REPORT NO. 1370863 a At. / 56 PROJECT TRIDENT TECHNICAL REPORT STRESS ANALYSIS OF SHIP-SUSPENDED HEAVILY LOADED CABLES FOR DEEP UNDERWATER EMPLACEMENTS — ARTHUR D. LITTLE, INC. — 3 CORN PARK _ CAMBRIDGE, MASSACHUSETTS ~ AUGUST 1963 O @205€00 TOEO oO MAC A A IOHM/18N REPORT NO. 1370863 PROJECT TRIDENT TECHNICAL REPORT STRESS ANALYSIS OF SHIP-SUSPENDED HEAVILY LOADED CABLES FOR DEEP UNDERWATER EMPLACEMENTS ARTHUR D. LITTLE, INC. 35 ACORN PARK CAMBRIDGE, MASSACHUSETTS DEPARTMENT OF THE NAVY BUREAU OF SHIPS NObsr-81564 $-7001-0307 AUGUST 1963 PREFACE This is one of a series of Technical Reports being issued by Arthur D. Little, Inc., under contract NObsr-81564 with the Bureau of Ships as part of the TRIDENT Project. Arthur D Hittle Ine. S-7001-0307 “98 postin Av Aes i sot ow : ee ane, * aininaaah ny HINO NORE Dh huiiate TABLE OF CONTENTS Page LIST OF FIGURES AND TABLES iv I. SUMMARY 1 A. PURPOSE AND SCOPE 1 B. CONCLUSIONS AND RECOMMENDATIONS 2 II. STATIC STRESSES DUE TO THE WEIGHT OF THE ARRAY AND CABLE 5 Ill. THE EFFECT OF OCEAN CURRENTS 7 IV. THE EFFECT OF MOTIONS OF THE SEA SURFACE 10 A. FORMULATION OF THE PROBLEM 10 B. THE STEADY STATE SOLUTION FOR SINUSOIDAL INPUTS 13 C. PARAMETRIC ANALYSIS OF THE MAGNITUDE OF THE MAXIMUM STRESS FOR SINUSOIDAL INPUTS 17 V. THE DYNAMICS OF THE VESSEL AND SPECTRAL CHARACTER- ISTICS OF A REALISTIC SEA 25 VI. BUCKLING 29 APPENDIX A - FRICTION ON THE CABLE 32 APPENDIX B - THE DRAG ON THE ARRAY 34 LIST OF SYMBOLS 36 PROJECT TRIDENT TECHNICAL REPORTS 39 DISTRIBUTION LIST 42 iii Arthur D.ULittle Ine. S-7001-0307 ii i Pea ew | ‘er ’ 3 v9 Washo ih TACT pores Hess riasee sin) is 13 O02 mn e ot ) er x ideal nO Ren TE NA nent WS PNRM ANCA one 0 4 ue nau) BA AACR UMTE, AON rannre wt! fie Agen 1 A a" ARAL ey ART Das ai aba non Ai) zl ai 1 aint ARS att i) hie ah oe i. VATS eer 4 ch Mo yon bigy * ju i RAM BY MO). tas AG nay HN y, Rel a r iy We Pail Ba aT Bd Figure Table II LIST OF FIGURES AND TABLES Page Definition of Space Coordinates 5 Bending of the Cable by a Horizontal Current 7 The Normalized Maximum Dynamic Stress Versus the Normalized Angular Frequency for Various Values of the _ 4apA|Uo| _/wL Parameters 8 (- eqn and uU = =) 18 Response Amplitude Operators for Cuss I and Maximum Safe Input Amplitude to the Cable as Functions of Frequency. (Cable is a Special Flexible Hoisting Wire Rope of USS.) 24 Amplitude Characteristics of Fully Developed Sea 26 Frequency Characteristics of Fully Developed Sea 26 iv Arthur D Little, Inc. S-7001-03 07 aoe rs aT) ; Ven ly oe : a) a ! eat Oh Renny om finite my Ret: lee a | Lok wb wane): whekne wu ie . Otis eullay! rn ? a” bay (latil fea ite A Ae 2 vn ee me : is biidigeh in ay, fhm" ) eat. a4 “vert ison ora niki va er un te arent tai Autidhy oy aah er ile (2a bait stat: ells qutiniott. apaela anal" ( tiphs na aa i eon ay : I. SUMMARY A. PURPOSE AND SCOPE This report relates to the operational aspect of lowering heavy array structures to the bottom of the deep ocean. Because of the weight of the array and the depth to which it will be lowered, the lowering cable can be expected to be under a high stress. Ocean currents, surface waves, and motions of the lowering vessel will also contribute to the stress placed on the lowering cable. Therefore, a critical design problem may exist for the lowering cable. The objective of this report is to develop a reasonable theory on which to base the ultimate design of the lowering cable. The study is limited to the case of a single cable lowering the array vertically. Each factor con- tributing to the stress on the cable is considered separately, and the conditions under which the analyses are valid and of practical significance are given. From the results of this theory some general conclusions are drawn as to the feasibility of the operation with regard to dynamically optimum array configu- rations, stability of the lowering platform in a rough sea and degree of rough- ness of the sea (sea state). The results of the study are presented in plot form involving a non-dimensional maximum (dynamic) stress versus a non-dimensional frequency for various mass and drag parameters. Other methods of operation, which are not treated in this report, but which will be considered in later investigations, are: (1) Cancel out most of the dynamic inputs to the cable by an auxiliary mechanical system on the vessel. There are two crucial problems here: the horsepower of the driver of such a system and the rapidity with which the over- all system can respond. (2) Introduce considerable damping along the cable, which will have the effect of flattening out resonant peaks. This can be effected by attaching bluff bodies to the cable at certain intervals. The drag of the water on them will give the desired friction. This idea seems quite attractive, since these bodies can also provide buoyancy to reduce the rather high static stresses in the cable, but its plausibility must be investigated quantitatively. Arthur D.ULittle, Inc. S-7001-0307 Oe é be iwi hal oyna te h bo ss o ‘ o Ain 0 a, 4 wey aa ete a oe ek Le es tt aeeneetic “5 Gblieg lt PR aM ST alike. jp iPae Wee caild Wy alge mama a la a oka aig wl AWA oy pein ri ss cah ape pepe anal Weta ay 4 timeapey Lorenstine Wb Ot 4 oR OT RES yt ; Rami AS) WHA iags pAaiieete bo Sud HY eR ieext an é ae he Weils hos Cunistioor pee ae) ges sieldles camer ‘aougitnts (ovis « OTiw WIAA ANED we Sen it Cel aa a wn nips hi Wea? Ce, wale cQey 1S NTRS Yl coal Hah aon - ens 1oki ab Ass ars it{ gts ' ere’ = he ow” esi Pe too ee ee ) foie? Up evodtibr ebay ie = ~ j iu ve pevire widy Ui Peewety tm he veolani we ae ; : eit . :( aay ” ee, f us han 7 ve Pratl eo 4 lire ime ~ ee x ; pe ay os je aes ie Will: stvliber ioe Wat hin. wn fone wecereyy ie dale Bee nada 2a ma _ _ a . ona adie % 7 LOY oes ee qe | AMM D “tial naa GP grva nl beeen’ ef novell ©06ii4 shale cul See ee , ay ‘Jj Wie alta” ilies see ange via bin @> we 9 7-4 ha hep onee Sells OO a 2 as (3) Use more than one cable. However, it should be borne in mind that each cable must actually be designed to withstand much more than an equal share of the total load because of some adverse conditions that can readily oc- cur. Also, the cables can be tangled, a situation which can lead to serious problems. B. CONCLUSIONS AND RECOMMENDATIONS The static stresses due to the weight of the array and cable can be taken care of by practicable cables. For a maximum depth of 20, 000 feet and 50- and 100-ton steel arrays, the maximum static stress in a steel cable of three square inches metallic area is 100,000 and 130,000 psi, respectively, when no buoyancy is introduced anywhere. For the same depth and a 15-ton aluminum array (which in water weighs about 10 tons), the maximum Static stress in a steel cable of one square inch metallic area is 100,000 psi. The ultimate tensile strength of such cables is about 220,000 psi; therefore, these cables can handle these stresses with a factor of safety of about two. However, the above static stresses must be reduced, because higher factors of safety (about 3) are usually required for such an operation, and in addition, there are other stresses (dynamic) in the cable. Static stresses can be reduced by increasing the metallic area of the cable or by making the array and cable more buoyant. Since a large portion of the static loading is due to the weight of the cable (the maximum length of cable which can hold itself without any factor of safety is only 62,000 ft), the most effective method is to introduce buoyancy in the cable. An equivalent way of effecting this is by using a tapered cable. Nylon type ropes are desirable in this respect, because they are much lighter than steel ropes while their ultimate strength is just about as high as that of the strongest steel ropes. The velocity of sound of nylon ropes is also much smaller than that of steel ropes. With respect to the dynamic stres- ses that can be induced by a rough sea, these two factors (small weight and velocity of sound) render nylon ropes undesirable. Of course, the small modu- lus of elasticity, characteristic of nylon ropes, is a favorable factor and can probably compensate for the undesirable effects of the above two factors. In any case, in this operation there are some environmental hazards which make steel ropes preferable to nylon ropes. The effects (stress and array offset) of reasonable ocean currents are negligible, provided that the vertical forces due to gravity minus buoyancy are much larger than the horizontal drag forces due to the currents. This will be the case in practice, because buoyancy must not be used to the extent of Arthur D Little, Inc. S-7001-0307 i iy ny Thin. Vivaah ‘ an F LN ayy ah hia r prem sl bie mn me | dade than Ce ye ! a Py epee Taine. 4% rte oh hy einer ibe | Ce — jf DRE OR a ee ein seamed PAE ieh i BR, e thi laupbaie f ce oe i apn anne ait in i ye é pik ened cbten hes ; 7 | un TT peer ¢ egos : , | Sa iin Y ; . a ii . Ve a att ey tials ey hy is We J : iD uly tkses Donen iow eta ; Palit hes uy canceling the gravity forces completely. It is highly desirable to have the cable in an appreciable static tension so that compressive dynamic stresses will not buckle the cable. Buckling of the cable is not in itself dangerous, but the formation of kinks, which can occur when the cable buckles appreciably, is. Some consideration of this problem is given in Section VI. Perhaps, the problem should be pursued further. The motions of the vessel in a rough sea (heave, roll, etc.) will induce dynamic stresses in the cable. Due to its great length, the cable can- not be considered nonflexible for the kind of inputs that exist in a realisticsea. In fact, for sinusoidal inputs, resonance can occur at frequencies as small as 1.5 rad/sec. These frequencies correspond to periods as large as four sec- onds, which are included in the spectrum of a disturbed sea. In order to obtain an accurate value of the maximum dynamic stress induced in the cable by a regular (sinusoidal) surface wave, the dynamics of the vessel, cable, and array must be analyzed simultaneously. The parame- ters necessary for such an analysis are too many, and the problem can become formidably long and difficult. However, it can be safely assumed that the dy- namic loading of the vessel by the cable and array is negligible, except possi- bly when the cable is attached to the vessel by a boom extended from the side. Thus, the problem can be simplified by considering the dynamics of the vessel and of the cable-array system separately. A theory is developed which takes into account the propagation of longitudinal elastic waves in the cable and the complete dynamics of the array in water. It is shown that friction on the cable by the surrounding water is small and can be neglected, even near resonance, since friction on the array is much larger. The drag on the array must be taken as quadratic because of the large Reynolds number involved. This is the only nonlinear term in the theory. It is linearized to a quasi-quadratic form. Thus, a formula is de- rived for the maximum dynamic stress due to a sinusoidal input. It gives this stress in a parametric form as a function of the input frequency and amplitude, characteristics of the cable, and of the weight and shape of the array. This formula is plotted in a dimensionless form with two parameters. It is shown that resonant frequencies as small as 1.5 rad/sec can induce very large stresses for input amplitudes of one or two feet; furthermore, that dynamically heavy arrays increase the dynamic stresses a lot, especially near. resonance. Entraining or displacement of large amounts of water in- creases the dynamic mass of the array. Therefore, the array should be a trussed open structure. Also, the velocity of sound in the cable should be as large as possible. This makes the resonant frequencies large so that they lie on the cut-off edge of the spectrum of a given sea state. Arthur D Little, Inc. S-7001-0307 nant sean wan made) 6 sarah m a ith ate oebaeeite Rey ea | ei Sob ve iS YPN sablego, 2 “ih . i WaNnieoyh ocr Ween 9 a tsln Hahei ete , + gphacstr ae att 6 GLU pie leaye) te Te oe | caas phe fess ae) feta, gh. QO are A Na me ul Oe eyh pitt sis laniyiee ral Fire ek ctcb , ah " =lpacy 1q¢eiex lade, iia ee ee peony ip | - ie di'aie: y fapkicdusyen lode 2: Wek jee “ty i palo a a “oe ‘i nt a ue = iphady =A} tire Sanne et), EA ek AL lope oe west ea aera: Gal ria 1 . yO Tenth aa sia iaaemhdl ro sical ron ball So Lak ve — a7 ‘ Ahk ety] Ai : iL aia } MP winipees te Celie ‘is “j mit Maen. (vl a t ts Pe i ts Wty hy 40 pomh' i Ace i - igcie wll tA eohoimive Gain wn Ne eee ag pis MAR Lestlaa Oh AE TT: at resin Steric wil D at Pe uit 4 aa 1h Wie” @ aaa i" ‘ NeOtbe All WOMEN Bou ro ead GOL ee ee By Pdr) aes a r hp GO ii hey 2 Oileet- At an fis i Yaw Ape reg Wy 4) hh ne 6 Tere | - mid. ah, (eee er vamalihitnadr Wilid ledh) ay aye ey tN ee Muse uyt 1/0 , ; in el et wlll B min T pAb) 2) BMA bs? via Rises 1 hve! FI . lqrvlta ; ine } a i y i ’ th : Tv: BI orig (i es eds fh ee teak 8 yl) Mopac 8 5 giv: Tie Aart tt ie. wasau) / Berio ias ed ‘Hii 7 Mi vve>. Ht ty, @ ; j ty meet } Ll ieee AGES? it is) Te) LO . Ba wail be = ih, tele Oke: @le ee aa A 15-ton trussed aluminum linear array, whose length is about 600 feet and whose cross dimensions are a few feet, and a 1-5/8-inch diameter 6 x 37 Special Flexible Hoisting wire rope (USS) are selected as a numerical ex- ample. A curve is computed giving, for each frequency, the maximum safe in- put amplitude for any length of the cable up to 20,000 feet. This curve shows that, as the frequency increases, the maximum safe input amplitude decreases sharply. Ata frequency of 1 rad/sec (six-second period), the maximum safe amplitude is 20 feet, while at a frequency of 2 rad/sec (three-second period), itis only 1.7 feet. This shows that the vessel must be rather stable if this oper- ation is to be carried out successfully in a moderately rough sea. Also, it indi- cates that ways in which the cable can be attached to the vessel are limited. Two obvious ways of attaching the cable to the vessel are: (1) from the center of gravity, and (2) from the side through a boom. In the first, heav- ing motions of the vessel in beam seas will be critical, while in the second, rol- ling motions in beam seas will be the most critical. Amplitude response curves for heaving and rolling of Cuss I (see footnote of page 27 for a description of this vessel) in beam regular (sinusoidal) waves are plotted. The problem now is that a realistic sea is highly irregular, with a continuous probabilistic spectrum of amplitudes and frequencies. To overcome this problem, we adopt the procedure, used by many other investigators, of rep- resenting an irregular sea by a regular one. In this way, we can draw some con- clusions as to the chances of the success of the operation, although much depends on the definition of the regular sea representing an irregular one of a specified State. It is shown that, by attaching the above cable to the center of gravity of Cuss I, the chances for the success of the operation are quite good in seas excited by a wind velocity up to 25 knots (beginning of sea state 6). It is also shown that, when the cable is attached to the side of Cuss I through a boom, the chances of success can be acceptable only in very small wind velocities. The above results lead one to the idea of using either an unconventional vessel or a lowering platform detached from the main vessel, in order to achieve higher stability. The use of a very stable platform for lowering arrays into deep water will be investigated further and discussed in a later report. Arthur D Little Inc. S$-7001-0307 _ , rr ‘aides. reer ai ae "4 A re a We 7 ee gp at ae vay en sf a j A A » Split j oe borate ao not ed . : - - 7 t } ok t se i F i * , i 7 am me rv — : 7 . ye ee ee ‘ “., 7 : Lis) 7 - ; Me 7 o -% " a = | 4 i” 7 ae 1 ca- Pon eh —— i we i] : ; ie on eae J whl’ 5 | ea 7 7 - > TF >> e 7 1 “Ri . @ eg 7 : — 7 ny aan R ie ) + ; errs ; if = ‘Wide heidi AA phd | ve ees ie sae reall Opis artes we) wanes i? rae ry iene od al Lee Ae Bins tli ta ii iWin sat rk ob sey sh Cee deh ine ia oY Lh cae ae mil ee 4 be aie oe ee en ams iat aae: os 7 a“ PNT wien We wid u, jell pe Wut RT BT BT pet br yu: a Oe tt + Lapel i 338 lity Nay PRL “yeti we ed ap) oe Mentioeteantl nie (dors ia area De ee a pee Pex eete gnctwines Pe? CSO ER KRW, THe aie mpage pit 8 Pap Wien dag tt hae to " Il. STATIC STRESSES DUE TO THE WEIGHT OF THE ARRAY AND CABLE The distribution along the cable of the static tensile stress, o, due to the weight of the array and cable is linear and is given by: ae é (L - x) (wb) + W, -B, | (1) where: x = Space coordinate measured as indicated in Figure 1. L = Length of the cable. Ss = Metallic cross-sectional area of the cable. Ww = Weight of the cable per unit length. b = Buoyancy force on the cable per unit length. We = Weight of the array. Ba S Buoyancy on the array. ae FIGURE 1 DEFINITION OF SPACE COORDINATES Arthur D Little Inc. This stress is maximum at the top of the cable and minimum at the array. For a steel cable with S=3 in? and L=20,000 ft, anda 50-ton array (without any buoyancy), the maximum and minimum static stresses are 103,000 and 33,300 psi, respectively. For a 100-ton array these values become 130,000 and 66,700 psi. This cable could be a steel wire rope of the Galvanized Bridge Strand type with a diameter of 2.25 inches. The ultimate tensile strength of this rope is about 220,000 psi. Therefore, this rope can take the above stres- ses safely. This type of rope has a ratio of ultimate tensile strength to weight per unit length equal to about 62, 000 ft, which is equal to the maximum length of rope that can hold itself. On the other hand if buoyancy is utilized, the static stresses can be decreased considerably. In this respect, nylon type ropes are preferable, since they are almost weightless in water while their breaking strength is just about as high as that of the strongest steel ropes. Arthur D Hittle Inc. S$-7001-03 07 Ill. THE EFFECT OF OCEAN CURRENTS Due to the frictional forces on the cable, a current will bend the cable as shown in Figure 2. (Such frictional forces will also act on the array and vessel if they are within the current.) Of interest here are three quanti- ties: the additional stresses in the cable, the offset (D) from the target, and the horizontal force (F) which must be exerted by the vessel in order to remain Stationary against the current. This problem can be solved very accurately by balancing the compo- nents of the various forces acting on an infinitesimal element of the cable. The results for all pertinent variables can be expressed in terms of integrals which have been tabulated (David Taylor Model Basin Report 687 and its Supplement). However, if the net vertical force (gravity minus buoyancy) is much larger than the horizontal frictional forces due to the current, bending of the cable will be very small, and the following simple analysis can be applied. ARRAY FIGURE 2 BENDING OF THE CABLE BY A HORIZONTAL CURRENT Arthur D Little Ince. presen ap? 9 eae a mate ys oy dan +t Avie 9 dl eli I poo) reat aah qa . (ese a as LBP ee ae a TT Series” © aes POM Te ae oe Consider a uniform current of velocity V and width X (see Figure 2). The horizontal force, F, which is equal to the total drag of the current on the cable, is given by: I Oh ents) W~ Ox (2) where: a = Drag coefficient. It is a function of the Reynolds 2aV number, Cele where y is the kinematic viscos- ity of water. a = Radius of the cable. 0 = Density of water. The stress at the top of the cable (which is the maximum stress) will be altered only slightly by the current. The offset, D, is given approximately by: ChEL o) W/E DL eye OF (3) a a which is an overestimate of the actual D. As a numerical example, let us consider a current with V = 1 ft/sec and X = 5,000 feet.* The cable described in Section II will be used. The drag of this current on the cable is 0.225 lb/ft, which shows that the above ap- proximation is valid. The horizontal force, F, is equal to 1,100 pounds. For the 50-ton array without any buoyancy in the cable and array and for L = 20,000, 10, 000 and 5,000 feet, D is equal to 18, 28, and 37 feet, respectively. These offsets are, indeed, very small. Even for lighter cables and arrays the offsets will be small as compared to the length of the cable. *A current with this velocity and width is about the largest that will be found in the Atlantic. Arthur D Little, Ine. S-7001-0307 7 vs | - ina ca w ir iD Rain Wy _ vibonty thane’ ma vipa ike ty ; i h ; a7 j mt "6 a oF ha TG (oderieauunlinen: of | cy AM >be : aq) rid x tw AT ed, acts Tha “Ae OAS YI Gt) Cees ys Sy j | Ayia © 1h. FR 4 ae y; if a , ab) og m4) way j it2 ee My a vail 7 be) jhe) fa wes? | vba . ee ae ee tA ine wal< iw ive ire oe ihn ral j ) oy 4 ‘ayia o a 5 i / mais tin’ ity! @ Pwhetonitaniy) deen) Of).,iote wt | ah Deere at very oo) i lala tinea fy tay th Lo TEL tae ee When the array is in the current, an additional horizontal force, due to the drag on the array, must be included. The magnitude of this force will vary for arrays of different shapes. Assume a linear array composed of three horizontal hollow cylinders each 600 feet long. Two of the cylinders are assumed to have diameters of two feet, and the third a diameter of one foot. For normal incidence, the drag of the current on the array is equal to 1.8 tons. Since this force is small, compared to the weight of the array, its contribution to the stress will be small. The offset and the horizontal force, F, will in- crease accordingly. Arthur DHLittle, Inc. $-7001-0307 oI ra .) IV. THE EFFECT OF MOTIONS OF THE SEA SURFACE A. FORMULATION OF THE PROBLEM A realistic wavy sea will cause the vessel to move in the vertical direction, and, thus, generate waves in the cable which will interact with the array. This Section deals with the analysis of the dynamic stresses induced in the cable by such waves. This problem can become as complicated as one wishes to make it. Here we will aim at a reasonable linear analysis. Therefore, we must, first of all, assume an elastic cable. Metallic cables remain elastic as long as the total stress does not exceed their elastic limit. For the cable considered in Section II, the elastic limit is about 1.5 x 10° psi. The maximum static stress calculated for the cable is below this elastic limit, and therefore, this cable is still elastic. Nylon type ropes are more plastic than elastic, unless they have been prestressed sufficiently. On the other hand, the dynamic stress wave propagating in the cable can be a compressive one. Then, if this com- pression exceeds the static tension in the cable, the cable will buckle, and the propagation of stress in the cable becomes a nonlinear problem. Buckling is treated briefly in Section VI. Since the present linear analysis assumes that the cable does not buckle, any results which indicate that the cable should have buckled are not valid. Consider the cable and the array in a vertical position. It has been shown in Section III that the deflection of the cable by a reasonable horizontal current is small under conditions stated there; therefore, even in the presence of such a current, the cable can be considered to be vertical for our present purpose. As the upper end of the cable moves vertically, waves will be propa- gating in the cable. If u(x,t) is the dynamic displacement at time t of an ele- ment of the cable Ax, which in the absence of waves is located at x (see Fig- ure 1), the governing equation for the propagation of waves is: o7u O72 u ou PD gr SSS seer SS ae i) 10 Arthur D Little, Inc. S-7001-0307 ay, ot ade Gale laser ‘a goenialied dapniativod oittasiinay Aydt itdas ht ai’ TF mane at 3 ne | Seen sr Aamay on vk Coase . 1 PaCONY et Hii erent iqalineny: eaves Oiikall grat ti 1 ND th So dtl hadornci geitubentce mom where: e. = Density of the cable S = Metallic cross-sectional area of cable E = Modulus of elasticity of cable K = Constant of friction on the cable by the surround- ing water. It should be kept in mind that K may not be actually a constant. In an actual operation, these waves can appear while the array is being lowered at some rate, i.e., L in general changes with time. However, if we assume that the rate of lowering is small as compared to the velocity of the dynamic displacement due to the waves, then L can be considered constant in time and the following dimensionless variables and parameters can be de- fined: ce alt Wess GA dene, eure © = (5) : E KL Sy wee = 6 c 0. B AGS (6) where c is the "velocity of sound” in the cable. Equation 4 then reduces to: o7u OM . OF m See Tle ae = ss (7) In order to solve Equation 7, one needs two boundary conditions (most likely one at each end of the cable) and, if transient solutions are sought, initial conditions as well. One of the boundary conditions, which we will apply here, is the specification of u at x’ = 0 for all times. Practically speaking, we should specify the motion of the ocean surface for all times. But, then the dynamics of the vessel and the way in which the cable is attached to the vessel must be considered simultaneously with the dynamics of the cable and array, which is a very complex problem. A discussion of the dynamics of the vessel is presented in Section V; here we will assume that we know u at x' = 0 for all times. 1l Arthur DHittle Inc. S-7001-03 07 ane iy Lise: ai alti shang wae sanhate | fh We TOWNE seri Ye Bis y hate | ‘ia a | a: an ae) Qulaolaw Bek of betaninve ae Uamere pee hain ‘t -. er) bantu bios od KA .! jit sboren ah © gah - on Act. nay a7" ae etna hrm walls man / seed v7 si = ¥ 4B) 7 tou deat wt iin | a 4 a> The other boundary condition is obtained at the array (x = L), where we must have: ou = = 0 (8) where: M_ = Dynamic mass of the array. In general, it will be composed of three parts: the actual mass of the materials of the array, an apparent mass due to the motion of the array in water, and the mass of any water trapped in the array and having to move with it. a = Drag coefficient of the array. This is a function of the Reynolds number, and for certain array configurations such as cylindrical or spherical, its value can be found in the literature. A = Area of the array projected in the direction of motion. Notice that a quadratic form has been taken for the hydrodynamic drag. (The two vertical bars in the first velocity factor mean "absolute value of," and they are necessary since the drag must always oppose the motion). This is absolutely necessary because the Reynolds numbers involved are high. Let us define the following two parameters: p SL Cc u = ae (9) a a Notice that u is the ratio of the mass of the cable to the mass of the array. Equation 8 can then be put in the following form: Te a 2 iF Xi at’ = 0 (10) J Arthur D Little Inc. $-7001-0307 : mi ae" be Oe Fi nodal iw K wi aur ch rc ‘Wie aft pr $b. tye sive a eh *. yer Lt why ee Kae atta. “rgel erggen talk aR teat, Mi) . oly BEd Oe liedesigbtes 6) boise ne Habe aad) weds: Ori = ic sts ie Lan a rs) a Tey nbc Tite-ulh ti Masoiceme ent ante tb & wi a bbe tre tah A GN ee, Ge ey eh haat Wit oe hy dinar h sii ate ne OUIGV Boal ted’ wedi ws whi ae 0S wht oh Wrist | esha at thy. sith Mi ; ltt 2c Te ee SAN SHA SE RN UCHR sala nope uid tin “all ’ arte ata havilov ra. wearin pte vied tt @ ste cae Rey ah ale So in | en wh Pe Minit 30 ney Ake ie Ie. ET of sil a ui LANE Aa Le 5c me te Giit Ate ah a tlt i e bs hk itt B. THE STEADY STATE SOLUTION FOR SINUSOIDAL INPUTS We will take u at x’ = 0 as sinusoidal, of amplitude U., andof angular frequency w and try to find the steady state solution of the problem. In Appendix A it is shown that, under conditions which are valid here, the friction on the cable is proportional to the velocity, and an expres- sion for the friction constant (8.) is derived. In Appendix B it is shown how the drag term in Equation 10 can be linearized and how good such an approxi- mation is. Once this linearization is introduced, the entire problem is linear and therefore, for sinusoidal inputs we will have sinusoidal waves. It must be emphasized, once and for all, that the approximation of the nonlinear quadratic drag on the array, developed in Appendix B, is not linear but "quasi-quadratic."' It replaces the drag, which is actually propor - tional to the square of the instantaneous velocity, not by a term proportional to the instantaneous velocity but by a term which varies in time as the velocity and whose amplitude is proportional to the square of the amplitude of the vel- ocity. In what follows a capital letter will be used to denote the complex amplitude (magnitude and phase) of the respective instantaneous sinusoidal variable. For instance, U(x') means the complex amplitude (magnitude and phase) as a function of x' of the instantaneous sinusoidal dynamic displace- ment u(x’, t’). Let us define a normalized displacement amplitude U' as being equal to U divided by Uo| . If U, is the value of U' at the array (x' = 1), then the solution for U’ asa function of x’ is: U = U, cos w'y' + C sin w'y’ (11) where C is a complex constant and x _ wl Te » W ae c (12) This solution satisfies Equation 7 if the friction of the water on the cable is neglected. As is shown in Appendix A, this is valid for the frequencies of in- terest here. Only at very sharp resonant peaks can the friction on the cable be important, and even then its role can be insignificant if damping of a greater order of magnitude exists somewhere else in the system (as, for example, in the array). 13 Arthur D Little Inc. S$-7001-0307 aaa it wey c Hoy Wi asia kL vi } ws Af mmole tm ants eislanniel Mf ‘et if weer bas toe tei ” ” f sel su ag int AW no sinatra wl sath the: a Jo A) Bedard VA ct tad vio re) x are wi fads wel tints Py Burd: ety soit . i rik . danniete ang orpad: ‘BG Jon ae TAN Saaanaica mal perens ; bi ee dley at te Set RabtEp Ae ehhh RD ye fuct Vat ati wes SND i ; stay. ont We shu tana’ id i) aTeUpe bit Biss Kyat ontoqe a nm: asia aa : it { ra ‘i ; , - ! nye curlapiet dy: Ti ie! ae vee i Py wr rity pu nyt { V5 4 t F sui a i aul ea. Ly seit Gi) or : ‘ ee iy 4 alice Site aie ‘iplioad) ; He Jeet oer Gsl BVI SS Pela) Ae (x sath Ce a: ca of) diirgn 4) seaplane eal(nice oo mirent (aed ein el ae | * apee PEs i ins sf i | al wh by Lab hoaninty by OT TE Ley Be itch: rhe a Be gas miterasryhy Sab oanity Beinn nANt 9K WO ® AORTA A A y rd ET ak 7 Chir. oon... The two unknown constants U; and C will be determined by the two boundary conditions. By defining the origin of the time (t') properly, we can take U; as being real and positive. Then, according to the linearization introduced in Appendix B, substituting Equation 11 in Equation 10 we obtain: - (w')? U; - wuc + iB(w')? (,)2 =O) (13) where: 8 4apA [Vol (14) 37M, This parameter is the ratio of the drag to the inertial force of the array. Therefore: t SC SEA aay at Se a U, (leu) (15) Then Equation 11 reduces to: U' = U, sec @ cos (w'y'+¢) +i8 (U,)? tan @ sin w'y’ (16) where: ian 6 = —, 0", as well as their locations, can be computed analytically. Denoting the magnitude of the maximum of ~%’ as (sea the location(s) of this maximum as y',,,,, the mag- nitudes of =" at x' = 0 and x' = 1 as |=}| and {=| , respectively, one can show that: 2 eas 2 = (w')? (U,) E + tan ¢ (tan ¥ + sec ¥) | (20) ad = ia = = ZO Man as fd ve Mm © iy Sy D coc (21) Z| = (HP (ye E + 2 tan @ cos? w' (tan ¥ + tan w)| (22) a = (w')? Wye E +2 tan @ tan ¥ | (23) where: 2 2 ’ Ve) ted an are) wy? _ _G9s2 (MO) (+ sin* w' sin 28) 1 (24) 28 sin? @ sin? w' cos- (w' +o) @ = arctan ron. (25) ¥ = arc tan 1 g2 (U')? tan @-cot2¢|, -~<¥<2 (26) 2 a =” =P 15 Arthur D.Hittle, Inc. S-7001-03 07 A Ae . + " ? a ‘(aia ae ay yas es a eon Kk : ‘ i pan. phecipeed @ ie atu ha a tuna ite te corer iat 7 ou i a idan Ay iy : bi “y p ‘a ‘qs ay me ae Ait hi y : yea iS Dati we me ened h F nee 7 ia '¢ sor es # a bal Beri ; ve i i Ls | ye ry < We eee poe For convenience, we repeat here the following definitions: La wL U Os peace ees | er x Ho sa er aw. IP UME 20) fe) © | 4 p. SL on 40 A|U,| ai S Ne 37M a a If the values of y',,,,» aS given by Equation 21, are outside the range 0 to 1 for all of the indicated n's, the maximum stress, as given by Equation 20 does not occur in the cable. In that case, the maximum and mini- mum stresses in the cable are the stresses at the top and the bottom of the cable, respectively. In order to have 0 $s Ymax * 1, the condition that 2w'+V¥2 5 must be fulfilled. 2. Special Limiting Cases Case 1 w' = mm, where m = 1, 2, 3..., andu< 1 so that @ = > Then U, = 1 and: 1 m= Li ’ = |r = ' = 2\3 | acl : Ba ee) (29) Case 2 L- 0 so that w’ and u- 0 but ¢ is a finite number other than zero. Then everywhere in the cable U and © are constant, and they are given by U= Ce and: Pye Eee) 2 ys m= = w UM, (1 +6?) (30) 16 Arthur D.Little, Inc. S-7001-0307 Wo, eM halide y LP Ber Ast C. PARAMETRIC ANALYSIS OF THE MAGNITUDE OF THE MAXIMUM STRESS FOR SINUSOIDAL INPUTS The magnitude of the normalized amplitude of the maximum stress, as given by Equation 20, is obviously a function of only three variables,namely w', u, and 8. Itis plotted in Figure 3 versus w' with u and 8 as parameters.* As it is stated at the end of Section IVB1 the maximum stress as given by Equa- : A T i yas tion 20 occurs in the cable if 2 w' + ¥ 2 7° When this condition cannot be fulfilled, which can happen only for small values of w', the stress at the top of the cable (Equation 22) is used in Figure 3, because it is greater than the stress anywhere else in the cable. Figure 3 then shows the dependence of the maximum stress on the input characteristics (t) and U,), the characteristics of the cable (c, E, andS), the length of the cable(L), and the weight and shape of the array. Let us see whether Figure 3 is compatible with the well known results of simpler systems. For 8 = 0, there is no damping in the system. In this case the resonant frequencies of the system are the roots of the equation tan w' = u/w'. The smallest of these roots lies between 0 and 1/2, the next between 1 and 31/2, and so on. At these resonances, the reflections of the on- coming stress waves by the hanging mass have the proper phase so that their individual contributions result in an infinite total stress. As the hanging mass is decreased indefinitely (u-~), the resonant frequencies approach the values 1/2, 31/2, etc., and we have the case of the free end spring. As the hanging mass is increased indefinitely (u-0), the resonant frequencies approach the values 1, 21, etc., and we have the case of the fixed end spring. On the other hand, when 8 # 0, energy is dissipated by the hanging mass, and the amplitudes of the reflected waves are diminished. Resonance can still occur but with finite amplitude. The actual value of 8 (amount of damping) should have a small ef- fect on the values of the resonant frequencies and a very profound effect on the amplitudes at resonance. One could expect that the amplitude at resonance will be decreased as the damping is increased. Figure 3 shows quite clearly all these expected trends, except one; that is, for the resonance occurring near w' = 7 (and 2n, 3m, etc., as well), the amplitude increases when the damping is increased beyond a certain value. *In computing these curves, the values of ae and | x have also been recorded. These values can be of use in the design of the joints at the top and bottom of the cable. IL7/ Arthur 7 Little Inc. S-7001-0307 tae eh ht oe un ae ~ pe une aac vi ‘ sult i ‘ in ay 5 rn : e ww: ud righ | Cs we vy. # to 4 Pr aH sl bil ‘il ‘ae i a te wi Mt, ’ Hh gi¥ 30.5 haere, eat ‘3 td a i aM als: Str Abagne Ph tt ai he sr) a i x i 4 hi ; ¢ al Ue oh! eh hy ipa ee reatst | Bee FEMA A iT) dey ie ey 48 ovr at aired ail 4 ergy “ | 190 ‘ “ . i, ; 1h nm " 7 i hi igi - i 4 ri i) uy a vA 7) iin a Lite * j Be yi fait i YG yf) wi : i , A ; \ ' Ah 4 j TOT wes TBE ai Biol Wass vy m f ; - . a tu y ' : 7, . D” . i | ny AR) bs Ne or eam LY rer = sary Ni 1 wa) uty wits ie Mh ' | | i ( Lai heh 0 1 2 3 4 >) py 1") NosmalizediAng Ularieirequency. wis cn) FIGURE 3. THE NORMALIZED MAXIMUM DYNAMIC STRESS VERSUS THE NORMALIZED ANGULAR FREQUENCY U 7_WL FOR VARIOUS VALUES OF THE PARAMETERS 8 (= 1 Alt ») and w=) 7 M, \ M, A 18 — é Fae | a eam derryl en ee ner eer” Ame Scena vnere 4 Paola nae fearan Nias ant oni et ia ro eine - i ois atthe a trai oak IRATE poradipas 2S es Gtk So een hs any sani A : Neper4 & far re ae) mw ~ spe AD nai ah a pee 1 om aman ? ~~ i i VO reins W ab i ‘“ ig LAT AN fais ee So eT Vt ve ee . ee eee bs, falas, li erring barbed cy sabe sii ate 8 iat vom a i ERR caRer shell ements owed uw ashi wr va dite ty i i. cient ila yi Ai I. SUMMARY A. PURPOSE AND SCOPE This report relates to the operational aspect of lowering heavy array structures to the bottom of the deep ocean. Because of the weight of the array and the depth to which it will be lowered, the lowering cable can be expected to be under a high stress. Ocean currents, surface waves, and motions of the lowering vessel will also contribute to the stress placed on the lowering cable. Therefore, a critical design problem may exist for the lowering cable. The objective of this report is to develop a reasonable theory on which to base the ultimate design of the lowering cable. The study is limited to the case of a single cable lowering the array vertically. Each factor con- tributing to the stress on the cable is considered separately, and the conditions under which the analyses are valid and of practical significance are given. From the results of this theory some general conclusions are drawn as to the feasibility of the operation with regard to dynamically optimum array configu- rations, stability of the lowering platform in a rough sea and degree of rough- ness of the sea (sea state). The results of the study are presented in plot form involving a non-dimensional maximum (dynamic) stress versus a non-dimensional frequency for various mass and drag parameters. Other methods of operation, which are not treated in this report, but which will be considered in later investigations, are: (1) Cancel out most of the dynamic inputs to the cable by an auxiliary mechanical system on the vessel. There are two crucial problems here: the horsepower of the driver of such a system and the rapidity with which the over - all system can respond. (2) Introduce considerable damping along the cable, which will have the effect of flattening out resonant peaks. This can be effected by attaching bluff bodies to the cable at certain intervals. The drag of the water on them will give the desired friction. This idea seems quite attractive, since these bodies can also provide buoyancy to reduce the rather high static stresses in the cable, but its plausibility must be investigated quantitatively. Arthur D.Uittle Inc. $-7001-0307 aie oo _—_ r4 a 1 = on ieee ts ue) is vie ae ¥ refit th ‘ ry Rie ae ype pau eink HD z re ; P re ts gue ome ib wits baer Kvieadh atrrny oonind: =) ig Vy, i dk vive ad Ja) ear til et : ane sigh adi aL Se fe at ws fi nai ita Fe 1 iP is il) Wivys J Aue ; 2 iin 7 “One ae i | viel - = 7 ity ; A } 4 9 fas , iv 4 ' . ‘se ; ra A rey. ty by & Anh we am iy gun rs i 4 day sah vib tay PY 2 ta ia are lan aaa m Vgas il uv a mys Ss Ill. THE EFFECT OF OCEAN CURRENTS Due to the frictional forces on the cable, a current will bend the cable as shown in Figure 2. (Such frictional forces will also act on the array and vessel if they are within the current.) Of interest here are three quanti- ties: the additional stresses in the cable, the offset (D) from the target, and the horizontal force (F) which must be exerted by the vessel in order to remain stationary against the current. This problem can be solved very accurately by balancing the compo- nents of the various forces acting on an infinitesimal element of the cable. The results for all pertinent variables can be expressed in terms of integrals which have been tabulated (David Taylor Model Basin Report 687 and its Supplement). However, if the net vertical force (gravity minus buoyancy) is much larger than the horizontal frictional forces due to the current, bending of the cable will be very small, and the following simple analysis can be applied. SURFACE — O56 ARRAY FIGURE 2 BENDING OF THE CABLE BY A HORIZONTAL CURRENT Arthur D Little Inc. : a a - _ a : i ry ; : : . gu | ee SS LAE _ Ln 7 pois Al =) 7 - ; a A — Mine me : iu - , it et - Eee. to —— ar ony hy Po ak at os: on a Septet Pw ere. A ae - - A a *- G0 us tied Lanny peaph.' cd a - bell rail | u rire x i hades sit ribet! vines AV ave 5 4 ee al y i s Lb oa _ a . : he, ‘ nga ey Rt ia « BTA RIAA EM TE) >. SS Cera This should be expected, because when the damping of the hanging mass be- comes very large, the mass cannot move very much, and we have again the fixed end spring with resonant frequencies at 1, 27, 3n, etc., and infinite am- plitudes. To put it another way, if a very large amount of energy is dissipated by the hanging mass, the energy propagating along the spring must be very large, since the spring is the only link between source and load. Two remarks are in order here. First, as Figure 3 shows, there are resonances (some of them highly sharp depending on the values of u) at w' =m or w=". For c = 10,000 ft/sec and L $ 20,000 ft, this corre- sponds to frequencies as small as 1.5 rad/sec or periods as high as four sec- onds. Since such periods can be found easily in a realistic sea, it is not cor- rect to consider the cable nonflexible when an accurate estimation of the stres- ses is desired, unless such frequencies are filtered out by the vessel very ef- fectively. Also, for small values of 8, there are resonances at values of w' much smaller than t, corresponding to periods much larger than four seconds -- which are quite frequent and prominent in a rough sea. Second, an idealized system, such as the free end or fixed end flexible cable, will give portions of the curves of Figure 3 rather accurately. However, this does not render the present analysis, which includes the dynamics of the array, superfluous. Our objective here is not to display a rough picture of the phenomena, which could be done by the free end or fixed end cable, but rather to find as accurately as possible the dependence of the maximum stress on the various quantities. In terms of a realistic and sound design of the cable, which hinges on the magni- tude of the maximum stress, simple models--such as the nonflexible and the free end or fixed end (flexible) cables--are useless, and a complicated model, like the present one, is indispensable. In order to demonstrate how these curves can be used, let us con- sider a cable and array as specified below: L = 20,000 ft M, = 600,000 lb ie 6: ..; re 2 E = 20x 10° psi A = 3,000 ft S = 3in2 @% 8 1.2 w = 10.6 lb/ft c = 13,600 ft/sec 19 Arthur D.Hittle, Inc. S-7001-0307 : aki Cart rl id ts aba ws wal pon zea eda nd. oa voto mF 0 pea \be ah% ox i witenn saul ae Mf veh at oth mu 7 i hie baa ait sxafie si ety | deh eta as raat il alee oa sah a tes) ulaw A i diene | ui oH Vay uf, et buat 1t wats cntes ony puibicnpeniny } pawn tis. d/o cy nod Hyo', gal, Nid tie? uy Deke Hie “i abbas li a elt Vit p NEE > Wlehdvee IY Paty Cone Te tart Cea et th wis. 4 one i Ania caves seit oe ee wink ve Laprciad i ot hee : : a Slat won mut weal AUTH Acie ere re’ al 1) Migk hairs ly fli ivy P > iia pie s VRE Det an ua% ri ay j isha ohn ctr ‘gee Lani ot raw) ‘Jut '; POO, hat cent ht sie 7 ad bin aw orks nO Canis aed aar ad Le iy ef teein y Bact al” tefay! split: thy th gis waatp: Ee a, ‘Nab ] bab nds StSea it thal? Whe 4 Leda pikue im esa, ple bi {ew lt) tema . ies ih We (ee wit.. we i ident, F The array described at the end of Section III has been taken here. This array has a weight of 50 tons. Because it is hollow with rather thin walls, approximately (600) (62)(2.25m) = 250,000 pounds of water are trapped in it and have to move with it. Also, the apparent additional mass of the array in water is about 250,000 pounds. Hence, the total dynamic mass of the array, M,,» is about 600,000 pounds. The value of A is equal to (600) (5) = 3,000 ft2 . The value of 1.2 for a has been obtained from the literature. Therefore, we have: p SL Wale 7s u = M mange e 0.353 a a 4apA|U | B = 3M = 0.16 |U | a pe SS Say Cc je oe ft oe PS oon fue max L o ma max Now, if we take | Us = 1 ft and w = 1.26 rad/sec (which corresponds to a period of 5 sec), then 8 = 0.16 and w' = 1.85. Then, from Figure 3, [pz | = 1.9 and therefore, > | = 1,900 psi. Also, for the same w, max max and|U_| = 10ft, |= _| = 20,000 psi. Similarly, for |U_| = 19 ft and O max fo) w = 0.63 (which corresponds to a period of 10 sec), eee = 23, 700 psi. The static stress distribution is linear, and for this cable and array without any buoyancy, the maximum static stress (at the top of the cable) is 103,000 psi while its minimum value (at the bottom of the cable) is 33, 300 psi. Stresses due to currents of the type considered in Section III are much smaller than these static stresses. Even without determining the locations of the above dynamic stresses, it is obvious that this cable will be able to withstand the maximum total stress with a factor of safety of about 2. If a larger factor of safety is desired, the metallic cross-sectional area of the cable can be in- creased and/or buoyancy in the cable and array can be utilized in order to de- crease the large static stresses. However, if the static stresses are de- creased below the dynamic stresses, the cable will buckle (see Section VI). 20 Arthur D Little, Inc. S-7001-0307 ‘ y i if Aa tei ay ee i i, ytay Ove) ihm Ean od +f 7 i p ‘ ma ¥ M /) +i oiT Oh et ey A Hie! ‘ a - im rie ay ey Is i | Fy) bag ead ae eee wa Magee ae a , iL M : re ee ey | 4 ¢ F hy “i 4 es 7 ie | ie - w Tite Link h a 1 eon ] bs i pe Pe yo wigs WAST | 73! : : vintr vee en 7 ‘ TA A I oO oad ‘ea OAL, odetht id Pe a ; “= ta » M hi: ays Par ay , ' j aa) ¢ Oo 5 h 1 re rn f Pi a | Parddi i habe ; ries ee Pee Wit tes Be tis 1) a AOC Td AAA tot * i iit ran ‘138 i ati iti Prey ie Pa wet aed be warts { et at a mi ees) Ro ea gat OD VE Uae La i ae ie ee 1h WEB Re MOP WAME Wot YC cat On the other hand, as Figure 3 shows, certain resonant frequencies can induce very large dynamic stresses. The frequencies of these resonances are close to w’ = 7, and their peaks are very high for small values of u. For the above cable and U, = 1 ft, we have (at resonance) w’ = 3.2 and a | = 90. Therefore, > = 90,000 psi and w = 2.2 rad/sec, corre- max max sponding to a period of about three seconds. If we take | Us = 3 ft, then [Bell = 195,000 psi at w = 2.2 rad/sec. Clearly then, resonant frequencies with rather small amplitudes can break the present cable easily. Depending on the state of the sea, the above resonant frequency with the corresponding am- plitudes can be found in a realistic sea. However, the input amplitude to the cable at this frequency can be much less than the amplitude of the sea, depend- ing on the stability of the vessel (or lowering platform) and the method used to attach the cable to the vessel. These aspects will be treated in the following Section, but the following two general conclusions must be emphasized here. : : ee mC Shoe First, since the resonant frequency is given by w = Te it is de- sirable to have as high values of c as possible so that this frequency can fall in the region of the input spectrum which is characterized by small amplitudes. Second, it is highly desirable to have a large value of u. This means that the dynamic mass of the array must be small. The present cylindrical array is poorly designed with regard to this aspect, because of the large quan- tities of displaced and entrained water. A linear array made of aluminum trusses can serve the same pur- pose as the 50-ton cylindrical array. The aluminum array does not entrain any water, it displaces a very small amount of water, and it weighs only about 15 tons. It appears that a reasonable value for the dynamic mass of this array in water is about 20 tons. (Compare this with the 300-ton dynamic mass of the 50-ton cylindrical array.) Let us consider with this array a 6 x 37 Special Flexible Hoisting wire rope (USS) with a diameter of 1-5/8 inches, which has the following characteristics: E = 11x 10° psi S = 1.1 in2 c = 10,000 ft/sec w = 4.1 lb/ft Notice that, for L = 20,000 ft, u is now equal to 2. 21 Arthur D.Uittle Inc. S-7001-0307 ths yi f , mm ra gd pri Oy hy. . sak amino vit crits Gh pat til vu % fiisn 7 ane a eT oh nr, porate ry er oo Na 2 | ‘ tis 4 AN ed ‘a V4 eee i hi iF ) fan cl 4 i psn uh »! ow The only unknown factor in the system is the drag coefficient 8 for the array. For the present array, we cannot obtain a value of 8 from the literature. For the cylindrical array, however, we found that B = 0. 16|U,| j Since the present array has the same dimensions as the cylindrical array, values of 8 a few times smaller than that for the cylindrical array seem to be reasonable. If we plot the value of | U,| versus the value of w which will give a specified maximum dynamic stress for various values of L, we will have curves like those shown in the following diagram. The envelope (dashed line) drawn through lUol S L, = 20,000' ~ the first minimum of each of these curves has the following significance. In- put characteristics | Wall and w corresponding to a point which lies on or below this envelope will never produce a maximum dynamic stress greater than the specified one for any value of L up to 20,000 feet. Then, if the specified maximum dynamic stress is the design stress for dynamic loads of a particular cable, this envelope is the boundary of safe (points lying below it) and unsafe (points lying above it) input characteristics. 730) Arthur D.Aittle, Ine. th ine j sivig ive ahr, im W ‘wa bay: a nye b bd gt EW at te cathand SUEY ee, has spine pis sae oe fins meta ‘at? eels ue? Left wo, nt 7 c ‘ OM. oti art ij ; iol aah hee lo Gh) Ta ie i. Bova lal te Hy", wise Pn th a ‘ j att Nhe iad pais a yas ae ihe, | AP t Ve le a bie iit, be Gert) rulbwty ir Hs ety ight) Wy patel, ae TW givenE High 8, edits cutie dab 8 pil Oe) 2 se aa Al RAY tere a a Toit Ate Day Ee eet ty ae 1 SD od eee eas, 1 ing eel qrceral tri ti) shih 1 Pert BE a) vedio, (4 : at * eT 19 13% ty) ee lh ote ita gee a 7 a Such curves are plotted in Figure 4 (dashed line) for the aluminum array and above specified Special Flexible Hoisting wire rope. The maximum tolerable dynamic stress is 32,000 psi, about one-sixth of the ultimate strength of this cable. The values of 8 assumed are stated on the curves. Notice that more friction moves the curve to the right. Assuming that 8 =0.05 |U,] 5 we see that at w = 2 rad/sec the maximum dynamic stress in the cable will not exceed 32,000 psi as long as | UL is equal to or less than 1.7 feet and so on for other frequencies. If B = 0.1 | U, , then for the same frequency the maximum allowed | U,! is 3 feet. 23 Arthur D.Little, Inc. S-7001-0307 aes ee TR, if wv : ery aaah tpl ’ . nuance bat’ oo snr? yatiRGlt mua gone MOUs Het WY pat eae ony Derhkerany: " ‘in LAG © # aii gtr AIH OEE spied aes wht Al slewrse vata ine ovdsrotbagn aidan ik |e Bike et A atl) wank a6 ct dandnit ot b ny wa geek aaa’ asl quinborpitey ratendie at dh pel oljats i \ ee eh a hewathl: — iy a “a - - , i : \ \ \ IUol FOR 3=0.1 Up! 1Uol FOR @= 0.05 is \ HEAVE ee RESPONSE AMPLITUDE OPERATOR FOR ROLL(10 ‘rad /ft) AND FOR HEAVE MAXIMUM SAFE 75 INPUT AMPLITUDE [Uo| (ft) 0.50 5.0 ROLL (CUSSI1) (0) () 05 1.0 1.5 2.0 25 3.0 FREQUENCY (rad/sec) 20 10 6 4 3.5 2.5 PERIOD (sec) 1500 500 200 100 60 40 30 WAVELENGTH (ft) FIGURE 4 RESPONSE AMPLITUDE OPERATORS FOR CUSS I AND MAXIMUM SAFE INPUT AMPLITUDE TO THE CABLE AS FUNCTIONS OF FREQUENCY. (CABLE IS A SPECIAL FLEXIBLE HOISTING WIRE ROPE OF USS.) 24 Arthur D.ULittle Inc. wid ‘ paw ADM da Ty he | 4 Lae \ ee) "acre i iA J Hd '} 5 yu ry 4 OT, , hia A ; aa iv : q Me ; ¥ ING san MRT Mee 210AO SY OF. COUT TIGR a “sayy Um Se ie LAO eee BR) ORR . bik. ly 1) RAE Se my i V. THE DYNAMICS OF THE VESSEL AND SPECTRAL CHARACTERISTICS OF A REALISTIC SEA It has been already pointed out that, in calculating the dynamic stres- ses induced in the cable by a disturbed sea, the dynamics of the vessel and the way in which the cable is attached to it must be considered simultaneously with the dynamics of the cable and array. Besides the elastic force exerted by the cable, other forces acting on the vessel are inertia, dynamic buoyancy due to the waves of the sea, viscous damping, and damping due to the "radiation" of surface waves by the vessel. Then, in an analysis of the entire system, simi- lar to the analysis carried out in Section IV for the system without the vessel, at least three additional parameters must be introduced. This will be a very complicated problem. We will assume that the dynamic loading of the vessel by the cable and array is negligible and, therefore, the motions of the vessel during the operation will not be influenced by the cable and array. Because of the relative dynamic masses of the vessel and the cable and array, this condition is well satisfied in practice, except possibly when the array is lowered through a long boom from the side of a vessel which is rather unstable in rolling. In Section IV, we have shown that near resonant frequencies, which can be as small as 1.5 rad/sec, the amplitude of the input to the cable should be about 2 feet or less. This definitely shows that the vessel must be very stable at such frequencies, and it limits the variety of ways in which the cable can be attached to the vessel. Let us first examine the spectral characteristics of a realistic sea where such an operation will be carried out. A realistic sea has a continuous probabilistic distribution of all possible amplitudes and frequencies. Depend- ing on the weather conditions (mostly the speed of the wind) and the degree of development of the sea (duration of the wind and extent of the fetch), amplitudes above a certain value and frequencies outside a certain range are not very prob- able, while amplitudes and frequencies within a certain range are most probable. It appears that separate probability data for amplitude or frequency exist in available literature, but data for the joint probability for a specified amplitude and frequency do not exist; i.e., we do not really know what is the probability for the occurrence of a certain amplitude with a certain frequency in a given sea state. Tables I and II show some important characteristics of a fully arisen sea (infinite fetch and wind duration) at various wind speeds. These data have been obtained from "Observing and Forecasting Ocean Waves," H.O. Pub. No. 603, U.S. Navy Hydrographic Office. The significant range of frequencies is defined as the range outside which it is highly improbable to find a frequency at the indi- cated wind speed. 29 Arthur D Little Ince. S-7001-0307 Ff 7 shes ac ak iooaion th $e ip sublha iy sah rahe fry wii oo vue pasty cline way wid Toeuenieny aft pace ott Sita Shag ihiynnt et biviteien 24 Re serene) ovat ty ban, qian mt yl be saalllne (At fou, {lyr nail Howse: ‘aditibgas gin)’ Yuri hay, han watt Hide’ teawel pakd Ye intact ne 1 yal 6 ieilo’s a hevgeyney)! Ai KA hh mh radiw EE cane fyb “Ko, dapsone i bral Bt {for ni otdnin is githaes Bh vigil sey i ey iain wh) Mapa Ed g ssdalw wins Apo semitones? TW ject sein Bd felt ‘irk adel al = Ediipite ehies eet od ujinh oft To petioles ay be Nba Baha Dian ep ms iat : Yeas ad teed fenuey ot jee mwa. Yi tala hele al Me R aper'h Lanta aS iy Ls sae i sy eee sails eld at) ban Lhd ere 1 tii ay iat ; ; San at 0 aieat alt at leper RS i tay it ait i i ee Cee aah ADORE 9 her cha MY has hs oo an ge 19 Sy. Sate haga, , ea pau jy 3) Wee Seb BLL nh sh Hw 3 fe ts art wih EU rites a . iY cla rl tare (oniw aH do ba vegan oar ae Petite), en QTD Mase ae HON iW atl gebintiqne gay sds Do Varies bom haute dlr ko) Hobie) aoe 1% ne “rere eRe 10% a Ouiley pleTied & ete aShtenps? bie OIE Minis ; yekdalderety PADI Alle SUNY, AiATLOS A ale Gervawipngt rie gota atte IW. pie BAe GR ata ictal Ata TT Lone FF 7. HE PAS Ylmetipirat a Igri LOT mts i ebiiqman by io? Moone a OL RRR MS jot FOL bit HAY uy YOM » con ehiieh Se mi vi, treherd at wilt ee livier wood # Uae Moo ab 4 CS Lat, Mes " " CAH . Loi ievig: Moat Yoweipos) ented on ate hamtalgeene Abie Ph TO OCG a rate Viel ofp eset eas: hep peotyann OLN, ‘iL crit SPN Sie i Ropd Grad wie wed? soronge bidiWy aiesbiay' se dejan Wate da MINOR § iia B08 CH til) Ol ai dis O nt ease 2 bats ert epee) ern) (haat ie ; famiiah di waconaiyper'to ages Dip anal ET WOUAEY O Duepeeyeayet Ital i ft Hah Wa) 3m Youensapendt ae bail <0 plclaclennepssil: Clohytitvl 1 Atay pti eRiO mgm wit i ; By ae Se aR ey OR AG “eh hao ae oiswilaet a tt | apialaet ey silo ee dull TABLE I AMPLITUDE CHARACTERISTICS OF FULLY DEVELOPED SEA Average of 30% Average of 10% Most Frequent Average of Highest of Highest Wind Speed Amplitude Amplitude Amplitudes Amplitudes (knots) (ft) (ft) (ft) (ft) 20 2 7438) 4 5 30 5.4 6.8 iit 14 40 ll 14 22 28 TABLE II FREQUENCY CHARACTERISTICS OF FULLY DEVELOPED SEA Most Frequent Average Wind Speed Frequency Frequency Significant Range of Frequencies (knots) (rad/sec) (rad/sec) Lowest (rad/sec) Highest (rad/sec) 20 1.4 el .06 Po al 30 0.9 243 . 38 es 40 0.53 50) 529) oy 26 Arthur D. Little, Inc. S-7001-0307 l ne ' it an . i ne oa id ve i it ;, i | | . if et of | aly | . 7 PAS Win a Buney | are i ‘mors + ieee i i a sy ts ane a x ’ Mig) ia aT iS hi ; Tables I and II show that frequencies as small as 1.5 rad/sec (the smallest resonant frequencies of the cable) are more likely to be found at smaller wind speeds, but the amplitudes are much smaller at smaller wind speeds. Obviously, as the wind speed decreases, the chances for the success of the operation increase. The largest wind speed that is safe will depend on the stability of the vessel and the manner of attaching the cable to it. Two obvious ways of attaching the cable to the vessel are from the center of gravity, and from the side through a boom. In the first, heaving motions of the vessel in beam seas will be critical, while in the second, roll- ing motions in beam seas will be the most critical. In Figure 4, the heave and roll response amplitude operators for Cuss I* in unidirectional sinusoidal (regular) deep ocean beam waves are plotted versus the frequency w. Scales for the period (equal to 2 T/w) and the wavelength (equal to 2ng/w*) are provided. For waves approaching the vessel from other directions, heaving and rolling are less. These curves were ob- tained from "The Motions of a Moored Construction-Type Barge in Irregular Waves and Their Influence on Construction Operation,"’ Contract NBy~32206, U.S. Naval Civil Engineering Laboratory. In the case of attaching the cable to the center of gravity of Cuss I, some meaningful conclusions as to the safe sea state can be drawn from Tables I and Il and Figure 4. Assuming that we can represent an irregular sea by a regular one with amplitude equal to the average amplitude of the irregular sea and frequency equal to the most frequent frequency of the irregular sea, then, when the wind velocity is 20 knots, the input to the cable is w = 1.4 rad/sec and |U,| = (0.42) (2.5) = 1.05 ft. This input can be tolerated by the cable. According to this representation of an irregular sea, a wind speed of 30 or even 40 knots can be tolerated by this cable. Since the input amplitudes that the cable can tolerate are small for high frequencies, perhaps a fairer repre- sentation of an irregular sea by a regular one is by the average of the 10% of highest amplitudes and the most frequent frequency or the highest frequency of *Cuss I, originally a nonpropelled freight barge, is 260 feet long and has a beam of about 50 feet and a draft of about 11 feet at 3,000 tons displacement. It has been converted to a sea drilling vessel, and in March 1961, it was used suc-~ cessfully by Project Mohole to drill in almost 12,000 feet of water at a site 40 miles east of Guadalupe Island, Mexico. Cuwuss I is used for our preliminary computations, because it is the only vessel, of the type that could be used in the present operation, for which we can obtain the amplitude response curves from existing literature. 27 Arthur D.ULittle Inc. S$-7001-03 07 } nae. or oe pi sipah ck Tee ws “sides ‘het sttiboed Prue | i maa coat EL at prnsora attri i ro Sule kasummieierdes aes nee Wien Lams dbisMhanamnic Mahal ee Pui ava a eohuhiajae ith bn cvind une ee er ‘yaa Rae EAM mated a Mpheate (anh ing ky Ate lentes lien ial ha AVE Leh lunges bok zmg: ony nuk WeAbiaR Me leet . Probar aidt Qillska nO LonpH secteN FO; ablVirnd hp, shag alta ath “wl Le ehen ty pean | erst ea ctl ib Hi; 1% Pc Lahiri ii po: Halrgeerey easy vr ual pinks Sonntag herd caey 4 he muah: eth" 9 pie Balas sul searseanen ia inlet sett iodine th ea Sa hbk ; i i | a Lae te NS nes Rhona l diel ieid Mond inl: Tih prey wisn! fy ae nanbiets it's ‘a bins gids mattiule ome “ | Sapte? any ve ty wit teat mi nl ato ahi pk ae oa mer haiti phan yi 2 Pe ee afoursiga, AD 4! LaDy ATRIA eon tis UE aah ~ wie, Baliigiy et) Bi i) NLR eae en OU TOE md, Se 7 oh et aes)) mae ee Sr A RR ee A fags a ieee ene oe das Ay eae faced nde eae OR ah ‘ygtaleahaa Rane an: aca ait yioattiekat ink ainda WARE MT RO Leh AR RE a ‘Se na... au key naeaea Wah bet ah. may 30: ra peice a dios) ean ‘well duel w Vite, welty natok) tlelug: HageY Ye Load acedeligt st bis vp OY fa eee lll) \odiatila bape ha FTN SERRE! 1) tie RR a ne wh ie, YAR, #602 eA AMR 4 to) ps ane , ; Peep pee gry 0 4 in rhe alata’ NSH ait Saal Mauiee 3 a iL He herein A ff 7 1 ie Bacrn © be H a {ptenniely doin py 7 ‘? oye sy Va Nai ey cer Hit aii Kogan Hint flee ee | eee pane seit ait “i md: by hid Hh, Vieinegeterets berg ON) 8 ty fOet Hf idan hes ay le ‘Tene 1H? 9 ae Me isky pains ani by LOS) RRA eh Poel Near Shen HEF at Pe a a pee, Oe tay bein wus Meee Le re ab Thess i “hint fae neetlial ae wi mea eilaeiihicwy Bro at Cha wT wen ite on Haves til! sings ita ie Ade HO if a er) ay fig Pe ey MPa at hh eee iE Ey) Wi, ‘f sagt iesatucd if 7 1 ae igh vou “er pal aid as pian) HAs, wr dasih aie . the significant range of frequencies. Under these conditions, a 20-knot sea is quite safe, while a 30-knot sea is marginally safe. Of course, in the final analysis the reasonableness of our argument depends on the value of the fric- tion coefficient 8 for the array. We only guessed its value here. Assuming that we have not made a gross overestimate of 8, we can conclude that the chances for the success of this operation are quite good for wind velocities up to 25 knots. If the cable is attached to the end of a boom extended from the side of Cuss I, a rather quiet sea is required for a successful operation. Since Cuss I has a beam of about 50 feet, a boom at least 30 feet long is required. Heaving and rolling can occur in phase, and, as seen from Tables I and II and Figure 4, only in a sea excited by a wind velocity of much less than 20 knots are the chances for the success of the operation acceptable. Indeed, it will take a vessel having much greater stability in rolling than Cuss I to carry this operation successfully in a sea excited by a wind speed of 20 knots. 28 Arthur D Hittle, Inc. S-7001-0307 ie: ATT ee ee ip a “Selamat Pas is o Dien BARE Gel NE Thao. cee BN, “da | Nes take T wala Cort mon my eo ynmeely at Zucce amp qaMelinn! . 7 me jooedeaidcepuahahaennetbiriG ped todlaxs es deli peta AAT ose py 1rOKN nicer tic} es Waerea NED OF lose HED ett oily quritox a dite nit ghpr it My nail DE Ne bison ht iy side cdanaei nit aoe y De VI. BUCKLING We assume that the cable buckles as soon as the total stress at some point becomes compressive. Since, without any dynamic input, the cable is under tension due to gravity, not all kinds of inputs will make the cable buckle. In order to find whether a given input will cause buckling, step by step we must trace in the cable the propagation of the input, as well as its interaction with and reflection by the array and vessel. Here we will investigate only one as- pect of this complex problem, namely, buckling within the time T = = after the onset of the input. In the absence of any input, the static elongation of the cable, us: is given by DS E (w-b) + W, - B, | (31) Now the simplest input dynamic displacement, ws , Which satisfies the condi- : : : : ; a A tion that at time t = 0 the vessel is stationary is Ness t®, where a is an acceleration. Therefore, if buckling is to occur within the time T, uo must become greater than Us in time equal to or less than T, i.e.: aS 78 z (w-b) + W, - B, | (32) where g is the acceleration of gravity. Thus, the acceleration a necessary for buckling decreases with increasing L and with decreasing net weight of the array. For L = 20,000 ft, w = 10 lb/ft and W, = 100 tons (and no buoyancy anywhere), a> 3g. For We = 0, a>g regardless of L and w. Therefore, it takes tremendous input accelerations to buckle the cable in this manner. Suppose, though, that the cable has buckled. Then, depending on the degree of buckling, the cable may or may not form kinks. When the vessel now moves upwards and stretches the cable, these kinks will make the failure of the cable easier. In order to determine the degree of buckling, let us assume that the array falls freely in the sea after buckling. 29 Arthur D Little, Inc. S-7001-0307 aa i a 6 x ie Me mist lig vd id j ei ‘chomp rT ivy BA wt * ' vn Mb a ey nn ay is} tei rhe # ey al el Heh waitin ai Mi fe “ arent td ne i ne. heya aye ¥ bs ‘ a4 al] Shima MMB Rstindren The forces acting on the array are inertia, gravity, buoyancy, and friction. The downward displacement of the array, uy, (t), must obey the fol- lowing differential equation: 2 sn) = Li Peal M me = WN, BOW B. Yapa | (33) The second term in the right side of the above equation accounts for the gravity on a volume V of water trapped in the array and moving with it. For the cyl- indrical array discussed thus far, goV will be approximately equal to B, > and the solution of this equation satisfying the conditions that at t = 0, — Go a) § Mis aaraee is: uy = V_ ty, £n cosh 2 (34) Oo t, z where Vi = (2W.,) /(%A)| is the ultimate velocity of the array and ty = 1.41 M, (a oN) . For the 50-ton cylindrical array, ve = 5.3 ft/sec and ty = 1 sec. To determine the degree of buckling, uy and u, 7 Us must be plotted as indicated below. VELOCITY e TIME 30 Arthur D Little Inc. Wallace wingibity ic emains: Poat ys anle cme OR yet hn Th, ou we on The fact that the cable has buckled means that the curve ee must lie above the curve uy for some time. Of course, after some time of downward motion, the vessel will move upwards, as indicated by the dashed line. The difference between Ue ewe and u is a measure of the degree of buckling, and can be used in some way to determine whether the cable will form kinks. At the time corresponding to point A (or at some previous time, if the cable has formed kinks), the cable will be again under tension. The failure of the cable by kinks should, perhaps, be investigated. 31 Arthur D.Little, Inc. S-7001-0307 nd Male Anns APPENDIX A FRICTION ON THE CABLE In order to derive an expression for the friction on the cable by the surrounding water, we will consider the mathematical model of an infinite straight cable surrounded by a viscous fluid of infinite extent. Furthermore, we will assume that the cable moves longitudinally like a rigid body with a spe- cified velocity. This rigid-body approximation is valid as long as the distance along the cable required for an appreciable change in the velocity is much greater than the radius of the cable. This requirement is easily met in the present case. The governing equations of the motion of the fluid reduce to: ) Lie (A-1) where w is the velocity of the fluid along the cable, vy is the kinematic vis- cosity of the fluid, and r is the radial coordinate. If the velocity of the cable is sinusoidal of amplitude Wp, and fre- quency ™, then the solution for the amplitude of the velocity of the fluid W is: Bl (eae) en vey SS. (A-2) 2 (agen 2) O v where a is the radius of the cable and HY) is the Hankel function giving out- going waves. Therefore, the amplitude of the force F exerted by the fluid on the cable per unit length is given by: HY) (a J-i =) ane ee) Reh cna rf F = 27 ap vw, fig tis (1) = (A-3) H (a /-i — ) fo) WV) where p is the density of the fluid. 32 Arthur DHittle Inc. S-7001-0307 Jk ie. aoe sf tS i Sean ‘4 Br Pies oa a OM he pind tO hi Pt nen 2 ny ca — aN a . 1 wip! Ny ; A i a ; - Th i) ae ai — ae le 7 “és : Fi cc tk ee One Prenre i f bil, Dae sedi { anes aus pry va ‘ ; be ; } al $y er yh aaa see tan 7 Ae “sbi x sono a une ih ie nang | ihc | i : ea : m4 m4 i j r f fi : aH f 7 Now, for water v = ‘lesa ee fees. Therefore, for a and w even as small as 1 inch and 0.16 rad/sec, respectively, a | = >>1 and the Hankel functions can be expanded asymptotically. Then Hg (a i?) —————— ar (A-4) q) (a al 42 ) fo) ( —® Therefore, a UN F = -27 ap Juv W, era (A-5) This equation shows that the viscous force is out of phase with the velocity of the cable. Thus, it has been shown that the viscous damping on the cable is pro- portional to the velocity of the cable. The coefficient B rs used in Section IV-A is obviously given by: Hy ie B/N 5 savaa/GM eS — sg (2 o (A-6) c p,Sc P. \ca? Now for a metallic cable p. = 500 lb/ft? and c = 12,000 ft/sec. Thus, with the values of p and v for water and L and a equal to 20,000 feet and 1.125 inches, |8| = 0.013 w' #. For smaller L, 1s! is even smaller, and it is hardly possible to have a of a smaller order of magnitude than 1.125 inches. The inertial term in Equation 7 is of order w'2 , while the friction term is of order w' | 8 | = 0.013 w'3/2, Therefore, for w' = 0.02 the friction term can be neglected as compared to the inertial term (except at very sharp resonances). 33 Arthur D.Hittle, Inc. S-7001-0307 } amy wip i tes hie a vn we cat a} i: is hf i: marty: any er) cot) ‘te esti eluate arte Fy NI aia "5, sage Bray palin nity alae y wht ™ eee Mga a} ie Aree ih iy ee oe i, - ret wed 7 ae waatt rol te i. WON) PAL Ves cout GON 04 . ei sidion nit eat ‘ is pe JE i he 1 M “ny ne r Hie ’ i aha ‘in ay ae fiat 2 pet mv iT bitin W in ee x wend Be a i } y may ; ‘o if ay fn ine gn i mins Gap RS Se! PRE Ong >) Ab) Sas earn Sega pe) ‘eatin wk eR ae : | a yc i | en PED Ged fiat pl eats H,4 aN: a me Wich noi hn en ay N sat Lasser anne ine ui i APPENDIX B THE DRAG ON THE ARRAY The problem formulated in Section IV is linear except for the term of the drag on the array. This term makes the entire problem nonlinear, and enormously more difficult than a linear problem. It is, therefore, expedient to linearize this term. For sinusoidal inputs, one can take the drag on the array, D,, as being proportional to the velocity of the array (not the square of the velocity) and then determine the constant of proportionality experimentally. For veloci- ties and arrays of the type involved in this problem, this constant of proportion- ality will be found to be a function not only of the frequency but also of the am- plitude of the velocity of the array. However, such an experiment is not prac- tical in the present case; no effective scaling, according to the principle of similarity, of the variables is possible, because of the enormous dimensions of the arrays and the appreciable velocities involved. We therefore propose the following analytical linearization. The drag D, on the array is given by (see Equation 8): ou. ot ou ID, 3 2G fo) AN vo (B-1) For sinusoidal inputs, it is expected that the motion of the array will be periodic. If the | du/dt| is replaced by a constant, then D, and the entire problem become linear, and therefore the displacement of the array will be sinusoidal of ampli- tude U, . We Set the constant which replaced |au/dt| equal to Ww, . This selection results in the same amount of dissipation of energy by the array when u is taken as sinusoidal in both factors in Equation B-1 and when only the sec- ond factor is taken as sinusoidal. Thus, with U, a real positive number, the amplitude of the drag D, will be given by: ee Be 2 2 x D, = an 0PM AU, (B-2) 34 Arthur D Little Inc. S-7001-0307 it: ae “ih ms 6 seme whet, © AY ON Ht ko wAhupe nity ron). Ye reer hey | Sy che (Mi lisinatin /oooeye (Cielo ar ogu‘tr Ye tea bainiD eee Pinegentg, ty is OS tay cnet: mikey: vat howl toa a RW OST Ae on tne dtind Waronjwrd wily to vido! Sa wel 2% Fe bad re oe PGE mi fnaitte: le Teva wor "gerne owt io Yitognae adi to ebintieg aieintig wt} os snibregoe Tearane atone oe Arnis eek oct _ Inoia, %e eicuob ncmnyvitt auan pit silt XO SKU se vidtt ee | ort herpes | bidval: aslo atte site ea my tl 9 stat eh. otf | ey arya ee Pa ea ae sabbath at aaa ete or b y ie fu pavinys' th ok), ei d nari at yanks ei neil a we ant - eo Aye ee oa . 4 a ‘ A (Le 6 : dilate ih Th gar TH er 10 DANG wii baat Labed reser ea #! ‘yiuean! ? abs aH), aha i wand erate Y ONis was its WA aS pods Wee iw! wa towiamlgrs a by 459) 5 4 ont, ft " ; : hinecti iG labtoasnid ey? 4 Lay ven i aay 3G tm ay ea hyithy ai wit sabe tina: aN (; 2 aa" he ae AE Miah us a6 senual [3 Wioketw itp iamntih ity io BY! yu uy cba! ' 5 | a beni ¥, al. ie ere wien 60) WP a sire bd nol seate athe u inna et ‘ally wi els ag le 2a i or iedld mnt} 4 LG FEN ai a TEAITAEYT mH ae Oo? we eed, ny Fisie ub Les ie Pent ait: AS Reakiritr ewe Lear pow ib au tT” taht mths Aga LD cae ob eta tio” A ik De s n Fits at on fae ny ed ebiiieia y : (2°) x a eed iy po ay a ‘a my mH ae Pas To estimate the error involved in the above approximation, we consider the array alone and a sinusoidal force acting on it (in the actual problem this force is the force exerted by the cable). If we denote the velocity of the array by v, then the dynamics of the array will be governed by the following differential equation: dv Tage coe plviv (B-3) where F and p areconstants. Then, according to the above approximation: v= Vcos (wt -¢) (B-4) where: 4 2 2 BNE 2 3m Ww 16pF WS: Wl -14+]1+ B-5 I aa ees) and @ can also be computed. (In obtaining the above result, \y| in Equation 8 ‘ : : : : d B-3 was replaced by ae V.) This solution can be considered as a first iteration toward the exact solution. Now we can compute a correction v’ through a sec- ond iteration, by solving the equation: a+ BY S jp) |) Vv (B-6) where v and V areas given by Equations B-4 and B-5. By expanding the term in the right hand side of Equation B-6 into a Fourier series, we find that v' is given by a Fourier series, the frequencies of the components being 3w, Sw, 7W,...(i.e., v' has no fundamental component!). This result is due to the above choice of the constant replacing |v | in the first iteration. The am- plitude B. of the leading component of v' of frequency 3w is given by: w? \2 B, = 0.2V(1+12.5 — 5) (B-7) > WV. This shows that the correction v' is not more than 20% of v, and, therefore, the error involved in the above linearization of the drag is at most of the same order. 35 Arthur D Little Ine. S-7001-0307 US, i ‘ rt ani ai = leks: rr vib ition. reat Gil fh rinse BLY an ee on :) 7 * hierar deal ah reds tasrdes. rom iLL oat i io. eit ae ie ve A) , : 4 4 y BS) a, eee e ay wil a i ELD We AMAIOD RAW BW Re sso as dite i) vi ony ‘Gehan aut Aula we \ lb pe wal pany uo) ‘(ie Hh ; y (Te! ne rhe holga) |, sues : na ; ies, ay : tog es fares’. Rd Bite Ak Ree Wal que ee aM, ana 8 Hes | ra yi : air yas 3 a ee Ke: re aes a tae ye TUT £' AA a ST a a “Doth wl ow Me hrew sok gob OFM a ribeye 40, tate asta, Maghy ait, i | aerROny oi i : 7 a ( Qn ot Mrriarynndky Sale LS nee Loe Nap ie sath Byres eo: eal teYPE:: ‘ wi erent: ui. * f wf! wee, al Pim ee Cisse Leal reece tartat a) BLT) 7 Rr) Ce oe , 2.5m Al He RAGA Waa Taiaays ‘ite PN bale witty ui eeaelat a ae th he, fi Fish | er eat ? . ~ i 7 U ; a ane: nity hg NY iy 7 f VO eel WE oe tines HpwE TO) ye DSHS ORD es, i) N - j ' ie H) Ei | py Neth lip oft ‘ i : i ‘ i , i r 7 ; tt i : } | ry nh | he hie! | peo a er SUS pale, OP yar al Gea WER aul apt sana a Py) TE BI MALT, PE? Reig eae ee wie ed ea wmed), Mads: EY! fl: Deoviewns Bhai i rt bay Nar i f Fi { Bhi woe i i De hale / f 7 al ‘ OP ke 1) ne Peal eee 1) LIST OF SYMBOLS Radius of cylindrical cable, except in Section VI where it stands for a constant acceleration. Area of the array projected in the direction of motion. Buoyancy force on the cable per unit length. Buoyancy force on the array. Velocity of sound in cable. Horizontal offset of the array from the target due to a horizontal cur- rent. Hydrodynamic drag on the array. Young's modulus of elasticity. In Section III total horizontal force of the current on the cable. Acceleration of gravity. Hankel functions of orders 0 and 1, respectively, giving outgoing waves. Length of the cable. Dynamic mass of the array. (See page 12.) Radial coordinate measured from the axis of the cable. Metallic cross-sectional area of the cable. Time coordinate. Normalized time coordinate (=ct/L). Characteristic time of the array in free fall in water (see Equation 34). Time (=L/c). Arthur D.Little, Inc. S-7001-03 07 he a a 3 si ae tb 8 ana inate wi 1 wes ‘ht ae Mare, et Wy ere blomste a svayid bisa wt at 3 " is ‘oles ‘ay to moi lhe weary Siva sei = iia hl tails a ‘oa hoster Dynamic displacement of an element of the cable due to dynamic (time varying) inputs. Dynamic displacement of the top of the cable. Dynamic displacement of the array. Static elongation of the cable due to its own weight and the weight of the array. Complex amplitude of u for sinusoidal variation with time. Normalized U (= U/| U4 Ne Value of U at the top of the cable. Magnitude of Ue . Value of U at the array. Value of U' at the array. Velocity of the array. Correction term for v. Terminal velocity of the array in free fall in water. Weight of the cable per unit length, except in Appendix A where it stands for the instantaneous velocity of the fluid along the cable. Complex amplitude of the velocity w for sinusoidal variation with time. Value of W at the cable. Weight of the array. Space coordinate measured along the axis of the cable. Normalized x (=x/L). Width of current. 37 Arthur D Little, Ine. $-7001-0307 ssn eit ics “< 6 Yo ebuihtame af, wet “ ak i as oft 0 wee ae a u oa nie i ey ane fio he aides oy hy sth a quits ht wey es aivalay Daenanne tit ‘oa tot ne “tie wali oFfabion 8 48 Ww eee ce py ais i ites at "y ies it tl arenes “ua me iy Normalized space coordinate (= 1 - x’). Value (s) of y' at which the amplitude of the dynamic stress in the cable for sinusoidal variation with time becomes maximum. Drag coefficient. Dimensionless drag coefficient for the array, equal to the ratio of the drag to the inertial force. Dimensionless friction coefficient for the cable. Parameter equal to the ratio of the total mass of the cable to the dynamic mass of the array. Kinematic viscosity. Density of water. Density of cable. Tensile stress in cable due to static loading. Complex amplitude of the dynamic stress in the cable for sinusoidal variation with time. Normalized = (=LE/E| U4!) Value of &' at the top of the cable. Value of &' at the array. Maximum value of =". w' Angle (= arctan ane, j Angle (see Equation 26). Angular frequency (radians/sec). Normalized w (=wL/c). 38 Arthur D.Little, Inc. S-7001-0307 | ts at inh Ms, fem ha sire w white xt ue he veo wu tout : 1 a A sits usa yuu oil Btetis iw tig ali Ant ee nal efiGnit' Tao iet on nia ai ef DAA sti yh mae 7 itt have Palace” ) r ee PL ne Oe ere / i'¢ 3 © bende: OY iglesia FO (anil ee My, nailed (ARES OR OTS Tate 4 oy gaa deiicninnty! Ce RNY: hii Ox Fake! Ua Jao? || yom at | eaavenubev % sinogeree 4 we Lear CO he 7 ae (reget nintnaid M : \ i Report No. 1011260 1021260 1031260 1041260 1051260 1061260 1071260 1080361 1090561 1100561 1110561 1120561 1130961 1141061 1150162 1160262 1170262 PROJECT TRIDENT TECHNICAL REPORTS COLOSSUS I, December, 1960 (C) THEORETICAL INVESTIGATION OF CROSS-FIX PROBLEMS AND CORRELATION EFFECTS, December, 1960 (C) THE SUBMARINE AS A SURVEILLANCE PLATFORM, December, 1960 (5) Title Classified, December, 1960 (S) AIRBORNE JEZEBEL, December, 1960 (S) SURFACE -SHIP SONARS IN OCEAN-AREA SURVEILLANCE, December, 1960 (S) Title Classified, December, 1960 (S) LOW-POWER ENERGY SOURCES, March, 1961 (C) SOLUS, May, 1961 (C) NONACOUSTIC METHODS FOR SUBMARINE DETECTION, May, 1961 (S) ARTEMIS, May, 1961 (S) COLOSSUS II, May, 1961 (S) RELIABILITY OF UNATTENDED ELECTRONICS EQUIPMENT, September, 1961 (U) NUTMEG, October, 1961 (S) DEEP JULIE, January, 1962 (S) METHODS FOR ANALYZING THE PERFORMANCE OF DIS- TRIBUTED FIELDS OF DETECTORS, February, 1962 (C) MAGNETIC ANOMALY DETECTORS IN FIXED SHALLOW WATER BARRIERS, February, 1962 (S) 39 Arthur D.HLittle Inc. S-7001-0307 fey ind ‘rodte000) gta, ie sca re Maver nic are eat ai ia ve! A nt _ “eats von Seer | ce pee eeoyads, iBBORUOF Hea AMOR a i. mi ie sas Vall edly, iy Lag i aaa } i> ‘ener ade HOA B10 Herat ari ase Laas © ee a7 (| ee ® sae oe ae | 8) a eat ENO A | ow weirs F etn terrain) WO Prhiorat lane. | ih oa nn. a MOO i Me a 7) Coot iin, ein one rN Ai ae 56 she dies How ST pcnlaress Avis: i ore a a, 4 a a a ears ORT AN TOE He g | ie, Masai Abe cada str zmcrrcirs yu aanegega OAADA: i | i — ) | me bi) neo ; felt ‘Sara ui am te Ry KOH) et i At} ne Pa oe iy Wer hi Mal ul ae Ji Pe en ae Report No. 1180262 1190462 1200562 1210562 1220562 1230662 1240762 1250862 1260862 1270862 1281262 1291262 1300363 1310363 1320363 ELECTRICAL CONDUCTIVITY, COMPRESSIBILITY, AND VISCOSITY OF AQUEOUS ELECTROLYTIC SOLUTIONS, February, 1962 (U) A FEASIBILITY STUDY OF THE PASSIVE DETECTION OF QUIET SUBMARINES, April, 1962 (S) Title Classified, May, 1962 (S) RADIATED NOISE CHARACTERISTICS OF DIESEL-ELEC- TRIC SUBMARINES, May, 1962 (S) DIRECTIVE RECEIVING ARRAYS, May, 1962 (C) ANALYTICAL BACKGROUNDS OF COMPUTATIONAL METHODS FOR UNDERWATER SOUND PROPAGATION, June, 1962 (U) MARINE CORROSION AND FOULING, July, 1962 (U) SURVEY ON AMBIENT SEA NOISE, August, 1962 (C) DEEP SUBMERSIBLE WORK VEHICLES, August, 1962 (C) THE EFFECT OF PRESSURE ON THE ELECTRICAL CON- DUCTIVITY OF SEA WATER, August, 1962 (U) ENGINEERING PROPERTIES OF MARINE SEDIMENTS, December, 1962 (U) AN INTRODUCTION TO MODULATION, CODING, INFORMATION THEORY, AND DETECTION, December, 1962 (U) SPHERICAL DIRECTIVE ARRAYS: A PRELIMINARY STUDY, March, 1963 (U) ESTIMATES OF SUBMARINE TARGET STRENGTH, March, 1963 (C) PHYSICAL CHEMISTRY IN THE OCEAN DEPTHS: THE EFFECT OF PRESSURE ON IONIC TRANSPORT PROCESSES AND EQUI- LIBRIA, March, 1963 (U) 40 Arthur D.Little, Inc. S~7001-0307 0 : Ua ‘i + “th i ale i Auaby ne a) pee i a me > ma Sa nan if speci a wet ni iter 0 KEIO ye Aa ashi 0) SBR ell, MOLT HOA QM ane TT 9 ny hs ite : i i) ma ei Tu ie Ki xt ih, wlatbonld sa vt ie 1 _ i Al : ai Wi tel ' yi oH 1 ; ; ecbun’y.’ ; wend a sere ah Report No. 1330563 SUBMARINE CABLES AND REPEATERS, May, 1963 (C) 1340663 APPLICATION OF ADAPTIVE SAMPLING STRATEGIES TO THE PLANNING OF SURVEYS, June, 1963 (U) 1350663 CURRENT OPTICAL DATA PROCESSING TECHNIQUES - ASW SYSTEMS, June, 1963 (S) 1360863 ACOUSTIC SCATTERING IN THE OCEAN, August, 1963 (U) 4] Arthur D.Litele, Inc. S-7001-0307 A ii : a : Ly u — DISTRIBUTION LIST No. of Copies 10 Bureau of Ships, Code 370 Department of the Navy Washington 25, D. C. 10 Chief of Naval Operations (OP-71) Department of the Navy Washington 25, D.C. 4 Commanding Officer and Director U.S. Navy Underwater Sound Laboratory Fort Trumbull, New London, Conn. ATTN: Mr. H. E. Nash 2 Office of Naval Research, Code 400X Department of the Navy Washington 25, D. C. 2 Director U.S. Naval Research Laboratory Washington 25, D. C. ATTN: Dr. H. Saxton 2 Chief, Bureau of Naval Weapons Department of the Navy Washington 25, D. C. ATTN: Mr. I. H. Gatzke 2 Commanding Officer U.S. Naval Air Development Center Johnsville, Pa. ATTN: Mr. R. I. Mason 2 Commanding Officer and Director U.S. Navy Electronics Laboratory San Diego 52, California ATIN: Dr. D. A. Wilson 42 Arthur D.Little, Inc. S-7001-03 07 r ; ips a ee yo ay ; : ui is ; : " ie ae een Fk ener a" a Fn ee aD Ba dia 0 O° pagbanaer baw snotiity wothidann in) ea ae (tite, 4 baviends grungy inh ee fl ee ae eet iahetd peters & Sul rear 4), thoetactore le — pale Mk wi rm bedahhid i Me peKoon otis’ ai iedash fait ha cates ml | vent wile 2G, va minrenien..,. 4 et ROnaNtnA, aes aul cob ol Wagaya davai. 8. he SR Repay Lata a i ti ee aw cava itd ue mutt’ Po auth VR mete Nay ema hayeny), 4 a 7 nl ey reveal bia a gy A hat ae win . | ‘ns i oo) aesiiacenanes me tae vif Wy Hts 4 TA Levi ey 7 | ae wae ame Y asd eee wet. oe ry ae | J Th: ‘ sodrEt() tenes “yey ithed ju ie cade ; i Wicoet nial corre Ave a a ALCO WD Ae we, he tae = wiellW 6 tig wel 4a ry A. No. of Copies 2 Commander, Antisubmarine Warfare Force U.S. Pacific Fleet Navy No. 128, c/o FPO San Francisco, California Marine Physical Laboratory of the Scripps Institution of Oceanography San Diego 52, California Technical Library (Code P80962) U.S. Naval Ordnance Test Station Pasadena Annex 3202 East Foothill Blvd. Pasadena 8, California Officer in Charge U.S. Naval Oceanographic Unit Building #1, Treasure Island San Francisco 30, California Commander . Oceanographic System Atlantic Norfolk 11, Virginia Commander, Antisubmarine Warfare Force U.S. Atlantic Fleet Norfolk 11, Virginia Commander Officer U.S. Navy Underwater Sound Laboratory Research Detachment Navy No. 138, cf FPO New York, N. Y. ATTN: Dr. W. A. VonWinkle Hudson Laboratories P. O. Box 239 Dobbs Ferry, N.Y. ATTN: Dr. R. A. Frosch 43 Arthur D Little, Inc. S-7001-03 07 ote 1 i) a onaraenix sabi teri ipa’ ! sie sy cra be twiadao' ut ; io baat f oT wrrnk ie, 3 Luvietd Po 6 ee a Tang Beit hy yabaant WORD HR Aven yaalt COKE, th alii Ls 2 aiehagi en eye ol ig tee 2 apet aly Larch yoann La an 28 “pana wapanins YF 8 yarhthitt anethbaly OP ans! via iL pal " eae atria’ wit uct hres ¥e shi io NaPoO sf aga Pl} Sletseh ene? arate? | ot gett miawh is Teh ane tay Bt Alsip dd ae renee) teohay ade rte | writeable) ( Hi. (MORAY weRbtcg ad | TAU ASLO RE wy CULT ve wil Kot mal I Oaerrae! ut cruel Reh ial Be 1 Nt NY, : “ty q 7 i - i F] sy Pravin nt a No. of Copies 1 President Naval War College Newport, Rhode Island 1 Commander Destroyer Development Group Two U.S. Atlantic Fleet Newport, Rhode Island 1 Director of Research Institute of Naval Studies 185 Alewife Brook Parkway Cambridge 38, Mass. 44 Arthur D.Little Inc. 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