DTNSRDC-81/039 ION OF MOTIONS OF SWATH SHIPS IN FOLLOWING SEAS 2/29 (2-80) 30/44) DON Seep Ciel ; OS7 Nov 14% DAVID W. TAYLOR NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER Bethesda, Maryland 20084 PREDICTION OF MOTIONS OF SWATH SHIPS IN FOLLOWING SEAS by Young S. Hong APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED SHIP PERFORMANCE DEPARTMENT RESEARCH AND DEVELOPMENT REPORT November 1981 DTNSRDC-81/039 MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS DTNSRDC COMMANDER 00 TECHNICAL DI RECTOR: OFFICER-IN-CHARGE ANNAPOLIS OFFICER-IN-CHARGE CARDEROCK SYSTEMS DEVELOPMENT DEPARTMENT AVIATION AND SURFACE EFFECTS SHIP PERFORMANCE DEPARTMENT DEPARTMENT 15 COMPUTATION, MATHEMATICS AND LOGISTICS DE UMENT STRUCTURES DEPARTMENT Pal?) PROPULSION AND AUXILIARY SYSTEMS DEPARTMENT NN 0 SHIP ACOUSTICS DEPARTMENT I CENTRAL INSTRUMENTATION DEPARTMENT SHIP MATERIALS D ENGINEERING DEPARTMENT ii 8 0303 OTN NDW-DTNSRDC 3960/43 (Rev. 2- GPO 866 993 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO.| 3. RECIPIENT'S CATALOG NUMBER DTNSRDC-81/039 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED Final PREDICTION OF MOTIONS OF SWATH SHIPS IN FOLLOWING SEAS 6. PERFORMING ORG. REPORT NUMBER » AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s) Young S. Hong 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS Program Element 61153N Task Area SR 0230101 Work Unit 1572-031 2 3 [9. PERFORMING ORGANIZATION NAME AND ADDRESS David W. Taylor Naval Ship Research and Development Center Bethesda, Maryland 20084 11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE November 1981 13. NUMBER OF PAGES j 87 SECURITY CLASS. (of this report) Naval Sea Systems Command Washington, D.C. 20362 - MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. UNCLASSIFIED DECL ASSIFICATION/ DOWNGRADING SCHEDULE 1Sa. . DISTRIBUTION STATEMENT (of this Report) APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED - DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES - KEY WORDS (Continue on reverse side if necessary and identify by block number) Small-Waterplane-Area, Twin Hull (SWATH) Ship Heave and Pitch Motions in Following Seas Unified Slender-Body Theory . ABSTRACT (Continue on reverse side if necessary and identify by block number) To predict motions of SWATH ships in following seas, especially when the encounter frequency is small, unified slender-body theory has been applied. The longitudinal interaction term is computed and added to the results of the strip theory as a correction term. The added mass co- efficients are computed to be much larger than those from strip theory, (Continued on reverse side) DD , 5 on", 1473 EvITION OF 1 Nov 65 1s OBSOLETE piss S/N 0102-LF-014-6601 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) (Block 20 continued) while damping coefficients are slightly less than two-dimensional results. There is improvement for heave motion for a limited range of frequencies, but the pitch motion is generally poor. To improve these results further, the two-dimensional potential should be solved by the multipole expansion method rather than by the Frank close-fit method. UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) TABLE OF CONTENTS UALS OM WICEWIRIAS 5 65.5 0 00,6 6 616 6 66.6 5 0/610. LIST OF TABLES INOWANIEILON go 0 66 16 0 1G 2 8 6 08 0. 6) Gro OO 0 0 ABSTRACT ADMINISTRATIVE INFORMATION INTRODUCTION BOUNDARY-VALUE PROBLEM VELOCITY POTENTIAL OF STEADY FORWARD MOTION . UNSTEADY POTENTIAL DUE TO OSCILLATION . SOLUTION OF THE POTENTIAL FUNCTION WITH THE SLENDER-BODY ASSUMPTION OMiEee We@ollem 5 o 6 56 6 o Inner Problem . MATCHING . INNER SOLUTION . IDIETSRVAG MILO JEXCHMONABIUN 6 o ag 6G) Ge o6o 5 6 oo © 6 JaNCIDINOIDNANVAMALG, IKOIRNGINS $6 6 6 6 6 6 0 6.6 0 6 0 40 6 6 ADDED MASS AND DAMPING EXCITING FORCES LHONOPAIPICOINS Ole WOMLON 6 6 oe 6 o Glo G6 oO d/o 0 wo RELATION OF TWO-DIMENSIONAL SOURCE STRENGTHS BETWEEN THE FRANK CLOSE-FIT METHOD AND MULTIPOLE EXPANSION METHOD DERIVATION OF THE KERNEL FUNCTION CASE 1 - 4(mx)/2 CASE 2 - 4(mx)2/2 CASE, 35 =" cos! 0) < 0 COSHOMS ey aes Sn ce atten cee COSMO Ge rae mene e coves RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS ACKNOWLEDGMENTS iii Page APPENDIX A - DERIVATION OF Cop (y>2) WITH EXPONENTIAL FUNCTION . GU ONG Wolo: Var O86) 0G 86 Xo APPENDIX B - DERIVATION OF Se) FOR SMALL ARGUMENT . . 619 66) 6 6 6 9 APPENDIX C —- FOURIER TRANSFORM OF THE THREE- DIMENSIONAL GREEN FUNCTION ....... APPENDIX D - DERIVATION OF CG, (x,0,0) FOR NUMERICAL LIST GF FIGURES imtes of winter raltaon™ iii) semlien ciire 2 cos 8 < 1 Integral Path when cos 8 <1. Damping Coefficients of SWATH 6A in aie 20) IGnNOES of) O16) G 6)\o O90 Damping Coefficients of SWATH 6C in Ale 20) IGNOES) o 6 0 6 O00 6 6 oo Damping Coefficients of SWATH 6D in at 20 Knots... and Moment of SWATH 6A in Following and Moment of SWATH 6D in Following Motion of SWATH 6A in Following Seas at 20 Knots Motion of SWATH 6C in Following Seas at 20 Knots EVALUATION . REFERENCES .. 1 - Coordinate System Die 3 - Sectional Coordinate System 4 - Integral Path when 4(mK) Sis 6 -— Added Mass and Following Seas 7 - Added Mass and Following Seas 8 - Added Mass and Following Seas 9 - Exciting Force Seas at 20 Knots . 10 - Exciting Force Seas at 20 Knots . 11 - 12 - 13 - Motion of SWATH 6D in Following Seas at 20 Knots iv Page 49 53 55 5y// 75 3 38 39 4l 42 43 44 46 LIST OF TABLES Page 1 - Added Mass and Damping ........ 9 20 2 > Pigalioyealoeul Walinemasiome oe Sys Soong 4 G56 5 56 6 6 6 6 6 6 oe 6 6 5 6 6 35 NOTATION Amplitude of incoming wave Added mass coefficients Damping coefficients Coefficient for longitudinal interaction Hydrostatic coefficients Exponential integral Excitation force and moment Fourier transform of function f Green function Gravitational acceleration Struve function Mass moment of inertia Imaginary unit Bessel function of the first kind Incoming wave number Wave number of frequency encounter Ship length Ship mass Normal vector due to the steady forward potential vi U mn = — g => n(n, ,n,,n3) P q S Constant value Unit normal vector directed into the fluid Pressure Three-dimensional source strength Area of immersed cross section Steady forward velocity vector Velocity vector due to oscillation Volumetric displacement Bessel function of second kind Heading angle of incoming wave with respect to the x-axis (8 = 0 is the following wave and 8 = 180 is the head wave) Euler constant Length of incoming wave Complex amplitude of ship motion Two-dimensional source strength Total velocity potential Potential due to steady forward motion Unsteady potential due to oscillation Two-dimensional potential for heave and pitch due to harmonic motion Two-dimensional potential for heave and pitch due to steady forward motion vii Encounter frequency Frequency of incoming wave Incoming wave potential Diffraction potential Velocity potential due to motion of the ship with unit amplitude in each of six degrees of freedom viii ABSTRACT To predict motions of SWATH ships in following seas, especially when the encounter frequency is small, unified slender-body theory has been applied. The longitudinal interaction term is computed and added to the results of the strip theory as a correction term. The added mass coefficients are computed to be much larger than those from strip theory, while damping coefficients are slightly less than two-dimensional results. There is im- provement for heave motion for a limited range of fre- quencies, but the pitch motion is generally poor. To improve these results further, the two-dimensional potential should be solved by the multipole expansion method rather than by the Frank close-fit method. ADMINISTRATIVE INFORMATION This study was performed under the General Hydromechanics Research Program and was authorized by the Naval Sea Systems Command, Hull Research and Technology Office. Funding was provided under Program Element 61153N, Task Area SR 0230101, and Work Unit 1572-031. INTRODUCTION An analytical method to predict the motions of SWATH ships has been developed by tees The numerical results computed by this method provide good correlation with model test results for moderate speed ranges. The present author has improved the prediction of heave and pitch motions in head seas by adding surge effect to the pitch exciting moment and by correcting the viscous damping WEI 0 However, in following seas correlation between the analytical method based on strip theory and model test results is not satisfactory. In particular, when the SWATH ship moves almost as fast as the wave celerity, the encounter frequency be- comes very small and application of strip theory is not valid. The fundamental assumption of strip theory is that the frequency is far larger than the product of the longitudinal gradient of the body surface and the forward speed. Megaman has recently applied slender-body theory to the problem of ship motions. He has developed the unified slender-—body theory which is valid for all frequencies. *A complete listing of references is given on page 75. 1 In this theory the longitudinal interaction term is computed by matching the inner and outer solutions. This term and the results of the strip theory encompass the solution of the unified slender-body theory. The numerical results for added mass are larger than those from strip theory alone and the damping coefficients are generally smaller than the results from strip theory. The heave and pitch motions generally compare well with experimental results when the encounter frequency is small. For high encounter frequencies, the results become extremely large, especially for pitch motion. The large discrepancies at high frequencies might be explained by the fact that in solving the two-dimensional potential (strip theory), we have applied the Frank close-fit method, while NeRTiane applied the method of multipole expansion. In multi- pole expansion, there is a clear separation of the source strength at the origin from that of the wave-free potential while in the Frank close-fit method there is no separation. These different approaches to the solution of the two-dimension poten- tial function is believed to cause these large discrepancies. BOUNDARY-VALUE PROBLEM We define two coordinate systems: the first, Oo%o% 0% is fixed in space and the second, oxyz, is fixed with respect to the ship which moves with forward speed, U, in the positive Oo%, axis. The oz-axis is directed vertically upward and the ox-axis is positive in the direction of the ship's forward velocity; see Figure 1. The oxy-plane is the plane of the undisturbed free surface. These two co- ordinate systems coincide when the ship is at rest. Figure 1 - Coordinate System Let the surface of the ship be specified mathematically by the equation Y= ehiGcsze t) (1) and let the free surface be given by z= C(x,y,t) (2) Then, the fluid motion can be expressed by the velocity potential $(x,y,z,t) wt -i0) d(x,y,2,t) = $) (x,y,z) + d(x, y ze © (3) where o) = -Ux + $ (X2¥22)> the potential due to steady forward motion, and >» = un- steady potential due to oscillation. Equation (3) must satisfy the following conditions: 1. The Laplace equation in the fluid domain: Da EN Ose ntihOe yi iG (4) 2. The dynamic free-surface condition: On + gf +3 (Ot 400) -+ U for z a Gxenyant) (5) 3. The kinematic free-surface condition: Oe ap Oiled oe O + ce = 0 for z GBGxo Waite) (6) 4, The kinematic body condition: i] oO h(x,z,t) (7) Dat ae Oe aN for y 5. The radiation condition; that is, the energy flux of waves associated with the disturbance of the ship is directed away from the ship at infinity. Equations (5), (6), and (7) are exact boundary conditions. By substitution of Equation (3) into Equations (5) and (6), and by linearization, the free-surface condition is given by 2 w 2iwU U -iwt _ g 2) g We g 2xx a (8) on vores : vee i (oem The body condition can be expressed as follows: (Vo +¥6,40 ) + n=U, +n (9) Got oe) fo) ane oe) e Ss where wu. = (-U,0,0), steady forward velocity vector Uy = velocity vector due to oscillation n = unit normal vector directed into the fluid domain The solution of Equation (4) with two boundary conditions, Equations (8) and (9), is nonlinear between - and 5 The usual practice for solving this problem is to separate Oe and P53 and evaluate these two potential functions independently. By substitution of Equation (3) into Equation (4) and by separation of $, and d, in Equations (8) and (9), we have two sets of partial differential equations. Dee a Vos a One 6) (10) UCT ese eel (11) g °oxx * %oz (Vo +0.) + n= 0 (12) Varese” Onan =O C) Ww 2iwU U id oe Tc %» i g Pox i g ese Tae ee) V U n= 0 1 Equations (10), (11), and (12) comprise the potential function due to the steady forward motion and Equations (13), (14), and (15) comprise the potential functions due to oscillation with steady forward motion. VELOCITY POTENTIAL OF STEADY FORWARD MOTION The solutions of Equations (10), (11), and (12) lead to Michell's integral in the thin-strip wave resistance problem. In the context of slender-body theory, however, this three-dimensional problem becomes a two-dimensional problem which has been discussed in depth by nak, The solution of two-dimensional potential is given by = & , wees > 5 logy 9 log T) Ux (16) Ua where a = a ; the strength of source i 2 ‘ , Ss = > ™rS55 area of immersed cross section =2 2 2 B = Ga)” % (es) i) 2 2 Bes (y-n)” + (247) Here, (y,z) is the point where the potential is solved and (n,Z) is the source point. UNSTEADY POTETIAL DUE TO OSCILLATION The solution of Equations (13), (14), and (15) is a three-dimensional problem. This can be solved by distribution of the three-dimensional source strength over the surface of the ship and by numerical solution of the integral equation. How- ever, this method requires considerable computer time in numerical integration. To avoid such a lengthy process, the slender-body theory has been introduced by Neuman” 7 ; and Newman and Tuck to solve this problem. Because the unsteady motions are assumed small, the potential >» in Equation (3) can be decomposed linearly ji b=", +9,+) , ?; (17) jal where ves = incoming wave potential Po = diffraction potential Y. = velocity potential due to motions of the ship with unit amplitude in J each of six degrees of freedom €, = amplitude of motion in each of six degrees of freedom The diffraction potential, P55 satisfies the following condition 0 ora ee 2) =O form y= hiGcazet) men @lte)) and the velocity potential, MG (j=1,2,...6), satisfies the body boundary conditions Y, = -iwn, +m, (19) jn al J Here, the components of the unit vector are defined as (n} m5 .04) = 4 (20) (ny sng,n¢) = (xn) (21) and me are defined by Ogilvie and meee as (m,,m,,m,) = m= -(@°V) Vo, (22) (m,,mz,me) = -(nrV) (xV9,) (23) where x is a position vector. In the slender-body theory, the length of the ship is assumed far larger than the beam and draft. With this assumption, the components in Equations (21), (22), and (23) reduce to 2») = 24 (xxXn) = (yn,-zn,, XN, xn) (24) ae re) rf) m= —- (a, Oy Tes az), (25) (m, >m,m¢) = CSB OH go m,n, xm,-n.,) (26) By substitution of Equation (17) into Equation (14), the free-surface condition for the potential, Oa Gott cod) is given by BO, s ae °; + 2iwU ain af wu? v7) = 0 for z 0 (27) Jz jxx With the assumption of Ba > a Y, = 0 for z 0 (28) Equation (27) will be applied in the outer region where the three-dimensional so- lution is expected and Equation (28) will be applied in the inner region. The two- dimensional Green function which satisfies Equation (28) is given by Wehausen and Tenicone- for a single-hull body as co kz hel e_cos ky . Kz Gop (y>2) ae PV | ae dk - ie ~ cos Ky (29) fe) Equation (29) is the potential function of a unit source at the origin where K = mien With a change in the integral path, the Green function can be expressed as 1 a cos k £ eee COSY: Gop ly. 2) = { a dk (30) fe) ay The contour of the integral path is around the pole from beneath. With a change in the variables Equation (30) can be written as okztiky,, i ekz+ik|y| @ =.= Re { - (Kz+iky)} - i (31) 2D The derivation of Equation (31) is given in Appendix A. The exponential integral is E, (u) and is defined as =t E,(u) = i erat (32) u The asymptotic properties of the two-dimensional Green function can be obtained from the corresponding approximation of the exponential integral. For small values of Kr, Equation (31) can be expressed as (see Appendix B) il ; Gop = [y+2n (Kr) -it] (33) where y = 0.577... which is Euler's constant and (r,8) are polar coordinates such that y = r sin 0 and z = -r cos 6. For large values of Kl y| the asymptotic approxi- mation of Equation (31) represents the outgoing two-dimensional plane waves in the form es _, eK2tik|y| ome for Kly| >> 1 (34) The three-dimensional Green function which satisfies Equation (27) is given by Wehausen and Detkpontec as 00 T zu : : d aes) — al, Ae gue exp (-ixu cos “ates sin 6) a6 (35) ant Le gu — (atUu cos 0) Equation (35) is the potential function of a unit source which is located at the origin and moves with constant velocity U. Equation (35) is multiplied by a constant value, -1/4mt, for convenience. With the change of variables, u cos 6 = k, u sin 6 = 2 and udud§ = dkd&, G(x,y,z) can be rewritten as oo Dai? 1/2 ; (k +27) -iy Sete pe —ikx aur e* e G(x,y,z) =- oF j e dk a ae 1/2 ; dg (36) —00 -~ g(k +27) = (a+kU) If we define the Fourier transform £*(k) = { eG oo ae (37) and the inverse Fourier transform (oe) a@e) = i { A) aoe ak (38) then the Fourier transform of the three-dimensional Green function is given by 1/2 a2 (k +27) aviy® * . S55 => G*¥(y,z;k) a ae 1/2 s dz (39) 2 GON) | (GHED The value of G* for k = 0 reduces to the two-dimensional source potential, Equation (30), as < 1 z|2| -iy2 x 10) aa Seas) 2m | S__£____ ag (40) An approximation similar to Equations (33) and (34) can be derived for the transform G*(y,z;k). An asymptotic expansion of Equation (39) for Kr << 1 is derived by Users in the special case U = 0. For the case U > 0, the asymptotic expansion is given by (see Appendix C) al * = EO eae G*(k,K) Gop = £*(k,K,K) (41) where Gon is Equation (33) and 2K : £*(k,K,K) = | -in + 7G* (42) k| SOLUTION OF THE POTENTIAL FUNCTION WITH THE SLENDER-BODY ASSUMPTION Under the slender-body assumption the length of the ship is far larger than the beam or the draft. We separate the fluid domain into two regions: the outer region where (y,z) is of the order of the length, and the inner region where (y,z) is of the order of the beam or the draft. In the outer region, the three-dimensional Laplace equation is solved with the free-surface condition, Equation (27), and the radiation condition. The inner solution is governed by the two-dimensional Laplace equation with the free-surface condition, Equation (28), and the body condition, Equation (19), on the ship hull. Then, the inner and outer solutions are matched in the overlap domain to determine the slender-body solution. 10 Outer Problem The outer solution can be constructed from a suitable distribution of source strength along the longitudinal x-axis. If we denote the source strength with q, (x); then the outer solution is expressed in the form a = { q; (&) G(x-€,y,z) dé for j = "3 and 5 (43) 1b, Here, we consider the solutions for heave and pitch only. The Fourier transform of Equation (43) is derived from the convolution theorem in the form “aS = alae G*¥(y,z3k,K) (44) where G* is defined by Equation (39). The inner approximation of Equation (44), for small values of the coordinates (y,z), can be constructed by substitution of Equation (41) il Of G2 Cx. = = eg 8 (45) Inner Problem The fundamental solution of the inner problem is that of the strip theory. However, the matching requirement with the outer solution will differ from the condition of outgoing radiated waves satisfied by the strip theory. Because the outer solution includes a longitudinal function of x which depends upon the three- dimensional shape of the ship hull, we solve the inner solution in the following 3 form ional) oa ?. = 2. + SCN aA).) (46) where C.(x) is a function to be determined by matching with the outer solution and yao) is the strip theory solution which can be expressed as follows dha (s)) a = % ap >, (47) On the body surface, uF and % satisfy the following conditions db. = ~ alin (48) and =m. (49) In Equation (46) ¢ is the conjugate of OF The solution of 3 for heave motion can be expressed by a cos 2m8 K cos (2m-1)86 Oy = sop +2 %y | 2m 2m-1 | m-l Cy where Con is defined by Equation (29), 0, is the two-dimensional source strength at the origin, and the terms under the summation are the higher order multipoles which form the wave-free potentials. Equation (50) was first introduced by Tegel, A similar solution for >, can be given by Bre tae. es cos 2m0 K cos (2m-1)8 v3 me sa2D a2, “im | 2m yal p2m-l on For pitch motion we can have similar expressions to Equations (50) and (51) by vas “A substituting 0 a O3, and Oo for -x0 SOU “X03, and Ped respectively. From a 32 wipe Equation (50) the outer approximation of the potential >, is $, = 0, Gy(y52) (52) 12 A similar expression can be given to the potential Fe $, = 9, Cop(y.2) (53) By substitution of Equations (52) and (53) into Equation (46), the outer approxi- mation of a is Y, = Gs Gh Gi, -~ db Sed (0,610 5,Cop) i Q (?) + Q a + o. (x) So exo) Gop + Go (x) (Gop-Gop ) oF (54) From Equation (34), the last term of Equation (54) is dix q okz-ikly|, Ketik]y]) § 98 Gj. - G ver i( ene cos Ky = 2i By substitution of the above expression into Equation (54), the outer approximation is finally given by Y ='[0 +o. +C1G) (Go) | C. 2INCnGo ron 55 J a haged eae j >! 2D j j o>) The Fourier transform of Equation (55) becomes Ps) =) [o.+oe-+C. (x) (ol+o.) 1* Ga. - 2uC) Ga. 1* 56 j [argrGaCs) KO.) oD aie orl (56) MATCHING The inner and outer solutions are matched in a suitable overlap domain to determine the unknown source strength, are of the outer solution and the coefficients C; in the inner solution. From Equations (45) and (56) we have the following relation 13 1 g = = £¥q* = [o,4+0.4C.(0.4+0.)]* G TT aie I Aleta gi j >! i o.)* D + 21(C,5,) (57) i 2D Equating the factors of Con gives a relation for the source strength .* = [o,40.4C, (o,40,)]* for) = oeandes (58) y eo iver tie in bi 8)/4 1 | : Equating the remaining terms in Equation (57) gives f*q.* = -2ni [C,0.]* 59 als [ j 3) (59) The inverse Fourier transforms of Equations (58) and (59) can be expressed as q; = [oi saxe) He, Wol-ar0/,)) (60) 2ni [c,o,] =- | a, (6) f (x-€)d& (61) L The kernel f(x) is the inverse Fourier transform of Equation (42). Elimination of ei gives an integral equation for the outer source strength (o +0.) i q.(x) - i | —1—__ q.(&) £(x-E)dE = (0,40,) (62) J 2n0. J al ie) J INNER SOLUTION By substitution of Equation (61) for Cakes) into Equation (46), the inner so- lution is given by Ge Oe) & ae ($.+6.) fe f (x-&)d& (63) j 2m Sy ei) j 14 Equation (63) is the unified inner solution which is valid for all wave numbers K. The details of the derivation and error estimations are given in Reference 3. In Equations (62) and (63), when j is 5 for pitch motion, the two-dimensional source strengths, 05 and Os can be expressed as a function of or and Sige respec- tively. The two-dimensional potential for pitch motion is $= - x by (64) OS fam Oa ees a (65) From these equations, Of, and 0. are given by On = -— x 04 (66) ae c iU GPS Wa a Ca (67) By substitution of Equations (66) and (67) into Equation (62), the integral equation for the outer pitch source strength is given by (05405) a iU 2710 3 The inner solution for pitch motion becomes eS) ix ae y= 9.) - Gs) fas £ (x-E) dE (69) 2710 The derivation of the kernel function, f(x), from f£*(k,K,kK) will be given later. 15 DIFFRACTION POTENTIAL With a similar method to solve the potentials Me LOGE) oirand)) they idiitstrsac tion potential can be solved. In the outer region we solve the three-dimensional Laplace equation and in the inner region we apply strip theory. Through matching, the unknown source strength and coefficient are obtained. In the application of strip theory, however, there are two different approaches. One is to solve the two- dimensional Laplace equation and the other is to solve the two-dimensional wave equation or the Helmholtz equation. The first approach has been applied by Salvesen et ale and others. Newnan masesehy and others have solved the Helmholtz equation for computing the diffraction forces. In this method there is a singu- larity in the solution when the angle of the incoming wave is 0 or 180 degrees. To avoid this singularity, Newman has introduced the long-wavelength solution. How- ever, there is difficulty in solving the equation for short waves. In this study we shall solve the two-dimensional Laplace equation to compute the diffraction force as the solution of the strip theory. The inner solution for the diffraction potential can be given in the form (see Reference 3) 0, = 99) 4c) (49,) (70) where C_ (x) = a function to be determined by matching with the outer solution os) = the diffraction potential solved by the strip theory O. = the symmetric function of ae) , satisfies the two-dimensional Laplace equation with The diffraction potential, ae the boundary condition given in Equation (18). If we express the potential of the incoming wave as o)) ae ae O. a, exp (Kj2ti Kx cos B-i Ky sin 8) (71) the boundary condition of ys) is given b y 7 / MED) n = Ko (n,-i n, sin 8) O (72) 7 2 16 where A = amplitude of the incoming wave 8 = heading angle of the incoming wave: £8 = 180 deg in a head sea and 8 = O deg in a stern sea 2 Ky = w/8 If we express the outer approximation of ene) as p(s) _ ; O5 Gop + om Gop (73) then the outer approximation of the inner solution, Po, which is similar to Equation (54), is given by Y_ = 07 Gop + C_ (x) (0540) Gop + Cl (Gx) (Gop Cayy) 55 (74) The second term in Equation (73) is asymmetric. Because the ship has a symmetric centerplane, this asymmetric term is not included in Equation (74). The outer so- lution for the diffraction potential is given by Equation (45) with j = 7. Then, taking the Fourier transform of Equation (74) and matching with Equation (45), we can derive the integral equation for the outer source strength and the inner so- lution similar to Equations (62) and (63) as (o540,) a(x) - 1] fae £(x-E)dE = - 6, (75) 2710 7 e, = vis) 4 OF) | a, £ (x-E) aE (76) 2710 7 : and Gee fae £ (x-E) dE (77) 27105 17 HYDRODYNAMIC FORCES The pressure in the fluid is given by Bernoulli's equation p=-. (rece 4 val?) (78) By substituting Equation (3) into Equation (78), the pressure becomes p = p(iuh,-VO,°V4,) e OF - paz (79) Then, the forces and moments with respect to the origin of the coordinate system are given by F ie ff p a, ds for j = 3 and 5 (80) S where S is the submerged portion of the ship hull, and n, is defined in Equations (20) and (21). By substitution of Equation (79) into Equation (80), the hydrodynamic forces can be expressed by Bais 0 {{ [i0},-Vb4-Vbo] n, ds rae (81) 5 The second term of Equation (81) can be transformed by means of a theorem due to Ogilvie and Tuck (Appendix A of Reference 8) ff (Vb, °V$.) a dS = - {J ie $5 ds (82) S S In Equation (82), the line-integral term along the intersection of the ship hull with the plane z = 0 is ignored as a higher order term. By substitution of Equations (82) and (17) into Equation (80), the hydrodynamic forces are given as 18 6 F bog ocase ey vit Ot > 6,9 ds en 35 (83) j qed gl k k S k=1 ADDED MASS AND DAMPING We write the integral of the term under the summation in Equation (83) as 6 By Sethi bce oa ht (84) k=1 where Bae O ff (ian ,+m,) [ot +0,, (x) ($,+0,) ] ds (85) Ss The terms inside the bracket represent the unsteady potential an which results from oscillation. Equation (85) denotes the hydrodynamic force and moment in the Bien direction per unit oscillatory displacement in the ean mode, The added mass and damping are defined by Bay = = Aag We o> at ng @ (86) where A., is added mass and B,, is damping. Both A., and B,, for j = 3 and 5 and jk jk jk jk k = 3 and 5 are given in Table 1. In Table 1, is used the following notation: Hy = Ue e ly >, = 2 + i, Wg SF PSEgE Og F Es ; eng v Mes > QU mesma) 8 Oe: S sag Va > esi The added mass and damping in Table 1 are those of the bare hull only. The added mass and damping due to viscosity and stabilizing fins are given in Table 2 and Table 3 of Reference l. TABLE 1 - ADDED MASS AND DAMPING 2 i) 4 (ou _ 2 i - offs: as +f [[msby 28 + 25 [mgr as al nj, 4s 20 20 m {I Hee 7) eer = ol{n,o, as +2 nO, 25 +2 [/ m6 as + off agi, as 3°R HN ase oJ) ™3°r 3°R 2 iG. dg & 22 iC. && o([ R-3R @ {fs Tsoi _ 9 pu Lion i jee all xn, aS + 5 {| mes ds mall xmyb, 4S ll xmyb, 4S pu i) A 29 _ 28 . {| nz, dS + alll xn,o, dS + = {| xNzh_C31 dS ell xmzbpCep dS pu © ‘ ) 2 =- of amsep ds - luce ds {fms ds al xm 05 ds x { n> ds ) x13 Pp ds 20 XN3O,Cap ds a I x3 >pCey ds p _ pu R Danone * 1 alll xn, ds : [Jrs%. ds 5 (f xm, as ail xno, as i) i) 4 20 _ 20U af ate dS + malar dS + A [[stetsr 4 > {I o. C, 20 2 {{ xMz>pCgp dS 20 TABLE 1 (Continued) _ pu ‘ _ & A _ ou =- 0 |{ x50, ds a {J 56 ds B [f ams6, ds a {[ 3° ds Pl sma. a3 = snail de & 20 (lead Gd so ime. as Sul p 3°R p 3°R-3R ® 3R-3R 2 Le OT eZ pu” pu 4 A oul =- alle n3>7 dS - 3 {[es%: dS + || 3 dp ds 5 xm 5 ds OW ees & O_ (2 nOuiezans 1 29 fr 2 mlb m3 >, dS + walk m3, ds all 13>p ds rs [[> 13, ds 2 ae ou a ou of{ x N3>p ds + If nate ds +o [fate ds + [ste ds aif? no, dS + anes m3, dS + off x" xn 3p ds + 20f{ x" x n3O,C. C ds ail nO REST aS) oe aa x"m3nC5q as EXCITING FORCES The excitation forces are given by Equation (83) as have , -iwt Ee of (tem seam.) vot 95] dS e (87) Pals The integral containing Pe in Equation (87) is the Froude-Krylov force. If we substitute We for 5 in Equation (82), we have one -{[ a, O. ds = -{fv ax ay ds (88) Here, we have neglected the higher-order terms. With Equation (88), the Froude- Krylov force is given by h an il I D — — =) (as oe (S} sales Ss o® a wn fa] € ct A 1wt = ~ipw, {J n Ve ds (89) Ss where ve is given in Equation (71). The heave excitation force is given by Kz 25) f, = 20gA l-e cos (K,x-wt) cos K,y dxdy (90) 3} Z4 1 D and the pitch moment is Koz 5) a 20a || -l+e x cos (K,x-Wwt) cos K,y dxdy 5) Z4 iL 2 K Kz oD 4 Doral |fe fe) (Reena) sin (K, x-wt) cos Koy dxdy (91) 2 fo) Zz K 1 where Ky = Ko cos B, K, = Ko sin 8, and the integral limits Z4 and Zo are given in Figure 2. 22 24 Figure 2 - Limits of Integration By substitution of Equation (70) into Equation (87), the diffraction force is given by hy = -0 {{ (on) +m;) [ e+, (¢ 46 .)] dS iopawe (92) S We first substitute Equations (48) and (49) for n. and m., and apply Green's second theorem to the term containing OS The diffraction force can be written 2) Yp wn ° fo) f > -iwt h, = |-o i Aaa ($,-,)dS-p {| lei et NES aH) ds e (93) S S where Cl is given by Equation (77) and %, is the symmetric part of one that sat- isfies the two-dimensional Laplace equation and the boundary condition, Equation (72). By substitution of Equation (71) for % o> the diffraction heave force is expressed by { ik,x { KZ im 3 = i2pw A e dx e n, cos Kjy-n, sin 8 ° sin Ky (6-04) dy Cc - 20 Co (x) dx i (ion,+m,) Re[9_] dy (94) c 23 and the diffraction pitch moment is iwt Di Gales Sore hee =- i2p0,a | x+ am || C dx { e nN, Cos Koy, sin 8 sin Koy $3 dz c + 20 | C(x) dx [ (ion,x+Un,+m,x) Re[>.] dz (95) c Equations (90), (91), (94), and (95) represent the heave force and pitch moment of the bare hull of the SWATH ship. The exciting forces and moments due to viscosity and the stabilizing fins are given in Table 2 and Table 3 of Reference l. EQUATIONS OF MOTION If we let A, = Eee, Oe = cman, the total exciting forces By, and the total pitching moment F then, the equations of motion can be expressed bye M | (M+A, 3) a, + B3 3%, + C4305 + Axo. + By 605 + C4505 3 (96) M | iy 1} Cee ore UA mT pes Gr CTA) 5H Boss * S55M5 1 45303 ¢ Boats 1 o53%5 = 5 where M is the mass of the ship and I the mass moment of inertia about the y-axis. The hydrostatic coefficients are given by 33 wp C35 = Ce3 =- Ce Sek oe) (97) ws 2 a a Ih Cee ~ pe [I +A (x =<) ] + (KG-KB) « A 24 where Ba = waterplane area = moment wp Be = longitudinal center of gravity ale longitudinal center of flotation LL = moment of inertia of A_ with respect to the longitudinal center of flotation KG = distance between the keel and vertical center of gravity KB = distance between the keel and vertical center of buoyancy A = displacement of the ship With substitution oe UW = Fy3 Mapes and EE = Fos ae: Equation (96) can be expressed as complex algebraic equations: - ar CED )+C - iwB - nos +C -— iwB E F 33} 35) 33 35 35 35) 3 38 a (98) 2 : 2 ; - Ww Az, + C,, - iwB,, - (I+A, <) + Ces - ioB,. Es Fes RELATION OF TWO-DIMENSIONAL SOURCE STRENGTHS BETWEEN THE FRANK CLOSE-FIT METHOD AND MULTIPOLE EXPANSION METHOD The potential function given in Equation (50) is the solution obtained by the multipole expansion method. If we apply the Frank close-fit method to solve the two-dimensional potential, we should have some relationship between the two methods. That means we should have a correspondence between the sources located at the origin in the multipole expansion method and those around the ship's contour in the Frank close-fit method. Because the radiation condition states that far away from the disturbance, the wave is outgoing, we should have the same potential from these two methods at a distance far from the body. For the multipole expansion method, the potential of the outgoing wave is given by Equation (34) okz+ikly| > = -id, (99) where on is the source strength at the origin. 25 For the Frank close-fit method, we have the potential as follows p= [e Gop dy (100) where 6 is the complex source density at the ship's contour and G D for the twin-hull 2 body is given by Wehausen and Demteonee oo ik(ytiz-nt+ic) 1 : ; : ; (Ree EPR NS om = Re oF log (ytiz-n-it) - log (y+iz-n+it) + 2PV | Kk dk (7) I oO ae ee fe tk (ytiz-ntit), : - qik(ytiztntiz) + Re 5 log (ytiztn-it) - log (ytiztn+it) + 2PV [ Re dk 0 Nh se fe iKk(ytizintic) | (101) where (y,z) is the point where the potential is sought and (n,7Z) is a source point (Figure 3) (n,$) e (y, z) Figure 3 - Sectional Coordinate System 26 when ly| is large, the logarithmic terms cancel out. Then, Equation (101) can be expressed in a manner similar to Equation (30) ° (240) k(2tt) eit e cos k(y-n) ak e cos k(y+n) Cas = { eee (aS Hh { Se ak Cu) ) 0 2 = — 20 Equation (102) can be further transformed with the exponential integrals as shown in Equation (31). As ly| becomes large, these exponential integrals become small. Finally, as ly| becomes large, Equation (102) can be expressed as oy ee tac’) tetk lyn a gall: (103) Then, the potential function is given by > = i fae , [1-2 (nk) ] xfexp [| - Ta Okx x enh op a Ras a ee at ay fo) [k(1-k) (6k+1) (6k+2) ] — 2s 1/2 [>| x exp -[6(1-k) (S6k+1) ] a (121) In Equations (119) and (120), the first numerator is for n = 3 and the second for n= 4. All other notations used in Equations (118) through (121) are as follows: R, (v) =i IK R,(v) = -i (mx) 1/2 - mv sign x 5 (122) m, = (K-mv’) - (4 mK-1) v" \eoiniéo ) my = 4 (mx) 2/2 Vv (ese) sign x 33 YD Reis 2(mx)!/2 _ [1-4¢mxy1/?] for (mk)1/2 < 0,25 = (mx) 1/2 for (mk) 1/2 > O25 1/2 1/2 1/2 y/D k, = 1 - 2(mK) + [1-4 (mk) J for (mK) < 0,25 BAKO for (mk)~/2 > 0,25 1/2 k,= 142 (mk) /2 _ [144 (mx)1/2] 1/2 eel 2 (mk) 1/2 4 [144 (mK) 1/2] G22) S) SAGOw 2 6 a N, (k) = Ke n,(k) = -+ (GS) CED? aien -4 [éket1-+2 (mk) 2/2} 2 es oa g 2 2 e U g As in special cases, when K = 0, G3 becomes (see Appendix D) 19a x : Ga G, a oS A 2) + (2+ sign x) ve aw (123) and when m = QO, 34 es -* [-H|(K|x|) - ¥,(K]x)) + 2i J, (K] x] )1 (124) where Ho is the Struve function, and J and Nes are Bessel functions of the first and second kind. RESULTS AND DISCUSSION To test the numerical results, we have selected three models of SWATH ships: SWATH 6A, SWATH 6C, and SWATH 6D. Experimental tests for motion have been carried out for all three models. All models have the same lower hulls and the same distance between the two hulls, but differ in the shape of their struts. SWATH 6A has a single strut on each side while SWATH ships 6C and 6D have twin struts. The thick- ness of the struts differs for all three models. The principal dimensions of these models are given in Table 2. Because our primary interest is the motion at low fre- quencies of encounter, the computations have been carried out when the forward speed is 20 knots. TABLE 2 - PRINCIPAL DIMENSIONS OF SWATH SHIPS* Displacement, long tons 25719 2602 2815 Length at waterline, m Length of main hull, m Beam of each hull at waterline, m 2.6 Hull spacing between centerline, m Sjodl 4.6 Draft at midship, m Maximum diameter of main hull, m Longitudinal center of gravity aft of main hull nose, m 34.7 bo oO ay Vertical center of gravity, m 10.4 Qodl Longitudinal cM, m 36 7/ 26.4 Radius of gyration for pitch, m Uo 7 19.0 181.4 25.8 (per strut) 8.6 oY) 25.8 (per strut) 16.4 Maximum strut thickness, m 2, 2 (0) 3}aak *Dimensions are full-scale. 35 2 Waterplane area, m Length of strut, m Strut gap, m Figures 6, 7, and 8 show the added mass and damping coefficients. The presently computer results are compared with those of strip theory. The results of strip theory were computed by the computer program MOT35 (Reference 16) which is based on the analytical method developed by eee The two-dimensional potential (strip theory) is solved by the Frank close-fit method. The effects of fins and viscosity are included in the computation of added mass, damping forces, and excitation forces. This computer program has been improved by adding surge effect to the pitch excita- tion moment due to the incoming waves and by correcting for viscous damping. The results for added mass by the present method are almost 70 to 80 percent higher than those from strip theory, while the damping coefficients are slightly less than those from strip theory. In the present method the results of the unified slender-body theory are added to those of the strip theory as a correction term (Equation (63)). For the added mass coefficients, these correction terms are too large. The large discrepancy may result from the different methods used to solve the strip theory. In the derivation of the unified slender-body theory, Newnan solved the strip theory with the multipole expansion method (Equation (50)). In Equation (50) o, is a source located at the origin and is used to solve the three- 3 dimensional source strength (Equation (62)), from which the correction factor G, is computed as in Equation (61). The only part of the source located at the origin is oO The other part of the source is On from Equation (50), which is included in the ae, of the two-dimensional added mass and damping coefficients, but is not included in the three-dimensional source strength. In the present approach for solving the two-dimensional problem, the Frank close-fit method (Equation (100)) was applied. In Equation (100) o is distributed on the contour of the section, and the corresponding source O5° which is located at the origin, is computed by the relation in Equation (105). To solve the three- dimensional source strength, 0, was replaced with O5° but 05 includes the effect of oO. and on It is difficult to decompose or. into 04 and O in the Frank close-fit 3 method. If Oo, is not computed accurately, the three-dimensional source q. in Equation (62) cannot be solved correctly, even though the kernal function f(x) has been evaluated properly. Therefore, the difference between 0, and o. could possibly cause the large discrepancy in added mass coefficients. 36 STRIP THEORY ———7—= UNIFIED SLENDER- BODY THEORY Figure 6 — Added Mass and Damping Coefficients of SWATH 6A in Following Seas at 20 Knots 37) STRIP THEORY -—--— UNIFIED SLENDER- BODY THEORY Figure 7 -— Added Mass and Damping Coefficients of SWATH 6C in Following Seas at 20 Knots 38 STRIP THEORY --—-— UNIFIED SLENDER- BODY THEORY Figure 8 - Added Mass and Damping Coefficients of SWATH 6D in Following Seas at 20 Knots 39 When Gre = wU/g = 0.25, there is a discontinuity in the computations. This singular behavior can be explained by the Green function. If (nk) 2/2 approaches 0.25, Equations (119) and (120) become logarithmic functions as shown in Reference 15. Figures 9 and 10 show the results of excitation forces and moments for SWATH 6A and 6D, respectively. The heave forces for both configurations are slightly larger than those from strip theory. The excitation moments are generally larger also. For SWATH 6D, the results of strip theory for the heave force show better agreement with experimental results than those from the unified slender-body theory. The pitch moments of SWATH 6D are scattered between the results of the strip theory and the experiment. The motion results for SWATH 6A are plotted in Figure 11. Heave amplitude shows a similar discrepancy with experiment as with strip theory. When the en- counter frequency becomes very small, the peak of the heave amplitude disappears; this peak is shifted to the higher encounter frequencies. The plus points show the results of the "mixed method,'"' in which the added mass and damping coefficients are computed by the unified slender-body theory and the excitation forces are computed by the strip theory. When A/L is less than 2.5, heave amplitude, computed by this method, agrees better with experimental results than with the results of either the unified slender-body theory or the strip theory. However, when \/L is larger than 2.5, the results of the mixed method underpredict the experimental results and those of both theories. Pitch amplitude shows the same tendency as heave. The peak is shifted to higher 4/L values. Pitch amplitude, computed by the mixed method, agrees well with experiment for all i/L values. The motion results for SWATH 6C are given in Figure 12. The heave and pitch amplitudes become very large as the encounter frequency becomes large. The results of the strip theory are better than those of the unified slender-—body theory and the mixed method. From the results of the mixed method, we can conclude that the added mass and damping coefficients computed by the unified slender-body theory in Figure 7 are too large. 40 STRIP THEORY ———— UNIFIED SLENDER- BODY THEORY Figure 9 - Exciting Force and Moment of SWATH 6A in Following Seas at 20 Knots 41 pgVA/L 0.75 0.5 Fs pgVA STRIP THEORY 0.25 — ——— UNIFIED SLENDER- BODY THEORY O EXPERIMENT 0.0 Figure 10 - Exciting Force and Moment of SWATH 6D in Following Seas at 20 Knots 42 = ~ TO See ae ee ee Wa |E3| A STRIP THEORY --— —- UNIFIED SLENDER- BODY THEORY EXPERIMENT MIXED METHOD Llés 2A Figure 11 - Motion of SWATH 6A in Following Seas at 20 Knots 43 STRIP THEORY ---—-— UNIFIED SLENDER- BODY THEORY O EXPERIMENT + MIXED METHOD i L Figure 12 - Motion of SWATH 6C in Following Seas at 20 Knots 44 Figure 13 shows the computed motions of SWATH 6D. The heave amplitudes pre- dicted by the slender-—body theory are much larger than those predicted by the strip theory. Compared with experiment, the results of the mixed method are best except for high A/L values. The pitch amplitudes computed by the unified slender-—body theory and by the mixed method are too large and do not show good comparison with the results of strip theory or the experiment. In contrast to the results for SWATH 6A, the mixed method does not compute the pitch amplitude properly. This indicates that the added mass and damping coefficients in Figure 8 by the slender-body method may be in significant error. This error might be caused by solving the strip theory with the Frank close- fit method and by replacing 04 with 85 as mentioned in the discussion of the added mass and damping coefficients. SUMMARY AND CONCLUSIONS In this report the unified slender-body theory developed by Neeman is applied to predict the motion of SWATH ships in following seas. Only for a limited range of \/L values is there an improvement for heave motion. For pitch motion, except small encounter frequencies, the results are worse than those of strip theory. When the encounter frequency is large, the pitch motion becomes extremely large. The reason for this discrepancy seems to lie in the fact that the strip theory is solved by the Frank close-fit method, and not by the multipole expansion. From the present study, the following conclusions may be drawn: 1. In the unified slender—body approach, the largest contribution to the hydrodynamic coefficients results from the solution of the strip theory itself. Correct computation of the source strength at the origin is necessary to compute the three-dimensional source strength exactly. Therefore, for the outer approximation of strip theory, the multipole expansion method should be applied instead of the Frank close-fit method. 2. A more careful analysis should be made in computing the excitation forces by the strip theory. Application of the Helmholtz equation is mathematically more correct. However, there is a singularity in this solution when the heading angle of the incoming waves is zero or 180 degrees. The two-dimensional Laplace equation computes the heave excitation force correctly, but the pitch excitation moment is computed incorrectly. 45 Lite! 2A O EXPERIMENT + MIXED METHOD STRIP THEORY —--— — UNIFIED SLENDER- BODY THEORY fe) Figure 13 - Motion of SWATH 6D in Following Seas at 20 Knots 46 3. According to the unified slender-body theory it should be valid for all frequencies. However, for large frequencies, the motion results become unacceptably large. This may be caused by the use of a different method for the solution of the strip theory. 4. Because of the above mentioned discrepancies, until further research mentioned in items 1 and 2 are carried out, the application of the unified slender- body theory to improve the prediction of motion of SWATH ships is not recommended. ACKNOWLEDGMENTS The author expresses his thanks to Dr. C.M. Lee for his useful advice and gratefully acknowledges the support of Ms. M.D. Ochi. 47 vy {i he ron Wes vel migeciovae bth ) nee Bat te tok el oe an Danae me | bey eae ‘lees te euch opto ane met ‘eqitite ae i i Tae anc as.oagi Te te toad chat x’) 2 ele Ee id pa ekoads send sepesiane “hes M i bain anton rn Oh i neta piano Boast invent ee pela ad ee en ee ee APPENDIX A DERIVATION OF Con ly>2) WITH EXPONENTIAL FUNCTION Equation (30) can be rewritten as r kz iky, -iky { eum (emmecte ) dk it Con ly) = - oF kK ie) ay For the first term of Equation (A.1), we change the variable as w = -i(k-K) (y-iz) and -dw OSGI) The contour of the integral path will be different depending upon y S 0. we take the integral path in the following figure iK (y-iz) k-PLANE w-PLANE (A.1) (A.2) (A.3) When y > 0, By substituting Equations (A.2) and (A.3) into Equation (A.1), the first term becomes Co jo ik(y-iz) -w SR Waea= {Se See Ae k-K Ww o ikK(y-iz) 49 (A.4) If we consider the following integral along the closed path in the w-plane Ba ——— (hig |) Gp + | = 2ni Ww Pee Me ai Os then, the integral over [ = Rea vanishes as R > ©, and Equation (A.4) becomes Loe) | okztiky ik(y-iz) anor e " Tika, dk=e 27 i- Sm dw SN Se = eiX(y-iz) [2mi+E, (K2+iky) ] (A.5) When y < 0, the integral path is given in the following figure Ir K iK(y-iz) k-PLANE w-PLANE and the first term of Equation (A.1) becomes ~ okztiky as oik(y-iz) .-w \ ins ae dk = Cnn an Ge dw (A. 6) re) ik (y-iz) Fy If we consider that the following integral along the closed path is in the w-plane, 50 then the integral over IT = ma vanishes as R> ©, and Equation (A.6) becomes iK(y-iz) i ikK(y-iz) e kk =-e Teal dw a otk (y-iz) E, (Kz+iky) (A.7) For the second term of Equation (A.1), we apply the following transformation of the variable w= i(k-K) (ytiz) (A.8) and zn dw dk > FG za) (A.9) With the same procedure for derivation of Equations (A.5) and (A.7), the second term of Equation (A.1) is given as | ok2-iky mae dk = eh YB (Kz-iKy) for y>0 (A.10) oO = ef 7KY [omitk, (Kz-iKy)] for y< 0 (A.11) By substitution of Equations (A.5), (A.7), (A.10), and (A.11) into Equation (A.1), Con is given by 51 Sele te aR z trlikKey ea Kz +iky =, Gop ly>2) = oT {e [2ni+E, (Kz+iky) ] +e E, (kz +iky) } for y S 0 a og okz+ik|y| T ne (oS E, (Ket+iky)} -i (A.12) a2 APPENDIX B DERIVATION OF Gop y»2) FOR SMALL ARGUMENT We expand the exponential function and the exponential integral in Equation (31) for small Kr where y = r sin 9 and z = -r cos 8 SPOS) ec gaa iky (1+Kz) (B.1) oketik|y| = 1 + Kz + ikly| (1+kz) (B.2) E, (Kz+iky) = E Get) 5 i -y - &n(Kr) -i(m-6) + Kz + iky (B.3) By substitution of Equations (B.1), (B.2), and (B.3) into Equation (31), Gop is given as Gop (y>2) = - i Re {[(1+Kz+iky(14+Kz) ] [-y-2£n(Kr)-i(1-6)+Kz+iky]} - i [(1+Kz)+ik]|y| (1+Kz)] == (14Kz) [y+2n(Kr)-Kz+Ky6-im] (B.4) 53 ae Ae Cara Yyit.S PAL at. sand) TY wit, the (ambe th | K Aas 1 fet dresoges eda ore Re ke. ee HK) enatieny a | at ey S Bee a) ie t oe an i ur APPENDIX C FOURIER TRANSFORM OF THE THREE-DIMENSIONAL GREEN FUNCTION Equation (36) can be expressed as foe) (oo) 9 9 1/2 G(x,y,2z) = - 5 { ema a die c 7p» Bee SS Laws +p =tyhI\ ae (6.1) ple meee woo (2427) G29) ate where 2 K = ue (C52) The first double integral of Equation (C.1) is known. Therefore, we rewrite Equation (Gsi)) as 1 G(x,y,z) eS 2nR = G3 (x,y,2) (C.3) where Ra = a + va + a and G3 is given as - Dee mabe -ikx i K exp[z(k +27) -iy] G3 (x,y,z) For e dk 5 { ngs 1/2 ae 1/2 dL (C.4) 29 —0 (k° +2") Leas) -K] If we assume y and z become very small, Equation (C.3) is expressed as aL G(x,0,0) = - - G, (x, 0, 0) (G55) YD Delos) 55 The Fourier transform of Equation (C.5) is 1 x Sen = oS G*(k,K) = K (kl xn) C4*(k,K) (C.6) and from the definition given in Equation (38), G,* is given miele K d& CeO) | GUIS Rees Gol) -o (k +27) [ (k7+27) -K] In Equation (C.5), KO) is the modified Bessel function. For small r, K(/kle) can be expanded |k|x K(klx) =) 5) Min a) i (C.8) where y = 0.577... is Euler's constant. By substitution of Equation (C.8) into Equation (C.6), for small y and z, G* is Om aie * = ae G*(k,K) Fs Qn Damien G.* S56. 62 Ben (c.9) oT at OS ; where Gop is Equaticn (33), G,* is Equation (C.7), and f* is given by £*(k,K,K) = Ln a -in + 7G, (C.10) k 56 APPENDIX D DERIVATION OF G, Gx, 0, 0) FOR NUMERICAL EVALUATION We change the contour of the integral path with respect to u in Equation (115) as shown in the following cases. Ls (eS a. First Integral of Equation (115) Evaluate the following integral in the complex plane of w p —__K_exp(ciws cos $) __ dw = + + { = - 277i (residue) (D.1) w- CK) 24 any t/ 2u0 cos 6] Cy Cy C3 w-PLANE Because of 1/2 1/2 : 9 es a w - [(K) + (m) cos) Oi) = "— m cosm. @ (w-u,) (w-u,) the residue becomes K exp (-iu,x cos 6) -ik exp (-iu,x cos 6) a Re 1/2 -mcos 0 (u,-u, [4 (mx) 2/2 Saeo=iy 57 where u, and u, are given in Equation (110). The first integral in Equation (D.1) aL 2 is the same as the first in Equation (115). The second integral along the contour c, becomes zero 2 -7/2 I @») py Lin a Psnoe g Rel’ 4 do no :, -¢)° The last integral of Equation (D.1) becomes f K exp(-iivx cos 6) i dv i ik exp(-vx cos 0) dv © ae Di D sy = 1 aq ae ees 8) CACO) SUE Oe Gos 8] Se Therefore, the first integral of Equation (115) is given by (oe) K exp(-iux cos 9) exp Gaul os ©) du =- 27K In SE : 1/2 me o u — [(K) +(m) os 6] [4 (mK) cos 6 -1] a ik exp(-vx cos 0) 5 ake Ste AGO Ae e eae 61 b. Second Integral of Equation (115) Evaluate the following integral in the complex plane of w (Reise ees eae [os [+ J aoa esiee) 19/22 1/2 w — [(R) +(m) w cos sa Cy Cy C3 58 (Do 2) w-PLANE The residue is x cos 6) iy 2 cos 6] K exp(-iu,x cos @) -K exp(-iu 2 2 2 -—m cos 9(u,-u,) [1-4 (mK) 2/2 becomes zero as shown in (a) and the integral along c, becomes The integral along c 3 2 ) : , is K exp(-iivx cos 0) idv =- SK Gap (ors CoS 8) dv Oats 2 1/2 1/2 a ys ive [ (KR)! 244 (ny 2! y anal) Q abePl(O) "SalGn) “Ay Cos [3 Therefore, the second integral of Equation (115) is given by K exp(-iu,x cos 9) r : | (.) du =- 2nKi = es a ik qwpleve G08 8) 5 av (D.3) L BLAAGIO? eae Bi 5 BGO! Saga 1 v cos 0] D9) c. Third Integral of Equation (115) Evaluate the following integral in the complex plane of w K exp(iwx cos 0) duke i a | a | = 2ni (residue) Co en w cos an c,=Llo -PLANE The residue is is K exp (iu, x cos 6) K exp (iu,x cos §) 4 -———- — = = , 2 -m cos 8(u,-u, -m cos 6 (u,-u,) -K exp (iu,x cos 8) K exp (iu ,x cos @) ; LD, 1/2 [144 (mK) /2 cos 61 [1+4(mk) 2/2 cos 6] The integral along Co becomes zero as shown in previous cases and the integral along c. becomes 3 fo) | Gindwie K exp(iivx cos 8) Fae 2 Cy co iv-[(K) 1/22 (my 1/2 iy cos 6] e ik exp(-vx cos 6) dv 1/2 1/2 2 o iv-[(R) -i(m) v cos 6] 60 Therefore, the third integral of Equation (115) is given by exp (iu,x cos 6) exp (iu,x cos 6) ( ) du = - 27Ki ae een ue + 27Ki 73 1/2 L, [1+4 (mK) cos 0] [1+4 (mK) cos 6] +f ik exp(-ux cos 0) : ae (D.4) iv-[ (x) 1/24 ny ty cos 6] By adding Equations (D.2) through (D.4), 8, is expressed 1/2 oo i K exp(-vx cos 0) dv See we oS 2 BUS 5 tee aa!) ess 6) 8 1/2 oo Ts = aL K exp(-vx cos 9) dv ill K exp (-iupx cos 6) = dé Pape ee seo pr, - a 1/2 dé 2 (e) fe) iv-[(K) tf 2am)! 2y cos 0] fe) [4 (mK) 1/2 cos 6-1] i inlie K exp (—iu,x cos 9) ; whe K exp (iu, x cos 86) ia | 1/2 Comat 1/2 do 9 [1-4 (mK) cos 6] [1+4 (mK) cos 6] aE : m2 K exp (iu, cos 9) “ghee dé @=) a 1/2 ye o [1+4 (mK) cos 0] Dyn 58x30) We can follow the same method to transform the integral with respect to u. But, in this case, we take different integral paths: for the first and second integrals, we evaluate the complex integrals given in Case 1 in the first quadrant, and for the 61 third integral, we evaluate the complex integral in the fourth quadrant. The ex- pression for 83 similar to Equation (D.5) is given by 1/2 00 B57 4 dé { K exp(vx cos 6) dv ; au fe) fe) iv-[ (K) 2/244 ny tf Ay cos 6] 1/2 00 a — aL, ik exp(vx cos 8) dv 1 Ks exp (—iu, x cos 0) sy) ae mae Va °° air fo) fe) ive (i(k) 2! 244 Gn) t/ 2y cos 6] re) [4@nx)*/2cos 8-1] : m/2 K exp (-iu,x cos 6) any 2 dd (D.6) T 1/2 L/P 0. [1-4 (mK) cos 6] For By» the same procedure can be applied. But, the numerator, K, should be W/Z Wf replaced by -i[mu cos 0+2(mK) ] or if[mu cos 6-2(mK) ] with appropriate variables and poles. If we let A. =K Ba Kk WD (7) A, = -i[mu cos 0+2 (mK) ] 1/2 By = i[mu cos 6-2(mK) ] we can express for n = 3 and 4 as follows : m2 " A (-iu,9) exp(-ux cos 6) i n @ 3 ——— dé —_ TT du a on? j 1/2 1/2 2 fe) o iut[ (RK) -i(m) u cos 6] 1/2 oo oF — - 2 B (iu,8) exp(-ux cos 8) du A _(u,,6) exp(-iu.x cos @) at n ill gh) 2 Sa a { ei Ol Dare eo pe a ea, Tay ee ea ean aa 277 } 2 iu-[ (RK) -i(m) u cos 6] a [4 (mk) 2/2 Gos GE (D.8) (cont. ) 1/2 W/2 : A (uy 9) exp (—iu,x cos 6) i BL (u,,9) exp (iu, x cos 6) += So — dl - = 6 oe ee di T 1/2 1/2 T 1/2 1/2 o. [1-4 (mR) cos 0] fe) [1+4 (mK) cos 0] 1/2 : . B(u,>9) exp (iu,x cos 6) ees ooo dd atone pe 22510) (D.8) 1/2 We fe) [1+4 (mK) cos 6] n/2 2 AL (iu,6) exp(ux cos 86) ete { af rece. Sayee wall p ene yf fo) GOs 12,5 (mg) u cos 6] 4 m2 rs B (-iu,8) exp(ux cos 86) ee a Sn a eee m7 Lf PD z fe) o iut[ (x)? Ang (Gs) u cos 6] 8 é A (u_,8) exp(-iu,x cos 8) m/2 A (u,,9) exp(-iu,x cos 6) iL ia Soak P il i naw P iL a= oo Gd. HS = a —,— _ 0 (0.9) T 1/2 WYP T 1/2 1/2 fo) [4 (mK) cos 6-1] om [1-4 (mK) cos §] fore 3 S O) The first two double-integral terms in Equations (D.8) and (D.9) can be reduced with substitution of u cos 68 = v (Reference 17) to: ine =) andy xa Oho A, = By =K 63 w/ 00 ao) AAG ae || 40 I K exp(-vx) sec 0 dv 3 P 1/2 1/2 2 Oo Oo v] iv sec 6+[ (K) -i(m) N 8 T/ ns cid { 40 K exp(-vx) sec 0 dv i fe) 7 RS Ceo a -S | = Se || dé steele ey er ae rie ae 4 Te 4 1/2 eS ov +1 (Ki! i(m) 2/24] eons fo) fer +[ (K) 1/2 ~i(m))/2y] } The denominator can be written as 2 1/2 1/2 : v + [(K) -i(m) vals m, - at m, (D.10) where 2 m, = (eon) - (4mK-1) we ily/ 2 2 m, = 4 (mK) v (K-mv) The last integral becomes 00 foe) 1/2 -vVx (m,+im,) ° K e eH) 1 2 -VxX oe | SMe we dv = Fe { ts 72° dv (ea) fo) bal Sa) o (m_+m7) Dae With the change of the variables m, = rT cos 9 m, = (sign m,) G sin 6 c @e= E os auie 1/2 (mj+m,) nies Sain 6) S A 2 (m}+m,) the numerator becomes LY/D W/2 1/2 Wy /D : 1 / 1 2 2 ps ls DED (m,+im,) = ue [(mj)+m,) +n, ] +i(sign m.,) [(m)tm,) =m, ] Finall © b d inally, g, can be expresse @ r 0D 1/2 2 se 9 L/D 1/2 Bane 3 = 12 (mj +m) +m, +i(sign m,) (mj +m,) —m, key dv 22) ap 5, (2+ ) m,+m, (D.12) Be mS 3) eho 3 < Following the same method as that of Case 1, we can derive Oo) —_ | ena a ee n m,) ney oe iy evens dv 83 1/2 ee) 1 SO ie? iL 1/2 212) atm (m24m2) ily 2 (D.13) where a 2 2 AGO s Ge) (D.14) 2 65 (1) With Equations (D.12) and (D.14), g can be written for all values of x as 3 follows - iW) I/D 2 1/2| -v |x| Gan. DD one DD e 83 = Si || es =| +i(sign ay) |enten?s ,| | eu dv TG (m,+m,) (D.15) where n, = AG? 7 Came) Gian» (D.16) When n = 4, by following the same procedure, the double-integral terms in Equations (D.8) and (D.9) can be expressed for all values of x a 2” sf 1/2 ee A Se R, (v) udm?) ‘a,| +i(sign w,) | azea3) (2) T a ov |x! —m ——— ee dv (D.17) : eee I 52. where R, (v) =) 2 (mk) 2/2 - mv sign x (D.18) The rest of the terms in Equations (D.8) and (D.9) can be further reduced to the forms that are more useful for numerical computations. For x < 0, let the last term of Equation (D.9) be m/e A (u,,9) exp (-iu,x cos 6) (CD). at ee eee dé (D.19) oaee Th yf wie 6 [1-4 (mK) cos 6] AG 66 where A, (u, 59) = K and A 6) = -i[mu, cos 6+2 (mk) 2/2} Gere With the change of the variable 2 mu, cos § = kk (D.20) where uy is given in Equation (108) and 1/2 1 — 2¢mK)!/2 —p-4 mx) /2} for (mk)1/2 < 0,25 a i 2(mk) 2/2 for (mk)-/2 > 0,25 (D.21) (2) Bn is transformed as ’ 1 kk (@y me oe ee s2¢a1)/?]| k,k & i a sete ee L — @XDp le —— «) dk (D.22) {2 | [k, k+2 (mK) 2 In the numerator of Equation (D.22), the first term is for n = 3 and the second for n= 4. For x > 0, to the fourth, fifth, and sixth terms of Equation (D.8), we sub- stitute the following variables 2 mu, COs 9 = kjk (D.23) 2 mu, Cos os k,k (D.24) 2 mu, COs C= kk (D.25) 67 where ko» ky and k3 are given by 1/2 ke 2 le 2GnK)e 2b flee 21) eerucnk 2! 2a ones = 2(mx)1/2 for (mk)2/2 > 0,25 1/2 Iy/2 1+ 2(mk) 1/2 + [1+4 (mk) ] = Il I/D yl/2, 1+ 2(mk) 1/2 — [1+4 (mK = Il (2) With these changes of variables, Bo for x > 0 becomes kik 2ik K or fea LG oy kak (2) = 2 2 ex -i : x] dk g. a oe Maa he ie 2m il [k,k+2 (mK) ] -4k5k , ok k,k au es eel k,k = ae Z 1/2 exp \i ose dk i rs k, k-2 (mk) ies ae kik 2ik, E K or ales 2 (mx) 1/2 | kk rs le a 2 ie Ae (: om */) ak TST thegee2¢mey?/2y 44242 3 a With the change of the variable ak COS 0 ei et? (mk) 2/2 68 (D.26) 27/9) (D.28) (D.29) (D.30) The third term of Equation (D.9) for x < 0 becomes : — (Siegal Tv A (u, 8) exp (-iu,x cos 6) iu a 2 1/2 2 [4 (mK) cos 6-1] I/D 1/2 il ae K or + (4mK-k") -i(* +2 (nx) 2] dk w - 1/2 1-2(mx) 1/2 (Fee? sa tee2000 7/7 ret? (mk) 2/23 =) exp I oe (AG) So? | =| es: The third term of Equation (D.8) for x > 0 can be transformed to the form similar to Equation (D.31). Because of the lengthy derivation, we omit the intermediate steps. With the change of the variable in Equation (D.31) i 1/2 ae “17 eS Ea ae 4 (mK) -l1 4K) -l ae can be finally expressed as follows for all values of x TL = i 1/2 = Oue 2f NGO exp lH qm [1-2 (mK) | exp )- 30 sia x ie gs! [e(1-K) (Sk+1) (642) 12/7 bilan hepa ile x exp { -[6(1-k) (6k+1) ] ona (D.33) 69 where N, =K (D.34) -4 (CLD CEMD I 2 Catea -i [ok+142 (mK) 2/2 Zz i ] Then, B3 and By, given in Equations (115) and (116) can be expressed with n = 3 and 4 Gl) (2) (3) BST enh ce anaen (D.35) where gee is Equation (D.15) for n = 3 and Equation (D.17) for n = 4; eg?) is Equation (D.22) when x < 0 and Equation (D.29) when x > 0; and BO) is Equation n (D.33). For the details of these derivations, see Reference 15. In this reference, there are some errors in signs. As in special cases, when m = O or K = O, we can integrate these equations explicitly. 1. Steady Forward Motion, K = 0 Because of K in the numerator, ao = eg 7 Es qo From Equations (D.10), (D.16), and (D.18), m, = PGA) m, = 0 R,() = - mv sign x 70 By substitution of these relations into Equation (D.17), g 2 In Equation (D.36), the second kind. When K = 0, kj = k, For x > 0, from Equation 2 = 0 and k, = k, = 2. Therefore, for x < (D.29), ca is given by DtO7 -ik 4 H—, ikx 1/2 eae (- m ) ws 1 Oi) AA] 2 Dried T de [ (2k) OMe 4k] S| A yr Tale Ny becomes ° |x| -v|x ae { [-mv sign x] (2m, yee ween e 7 dv 2 (21) iL fe) [oe) -v |x| - sign x dv 277 ae 7/2 0, 8) Qyas (D.36) HG) is the Struve function and Y,@) is the Bessel function of (D.37) in Equation (D.31) is transformed from the variable 8 1/2453 te 1/[4(mK) 1/2 e e) becomes zero. Therefore, when K = 0, By, ) becomes zero. By adding Equations (D.36) and (D.37), By, is given by The integral limit of Ae that is between O and 0. = aoe o {1/[4 (mK) ] is larger than 1, 1 SN Fal er ag ee (=) RR [2+ sign x] Mee ar (D.38) Finally, from Equation (114), G(x, 0, 0) is given 194 x : (al) 8 (x, 0, 0) bana Enea H, (3) +(2+ sign x) Yo ey (D.39) 2. Pure Oscillation, m = 0 Because Ry» Equation (D.18), is zero, Sn becomes zero. When m= QO, ky = k, = 0 and k, = k, = 0. Therefore, gpa Equation (D.22), is zero when x < 0. When x > 0, as the complex argument of the exponential function becomes infinitive in the first and second terms in Equation (D.29), 7 (3) th becomes zero. The upper limit of the integral for g in Equation (D.33) origined from o. which becomes zero in this case. Therefore, there is no contribution from g, to g, when m= 0. QD Ge ie With R, = K, m, = K" + v', and m, = 0, 83 is (LAY ake 1/2 env ls Sees eles (2m,) SS co le i — a5 exp (-Kt |x|) dt 2, A (t +1) 5 [a Cclx|) - x (K|x|] (D.40) YZ With A, = K and we K from Equation (115), as m becomes zero, Bsr is evaluated from Equation (D.9) for x < 0 1/2 (2) at ; B3 som K exp(-ikK cos 8) d0 for x < 0 (D.41) fo) Because uz = K, and uy and uy, disappear, a for x > 0 is given from Equation (D.8) 1/2 (2) al ‘ 83 ah K exp(ikx cos 8) dé for x > 0 (D.42) fo) Ce uses For all values of x, 83 is given by 1/2 es = [ exp (iK|x| cos 6) dé T fe) en (K|x|)+ig_(K|x])] (D.43) 2 fe) fo) ; (3) Because of o. =O) 83 LS ZeTOR a tanailala. 83 is given G, = 8, - = (-H,(K|x|)-¥ | x|)424, |x|] (D.44) 73 " war = alae sua! a endif | OMe pier ver a oy is rf we ef Sa va ve i ‘ Ad ni vers } ? Oe if be Sealy i THE ORE LD ie aed aR ie Oe kt a a at Nn ae by , yu y ; R Be la a ; i * ; f LAr A ae ee cettern ty FE TT) te [oe at Ra We ‘OO ie ‘| | _ yee a i a - ' 7 beh &, i at ae ¢ A Tut By | x E y ; ' bf ‘i ys ides om * - i“ Tha ah : Pa ' ; A oye ‘ ipe,) : i ie i ‘i REFERENCES 1. Lee, C.M., "Theoretical Prediction of Motion of Small-Waterplane-Area, Twin-Hull (SWATH) Ships in Waves," DTNSRDC Report 76-0046 (Dec 1976). 2. Hong, Y.S., "Improvements in the Prediction of Heave and Pitch Motions for SWATH Ships," DINSRDC Report SPD-0928-02 (Apr 1980). 3. Newman, J.N., 'The Theory of Ship Motions," Advances in Applied Mechanics, Vol. 18, pp. 221-283 (1978). 4. Tuck, E.0., "The Steady Motion of a Slender Ship," Ph.D. thesis, University of Cambridge, Cambridge, England (1963). 5. Tuck, E.0O., "The Application of Slender Body Theory to Steady Ship Motion," DINSRDC Report 2008 (Jun 1965). 6. Tuck, E.0O., "Lectures on Slender Body Theory," University of California, Berkeley (Jul 1965). 7. Newman, J.N. and E.O. Tuck, "Current Progress in the Slender Body Theory for Ship Motions," Proc. Fifth Symp. Nav. Hydrodyn, ACR-112, pp. 129-167, Office of Naval Research, Washington, D.C. (1964). 8. Ogilvie, T.F. and E.O. Tuck, "A Rational Strip Theory of Ship Motions: Part I," Report 013, Dept. of Naval Architects and Marine Engineering, University of Michigan (1969). 9, Wehausen, J.V. and E.V. Laitone, "Surface Waves," In Handbuch der Physik, Vol. 9, pp. 446-778 (1960). 10. Ursell, F., "Slender Oscillating Ships at Zero Forward Speed," J. Fluid Mechanics, Vol. 14, pp. 496-516 (1962). 11. Ursell, F., "On the Heaving Motion of a Circular Cylinder on the Surface of a Fluid," Journal of Mechanics and Applied Mathematics, Vol. 2, pp. 218-231 (1949) . 12. Salvesen, N. et al., "Ship Motions and Sea Loads," Transactions of the Soc. of Naval Architects and Marine Eng., Vol. 78, pp. 250-287 (1970). 75 13. Troesch, A.W., "The Diffraction Potential for a Slender Ship Moving Through Oblique Waves," Report 176, Dept. of Naval Architects and Marine Engineers, University of Michigan (1976). 14, Lighthill, M.J., "Fourier Analysis and Generalized Functions," The University Press, Cambridge (1970). 15. Joosen, W.P.A., "Oscillating Slender Ships at Forward Speed," Netherlands Ship Model Basin Publication 268, Wageningen, The Netherlands (1964). 16. McCreight, K.K. and C.M. Lee, "Manual for Mono-hull or Twin-hull Ship Motion Prediction Computer Program,'' DINSRDC Report SPD-686-02 (Jun 1976). 17. Gradshteyn, I.S. and J.M. Ryzhik, "Tables of Integrals, Series and Products," Academic Press, New York and London (1965). 76 Copies 13 12 INITIAL DISTRIBUTION Copies CHONR/438 Cooper 1 NRL hh 1 Code 2027 1 Code 2627 USNA 1 Tech Lib 1 Nav Sys Eng Dept 2 1 Bhattacheryya 1 Calisal NAVPGSCOL 2 1 Library 1 Garrison NADC iL NELC/Lib il NOSC 1 Library 1 1 Higdon NCEL/Code 131 1 NAVSEA iL 1 SEA 031, R. 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