005 and 006). As was stated above, the heave motion is not importantly affect- ed by GM. On the other hand, the pitching motion is greatly affected by stability with excessively low metacentric height giving rise to large angular motions which are not characteristic of well-designed float- supported-platforms. 804 _— ee ee Analyses of Multiple-Float-Supported Platforms in Waves The results have been given for a range of dimensionless sea states Hy/3/V* 3 upto 1.0, so that for a craft of 30000 long tons displacement the corresponding significant wave height is 102 feet. This corresponds to a high Sea State 9 and, according to the recom- mendations of the ITTC (16] , the approximate wind speed in the open ocean would be well above 80 knots. For a 10-ton craft, however, the significant height would be 7.05 feet, or Sea State 4. Dimensionless statistical response data are presented in tabul- ar form for all of the model tests reported by Mercier [5] , in Tables da-oy. Large values of Hy/3/v" 3 soupato. 45 0) (Hay ei 7o it forve 3000-ton craft), are covered in these tables but because the transfer functions derived from the basic tests are not precisely defined at low frequencies, which are ae excited by extremely high sea states, the results above Hj/3 1,0 are somewhat less reliable. This is especially true for cy pitch motions which have very low but un- defined eens frequencies. The transfer functions were extrapolated "by eye. 'to lower frequencies to permit these calculations for 1.0 < Hy/3/V° 3<4.0 to be undertaken. Extreme Response Ochi [2 1] has adapted order statistics with a given probability of being exceeded, for evaluating long-term statistical behavior which may be more appropriate for evaluating limiting sea state operating conditions. His equation, which is applicable to Rayleigh probability distributions, may be expressed non-dimensionally as a ~ 1 a = 4-2 EB fn {1-(4-Prob) INo} (9) 1/8) x V where & = extreme value (amplitude) (for Wacike : Lee Lave pitch, vi ; acceleration, (A/g) Siar — &60U f g ~ N, Cae Time g/3 B,/E. Time = Duration time in hours Prob = Probability of computed value, & , being exceeded Be = Dimensionless response variance, Eq. (6) in = Dimensionless response velocity variance x 2 33 >) f = [2 joo (ay] SORE Wher) 1/3! as (10) (69) ° 805 Kim and Mercter For example, if a probability of being exceeded is 0.1, the expression implies that one platform in ten floating in the same area or statistical environment may experience a larger response than the estimated ex- treme value (&) during the specified duration. Because this type of response information is of interest for certain investigations the second moment of the response spectra, B have been calculated according to Equation (10) and included in Tables 5. It may be appropriate to note that a recent study by Dalzell [22] of the statistics of responses of systems with non-linearities suggests that applications which assume the Rayleigh distribution and involve order statistics and/or the estimation of extremes tend to over-predict. Since most platforms are subject to some extent of non-linear damping, at least for extreme sea states, Dalzell's observations should be borne in mind. 3 SOME PROBLEMS OF THE HYDRODYNAMIC INTERACTION BETWEEN TWO FLOATING BODIES IN BEAM SEAS BACKGROUND Ohsusu cp 24] has theoretically evaluated the hydrodynamic forces and moments on two or more cylinders heaving, swaying and rolling on a calm water surface. Ohkusu and Takaki [25] have applied these analyses to evaluate the motions of multi-hull ships in waves. Wang and Wahab [26] have also studied the hydrodynamic forces on twin heaving cylinders on a calm water surface. These investigators used the method of multipole expansion to determine the unknown velocity potential. In these analyses, the in- dividual section must be symmetrical about its own vertical midplane and the two cylinders must be identical. Lee, Jones and Bedel [2 7] have reported theoretical and ex- perimental evaluations of the hydrodynamic forces on twin heaving cylinders on a calm water surface. Their theoretical analysis follow- ed the method of source distribution over the immersed contours [28] of the cylinders. Consequently, cylinders of arbitrary cross-section, not necessarily symmetric about their vertical midplanes can be dealt with. This investigation also assumed the two cylinders to be sym- metrically disposed with respect to each other. All the above investigations [23-2 7] of multihull cylinders as- sumed that the cylinders are rigidly connected. 806 Analyses of Multiple-Float-Supported Platforms tn Waves The present study also applies the close-fit source distribu- tion method pioneered by Frank [28] . We assume two arbitrarily shaped bodies, which do not have to be either symmetrical about their vertical midplanes or to be symmetrically disposed with respect to each other. Furthermore, the two bodies may be unconnected or con- nected either rigidly or elastically. This investigation, based on two-dimensional linearized theory, considers both the radiation and diffraction problems for two arbi- trarily shaped cylindrical bodies floating in a train of beam waves ; the hydrodynamic inertial and damping forces and moments due to swaying, heaving and rolling of the cylinders on a calm water surface and the forces and moments induced by beam waves on the fixed cyl- inders are evaluated. Brief descriptions will be given of the methods used to evaluate the unknown velocity potentials for the radiation and diffraction pro- blems. Since the fundamental velocity potential of a two-dimensional pulsating source of unit intensity located below the free surface and satisfying the required hydrodynamic conditions inside and on the boundaries of the entire deep-water domain is a well-known solution stated by Wehausen and Laitone [29] , it is only necessary to discuss here the kinematical boundary conditions on the body contours. Two applications of the theory will be considered in detail : 1) the analysis of the hydrodynamic forces on two rigidly connected cylinders and, 2) the relative heaving motions of two unconnected bodies in close proximity. Results of calculations for these cases will be presented and discussed. The case of a pair of three-dimensional, vertical body-of-revolution floats, in close proximity will be discuss- ed on the basis of experimental results. KINEMATIC BOUNDARY CONDITIONS The Radiation Problem Consider two arbitrarily shaped parallel cylinders oscillating in prescribed (arbitrary) modes of motion, with given amplitudes and phases, on or below the calm water surface in the form (m_) a [s]_ SS) a: (11) ==) ees ion " (op) co} 807 Kim and Merceter where acd : Sie = amplitudes of displacement in the mode of motion m, and My » respectively (m, or my, = 2,3,4 correspond to sway, heave, roll) Eg Sp = phase difference between the motions of bodies a a and b For certain kinds of problems such as the relative heaving motions between adjacent bodies, it may be convenient to refer the phases of the motions to, for instance, the wave-exciting force. The space coordinate system is defined in Figure 14a ; the y-axis lies on a calm water surface, the z-axis points vertically up- ward and the origin O is taken at the midpoint between the two walls of a oand’b. The body contours C, and C, of a and b are approximat- ely represented by polygons with a finite number of segments. A puls- ating source of unknown strength is uniformly distributed on each se- gment to represent the flow induced by the motions of the two bodies. The velocity potential for the source of unit strength at (n,§) may be written [29] : ot -i Gly,z37,$)e (12) where (y,z) is the coordinate of a field point. The resultant velocity potential is represented as a sum of all of the discrete source se- gments of the polygonal approximation to the contours C, and Ci is (mg, Mp» €S,Sp) > e(y,2) = ££, Q, G(y,z37, ¢)dS — al M Fa Oe footy.2in, Das (13) Sky where Te is Sky, = i and re polygonal segments of a,b, respect- ively th ee Qi = (uniform) complex source intensity of the r and J kth polygonal segments of a,b, respectively The unknown source intesities Q are determined numerically, satisfying the kinematic boundary conditions, 808 Analyses of Multtple-Float-Supported Platforms tn Waves (m » Mp» €g S ) (m ) oe ‘a ab wi Pek rt ae) SS 2H jwu aa (m m ) € S ) (m ) -1€ o¢ ae eemeal Ly: b SaSb sn bp? Yb) -ieu (y»2,)€ (14) (m,) ; 2 ; 5 with uw, (y, Zz.) sina, for ms =cos Ot - 3 4 =O are + 7 (y,,cos a, +2 sin a) for body a, and similarly for body b. The normal velocity aoe in Eq.(14) is taken to be the velocity induced on the rao segment by all of the other segments. %, = the orientation of the segment 5, in accordance with Ref. 28. Two special cases of interest may be mentioned. If the bodies a and b are rigidly connected twin cylinders, Eq.(14) takes the form sau iaet (y,22,) = ~iwu (y,°2,) (m) Bead (Giles souk ayelze) (14a) while if body a is oscillated while body b is fixed, Eq. (14) reduces to ; Se (Y,2 Zz) on 0 (14b) The Diffraction Problem Consider an incident wave i Ee aie ee (15) where a = wave amplitude vy = wave number 809 Kim and Mercter and the time factor e7!“t is omitted in this and subsequent sections. This wave encounters the fixed arbitrarily shaped bodies and is dif- fracted. The velocity potential corresponding to the incident wave is ga tote Si ene ere (16) | w which can be expressed as odd and even function of y , Cop head tga, Zz ? x Bap e sinvy ge) he tem e” cosy (17) I w y (o) The odd function corresponds to the part of the flow which is asymmetric about the z-axis , while the even function 9 ©’ corres- ponds to the symmetric flow. The flows represented by the potentials go) and (e) are disturbed otal pa ag the bodies. The disturbed flows correspond- ing to and are described by diffraction potentials, denot- ed ep? 0) and ype » respectively. These potentials are represent- ed in the same form as the radiation potentials [Eq.(13)] , but with different source intensities, Q. The unknown source densities are determined by satisfying the boundary conditions ae (8) ay, 8) on y +23) whe on (y, 444) oe {8) 308) Tee meee ant RR! (ae where 6 = o or e, onthe straight-line segments representing the contours C, and C,. Thus there are two separate boundary con- ditions for the asymmetric and symmetric flows. 810 Analyses of Multtple-Float-Supported Platforms tn Waves HYDRODYNAMIC FORCES AND MOMENTS Relationship to Potentials The linearized hydrodynamic pressure is given by [ol = apse” (19) where, for the radiation problem (7) = Pieceyaaioy €SaSp) , and for the diffraction problem ¢'”? = glote) a er) . Using the single superscript notation for the radiation or diffraction problem, the hydro- dynamic forces and moments are given by Game (y) Bie = Joe a pi . bi Malia ae = i Dacre eee) (20) where C = body contour C, or CG. Non-dimensional hydrodynamic moment arms for the sway and heave components of force, taking account that the pressure is complex, p VY) = Dye Wd) te ip; Y » can be written gi) ie habe S. a ib ae an (Y) [Drea ri emennere if S. vgs i since dz wed ydy yeCitinGs ab ds vue He ar house As G (Y) ea eycly, PQ ous apron L Isls ae fo” ay (21) Ktm and Mereter where T = the maximum draft of the two bodies. The subscripts S,H,R indicate swaying and heaving forces and rolling moment, respectively. The hydrodynamic force has real and imaginery parts, for instance (22) In the radiation problem, the real part of the hydrodynamic force (moment) is called the inertia force (moment while the imaginary part corresponds to a hydrodynamic damping force (moment). In the dif- fraction problem, the complex force defines the amplitude and phase of the wave-exciting force relative to the passage of the crest of the incident wave by the origin 0. The hydrodynamic moment arms also consist of inertial and damping parts in the radiation problem. In the diffraction problem, however, the moment arms of the real and imag- inary components are identical because the force and moment maxima occur at the same instant. Force and Moment Coefficients Hydrodynamic interactions affect the pressure distributions and forces acting on the individual bodies. Certain components of these resultant forces and moments may be considered as internal forces for the evaluation of dynamic structural loading on rigidly con- nected cylinders. The hydrodynamic forces on individual bodies are useful and the individual bodies when they are elastically coupled or unconnected. Table 6 defines the various force and moment components, in accordance with Eq.(20), and their corresponding dimensionless coef- ficients. The first subscript denotes the mode of motion (in place of the parenthetical superscript used previously ; the forces, moments, and coefficients do, however, depend on the -details of the modes and phases of motions), while the second subscript denotes the components of force (moment). For example, aie = roll hydrodynamic inertial moment induced by swaying SR ; b : : motion of unit amplitude of displacement oNoy = the heave damping force induced by rolling motion of unit amplitude of displacement 812 Analyses of Multtple-Float-Supported Platforms tn Waves The hydrodynamic moment arms, according to Eqs.(21), are presented in Table 7. The subscript notation is analogous to that used for Table 6, so that, for example, Log indicates the moment arm for the hydrodynamic inertial sway force induced by sway motion. The roll moments may be expressed in terms of the heave and sway forces and their respective levers as indicated in Table 8. The wave-exciting forces and moments may be expressed in dimensionless form as shown in Table 9. The wave-induced moment can, of course, be represented in terms of the sway and heave forces and corresponding moment arms in accordance with Eq.(21), in a manner similar to the motion-induced moments (Table 8). The separation of wave forces into odd and even parts will be shown to be useful for the case of the catamaran-type ship. Rigidly Connected Twin Cylinders All of the above formulas may be applied for bodies a, b and at+b, regardless of whether the bodies are similar (some care and consistency must be used in choosing and using relevant dimensions, draft and beam, for use in the dimensionless coefficients if the bodies are not twins). Based on a limited number of numerical calculation of twin cylinders, it is observed that the hydrodynamic forces, moment and moment arms due to oscillatory motions are entirely identical in magnitude for bodies a and b (see Table 10). Consequently, the resultant hydrodynamic forces and moments may be given as shown in Tables 11 and 12, where the subscript (a+b) for the rigidly coupled bodies is omitted. Since the bodies are twin and the incident wave flow consists of part which are symmetric and asymmetric with respect to the vertical z-axis, the odd and even components of the wave-exciting forces and moments on bodies a and b are equal in magnitude, whereas some forces on a,b act in opposite directions. (Table 12) These relations which arise due to symmetry afford consider- able simplications for evaluations of catamaran-type configurations. They are exhibited graphically in Figures 1l4a-d. For convenience, the previously defined force and moment coef- ficients C, 6 (Tables 6,7) will be called the forces and moments. Figure 14a illustrates a typical force system, induced by the heaving motion of the twin bodies. The heaving motion induces both heaving and 813 Ktm and Mereter swaying forces, [Con] : [8,55] and [Cus] ; Le ] , respectively, on the individual bodies a and b. The swaying forces [Cus ; Lous on a and b are equal, opposite and collinear, hence, the resultant forces on the twin bodies (a+b) are only the heaving forces 2([Ciy], and 2[5411, (see Table 11). Figure 14b represents a typical force system induced by the swaying motion. This motion also induces both heaving and swaying forces on each body. The sway-induc- ed heaving forces [Cg,;] and [6,,] on a as wellas b set upa couple which contributes to the resultant rolling moments 2[Cop | x“ and 2 [oop | , (see Tables 8, 11). The sums of the swaying forces on the twin bodies (a+b) are equalto 2[Cgg], and 2[bo], . | which also contribute to the resultant rolling moments 2[Cop x a 2[5cp | , (Tables 8,11). Another typical force system is that induced by the rolling mo- tion, as illustrated in Figure 14c. The rolling motion induces the heav- ing forces [Cay! ; [Ppa and the swaying forces [Cas 5 Ling] on each body. The heaving forces set up a couple and hence contribute to the resultant rolling moments Cpr » ORR on the twin bodies ( atb) (Tables 8,11), The swaying forces on a and b are equal. Their re- sultants on (a+b) are equal to 2[Cag] : and 2[5n 5] q+» and also contribute to the rolling moments Cor a oaR (Tables 8,11). The non-dimensional expressions of the wave-exciting forces and moments are defined in Table 9. Referring to Figure 14d, first let us observe the typical wave-induced force system. The even and odd wave potentials induce both Way - and peeve ne aw Ee forces. The Sway-exciting forces on a,b [tg° ui ‘ [26°] are equal, Wage e and collinear, while the heave-exciting forces [4°], . (£6? h set up a couple, We see from the figure that the resultant roll-exciting moment and the resultant sway-exciting force are due only to the odd wave po- tential, whereas the resultant heave-exciting force is due only to the even wave potential. (See Table 13). The Radiated and Diffracted Waves We consider at first the evaluation of the radiated and diffract- ed waves generated from a rigidly connected twin-cylinder floating in a regular beam wave, The radiated and diffracted waves generated by a monohull cross section floating in a regular beam wave were evaluated in the previous work [30] . The radiated or the diffracted wave is the vector 814 Analyses of Multtple-Float-Supported Platforms tn Waves sum of the far field waves induced by the pulsating sources Q\Y) di- stributed on the sectional contours, where y = m (mode number) for the radiation, and y= 8(0+e) for the diffraction problem, The asymptotic expression of the velocity potential yg) at y> ae for both radiation (y= m) and diffraction (y=8) are identical in their forms f i(tvy Fela) Prt yam eiee? baa tes r (23) where * suffix refers to y too It is to be noted that o + denotes both the potential per unit amplitude of displacement in forced oscillation for y= and that per unit amplitude of the incident wave for y=6 , AMY and a (7 are evaluat- ed in a fashion similar to that given in Ref. 30 where the arbitrarily shaped geometry and both symmetric and asymmetric flow conditions will require that the term with (-1)™ in the formula Eq. (25) of the above reference should be taken as zero. Hence, (Y) (y)? (7)2 — = Ce + De (Y) hays eee oe = tan (y) ae Gn ae @ cra ae sharhalle tht bite Bio l? £Q)" by - Qn Ks s d Owe Ore Ki (24) Ko ree cos(yn .,, +a.) A cos(vn, + 4.) 1h) Kim and Mercter vt. ; x peiltrs’. Nut L. =e sin(v7,,, +a.) -e ° sin(vn, +a.) j Frans j where a,b = the suffixes indicating the terms of the bodies a and b, respectively (a) cohiis Q; » Qn; = the real and imaginary parts of the complex source strength, uniformly distributed over the elementary j segment of bodies a or b (n.» ¢.),ete = the coordinates of the end points of the jh Jou 4 segments 7 = the slope of the jh segment The far field wave ne” is derived from the far field potential v\Y) (Eq.23) as 5 t (7) (Y) iA'y ifkeg:t-e.-—} (25) Aas MR Let the complex amplitude ratio be Y ; ay!) Fi et | ) a Gate es ee i eon (26) = i) = The energy conservation law leads to the well known relation between the hydrodynamic damping coefficient N ™/ and the radiated wave amplitude ratio vay 2 2 3 + = z where + = suffix indicating the radiated wave at y-—+teo or - oo e It is to be noted that La! | has the dimension of length ; in other words Ay | is nota non-dimensional, whereas the Jat? | ,|as?)| are. 816 Analyses of Multtple-Float-Supported Platforms in Waves By the Haskind-method [3 1, 32] , one can also evaluate the wave-exciting forces and moments by the radiated wave amplitudes ; for the wave progressing to the positive end of the y-axis (Eq.15), the wave-exciting forces are _ As Es in vB Tak oa H vB ; ¥ visual 22d R vBT (28) It is bo be noted that the wave-exciting forces and moments presented in Eq. (28) are the resultant values. In other words, we see by the Haskind-method that the exciting forces and moments on indi- vidual bodies a or b cannot be evaluated. According to the theories by Haskind [33] and Maruo [34] Sane mean wave force K is determined by the diffracted wave 2 a ls (ote )| (29) NUMERICAL CALCULATIONS Added Mass and Damping Coefficients for Twin Cylinders Figures 15, 16 and 17 exhibit the various hydrodynamic forces and moments on one of a pair of half-immersed circular cylindrical bodies as functions of the frequency parameter ae . The definitions of the various forces on this Body a are given in Tables 6, 7 and 8. The hydrodynamic forces on Body b are evaluated by employing the formulas in Table 10 with the aid of the information given in Fig. lo, 16 and 17, The resultant forces and moments on the twin Bodies (a+b ) due to the swaying, heaving and rolling motions are evaluated by employing the formulas in Table 11 with the aid of the data in Fig.15 1G and) dif. 817 Ktm and Mercter The Wave-Exciting Forces and Moments for Twin Cylinders Figure 18 illustrates the behavior of the sway- and heave- exciting forces on Body a due to odd and even wave potentials as func- tions of the wave frequency. The vectorial sums of the forces such as plote) = slo)+ gle) on a,b, and (atb) are shown in Figures 19 and 20. In this connection, refer to Tables 12 and 13. The Haskind-method was applied to the evaluations of the re- sultant wave-exciting forces on the twin bodies and the results confirm exactly the corresponding values obtained by the present method. (See Figures 19, 20 and 21). A relevant discussion of this matter will be given in the following section. It is interesting to observe the behavior of the sway-exciting forces on a and b (Figure 19). Ohkusu [35] recently reported on the hydrodynamic interaction between three vertical cylinders. His Fig.13 on page 108 of Reference 35 shows behavior similar to that of the sway- exciting forces on Bodies aandb. A relevant discussion of this mat- ter will be given in the following section. The Radiated and Diffracted Waves For the twin bodies, the radiated waves a) and the dif- fracted waves A‘°*©’ were evaluated according to the formulas in Eqs. (24 and 26) (see Figure 21). The radiated wave amplitudes A (m) are very useful in evaluating the wit ee force coefficients N(™) or 6\™/ and the wave-exciting forces f ote) (see Eqs.27,28). By compar- ing Figures 15,19,20 and 21, one can readily confirm the validity of the formulas in Eqs. (27),(28). The diffracted wave amplitude ratio A (ote is applied to estimating the mean wave force on the fixed twin bodies in the given incident wave, Eq.(15). The sudden drop of the value of A | ©’ at frequency around 0.45 may be ascribed to the ef- fect of the hydrodynamic interaction between the bodies in that close proximity. The Wave-Exciting Forces on Some Semi-Submersible Cross Sections We chose two simple cross sections : one submerged and the other surface piercing. Figures 22a and 22b represent the sway- and heave-exciting forces on the submerged circular cross section. It is seen from these figures that the interaction effects are negligibly small for the given frequency range. The wave-exciting forces on surface-piercing twin bodies are plotted against the wave frequency in Figures 23a, 23b. It is readily seen that the influence of the interaction effect on the sway-exciting forces is remarkable while the influence on 818 Analyses of Multtple-Float-Supported Platforms tn Waves the heave-exciting forces is relatively small. Similar behavior was pointed out by Ohkusu [35] as described previously. In connection with the present investigation, force measure- ments were made at Davidson Laboratory on twin Motora-type floats. These measurements are plotted in Figures 24a, 24b. Since the tests were carried in a relatively high frequency range, even a rough comp- arison with the present two-dimensional calculations cannot be made. However, the interaction effects appear to be quite small compared to the two-dimensional case, especially considering the very close spac- ing of the floats. Interaction effects and the occurence of dramatic variations of force coefficients for particular, characteristic, frequencies should be expected to be weaker for three-dimensional than for two dimensional bodies. We have, however, noted an unusual flow pattern for waves passing a fixed toroidal body ; at a particular wave frequency a pulsat- ing jet erupts in the center of the torus. Of course, the jet also exists for forced heaving oscillations in calm water at the same frequency. The shape of the torus corresponds to that of the profile of the twin cylinders of Figures 15-20 rotated about a vertical central axis, with 2S/B = 2. The characteristic frequency described for this three- dimensional case corresponds to vB/2 = 0.42, rather close to the frequency shown in Figure 15 where Cy, and Or vary dramatical- ly. Some Hydrodynamic Characteristics of Two Different Cylindrical Bodies Floating in Beam Seas Some aspects of the hydrodynamic interaction between two ar- bitrarily shaped cylindrical floats were investigated numerically (see Figures 25a, 25b,25c). First we evaluated the heaving added mass and damping coefficient C , 6 on bodies a and b for heaving motions with unit amplitude and different phases. The results are plotted against the relative phase angle ¢, at two different frequencies »(B,+Bp,)/2=0.45 and 0.26 (Figure 25a). The results show significantly large effects on the hydrodynamic forces due to the hydrodynamic interactions between the two different cylinders floating freely with different phases. As a special case of the above, we calculated the hydrodynamic forces on the swaying and heaving Body a in the presence of the fixed Body b (Figure 25b). Two draft ratios gee ie = 2and15 were taken in order to observe the false wall effect on the hydrodynamic forces. The figure also shows the hydrodynamic inertial forces Cyy , Cog on an isolated swaying and heaving Body a. It is seen that the increment of 819 Kim and Mercter the swaying added mass Cgo due to the increase of the depth of the false wall is nearly independent of the frequency. The heaving added masses Cy;, are remarkably dependent on the depth of the Body b , T,/T,, and the frequency. The resultant heave-exciting forces F,fote) on bodies a and b were calculated and plotted in Figure 25c. For comparison, the neave-exciting forces on isolated aandb are also plotted in the same figure. SLENDER VERTICAL FLOATS : ISOLATED PERFORMANCE AND INTERACTION EFFECTS FOR LARGE ARRAYS BACKGROUND Column-stabilized floating ocean platforms used for oil ex- ploration, research, or other applications, achieve low motions re- sponse to wave action by proper shape and spacing of rigidly-connect- ed flotation elements. Another type of platform, which can also afford a useful working surface, might consist of an assemblage of resilient- ly-connected flotation elements, each of which individually has suitable motions response characteristics. Conceptual design studies for such a platform have been undertaken by engineers of Goodyear Aerospace Corporation (GAC). Davidson Laboratory of Stevens Institute of Techn- ology has conducted a variety of hydrodynamic studies in connection with the design developments, some of which will be described in the present part of this paper. An illustration of a concept for the array of floats and deck structure to be considered is depicted in Figure 26. The details of the deck panels and truss system are not important for the present hydro- dynamic study : the structural restraint on body motions is assumed to be negligible (or, in a particular case, to be approximated by simple mechanisms). The effect of hydrodynamic interactions on the relativ- ely closely spaced floats may, however, be important. Much of the presently described work, and work currently in progress, deals with these interactions. An interesting feature of these multi-float platforms is that they are expandable. Modular construction is intended to permit con- venient deployment of arrays with any number of floats from about 50 820 Analyses of Multtple-Float-Supported Platforms tn Waves up (a minimum number are required to afford stability against cap- sizing). The resiliency of the structural components, including the floats themselves, which may be inflatable reinforced fabrics, pose certain interesting hydroelastic questions. Several hydroelastic aspects have been and are being studied but these will not be treated in this paper. ANALYSIS OF VERTICAL FLOAT HEAVE MOTION Elementary Equation of Motion In the configuration depicted in Figure 26, the net buoyancy comes from the portion of the float above the hinge, which is located some distance below the stillwater level. The float shape, which is enlarged below the waterline, is intended to minimize the wave-induc- ed vertical forces transmitted to the deck and structure, and thus minimize deck motions over a ''sufficient'' range of wave frequencies. The hinge is introduced to alleviate lateral loads in the float and in the structure : the wave-induced forces produce pendular motions of the lower part of the float (which is called an ''attenuator'') which relieve the elastic stresses and transmitted loads which would develop without Lie gninice:. Since the deck structure is assumed to be quite flexible, the (linearized) equation for the heave (z) motion, neglecting elastic re- storing forces, can be expressed simply as DVRS Sere ELL AN DTS AC (30) Zz Z ¢ where V = total displaced volume of float and attenuator m’'' = added mass 23 = damping force rate 2, = buoyant restoring-force rate = pga oe = waterplane area a. = wave-induced vertical force per unit wave elevation ¢ = wave elevation If the deck's elastic restoring-force is to be taken into account, a term such as Elva must be incorporated to describe motion in 821 Ktm and Mereter regular long-crested waves, such that the deck structure behaves like a beam (the more general case would have to represent the deck's elasticity as plate-like, or describe the details of the deck-truss structural behavior). These partial differential equations are consider- ably more complicated than the simpler equation (30). The deformat- ion of the deck of a rectangular array of floats in regular ''head'' seas, assuming that no hydrodynamic interaction effects occur on the vertic- al forces on float elements, will be (at least in the case of linear re- sponse) a traveling wave with the same frequency and celerity as the incident water wave : ¥, .. lame 2 > 2 oein ee »*) (31) then ata, a 4 = |—) 2z (32) or, since for water waves 27/\ = w*/¢ : ae 8 Bn a yaleueds a Ox g The importance of a restoring force term like EI-—q_, compared to pgA,,z , increases as the frequency increases, othe wave length decreases. The design of the float-attenuator geometry is intended to produce vanishingly small wave-induced vertical force as frequency is increased. GAC structural analysts decided that the simple Eq. (30) is appropriate for analyses of the motions of the deck supported by a resiliently-connnected array of floats, such as is shown in Figure 26. Coefficients of the Equation and Forces for Regular Waves Elementary hydrodynamic theory can be applied to the estimat- ion of the hydrodynamic coefficients and the wave-exciting forces for the equations of motion of slender vertical floats without external ap- pendages. In the present instance, it is anticipated that external ap- pendages will be required to assure sufficient heave damping and analyses of the hydrodynamic effects of the appendages must be ap- proximate and quasi-empirical. The buoyant force rate has already been expressed as Be = pgA., 2 The vertical wave-exciting force can be expressed as 822 Analyses of Multtple-Float-Supported Platforms tn Waves O o¢ kz 2 2,8 = pgfS(0) - pat [ (wel) Sea - Ze, - Dum! w fe © ar ‘s sbuad Seeds See atl C ommection sitet ee a Froude-Krylov (34) where S = sectional area of body k = wave number = wie = 2n/ A = wave length Z,. = damping coefficient = wave elevation = wave amplitude = wave motion evaluated at depth corresponding to as- sumed damping source = element of effective added mass in vertical direction effective depth for evaluating wave acceleration for element of added mass The Froude-Krylov force corresponds to that predicted by slender body theory and is the same as predicted by assuming that the presence of the body does not influence the wave's pressure field. For finite draft, T , the Froude-Krylov force decreases with increasing wave frequency (because of more rapid attenuation of wave pressure with depth) and, if the attenuator is shaped as in the sketch, may become opposed in sense to the wave elevation for sufficiently high frequencies. The response of slender spar buoys to waves has been studied by Newman [3 6] ,» who performed a detailed slender-body analysis and by Rudnick [3 7| » who derived equations similar to Newman's [36] on the basis of a more elementary analysis, and who compared results of calculations with field measurements of the motions of the Flip platform. Newman notes that the slender-body theory applied to floais in waves loase its applicability at higher values of slenderness ratio than is the experience for aerodynamic analyses. Adee and Bai [38] have conducted experiments with cylindrical models having either flat or conical bottom ends and various draft-to-diameter ratios. They find that it is important to account for added mass effects even for quite slender floats. However, while they include added mass effects 823 Kim and Mercter on the inertial force (proportional to Z), they appear to have neglect- ed its effect on the vertical wave-induced force. For non-cylindrical floats, such as shown in the sketch, the inclusion of added mass ef- fects is still more significant. The added mass wave force results because the flow is un- steady and the presence of the body does ~nodify the fluid acceleration patterns (contrary to the Froude-Krylov assumption) resulting ina pressure force in phase with the vertical wave acceleration. The add- ed mass may be associated with two principal sources, the primary one being the enlarged attenuator at the lower end of the float. Extern- ally-attached damping devices will also have associated added masses. The effective added mass from the primary source, the enlarge at- tenuator, may be estimated by assuming that the attenuator is similar to a prolate spheroid with a ratio of semi-major to semi-minor dia- meter, a/b equalto L,/2R, . Lamb [39] gives theoretical added mass coefficients for translation ''end-on'' that can be expressed as i ee te pre (35) where a can be taken from Table 14. It has been found, of course, that the ideal fluid theory added mass is insufficient for slender craft such as airships and surface ships in monotonic rectilinear motion,presumably because of boundary layer influences (cf. Thompson and Kirschbaum [40] and Smith [41] ). The reasons for the differences between theory and experience for these craft may not be relevant to the float-attenuator in the wave flow field so the tabulated theoretical values are recommended for use pending more complete experimental results. The wave acceleration can be evaluated at an effective depth, z, = T - L,/2. The added mass associated with the damping devices, which probably would be attached to the float at the upper end of the conical transition above the attenuator (see sketch), is not derivable from familiar simplified cases. The interaction of the flow about the damp- ing ''collar'' with the flow about the cylinder may be important and ought to be studied experimentally. Fot the purposes of the present analysis , the added mass of damping devices will be assumed to be a fraction of the added mass of the attenuator m) = c'm! (36) where approrpiate values of c' should be obtained experimentally or simply assumed. The value of zg to be used for evaluating the wave 824 Analyses of Multtple-Float-Supported Platforms tn Waves acceleration for this component of force should be the depth correspond- ing tothe damping source. The total added mass, m'' , is the sum of that associated with the attenuator shape plus that due to damping devices. The heave damping force rate, Z. , is partly due to generation of radiated surface waves and partly to viscous influence associated with turbulent eddies around the float and damping plates and skin friction drag. It will be shown in the following section that the damping due to wave generation, which is strongly dependent on frequency, is quite small for slender vertical floats and, therefore, itis advantageous to provide additional viscous damping, which is likely to be independent of frequency. The damping coefficient will be expressed in terms of the ratio of the damping to the critical damping coefficient, (e/en), in the form yy Sot S)x Oa oi seas (37a) Z c n c c = (_—— 2pA ANG 1+C 37 C= eaep a) VeTC (14C 5) (37%) where ° ey = vertical prismatic coefficient, AT 1m"! Cr added mass coefficient, OV The viscous drag due to external damping devices will not, in general, be simply linearly proportional to velocity (although for small waves and motions it will be approximately so). The use ofa linear coefficient may be justified on the bases that calculations based on such a simplification are instructive and that ''equivalent'’ lineariz- ed coefficients may be derived for drag which is proportional to some other power of velocity in the way that Blagoveshchensky [42] and others have dealt with square-law damping. Particular values of drag coefficients may be estimated for plates oscillating in a direction normal to their surfaces from results of significant investigations by Keulegan and Carpenter [43 ] , McNown [44 ] » McNown and Keulegan [45 | , Paape and Breusers [46 ] » Martin [47] , Ridjanovic [48] , Brown [49] , Henry [50], Woolam [51] anda Tseng and Altman [52] . Additional investigations of the oscillating drag of ring-type damping collars around bodies of revolution will be 825 Kim and Mereter needed to provide information on the type of configuration being con- sidered for these floats, as shown in Figure 26, where damping plate- body interaction effects may be significant. The wave-exciting force associated with the damping devices, 2 SE » may be estimated by taking sy = 2; and vi is the wave motion evaluated at the depth corresponding to the damping plate. Of course, since the oscillatory drag force on the damping devices is, in general, nonlinearly related to the relative velocity between the fluid and the plate, the detailed analysis of the motions would be rather more complicated than the simplified treatment given here. The effects of the nonlinearity of the drag may be expected to be important only for frequencies near the resonant frequency. Responses Although analyses have been presented by Newman [36] and others for wave-induced forces and motions of isolated spar-type floats, no results of systematic evaluations of the dependence of the forces and motions on geometric characteristics of floats are known to be available in published literature. Some results for the special case of floats like that shown in the sketch accompanying Eq. (34) will be presented here. The influence of the ratio fo the diameters of the lower and up- per cylindrical parts R,/R, , the ratio of the length of the lower cylinder to the overall draft, L,/T , and the ratio of the draft to waterplane radius (slenderness ratio), T/R, ,» will be shown. Wave forces and motions due to regular waves will be presented as a function of frequency, and spectral response information will be given as a function of significant wave height. The influence of the degree of damp- ing on the responses will be described in a subsequent section. Wave-Induced Force The wave-induced vertical force, Z,¢ , expressed as a function of the buoyant force, pgS(o)f , is exhibited in Fig. 27 as a function of the dimensionless frequency parameter w*T/ , showing the influence of Ry/ Rs . Other geometric parameters were held fixed for these results, viz., L,/T = 10), (ders aR, = 30 ; the assumed damping coef- ficient corresponds to a value of c/cg = 0.07 . Figure 28 shows the influence. of, La/T for Rg/Rg sl. 8:4! T/Rg = 30); again, « ¢/e, 30207. The influence of T/R, is presented in Figure 29 for Rg/R)= 1.8, Laf't 10), 5.5 and e/cg = 0.07. For all cases presented the damping 826 Analyses of Multtple-Float-Supported Platforms in Waves plate added mass, my , is assumed to be 0.3 m! and its effective a depine ze = Lal, The results presented are typical : as frequency increases, the vertical forces decreases at first until it reaches a minimum value (which corresponds very closely to the ''damping'’ component of the wave-induced force, Eq.(34), and then increases again when the com- ponents of force which are out-of-phase with wave elevation (due to pressure gradient and added mass) become important, followed by asymptotic attenuation to zero force for very short waves. Both F/R and ae are seen to have important effects on the wave-induced force, T/Rg ; T/R, is less important, in fact, the simplified theory (Newman [36] ) ) neglecting added mass and damping indicates no de- pendence on T/R, Transfer Functions The ratio of heave motion to wave elevation can be derived from the solution of Eq. (30). This may be re-written in a form similar to that for the familiar simple harmonic oscillator. Z. £ | pgS(o)f z ernie | Wihied off Mie Mee dak fife ee (38) where 9 o, oe -1 : ———- = (Crusreal tl beam G: 39) - : al Sig (39) Only one set of transfer functions, exhibiting the ear aan on Ra/ Re for 1 jel "Un om, sates = <3) (0 hans o/kon =U Oia 0.3m) and z, = 0.5 (for damping plates), will ae given in Pienes aah These results are, again, typical : the trends of the variation of motions with frequency follow the wave forces modified by the dynamic amplification factor. Note that the damping coefficient assumed, eyes =O 0H. results in values of the transfer function around 2.0 at resonance, and that the maximum value depends on the float shape as well as the re- lative damping. 827 Kim and Mercter Spectral Response The statistics of the heave motion response may be derived for slender vertical floats using the transfer functions and the (di- mensionless form of the) Pierson-Moskowitz wave spectrum, as was done in the first part of the paper for three- and four-float platforms. The significant dimension for use jn non-dimensionalizing will be the draft for this case, instead of ou : Some results showing the effect on "significant heave motion", Z4 3/T 78 OL R,/R , are given in Figure 31, with other particulars, the same as for Figure 27 . The influence of w/e on the significant heave is shown in Figure 32 for the same cases as are considered in Figure 28. The effect of Tyee on the statistical responses is small, as might be expected from the results for forces shown in Figure 29 - at least for floats which are sufficiently slender. According to Figures 31 and 32, the ''best'' float shape is evidently a function of the design sea state or significant height : slender floats with displacement relatively uniformly distributed be- ing better for mild sea conditions while higher values of Rape with the displacement concentrated near the bottom are better for more severe seas. An irregularity, or ''bump", is discernible in some of the curves for values of Hy 3/T at which an increase of sea state introduces a large increment of wave energy at the resonant frequency of the float. It is interesting to note that the dependence on significant wave height of other heave-related spectral response characteristics may differ from that of heave. Figure 33 shows significant values of heave motion, vertical acceleration, and deck curvature ,» for a partic- ular float having R,/R,= 1.8, L,/T=0.5, and™T/R, = 30. Since the transfer functions for acceleration and deck curvature depend on frequency in a different way than does heave, weighing high frequency more heavily, while attenuating low-frequency input, higher sea states do not produce as much increase of response as for heave. This is because an increase of sea state (according to the Pierson- Moskowitz spectra) adds significant energy in the low frequency range but not much at higher frequencies. For the case presented, the deck curvat- ures (and therefore the deck stresses) are very nearly proportional to the significant wave height, since OZ T— / (H,,.,/T) 0.3 over the range of H,,,/T values Ox 1/3 1/3 presented. 828 Analyses of Multtple-Float-Supported Platforms tn Waves The Importance of Damping The primary effect of an increase in the damping coefficient on the heave transfer functions is to reduce the maximum heave re- sponse, which occurs at a frequency slightly lesser than the resonant frequency, and to increase the minimum value which occurs for the frequency when the wave-induced force is due to the damping devices alone. For design-analyses, the spectral response is most useful. Figure 34 shows effect on significant heave motion of damping coefficient, Cee , for a particular float geometry. It is evident that damping is very useful to control motions in high sea states where some wave energy exists at frequencies corresponding to heave natur- al frequencies. The damping which is available due to wave radiation can be obtained from the Kaskind-Newman relations [32] » which gives the same result as Newman's slender-body analysis for the forces at re- sonance [36] nese nove Dierpepin SE 2) (40) Zz pow thus, at resonance w = w, » the ratio of wave-damping to critical damping can be obtained as is hee ee 2eV(1+C eo, 2 2 2 Z Cera A & Save ly teaein Ww salpoX 4 2 (41) pgs w g T and the corresponding transfer function Ce eee caw (42) & Zile/ ce) w contd. \ 829 Kim and Mercter 277 /a Sere (42) (iy IZ, /egA I : where Z is the wave-induced vertical force evaluated without taking into account the damping term. BCR at Pesonane’s fokw! PBA |< 1, and for slender vertic- al floats T 274. S>1 and & then c/c, <1 and z/t>l. For example, for T/R, = 30, epee = 1.8 =f La/T =, 053.55 the calculated ''wave- ne resonant heave motion z/f serial be almost 8000 £ Wave-associated damping is inadequate for slender floats and viscous damping controls the resonant motion. Results of tests ona 1/13th-scale model of a Manned Open Sea Experimentation Station (MOSES) reported by the Oceanic Institute, Waimanalo, Hawaii [53] , showed that a ratio of damping to critical damping of 0.075 could be achieved. This was discovered in tests of the model with about 18 external rings attached to a slender shaft. The rings, which are intended to provide structural stiffening, have out- side diameter about 15% larger than the shaft. Complementary tests were carried out with acetate sheet wrapped tightly around the rings to present a smooth, unbroken surface. With this shroud the damping was about 1% of critical. Damping coefficients will, in general, be obtained most ef- fectively by experiments, or will be estimated on the basis of empiric- al results. It has been known since Froude's investigations in 1874 [54] that the drag coefficient for an oscillating bluff body can be very differ- ent from that for the same body in steady flow. As in all model ex- perimental work, it is important to be sure that the model conditions correspond to the full scale : thus, for dynamic similarity to exist, the model should be geometrically similar and the flow kinematically similar to the full size. Certain experience from investigations of roll damping of ships with appendages can give insight into questions relevant to damping of platforms oscillating in waves. It is important to recognize that a small ambient current, due to oceanic circulation or induced by wind, can have an appreciable effect on energy dissipation, as has been found by many investigators into the rolling of ships [54-53] . This is because the model, in the course of its cyclical motion, must impart motion to fresh, previously unentrained water. Consider that for a current speed 830 Analyses of Multtple-Float-Supported Platforms tn Waves of only one knot past a moored platform, the fluid which is 'entrained'' by the motion of a platform may convect about 17 feet during a 10-sec period ; such a period is common for ocean waves, and the distance is about half of a typical column diameter for a large semi-submersible drill rig. Entrained fluid energy can be convected away at an appreci- able rate by modest currents producing important effects on damping and, hence, resonant response. Indeed, when the damping force is non-linear and, hence, superposition cannot be applied, it may be inappropriate to apply an oscillatory drag coefficient obtained for a particular structural element from tests with rectilinear oscillatory motion (43-45, 47-51) to the somewhat different kinematic conditions of the orbital velocity pattern of waves. The differences may be modest but the question should be posed and, hopefully, investigated. The question of scale effects is persistently present and model experimenters must be alert (and somewhat intuitive) to recognize when it may be appreciable. When phenomena are recognized to be predominantly viscous in origin, we are likely to suspect the possibi- lity of scale effect. This is, of course, due in large part to the hi- story and experience of testing ship models for resistance. Very little is known about scale effects on oscillatory hydrodynamic forces which may be relevant to platform motions testing and analysis. Some years ago, however, a program of experiments to study scale effects on roll damping of circular cylinders with and without appendages was undertaken by the Naval Ship Research and Development Center and Davidson Laboratory. While these studies were not directed to plat- form motions, the results are relevant to the phenomena of oscil- latory damping in general and since they are the only results with which we are familiar which show the effect of model size, it may be useful to discuss them. Three cylinders with diameters of 6-in, 12-in and 24-in were suspended vertically in water by torsion springs. Three kinds of appendages were symmetrically attached to the models, as shown in Figure 35 for the smallest (6-in-diam) cylinder. Curves of decaying oscillation from various initial angular displacements were recorded and analyzed to obtain ''square-law'' damping moment coef- ficients of the form damping moment Gc = ores (43) m (p/2)A R. a 6161 where A is the frontal area of both appendages and R, g_ is the radius from the axis of rotation to the center of area of the appendage. The results are tabulated in Table 15. 831 Ktm and Mercter The 6-in and 12-in diameter models were tested at Davidson Labor- atory by Mercier [59] ,» while the largest model was tested at the Naval Ship Research and Development Center by Gersten [6 0} 2 The lessons of these test results are not entirely unexpected : sharp-edged geometric details produce high drag and little scale effect while well-rounded geometries produce lesser drag and are susceptible to perceptible scale effect. These results may provide qualitative guid- ance for other applications and configurations, such as for choosing a suitable scale ratio for a wave test of a floating platform with buoyant caisson-and-footing floatation elements. Damping effects, while of principal importance for a fairly narrow band of frequencies near resonance, can have an appreciable effect on spectral response when there is appreciable wave energy at the resonant frequency. The nature of the damping and its dependence on geometric and flow features is as yet only imperfectly understood adn needs to be studied more vigorously. ISOLATED FLOAT EXPERIMENTS An experimental program was undertaken to determine hydro- dynamic forces and moments and certain other features of perform- ance for several floats. The results of these experiments may be used in conjunction with analytical work, such as described in the last sect- ion, to obtain empirically adjusted analytical procedures for perform- ance evaluation. Certain results may also be compared with results of interaction tests, where several floats were placed in close proximity, to determine hydrodynamic interaction. Models Several types of models were built to evaluate the influence of various features on performance. Some of these models were selected on the basis of the analyses described in the previous section of this report. Some were also tested as part of an array of floats to evaluate interaction effects. The scope of isolated float testing was curtailed when the extent and significance of interaction effects on wave-induced vertical floats was discovered. All models investigated in this test program were built to a scale ratio of 1:57.6. Figure 36 illustrates the models which were tested. The deep cylinders (Figures 36-36) were planned to investigate the effect of 832 Analyses of Multtple-Float-Supported Platforms tn Waves bending rigidity of the fabric attenuator on float forces and other per- formance features. Using classical vibration theory [6 1] , with an as- sumed bending rigidity E = 15.268x 10° 1b-ft2 for a six-foot diameter pressurized inflated float tube constructed of 3000 lb/in fabric, the computed lowest lateral bending frequency of the attenuator is 0.11 Hz, assuming hinged-free boundary conditions (corresponding to tests with the hinge, if the hinge is omitted, the lowest two cantilever natural frequencies are 0.025 Hz and 0.16 Hz). Since these frequencies are within the range of significant wave energy, it appeared to be impor- tant to study flexible model performance. The scale-equivalent flex- ural rigidity of the 1/57.6-scale model was provided by a central 0.45''x0.06'' plexiglas flat strip. The external shape of the float was simulated by cylindrical segments secured to the flexure strip by ple- xiglas bulkheads. A system of baffles and flow passages were used to inhibit flow passing from outside the model to inside it but which per- mits flow communication within the float. For shorter floats, the e- lastic laterla natural frequencies are higher and consequently, the in- fluence of rigidity is lessened. The slender float of Figure 36c was used in one set of inter- action tests, while the full float of Figure 36d was used in the other set of float interaction tests and for the dynamic island tests. The float of Figure 36e is the same as that of Figure 36d except for the rounded bottom. Apparatus The apparatus used for these tests was an improved version of the equipment described by Mercier in published references (62, 63] : Existing force balances and motion transducers were adapted for these tests. A new data reduction procedure was applied for these tests. Electronic signals for forces, Moments, waves and/or forced motion (heave, surge, pitch, sway, roll ; but not yaw) were recorded on analog magnetic tape as well as on oscillograms. For some of the tests, the signals were immediately processed by an on-line computer- controlled analog-to-digital converter and further evaluated by the computer (PDP-8E) to determine the Fourier coefficients of the signals, Other tests were ''played-back'' later during off-line processing. Sampling was carried out at the rate of eigher 20 or 50 samples per second, permitting accurate (and easy) interpretation of electronic- wave-forms which were sometimes relatively ''noisy''. Usually only the first harmonics of the wave forms were determined, but for sever- al tests the second and third harmonics were also computer. Higher harmonics are of interest primarily for large amplitude waves or S15)9) Ktm and Mercter motions where linearity of forces and moments is in question. Five to ten complete cycles of the wave forms were "averaged" in determining the Fourier coefficients except when the test period was very long, when no less than three cycles were used. Forces oscillation tests were carried out by adjusting the crank offsets of the several motion-producing linkages so that either pure heave, surge, or pitch were produced. Yaw motion was not tested. A cosine potentiometer was coupled to the drive shaft of the mechanism, A constant (battery) voltage was applied to this potentiometer whose output was consequently proportional to the cosine of the shaft rotation angle and thus was suitable for use as a phase reference for the motion. For some tests, both in waves and with forced oscillations, a hinge was used to permit the lower. end of the attenuator to oscillate like a pendulum under the action of waves. In this way, the periodic side loads due to the waves are not completely transmitted to the float- ing base connecting structure by way of bending moments in the float but are rather absorbed by the pendulum-like motion of the attenuator. This reduces the strength requirements of the inflated float and, con- sequently, weight and cost. A rotary variable differential transducer was connected to the attenuator by a system of strings to permit the measurement of ang- ular motion. Sealing The periodic hydrodynamic forces and moments are assumed to follow Froude's scaling law. Thus the full-size force and moment are related to model quantities by Prull size 3 sss é K F x full size model on aes Prull size 4 pase teat Se eh = ! 4 Mild size MM added p x ($5) model where F = Force M = Moment p = Fluid density X = Seale ratio 834 Analyses of Multtple-Float-Supported Platforms in Waves The corresponding frequencies f (cycles per second), for full scale and model, are also related by Froude's law f.s =x, «Se Vx f (46) Results Vertical wave forces for the floats of Figures 36d and 36e are shown in Figure 37 along with a comparison with theoretical results according to Eq. (34) (except Ze was assumed negligible) and with the simplified Newman (or Froude-Krylov) theory which neglects that added mass type force. Model results are corrected to the full-size float in this and all subsequent figures. The effect of rounding of the bottom end of the float is insignificant and cannot be detected within the ac- curacy of the experiments. The phases between the wave and the heave force are not shown in the figures but, in general, for low frequencies the heave force is almost in-phase with the wave (maximum upward in way of a wave crest) while at high frequencies the force is nearly 180° out-of-phase with the wave. It may be noted that the agreement of test results with the complete predictive theory is quite good for this case. A comparison for a fuller float, which had been tested by Mercier [64] on a previous project, is shown in Figure 38. A computation procedure applied by Ochi, [65] based on two-dimensional strip theory is also shown. It is seen that the calculations are not as satisfactory for this rather fat float. Tests with the attenuators attached to the upper part of the float through the hinge indicate that the vertical forces due to waves are not affected by the hinge, within the limits of experimental accuracy. The side forces measured in waves are exhibited in Figure 39 for the full float of Figures 36d and 36e. Again, no influence of the rounded bottom is discernible. The phase of the force is approximately 90° with respect to the wave, indicating that the force is predominant- ly due to pressure gradient and inertia forces. The influence of wave amplitude on the forces and on the phase of the forces relative to the wave has been found to be small for the range of wave amplitudes used in the tests (corresponding to 1.1-ft to 10.0-ft, full size). Side force due to waves for the slender float of Figure 36c is presented as a function of wave frequency in Figure 40. This infor- mation is presented in Figure 41 for the rigid and flexible cylindrical models of Figures 36a,b. Results for all models with the hinge in place are given in Figure 42 and the corresponding angular pitching 835 Kim and Mercter motions are given in Figure 43. The side forces are seen to be remark- ably lower than the results without the hinge. The amount of angular motion, which is greatest for low frequencies of course, is about 3° per foot of wave elevation for a frequency of 0.07 Hertz. The resi- lient model exhibits a resonance-like behavior at a frequency of about 0.12 Hertz, somewhat higher than the calculated value of 0.11 hertz, as evidenced in both side force and pitch motions. This relatively modest amplification of response occurs at a frequency at which signif- icant wave energy exists and is consequently considered disadvantage- ous. A variety of other tests also were carried out, using several of these floats models and with various damping plates fitted. Besides wave tests, forced heave and surge oscillation tests were carried out. Results of these tests have not yet been analyzed in detail but are stor- ed on magnetic tape for future processing. INTERACTION EFFECT TESTS Preliminary Wave Tests Although in previous experience with floating platforms such as for drill rigs, hydrodynamic interaction between adjacent floats was found to be negligible, the large numbers of floats planned for the Floating Expandable Base are so closely packed in relation to their size that it was considered vital to investigate the interaction between floats at an early stage of this program. Wave-induced forces were measured on individual floats in an array consisting of five rows of five floats each. The force-measuring balance could be moved so that forces on any one of the floats could be measured, as desired. The spacing of the floats was either three times the water-plane diameter or five times. Two sets of floats, shown in Figures 36c and 36d, having different proportions and drafts, were tested with and without damping plates. Certain general findings of the tests can be given : the side forces and pitching moments acting on the floats due to waves are prac- tically un-influenced by proximity ; the vertical wave-induced force is modified by an apparent increase in an added-mass type force com - ponent which is significant for higher frequencies and accounts for about a 30% increase above the isolated float results for the fatter of the two floats studied. Further, damping plates attached in way of the 836 Analyses of Multtple-Float-Supported Platforms in Waves fat lower parts of the floats may result in severe interaction influence on the drag-type force component but plates may be attached to the slender upper part of the float without important interaction. When the spacing of floats was five times the water-plane diameter, no measur- able interaction influence was observed. Some results for vertical (heave) force on the full float of fig- ure 36d, both in the middle of the 25 float array and isolated, are exhibited in Figure 37, which shows the increase in force at high fre- quency. The introduction of a ten-foot diameter damping plate at the junction of the conical transition piece and the upper float produced only a minor increase in lift force at high frequency. Rather similar results were obtained with the deeper, more slender float of Fig. 36c; but when damping plates having 13.5-ft diameter were fitted to the lower end of the floats, the wave forces were dramatically increased, evidently because of a drag-type component in phase with the vertical wave velocity. The forces on the interior float elements of the array were found to be virtually the same as one another while the floats in the forward row (near the wave generator) and in the aft row were close to the results for isolated floats. Horizontal forces measured on the full float of Figure 36d, isolated and in array, are shown in Figure 39. It is found (somewhat surprisingly) that little or no interaction occurs for this component of force and the floats in array experience essentially the same side force as the isolated floats. Large-Array Investigations A freely-floating model of a substantial segment of a Floating Expandable Base, having 35 rows of 6 floats each like those of Fig. 36d, with rows connected to each other by parallel motion linkages which are intended to permit freedom to heave while restraining against pitch motions, was tested in regular waves in Davidson Laboratory Tank No.3, which is 300-ft long x 12-ft wide x 6-ft deep. A particular, unexpected result of this model test program with the 35 by 6 array of floats (which have a nominal scale ratio of 1/57.6) was a ''tail-wagging'' phenomena where the heave motions in- creased from front to rear of model. This is an especially significant feature of the performance of arrays of large numbers of such floats. A variety of experiments have been carried out in order to study cer- tain aspects of the hydrodynamic interaction observed in the motions response tests of the 35 by 6 array of floats. These include wave force 837 Ktm and Mercter measurements on individual element rows of this large array to de- termine if the variation of wave force, with the model held fixed, is sufficient to produce the motions obtained in the previous test. Possible scale effects were investigated briefly because of the possibility of viscous wake interaction due to vortex shedding, separat- ed flow, etc., being dependent on Reynolds number. Since large scale model investigations are liable to be quite expensive, smaller scale tests were undertaken. Although it is not at all clear in what way the interaction effect in this unsteady flow situation depends on Reynolds number (this dependence can only be established by extensive ex- perimentation), it has been found that for many flow situations a mo- dest reduction in size, or Reynolds number, can have as much effect as a substantial increase in size. A model approximately one-third of the size of the 1/57.6-scale model, resulting in about one-fith of the Reynolds number, was employed. Review of the wave-induced force measurements with the 1/57.6-scale model indicated that the variations in forces correspond reasonably with the variations in motions over the forward and middle part of the island but do not exhibit a continuous increase toward the trailing edge, which was felt to be called for to explain the tail-wagg- ing. Since a suitable explanation in terms of elastic interaction is not presently available, it was decided to re-test the articulated 6x35 array in the 75'x75'x4.5' deep wave test tank (No.2) in order to as- sure freedom from tank sidewall influences. An approach to an analytical description of the deck motion, taking deck elasticity into account, is discussed but an explanation of the tail-wagging does not appear to follow from this analysis. Plans for a comprehensive test program to determine the ef- fects of variations in parameters such as float spacing and shape, wave frequency and height, deck rigidity and number of floats on the motions of the platform are described. Models Articulated Model A-preliminary design for float-attenuator shape was developed on the basis of a simplified hydrodynamic analysis and a particular limiting vertical motion criterion. The selected float had a relatively shallow draft and large diameter near the lower end. No interaction effect was anticipated in selecting the float shape. A scale ratio of 1/57.6 was selected ; the full-size float has 6-ft diameter at the water- line while the model was fabricated with 11/4-in O.D. plastic tubing. 838 Analyses of Multtple-Float-Supported Platforms tn Waves The 210 float elements were made of plexiglas tube and sheet accord- ing to the sketch shown in Figure 44. Solvent-bonding was used to as- semble the parts in a watertight fashion. The float elements were connected in sets of 6 to an aluminum channel. The channel was lightened considerably by drilling holes and the tubes were ballasted with brass weights and lead shot so that a row floated at the correct draft and roll angle, with a small positive roll stability. This was checked by floating the sets in a fish tank while lightly restraining them against pitching (the rows are very unstable in pitch). The rows are connected to each other by linkages consisting of 3-3/4'"'x1/2'"x0.050"' aluminum strips with 1/8'' diameter reamed holes spaced 3-1/4" center-to-center. The linages roll on 0.1245 diameter x 1/8'' long shoulder screws which are secured to light posts at the ends of the rows of floats. The vertical spacing of the linkages is 4 inches. The float are arranged in an equilateral triangular fashion, as indicated in the sketch of Figure 45. Although roll stability of the articulated model is present be- cause each row is suitably ballasted, the pitch behavior is unstable because each row is unstable and the 4-bar-linkage connections provide no restraint unless one row is held so that it can move only vertically. The center row (number 18) was restrained by a vertical tube which slides in a pair of linear-motion ball bearings, as indicated in the sketch of Figure 46. The linear motion bearing is secured to a light weight (approxi- mately 3 lbs) carriage which rides on low-friction wheels on a mono- rail about 12 inches avove the water, permitting effective freedom of surge. The vertical motions of five locations along the lenght of the model, at rows 1,9,18,27 and 35, were measured by systems consist- ing of a long (approximately 8 ft) vertical string between the measure- ment point on the model and quadrant connected to the shaft of a rotary variable differential transformer (RVDT). These RVDT's have very low friction ball bearings and the quadrants are very slightly counter- balanced to assure that the string remains in tension. Motions tests were carried out in November 1971 in Davidson Laboratory Tank No. 3, which is 300-ft long, 12-ft wide and 6-ft deep. Additional tests were carried out in June 1972 in DL Tank No.2, which is 75-ft long, 75-ft wide and 4.5-ft deep, to check whether tank 839 Kim and Mercter sidewall influences were appreciable. These tests were undertaken after measurements of wave-induced forces were carried out with the same floats, secured to a different mechanism, so the floats and ar- ticulating linkages were completely disconnected and reassembled be- tween the two sets of tests in the two tanks. Wave elevation measurements were made at three locations during these tests : a) at a location about 10-ft forward of the ''bow"' of the platform model and about 1-ft abeam of the model centerline (the bow of the model was situated 35 feet from the wavemaker; b) at a location about 1-ft abeam of the side of the model at its midlength and, c) at a location about 10-ft aft of the model's stern and about 1-ft abeam of the model's centerline. Wave measurements were made without the model in the test tank for all wavemaker settings used for the test program so that reference measurements without possible re- flection effects would be available. Spring lines were connected to the bow and stern to restrain the model against yawing or excessive drifting. The bow line was connected, through two ordinary rubber bands in series, between the deck of the model and a fixed point at the same elevation about 10-ft forward of the bow. The stern line was connected at the deck and over a pulley about 10-ft aft of the stern, to a 0.10-lb weight. The light tensions in these lines are considered to exert very little influence on the motions of the model. Wave-Induced Forces Large Model In order to study the effect of interaction on wave-induced forces to correlate with motions measurements of the articulated mo- del, the same floats used in the motions tests were adapted for use in a restrained model rig. The sets of six floats were disconnected from the transverse channels solvent bonded to l-inch square times 30-inch wide plexi- glas bars. The bars were secured to a pair of 2''x4'"'x12' aluminum strongbacks which were, in turn, connected to the bridge spanning the 75-ft length of DL Tank No.2. The spacing and staggered array of the floats is the could be measured for any desired row of floats. Wave elevations were measured at three locations : a) 10-ft forward of the row of floats nearest the wavemaker and about 1-ft abeam of the model centerline (the first row was situated 35 feet from the wavemaker for the small model tests also), b) atalocation 14-in 840 Analyses of Multtple-Float-Supported Platforms tn Waves from the model centerline directly abeam of the row of floats in which measurements were being made, and d) at a location 4-ft aft of the last row of floats and about 1-ft abeam of the model centerline. Again, wave tests were done at wavemaker settings for which wave measure - ments with the model present were available. Results Motion Tests Articulated model motions tests were carried out with regular waves in the 75-ft wide Tank No.2 to compare with results previously obtained in the 12-ft wide Tank No.3. Presumably if any effects of tank wall interference were present for the first series of tests, they will not be present in the tests in the wide tank. Results in the form of heave amplitude divided by wave ampli- tude, Z/¢(in/in) for Row 18 are given in Figure 47 as a function of model-scale frequency. The repeat test results of Tank 2 are given with different plotting symbols to distinguish them from the previous Tank 3 results. The differences between the results are not great and similar agreement exists for measurements for Rows 1,9,27 and 35. The results have been cross-faired by means of ''carpet-plotting'’ and the smoothed curves are presented in the composite Figure 48 which shows the substantial tail-wagging. Force Tests Large Model Force measurements results, in the form of oscillatory heave force amplitude divided by wave amplitude, Z/f (lb/in), for Row 17, are given same as for the articulated motions model. One row of lfoats is not connected to the strongbacks but is coupled ot a force balance system for measuring vertical and horizontal wave forces. This row may be located in any desired position, while the ''fixed'' rows are also relocatable so that the forces acting on any one of the 35 rows of floats can be measured. Wave elevation measurements were made at several locations during these tests : a) at a location 10-ft forward of the row of floats nearest the wavemaker and about 1-ft abeam of the model centerline (this first row was situated 35 feet from the wavemaker) ; b) at a lo- cation 27-in from the model centerline directly abeam of the row of floats on which measurements were being made ; and c) at a location 4-ft aft of the last row of floats and about 1-ft abeam of the model 841 Ktm and Mereter centerline. As for the motions tests, wave measurements were made without the float models present for all wavemaker settings used for the test program. Small Model In order to study the effect of model size on hydrodynamic forces, an array of 35 rows of 6 each of smaller scale models was constructed. The scale ratio relative to the larger model was 7/20, corresponding to the ratios of the diameters of the surface piercing tubes, 7/16:5/4. The shape of the floats, similar to that of the large models, was produced by thin-walled wax castings. The damping col- lars at the top of the conical part of the float are stiff mylar film glued to the tube. The floats were attached in rows to 30-in wide bars (except for the row used for measuring forces) which were, in turn, attached to a pair of 2''x 4''x5' aluminum strongbacks which were connected to the bridge in the same way as for the large model tests. The spacing ratio and staggered array are the same as for the large model. The measurement row of floats was connected to a force ba- lance system for measuring vertical and horizontal wave forces. This row of floats could be moved, as in the tests with the larger model, so that forces in Figure 49 as a function of frequency. Surge force results are given in Figure 50, Cross-faired results in the form of carpet plots are given in Figures 51 for heave and 52 for surge force. It is interesting to note that the heave force shown in Figure 51 does not increase dramatically at the stern as does the heave motion, as shown in Figure 48, Small Models The force measurements data obtained in the tests in the large tank are presented in the corresponding figures for the large models. Results have been expanded by Froude’s Law so that small model data are expressed in large-model-equivalent forces and frequencies. Dif- ferent plotting symbols are used to distinguish small model results from large model results. A composite carpet plot of the heave force is given in Figure 53 for these small model results which are some- what different from the large model results. 842 Analyses of Multtple-Float-Supported Platforms tn Waves Discussion of Interaction Test Results Comparison of Articulated Model Tests The results, examples of which are given in Figure 47, de- monstrate that tests with this model are repeatable. Although the "seatter'' of the data points even within a given test program (Tank 3 or Tank 2) is rather large, there is sufficient consistency between the tests to say with confidence that the measured motions - in particular the unexpected ''tail wagging'' - are characteristic of this articulated model (and its associated apparatus, viz., pitch-restraining heave most at Row 18 and low-tension spring lines at bow and stern). The motions recorded in the tests in Tank 3 were not importantly influenc- ed by tank sidewall effects. The carpet plot of Figure 48 exhibits the dependency of the deck motions on position along the length of the model and frequency. The tail-wagging phenomena are shown clearly for all gigher frequencies. The frequency range covered here corresponds to full-scale fre- quencies for which significant wave energy exists for sea states with significant wave height iy te <15 feet. Correlation Between Heave Force Measurements and Motions The heave force results obtained in the large model (an ex- ample of which is shown in Figure 49 and a carpet plot in Figure 51) show important interaction effects on the vertical wave-induced force. For instance, for f= 1.2 Hz, the force in the middle of the model is 42% greater than that at the bow, while the force at the stern is 36% higher. It may also be shown that, for this frequency, the force at the bow is 20% higher than those reported for the tests with the smaller (five rows of five) array except that the outside rows (front and back) are different from each other and different from isolated float results. The fore-and-aft asymmetry of the wave-induced heave force suggests that the interaction may be influenced by either free-surface- type frequency dependent effects or, perhaps, some viscous wake ef- fects. If the interaction were purely potential in character and unaffect- ed by wave diffraction effects (as is expected for slender bodies in re- latively long waves), a linearized representation of the interaction ef- fect on the vertical force due to waves on the jth float due to the pre- sence of the other floats might be expressed, formalistically at least, as iedoln te aa rman sats se oyltyays =) (47) Rake ain lee kf j 843 Kim and Mereter where Z th (=) is the vertical force due to waves onthe j_ float as he: é : j,o though it were isolated, : ’ th and WwW. expresses the interaction effect of the k float on the vertical force due to waves on the j float. The interaction is expressed simply as a function of the distance be- tween the two floats, which would be valid under the assumptions stat- ed of negligible free surface and viscous wake effects. This repre- sentation indicates that the interaction effect should be symmetrical, fore-and-aft. It is not possible, at this time, to say whether the asym- metrical characteristic of the force is due to free-surface or viscous wake influence. An investigation of the effect of wave diffraction ac- cording to a simplified slender body analysis is presently being plann- ed. Dynamic Motions Analysis The response of multi-degree-of-freedom dynamic systems to constant frequency exciting forces can, in general, be expressed as a sum of normal mode components (cf., Biggs 66 or Timoshenko 61 ), which can be expressed for a beam in the form n z(x,t) = 22 A, . (DLF) ¢ (x) (48) st n a where th g(x) = is the normalized model shape of the n mode of oscillation of the structure A a sient $y (x) dx Nst D) 2 w m og “(x)dx n n P(x) = distributed exciting force m = mass per unit length of beam 2 2 th w = (natural frequency) of n normal mode 1 DLF mo Tage NG for simple harmonic exciting force with 1-(w/w ) frequency w , neglecting damping Normal mode shapes, ¢, , may be characterized as symmetrical and asymmetrical about the midlength of the deck (beam). 844 Analyses of Multtple-Float-Supported Platforms in Waves A set of normal modes and natural frequencies for a particular assumed island structure and float-attenuator size have been calculat- ed by J.Rice of Goodyear Aerospace [67] . The first eight elastic "free-end'' modes of oscillation were found to have natural frequencies corresponding within 0.23% of the pure heave natural frequency ! While these results are applicable to the particular large is- land which Rice considered, it seems probable that the articulated model, with its essentially negligible elastic interconnections, will also have normal modes whose frequencies correspond to the free heaving frequency of the float elements. Thus the dynamic load factors (DLF's) for all modes, symmetric and asymmetric, will be essential- ly the same. Then, according to the definition of the amplitude func- tion, An; , the motion should correspond closely to a weighted sum of the distributed load. A detailed evaluation of the response would require significant numerical work but, intuitively, it does not seem reasonable to expect the modest asymmetry of the wave induced heave force (Figure 51) to produce the pronounced asymmetry of the heave motion response (Figure 48). Surge Force Interaction Results given in the carpet plot (Figure 52) indicated virtually no influence of position in the array on surge force due to waves at low frequencies, but as much as 43% increase (monotonic with di- stance from the bow) at f= 1.4 Hz. Seale Effects Force measurements shown in Figures 49 and 50 include the small scale model results. They are seen to be somewhat lower, in general, than the larger model results, but the trends of the results are quite similar. Differences may be partly attributable to experi- mental error. The magnitudes of the oscillatory forces being measur- ed on the small models are of the order of 0.001 lbs : such small measurements are not routinely executed in hydrodynamic laboratories such as Davidson Laboratory. The scale effect exhibited may be due to either viscous effect (Reynolds Number) or surface tension effects (Weber Number). Wave Measurements A few wave elevation measurements were made at locations within the array of the rigidly-held large models. The results show that the wave amplitudes are significantly higher (about 10 to 20 per 845 Ktm and Mercter cent) than in the absence of the model, probably because of the block- age of flow within the nest of obstacles. Plans for Comprehensive Test Programs The results of the exploratory tests have not yielded an explan- ation of the interaction effect, specifically the ''tail-wagging'' pheno- mena, associated with the articulated model motions tests. The force measurements reveal that a significant hydrodynamic interaction ef- fect exists which would be expected to importantly influence the heav- ing response even if the tail wagging behavior were not observed. Con- sequently, it is necessary for design purposes to systematically in- vestigate the effect of hydrodynamic interaction on the heave motions response of resiliently connected arrays of floats. An experimental program to study the influence of float center- to-center spacing, expressed in terms of waterline diameter, float slenderness, deck rigidity, size of array, externally provided damp- ing, and yaw-restraining spring-line restraint, has been developed and will be carried out during August 1972. Three sets of floats, having different spacing ratios (3 to l, 3.75 to 1 and 4.5 to 1) are being built. Each set will consist of seven- teen rows fo ten floats each. The ten floats in each row will, in this case, provide ample roll stability : these rows consist of sufficiently stiff, yet very light weight, T-sections connected to the cylindrical floatation tubes. The seventeen rows will be connected by two sets of plastic splines, one pair at each gunwale, which provide sufficient pitch res- traint for the otherwise unstable rows of floats, and which simulate a specified deck elastic beam-like behavior. The full-size deck stiffness is assumed to be equal to a plate El of 80x 106 ib-ft2 per ft of deck width, a value recommended by GAC engineers. This stiffness scales according to the 4th power of the scale ratio, which is taken to be 1/48, with 11/2-in diameter model floats corresponding to 6-ft diameter full-size (waterline diameter). For the middle-spacing set of floats, an additional simulated deck rigidity, twice as large as the nominal value, will be tested by using plastic splines of the same thickness and spacing but each twice as wide. The float attenuators to be used are thin-walled wax castings with aluminum tubes at their upper ends. The shapes were selected so that their calculated heave responses, assuming no interaction effects, would be the same. The maximum diameters are 1.5 and 1.8 times 846 Analyses of Multtple-Float-Supported Platforms in Waves the waterline diameter, while the corresponding full-scale drafts are 96 ft and 78ft, respectively. Since the full-scale floats are expected to require a hinge to alleviate the bending loads due to wave action on the upper part of the float, it has been decided to simulate these hinges for the present comprehensive test program. The hypothesis that the attenuators will oscillate in harmony under the action of waves will consequently be tested at this early stage. The hinges are made of very flexible sili- cone rubber. The attenuators are to be ballasted so that when flooded with water, they have effectively neutral buoyancy and the center-of-gra- vity slightly below the center-of-volume so that a small positive pen- dular restoring moment exists. The attenuators and hinges can be in- terchanged from one row of floats to another. Tests in regular waves to determine the heaving motions will be carried out with all possible combinations of floats and attenuators, plus the increased deck stiffness for the intermediate-spacing floats with one of the sets of attenuators. An auxiliary investigation will be made of the motions ofa smaller array of floats, 10 rows of 10 each, to explore the effect of extent of the array on the interactions. Other auxiliary investigations will include a brief study of the effect of externally applied (not from appendages immersed in the test tank) viscous damping for a range of frequencies, including as nearly as possible the heaving natural frequency. Some tests will also be earried out without the yawrestraining spring lines in place. Results will be compared with theoretical calculations and, it is expected, sets of interaction coefficients derived. 847 Kim and Mercter DISCUSSION The chief findings of these investigations may be given as : i The vertical wave-induced forces on relatively slender, ver- tically-oriented, isolated floats can be evaluated with adequate ac- curacy according to available analytical procedures (Eq. 34). 2. The introduction of a hinge to permit lateral pendulum-like motions of the lower (attenuator) portion of the float produces a large reduction of the lateral wave-induced load. a This hinge does not have an appreciable effect on the vertical wave-induced forces. 4, Articulated Model Island tests indicate that an important inter- action effect on heave motion occurs. This may be due to hydrodyna- mic effects, or to some kind of elastic or connecting mechanism ef- fects. A simplified elastic normal mode analysis does not appear to indicate ''classical'' linear elastic effects leading to the interesting "tail wagging'' phenomena found in the tests. 5. The vertical wave-induced forces acting on rigidly-fixed floats in an array like that of the articulated model island show an appreci- able effect of interaction, principally on floats in the interior of the array. There is, however, a small fore-and-aft asymmetry of this heave producing force, with the stern row of floats experiencing slight- ly higher forces than the bow but still less than the middle floats. It is felt that this asymmetry of the exciting force is not sufficient to pro- duce the substantial ''tail wagging" of the motions tests. 6. The longitudinal wave-induced forces acting on the rigidly-fixed array of floats show only a modest effect of interaction, which is greater for high frequencies than for low. ie Scale effects on wave-induced forces acting on a large rigidly- fixed array of floats were investigated by testing a model whose size is much smaller than would ordinarily be selected for hydrodynamic testing. The forces measured on the small models were somewhat lower than those obtained with the larger model. The trends of va- riations of forces with frequency are the same for both sizes of models. It is considered that models sizes ordinarily selected for platform motions tests are generally satisfactory and free of important scale effects except possibly for some viscous effect on damping forces for 848 Analyses of Multtple-Float-Supported Platforms tn Waves well-rounded structural elements. This exception is not expected to be particularly unfavorable unless resonant behavior is of special in- terest. ACKNOWLEDGMENTS The authors must express their gratitude to the members of the staff of Davidson Laboratory who have assisted in the investigations which have been described. Mr. M. Chiocco has been especially help- ful in carrying out the experimental work of the large-array float inter- action tests. Dr. D.Savitsky and Mr. E.Numata have given helpful advice on all aspects of the work. REFERENCES HANDLER, E.H., ''Tilt and Vertical Float Aircraft for Open Ocean Operations,'' Journal of Aircraft, Vol.3, No.6, pp.481- 489, November-December 1966. DEWEY, D.B., 'Open Ocean Demonstration of Vertical Float Sea Stabilization Concept'’, General Dynamics/Convair Report 64-197, June 1964. TESSITORE, F.A., ''Tests of the Gyrodyne Tilt-Float Helicop- ter-November 1964", Gyrodyne Co. of America Final Report Y60-313400-1, November 1964. LUEDERS, D.H., ''Motions of a 1/4-Scale Model of Gyrodyne Tilt-Float DSN-1 Helicopter in Regular and Irregular Seas", Davidson Laboratory Report R-881, June 1962. MERCIER, J.A., ''Motions Response of a Related Series of Vertical Float-Supported Platforms in Irregular Seas", Davidson Laboratory Report SIT-DL-69-1334, November 1969. BARR, R.A., ''The Hydrodynamic Design of Float Supported Aircraft II Method for Selection Optimal Vertical Float Support Systems'', Hydronautics, Inc., Technical Report 513-6, October 1968. 849 10 11 12 13 14 15 16 17 Kim and Mereter CARRIVE, F. and JULIEN, B., ''Designing Highly Stable Floating Platforms'', Ocean Industry, Vol.4, No.8, pp.48-52, August 1969. HOOFT, J.P., ''Hydrodynamic Aspects of Semi-Submer sible Platforms'', Doctoral Dissertation, Delft University, 1972. MARSH, K~-R., ''Feasibility Study, XC-142A Modified for Open Ocean Operation’, LTV Yought Aeronautics Division, Report 2-55400/49-963, February 1965. DEWEY, D.B. and FISHER, R.R., ''Inflatable Float Design Study", General Dynamics/Convair Final Report 65-193, Sep- tember 1965. GONSALVES, J., ''Design Study for Establishing an Open- Ocean Tilt-Float Configuration Helicopter'', Vertol Division, Boeing Report R-452, January 1966. NUMATA, E., ''Hydrodynamic Model Tests of Offshore Drill- ing Structures'', Marine Technology, Vol.3, No.3, July 1966. DALZELL, J. and YAMANOUCHI, Y., "Analysis of Model Tests Results in Irregular Head Seas to Determine Motion Amplitudes and Phase Relationships to Waves'', Lecture Notes on Ship Behavior at Sea Second summer Seminar at Stevens Institute of Technology, June 16-20, 1958, Experimental Tow- ing Tank Report 708, November 1958. FUCHS, R.A. and MacCAMY, R.A., ''A Linear Theory of Ship Motions in Regular Waves'', University of California, Institute of Engineering Research, Series 61, No.2, July 1953. CUMMINS, W.E., ''The Impulse Response Function and Ship Motions'', DT MB Report 1661, 1962. ABKOWITZ, M.A., ''The Interim Standard Sea Spectral For- mulation of the 1lth ITTC - A Review and Evaluation of Its Use'', Appendix V of Seakeeping Committee Report, 12th In- ternational Towing Tank Conference, Rome, Italy, Sept. 1969 (see also Technical Decisions and Recommendations No.7-4). PIERSON, W.J., Jr., and MOSKOWITZ, L., 'A Proposed Spectral Form for Full Developed Wind Seas Based on the Si- milarity Theory of S.A. Kitaigorodskii'', New York University, 850 18 19 20 21 22 23 24 25 26 27 Analyses of Multtple-Float-Supported Platforms tn Waves School of Engineering and Sciences, Geophysical Sciences La- boratory Report 63-12, October 1963, BENNET, R., ''A Method to Determine the Response of Ships in Irregular Waves'', Chalmers University of Technology, Division of Ship Design Report, September 1966. LINDGREN, H. and WILLIAMS, A., ''Systematic Tests with Small, Fast Displacement Vessels, Including a Study of the Influence of Spray Strips", Society of Naval Architects and Marine Engineers, 1968, Diamond Jubilee International Meet- ing. FRIDSMA, G., ''A Systematic Study of the Rough-Water Per- formance of Planing Boats (Irregular Waves-PART II)", Davidson Laboratory Report SIT-DL-71-1495, March 1971. OCHI, M.K. and VUOLO, R.M., ''Seakeeping Characteristics of a Multi-Unit Ocean Platform'', Society of Naval Architects and Marine Engineers, Spring Meeting, Honolulu, Hawaii 1971. DALZELL, J.F., ''A Study of the Distribution of Maxima of Non-Linear Ship Rolling in a Seaway'', Davidson Laboratory Report SIT-DL-71-1562, September 1971. OHKUSU, M., "On the Heaving Motion of Two Circular Cylin- ders on the Surface of a Fluid'’, Reports of Research Institute of Applied Mechanics, Kyushu University (Japan), Vol.17, No. 58 (1969). OHKUSU, M., ''On the Motion of Multihull Ships in Waves (I)", Reports of Research Institute of Applied Mechanics, Kyushu University (Japan), Vol.18, No.60 (1970). OHKUSU, M. and TAKAKI, M., ''On the Motion of Multihull Ships in Waves (II)'', Reports of Research Institute for Applied Mechanics, Kyushu University, Vol.19, No.62, July 1971. WANG, S. and WAHAB, R., ''Heaving Oscillations of Twin Cylinders in a Free Surface’, Journal of Ship Research, Vol. 15, No. 1, 1971. LEE, C.M., JONES, H. and BEDEL, J.W., ‘Added Mass and Damping Coefficients of Heaving Twin Cylinders in a Free Surface'', Naval Ship Research and Development Center, Re- port 3695; April L971. 851 28 29 30 31 32 33 34 35 36 37 38 39 40 Kim and Mercter FRANK, W., "Oscillation of Cylinders in or Below the Free Surface of Deep Fluids'', NSRDC Report 2375, October 1967. WEHAUSEN, J.V. and LAITONE, E.V., ''Surface Waves", Encyclopedia of Physics, Vol. 9, Springer-Verlag, Berlin, 1960. KIM, C.H. and CHOU, F., ''Prediction of Drifting Force and Moment on an Ocean Platform Floating in Oblique Waves", Ocean Engineering Report 2, Stevens Institute of Technology, December 1970. HASKIND, M.D., 'The Exciting Forces and Wetting of Ships in Waves'', DTMB Translation 307 by J.N. Newman, November 1962. NEWMAN, J.N., ''The Exciting Forces on Fixed Bodies in Waves", Journal of Ship Research, Vol.6, No.3, Dec. 1962. HASKIND, M.D., ''The Pressure of Waves on a Barrier", lm@hen. Sb.4; No.2, 1472160) 1948. MARUO, H., ''The Drift of a Body Floating on Waves", Jour- nal of Ship Research, Vol.4, No.3, December 1960, OHKUSU, M., ''Wave Action on Groups of Vertical Circular Cylinders'', Presented at the Spring Meeting of the Society of Naval Architects in Japan, May 1972. NEWMAN, J.N., ''The Motions of a Spar Buoy in Regular Waves'', David Taylor Model Basin Report 1499, May 1963. RUDNICK, P., ''Motions of a Large Spar Buoy in Sea Waves", Journal of Ship Research, December 1967. ADEE, B. and BAI, K.J., ''Experimental Studies of the Beha- vior of Spar Type Stable Platforms in Waves'', Report No. NA-70-4, College of Engineering, University of California, Berkely, July 1970. LAMB, H., Hydrodynamics, Cambridge University Press, 1932, republished by Dover Publications, N.Y., 1945. THOMPSON, F.L. and KIRSCHBAUM, H.W., ''The Drag Characteristics of Several Air Ships Determined by Deceler- ation Tests'', NACA Report No. 397, 1931. 852 41 42 43 44 45 46 47 48 49 50 51 52 Analyses of Multtple-Float-Supported Platforms tn Waves SMITH, S.L., ''BSRA Resistance Experiments on the LUCY ASHTON Part IV, Miscellaneous Investigations and General Appraisal Appraisal'', Transactions, Institution of Naval Ar- chitects, Vol.97, 1955, See especially Fig.12, p.542. BLAGOVESHCHENSKY, S.N., Theory of Ship Motions, Vol.1, Dover Publications, New York, 1962, pp.141-143, KEULEGAN, G.H. and CARPENTER, L.H., ''Forces on Cy- linders and Plates in an Oscillating Fluid", National Bureau of Standards Report 4821, September 1956. McNOWN, J.S., ‘Drag in Unsteady Flow'', Proceedings of IX International Congress of Applied Mechanics, Brussels,1957. McNOWN, J.S. and KEULEGAN, G.H., ''Vortex Formation and Resistance in Periodic Motions'', Proceedings of the American Society of Civil Engineers, Engineering Mechanics Division, January 1959. PAAPE, A. and BREUSERS, H.N.C., ''The Influence of Pile Dimensions on Forces Exerted by Waves'', Proceedings of the Xth Conference on Coastal Engineering, Tokyo, 1966. MARTIN, M., ''Roll Damping Due to Bilge Keels'', Ph.D. Dissertation, State University of Iowa, June 1959. RIDJANOVIC, M., ''Drag Coefficients of Flat Plates Oscillat- ing Normally to Their Planes'', Schiffstechnik, Bd 9-Helf 45, 1962. BROWN, P.W., ''The Effect of Configuration on the Drag of Oscillating Damping Plates'', Davidson Laboratory Report 1021 May 1964. HENRY, C., ‘Linear Damping Characteristics of Oscillating Rectangular Flat Plates and Their Effect on a Cylindrical Float in Waves'', Davidson Laboratory Report 1183, June 1967. WOOLAM, W., ''Drag Coefficients for Flat Plates Oscillating Normal to Their Planes-in Air'', Southwest Research Institute Final Report 02-1973 (NASA CR-66544), March 1968. TSANT, M. and ALTMANN, R., ''The Hydrodynamic Design of Float-Supported Aircraft. I-Float Hydrodynamics", Hydro- nautics, Inc. Technical Report 513-5, October 1968. 853 53 54 59 56 57 58 59 60 61 62 63 Ktm and Mercter NORRIS, K.S. and HANSON, J.A., ''Manned Open Sea Exper- imentation Station (MOSES) A Feasibility Study'', Oceanic In- stitute, Waimanalo, Hawaii, June 1971. FROUDE, W., ''On Resistance in Rolling of Ships", Naval Science, 1874. HISHIDA, T., ''A Study on the Wavemaking Resistance for the Rolling of Ships, Part VI, Effect of Motion Ahead on Wave Re- sistance to Rolling'', Journal of Zosen Kiokai (Society of Naval Architects of Japan), Vol. 87 (1955). BAKER, G.S., ''Rolling of Ships Underway. The Decrement of Roll Due to Hull and Bilge Keels'', Transactions North-East Institution of Engineers and Shipbuilders, Vol.56, pp.25-42, 1939-1940, GERRITSMA, J., ''The Effect of a Keel on the Rolling Cha- racteristics of a Ship'', International Shipbuilding Progress, Vol.6, pp.295-304, 1959. GERSTEN, A., "Effect of Forward Speed on Roll Damping Due to Viscosity and Eddy Generation'', Naval Ship Research and Development Center, Report 2725, June 1968. MERCIER, J.A., ''Scale Effect on Roll-Damping Forces at Zero Forward Speed'', Davidson Laboratory Report 1057, February 1965. GERSTEN, A., ''Roll Damping of Circular Cylinders with and without Appendages'’, Naval Ship Research and Development Center Report 2621, October 1969. TIMOSHENKO, S.P., Vibration Problems in Engineering, 3rd Edition, D. VanNostrand Company, Inc. Princeton, N.J., 1956. MERCIER, J., ''A Method for Computing Float-Platform Motions in Waves", Journal of Hydronautics, Vol.4, No.3, July 1970, pp. 98-104, MERCIER, J., ‘Hydrodynamic Forces on Some Float Forms", Journal of Hydronautics, Vol.5, No.4, October 1971, pp. 109- ity ihe go 854 64 65 66 67 Analyses of Multtple-Float-Supported Platforms tn Waves MERCIER, J.A., ‘Hydrodynamic Characteristics of Several Vertical Floats in Waves'', Stevens Institute of Technology Report No. 1481, 1970. OCHI, M.K. and VUOLO, R.M., ''Seakeeping Characteristics of a Multi-Unit Ocean Platform", Paper No.2, Spring Meeting, Society of Naval Architects and Marine Engineers, May 25-28, ali BIGGS, J.M., Introduction to Structural Dynamics, McGraw Hill Book Company, New York, 1964. GOODYEAR AEROSPACE CORPORATION, ''Phase I Technical Report - Expandable Floating Bases'’, GER-15048, 15 Nov. 1970, pp.206 et seq. 855 Kim and Mercter NOMENCLATURE FOR MOTIONS RESPONSE OF THREE AND FOUR FLOAT PLATFORMS coefficient in interim standard sea spectral formulation, Eq. (2) coefficient in interim standard sea spectral formulation, Eq. (2) variance for xt) mode of response of platform to irregular wave excitations, Eq. (6) dimensionless response variance, (e.g., Eq.7) dimensionless response variance, Eq. (10) metacentric height acceleration of gravity significant wave height impulse response function variance of wave spectrum first moment of wave spectrum about w=0 axis number of occurrences of irregular response in given duration of time probability of computed value of response being exceeded in a given duration of time wave elevation spectral density 2m m,/my , or observed characteristic period of waves time, sec dimensionless time, t Ve/yt/3 average apparent response (crest-to-trough) in th mode average of one-third highest responses, or ''significant'' re- sponse average of one-tenth highest responses calculated extreme value of response in xth mode, Eq. (9) heave motion th dimensionless response (transfer function) for x’ mode, Eq. (8a-f) craft displaced volume wave elevation 856 Analyses of Multiple-Float-Supported Platforms tn Waves 0 pitch motion N wave length vr) roll motion w frequency, rad/sec o dimensionless frequency = oN! ae FOR PROBLEMS OF HYDRODYNAMIC INTERACTION IN BEAM SEAS AY waterplane area ey complex radiated wave amplitude ratio ae! diffracted wave amplitude ratio a incident wave amplitude B beam Cog ,ete added mass coefficient defined in Table 6 (C section contour F force or moment F(™) ete wave-exciting force (moment) derived from the radiation of mode m Lear) DM —~ } et Q non-dimensional exciting force defined in Table 9 Green's function (source potential) gravitational constant wave elevation hydrodynamic moment arm mean wave force or constant defined in Eq. (14) integer, moment or inertial mass number of mode of motion or sectional mass two-dimensional added mass integer or two- or three-dimensional damping coefficient origin of the coordinate system hop ay ee tele ee) ee ee Use @) hydrodynamic pressure 857 Kim and Mercter Q source intensity r field point NS) amplitude of displacement, surface or spacing ae ,ete surface at y—~-o, etc. s length of contour, segment or spacing r draft of hull or period t time x,y,z Cartesian coordinate system X,Y,2Z components of force SUBSCRIPTS indicating body lon |@ indicating body indicating diffraction indicating force indicating heave or heaving force indicating wave aes pes 6 Se indicating incident wave i V-1 , or indicating the imaginary or hydrodynamic damping part j indicating jth ,th segment k indicating segment fe) indicating origin r indicating the real or the hydrodynamic inertial part or relative motion indicating swaying indicating rolling indicating waterplane or waterline kh 2 a a indicating y~too 858 Analyses of Multtple-Float-Supported Platforms tn Waves SUPERSCRIPTS e indicating even function fe) indicating odd function m indicating mode of motion GREEK LETTERS a slope of a segment B suffix standing for e,o or (eto) Y suffix standing for m for radiation and 6 for diffraction 6 non-dimensional damping coefficient € phase angle ~ velocity potential r wave length v wave number p water density n the y-coordinate of source distribution, swaying motion or suffix denoting swaying E the z-coordinate of source distribution, heaving motion or suffix denoting heaving w circular frequency FOR SLENDER VERTICAL FLOATS AND LARGE ARRAYS A frontal area of appendages, Eq. (43) Anst model static deflection, Eq. (48) Be waterplane area Coy added mass coefficient for heave Ca damping moment coefficient, Eq. (43) Cyp vertical prismatic coefficient, Vj ee c! ratio of added mass of damping devices to added mass of at- tenuation, Eq. (36) 859 Bes a= yo a iv) md e a 0) fe) 44 ~ F Pa ° o. Ktm and Mercter ratio of damping to critical damping coefficient dynamic load factor deck rigidity force frequency, cycles per second acceleration of gravity wave number = w*/g = 2n/r added mass coefficient for float-attenuator in heave, Eq. (35) length of attenuator moment total added mass in vertical direction element of added mass in vertical direction added mass of attenuator in vertical direction added mass of damping devices in vertical direction radius of attenuator radius to center of area of appendage, Eq. (43) radius of float waterplane sectional area of float body draft of float interaction effect of kth float on vertical force due to waves on jth float of array, Eq. (47) horizontal coordinate, along deck vertical force vertical coordinate effective depth for evaluating wave acceleration for element of added mass, Eq. (34) displaced volume wave elevation wave amplitude wave motion evaluated at depth corresponding to assumed damping source 860 Analyses of Multtple-Float-Supported Platforms in Waves angular coordinate, Eq. (43) wave length scale ratio, full-size length/ model size length fluid mass density frequency, radians per second natural frequency 861 ercier i“ Kim and I 1 €0 gh £00 700 Lo C10 €00 499 990 e590 850 090 2£0 L+0 900 $00 gid Z10 200 uny S91 26°4 02°9 92°9 6€°S 65°S 65°S 6S°S gz°e 3 di°t €9°¢ €2°& als eL°€ gz°e elt Livy eg°e el°€ eZ°¢ Gis e/t4 uroed§ 3e0|4 [19H UOTPeANSTJUOS (UIAOJIONAD) yeOTJ-ANOJ YM STSpour JO SOT}STABjOeIeYO SuTTOjye[d yeoT] Jo selszas poyveTey = We!lg Jeo,y Buimsweig yeoly [INH seos weeg = W21G 2eo[4y Bulm/weig 3eOl4 {NH sees pesy 052°0 L£Lz°0 Zgl"O gin 0 Lyl‘o 0£3°0 ,(50°°) 40i°0 " gol*O LLz"0 861°0 LLZ°0 861 °0 iLé"0 Tow | e(ec*~) 0£3°0 piel ~~) 0£9°0 / cee 0 LLZz°0 Z€Z°0 Llz"0 Z€Z°0 LLi°0 HOLId Sz" 459°0 tLe Zg1°0- LL2"0 7z9°0 0£g'0 5 Cais ~) 622'0 ” 102'0 LL2"0 102°0 LL2"0 10Z°O L120 HLId S*t Ppeinsesy poe ynduo) ¢ po/WS = WeIg yeojy Bulmsweig yeoly | [9H sees peoH wowtTnmnwowowov”o ° wowstaowowowvwo . — ee ee ee \O fF 00 t+ CO weig .eol4 weig e3e{dg buidweg T eTaeL °8 dNGud “VV dN0ud weig 3eo;3 374249 2eO|4 862 Analyses of Multtple-Float-Supported Platforms in Waves Table 2 : Related series of float platforms characteristics of models with three-float (triangular) configuration GROUP D: Head Seas, All Floats Equal Float Draft Damping Plate Diam Pitch eu/vl/3 Radius to Float ¢ Test Float Diam Float Diam « Computed Measured gl/3 Run 4.0 a 0.366 0.271 3.16 074 4.0 1.4 0.366 0.271 3.16 091 4.0 ee 0.366 0.271 3.16 087 4.0 1.6 1.098 (~1.0)”* 355 075 5.0 1.6 0.366 0.300 3250 096 3.0 1.6 0.366 0.716 2.76 083 “not measured 863 Ktm and Mereter. Table 3 : Transfer functions for heave and pitch, Run No.002 Four-Float (cruciform) configuration head seas tests Hull Float Diam/Wing Float Diam = 1.5 Float Draft/Diameter = 2.0 Damping Plate Diam/Float Diam = 1.6 Metacentric Height/v/* = 0.201 WAVE-HEAVE RESULTS ® = wWy/g Izt/ICl Phase Coherency 0.0 1.000 0 --- 0.244 0.867 22 0.97317E+00 0.305 0.990 -. 0.98395E+00 0.366 1.011 h, 0.97982E+00 0.427 1.001] qe 0.98670E+00 0.488 1.015 Wire 0.98358E+00 0.550 1.037 18. 0.98095E+00 0.611 1.051 26. 0.96360E+00 0.672 1.047 38. 0.93955E+00 0.733 0.992 56. 0.90799E+00 0.794 0.700 83. 0.73138E+00 0.855 0.378 119. 0.71285E+00 0.916 0.223 136. 0.74496E+0C 0.977 0.120 156. 0.83387E+00 WAVE-PITCH RESULTS ePIC 0.0 1.000 90. --- 0.427 0.289 112. 0.49342E+00 488 0.273 big. 0.77833E+00 0.550 0.241 115. 0.8971 1£+00 0.611 0.206 116, 0.93136£+00 0.672 0.177 120. 0.94658E+00 0.733 0.147 126. 06.95218E+00 0.794 0.103 135. 0.90537E+00 0.855 0.065 146. 0.89384£+00 0.916 0.046 153 0.88248E+00 0.977 0.029 165. 0.69802E+00 1.038 0.011 251 0.16130£+00 1.099 0.026 315. 0.65058E+00 1.160 0.027 320. 0.58830E+00 "4 positive phase angle indicates that the motion lags the wave trough (max. positive amplitude) passage, 864 Analyses of Multtple-Float-Supported Platforms tn Waves Table 4 : Impulse response functions for heave and pitch, Run N.002 Four-float (cruciform) configuration head sea tests Hull Float Diam/Wing Float Diam = 1.5 Float Draft/Diameter = 2.0 Damping Plate Diam/Float Diam = 1.6 Metacentric Height/V¥* = 0.201 z -25./2 0.0009 -.0001 -24.01 0.0000 -.000] -22.29 -.0004 0.0006 -20.58 -.0014 ~.0004 -18.86 ~,0006 -.000! -17.15 0.0020 0.0003 -15.43 0.0000 0.0002 -13.72 -.0053 0.0002 -12.00 -.0037 -.0007 -10.29 0.0050 0.0009 - 8.58 0.0081 - 0006 - 6.86 -.0002 0.0004 - 5.14 -.0073 0.0046 - 3.43 0.0183 -,0073 - 1.72 0.1066 -.0130 - .00 0.2208 -.0026 1.71 0.2428 0.0028 3.43 0.1108 0.0132 5.14 -.0583 0.0102 6.86 -.0968 -.0081 8.57 -.0100 -.0028 10.29 0.0533 0.0040 12.00 0.0250 -.0017 Wo 7/2 -.0217 0.0003 15.43 -.0203 0.0003 17.15 0.0063 -.0009 18.86 0.0141 0.0005 20.58 0.0023 -.0004 22.29 -.0070 0.0005 24.01 - 0045 -,000] 25./2 0.0020 -.0008 865 Kim and Mereter O8+ILLv*gDg 2O+3z2Tb*aQ O6+I6ve'D 00+3062'@ 08+3L92'9 O9+IT9T'g O9+36Tt'yg 89+390T'S T9=3686'O Tg-3TTe'a 39-3229'@ tas36e6'g Ta-3e8ee'g Ta-38¢2'g TQ=3TOT'g@ 20=36T2'S €/l 20-3S0T'S £0-31506'°D £2-3S92'D Sy-3IlL9'B O-3E2S'°O £G-IL9C'D So9-3ptc'a SY-358T's S2-396T'D £6-392T'S y2-3796'2 94-3049'°D b2-3£00'O bU-328T'D GJ-3S9Sp'a 99-3LE>'D TO-3ebT'Z TO-390T' os 20-3094 20-3926 4 CM-3uGe' ad 20-3vAzZ Ss CO-3C84°S SB-39az' EA8-3tSs’ ED-jtTr’ €O=-3992' £B-J6Lt’ ¢O-36f6' pA-jrse* G@=3bpy’ 9f=35%¢° es t. Beas uaek&t Ga ° =n =. 1c il Tg+5268'°O Ta*76fk a Tgo+3pe82°D Tor+aced'd TH+ 3267'S Ty+aapta De*32L6'°A ZE+I298'°S QB+5~9l's @2+36S9°2 20+3TSS'°G G2+38fb'°D OB+jTCL'* YZ GB+566T'2 TQ-3262°D 2H-39Lb'°S Ina) a. > (aa) ~ N dazoweig/zse4q 7zeo|4 we!q zeo,4 Burm/weig yeol4 1LOH bO-4ped'w bO- Beg" wn tO-FaAo*u bO-Je2y'u b2398e as ber3k¢)!'y BaA~3lb tt a ArILS hw bA-3LOT cO-F98e'a CO" Ftp9' a CO~W6bb' wv CB-3tle'a CO~3uTT ia ae Ei Ae S@*36TT sé ¢ A/7y618eH D14}UeDeaW AP zeojs/weig e3e,d bul dweg $180} Sees peasy uolyeundsijuoo (uItOjJIonAd) JeoTf-4n08 F ZOO°ON uny ‘sesuodsed uinajoeds QL*°3296'B Oe+3ZtL'B @4+352S'°0 Biiracrs's Outjer2'o Zi+3Zcot'a Tw-3066°0 Tu-39dr'o Tu-39S92'D To-3TL2'D Te-56nT'O Ta-3dct'D 2d-37v9'°D 2g-3lye's Sv-3Iule'D Gu-3sSvt'so eg eTqeL, mn — Seucqgocqaeqgqw4ditonnmn Ss am 866 Analyses of Multtple-Float-Supported Platforms tn Waves O9+36Gr'g 23+396o'¢ GO+IVES'g’ 00+3692'9 Q8+3rt2'y OG+3SST'p 1O-356T6'G Ta=3862'o Fg-32989'2 Tg-3226'g TH-399b*2 Ta-319e'g 19-3662'Q Tg-309T DB CO-382L'¢ cQ@-ijttz'a e/t £¥-3698'°D £4-38EL'°@ Sb-53r6S'D $2-369r'D SY-3JOSe'B 2-3822°D S2-3avyTtt's? p2Z-3Lp6'D pd-3L9L°S bd-3TZ9'D bd-3lor'se bii-3LG2'°O bY-398T°S G69-3S88°d 6£-3TL2's 92-J6e8o'¢ e/ 1 102°0 TO-3ST’ 24~-3646" cB=39re89° CQ=-3£G7° 20-4992" c@=3tgt® SA-38¢5° Sh=326¢° SA-3c6c' £9-35¢e° £O-39F7° vy2-392TeE’ pda=-JoTy’ pZ=Ft9T GM=-3bos" 9G-324¢' e/t it dé SI Il Se Ba a & & cg, TE+ipty32 TE+59G2 2 Ta+3Toeea Te+rd6ve'a Tergece's TE+3¢ST°¢ CO+3866°S Pg+ 4688" ¢ Gg+jVyl£ a EBt+I9IL9I'*A O2+492L99°3 GZ+3psr's Cur jJCl gs Cg+30ce's T¢-35966'% Ch-59C6°2 bO-3082 i FO7-969y'u TO-3b9g hu TA-3c9v wu BOWS 9e ec bO=329? "a TO-9eSb iu Riese bls beesett as e@-3rte6'u ARNE 5 ys eEveaers ' 3 cyesaces* Z20-386T! r@-399¢° Sa-38Gy7' be st ISS its: on ™~ °N A/iyHreaH Bl4zUa.e.aW welg Jeoj4/werg aye,dq Bbulidweg Jozawe:g/z4se4qg yeol|4 weig jeo,4 Burm/weig zeo|,4 | {NH S}S9}] SeoS peoy UOTLEINSTJUOD (UIAOJIONAD) yeOT{-1n0 | ZIO°ON Uny ‘sesuodses uintjoedg « © Ti+adut 2o+I208'SD Gar dL9G°C Cet3leK'e Baral Byiet36vt* Ti- 3229 * To-4G6P' Tim aSul’ Ta-49ud° Tu-35Tu2' td=36en" w= J6EL'S Cur Iewe Sur ITI SO Gv-4a9rS'B ag eTqeL Seve Soaqanvsss 867 Kim and Mercter 0G+39¢S'A QO+3Ser'o 00+322¢'g 00+3098'¢ QG+iITaS'D QB+3ve7?'gD Q0+396T'g Q8+3IB¢T* gy MA+3o2T'gS 0G+390T'@ Tg-3r8e'D 18-3969'@ Ta-3tag'g Ta-3ras'g Tg-392T'@ Z2O-3LLT* Oo E/1, sj1so}] seas peoy 2O-325T'D Cb-ADv T'S CH-3TCT'O CB-3LOT'O £6-3228'°D SY-3yT9'°S £V-I3I9IIE*D £G~3ASTte'o ¥O-3692°O £B-IALt?'D £9-3T9IT'O f2-3¢TT'D bY-3659°B pd-3882'°O G$2-38S59'2 90-3802'°D L0Z* TA8=3998T'v TO-JLbT' oD TH-4tTt sé 29-32TE' w 20-39596'¢ CB-3bos' ed CA-3cSt 2¢s3cZzh SA-36b6 E2-35a~¢'° 2O0-3v8r' SP-3ree SO-32L5t pB-362¢' GA-3u66' 90-3487 ° a er @ aSee& ¢ & Sa & E& & ive €/t Ul 0 l (4 l ad Moo TG+396L'A TO+4Z2v20°¢ TB+IaA9dVse'D TH+3agec' Tg+4daot® TO+abrt sé OB+I32L26' @2+39¢8" Zua+jvcel’ Zg+3tc9's OB+4b~TS° Z@+ iter? Ab+j28e° 2y+3,9T° Te-32¢S° 22-3622" SS GS QBVeyvsewvwavseas al ss > oy ~ N A/3y461eH D!14}zuUe9e79W we!qg zeoj4/weig e2e|dq Bbuidweg Jazoweig/z4e4g 3e0|4 = weig zeo|4 Bulm/we:g 3ze0(4 LILNH te-3tés' TO-Fves! tZ-36ep! T2-386L° TA-3eTe! bO-36TC" b2-3pet! be-39aT° e@-39Ly° Ca- 3969! cg@-3cte! CQ Avpe® 24-3681! e2g-3b69! pO-3tTe! 98-3¢9c' ct. tc. Loy Geen sae GSS ef & se & oTjeanstjUuoo. (WITOJIONAD) }eOTJ-1N0 F 810 °ON uny ‘sasuodsea urnajoeds OurIjBu6'C ZurjAGzrlL°S MurIjZ2ts’so BitIGve'B Biurilee'@ Gu+3AIet’s Tur 3lZeg'a Ti-3Scer'D Tumalce£'@ Ti-3tr2'o Ts-3S9T' A G-38wt'o Cu -4196b'°D 2u~46gT'? Si-sapdt’@ 9¢S4z2CE" 6 og eTqeL 868 Analyses of Multtple-Float-Supported Platforms in Waves 2G+3629'¢ 00+399G'g OB+326b'D OO+38Trv's QG+I8be'g 20+3892'g QG+3TLT' os GAsItSt'g QG+ACETt'gs 00+360T'@ Tg=3088*2 Td=3599'g TO-349b'g Tg=+3622'3 TO-3LTT po CO-3ILET'S e/l, €d-3502'O tO+399Se°sh TO*+3000'S tO-dueL'u 2O-308T*S W-340C'v TQ+5SpS*4 a-4S5T9*& 26-39ST°G TWa-9nSb's Ta+3682°W Ta-dpts ti 2O-J9TST*A Th-360t'w To+3eee*4A ba-3e2p%u CU-3bVT*D CH=-39S2'°u Te+af6t oe Taraaes *u SG-3vSL£'D cB=-3lov'u Ta+a4vt og te-atpe'v EG-3862'°H CH=-3y87T's Zo+d696°O FAa-3sbt is SP-ALEL* BA CB=-39LHT s OG2+3298°S tbe-3r2t ev EG-3692°HA c2e=499HT'S GGV+sG9L°A VOrsavpat sy SO-3202°8 &A=-J6¢L'°2 G06+3T99°AZ cA-aury su PG-F6bT°A LA-3Eep's OG+498SS°h ch drpgiu p6-3626°@ LB=399Le2°D OQ+56LP°A cB-3sSy'u vZ-4309G°D LA=3vet's BO+stcl*A cb3elcetu p2-3Spe'G vo-3eer’s BO+3I8@6T°A CUAL TT sh GU-3096'@ sSO-3cSGu'u DSI L/L yh WS ElA Ie oid 92-3682'D 9M=30ec'% Ci-4b8b°B sa-deet a B e, e/14 eo a“ €/\ e J z q ed SL°07= gj o/ tue eH 214} U80e7 OW g°| = weig jeoj4/weig eze{d burduweg O°Z = 4e}z0wWeIGg/iJe4q 7eO[4 G°| = weig yeo,4 5urmyweig 3e0O14 11H $189] Ses peoy UOTEINSTJUOD (UIAOJIONAD) VeOTJ-1nNO YF coo*ONUNY ‘sesuodses uInaj}oOed¢g 2+3666°D Qi+Ipvd'a w+ dees’ B Out JLo * B- Baralre’s Qi+j9eT’S Tv-390G°S Ts-4690'°D TE-3S9L°SB Perdaele°o@ Tu-dTot’@ bie 3tet'e Cu-41S79'C Cu-jASrd’s Sir IOvl’B w-A9pT'D Pg 98TAeL 869 on ~ Kim and Mercier 20+I6TS'D O0+3¢Sp'a 09+329°'O 09+3c7o's 09+3292'°8 00+169T'G 20+360T'O TG-3606'O BO=3L6L'D 1g=3969'¢ 1g-3926'9 TO=-3000'o Ta-3682'2 Fa-382T*o CO-32T8'D 20-3602'H E/l, ZO-3ASTtt’Se 2Y-3ITAT'D Sb-3Ttyse'so £2-35929'°@ £6-3905°O S6-3S2e'D FB-3LST'A £2-3SeT's pi-3T66'O b@-3092'°O pQ-3fS5'°S pZ-3rle'*sD bd-39CC'D pd-ABZT'D GG-F6TL'D 93-3eS6e'OC TO=-3097T'D TO=362T SG 20-3526'd CH-3Lb9'°u CO-3t2p' hs 2G@=4v2e2'u EM-3LbL'o SA-31G5'¢A SO-3L60'0 S2@=369¢'9 CO@+3CLt'°s pO-3¢66's pO@=322Go'2 pA-3661° 4 G@=-3uTp' Ss 98-324¢e'4 t/t iNO.w™)O sO — tS Me) i} TG+*390b'°S Ta+4gce°D To+af6c'°D TO+3GV2°SD TQ+3S6T'D Tg+a8yT'sd A9+35696°23 QOZ+IZ268'S 20+ 3¢v2°s ~OZ+*sIgl9'*D 02+39c5'D @Q+ibto'a Z2+3862°D ZG+I6LT*d TE-5699°S 22-356S2°D nal ™~ > oY > N T@-342e2' bO-3yTg! TQ-3yTs' Ta-36Tp! TO-Jo2e' Te-3rec' Ta-jeet’ tOraptt oe CBr IL 76% ui COILS L 22°73599S'w CEOMALEE SS 20-3S22'u SB" 326 wv $A-3S5Gt'u JP" F3OSy'u G: & S&es A/7YyH19H Dl4Azusse ew weig zeo|4/weig e3e|d buidweq dazowe:g/7ye41g 2e0|4 we!qg 3eo,4 Buym/we!g 32014 11H $j1s0] seas peoy UCT{eEAINSTJUOO (WIAOJIONAD) JeoTf-1znNo0Oy 900°ON Uny ‘sasuodses windjoedg Tut geet ’D AwrIGul'D Co+3jSrSG°o Air 5GVye’a Buorjl07'sB Qstjaze¢Tt'oD Tu-3229'°0 Teratsoy'@ Tu-i9pe'é Tu-3p52'°B Tu-ardt‘'o Ty-32ut'D Zu 59659'°C Cort avde’s S¢é-3262°H 9u-48sl'°D ag eTIeL 870 Analyses of Multtple-Float-Supported Platforms tn Waves UG+328r'gQ QO+I8Trv'g BDA+stce'g 00+3682'¢ OJ9+ICE2‘*D 2B+IAZLT'g” OH+I72GT'*D tO-3ela's Ta+32oz'@o Tg@-412T9*g Ta-398r'o Tg-399¢'g Ta=3bSGz2'g TgZ-3TGT’*g €2-3689'¢ 2O-3f6T'D e/t £-3686'DO fy-38es's S9-3T69'D £G-3f66°A So-3ebe’D fY-3lL2'°B SU-3Tet'D £B-390T'D bQ2-30>8'°SD bG-39£9'°¢ bH-3ALGH'D y0-3960°S b2-3269T'D GU-3Sr2'°S S2-3602°C 92-Aceo’D N © iS) i] LN LN WO —wN — Il TO-3Sbt'S TQ-362T'o 22-9229 CO-4382o'd CO-AL vee CO-39Bi'S SO-39b9'a LO0=-3e8y'o SI9-3rvc’s SG-3pec'z CO-Jebi'’Sy bQ-35Su'°2 p2-320p°s pA-Jcri's G8-396G62°« 9A-38E2°s e/1 T2+3268'°D TOrIGee a TO+39L2°S Te+58ee°D Ta+392T°2 TZ+agpetd 04+3828°o YOZ+ived’e Ev+3T69°D Gg+3¢6S°¢ OZ+43G6v°2 Z2H+i2ot'sY ZB+sAstc'D QG+43oct'*s Tz-3pd9°2 CY-ISSL GD ile e/1, A/7yub1aH D14}Uuase {OW T2739 by® TO-adcps’ TQ-alpp? bOr739Ge° To-362c° Te-3u6t? t2-49TT CO73b66'; COT ILS! eg73atzy? c@-3auTs? égr-3ascce? ce=attz2* rd-392%6° v2-36ct" yg-3d¢c' weig 2eo,4/weig a3e{d Bbuidweg doJowe!g/zse41qg VeO|4J weig 3zeo|4 Buymsweig yeoj4 1{1NH $189} Seas peasy UOTeInsIjuoo (uUITOJIONAD) yeoTJ-1n0 WF L¥O°ON Uny ‘sesuodsed wnajoedg Juritso'D Bi+acvl’@ Ouiraddrv’p Ode TIC’s. Ou+iv6Tt’s Zo+zeit's Ti-3dub'D Tu-3Suet’"O To-3842'@ Ta-ate2'*g Ty-3f45T SG 24-3656'D C47 38uS' SD é-jtot’eg Sur-Abec’s 94-3569 '°B C7z> Zz 3 > JG STdeL BH nvnvnoanoow ©) 871 Ktm and Mercter 60+3S6¢'D QD+ivSp'a 0O+I0LE'*S 0G+3ITTS'*g 00+31962'G 68+3S6T'S 00+392T'S OG+3CTT'S TQ-3766'@ Tg+31298'@ Tg+1222'o TH=362L6'2 O=362r'S Ta=-3¢b22°o Ba=3S52T'S 20-39LT'O E/l, CQ-3I9tt'S CU-3L0T'O £2-3088'°o £0-3r8L'D £0-358S'°D £2-3I2cv'D Sd-3alpc'D £B-3LT?'L £0-30eTt’sS £Q-ABHyT’D S2-ISTT'D bO-3¢c8'D p2-32TS'S pO-3672'° SG-3859°O 92-3002'°D TO-39LST' TQ+396£°S TH=3LTT' Ta+3622°O CO-39LS5'°h Tg+3te2'¢ CO-IvB9'L TG+39C2°D 20=3887'v To+apet'sa CO=3ul2's¢ Tg+3Tbet' a 2G=-3JuOTt’u 22+39b6'°D SB-3962'0 20+43TSB'sD S2-JuTy'o 00+3¢pS2° SB-329p'u 2O+i1299'°D EB-3925'° 22+397S°d S0=3600'h2 2O+I18lb'D CB=-3jSTE'w QZ+34Scl'D p2=369p'o QB+ALe~e"|D GQ@-37246'6 TQ-3288°2 96=3L6t'2 22-3TLL'°D A : z Zz, 459°0 = 5/4/3248 12H D14}zUa.e OW b4=3989%u TO-3t8g' we TO-328b wv TO-3f6E 8 TO-3eto ty TO 3622'& TO=3obT'w FO-322b we FOr Alar ie C= 3aroe'i CO"3LS9'w 20-399 % wb C2" 3u6c's CO-3Sft ye S8-3vee'a 6273982'u | = welg jeoj4/weig 93e1d Bbuidweg S°| = 4e}ewe!g/i4e4g 3eO[4 G*| = weig zeo,4 Bulm/weig yeoj4y | |NH $}]S93} seas peoy UOTIeAnSsTJUOD (UIAOjTONAdD) yeOTJ-1Nn0 YF ZE0°ON Uny ‘sSesuodsaz uinajoed¢ 24+39826'°O QeEr398td'a Bux5i26p'D Oe+35oeo'B Zarjtt7e'o Gd*ipet’s Tu-309S5'O Terde2ap's Te- 35528 'SC TeE-3L9C'6 Te-48ub'G Te-30cT'O Z2¢-3099'°D 24° 35L92'B So-326b'D Gt-3spee'’s fice 3 Zz 8g eTqeL, (ya) “~ 872 Analyses of Multtple-Float-Supported Platforms tn Waves 8+320¢6'@ OB+3e¢e'D 23*36LE'O BA+I6TES'D 2O+3£92‘*D 08+3082'o QB+IZ2T'*o BA+IOTT'S Fg=-31T86'S Tg-39f88'Do g-38989'S Tg=3986'S TG=368e8'S Ta-+3Tr2'a TA-3Sat's Z2O=3Spt's g/l, S}S9}] seas peoy UOTLeANSTjuOD (WIZOJTONAD) JeOTJ-4n0 q 090°°ON uny ‘sesuodsey urntjoedg 2Q-32CT'D c2-390T'D £B-3586°D £Y-32G2'°B £9-376S5'° £4-38Ttb’ so fZ-3le?°oD £6-3B6T'O £4-3r9T*D f2-3det’o b6-3086'°D pG-3929'°C@ 96-3800 'S bU-3LE8T'S SY-30S¢°D 90-3LST'D SE" TA-346t'd TO-3c72T'D 20-3468°0 23-3492 2O=3C8p'o 2D=Jb6ve'o CO=3tOt'u SA=39262'2 SO-9109's¢ SB-ILEy°s SG=3962'oD EGg-3tet’o pO-35b6's pO-379e' 2 GO-3289's 98-3thT's egy 9 z 9° 0° TG+386S°B Ta+3crve'o Ta+iv9se'o Ta+aTS2's Ty+398T'O TO+3Gbt'D Ag+jpl6°g Q6+39T8°d GZ+3LtlL°o ZH+4Bt9°s 20+351S°@ OB+39tb'D @o+32ee02°B GQ+43T6T'D TZ-39v8°S 20-3245 °9 on ~ b> ia) ~ N bQ-3S69'2 Ta~-362o'u ta-3elp'o TA=3L8E's4 0 Aan bo tO-39TC oO tO-aLet'e TO-360T'd cB-35TG6' se 2O-3TbL od 28-3696'b ce 3ldr'v 22-38b2'o ZO°3STtL'o EB-3A2¢e'wv SQ@-43LLT'h A/34b 18H D14}Uade7 EW | = weig jJeo,4/weig aze;d buidweg Z = 4a}owe:qg/i4je4q .eo|4 [| = welg 3e0,4 Bulm/weig 3e014 [INH Gurtj266'°B Ag+3022°S BurtIlso’D ne ha ab a 4) Cer+aotre’so 2d*3ecT'o: Tu-320S9'°C Tu-49Tb'D Tu-3accft'@ Tu-3622°O Tu 59ST'O Tu-3S5T'D 2H-30LS5'°O. 20-3600'@ Si=-390b'°O Gu-32te'o : Ug eTqeL 873 Ktm and Mereter QO+I3LLb*D OG+3ETo' gs @G+39So'D 28+3962'¢ Gd+36E2'B 2G+I9LT'g GO+3rat'ga Tg-3£68'¢ Tg=3262'0 FG-3bT9'o TG=36L>'o Tg@s306¢'@ Fg-38f2'o Tas3be t's Za-3ef9'¢ 2O-30bTt a t/t 22-300T'S T-3cbT's TO+3e6l°S@ W369" wv £2-3669°O ttH-360t'S Tg+3eve's Tta-3seste £Q-3STL°S 2H-3t6L'd To+spec*s te-3S8r's EQ-3SL56°O cO-F¥bG*y Te+3Z02°h Wraps 'e EV-ILLh'S 2d-3LSE'v TO+3LGT*D TO-F60e'w f2-3t82°'A 2H=3vbT*e Tg+3tot ds ta-3bec'yv SO-32OT'D CH=3849° GG+3pt6'S@ FAaTITEI' sd £G-3A9ZT°' OD LB-386y's @B+39TE*S TTErFJeTtt wo bO-3L28'O LM-Arse’d OB+36TL*S 2H-3v96'w pa-3TT9'A Le=-3982's OE+3L29°S CB-3eSL' uo b2-322b'D LH-3LbT' a OBealeS*B cB-3ALtgo'e PB-3292°H pO=329L°S GBZ+3TCho* Ss CO-3uSe'u PO-3BST'D PA-3PSe'’a S6eiOtS*S CB-3Ab6c'u G@-3289'O pasqelt’s @2+360C°S cOr3LbTt wy Gd-3T8T'S SO=30SC°% TO-3e246°O LArAces'e 90-3b9T°D 90=322T's TO-3OET'D SB3t9st« 6 85 e/i4 eo AP g/l, 25 e/t ZEZ°O = ¢/pO/248 19H DF4qzusseJoyW H° | = weig zeojs/weig aze,d Buidweg O°Z = 4e}zOweIg/je4g Jeol[y SZ°b = weg yeo;4 Gusmswerg yeot4 {INH $j1S9}] Seas pesy UOT{eXNSTJUOO (WIIOFIONAD) yeoTJ-1n0 ¥ 8S0°ON Uny ‘sasuodsaz urndjoed¢g O4*3806'°O Gd+3GEL°D @2+390S'°A GBitzZg9ee °C Burzebe'*o Ber+jpet's Té-32eS'°H Te 39Tb'O Fe-3fcf8'D Tu-32r'e'o Tég-atat'o Té-3Tttt'so 24-39¢9'°D fu-3242°SB So-326S'°2 62-3929°D Tg 8T9eL 874 Analyses of Multiple-Float-Supported Platforms in Waves 0@+39G9'g@ 2d-398T°D TH-3vE2%'s TO+3Z6L°D TA-F7094'*w OB+38es'g Zd-3v9T*D TO-3Tet yv Te*3dZ20°A TWe-3eao'e 00+399b'g 2yu-AToT'D TTH=39ET'D TQ+3982°'S O-3t9y'wu O8+3l6E'G 23-36TT'A 2O-3P86' TQ*36e2°O@ 9 ta-aedse 2@+3628'g C2-3Sb6'D 28-38L9's Ta+apet 's tB=Ju62'w O32398G2'H C4-30L9'D 2H-366e's TO+96ET*O TA-dvaz'w OO+329T'g LCO-3bdE'D CH-9SIT 00*3p68°O te-Astt ic WAeIbot*g LCB-39TE*D 2-362 O0+3662°D 20-3Ll6%d O9+3IG2T*g LCH-3092°D CH-3L4'vV 80+3969°O cO-F805'o O0+39@T'o CB-3SH2°'D LCO-3eIL'w 22+396S°9 cP-3ub9'e TQ-35998'g LO-326T'D LN=-389y'sD Q@B+3IT6b°O@ cAar-3cldy's Tg2=3029'f FB-3bgTt’s CH=-3082' OG+3A268°O cH -3ste'u Tg=-36ly'Qg b8-3229°D CH-3Lbt’s O@G+3720°9 COr-3edt'«u 39+3962'G bv@-3682'O 8=360G'v 2O4+32ST°A LU-jrOo'wv TOH+*3ICOT* AS SU-3292L'D pO-3eTt's TO-3byS°'2 vyO-3tb6'wu 23=3T22*y I9H-3dpe'’D Y9G-3Sso'd CO-3te?°’A 3 8O8-3yhc wy E/t 6 Q eyih egi2” 8 ei ycrs : e/1, 25 eit 3 ZEZ°0 ¢/O/ 248 19H 2}4}uU209e20W g°l = weig yeojs/weig e2e1d Buidweq O°Z = 49}0We1G/24e4Gg }eO|4 SZ7°1 = weig yeoj,4 Burmjweig yeold 11H S$j{S9] Seos pedy UOT}eANSTJUOSO (WIAOJIONAD) }eOTJ-41Nn0 WF Z90°ON uny ‘soesuodsey vuinajoeds 0i*36S56°O 04+ 3602°O Bé+3T6'O O+3LEL°D @w+a2te’s@ Q2*jTCT’O Td-3089°A Tu 5568'S Td 3208 '°D Té-32c2'@ Tg=3TST'O CH= 3ILT6°O 24-3950°0 2é-3LST'D LL-3ISsT'D 92=3b9f°D : [gc aTqey, KOL'y 60S°s wOe's ZOS°zZ cBu'2 COS°T oBG°T EO6°2 £Oe*y eBL*S 689°G uGS'°2D dOp'g wBl*g wBe*g eOb gy 875 Cter mand Mer . 4) v K 08+32L26'¢g 00+I19p'g QO+3L69'D B4+392o'@ OG+3I£97%'g GA+Icb6T*” OO+I3ITTT'*g Td-38b6'D bO-3reL'B Ta-3829'g Td=3¢blo'o Ta-3420'2 TA=3002'2 T@-32aT'g CO-386>'@ CO-ISST'*S e/l, C2-3B2T'D 2@-3f0T'OD £3-3069°@ fY-3829'o £0-3S05'° Sg-38TeS'D £0-F36ET'D £8-F60T'O 90-3578 'S bd-3TLS'°D pD-3f9f'B pU-3202'O GU-3896'O $3-3562'¢ Ga-ABEST'O 90-4b22'°DB TO=3r Lt ao TO-FLLT' SY 2Q=-3296'D 20-3999'2 CO-3Ch y's TA es aa SA-399L" SH=-39Ss° S@=3v98s° EO-39pc' EB-Juvt’ p0-3299° pO=31G2° G¢@=-3uTzZ’ G@-3SSt" 90-3uSt*u &. qe & Boone RY Gs €/t NOt ™ a Be a TZ+38l La) a N FA-3L09'u FO-3L2g*u TO~ 3ST pha TAcdvee te bO-3eGc'u TE-F6Lb iw bA-9e@T iv CO ICSE 20736025 C8" 3696'u COT ILS GG CO"3SAE Su CA-368T wu EO"326E'u PO"3L4AZ'w GO73861'u 6 1° E| ‘O~= A/7y4618qH D14}Uaede AW = welg Jeo|4/we!g a3e{d Buidweg 4azowe!g/i4e4g eo 4 weiqg 3eo,4 bulm/weig yeo,4y | [NH $}S9} Seas peasy UOT{eAINSTJUOD (UIAOjIONAD) yeoTJ-an0 7 990°ON Uny ‘sasuodseu wnajaedg Bur4loR'D @v+4lZS9'D Our3Ie2orv’s Gur+ssoc'o Gu*jTot’s Ais+3lut’@ Te-3L2¢'°OD TL-Fpee'D Tu-3po2'o Té-498T'A Ter~adet'd CU-326L'°D Cur 3Lev'D Zur ILLT'D Sd" ASLO°D w-3282'O MG eTqey 876 Analyses of Multtple-Float-Supported Platforms in Waves 00+12406'O OG+ISbb'a QG+I3Itel's B+sIZ72eo'so QG+I1L92'°B Q9+I00c'D GG+iIG8T's OB+AI9TT* as 06+320T'@ Tg-3628'O TO=-3PLL'D Ta=3T86'@o TG=3c2r'D ta=3092'o Ta-3TTT ‘so 2G-3f0z2'D e/l, 2u-3bcT’o c0-38aT'S SO-3>26'D $9-569L°B £$2-30819°O S4-39fb'°D Sb-34S52°D SV-38Tc'*so S2-3f88T'D Spg-jAsrt’D Sy-ALTT's bo-3f6L°O po-ALLb'D p3-39T2°SD G2-392S°C 90-J6oe'B S TO-3w9t's TO-3¢et'u 20-3906'0 CO-3e09'« ZO-3bep 20-3052 20-3501 SB 3Loe'u S2-36b9°L CH-3zOr's a he es SG-3ITZ's SG-3tTT ie pi-dcecb a GG=-3v9L'°S 9f=3L6e'u L° g° 0° —N — i} Zz TZ+a6LF°O TE+4agce A Te+39le°so Tg+da¢e'9 Ter+acdt?a TE+3eeT'a 2G+IjL-B'sA CQ+39¢l sD 22+44S9°D GB+I3ISSSo'°s 2o+I2Sb'°D CB+I6¢v'o OZA+3l6E*G Q9+3IG2E'C 20+3S6S2'Q O60+32lT'@D OB+I-PsSt'gs GB+ILET*gs GA+Iv2T* oD 2J+3I2BT*g Fd=-30fe8'g TQ-3069'@ Ta-3Sbr'g Ta-3Lb2'Q C€2-36b69'SD E/ ls CO-326T'D CL-3CLT'D 22-3GST'D Cd-F6cT'D CO-3GUT'D £3-306L'°D fYV-386b's LE-368h'o SE-F6LE'°B £Z-A8Ttel’s SC-3252° SH-396T'D PAS AIS) Sea b2-3728'°D bpdv-3CCE'O G2-38c2'O T@=3¢e¢c'9 Ta+3dcr'o Te-3teg*w T@-3S2L1°0 Ta+3p9f° Sg 18-3289'v TO-3c81's To+3d¢0°O@ te-32do'u 28-3596'°¢ Te+3d>Se°O ie-ardp'u 2O-3r19'L Te+ad2e°2 TWO-aele tu CO-IeApy’zZ TO+3ZST*O O-399¢'u Z2O=3vET's To+3teT?o Te-3cstiy CQ-J6pT OU+3c26°s terseet eu CB-38TT’s GG+3262°@ WW-3egt'sa 20-356" °u Oé+3269°S cg-3e9u's SW=3S5py°2 ME+399S°A cO-3269'K S8-3TSp's QG+3Srb*s cO-3rSy'wv S8-39Sc°o Of+3228°O cB-sL9e2' SG-3var's OE+396T°A Chr-ASTT'yw pO-31e¢'2 TE-3¢SL2°AZ CB-3tac'v GM=329c'C CUS aS Cy 0 aii dosu nk ob, eit ae Wi lle 25 S0°0~= gp o/ tub 12H D14}ua09e28W g*| = weg zeojj/welig e3e|d Surdweq O°Z = d4ezoweig/7jse4g 7eO|4 G*| = weig zeoj4 burm/werq yeOld LLMH S1S01 SeOSs UIeAq UOTLEINSTJUOD (WUIAOJIONAD) JeOTJ-A1N0 | FO00°ON Uny ‘sesuodsey uintjoedg Tut isOtt’s Oi*s6ce'o Bi+38u5’'O Ouran’ Bo4+3lyc'so Ourt3bat'D Tur dfv9'D Té-5460S'SC Ti-4sz6e'’D Ti-36uc'D Ti-J66T'DO Té-4bcT'@ Ju 5609'°D Cu - 3622'°@ Si-4aSSF°D GL~539vT'D dg eTqe 1 881 ercter Kim and M OB+3boG'g 00+39Lb'o 03+398r'g O0+36e¢9'g 00+3922'°@ 00+39802'¢ BA+3ICe tg” QO+I8TtT's GO+Ivbat's Td-3806'¢ TO-38L2'D Tg=32969'9 Fg=3086'9 Td-3f0b'g Td=3TS2'a 2O=3¢82'B E/\4 cU-JIbvT'D 20-35eT'S CU-3LUT'S £2-3288'S £3-3901'°S £GZ-3ATTS'D £O-39TL'S £2-3AW8c' sD £B-3S¢2'°B £g-3TT2'sS £B-38LT's £2-3ASpt’s fg-3TTT'! bO-3cG2'°D 72-3652 '°D G@Q-352>'°S Lh Te=39S8T' a TO=3CoT a TO-3L0T ‘wv CO=J6TL‘Y C0-3LLp'¢ CO=-Jule'u CO-3vTT a ee a £G-3649'S SG-3STs'y SG-Jele'v £G8-359¢'U EB-J9LT' SA=3VT a bpM-3S6e G@=43Le¢'¢ adiewsk 2 | ls 9° rays S° —-N- i) Te+astoa Tg+39gea T2+3662°C T2+3G¢ve°o TO+396T'D To+456bT DO C2+3TS6°a AS+I3ISb3°D OB+GLEQL4'2 QZ+3AZlQI*| 22+49CS'D O2+i80°D CB+IL6C'°D OZ+3ZLT'B TO-3LL9'D 20-IGLb' a TO-3uS2'h Te-39eG%e TO-322s'u Te-3Ley a ber3eey ly Ber aee? $B T2-5ToT a TEATTT a CBr 3beo'i CA 3vl Le C@73bGs'e COr3dlZe's eB-3n2e'oe £2-39d26'u £O=36ST! 92-3pse! S & 6 se Zz - A/346 19H 214}3Ua9e7 aW weig zeoj4/weig e3e{d Bbuidweg dozowe!g/7je4g 3eo|4 we!g 3e0,4 B5urm/weig 3e014 LINH $189} seas Uledq UOTJEANSTJUOD (UILOJIONAD) JeoT{-1n0 FT LOO‘ON uny ‘sesuodsea urnajoedg Ti*3euTt'o QLb+33C8°R O@v+38oS'D G@er+3oLe°o Scr3Apv7e's Oi+a6rT'O Te-3S9S'O Tordppo's Tie J6LE'°D Ti 382°C 6-469T'D To-3rut'o 257 38LS°B Zu-3A96T'O Ss-3Ssb2'°B GZ-j20T'O bg eTqey, Keeecayqaa cn nwnmmm 882 Analyses of Multtple-Float-Supported Platforms tn Waves GA+I3LSG2'B OY+I3IV22'G G0+I98T'S OB+3brcot'g OB+I5SeT'D TG-35p6'@ Fg-3TT9'D TA-3L>G'°G TQ-3S8b'o Td-3b2p'g Tg-329¢'9 t9=-3208'D Ta=<3992'g G-369T'S CB=-3586'D 20-3282'G E/\4 fg-3TBe'sD £)-3092'°O £0-3AC?S'D Fd-3cet's £0-3Sbt’s £8-3S0T'D bH-3879'@ bé-32LS'°O be-AL6b' sD bu-4aScr's bs-JBPS DY bO-3EL2'°D bY-3B6T'O bY-3LCT'D Gd-3925°2 90-30>9°2 N (oe) oS ll 2O-3bey' CO=-3E2E' 20-34LTC' CO-Févt' f8-3646' EB-Jygg' QG-3eEc' £o-32eT' S¢=—3Lpp° 2B-3cTtt’ pA=Jucg’ pg-3T9G' pe=J8ps' ba=-362T° SQ=3929' 98-396r' LN LNWO ° —-—N— Hl e/ 1 weig zeo|4/weig e7e|d Bbuidweg Jajzoweig/zjes1g 3e0|4 weiq zeo,4 buym/weig yeol4 LLNH B2BaGOQBacans ts & G &. ry e S&S Ter 36G8°A TO+ALEL OD Ta+35pS2°C To+4gu2'Q Tg+4ap9Tta To+4ecTg QZZ+3598l°o QE+I38969°B QO+Itt9°g Geripcs's BY+IGeb'gD QWS+5love dD Q2+36r2°sB ZE+IKST*D Ta-398S° CE-3bse°D c be-dsees' TA-3TGp! FOr I69e 8k bO7 3962 '« be~-34622' be=43u9T! e2-3986° e@=3nLd* cO-3Ssp9' c@=3Lts' c@-3bés! CO73bLe' a oO ITI u $2-368G9'2 ei Pot 9A8-3S6¢'b eer oe Gm & & e. & ey A/3y4619H D14}zUsDe7 EW $189} Seas WIedq UOleINnsTJUOO (UIAOJIONAD) YeoTJ-anoW 970°ON UNY *Sesuodsei wnajoedg Oe sl. SC O5+4esS'D Qsrjcsv’s QZ+ {lS *D Qur399t's Tu- 3206'S Tu-49ue' Tu-dbut'@ Temrdyee'@ 0 Pe Ay at Sa Tu-JOTT’o Cb- Ape L'D wr AlZee'’s Ze ~3Gvt‘e SL- 3912 '°D 94°4905°0 €/Z > 2g eTqed, 883 Kim and Mereter OG+39LG'°G 20+380¢6'9 28+I38fb'D OO+3T LES OO+I8GS'gA 03+38E2'¢ O8+309T'O O8+I3ISbT'g QO+3I0ET'” JO+3¢~TT'g” 19=3286'9 TO=-3088'o 1g-3699'g¢ Td-3246b6'2 Ta-3662'9 2O-3762'S e/a C2-3LLT'D CO-3LST'°D CO-3ILET'O €@-3LTTt'D £0-3£96'°D £d-3eLL'°O £2-3Teb'S £B-3Teb's $G-308f'°O £d-F6cL'D £O-3L142°B fb-382e'D £2-389T'S £U-36GT'D pO-3S8b'S GSB-38Lb'°S 052°0 7th cel ball TO=3L02'2 TO=-3TIT' TO-302T' 08-3656 '°sS 20=3465'2 20=35Gr'u 2O-39T 2 CO=3TET ob 2B-3S06T's E9-39Te'Z £8-3689'°S £O-3bLy's SOS392' SA=3rSt'u pO=3ugc’s GO=3v6e'Z e/t To+3Tibo's Ta+3Sre’o TG*#3462°O TO+3Gf2°O T2+3G6T'S TO+3bvT's @2+39b6°D 20+38r8°a 02+30¢02°@ ZH+Ilb9'°S O2+3cvS'D CQ+aplrv'ga OQ+jtce'gy CH+sgse'g TO-ALLE'D 2O-3b9L'D A/7461aH Di4}zUuaseJOW weiqg zeojj/weiqg ajze,d Bbuidweg Jazoweig/i4ye4g zeol4 weiq 3e0,4 Bulm/weig zeo,4 | INH TH-3TTL a TO=3pG9'u TO-320G'hb TO*3UTb TO-3p2s'su TO-3eo7 by FB~368T ss VOr3a2e "bw TO-3tOt' a CO=392¢6'u CO2-3689'w CB-39Sp'a CO-386C'u c@=3cSt EE-36A2'w GO"3J6pe'b $}1S9} seas Uleaq UOT}euNSTJUOO (UITOJIONAD) yeoTJ-ano0 4 TES0°ON Uny ‘sasuodsea winajoedg Teratut’s Ga+3G02'D Qv+39TS'O Od+3Sb~l'S Q2+3S¢2'D Qe+30zrT'e To-35296°C To-J6vb'D T2-305e'C Tu-3292'°o Té-3b6T'O Te-3LTT'°S@ 2u73509'°D Cu" ITI? 'L Sei-3llo'°B Gu-3sG9e'D Sg eTqeL 884 Analyses of Multtple-Float-Supported Platforms tn Waves OJ+ITGL'*g 20-3962'°R WH-3eSe*w Ta+ae62'°O tOr3tS9'w 0g+3f99'o 20-30220°D TO-3r4e'0 Ta+az2pe's a=3sp5'v BO+deLs'g 24-398T'D TO=-3L02' 4 To+aeee'd TW-3tbp'd OO+36Lb*O 2e-3TST'S TI-3ervt's Ta+g0ee'O@ bOr3cSe’u OG+3698'G Cw-jvTt’s Z0+39b6'v To+apst od b8-4dL9C'u OG+3Ge2'g F0-32221°B 26=3406°9 TO+59eT*D THrawlT oY OG+3Ip9t'ga Lg-I3Lee’sD 2B-43L94°9 Pe+seees's ce-jAvlece QB+3IGvt'g L2-3220°O 20-3cet'D Go+seed?'s ecg-3ebd'o QG4+aBTt'g L¥-3St2°S LO-3298'o gg+96c9'O c@-avé6s'eu Ta-3296'g@ SB-3L9TO SG=J6LG'u Go+3G0S'S ce -3s6bp'w Td+3T92'o Sd-38TT'D LB=3t9e 2 OB+36Tv'O cersete's Ta=32lq'@ vv-3000'O f0+3S0c'u @o+zete' A cO-av6t'u Fd-3240v'G vv-3005'°D SAavar's @g+jeue'o Ler3e86'w Tg-3£92'@ b0-3292'O pe=3Lfy'o Zu+actt sa LO-3See'o Ta-3eet's ve-icut’s p@-36TT' wu TO-d6E8°O pO rAasdéy's Za-3ie2e'g GB-abTT’S 98-3468" u 2¢-3262'O 96TACb9'w £/l, 6 05 eh enn €/t [4Z°0 = ¢po/tuP 12H 9142uU99e7 OW g°{ = weig zeo,s/weiq 23e1d Buidweqg Jojzowe:g/3se4q 3eOl[4 0° $189] Sees peoy UoT{eInsTjuoO (Te[nNsuelst}) yeOTJ- 92914, L, £),Q0°ON uny ‘sesuodses wuintjoeds d+ 3066'D Q@i+36CL'D Ze*32nS'°D @J+jtee'o Qurgttfo Zit 39tt'O Tu- 362° Ty-39ES'D Tu-3alpe'o Te - acd '@ ae a 2738u9'°C 2)-3242'°2 S2-3TEl’@ pd-3LtlL'@ Cespes'D > 4G STAeL oOu'’p wOs's wBu's wOo'? w@u'dZ whS*t tZe°s wO6*Z WO’ s bZL°s 209° rOS*% wle’s KcBl’Z we? y w@b?sZ 885 Ktm and Mercter 00+3L26'9 Q+IS9b'DA 00+320b'g 20+30ve'g 20+3tTE2'@G 29+3IST2'g OG+30rbT'g 20+3S2T'g OO+3VTt'a T9-3266'2 td+3pde'o Tg-3TG9'o9 TQ=396b'g Ta-3Tbe'g TO=+3L6T'D 24-3L26'B e/t sj1s90}] seas peoy UoT{eEANSTjuoO (AeTNsuels}) yeoTJ-sae1 J, 160 °ON Uny ‘sesuodsed urndjoadg fd-3TbT'S c8-352T'D cd-3L2T'D £U-3506'°O £O-3L2L'O £B-3226'O £2-3LTe's £6-39L2'°D $0-3222'D £4-3L6T'B £0-38sT'S £B-F6TT'D 6¥-367T8°S bpH-IT8b'D bU-3202'D GJ-32T2'D THO-3L2T‘ TH=35ET eZ TO-3IOT's 2O-49S24'u 2B-356b's 20-3682'2 2HO=4eet'e E@-3UL6'C RA=-3rSz'4 RA=3L9S'2 LB=3pdv'w 22=3592'2 LB=-3IvGt's y@=34L2L'0 bA-3CL7' wv GJ-3vdt'u \2Z°0 ua | &/ | TO+45L6L°R TU+t3DvC'a To+3z2ec'y Ta+3922'°f To+azutey Te+59CT OD 20+3578°O Od+saprl'a 0¢+5979°@ Q2+I3L7S'°D 2G+ILbo'°D Od+3Srl'a Qu+3fbe'D QAr+Ilvtgs TR-36uS'°D CU-AGEL'D VO732S9 TO-39bG' wu bO=3Sby we TO-3pGe' TOr atl? bOravety 2O-Ju86' cO-3tfatn cf" 30l9'u CRATES *i 207 3H68* vi CO" 309% 9 COM3G dT LM-“39LS's pir 3288's y@-3966'h A/346198H D14}zUua De EW we!g }yeo|j/weig aze{d Bbulidweg O° = 4azowe!;q/zye41g yeol4 Bursls6'@ Bér30cL'B Birilorv’s 25+39CL'D Gir jBc2'*D Ov+3IStt’a TL-397b°D Ter 39re'D Ti-4G92°O Tu-4dL6T' A Tu-IiSGcT’? Cum ILHL'D cu- sec 'O eg~3Let'A Su-3Zy9T'D Gur ILuG'B > ng eTqey, 886 Analyses of Multtple-Float-Supported Platforms tn Waves OG+3I8bZ'B Bd23Ir90'g BB*3ILLG'o Q0+3c6ry'gG 00+3L08'aQ OG+389"'D Od+399T'g OO+4Ip9t'g OB+3IBoTt'g 00+39TT 9 3G+3826'Q TQ=3baz'g 3G+386b'D Tgs3acte'g TO*+389T'D 20-3800 'o g/l, C0-3S22'D CO-3erv2'D 28-3902'O CQ-ILLT'D f8-3SET'D £B-36T6'D £2-3696'@ £d-328e'B LG-3TTE'D £G-3682'D £G-3IPct'D £V-3stt'a b2-3024'@ pO-389L'°D bO-3CLT'O S2-3GET*A Ta-3uSe" TG=-39L2° TQ=38Ge2" Ta-9tGT’ TO-3raT' 222376s' 28-3122" 2O-3L9T° 2B=3ect° Sd=3e¢pg'2 PB=J6ES'v SH=JoTS se CO-3SS1's p@-3u29' pO=-3v9T'S GO=3ve@t'd Ge & BQeaewg & 12£Z°0 Bl e/ Te+7f62'O To+59e2'¢ TQ+38L2°h To+3pc2°2 Ta+5ZZT°@ TG@+3TST oO O2+4126L°D 20+1689°A O00+3289'°O ZQ+3ASen'°D OG+3AITSl'g” 02+3922°O B2+36LT'°D TZ-3yT6°@ T2-39S92°9 S2-3868°D BO-3LT9' wv Ba-3eTs'w TO-3LTp wy TO-3e!ee'e ba-32¢c'u FO 36Sh* w C2-3ATEL ow CO-Fpe9'w CO-396p'w 20-3998 'u CO*ILbe'y ZO-39bTs EO=3T69' wv £27 368c'w pO-3tB2's LO-32Gy'¢ A/7Y4619H Di4}zUuaDezEW weiqg zeoj4/weig a3e,dg Bbulidweg O°} = JozOwe!g/i4se4g eOLY $j1S9} seas peoy UOT}YeEANsSTJUOD (Je[NSuets}) LeOTJ-9eIU J, L80°ON uny ‘sesuodses windjoeds O2+3L96°D Q4*382L°O Bur+icsrn’sB Oral ts's. Bd+A7Lot'D Ds+slot’B TS-3268 OB bé-3262'O Te-39T2'e Te-slot’o Cu-3606'DO Cv- ster’ so C0-30u2'D £6-30cS'°B bu-jdty’D Li-396b'O AG eT9IeL wOW'b WBS" “Wes LOSG°2 0O2°2 -OS'T wee? T w06°2Z «Bb°¢” “vOL gs 009'°¢g KOS*g uUBr gs VOL? ga KwOe°g car's 887 00+36Sa'¢ QO+IL2LL'°D 09+3£99'@ QO+3L6S'*D OB+IZ8S'g”D 00+328e'S Q8+Ibo72'gD Q9+I3ILT!]'*|s 0B+I3TET'@ QB+IGpot'gs QB+3ISTt'e Tg-itze'o TO+ILbG'gD TO-3r2o'Q TO=3S66T'G CB=J6b9'S E/ly €0-356E'O fY-3LSE°D 28-3602 '°S CU-3£92'°D CO-30T2'°S 2O-39bT'D £B-3S5b2°OC £6-3909'O Se-3TLo'o £O-3Svf'°OB £B-3vE2'°o LO-3L&yT'o bB-3282°S bB-3998'°R bO-ALTT'@ 90-3828°O TO=3T9b'o FO=3rl2e'v TO-3062'S TA=3u22'0 TO-39Sb'2 20-3586°o CQ-322E' oh CO-3r8c'u 20-3950" 2O-3LET'b O=-3n28'0 SB=322p'°D SA=3JrOT'o bO-39G9'¢ p9=J6yt'*su 90@=-3u9L'S oO i] e/l a To+3gee'D TO+3Scl'” Tg+3892°9 Ta+3Lte'd Ta+azdt*a Ta+adc2t'o Ov*I12LL°D ZBrsSL9°S 02+3S2S'°2 GB+IeLy'°sD A2B+3082°D AB+3ITe2'sm CB+3I98T'D TE-32L6'A T@-3662°O 20-1952°SG on ~ > (aa) ~ N BO-362S'u FO-328r hu TO-368S wu TO-390e wy TO-Jae7'o FOr32othi CO3BGL' a CO" 4u29%K CO 3d6p' wo CO- 3898 u CO-3pSe'v CA-3SSt'wv FV-FIL9L wv CW-3S¢e' wh pB"3962'u 9B73vdo'v oN A/1iy61eH D!14zUEeDe OW weiq }zeo,4/weig a3e,d Buidweg O°} = s4ezowe1g/34e4g yeo|4 $189} Seos peoy UoT}eEANndsTjUuOO (Te[NSsuelI}) yeOTJ-seIU,], GLO°ON uny ‘sesuodsadt urnajoadg G4+32v6'D 2d+3659'°O Au*I0Sr'D Gurt3l60'C- BL+4Sut’s OU+3TuT'D Te-32L2'°O Tu-3fu2'°o Ti-3lue°e To-slrt' se Zw IL 46'D 2w-3560'O 24-35b2°O £u-3265'S pur 3tps'@ 9U-39195°8 MG ede L 888 OG+3Tez'g 2e-3Pe2e*s Ta-3tec'y Ta+aZ6f'O FOr3aLe9'w Bwr+dbs6’'DB O0+3T69'g 2G-3L~2°D T0-3662'A TO*+a0vlO°S Ta-32e8oq's Byr+aecL'D QO+32466'g@ CU-36020°D TA-3kec's Ta*+3282°'O TA-julyo'w Bv+5960's O8+3bac's 2O-30LT°O 10-3667 'w To*F9CC°D TO-3o8e's BdtI9ICL'D QB+ITTo' ga CV-362T'O TH-39ATSS Te*+32BT°O F$O-39Gd's Ov+9lse°OD OG+spac'g LA-3Lp8'O C2A=-99LG'¢ TO*3pET*S TO-Fe9L' wy But+gett’s OAG+3SLZT'o@ LCO-3TAr°SG CH-386T'v O2+99TB*O cecO-3d42u'wu To-J3lto'D OWB+3lGT*g CPG-3SCL'D Ce-39b1's Z2O+3ttL*A CHAKe3tL9'uw Tw-G9te’sB Q03+362T'g LB-399SG2'O 28-J3F0T' Q@0+*3909°S cO-3res's To-asr2'so O0+3SOT'g LB-3r6T'H FO-3569's OB+3TR2S°*O cO-39ee'Y Te-345T'°O BO-3Tee'g LG-B6ET*A LCH-3bSpy'v @o+3p68°A cO-JjuSce'w c#-3TL6'O TO=3E29'G vU-32v6'D LCHO=-3cvc's O@B+AIBS*D CO-36bT*'s Cw-3dts’D TG-30pp'o vO-328S'O SO-RbCt‘é OB+*97BBT°A F£H-3L99'se Cw-jslwe's TO=3682'g bH-39TES*'O vpO-Je0G*u T@-3T98°O SHR3LLb'w LCw-Al9H ‘eS TO-326T'D bvU-322T'H pO-3SbT's Te-3Te2'H pyOr3totte bvd-abel’e CO+3ves' os SU-360T'D 90-39826'h ZO-38te’Ss GPH-Astc'w Yw-jabvu2'a 6 e/\ 6 8, e/a 7A e/ge : g €/1 Zz z A°3 z ae z| S| OO€"0 = 50/248 12H 214}us9e708W 9°| = welg zeo|4/weiq e3e\{d B5u:dweq 0°S = 4eazeweig/iye4g 3e0|4 s1s0}] sees peoy UOT{EANsTJUOD (Je[Nsuels}) yeOTJ-99.1U J, 960°ON uny ‘sosuodset urndjoedg : xg 9eTqQe J, Analyses of Multtple-Float-Supported Platforms in Waves 889 Kim and Mereter 00+3£98'2 08+3SZL'¢g 23+3£99'O 20+306S'@ O0+Iv6b'a O0+361L0'9 Od+3982'B 08+3L02'o OO+3LLT'o B3+I39¢t'o OO+3ISTtT'gs Tg«3tGea'@ TA=-3£6G'@¢ TO-3S2¢'o TA-3T6T oO 20=3T9G'¢ e/l 20-3268'O CB-IWSE'D C0-3S0f'°O 2B-38S2'°S@ CU-3S02'D C2-32yT'D SA-F9EL'S £2-3509'°@ £0-308b'O £P-ILIL'D £Q-3LS¢'O £O-30LT'S fa-3f0T'D pa-3Tes'D bO-386T'S G0-3292'D 91Z°0 TO-399p° TE-39L5" TQ@=3t6z° TO-36T2’ TW=-3ESt° 20-3006" cO=3fSe° 20-369c' 20-3564" 20-32eT" ¢G+3872g' SB-3cSGp° SO=3Je2c" pO-392e° pB-3222" GU-3561° 951 o°e €/\ weig zeoj4/weig a3e,d Bbuidweg dozewe!g/3je4d 3e0|4 3 > SGWSS SEES Bean wt TO+4L6£0'°D Tg+atved TO*3b82°so Tg*3Z2e2'D Tae3Zeted TQ+3TyTt'a 00+3206°D 02+3THB°d O2+43cvL°D G@+3T29'°2 00+386¢'°SC ZO+3Tol’a WG+IlSe'yD OQ+3TLT'a TZ-3979'°O 2g-3ste’*s on ~ > on ~ N 10-3615 °c TOr3rlo'a BOr3pdo tis TO-3pex ui FO=3062'u TO-3utete TO-3oTt TO-3bOt ie CO°3ebe tu CO-3ELG%K CO-380S'b CO~F60e su CB-9Shd'u SO 39LS6°u £B-3fyT! ST tet hs ees TH! °N A/3461eH 914} UE9e7 EW $1891 Seos peay UolT}yeAnsTjJUuoO (JeTNsuUelA}) yeoTJ-s97yL, €80°ON Uny ‘sesuodsaa urnajoads Cirabu6'e 00*38¢2'°O 02+35uS'°C Ber Jove’ a @j+3Vb2'°D ZJ+3vct's Tu-385S°o Tu-4Twb'Q Te-38ul'D Ty-39¢2'°O Te-3SST'O CL -3856°SC 2u-4660'D Zur Tot’ Pm Sé-4A9?'°D 9u-sph9'A : Ag aTqey, 890 Analyses of Multtple-Float-Supported Platforms in Waves TABLE 6 : ADDED MASSES AND DAMPING COEFFICIENTS ADDED MASS DAMPING COEFFICIENTS Mode COEFFICIENTS 891 Kim and Mercter TABLE, 7 THE HYDRODYNAMIC MOMENT ARMS MODE | INERTIAL PART | DAMPING PART TABLE 8 Ss Fa } Ss “SS; Cus 4 HS,” ee J HS “HS; Cx sue RS RS, af RS RS; 892 MOMENT COEFFICIENT RELATIONSHIPS Cs Su, +5. Jf se’ Sr. (HH fon +5 J HH HH, +c Jf RH RH, ae TS RH RH; Analyses of Multtple-Ploat-Supported Platforms tn Waves TABLE 9 TABLE 10 THE NON-DIMENSIONAL EXPRESSION OF THE WAVE- EXCITING FORCES AND MOMENTS sway-exciting force heave-exciting force = Don roll-exciting moment beam of the body (a or b) o or e corresponding to the odd or even potential fo ve THE RELATIONS BETWEEN THE FORCE COEFFICIENTS DUE TO MOTIONS FOR TWIN BODIES a AND b = Lbedhso [aad EGsedh, 5 Sale usb ; = ornare ; =-[Cupd, =< Cede Le rs ob Leeds [Spall = LOacd., = bead. = [eed = =~ boyd, el= Esse, Cus Ja Meus J, I ee oT Corn Jaen I, [fps Ja= [ers 4, ie Ee (eh2) gfe), 893 Ktm and Mercter TABLE 11 : THE RESULTANT HYDRODYNAMIC FORCES AND MOMENTS DUE TO MOTIONS FOR TWIN CYLINDERS TABLE 12 : THE RELATIONS BETWEEN THE WAVE-EXCITING FORCES AND MOMENTS FOR TWIN BODIES a AND b 894 Analyses of Multtple-Float-Supported Platforms tn Waves TABLE 13 : THE RESULTANT WAVE-EXCITING FORCES AND MOMENTS FOR TWIN CYLINDERS TABLE 14 ; VERTICAL ADDED MASS COEFFICIENT L/ 2k a a 0 POs 1,5 0.458 2.0 0.418 220k On so2 Ae) Uses 3,88 OO. S27 Ame9) (0294 TABLE 15 : EFFECT OF MODEL SIZE ON ROLL DAMPING MO - MENTS FOR CYLINDERS WITH APPENDAGES. vce values Eq. (43) ) Cyl. Bilge Sharp-Edged Streamlined Diameter Keels Fins Fins 6 in ny 16 4:52 2 in 16 Nt / 3,1 24 in 15 Not Tested 2.9 895 Kim and Mereter CENTER OF GRAVITY Sa Ne aw 38" oO * = |< = 4 Pa Pe ie { 44, AY Or 'N, ey, ALUMINUM ~~ (SS Oiifh 4 ; FLOAT FRAME < SS 66" Fk <——— On 4 11.80 Ow STYROFOAM WING FLOAT DRAFT OF HULL FLOAT=T,, “N= PLEXIGLASS DAMPING PLATE DRAFT OF WING FLOAT=Ty MODEL WEIGHT = 20 LBS PITCH GYRADIUS =16.50 IN ROLL GYRADIUS =14.121N Figure 1 : Sketch of crudiform float platform model 896 Analyses of Multtple-Float-Supported Platforms in Waves CENTER OF GRAVITY ALUMINUM FLOAT FRAME @Q@ . “a t aK FLOAT DIAMETER =D —* 11.50" I Ss STYROFOAM FLOAT DRAFT OF FLOAT = T =F PLEXIGLASS DAMPING PLATE MODEL WEIGHT = 20 LBS PITCH GYRADIUS =16.68 IN ROLL GYRADIUS 9.0 IN Figure 2 : Sketch of triangular array float platform model 897 Kim and Mercter °200 UNY ‘Sees pedsy UI [spout UIJOJIONID ‘seas aeTNSeaa1 ut SUOTJOUI UTJOF}eTd JeOT JO pOdeA UIeAZOTIIOSO Jo atdurexq| : ¢ dINST (SQNO93S) 3WIL 93S al i 3AVM NI | 3YIM JAVM AYVNOILVLS H NOILOW 3AV3H BAV3H NII 19 4 T — eS TO SO Se See J9YNS NI 2 NOILOW 39YNS NOILOW HOLIid if ———— eee BI HO1Ild S30 € 3AVM NI | 3YIM 3AVM ONIAOW 898 Analyses of Multtple-Float-Supported Platforms in Waves WAVE -PITCH 260 AMPLITUDE PHASE COHERENCY PHASE, DEGREES rw ~ Io wu 3 Ww SS re @ wy Pir, ran) > ui Oo w = < Ww x a a r [o) oO 12 WAVE -HEAVE ao 180 160 140 wy 120 SS N 100 A; 80 z WwW a 60 J Se o 40 20 ie) = 20) ie) 0.2 : 0.4 0.6 0.8 1.0 te. 1.4 1.6 FREQUENCY PARAMETER,W:wv V'/?/g Figure 4 : Examples of transfer functions : amplitude, phase and coherency. Cruciform model in head seas, Run 002. 899 Kim and Mercter zoo UNY ‘Sees pesy UI Tepour wOjfonID “yo ytd soy y pue aaesy Jol 4y : suOoTjOUN} asuOdsaa astnduit Jo setdwiexq : Gg eunsT |g sn d/Pficy MYSLIWVYVd JWIL 02 Ol ; 0 Ol- O2-. 010 0— {0} [oxe) 900 Analyses of Multiple-Float-Supported Platforms in Waves Z200 UNY ‘onbruyde} ssuodsaaz estndurt Aq paiotpead Sees pest Ul Tepow WAOJfonao OZ SuoT}OUL Yoytd pue sAea_T : g ounsty (SONODA3S) ANIL 61 gl Z| 91 S| 0A] ‘ell 4 W 0! 6 8 Z 9 GS v £ LNAW3YNSV3W 3YIM 3AVM ONIAOW as 1—+| (See NOILONNS 3SNOdS3¥Y 3S71Nd WI WOY¥S G31NdWOD HOLId LNAWSYNSVSW HOLId NOILONAS 3SNOdS3Y¥ 3S1NdWI WOY¥S G31NdWOd ey ANSW38F.SV3W 3AV3H 901 Kim and Mercter Iz|/|o | 1@) 0.2 0.4 06 0.8 1.0 1.2 1.4 FREQUENCY PARAMETER, w:w Vv '’%9 Figure 7 : Transfer functions (amplitude only) for heave of some cruciform float arrays in head seas Group A, hull float diameter/wing float diameter = 1,5 902 Analyses of Multtple-Float-Supported Platforms in Waves WAVE-LENGTH/V> = yy? te) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 FREQUENCY PARAMETER,W -w J/V'/5q Figure 8 : Transfer functions (amplitude only) for pitch of some cruciform float arrays in head seas group A, hull float diameter/wing float diameter = 1.5 903 Kim and Mercter oO rs SIGNIFICANT PITCH MOTION, 81,3 (RAD) SIGNIFICANT HEAVE MOTION, (Z3/V >) Figure 9 : Effect of damping plate size on spectral response of cruciform array of floats Hull float diam/ wing float diam = 1.5 (head seas) 904 Analyses of Multtple-Flyat-Supported Platforms in Waves SIGNIFICANT PITCH MOTION, 9173 (RAD) SIGNIFICANT HEAVE MOTION, (Z43/V"/°) Figure 10 : Effect of damping plate size on spectral response of cruciform array of floats Hull float diam/wing float diam = 1.25 (head seas) 905 Ktm and Merecter SIGNIFICANT PITCH MOTION, 6173 (RAD) SIGNIFICANT HEAVE MCTION, (Z4,3/V”) Figure 11 : Effect of damping plate size on spectral response of triangular array of floats (head seas) 906 Analyses of Multiple-Float-Supported Platforms in Waves CRUCIFORM ARRAY HULL FLOAT DIA _ |, WING FLOAT DIA 20 25 TRIANGULAR ARRAY ~ SIGNIFICANT HEAVE MOTION, Zy3/V "9 Oo Figure 12 : Effect of slenderness on heave spectral response of crudiform and triangular float arrays (head seas) 907 Kim and Mereter SIGNIFICANT PITCH MOTION, 613 (RAD) SIGNIFICANT HEAVE MOTION, (Zy3/V"*) Figure 13 : Effect of metacentric height on spectral response of cruciform array of floats Hull float diam/wing float diam = 1.5 (head seas) 908 Analyses of Multtple-Float-Supported Platforms in Waves Figure 14a : A typical system of forces induced by heaving motion Figure 14b : A typical system of forces induced by swaying motion 909 Kim and Mercter Figure 14c : A typical system of forces induced by rolling motion Figure 14d : A typical system of forces induced by waves 910 Analyses of Multtple-Float-Supported Platforms tn Waves 6 2) Cer 4 SHH 3 2 Sep ie ) Sia , ee 0.1 0.2 0.3 0.4 0.5 yYB/2 =) Figure 15 : Heaving, swaying and rolling added mass and damping coefficients for cylinder a 911 Ktm and Merecter Cus 2 ] Cy 5 or apeaes seca by op fo) an = a ol 0.2 0.3 0.4 0.5 0.6 vB/2 Bi ae 5 ee SRH =o N44 CRH -3 — -4 Sus Figure 16 : Heave-induced swaying, sway-induced heaving, and roll- induced heaving and swaying forces on cylinder a FhZ Analyses of Multtple-Float-Supported Platforms tn Waves -40 -3.5 -3.0 Qeur =25 -(o) Doni Lau = in 5 We SHE? RBI Daur USMC. ven ee -1.0 0.5 = (e) Gout lsu 0.4 Dusr, Les, Lrsiz Qss; =2< (o) VB/2 Figure 17 : Hydrodynamic moment arms on cylinder a 9¥3s Ktm and Mereter 0.5 (e) fs O.l _———_| fo) 0.1 0.2 0.3 0.4 0.5 VB/2 Figure 18 : The sway-and heave-exciting forces on cylinder a induced by even and odd waves 914 Analyses of Multtple-Float-Supported Platforms tn Waves SS [Lf /s\\0) YVB/2 Figure 19 : The sway-exciting forces on cylinders a,b and atb 915 Ktm and Mercter 1.6 atb (o+e) Fy I PgaB l.2 a 0.8 0.4 fe) 0.1 0.2 0.3 0.4 0.5 YB/2 bee Figure 20 The heave-exciting forces on cylinders a,b “and atb 916 Analyses of Multtple-Float-Supported Platforms tn Waves 160 SWAY (y=2) 80 HEAVE (¥=3) =| = LEAD ro) 80 ROLL (y=4) oO a — iGO DIFF. WAVE (y=0+e) 2.0 16 HEAVE (Y=3) 12 ROLL (y=4) 0.8 SWAY ( Y=2) 0.4 DIFF. WAVE ( Y= 0+e) fo) 0) 0. 0.2 0.3 0.4 0.5 0.6 yB/2 Figure 21 : The radiated and diffracted waves generated from the twin cylinders 917 Kim and Mereter 08 (o+e) lFs | PgaD a+b 0.6 a ISOLATED 04 : a,b 0.2 0 fe) 0.1 0.2 0.3 0.4 0.5 0.6 vt Figure 22a : Sway-exciting forces on the submerged twin circular cylinders 918 Analyses of Multtple-Float-Supported Platforms in Waves Ke) a+b 160 b 140 (o+e) faa) Cr h S120 H WwW a) (0) LAG <}+—_ Sy) ro} 140 160 180 e 0.8 a+b 06 GQ ISOLATED (o+e) 0.4 IF | a,b PgaD 0.2 fe) fe) 0.1 0.2 0.3 0.4 0.5 0.6 VT Figure 22b : Heave-exciting forces on the submerged twin circular cylinders eee) Kim and Mercter LAG SSS) - LEAD cla of O|M O 0.1 0.2 0.3 0.4 0.5 0.6 VT Figure 23a : The sway-exciting force on cylinders a,b and atb 920 Analyses of Multtple-Float-Supported Platforms tn Waves 120 ——> LEAD + (oe) LAG (o+e) ley | aB x6 Pg a+b S D Le p75 B D 0.8 04 b 0 0.1 0.2 0.3 0.4 0.5 0.6 VT Figure 23b : Heave-exciting forces on cylinders a,b and atb 921 Kim and Mercier O a ISOLATED 4a IN THE PRESENCE OF b Ps 40 O a+b } a 100 4.5 4.0 3.5 3.0 25 2.0 4jo olw 4ln ost /® Figure 24a : Sway-exciting force on motora type twin floats 922 Analyses of Multtple-Float-Supported Platforms in Waves 180 160 140 120 100 Ss Ti B D 8 ii LEAD © a ISOLATED Q& a _IN THE PRESENCE OF b fl) “aseb [Fi pa aAw Ae eS Figure 24b : Heave-exciting forces on motora type twin floats 923 Kim and Mercier C,coSiwt+é,) ce : : C, coswt +€,) =i i) b | B Na Nb all ae 8q= 7 8 = = Zle 2T, 1.25 2 2 Pe tp oboe 'b By —- = we Mo my" Bp Ca: Ch = Le wee Ss =O! Po a P> b By y( By +B,)/2 20 Cg(0.45 ) Cgl0.26 } 16 12 8, (0.45) -4 8p (0.45) § (0.26) -8 -12 -16 0 18 36 54 72 30 108 126 144 162 180 Ep (DEG) €a=0 Figure 25a : Hydrodynamic characteristics of two heaving cylinders as function of phase difference 924 Analyses of Multtple-Float-Supported Platforms tn Waves Z —-—a ISOLATED ae fe S Tay acre +8, ---- Ty/ Ty 215 (0) 0.10 0.20 0.30 0.40 0.50 0.60 (0) 0.10 0.20 0.30 0.40 YBp/2 O 0.05 0.10 0.15 0.20 VB, 72 Figure 25b Swaying and heaving hydrodynamic forces on cylinder a influenced by fixed cylinder b 925 Kim and Mercter — Oo aq = 40 . (o +e) Fy | pgaB 1.0 08 a ISOLATED b ISOLATED 06 04 fo) 0.10 0.20 0.30 0.40 0.50 0.60 y(Bg+B,)/2 fo) 0.10 0.20 0.30 0.40 Y Bp /2 fe) 0.05 0.10 0.15 0.20 VB,/2 Figure 25c Heave-exciting forces on cylinders a andb 926 Analyses of Multtple-Float-Supported Platforms in Waves DECK AND FLOAT eS DECK INTERFACE . a Na ie : po SS ae ees 2 ie ix om oe ae ‘ FLOAT ey fe (FILLED WITH PRESSURIZED a AIR) ~~ \ by Sa ; Vas S « TRUSS SYSTEM \ HINGE ATTENUATOR (FILLED WITH PRESSURIZED WATER) y A DAMPING “ COLLARS i Figure 26 : Illustration of float configuration Be ty 927 Ktm and Mereter oO QOIOJ [TWOI}ABA PaoONpUT-aAeM UO Ht JO aoueNTJUI eu, : 2Z sans a ade JoqJoweteg Aduenbes4 In 3 d L 9 S 4 € @ l 0 z0°O e 1WE°O = Pu 25 a 10°0 = = aoe ey 6°0 = > L S0°O F fo) Ca pte we LE *S1VO14d 11V YO4 ' g0°o mal L°o gl 7 6° ( 2 Z°0 0°? ¢] e 1 LA se} [09 fi =Hu!dweg S700) uy \ g°0 O*l —— 3 (0) S6d a2404 [29!}4a poonpuj—-oAem "7 928 Analyses of Multtple-Float-Supported Platforms tn Waves VIDIO] TBOTIZIA PXONPUT-S9ASM UO No) Ai }awWoay JO uoljdisosaqg 404 Ja aeunbiy ul YD}a8xS 9aS S°0 iv ea abe da}zaweseg Aduanbai 4 aE wat € 9°0 JO 9OUSNTJUL OU], i I Twig + = fu 5 LO) °" Sl = s- uy ou Ob =a gz eunst 7 se) S0°0 8°0 O° 1(9)see ad404 | eD!249/fA Poonpul|-saen mm 929 Kim and Mercter | DddIIOJ [TVOLJABA PSONpUT-s9AeM UO — QOUSNTJUT 9 J, 3) —— Jaj}oweseg Aduanbas 4 15 S 4 € oc l | weg = Pu , ay LOGQ- Es. c*o.= = ; mh o, SN Cink =e Bee e / \ oes SivOl4 Tv yO4 oe So ~ So Le m if Of ——— uy Od Sei Ai }oawosy Jo udiqdissseag 404 Je osnbiy $O YyD}9%S 99S 6% eAnst 7 1°O = je) < © ! = a & (@) 2°70 Blo nla ol< 0 gut ct (@) w@ TI ° ba | @) G°0 © i N oN g8°O 930 Analyses of Multtple-Float-Supported Platforms in Waves Heave Amplitude Z mplitude 7” C€ Wave Transfer Function, 1.0 0.8 0.2 0.05 0.02 0.01 See Sketch of Figure 27 for Description of Geometry FOR ALL FLOATS: 1.8 T 1.6 zn 30 fo) CS 0.5 c = 0.07 c m4 0. 3m5 ] 2 3 4 uFT Frequency Parameter, aa 931 on the heave transfer function Ra Ro The influence of Figure 30 Ktm and Mereter See Sketch of Figure 27 for Description of Geometry 0.5 (0) 72 a MGS Us CO) kD: Significant Heave Motion Significant Wave Height = 30 0 0.1 0.2 0.3 0.4 0.5 « Significant Wave Height ae 3 Draft oy oe Figure 31 : The influence of R on heave spectral response fe) 932 Analyses of Multiple-Float-Supported Platforms in Waves 1.0 0.8 See Sketch of Figure 27 for Description of Geometry 0.5 Sys ny/3 TT oe // a We 0.5 Wi ef 0.1 FOR ALL FLOATS: ave Height Significant Heave Motion . .. t Wave Height ” Significant 0.5 0) 0.2 0.3 0. Ons Significant Wave Height _ 41/3 Draft Ree 7 La Figure 32 : The influences of aaivea et heave spectral response 935 Kim and Mereter 1.0 0.8 0.5 3 T “. Acceleration, es 7 "1/B w & c=) & Deck Curvature, ® 0,2 ee a H a 1/3 wn v a Onn FOR ALL FLOATS: See Sketch of 0.0 Figure 27 for = 30 Description of Geometry 0.0 1.8 O 0.1 0.2 0.3 QO. 0.5 Significant Wave Height Draft Figure 33 : The effect of sea state (significant wave height) on thre heave-related spectral responses 934 Analyses of Multtple-Float-Supported Platforms in Waves 1.0 0.8 See Sketch of Figure 27 for Description of Geometry 0.5 ~m = ES Cc Cc ——— = P2038 = ce ° 0.05 = i 0.2 j 0.07 > o 0.09 ke w 0.11 Cc 0 2 be 0.1 = a 0.08 FOR ALL FLOATS: i z= 2 o 0.05 R, Re = eG L qi m* 0) 0.1 0.2 0.3 0.4 0.5 Significant Wave Height _ 11/3 Draft rey ; " os Cc Figure 34 : The influence of damping coefficient fo? on heave spectral response 935 Kim and Mercter soesepuedde snotsea Y}IM JaputtAd Ja}eurerp SNts 03903-d¥VHS (9) SNi3j 4O 3903 SNIGV37 —,00 > SNIZ O3NIIWV341S(q) 19 3° YONG =: GE BINS WT $133» 39118 (0) ane 936 Analyses of Multtple-Float-Supported Platforms tn Waves 23 23 TESTED WITH AND WITHOUT HINGE TESTED WITH AND WITHOUT HINGE SEGMENTED MODEL CENTRAL SPLINE HAS EI=3.5 LB-IN.* SIMULATES FULL SCALE EI=2.2x10°LB-IN* (dQ) (Db) TESTED WITH RIGID CYLINDER FLEXIBLE CYLINDER AND WITHOUT HINGE TESTED WITH AND WITHOUT HINGE Figure 36 ;: Models used in test program, scale ratio: 1/576 937 Ktm and Mereter JeOT] TIN] UO BdIOF TeOI}ASA paonpuT-sAeM 020 sto SSVW G3qaQqV SN1d AYOSHL NVNMS3N AMYOSHL NVWMSN a31V10S! (@°p-ge saansqj) ZLY3BH ‘ADNINOSYS AVYYV NI soo \ K 39404 SILVLS S, LE eanst 7 ss 002 OO 009 008 ooo! 002! OOvl oos! oos! 43/87 '3QNLINGNVY BAVM/ 39403 3AV3H 938 Analyses of Multiple-Float-Supported Platforms in Waves VERTICAL FORCE/WAVE HEIGHT IN TONS/FT. Figure 38 ae NEWMAN THEORY PLUS ADDED MASS =a PANS a alia’ PNY sent go FLOAT SHAPE AN ee a 0.8 ne) tee 1.4 a ee: RAD/SEC Wave-induced vertical force on a very full float (references 63, 64) 939 Ktm and Mercter (@2‘p-g¢ S2INSTJ) JeOT] TIN] UO ad10f SpIs peonpul-aAeM : GE 2ANST YT Z143H ‘ADN3SNO3ZYS SzO 020 7 se) oro eke) ooo! 0002 ZSONIH HLIM ‘G31V 10S! lolete} 3 jolele} 0000S 0009 O @31V10S!I GNV { Q AvuUV NI 0002 14/981 ‘3QNLIIGWY SAVM/ 39404 3301S 940 Analyses of Multtple-Float-Supported Platforms tn Waves (9-9¢ SAINSTJ) JeEOT] [[NJ UO 9dIOJ apis paonpul-aAeM : OF VANISTY ZLUIH ‘AONSNOSYSA Szo 020 Slo oro soo coforey JONIH HLIM ‘G3LV 10S! 0002 000¢ O0O0b OO0OS a 931V10SI ONV DO AVYYV NI 0009 ofeleyra 13/87 ‘30NLIIdWY 3AVM/39N04S SQIS 941 Kim and Mereter (q‘e-9¢ SaunsI}) stapour Teotmtputytso PeTTeos-ATTeoI1jseTa pue pIsIa uO adu0] apts Peonpur-sAeM : TH dINST ZLYSH ‘ADNANOSYS SZ0 020 S10 oro SOO 0002 TA00W eek OO00€ O00 T30OW GISIY O000S 0002 13/81 ‘3QNLIIdWY 3AVM/ 39404 JaGIS 942 Analyses of Multtple-Float-Supported Platforms tn Waves Seo STepoul [Te : (9*p‘o‘q‘e-9¢ SaunsTy) asuly youd YIM 9d10J SPIS paonput-eAeM : zP 9ANnsTT ZLY3SH ‘ADNANOSYS 020 SIO (o} Ke) (O'2= U7) 92! BUNDI4 JO YOLVANSLIV -H-- (91=%7/'S) 921 JYNOI4 JOYOLVANILILIV SW (Y3SGNITAD X3145) g 2! JUNO! JO YOLVANSLLV == (YSONITAD GIS) D 2! JUNDI4 4O YOLVANSLLIV —-D— YOLVANALLV ON ‘AINO LVO1I4. —O— 000! 0002 oxele} 3 COOb oxeto}s 0009 14/87 ‘SQNLITdWY 3AVM/394804 Jas 943 Kim and Mereter (a'p‘oq‘e-gg sainst 4) STepour pesuty 1oOj suoTjour yo}Id paonpur-sAeM : EF ainst 7 ZLY3H *ADN]ANOAYS S2'0 020 slo (e}Ke) soo (o) 14/930 ‘LHOISH SAVM/319NV SONIH HOLId (O2=%/'1) 921 JNNOI4 JO YOLVANALLVY = --O-- . eo (912%/'S) 921 JNNDI4 4O YOLVANSILY = =—y— 0 \ (MYSONITAD X314) QZ! 3HNDI4 JO HOLVANSLIV = —AR— (YSGNITAD GIDIN) D Zl JHYNDIS JO YOLVANSLLV -D- 9 944 Analyses of Multtple-Float-Supported Platforms tn Waves 15.6"! 2.18'' Diam x 1/8! 10.82"! Thick Damping Collar Styrofoam Ballast 04 ae 3"! 0Dx1/8"' Wall 3.78 12" opx1/8" wall 1/8'' Lucite 945 Sketch of float fabrication method Figure 44 Ktm and Merecter s}eoT] Jo Sutoeds pue juswesuetsze : Gp sunst yg 1 B/E-62 L 1g/L-~¢ ya cls | oy Fb a = ESE Eh Burseds \,$z"¢ ere See z Hoy oo 6 ey, FP Jey nGl°€ — = 6-5-6 o— 7} Z MetIDS AaplNous sobeyuiy / ere 350d [29!137429/ SMOU SE 946 Analyses of Multtple-Float-Supported Platforms in Waves 2dTASp SUIUTeI}S9I YO}Id-jseur aAeay jo Yo.eygG : OF dans 7 a,e9S OL JON a 0N Aestiy $0 moy 4Ja}ua) oO] pounsas eqnl peausS!{og buoq 51X00 ng/F Ql Moy Jo jouueY) q4edy ,,¢ peseds sbuideog {jeg Jeou!7 isl AANA, [1ey — ebeisse) eet Kim and Mercter om, © TESTS IN TANK 3 O (0 TESTS IN TANK 2 1.0 QO ins O84 vo cjlvo fo} B=} |e w]e 2a of & > Ol o ol|> xj = => 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Frequency, f , Hz Figure 47 : Heave motions due to waves on row 18 of model island 948 Analyses of Multtple-Float-Supported Platforms in Waves a foe} 5. 5 > ron _ C) ipo) oO wv a Za V: / ii / \ \ Heave Motion z Wave Amplitude iC aakcs aoe te lew oo toa iio cowe lara! ro) (=) o>) — ia ron ro ro) ee | la \ ] / ~ =< ! | Nos We ~ ~S =< « 3 ae ”~ ~O ay 5 = De Sh Figure 48 ;: Carpet plot of heave motion due to waves for articulated model 949 Ktm and Mereter PUB[ST TepOwW JO ,T MOT UO S9AeCM 0} BNP BdIOFJ DALOTT ZH ‘ 43 ‘Aduenbaa4 gl "0 910 1°0 Z1°0 (ore) g0°0 90°0 qa00oW TWWWS 1300W 39¥v7 O 6p PInsTy ) , epnzitdwy aaem 22404 SAedH 950 Analyses of Multtple-Float-Supported Platforms tn Waves pUBTST TOPOU WOT] PT MOT UO S9ABM 0} ONP 9d40} adan¢g + 7900W W1ws [ 7a00W 3DUv1 CO OG emnst 7 ooo! 0002 O00€ 0004 --—|——_}|_..-_}|__} 000s 4 ee be ous ae Le « Spnzi,dwy eaen 20404 abuns 2 (33/91) x 951 Ktm and Mereter syeoT] Jo Aerie ce xg peuTesjsat UO 3010} [BO1J19A padnpuUl-sAeM Jo joTd jadued : TG eanstz 2 « PN, duy svem 29404 aAedH z (3/91) 952 Analyses of Multiple-Float-Supported Platforms in Waves SO8ACM 0} BNP 9010} adans jo joTd yedaea ZG eanst wy 000L 0002 O00€ 0004 0005 0009 53 . 2pnzijdwy svAem a9404 abans X (34/41) 953 Ktm and Merecter [T@pour TTeuUIs JOJ So9AeM 0} BNP 9dI0j aAVeaY jo joTd Jedueg €S aINST 7 a2 « 2pnritdwy aaem 30104 oAeoH y4 (33/91) 954 Analyses of Multtple-Float-Supported Platforms tn Waves DISCUSSION Michel K, Ochi Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. My discussion is directed toward the evaluation of under- water float configuration which is one of the most difficult deci- sions to make for the design of a float-supported ocean platform. The authors have derived a significant conclusion that a slender float with displacement relatively uniformly distributed along its length appears to be superior in mild seas, while a float with displacement concentrated near the bottom is preferable in relatively severe seas, This writer concurs with this conclusion, It may be well, however, to call the authors' attention that this con- clusion is true from the view point of platform motions, but the va- lidity of the conclusion should also be confirmed from the view point of wave-induced forces and moments of the platform. To clarify the point of discussion, let us consider the follow- ing example : suppose a platform of proper size is floating without any restrictions, the minimumization of motions is of utmost im- portance. Suppose the platform is moored, on the other hand, a float configuration which yields the minimum wave-induced forces and moments of platform is highly desirable. The same way be true for a platform of relatively large size consisting of several element plat- forms which hawe been connected into a single unit. This is because the minimumization of the hydrodynamic forces and moments is ne- cessary for safe connection of each element platform, Selection of the best float configuration from the view point of motions may be rather difficult ; however, the best configuration from the view point of wave-induced forces can be achieved by adjust- ing the waterplane area of floats, since the wave-induced force ona floating body can be reduced to almost zero at a certain frequency if the underwater configuration is properly selected, This property was discussed by Motora and Koyama at the 6th Symposium in 1966 and is also demonstrated in the authors'study. 955 Kim and Mereter In short, this writer would like to suggest that the evaluation of float configuration should be made from two different view points ; i.e. motions and wave-induced forces and moments of the platform. REPLY TO DISCUSSION John A, Mercier Stevens Instttute of Technology Hoboken, New Jersey, U.S.A. Doctor Ochi is, of course, right. The proper design charac- teristics must depend on the design problem and in the case of study- ing structural characteristics and the behaviour of rigidly connected floats one must be able to calculate the forces and minimize them, rather than the motions responsible, which I studied at that time. I found that our situation in regard to calculating the vertical forces on such floats was pretty satisfactory so long as the interaction effects could be safely ignored, that is, so long as the floats were sufficiently spaced - and I hope to be able to say what I mean by "sufficiently'' at some time in the near future, I know that if they are too closely spaced we cannot calculate them satisfactorily. The horizontal force, on the other hand, is more complicated, as Weigel has just reported, and we must await further results for this case, In fact, the calculations may be influenced by the presence of ambient currents and other complicating factors. 956 SOME ASPECTS OF VERY LARGE OFFSHORE STRUCTURES G. van Oortmerssen Netherlands Shtp Model Basin Wageningen, Netherlands ABSTRACT Due to the fast development of the offshore industry, there is a rapidly increasing demand for very large unconventional offshore structures, both floating and fixed tothe bottom, to be applied for storage and pro- duction purposes. The general hydrodynamic aspects of these big objects will be summarizedin this paper. In the case of floating structures, the drift force is relatively important and consequently resonance phe- nomena can occur in the anchor lines. Therefore, in rather shallow water a structure fixed to the bottom will be preferred in many cases. From calculations and model experiments itappeared that the wave loading on a large object and the wave pattern around it can be calculated with great accura- cy with a diffraction theory. _ As an example a cylindrical storage tank - 96m in diameter, fixedtothe bottom in 50 m deep water and extending above the water surface - will be discussed. This example is hardly hypothetical, since structures with comparable dimensions are in the design stage or under construction at present. The wave pressure on the tank and the wave diffraction as calculated with the potential theory are compared with measurements. The agreement is very good. From the wave pattern around the tank it was found, that it can be advantageous to moor a tan-., ker immediately to the tank. Model tests were con- 957 Van Oortmerssen _ ducted with a tanker moored behind the tank in ir- regular seas, while the tanker motions and the for- ce in the bowhawser were measured. The results of these tests will be compared with the results of tests conducted with existing mooring systems. I, INTRODUCTION The increasing importance of remote offshore oil fields has created a need for very large unconventional structures for production and storage of oil or liquid natural gas. Some very large structures are now in use, as for instance the floating oil storage 'Pazargad' and the submerged tank in Dubai, while others are under construction, as for example the large concrete tank for the Ekofisk field in the North Sea. Besides structures for exploitation and storage of minerals, the use of very large offshore structures is considered for a variety of future purposes, Plans exist to build polluting or dangerous plants on artificial islands, far from the living areas, to prevent a deterioration of the environmental conditions in densely populated industrial coun- tries. Fear for calamities and a need of plenty of cooling water was the reason to study the possibility to build offshore nuclear power plants, and there is even talk of constructing a floating intercontinen- tal airport. With regard to the design and construction of a large uncon- ventional offshore structure, a lot of problems arise. The structure has to be strong enough to survive the severest weather conditions. In the case of floating structures, it is a problem to design a proper anchor system. When the structure is fixed, the entire construction has to be stable. In most cases, such artificial islands require trans- shipment of goods from ships to island or vice versa. Consequently, attention has to be paid to the mooring of ships to the island. If a cons- truction on the sea bottom is considered, its behaviour during immer- sion has to be studied carefully. In order to be able to cope with future developments, a re- search program has been performed at the Netherlands Ship Model Basin, A computer program has been developed for the calculation of wave loads on objects of arbitrary shape, using a three-dimensional source technique, while the effects of the free surface and of finite water depth were taken into account, With this program it is also pos- sible to calculate the wave pattern around the structure. Subsequently model experiments were carried out to check the theoretical results. 958 Some Aspects of Very Large Offshore Structures Also the mooring of a tanker to a large circular storage tank was in- vestigated by means of model tests. In this paper the following topics will be discussed successively : - the calculation of wave loads and wave diffraction, with a compari- son of theoretical and experimental results; - anchoring of floating structures; - mooring of a ship to an artificial island. The object is not to give practical solutions, but to scan the problems and possibilities which occur in the field of hydrodynamics. II. WAVE-STRUCTURE INTERACTION We shall consider the following aspects of the interaction bet- ween waves and a structure : - the pressure distribution on the surface of the body, which has to be known for the structural design; - the total wave excited forces and moments, which are important for the design of an anchor system in the case of a floating structure, or, if the body is fixed, for the stability of the structure : the ampli- tude of the vertical force, for instance, must be smaller than the ap- parent weight of the structure in the case of a submerged structure fixed to the bottom; - the wave diffraction : if ships are to moor to the structure, itis important to know in which way the incident waves are deformed by the peesemce of the structure: The interaction between waves and a structure is governed by inertial, gravitational and viscous effects. The relative importance of each of these effects depends on the ratios of the wave height and the wave length to the body dimensions. In figure 1 the regions of influen- ce of the different effects are indicated for the case of a vertical cir- cular cylinder (See ref.[1]). From this figure it appears, that gravita- tional effects must be taken into account if ka is larger than 0. 6, or in general, if the wave length.is smaller than approximately five times the body dimensions. This means that, for the structures with which we are dealing here, both the inertial and gravitational effects must be considered. These phenomena can be described adequately by means of the potential theory; this theory, however, presupposes an inviscid fluid. Fortunately, it can be stated that for large structures the poten- tial forces are predominant to such a degree, that the viscous effects can be neglected. 959 Van Oortmerssen II. 1 Potential theory approach Consider a fluid, bounded by a partially or totally submerged rigid body, a fixed bottom and a free surface. The undisturbed free surface will be taken as XOY-plane of the co-ordinate system, with the z-axis pointing vertically upwards. The fluid is assumed to be inviscid, incompressible and irrotational, All motions will be infinitely small, At infinity the fluid motion behaves as a single harmonic wave, travelling in the positive direction of the x-axis. If the undisturbed wa- ve has a frequency W, the velocity potential may be written as = Rel ae (1) The function ( has to satisfy the Laplace equation : 2 VO =0 (2) and the boundary conditions : - at the bottom gis 0 for z = -d (3) - in the free surface ot. rv" for’z ='0 (4) é Oafbs a - at the body contour va 0 forx=s (5) in which d = water depth y = W)?/g g = the acceleration of gravity s = vector which describes the body contour n = vector normal to the contour The function ( can be split into two components : P= 0.40, (6) in which Y A = the wave function of the undisturbed incident waves Os the wave function of the scattering waves Both components have to satisfy the Laplace equation. The function for the incident wave, including the boundary conditions in the free surface and at the bottom, is given by eee cosh k (d + z) ikx Oo) cosh kd ¥ (7) 960 Some Aspects of Very Large Offshore Structures in which : Hi incident wave amplitude wave number = 2 TN/, wave length Si SS Liat The relation between wave frequency and wave length is given by the dispersion equation : oe =kg tanh kd (8) The wave function Y.z , corresponding to the motion of the scattered waves must, besides the boundary condition in the free surface and at the bottom, also satisfy the radiation condition. This condition requires that, atinfinity, ((, behaves asa radially outgoing pro- gressive wave and imposes a uniqueness which would otherwise not be present. In a system of local axes with cylindrical co-ordinates r, 6 and z, the radiation condition can be formulated as: j W/Z Osean - ee ( eae eerily) = 0 (9) in which : 13 = jes + a V/2 Q = arctan (y/x) II, 2 Analytical solutions An analytical solution of the potential function can only be given for certain bodies of which the geometry can be described by means of a simple mathematical formula, such as the cylinder, the sphere and the ellipsoid. Havelock Pa for instance, has given the solution for an infinitely long vertical cylinder of circular section. This solution has been adapted for a cylinder fixed to the bottom in shallow water by Mac Camy and Fuchs how and Flokstra [ 4 ] “ According to Flokstra, the analytical solution of the potential in cy- lindrical co-ordinates is - for this particular case - given by : g e On(eOeizat) 2 a 28, cosh k (z+d) e ae + x, E. C. (i) 7 coon 0 (10) 961 Van Oortmerssen in which : A Pe J, (kr) ae i (ka) - a f (ka) x (kr) nalde (ka) +i Y (ka) n, © a, £ ‘a = baforva= 6 n Cantint Z ior n. 7.0 For the case that the cylinder does not extend to the bottom, Garret [5 ] has derived an analytical solution, using variational principles. II. 3 Numerical solutions For the body of arbitrary shape, the velocity potential can be found from numerical methods. At the Netherlands Ship Model Basin a computer program has been developed for the numerical calculation of the velocity potential, using a source distribution over a surface inside the body. According to Lamb [6 | the potential function can be found from : , &) =ff a@) ¥ wa) aa (11) A in which : % (x,a) = the Green's function for a source, singular itt say = vector which describes the surface A, on which the sources are located. q (a) = the unknown source strength. The Green's function represents the contribution to the velo- city potential in x due to a unit wave source located ina, A Green's function which satisfies the boundary conditions in the free surface, at the bottom and the radiation condition, has been given by John [ 7] 2 2 ¥ (x, a) = es cosh k (c+d) coshk(z+d)) Ya (kr.) - id | (cx) | k“d -v"d +) J J (12) ss 4 (k,” + v.7) gy a cos k (zt). cos k (ctd) Kee N=1 ak Say" -v ? on J n in which : 962 Some Aspects of Very Large Offshore Structures r = fag a + ese Ic, tan (Ik) 2d) Yeo = 10 n n The source strength q(a) can be obtained after substitution of (11) in the boundary condition at the body surface : oY” dp; oY, Ones Onn a On . (x) i q(a). Y (x,a) dA} = - ott for x=s (14) A For a restricted number of discrete sources, this integral equation changes into a set of linear equations in the unknown source strengths. For an infinitely great number of sources, the numerical solution approaches the exact solution. It will be clear that the accu- racy obtained in the calculations depends on the number of sources applied and on the location of the sources. = 0 for x=s5 (13) or: II. 4 Pressure, forces and wave diffraction Once the velocity potential is known, the different aspects of the interaction between structure and waves can be calculated without much difficulty. According to Bernoulli's theorem, the pressure is given by : p= F(t) - pgz + pass p{(S%? 4 (22)? 4 (29?) (15) The dynamic wave load on the structure is given by the linea- rized pressure : p=p 22 (16) The total wave excited forces (and moments) can be found by integration of the pressure over the surface of the body. The total force is composed of a periodic and a constant part. The oscillating part of the wave force is found from the linearized pressure : E=ff p@.n. aa (17) A ‘Similarly we find for the moment : 963 Van Oortmerssen M=ff p(x) {xx nbaa (18) The constant part of the wave force or drift force can be found from : FE. =sPff{iaor oe ——. + (2o)* \ eM ope (19) A Evaluation of this integral results in a constant term plus higher har- monic components, which can be neglected. Although the constant for- ce is a second order effect, Havelock [2] has shown that this force may be determined, using a first order approximation for the veloci- ty potential. In general, the constant force is small in comparison with the oscillating wave force; for large structures, however, it may become of interest. The wave pattern due to the diffraction of waves by the object can also be found from Bernoulli's theorem. In the free surface, the linearized pressure has to be zero, hence : p=-pgztp 22 = 0 (20) Consequently we find for the surface elevation : Faik fae) : ge lot a II, 5 Comparison of theoretical and experimental results. Model tests were performed at the Netherlands Ship Model Basin in order to check the theoretical calculation of wave forces, pressure and wave diffraction. In figures 2 and 3 the oscillating horizontal and vertical wave forces on a circular cylinder, as calculated with the computer program of the Netherlands Ship Model Basin, using the three-dimensional source technique, are compared with experimental results. The ex- perimental values, which are given in these figures, were obtained from cross-fairing of the results of a great number of measurements, which were performed with systematically varied cylinders. Also given in these figures are the values according to the analytical solu- tion of Garret. The results of the numerical calculations, which were obtained using only 42 sources to represent the cylinder, closely ap- proximate the analytical results of Garret, while there is also a good agreement between the theoretical and experimental results, 964 Some Aspects of Very Large Offshore Structures From the measurements of the total horizontal wave force on the cylinders, the mean value which represents the constant resis- tance or drift force, was also determined. In figure 4 the results are given for a particular case, together with the calculated values. In order to check a more extreme case, calculations and measurements were performed for a pyramid-like structure, of which the shape is given in figure 5. Due to the sharp edges, it is difficult to represent this object by means of a source distribution. The number of sources, applied in the computer calculations, amounts Toye ane The results of the calculations and the measurements of the horizontal wave force on the structure are given in figure 6. Even in this case the agreement is reasonable. Some aspects of the interaction between structure and waves were studied in greater detail for a circular model, which - at a sca- le ratio of 1: 100 - can be regarded as the representation of a cylin- drical island, for instance a storage tank, 96 min diameter, fixed to the bottom in 50 m deep water and extending to above the water surface. The pressure distribution on this model was determined in regular waves with varying periods. To this end the model was pro- vided with four very sensitive pressure gauges. These gauges were placed on a vertical line at regular distances, to obtain the distribu- tion of the pressure over the water depth. The measurement of the variation of the pressure along the circumference of the cylinder was established by rotating the model. In figures 7 and 8 the results are given for ka = 2 and ka = 3, which for a scale ratio of 1: 100, corres- pond to wave periods of 8 and 10 seconds, In general, the measured pressures closely approximate the calculated values. The diffraction of the waves by the cylinder was calculated with the potential theory and also measured in the basin in a large number of points around the model. Figure 9 shows the calculated wave pattern for ka = 1.4. The lines in this figure connect the points with equal values of the ra- tio of resulting wave height to incident wave height. In figures 10 and 11 the results are given of the calculated and measured wave height behind and in front of the cylinder for ka = 4. Again, the experiments confirm the theoretical calculations. II. 6 Wave loads in high, irregular and breaking waves. Up till now only sinusoidal waves of low amplitude were taken into consideration. However, for the design of offshore struc- 965 Van Oortmerssen tures, the maximum wave condition is important; such a condition usually is an irregular sea-state, consisting of high waves, among which sometimes even breaking waves will occur. High regular waves are not sinusoidalany longer, the distance ofthe crest to the still water level becomes greater than the distance of the trough to the stillwater level. However, a steep regular wave can always be splitupinto a number of harmonic components. From various experiments the experience was gained, that the forcesand pressuresin high waves canbe found by summation of the forces and pressures, as cal- culated for the different components according to the potential theory for sinusoidal waves of lowamplitude. In non-periodic waves, as far as the linear phenomena are concerned, force and pressure spectra can be calculated, departing from the wave energy spectrum and the force and pressure response functions. In sucha statistic approach, no data can be obtained with regard to drift forces. Since the magnitude of the drift force is pro- portional to the square of the wave height and also dependent on the wave frequency, this force is no longer constant in irregular seas and is thus known as the slowly oscillating drift force which has a pe- riod of oscillation in the order of magnitude of ten times the mean wave period. For an estimation of the drift force a deterministic ap- proach can be applied (see Hsu and Blenkarn [38] and also Remery and Hermans [9] ). In this approach the point of departure is not the energy spectrum of the waves, but a record of the wave height to a base of time, which can be obtained either by field measurements, or by calculations, in which case one of the possible realizations of a spectrum is generated by a computer. The wave record can be regarded as a sequence of separate wave crests and troughs, each with its own period and amplitude. For every part of the wave record the drift force can be calculated, re- sulting in a record of the drift force to a base of time. The drawback of this method is, that no indication is obtained about the chance of exceeding a certain force. The maximum force, encountered ina certain wave train, will differ from the maximum force in an other wave train with the same energy distribution. No theoretical approach is available for the determination of peak loads, which can occur in breaking waves. In [10] Wiegel gives a review of experimental work performed on this topic. Most of the investigations were related to the phenomena which occur when a wave breaks against a vertical barrier; a smaller part was concer- ned with cylinders in breaking waves. From the laboratory tests with vertical barriers it appeared, that when a breaking wave hits the 966 Some Aspects of Very Large Offshore Structures wall, the chance that a peak load occurs is about two per cent, Wave induced impact forces only occur, when the wave breaks just at the wall, while trapping a thin lense of air. Apparently, the energy of the impact is stored in the compression of the air cushion. Therefore, it is very unlikely that peak forces will occur if the surface of the object is curved. In the case of large structures with flat or practically flat walls, the possibility that peak loads occur due to breaking waves, must be taken into account. The magnitude of the peak loads can only be found by means of experiments. Ill. THE ANCHORING OF FLOATING STRUCTURES The anchoring of very large floating structures involves tre- mendous problems, since the anchor system must be able to survive the severest weather conditions. In high waves the drift force becomes very important and causes a high mean load in the anchor lines. Due to the non-linear characteristic of the anchor system - which is sche- matically shown in figure 12 - the spring constant increases conside- rably by this mean load and consequently the oscillating motion of the structure induces high oscillating forces in the anchor lines. Let us consider, as an example, a circular storage tank - 120 min diameter, with a draft of 25 m and a displacement weight of approximately 290, 000 ton - which is anchored in a water depth of 40 m. It was calculated that, in a design wave with a height of 20 m and a period of 19 seconds, this structure is subjected to a drift force of 4, 730 ton and an oscillating force with an amplitude of 58, 900 ton. If it is assumed that the motion of the structure is a pure surge mo- tion and that the damping can be neglected, the motion can be descri- bed by: bs -iWdt mxtex=F .e % (22) Vv xa in which : m = the virtual mass Vv : ; 5 4 c = the spring constant in x-direction of the anchor system Fa = the amplitude of the oscillating wave excited force in x-direction Since the relation between the force and excursion of the anchor system is non-linear, this equation has no simple analytical solution, Due to the drift force, the motion of the structure will be 967 Van Oortmerssen an oscillating motion around a point which is situated in the steep part of the load-excursion curve, as indicated in figure 12. The relevant part of the curve may be regarded as linear with an inclination c. Consequently, the resulting surge motion is given by the linear appro- ximation of equation (22) : AES 5 (23) in which : x, = the amplitude of the motion After substitution of (23) in (22), we find that the amplitude of the surge motion will be : 7 =a ip whi wae) ch =smiwGd Vv The resulting maximum reaction force in the anchor system becomes : ys eh ah reer teen Ot tte ons (25) In figure 13 the maximum reaction force in x-direction is given to a base of the spring constant. From this figure it becomes obvious that it will be very hard in this case to design a proper an- chor system. Resonance will occur if: Ps Migs G) (26) and, since most of the wave energy is related to wave frequencies between W= 0.2 and W = 1.0, values of c between 2, 400 and 60, 000 ton/m should be avoided. A value of c higher than 60, 000 ton/m means an almost ri- gid connection to the sea bottom, which must be able to absorb a ho- rizontal force of over 60,000 ton; this does not seem to be a practi- cal solution. On the other hand, if c is chosen to amount to less than 2,400 ton/m, the risk exists that in irregular seas the slowly varying drift force induces resonance phenomena, In reality the problem is much more complicated than was assumed in this simple calculation : besides the surge motion, also heave and pitch may be of importance, and due to the high waves, the 968 Some Aspects of Very Large Offshore Structures drift force and the characteristics of the anchor system, the motions will be non-linear, Therefore, model tests are indispensible to in- vestigate the anchoring of large structures. The above example has shown, however, that enormous pro- blems are involved with the anchoring of very large structures with a small length to breadth ratio. Therefore, in rather shallow water, a structure fixed to the bottom, will be preferred in many cases. If a floating structure is required - for instance because there exists a risk of earthquakes - or if the structure has to be more or less mobile, it is desirable to choose a shape with a minimum drift force, as for example a ship-shaped structure moored toa single point moo- ring system or a semi-submersible structure. IV. MOORING OF A SHIP TO A LARGE STRUCTURE For the oil storage tanks which are now in use or under con- struction, a concept was selected by which the loading tanker is not moored immediately to the storage tank, but to a separate single buoy mooring system. If we consider the wave pattern around the circular tank, as given in figure 9, regions where the waves are higher, as wellas regions where the waves are lower than the incident waves, can be observed, For other wave lengths, the wave pattern changes, but the- re is always an area behind the structure where the waves are lower than the incident waves. It can therefore be expected, that the diffrac- tion of waves by a large fixed structure will be advantageous when a ship is moored immediately behind it. In order to investigate the behaviour of a tanker, moored to a storage tank by means of a bowhawser, a model test program was performed at the Netherlands Ship Model Basin with the cylindrical model - discussed already in a previous section - and a model ofa tanker with a displacement of approximately 100,000 ton. The main particulars of the tanker are given in Table I, while figure 14 shows a small scale body plan. The weight distribution and stability charac- teristics of the tanker were reproduced to scale. The tanker was moo- red to the storage tank by means of a single bowhawser, represen- ting a nylon mooring line with a breaking strength of 150 ton anda length of 50 m. The load-elongation characteristic of this bowhawser is given in figure 15. The following tests were preformed : a- Measurement of the wave height in regular and irregular seas 969 Van Oortmerssen behind the structure, at the position of the midship section of the tanker; b- Measurement of the mooring line force and of the surge and heave motions of the bow of the tanker with the tanker moored to the cylin- drical tank in irregular seas; c- Measurement of the mooring line force and of the motions of the bow of the tanker with the tanker moored to a fixed pile of small dia- meter, in the same sea-states as tests b. These tests were perfor- med in order to determine the influence of the wave diffraction on the behaviour of the moored ship. The different test arrangements are shown in figure 16. For the measurement of the wave height a wave transducer of the resistance type was used. The force in the bowhawser was measured by means of.a strain gauge transducer and the surge and heave motions of the tanker by means of a pantograph. The measurements in irregular seas lasted 210 seconds or 35 minutes for the full scale, which is regarded to be long enough to obtain reliable statistic data. Besides the measurements, the wave diffraction at the posi- tion of the midship section of the tanker was also calculated with the potential theory. In figure 17 the calculated ratio of wave amplitude behind the cylinder to incident wave amplitude ¥ 3/ ¥ 4 is given toa base of the wave frequency W, together with some experimental va- lues. With the aid of this curve of & Py 7 q the energy spectrum behind the cylinder can be calculated for any incident wave spectrum, The spectral density Sy of the incident waves is defined by : 1 ie Sip ( W) d W => S ba (27) in which : 4 pie ayh= the amplitude of the n th component of ¥ (t) with cir- cular frequency \) 3 Consequently, the spectral density of the waves at the posi- tion of the midship section of the tanker can be found from : ‘é 2 Si (w,) aw=5o |? z= (W) ] (28) 970 Some Aspects of Very Large Offshore Structures S% (W,)=S 9 (W,) a (w 1 (29) n fan In figures 18, 19 and 20 the spectral densities of the sea-states applied during the tests, are given together with the predicted and measured spectral densities behind the cylinder. There is a good agreement, The tests with the moored tanker were performed in the spec- tra 2 and 3, with significant wave heights of 3.36 mand 5.05 m. The most important test results are stated in Table II. The most remarkable outcome of the experiments is the considerable reduction in the mooring line force, due to the presence of the cylindrical structure. The reduction in the force is relatively much higher than the reduction in the wave height. This can possibly be explained by the fact, that the drift force plays an important role in the behaviour of a moored ship, this drift force being proportional to the square of the wave height. If, for instance, we have a wave with frequency & = 0. 8, it follows from figure 17, that the wave height is decreased by 20 per cent. at the position of the moored tanker, and consequently the drift force is decreased by 36 per cent. cornpared with the drift force in the undisturbed waves. In figure 21 the results of the present tests are compared with results obtained from the statistics of tests performed at the Netherlands Ship Model Basin with different single point mooring systems. For this comparison the following dimensionless coefficients were applied : - for the mooring line force mY Me pia as P sv Cw 1/3 and for th f£ a L - for the wave frequenc ea V “pp/g in which : V = the displacement volume yee = the length between perpendiculars. 971 Van Oortmerssen Due to the non-linear characteristic of the bowhawser, the significant force is not proportional to V?/3and $w'!/,, and therefore only results of tests with tankers of comparable size in comparable sea-states were selected. From Figure 21 it appears, that the results obtained with the tanker moored to a fixed point represent approximately the lower limit of the results of conventional single point mooring systems. The forces occurring in the mooring line when the tanker is moored behind the cylindrical storage tank, are much lower than those for all other considered systems. These model tests have shown that it is advantageous to moor a ship immediately behind a large fixed structure, though it should be admitted that a rather simple case was considered, since the additional effect of current or wind from a direction different from the wave direction was not investigated. V. CONCLUSIONS 1 - The wave loads on large structures due to non-breaking waves can be predicted fairly accurately by means of a three-dimen- sional source theory. 2 - For the study of the anchoring of large structures or the mooring of ships to large structures, an entirely theoretical approach is not feasible and consequently model experiments are required. 3 - Very large floating structures anchored in exposed areas should preferably be either slender or semi-submersible; large structures with a small length to breadth ratio will require extreme- ly heavy anchoring equipment. 4 - Mooring a ship on the lee-side of a fixed structure can be of advantage. If the ship is moored to the structure by a single bow- hawser, the force in the latter will be smaller than the force which would occur in a conventional single point mooring system. 972 Some Aspects of Very Large Offshore Structures NOMENCLATURE : a = cylinder radius c = spring constant of the anchor system d = water depth iy = oscillating wave excited force ee = drift force FR = reaction force of the anchor system cae = amplitude of the horizontal wave excited force ae = amplitude of the vertical wave excited force g = acceleration due to gravity h = draught Jn = Bessel function of the first kind of order n “id Rey he derivative of a with respect tor k = wave number Ko = modified Bessel function of the second kind of order n oe = lenght between perpendiculars M = oscillating wave excited moment Pp = pressure q = source strength Sp = spectral density of the waves ws = Weber's Bessel function of the second kind of order n ss) wn ay derivative of a with respect to r A = wave length WO = circular frequency Oo = mean circular frequency in irregular waves e = fluid dens ty vy = Green's function 0) = velocity potential e = wave function DHS Van Oortmerssen wave function of the incident waves = wave function of the scattering waves Z =W" /g¢g = wave elevation incident wave amplitude local wave amplitude wave height (crest to trough) HW volume of displacement REFERENCES OORTMERSSEN G. van "The interaction between a vertical cylinder and regular waves"! Symposium on ''Offshore Hydrodynamics", Wageningen (August 1971) HAVELOCK T.H "The pressure of water waves upon a fixed obstacle! Proc. of the Royal Society of London, Series A - N° 963 Vol, 175 (1940) Mac CAMY R.C. and FUCHS R.A. "Wave forces on piles : a diffraction theory" Technical Memorandum N° 69, Beach Erosion Board (1954) FLOKSTRA C. "Wave forces ona vertical cylinder in finite water depth" N.S.M.B. Report N° 69-107-WO, Wageningen (September 1969) GAREE L G..1., ik. "Wave forces ona circular dock" Journal of Fluid Mechanics - Vol. 46 (1971) LAMB H. "Hydrodynamics" Sixth Edition (1932) 974 [ 10 ] Some Aspects of Very Large Offshore Structures JOHN F. "On the motion of floating bodies" Comm, on Pure and Applied Mathematics, 3 (1950) HSU F.H. and BLENKARN K.A. "Analysis of peak mooring forces caused by slow vessel drift oscillation in random seas"! Offshore Technology Conference, Houston (1970) REMERY G.F.M. and HERMANS A. J. "The slow drift oscillations of a moored object in random seas"! Offshore Technology Conference, Houston (1971) WIEGEL R.L. "Oceanographical Engineering" Prentice-Hall, Englewood Cliffs (1965) SN fe) Van Oortmerssen Table I Main particulars of the tanker 249. 38 37.41 13. 85 106, 792 109, 462 0. 826 0. 985 58. 61 ley Length between perpendiculars 42) ue) Breadth Draft (even keel) Volume of displacement > =] Aa Displacement weight in sea water Block coefficient Qa 4 00) Midship section coefficient * (<>) > 1) - Oncoming wave is Airy wave - The exciting forces are computed using Froude Krilov hypothesis With these assumptions , the equations of motion with six degrees of freedom are obtained reasoning as in [2] (m+Q) —-[v] +[N][v] +[r] = [F] + [P| with | 1031 ge p g m A Lv |= col M,O,N vu Ava Cc ao 7 (5°%)- Pp ce — x Dern —»> ——b 0 ie a iis coe oi a a Za — 2f Pe Bei ve Ra ated 4u m g O specific mass of water acceleration of gravity mass of the buoy BE: , OY ede a aes a dt dt dt dt dt Tensors of inertia , added inertia and damping Immersed volume at time t Acceleration of fluid particles due to orbital motion of the waves imperturbated by the buoy where x is unity vector of Ox axis Current point on the buoy axis of revolution The tensors M, Q, and N are considerably simplified due to the fact that the buoy is a slender boy of revolution iM = diag [ m, m, mi, ' hee, Tyy, 0 | +0 (R*) m O O O m O xX xO O m O m O O yy y¢ O O m O O O ZZ O m O a O O py xx m O O O J O Ox yy O O O O O O 1032 Unstable Motton of Free Spar Buoys tn Waves N O O O N O xX bdo) O N O N O O yy y¢ = O O N O O O ING ZZ O N O N O O ey ee N O O O N O Ox 6e O O O O O O II - EOUATION OF HEAVE MOTION - The equation (II-1. 1 ) gives for heave 2 duZ dz ie Lf Sie + = (II-2.1) (m m_) ne jE Vrae U tee e chy = pay ieay + mg if the potential of on-coming waves is 2% Ni Zo h k D (x, rt) leer ge Sika (Grate = x, ) = k Zo then = 5 2 no kg . e coso t Let S(z) be the area of transverse section at station z pef avi pe fst) ang af le) da + me In type 1 buoy , S (z) is roughly : z +h, OF igi Zi 0 with 1(z) = aah ize. 10 h, and h define the form of upper part of the buoy as indicated on the following figure ho3S Dern Type 1 and 15 buoy Type 12 buoy (approximate shape) (exact shape) For the type 12 buoy the exact expression of S (z) is zt+hy 2 hy -h Mea] el (z +h.) For type 1l or 14 S (z) = S (o) 1 (2 + h,) where h, is very large . As an example , for type l h, > 0 1034 Unstable Motion of Free Spar Buoys tn Waves Ss AP Fo TBO. 3” es ont wolmie aw hi Bails SO gen +P Beh h, 2 Ww hy ee 2 rs f dU = me +p g Slo) | cz - bales we with (II-2. 2) f(s) eee 5 The calculation of the integral in the right hand side of the equation (II-2.1) is much simplified by the hypothesis of slender body theory and if we suppose that h. > =4 OR) 2 and k Ro _ o(R) then ks =)" o\(R) = ~oz0 dl = =H ge Seo) VC kH Q, (KF + o(R) Sf hy p with the notations displayed in the nomenclature . 1035 Dern Finally , the heave equation is 2 (1-2.3) (mt+m,)S2 +N, 2 4 5 g s(0)[Z + 4s) ] ‘ dt = pg S(0) [ = C, kH Q. (k)]¥ (t) . where f (s) is given by (II-2.2) for type 1 and where This equation completely determine the buoy heaving beha- viour if the values of m, and Note known 4 3 . ae Gh from ([ 5] p. 200) 2 N = gh alee: lof : g. [4 ae k H Qo (k) | from [1] ca = H . Pp Thus , in this theory , the added mass is independent of the frequency but the damping is frequency dependent . The heave equation in J.N.Newman's theory is obtained by setting f (s) == 0 and mo = 0 . Infact , in this paper , we mean by J.N.Newman's theory , the theory where f (s)==0 but where my is given by the expression above . III- EQUATION OF SURGE AND PITCH MOTIONS - Equation (II-1.1) becomes for coupled surge and pitch motions Z 2 8 (mtm, |) Set mS tN SX en 5 oo = 2p facx ae elt anc: ee ee ae PB a°o a°x do ax au (ita) tea ge gl dee g) dV =2p) (GP aox) Bea gan He Go SOP OR Aa Z n 8), i noon where ( )x designates the projection on the Ox-axis . 1036 Unstable Motion of Free Spar Buoys tn Waves The pitch equation obtained by elimination of X in the above two equations is of order 3. In order to simplify subsequent calculation we shall make use of the fact that the coefficients N and N are small . Neglecting them we obtain a second order equation in®. bas is) a“e dé@ mx9 sie2 Se —= ——P_ a —P sp oct Did tyqgsdyy ee | 2 tNeo at ° he i [GP,aox), m+mxx ]av wv The calculation of both integrals is performed in the same way as for the heave equation . Neglecting second order terms in s one gets : (11 - 3.1) maa a’o de ZG “'’ \iyytIyy- at2 +NOO 3, + mg| t+at (s) m+mxx H 8 mx®@ 1 G+3C d =2 mela (i) + ———Qo0 (k) +. 2# s : m+mxx 2G H dt Pp where the symbols introduced are defined in the nomenclature . The function g (s) is for all types : a7 (Wee. @) G(s) Shas) lit, es ahs WHES 95 oo) fom ihe hype Il amd 14 The coefficients of equation (II-3.1) are the same as n [1] J = ig Je yy 2 a = P, x8) = 5 2), ian m = 1h xX TI R4 a ee 4W Nae =2 p2 c2 H? [sire [2, Ole 1037 Dern IV - EQUATION OF ROLL MOTION - The roll equation is obtained from the pitch equation by removing the forcing term located in the right hand side of the equation : gre : 4 a . my p Z sf im cae —— = (II 2 4, 1) Ixx+Jxx -——— ah + Nee 4 ah ¥ = P 3 0 ; bi 2 dt H mtmyy/dt with J iH qy we yy and NPP =Nog m and mm =m yy x0 yy xX | Experiments have shown (cf.Section I) that the roll am - plitude may be much higher than the pitch amplitude . This fact suggest the introduction in the equation (II-4.1) of a new viscous damping term . This term is obtained using the Morison O'Brien equation which gives the force d F (z) acting on a strip of length dz for an infinite length cylinder in an unstationary flow [27] TD av(t) P d F(z) = Zzlp:P lure). u(t) + Ce as where D is the cylinder diameter , U (t) the velocity of the fluid relative to the cylinder , Ch and Cur are constants The second term of this equation was already taken into account : the first one corresponds to a non-linear damping . Equation (II-4. 1) becomes (11-4. 2) Z rs OV Ee tah ee een ot 2G Ixx+ Jxx- Reais ate + Ny a Nyy at | at t™m8|ttA+H g(s)IP = 0 (1) with Ne - NY (2) p 2*G (H- “er Nyy oo. Cp: ye, + 4 withD=2R. 1038 Unstable Motion of Free Spar Buoys tn Waves Equation (Il-4. 2) is only approximate for roll-sway cou - pling terms have not been taken into account (cf. previous paragraph for linear terms and Morison - O'Brien formula for non linear ones). Moreover Morison - O'Brien formula is only approximate . The coefficient C_ is actually time-dependent and the value to be attribu- ted to this coefficient varies in a large range [6] . Lastly , the finiteness of the buoy slenderness (H/D) is not taken into account (end effects) . One should note than equation (II-4. 2) does not include the non linearity due to the static restoring moment [ (y ) . In fact , the exact expression of, [* ( ) is: TR Vv 2 " (p) =-mg [ ta aS + tg yp ] sin ~ with PA) v= aR tg Y an There is very little difference between this expression Bae -mg (r+ a) sing even for angles approaching 'P = 60° (relative variation is about 3/7 LOO): V - APPROXIMATE EQUATIONS FOR HEAVE-PITCH-ROLL MOTIONS - Figuresn° 11,12, 13 relative to the frequency respon- se of version n° 11 show that, in the linear case,the dynamic system associated to the buoy (in the sense of paragraph 1-6, 2) may be approximated by a second order differential system . This result suggests that we may substitute to equations (II-2.3) , (II-3.1) and (II-4.2) the equations (II-5. 1) 2 Z 1) Sr evesto) [2-110] nso -c,as1000f ate +N2z 4, +pg S(o) |Z + £(s) |= pg S(o) Ty nat) (t) Qu (m+mzz) 1039 Dern (IL-5. 2) m xe d 06 aan d@ ZG - (1y¥+9y~ ma ate + Nog at +ms | stat as g(s) [© mxe lo 2G ae ay. —_ me] 0, 06 4 Paccnantig eae 2Cy H dt (1-5. 3) my hey (0) *¥ A ee ee Ixx+Jxx - evtim Fa ee +Npe ar * Nee at] ae t™mslrtat g(s)I? =e where the coefficients in the left hand sides of the equations are constants and where f (s) and g (s) are given by (IL-2. 2) and (II-3. 2). In the sequel this particular form of the equations will be used . VI - EXPLANATION OF THE DOUBLE REGIME IN THE HEAVING MOTION - The presence of double regimes in a rolling motion has already been investigated by various authors . The experimental finding of this phenomenon was made by E.G. Barillon who explained it by considering a non linear damping proportional to Op" (oS are a forcing moment proportional toh (m< 1) and a restoring moment in the form I'(p ) = - A@ - Bg? (A and B are constant) [7] . The rigorous mathematical explanation of the phenomenon was given by R. Brard , starting from equation of the form : 2p (II-6. 1) — + B @(se Hy (9) = F cos ;re where B is aconstant , andg(. ) andh(. ) are analytical functions [8 ] , but numerous approximative methods exist [9 ] : [10] The double regime in the heaving motion of the buoy is an analogous phenomenon , which can be explained at least qualitatively by equation (II-5.3) . Taking s (t) as the unknown in this equation , it becomes : Zz ds prea = at” + nas at ds (mt+mzz) + pg S (0) F (zs) = [ (mtmzz) ~ pg S(o) C. k H Qo (k) I A coset toNzz f ,sin ot 1040 Unstable Motion of Free Spar Buoys tn Waves s lity tore 0 2 s where F(s) sila: _ Ou s 3mh., TYPE 1 BUOY - From equation (II-6. 3) it is easy to see that for a given frequency and a given wave amplitude , there exist one or three positive roots s, . By a method identical to that used in the following paragraph , it can be verified that when there are 3 roots , one of them is unstable . This explains the jump and hysteresis phenomena observed when solving equation (II-6.2) on an analog computer . Figure 17 presents the gain curve obtained (25 versus f) , and shows a good agreement with experiment . fa LP ie 2 BUOY — Equation (II-6.4) is of degree 6ins, . It has therefore six roots but it is difficult to see whether they are real positive . Rather than solving directly this equation on an analog computer , we solved it graphically , which permits to understand better the jump in the gain curve of the relative motion (Figure 15) . The notations and terminology of i Bil ] are used here, and it is assumed that the wave amplitude is small enough for s (t) to be always less than h, . In order to simplify the computations it is also assumed that h, =0 . In facth, = 0.08mandh, =0.42m. Setting h, to zero al thus slightly modify the results . However the general shape of the phenomena will be kept . The describing func - tion of equation (II-6. 2) is 4Aa_ a 2 (o) Fiona) —s. pie4S(o) sme + 8h, -o (m+t+mzz)- io Nzz where it has been seta A s for a while . 1043 Dern The characteristic equation of (II-5.2) reads 2 A fo ey OX Hie (a, ©) ’ tase = 0 ©)2 h ee Ae en ey ees Nig Bags ie = nO pg S(o) (2a- 4 ae fa-4 Sones as |e s(o)(1- 8a ss + mate yom.) 3h, = Bh For a given frequency and a given value of a, the quantity a H (io, a) defines a point in the Nyquist plane . When a varies , this point describes a curve called an equifrequency curve . The solutions of equation (II-6.2) are obtained as the intersections of this equifrequency curve and a circle centered at the origin and of radius equal to the numerator of expression (II-6.4) times f (This construction is but the geometric interpretation of equation IIl-6.4) . These solutions are stable if the real parts of the roots of the characteristic equation are negative , they are unstable otherwise. When the wave frequency is small , equation (II-6.2) has only one solution (Figure 18) . For a slightly larger frequency , equation (II-6.2) has two stable solutions and one unstable solution (figure 19) . Only the smallest solution has been obtained experimen- tally , because during the experiments we did not attempt to see if a second stable motion was possible . When the frequency ‘s 0.182 Hz there are only two solutions both stable (Figure 20) . Figure 21 clearly shows the jump phenomenon . Beyond i = 0.182, there is only one stable solution (Figure 22) . From the quantitative point of view , Figures 18 to 22 lead to solutions slightly different from the experimental results . In particular ; the jump phenomenon occurs for f = 0.182 Hz instead of f = 0.190 Hz . It is believed that this discrepancy is due to the 1044 Unstable Motton of Free Spar Buoys tn Waves difference between the real value of h (h, = 0.08 m) and the value actually used in the computations (hy 0) a From figures 18 to 22 , it is easy to determine the in - fluence of wave on the value of a=s, . The equifrequency curves are indeed independent from the waves amplitude , and the circle radius is proportional to it . One can see , for example , that for f = 0.182 Hz the curve gain VS half-wave-amplitude must exhibit a jump att, =0.128m. In conclusion to this investigation of the double regimes , figures 23 and 24 show the effect of h, and f , on the shape of gain curves computed for the type 1 buoy . If we turn back to figure 15 we see that the above theory does not explain the double regime in relative motion in the vicinity Ont = 2%. . ae reason for this discrepancy is that we have assumed that 4 (t)< (see beginning of this paragraph) . No attempt has been ne at explaining the double regime in the vicinity of f = 2 fy by discarding this hypothesis . VII - EXPLANATION OF ROLLING IN REGULAR WAVES - It has been known for many years that a ship moving in longitudinal regular waves can perform rolling motions of large amplitude [12] . In 1955, Kerwin [13] explained the motion by the periodic variation of the restoring moment in rolling due to the on-coming waves . The roll appears as an unstable solution of a Mathieu equation . In 1959 , Paulling and Rosenberg [14 ] have shown that instabilities in the ship motion could be explained by the effect of second order coupling terms in the equations of motion (see also [15 | and [ 16 )). This latter work was pursued by M.R. Had- dara for the case of a ship in oblique regular waves [iv] In the case of a spar buoy , Kervin's approach does not apply , since the wave length is considered as large as compared to the buoy diameter . However , the rolling motion can be explained by the presence of a non linear coupling term between heaving and rol - ling ( equation II- 3.3) . 1045 Dern os! 00! uoljow ajqnig O wzvos*y sg ety AONE TL adAl ( 7H091'0=4) JAYND ADNINOIYIINOA -— gl JNO os 0 00l i os~- 0oI~ 1046 Waves mn Unstable Motton of Free Spar Buoys uoljow ajqoisun OI uoljow ajqnig © wzpos?y = gt ty AONE Tl AdAL ( 7H9ZV0=4) JAYND ADNINOIYIINDA ~6I INNO! 1047 Dern (7H zet0=4 ) 3ANND wzz'o =%9 ADN3INO3IINOA ~ OZ INNO! 0s- UOI}OW 3/GD1S © wzpory oly AONE Th AdAL 1048 Unstable Motton of Free Spar Buoys tn Waves "x (7H psiO=} ) JAYND ADNINOIYIINOA ~1Z JYNOId 001 0S 0S - 001- sr’o wizio: ¥ 080 Olt uoljow ajgqois © yh wzpo2-y O= ty AONE TZ adAl 1049 Dern 00! (7H 0zz'0 =} ) 3AYND 0S ZzVo Bye, ADNINOIIINGI ~ TZ JUNO Id 0 0S~ Mh 0017 Uol}OW ajqDis oO w2po=Zy oFty KONG TZ} AGAL 1050 Unstable Motton of Free Spar Buoys tn Waves (V5/Vg) seamo ures paqefnoyeo ey} Jo edeys ay} uo Sy JO anjea ay} Jo Wojjq-EZ onsigq as Oe ort ON aoe ea we RE Mv 3 (z4)senom ipjnBas yo Azuanbai4 “GZ0°0 =V9 wG9'O Seu ® 9 wor'O =%y © 4oeo ="4Y @® ; : =7Z “sro =“4Y @ XY 1051 Dern - (V5/Vg) seamo utes poyemoreo ay} Jo adeys ey} uo VJ} Jo onjea oy} JO ejJq — FZ amMsiq ‘0 v0 eo AC) Ke 3 (ZH) s@anm spjnBas yo Aduenba.4 _—e “ozo = zy “0070 = V9 ® “szl0 =V¥) © 9 “GZ00 = V9 © “OE00 = V9 © 1052 Unstable Motton of Free Spar Buoys tn Waves Zz If it is assumed that No 0 , equation (II-5.3) becomes 2p m afi. EA ee 5g | s)] | ¢ = 6 dt@ dt © ne N B = Ixx + Jxx = m2 yp (37.1) m+ myy mg (rt+a) 6 = Ixx + Jxx - meyp = ant be m+myy i pl Zi Gr mg (ie — aha Aoi | Ixx + Jxx - m2y/? m+myy Seis piventbyaequation, ((l-= 5.1) ithusa(t)—w4e(t)) ela it) Let us assume now that s (t) is a sinusoidal function . This assumption is rigorously verified for the type 11 and 14 buoys., since equation (II-5.1)is then linear . It is only an approximation for the type 1 and 15 buoys , since equation (II-5.1) is then nonlinear Furthermore , let us assume that s (t) does not take too large a value , so that s (t)=s, cosa t < h, Equation (II-7.1) can then be written as : d2y age) Pe cuaueos 24) vy = 0 ast 2 y =e Q (t) a o t a2 2 A 2: tao pe DM GTS ABE S EE Oreo 2 4 aes iz o o f A q = bs, 1252 1053 Dern Equation (II-7.2) is a Mathieu equation [18] . Its stability chart is recalled in Figure 25 . The values of B , & and 6 for the buoys type 1, 11, 14 and 15 are given in the following table . Except for type 11 and 15, the values of P are quite small , and the stability map of equation (II- 7.2) is thusas a first approximation the stability map of equation (II- 7.1) . The experimental points obtained for buoys type 1 and 14 are shown on figure 25 , where it can be seen that they agree rather well with the theory . When the wave height stays constant and that the frequen- cy varies , the point of coordinates (p , q) moves in the stability plane (p - qplane) : when f increases , decreases and the point moves to the left . To have an idea of the phenomenon let us replace the curves (a,) and (b,) by their tangent at the origin . Instability occur when : 2 2 q ={p-1) 20 i.e. when os 58 4 °° =F = < if > f Then S, =Jap for f=2 fg and the condition for no rolling reads : P% (@i=7. 4) Se < = or ZC (II-7. 5) ae Q del 1/3) (: - = 28 Unfortunately B isnottheoretically known since oo is determined experimentally . Nevertheless for a given value of Sn , (Ii-7.5) shows that there is no rolling if the center-of gravity is located well below the center of buoyancy . Vill - EXPLANATION OF THE UNSTABLE PITCHING PHENOMENON- The unstable pitching observed with the type 1 buoy can be explained qualitatively in the same manner as for rolling , by reducing the problem to an equation where the left member is a Mathieu equation . However there now exists a right member which is a sinusoidal function of same frequency as the perturbation term in the left member . It is known [ 19 | that the regions of stability and unstability are the same as for the equation without a right mem- ber . The qualitative results established in paragraph VII are thus valid . Figure 28 shows that the theoretical domain of instability does coincide with the domain of instability determined experimentally . POST Dern SAJAVM YVINDOIY JO ADNINOII SA ONII1OU JO JANLIIdWYV 41IVH ~ 97 JUNOI4 70 (7H)J 0 SNI110¥ ON ONITION ON OL oe 47710 * 2 (0 = 44N ) pasoin2}05 dxgj + syuawisadxy = boN adAL = (soiBap) Up 1058 Unstable Motton of Free Spar Buoys tn Waves om @) le (sajow) V5 SSAVM =YVINOIN JO JANLNIdWY “4JIVH SA ONITION JO Ae) €0 _ ad (os9= GyN ZH OVEO= J JaNLNdWY 4IVH ZZ AYNOIS ZO Ke) d3LVINDIV5 SLNAWIdSd X43 +f LN JdAL Ol Oc (0) ( saiBap ) Vd 1059 .Dern (2H) SJAVM «=YVINOIY JO ADNANDIYS SA ONITION JO FAGNLIIdWY 41IVH 8Z JNO! v0 Burjjos ou woZlo =V9 (o= q)N) dauvindivw>—— SLNAWIYadX3 PLN 3dAL mi Z, rae) UO Buijjou ou Oc (seaBap) vy h 1060 Unstable Motton of Free Spar Buoys tn Waves SSAVM = aVINO’Y JO JGNLNdWV ~4IWH SA ONITION JO SGNLIIdWV 4IVH 6% AYNOlJ 10) (ssajau) v9 44 7,0 £0 ZO ee) 0 = + 1 ol oz (2st =(N) Galvinolvoleeme ZH OVZ'0 = 4 () a SLNAWINadX3 VL.N ddAl of ( saiBap) vt 1061 Dern AONG + 3dAL YOS SINIWIYIdKI~AYOIHL NOSIYWdWOD:3SNOdS3Y HDLId ~ Of 3YNOI4 (ZH) seaom 4pjnBas 0050 OSv 0 0060 ; yo Azuanbasy iavis NN ° INNS 6j OSF0 00z0 osto ono OS00 0 = 1 4 + + oF oot wozo=”9 @ ------- wsioz *) + syuawisadxa Asoay) ost "S w/bop) Ve 1062 Unstable Motton of Free Spar Buoys tn Waves SECTION III ROLL BEHAVIOUR OF A SPAR BUOY IN MONODIRECTIONAL IRREGULAR WAVES I - INTRODUCTION It seems that no paper has been published to-date on the rolling motion of bodies in monodirectional irregular waves . At least the author has been unable to locate any . One may quote Kastner's study [20 | about the righting arm of a ship in a longitudi- nal irregular sea . In a sense , his study is a continuation of Kerwin's theory but it does not deal with the effect of the irregular variations of the righting arm on the rolling motion of the ship . From a mathematical point of view , the problem of the rolling motion of a spar buoy is similar to various other problems of mechanics (for example : simply supported beam subjected to stochastic axial load ; vibrating string when the distance between its ends varies stochastically) . All these problems are related to the study of non-autonomous stochastic dynamical systems . Many results exist for these systems but the most interesting one seems to be that of G.F. Carrier| 21 | . In the present Section , the rolling motion of the spar buoy is studied by using Carrier's theorem whose statement will be given in the next paragraph . The validity of rol - ling criterion is verified experimentally . II - HYPOTHESES FOR CALCULUS - In this Section we use equations (II -5.1) and (II -5. 3) with the following simplifications : (2) Nee ii} So = (oe) ae 1063 Dern The approximation h, = o is valid only for type 11 and 14 buoys . For type 1 and 15 buoys , this assumption is very se but it avoids intricate computations . We must recall that Na is experimentally determined (damping tests) . We assume that the wave elevations (t) is anormal , strictly stationary random function . We also assume that the sea is monodirectional and that its spectral density is given by the mo - dified Pierson-Moskowitz formula| 22 |, namely . 2 2 : SK 29 eas 2a ie a a a i exp E 0, 44 (Tvf) | It follows that signal s (t) is also gaussian and strictly stationary . III - ROLLING CRITERION - With the above hypothesis the rolling motion equation takes the form (see paragraph 7 of section II) 42 d (III-3.1) —* 0 when t-++ © where <-> denotes ensemble average . Now equation (III-3.1) may be written as | sp ge (,) | p, (t)= 0 where e, (t) ae 1064 coar- Unstable Motion of Free Spar Buoys tn Waves (il ates 2 t JN ay a t l= 4 26 34 (t) a ae ae Be s (t) 4 Equation (III-3.2) has been studied by G.F. Carrier [ 21 7 who has given an expression for which is valid under certain conditions (see also Keller ea) ) . Carrier's expression and relations (III-3.3) permit us to give the expression of by the following assertion : Assertion - If , in equation (III-3.1) , the coefficients BS 5,6 are constants and if s (t) is a gaussian white noise , then , the asympto- tic expression for<@ 2(t) > is 2 2 t 6