AD-AL07 723 DAVID W TAYLOR NAVAL SNIP RESEARCH AND DEVELOPMENT CE-ETC F/9 13/10PREDICTION OF MOTIONS OF SWATH SHIPS IN FOLLOWING SEAS. (U)NOV W1 Y S HONG

UNCLASSIFIED DTNSRDC-81/039

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DAVID W. TAYLOR NAVAL SHIPRESEARCH AND DEVELOPMENT CENTEA

PREDICTION OF MOTIONS OF SWATH SNIPSIN FOLLOWING SEAS

b" "DTICELECTENOV 2 4 1981

C, E

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COMMANDERFTECIHt4CAI. DIRECTOR

01

SYSTEMSDEVELOPMENTDEPARTMENT 1

COMPUTATION.

SHIP ACOUSTICS AUXILIARY SYSTEMS

DEPAfRMENT DEPARTMENT

F 19.

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DTNSRDC-81/039 1,>-A/o ',10 .4 TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED

PREDICTION OF MOTIONS OF SWATH SHIPS IN Final

FOLLOWING SEAS 6. PERFORMING ORG. REPORT NUMBER

7. AuTHOR(s) B. CONTRACT OR GRANT NUMBER(*)

Young S. Hong

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT. TASKAREA A WORK UNIT NUMBERS

David W. Taylor Naval Ship Research Program Element 61153Nand Development Center Task Area SR 0230101Bethesda, Maryland 20084 Work Unit 1572-031

11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

Naval Sea Systems Command November 1981

Washington, D.C. 20362 13 NUMBER OF PAGES

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18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse aide if necessary antd Identify by block number)

Small-Waterplane-Area, Twin Hull (SWATH) ShipHeave and Pitch Motions in Following SeasUnified Slender-Body Theory

20 ABSTRACT (Contitsue 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 JAN 7 1473 EDITION OF I NOV65 IS OBSOLETES'N 0102-LF-014-6601

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(Block 20 continued)

while damping coefficients are slightly less than two-dimensionalresults. There is improvement for heave motion for a limitedrange of frequencies, but the pitch motion is generally poor. Toimprove these results further, the two-dimensional potential shouldbe solved by the multipole expansion method rather than by thePrank close-fit method.,

fAccession For

17TTS

UNLSSFE

-SEURTYCLASIICTINCLSF TIAE"Dt Etrd

TABLE OF CONTENTS

Page

LIST OF FIGURES . iv

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

NOTAT I ON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

ADI NI STRATIVE IN FORMAl' ON . . . . . . . . . . . . . . . . . . . . . . . . . 1

I NTRODUCT I ON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

BOUINDARY-VAIAUIF PROMFB . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

VEIOCI1TY POTENTIAl OF S'TEA\Y FORWARD MOT'IION ....... ... ................. 5

UNSTEADY POTENTI Al, IlE ITO OSIL.ATION ......... .. .................... 5

SOIUTION o1 TIE POTENI'II F'UNCTI ON WITIl THESlENDER-BODY ASSUMPTION ........... ......................... 10

O1t tr Prob I . ............. . ............................. I1

Inner Problem ............. ............................. 11

MATCIl Nt; ................ . . ................................. 13

INNER SOLTI, ' ON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

DI FFRACTION POTENI'A1 ............... ........................... 1 (

IIYDKRODYNAMI C FORC:.. . . . .. . . ........ ........................... 18

ADDED MASS AND DAMPINt; ............ .. .......................... 1q

EXCIT'll NG FORCE... ............. ............................. 21

FoU T' I'ONS OF: MOTION ............ ........................... 24

R:I.AI'ION of.' TWo-DI ,FN: IONAI. SOL RCE STRENGTIIS BETWEEN TiltE,

FRANK C1.OSE- , IT METHOD AND 1,l.TI Ptl0,F EXPANS ION METHOD. ... ............ 25

P'RI\VATION OF" TIlE KERNI:l, FUNCTION ......... ...................... 28

A I 1 (m / ' 1 . ........... ....................... 29

CASI ' - 4(mK'/) ..

CASE I - co.; 0 • .. ............. . . ............................ 2q

R'SII['S AND) DISCUSSION ........... ... . ........................... 35

SUMMARY AND CONCISIONI. ................ ........................... 45

ACKNOWI.EID;IMENTS. .............. ............................... 47

ill

Page

APPENDIX A - DERIVATION OF G 71)(y,z) WITH EXPONENTIALFUNCTION..............................49

APPENDIX B - DERIVATION OF G 2D (y,z) FOR SMALLARGUMENT..............................53

APPENDIX C - FOURIER TRANSFORM OF THE THREE-DIMENSIONAL GREEN FUNCTION ..................... 55

APPENDIX D - DERIVATION OF G 3 (x,0,0) FOR NUMERICALEVALUATION... ..........................

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES

1 - Coordinate System..............................2

2 - Limits of Integration...........................23

3 - Sectional Coordinate System.........................26

4 - Integral Path when 4(mK) 12Cos 0 < 1.....................29

5 - Integral Path when cos 0 < 1........................30

6 - Added Mass and Damping Coefficients of SWATH 6A inFollowing Seas at 20 Knots...........................37

7 - Added Mass and Dam~ping Coefficients of SWATH 6C inFollowing Seas at 20 Knots.........................38

8 - Added Mass and Damping Coefficients of SWATH 6D in

Following Seas at 20 Knots...........................39

9 - Exciting Force and Moment of SWATH 6A in FollowingSeas at 20 Knots................................41

10 -Exciting Force and Moment of SWATH 6D in FollowingSeas at 20 Knots..............................42

11 - Motion of SWATH 6A in Following Seas at 20 Knots. ............. 43

12 - Motion of SWATH 6C in Following Seas at 20 Knots. ............. 44

13 - Motion of SWATH 6D in Following Seas at 20 Knots. ............. 46

LIST OF TABLES

Page

1 - Added Mass and Damping............................20

2 - Principal Dimensions of SWATH Ships.....................35

V

NOTATION

A Amplitude of incoming wave

Ajk Added mass coefficients

B.k Damping coefficients

jkC.j Coefficient for longitudinal interaction

Cjk Hydrostatic coefficients

E1 Exponential integral

F. Excitation force and moment

f* Fourier transform of function f

G Green function

g Gravitational acceleration

H Struve function0

I Mass moment of inertia

1/2

i (-1) Imaginary unit

J Bessel function of the first kind0

2

K = - Incoming wave numbero g

2

K = Wave number of frequency encounterg

L Ship length

N Ship mass

m(ml,m2,m 3 ) Normal vector due to the steady forward potential

vi

m Constant valueg

n(nl,n 2,n3 ) Unit normal vector directed into the fluid

P Pressure

q Three-dimensional source strength

S Area of immersed cross section

U (-U,0,0) Steady forward velocity vector

U2 Velocity vector due to oscillation

V Volumetric displacement

Y Bessel function of second kind0

Heading angle of incoming wave with respect to thex-axis (3 = 0 is the following wave and = 180 isthe head wave)

y = 0.577... Euler constant

ALength of incoming wave

Complex amplitude of ship motion

a. Two-dimensional source strength

=l + 2 Total velocity potential

Potential due to steady forward motion

2 Unsteady potential due to oscillation

$., j = 3,5 Two-dimensional potential for heave and pitch due toharmonic motion

., j = 3,5 Two-dimensional potential for heave and pitch due tosteady forward motion

vii

Encounter frequency

(Ij 0Frequency of incoming wave

Incoming wave potential

SO 7 Diffraction potential

Pj=1,2,.. .,6 Velocity potential due to motion of the ship withunit amplitude in each of six degrees of freedom

viii

ABS TRACT

To predict motions of SWATH ships in following seas,especially when the encounter frequency is small, unified

slender-body theory has been applied. The longitudinalinteraction term is computed and added to the results of

the strip theory as a correction term. The added masscoefficients are computed to be much larger than thosefrom strip theory, while damping coefficients areslightly 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. Toimprove these results further, the two-dimensional potentialshould be solved by the multipole expansion method rather

than by the Frank close-fit method.

ADMINISTRATIVE INFORMATION

This study was performed under the General Hydronechanics 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 developed1*

by Lee. 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 terms.2

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 fa:-t as the wave celerity, the encounter frequency be-

comes very small and application of strip theory is not va2lid. 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.

Newman 3has 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.

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 Newman3 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 PROBLEMWe define two coordinate systems: the first, 0 X Y Z is fixed in space and

the second, oxyz, is fixed with respect to the ship which moves with forward speed,

P, in the positive 0 X -axis. The oz-axis is directed verticallv upward and the0 0

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.

z

Figure I - Coordinate System

2

Let the surface of the ship be specified mathematically by the equation

y = h(x,z,t) (1)

and let the free surface be given by

z = (x,y,t) (2)

Then, the fluid motion can be expressed by the velocity potential 4(x,y,z,t)

t(x,y z,t) = l (x,y,z) + 2 (x,y,z)e'"it (3)

where = -Ux + o(X,y,z), the potential due to steady forward motion, and 2= un-

steady potential due to oscillation.

Equation (3) must satisfy the following conditions:

1. The Laplace equation in the fluid domain:

xx yy ¢ zz (4)

2. The dynamic free-surface condition:

1 2 2 2 1 2+ gr + 2 +( + ) = U for z = ((x,y,t) (5)

3. The kinematic free-surface condition:

x + y - + t = 0 for z = r(x,y,t) (6)

X'x y y

4. The kinematic body condition:

h y + zh + h for y = h(x,z,t) (7)

3

5. The radiation condition; that is, the energy flux of waves associated with

tho 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

coudition is given by

U2+2 2,,)g oxx + oz + 2z g '2 g 2x+ g 12xx e2ThU 0 (8)

l1' body condition can be expressed as follows:

(V o 2 (9)

0 2 + U o U 2

where U (-U,0,0), steady forward velocity vector

U, = velocity vector due to oscillation

n = unit normal vector directed into the fluid domain

The solution of Equation (4) with two boundary conditiors, Equations (8) and (9),

iS nonlinear between o and ,z. The usual practice for solving this problem is to0Z

separate 5o and 2 ; and evaluate these two potential functions independently. By

: ubstitution of Equation (3) into Equation (4) and by separation of to and 2 in

Fquations (8) and (9), we have two sets of partial differential equations.

oxx oyy ozz

+ = (11)g OXX oZ

(V:o+Uo ) • n = 0 (12)

2xx +0;jy + -- 0 (13)4X 2ZZ

4

2 22iU U2

- + +- = 0 (14)2z g 2 g 2x g 2xx

(V 2 -U2 ) n = 0 (15)

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 has4-6

been discussed in depth by Tuck. The solution of two-dimensional potential is

given by

2 log r + - log r I - Ux (16)

Uwhere a = - S'; the strength of source

7T

1 2S = I nr ; area of immersed cross section

2 o

-2 2 2r = (Y-0) + (z-()

-2 2 2rI = (y-n) + (z+)

Here, (y,z) is the point where the potential is solved and (],&) 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 Newman3

5

and Newman and Tuck7 to solve this problem. Because the unsteady motions are assumed

small, the potential 2 in Equation (3) can be decomposed linearly

j=6

2 = Po + O7 +, jF i (17)

j=l

where P = incoming wave potential0

07 = diffraction potential

SO. = velocity potential due to motions of the ship with unit amplitude in

2 each of six degrees of freedom

= amplitude of motion in each of six degrees of freedom

The diffraction potential, SP7 ' satisfies the following condition

( So+ ) =0 for y = h(x,zt) (18)~n o 7

and the velocity potential, S.j (j=l,2,...6), satisfies the body boundary conditions

P. = -iwn. + m. (19)

.r, t Lt' ComponentS of the unit vector are defined as

(nl,n 2,n3) = n (20)

(n 4 ,n 5 , n 6 ) = (,xn) (21)

8and m. are defined by Ogilvie and Tuck as

(mlm2 ,m3 ) m = -(n') V 1 (22)

6

(m4 ,m5,m6) = -(n.V) (X×V 1 ) (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

(xxn) = (yn3 -zn2 , -xn3 , xn2) (24)

M - n2 -y+n 3 -zz @I (25)

(m 4 ,m5 ,m 6 ) = (-n 2 iz+n 3 ly+m 3-zm 2 , -xm3+n 3 xm2-n 2 ) (26)

By substitution of Equation (17) into Equation (14), the free-surface condition for

the potential, pj(j=O,l,...7), is given by

2 " 2g p.z 2 ci ' + 2iwU p. + U 2 = 0 for z = 0 (27)g jz I jx JXX

With the assumption of

2

Sjz-W 'P . = 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

Laitone 9 for a single-hull body as

G2 (y z PV e cos ky dk - ieKz cos Ky (29)G2D(Z Tr f K - k- coKy29

0

7

Equation (29) is the potential function of a unit source at the origin where K =!9

2 /g. With a change in the integral path, the Green function can be expressed as

kz

G(yz) e Cos kydk (30)2D(' = J K - k

0

The contour of the integral path is around the pole from beneath. With a change in

the variables Equation (30) can be written as

1 Kziy.KiKYi 2 Re T e +iKYE (Kz+iKy)) - Kz+iK y l (31)G2D =T-

The derivation of Equation (31) is given in Appendix A. The exponential integral is

E (u) and is defined as

~C1

E1 (u) = e dt (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,

Fquiation (31) can be expressed as (see Appendix B)

G2D =1 [y+Q.n(Kr)- i] (33)

, r,,re =0.577... which is Euler's constant and (r,O) are polar coordinates such

that v r sin * and z -r cos 0. For large values of Klyl the asymptotic approxi-

mation of Equation (31) represents the outgoing two-dimensional plane waves in the

formIf

G2 = ieKz+iKlIy for Kly! >> 1 (34)_D

The three-dimensional Green function which satisfies Equation (27) is given by

Wehausen and Laitone9 as

7TG(x,y,z) = -i du gu ezu exp (-ixu cos e-iyu sin e) dO (35)

412 gu - (W+Uu cos e)2

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, -1/47, for convenience. With the change of variables, u cos 0 = k, u sin 0 = 9

and udud6 = dkd, G(x,y,z) can be rewritten as

gez(k 2+2 ) I/2 yG(x,y,z) = - f e dk 2 12 (36)2-7 ..... -dk2 g(k2 +Z2 ) /2 (w+kU)2

If we define the Fourier transform

f*(k) = f(x) e ' dx (37)

and the inverse Fourier transform

f(x) = _2 f*(k) e- ik dk (38)

then the Fourier transform of the three-dimensional Green function is given by

3 2 2 - 2/1 -gz (k 2+q, 2) 1/ -iyz-C*(y,z;k) f -~ 2-Q 2 )1/2 -(+k)2dZ (39)

- g(k ) - (-kU)

9

r°

The value of G* for k 0 reduces to the two-dimensional source potential, Equation

(30), as

i f eZ eiYz

C*(y,z;O) = - -e dZ (40)

JR1 -K

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

Ursell I0 in the special case U = 0.

For the case U > 0, the asymptotic expansion is given by (see Appendix C)

G*(k,K) = G - 1 f*(k,K,K) (41)2D 'ii

where G 2D is Equation (33) and

f*(k,K,K) = n(2K)-i7 + 7G3 (42)

SOLUTION OF THE POTENTIAL FUNCTION WITH THESLENDER-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 qW(x),

then the outer solution is expressed in the form

'J = fL qj (E) G(x- ,y,z) dE for j = 3 and 5 (43)

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

.= qj* G*(y,z;k,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)

q G 1 f*q (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

form3

s. = + C((x)S()j+j) (46)2 j 21 12

where C.(x) is a function to be determined by matching with the outer solution and

p(S) is the strip theory solution which can be expressed as follows

J1

(s) = + , (47)

On the body surface, 4. and 4. satisfy the following conditions

jn = -ion. (48)

and

4j m. (49)Jn 3

In Equation (46) P is the conjugate of 4.. The solution of 43 for heave motion can

be expressed by

C3 Gc +Z ,~ [cosm9 K Cos (2m-1)6 (50)3 3 2D + #m L r2m 2m-l 2m-1 Im= r r

where G2D is defined by Equation (29), o3 is the two-dimensional source strength at

the origin, and the terms under the summation are the higher order multipoles which11

form the wave-free potentials. Equation (50) was first introduced by Ursell. A

:iimilar solution for 43 can be given by

0o

[2 Fcos 2me K cos (2m-l)e(

3 = 2D m L2 m -2m-I 2m-i j (51)

m=1

For pitch motion we can have similar expressions to Equations (50) and (51) by

substituting , 9rm a3, and am for -xo3) -XXm, -xa and -xae, respectively. From

Equation (50) the outer approximation of the potential 4. is

4j = . G2(Y,Z) (52)

12

A similar expression can be given to the potential 4j:

= j G2D(Y,Z) (53)

By substitution of Equations (52) and (53) into Equation (46), the outer approxi-

mation of P. is

G. 0. G2D + 0j G2 D + C(x) (ajG2D+-0G2D)

j 2D + + 2D + W j 2D + C(x) (G2D-G2D) (54)

From Equation (34), the last term of Equation (54) is

- Kz-iKjyj Kz+iIly[ KzG - G = i(e +e z ) - 2ie cos Ky - 2i2D 2D

By substitution of the above expression into Equation (54), the outer approximation

is finally given by

S= [o+o+CW(x) (a .+.)] G2D + 2i C.(x) o. (55)

The Fourier transform of Equation (55) becomes

[a* = [oj+oj+C.(x) (oj+o)0 * 2D + 2i[Cj(x)QjJ* (56)

IMATCHING

The inner and outer solutions are matched in a suitable overlap domain to

determine the unknown source strength, q., of the outer solution and the coefficientsJ

C. in the inner solution. From Equations (45) and (56) we have the following

relation

13

q G f*q* [(.4c .+C (aY +0 2DG~~1 2 + 2i(C CT) (57)

Equating the factors of GD gives a relation for the source strength

q [0 .+9.+CG.(a. i+0.I for j 3 and 5 (58)

Equating the remaining terms in Equation (57) gives

=~ -2nfi [C C..* (59)

The inverse Fourier transforms of Equations (58) and (59) can be expressed as

q. +0 fc +C (C) .+0.) (60)

27i[ ' - F) f(x- Ddi. (61)

L

The kernel f(x) is the inverse Fourier transform of Equation (42). Elimination of

C.i gives an integral equation for the outer source strength

qWx V .) f (x-,-)d,- (ao .) (62)

INNER SOLUTION

By substitution of Equation (61) for C.(x) into Equation (46), the inner so-

lution is given by

S(s) _ i (q.+. (F f(x-FDd (63)

27, L

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, a5 and a5, can be expressed as a function of 03 and o3, respec-

tively. The two-dimensional potential for pitch motion is

5= - x3 (64)

Ut5 =- -x' (65)5 3 3

From these equations, a 5 and 05 are given by

(1 =- x P3 (66)

iF

- 1 - (67)5 3 3

By substitution of Equations (66) and (67) into Kquation (62), the integral equation

for the outer pitch source strength is given hy

q x q f (X- )d~ x( ) + C, (68)q5( - ' ,] J 5 K 3 33

The inner solution for pitch motion becomes

_ = (s) ix ¢3 + 3 ) q5 (.) f(x-VI)d, (69)5 = 5 2 - 3 3f5

2703

The derivation of the kernel function, f(x), from f*(k,K,K) will be given later.

15

DIFFRACTION POTENTIAL

With a similar method to solve the potentials P. for j = 3 and 5, the diffrac-3

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 al. 1 2 and others. Newman, 3 Troesch,1 3 and others have solved the lelmholtz

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, Newman3 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)

7 () 7() (s+) (70)

where C7 (x) = a function to be determined by matching with the outer solution

= the diffraction potential solved by the strip theory7

(s))S the symmetric function of is)°

The diffraction potential, V s), satisfies the two-dimensional Laplace equation with

the boundary condition given in Equation (18). If we express the potential of the

incoming wave as

= - exp (K z+i K x cos P-i K v sin )(71)0 ,0 0 OO'

0

the boundarv condition of is given by

7n = - K (n 3 -i n2 sin ) o (72)

16

where A amplitude of the Incoming wave

- heading angle of the incoming wave: = 180 deg in a headsea and 0 deg in a stern sea

Ko = 2 /g2

If we express the outer approximation of P as7

p(S C + o ' (73)7 7 2D a 2D(

then the outer approximation of the inner solution, %07' which is similar to Equation

(54), is given by

' P7 = 07 G 2D + C7 (x) (o7+o7) G2D + C7(x) (C 2D-G2D) .7 (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

7 2 T 7 j~ 7 ~

() - i)fx~)C

7O 7= (s+ s q7 () f(x-)dE (76)

21 7

and

C _ T0i f 7 () f(x-r)d- (77)7 172rc 7 -f

17

HYDRODYNAMIC FORCES

The pressure in the fluid is given by Bernoulli's equation

P= - 0 t+gc+1 (78)

By substituting Equation (3) into Equation (78), the pressure becomes

= e (io2- h) t - pgc, (79)

Then, the forces and mon'ents with respect to the origin of the coordinate system are

given by

F. = p n. dS for j 3 and 5 (80)J JJ J

S

where S is the submerged portion of the ship hull, and n. is defined in Equations1

(20) and (21). By substitution of Equation (79) into Equation (80), the hydrodynamic

forces can be expressed by

F. = - p [iw4 -Vt . ] n dS e (81)

S

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 (Vt1 ' V 2) nj dS ff mj 2 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

r6

F j J (iin.n.)[ S i + SP7+ 'f k P dS e - i wt (83)

S k= 1

ADDED MASS AND DAMPING

We write the integral of the term under the summation in Equation (83) as

6

gj = t e-t (84)

k=1

where

tjk =- ff (iwn.+m.) [ k+ k+Ck(x) (k+ k)] dS (85)

S

The terms inside the bracket represent the unsteady potential Pk which results from

oscillation. Equation (85) denotes the hydrodynamic force and moment in the jth

direction per unit oscillatory displacement in the k mode.

The added mass and damping are defined by

2

tjk = - Ajk W 2 i Bjk W (86)

where Ajk is added mass and Bjk is damping. Both Ajk and Bik for j = 3 and 5 and

k = 3 and 5 are given in Table 1. In Table 1, is used the following notation:

3 R+ iI

3= R+ iIC3 = 3R + iC31

C5 =C5R + it51

n5 -xn3 ; 5 = -0 3

Um5= -(Un 3+xm 3); t5 = -i 3 X 3

19

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 1.

TABLE 1 - ADDED MASS AND DAMPING

=A - ffn3 +1 dS + ffmR dS + P- m dS - k n ; dSA33 3 + mA)fm 3 R n f 3 1

2off C dS- (12J n3nis3i dS+ Z2j-fm3tRC3R d S

B33 = PJn 3 i R dS + L m3¢ I dS + PLffm 3 I dS + p n3R dS

+ 2p n3 C dS + 2rfm4 c dSjj 3 R 3R W m3 131

A3 5 = -ff xn3 dS + lff m3 I dS - Lff m3;R dS - jfxmf,3 dS

+ jn 3 * R dS +Qf xn 3 dS +- 2f xn)iRC3 i dS xm3 tRC5R dS

2 f3Ol f f ff 3 1 ff 3R5

B = 4 f xn dS -ds xm3¢ dS - £ xm I dS

35 22 fJ3 R 2ff 3 1 ffJ3R

xn dS - 11- n3 d S -2fxm¢ d S - dS53 3~ 1xn3l R 2I 3 Rn3 R

A ff ff ffPe 2ff =

- xm 3 R d S + ndS + Sffxn dS + }Jxn 3 RC3 1 dS - JJ n3 RC3R dS

2,J xm3RC dS

20

TABLE 1 (Continued)

B53 Pffl3n dSl :*f~n; dS - P dS n P!jn~ dS

53 f x 31 d - pf Xnt S-2ff Xn~CRd Pff f3GRd

55=-pf2~dS - 2 n dS 2k ~xn 3 I dS - 1 xm n dS

+41- f=~ 3 R dS + R ff 3~yt dS 3R dS -f f3 R 3R

2pLff ( CR dS +2~ ~CRd+2 3l XlR 31 ~ x

= ~J 2 ~~ dS +2ff n3~ dS + 2-a-ff xfln dS + UIIc~ dS

+-'f xm 3 d S + JJx~ dS+ f x nR d S 1-- m2 pf d~RS

+2,Uff xn dS + 2pf x2m3 RS dS

TB e xcittin foce are ive-2 n byd Eutin (83 as +P m

=f x -d 2f (iom [ + ~pff x a S +t (87)2n

3 ff 3 ~~13R f 3R5

The integral containing p in Equation (87) is the Froude-Krylov force. If we

substitute ' 0for 02 in Equation (82), we have

-ffni.o ds = - U D n. dS (88)

Here, we have neglected the higher-order terms. With Equation (88), the Froude-

Krylov force is given by

f ' = - n ( i( -U 2-) p dS e

s

= -iPo ° ff n. 'P dS e (89)s j 0

where P0 is given in Equation (71). The heave excitation force is given by0

f 2pgA e 0 cos (KlX-wt) cos K2y dxdy (90)

and the pitch moment is

2pgAj +e x cos (K x- jt) cos K dxdy5 jf z 1j1

(K z-1) sin (K x-wt) cos K2 dxdy (91)

20

where K = K cos , K = K sin , and the integral limits z1 and z2 are given in

Figure 2.

22

-2 . .- - .. . . . .

zi

zi

Figure 2 - Limits of Integration

By substitution of Equation (70) into Equation (87), the diffraction force is given

by

h. = -0 I (icnj ) +m 4(+)+c ] dS e-iwt (92)jj jf j 7 7s sS

We first substitute Equations (48) and (49) for n. and m, and apply Green's second

theorem to the term containing (s) The diffraction force can be written

h. [ 0 ff (j- ji)dS-P f (iwn i Mji)C 7( s+ s) dS e - t (93)

S S(s)+ etts-

where C is given by Equation (77) and s is the symmetric part of that sat-7 5 ' 7isfies the two-dimensional Laplace equation and the boundary condition, Equation

(72). By substitution of Equation (71) for P0 , the diffraction heave force is

expressed by

(OKzh3eimt 12rw 0 A dx e n3 cos K2 Y-2 sin g sin K2 ) 03-;3) dZ

c

- 2p C7 (x) dx J (iwn3+M3) Re[ s] dZ (94)

c

23

and the diffraction pitch moment is

h e ii2Pw A f(+ e dx e n3 cos K2 y-n2 sin B sin K2Y 3 dk

C

+ 2pfC 7 (x) dx f (iwn 3x+Un3+m3x) Re[ s ] dR (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 1.

EQUATIONS OF MOTION-iwt -iwtF3

If we let a 3 = 3 e a = 5e , the total exciting forces F3, and tht

total pitching moment F then, the equations of motion can be expressed

(M+A3 3 ) 3 + B33 3 + C3 3 a3 + A35 5 + B 355 + C 35a = F3

(96)

(I+A 5 5 ) 5 + B5 5 5 + C5 5 %5 + A5 3 3 + B53 3 + C 53A3 = 5

where M is the mass of the ship and I the mass moment of inertia about the y-axis.

iTe hydrostatic coefficients are given by

C33 = pg A33 wp

C35 = C53 - Pg[M wp-xgA wp (97)

2C55 Pg[IL+Awp (x g-x) + (KG-KB) A

24

where A = waterplane areawp

M = momentwpx = longitudinal center of gravityg

x = longitudinal center of flotation

L = moment of inertia of A with respect to the longitudinalcenter 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

F3 -icit F5 -i(JtWith substitution of F = F33 e and F = F55 e , Equation (96) can be

expressed as complex algebraic equations:

2(M+A 33 ) + C 33 - iwB 33 - (2A35 + C5 -35 35 3 F33= (98)

A53 + C 53 - 2 (I+A55) + C55 - iWB5 5 FS} 5 F 5 5 (

RELATION OF TWO-DIMENSIONAL SOURCE STRENGTHS BETWEEN THE FRANK CLOSE-FITMETHOD AND MULTIPOLE EXPANSTrN METHOD

The potential function given in Equation (5u) 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)

= Kz+iK-y (99)

where o is the source-strength at the origin.0

25

For the Frank close-fit method, we have the potential as follows

= a cG 2 D d9 (100)

c

where a is the complex source density at the ship's contour and G2D for the twin-hull

body is given by Wehausen and Laitone9

2D = Re - log (y+iz-n-iC) - log (y+iz-n+iC) + 2PV e Kk dk

- i Re [e- iK(y+iz-+i )]

+ Re-1 log (y+iz+n-iC) - log (y+iz+n+ir) + 2PV K-k dk

- i Re [e- iK(y+iz++i )] (101)

where (y,z) is the point where the potential is sought and (p,C) is a source point

(Figure 3)

t 2Y

1-77. 0} c- -_ c + U u(17 0

0 (y, z)

Figure 3 - Sectional Coordinate System

26

when jy( is large, the logarithmic terms cancel out. Then, Equation (101) can be

expressed in a manner similar to Equation (30)

e + cos k(y-) ek(Z+)cos k(y+)dk (102)

Ckz~)o dk + dk K-k102)G2D =7T K-k 7 K-k

0 0

Equation (102) can be further transformed with the exponential integrals as shown in

Equation (31). As jyj becomes large, these exponential integrals become small.

Finally, as iy becomes large, Equation (102) can be expressed as

G2D = iK(z+ [e iK y- i + e iK ]y+r1 (103)

Then, the potential function is given by

= 7[-ie K (z+ ) e iK y-r d

c

+ f c [-ie K (Z+ ,) eiKly+,n ] d

c

Kz iKy f K= -2i e e f (I e cos Krv d (104)

c+

Equating Equation (99) with Equation (104), we have the following relation between

and

2 o e cos Kn d (105)

c +

27

This equation simply states that the corresponding source at the origin is obtained

by integrating the distributed source along the contour with the weighting function

of 2e K cos Kp.

DERIVATION OF THE KERNEL FUNCTION14

The inverse Fourier transform of Equation (42), we can express f(x) as

f(x) = n(2K) 6(x) + + T G (X,0, O) - iT c(x) (106)2jx

3

where $(x) is Dirac's delta function and is defined by 1 where x 0, and 0 when

x # 0. From Equations (C3 through C4) and (35), G3 (x,0,0) is

JF

G3(x,0,0) 1 -ikx K d14.2 k o91/2 2 1/2

47 f ? 2 2(k-+q-) ((k2+ ) -K]

c ,/2

1 2f 1/ (K 1 2+m I / Co 2 exp(-ixu cos ((du dO (107)

0 0 u - -K+m u Cos ()

9

where m = V/g.

The poles of Equation (107) are

1I 1/2= 1 - 2(mK)11 2 cos 0 [1-4(mK) /2 cos (108)

Il 2u 1 2 m cos 0

The integral path of Equation (107) is determined by the value of cos as follows:

28

CASE 1 - 4(mK)I/2 cos 0 < i

The integral limit in 0 is between 0 and 7/2, where 6 is given byT T

= i L /21 = cos- (109)T [4(mK /2

The integral path of u is given in Figure 4.

UI1 L1

U 2 U/u2

Figure 4 - Integral Path when 4(mK) 1 / 2 cos < < 1

CASE 2 - 4(mK) 1 /2 cos 0 > I

The poles in Equation (108) become

u) - 2(mK) cos i[-dmK) cos -1) (110)

2 m cos

The integral limit in " is between 0 and .

CASE 3 - cos 0

With the chiange of the variable - 3, the integral with respct to

becomes

""/2 21/2 1_ 1/2 u o )2eDiucos 0)

ff 12 112 2 V

r/2 o u - (K -m/u Cos

The poles of Equation (111) are

29

[1+41/2)1/2 os 1/21 + 2(mK)/ cos + [+4(mK) Cos

u 3 2 m cos

The integral path of u is given in Figure 5.

u3 u4 12

Figure 5 - Integral Path when cos 0 < 1

By substitution of Equation (111) into Equation (107), C3 (x,0,0) is

1 f T 6/2 (K 1/2+M u o 2 exp(-ixu cos )

G3(x,0,0) -' dQ du + dq du -( - 23 27 2 u (1/2+ 1/2 os )

/ 2

1 T/2 (K1/2mI/2u Cos c cxp(ixu cos

+ dc du ( m u cos (113)2- f. J 112_ 112 2

L o u - (K -m U Cos

With the following expansion

(K 1/2+m /2u cos 0) 2 K + u cos 0 [mu cos C-12(mK) 1/ ]

Equation (113) can be rewritten

30

6 r/

21T2J J { u K exp(-ixu cos (3)G3 2f d8 du + f deO du 1/2/2a 0 L u-(K 1/2+n /2u cos 6)oT1

! T/2

+ 1 2 C f u K exp(ixu cos 6)

22 J 1 1/2_ 1/2o L2 u-(K -m u cos )

6 /2 1/2

dO du +f dO du u cos [mu cos 0+2(mK) I/ 2 exp(-ixu cos :.)

(f1/2 1/2 20 0 L. -(u-K -r u cos 6)

7/

/22

1I dO du u cos O(mu cos '3-2(mK) l]+P 1/2f uu o 2 exp(ixu cos2 _(K1/2 1/22

o L U-(K --m u cos 0)

D2

= 3- 4 (114)

where g3 and g4 are given by

$ dOfduO+F f K exp(-ixu cos 0)g3 2 7T- fu f 112 11

0 OfT L1 u- ( M u Cos q

(115)

7/2

+ 12 [ dO du K exp(ixu cos n)

2T2 f f 1/2 1/2 20 12 u-(K -m u cos 0)

and

31

1 du dO d) i[mu Cos 0+2(mK) 1/2exp(-ixu cos 0)g4 27 d2 du f d f 11d1

0TL ) 1u(Kl/2 +ml/2u Cos 0)2

(116)

l 2 de du i[mu Cos 0-2(mK) /2 Jexp(ixu 2cos 0)

2 f2 f 1/2 1/22 u-(K -m u cos 0)

With the change of the integral paths and the change of the variables, we can

derive the equations suitable for the numerical computation from Equations (115) and15

(116). Following the derivation of Joosen, we can express g3 and g4 as follows

(see Appendix D):

(I g ( + + (3) for n = 3 and 4 (117)

where

1/2 1/2 1/2f 1/2 1/2 }

(1) 21 Rn(v) [(ml+M2) +m] +i(sign m 2) [(m2+M 2) -ml

OI2(2 7mi-T 1 f

-v xldv (118)

2 2 1 2

(m +in )

21kl K or -L +2(mK) klk

gn - f 1 'k+4_ 2 2 1/2 exp 2i --- x) dk for x < 0 (119)

0 [[k k+2(mK)' 4k k3

32

0k 2k 1121n(2) 2ik2 K or -i _ +2(mK)/ x k2k d

1/2 2 exp i 2- x dkSUk 2k+2(mK) ]4k 2k I

2k [Ko2i 2(mK)) /k2]4 4 221/2 Lexp i k x dk

o {[k k-2(mK)I / 2] -4k k 2 }

2 ik3 1 K or if k 3-k 2(mK) 1/~2 k kk\+ J 2 J- - exp xJ dk for x 0 (120)

o (k 3 k-2(mK)lI-I4kk2 3m /

(3) 1 N(k) exp {-- [1-2(mK) xIexp 4ko [k(l-k) ( ik+1) (6k+2)]

× exp {-[6(1-k) (6k+l)]1 1/ 2 xl 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:

R3 (v) K

R4 (v) -i (mK) 1/2 - mv sign x

2 (122)

m I = (K-mv2 ) - (4 mK-I) v(cont.)

=4 1/2 v (K-mv2 ) sign x

33

1/2 1/2

112 [4(mK) /2 for (inK)1 / 2 < 0.25

k 1 - 2(inK) 11 +-14m)' / fr(K1 K02I2 (mK)/2

for (inK)I / 2 > 0.25

k2- = I - 2(mK)I11

2 + [i-4(m K ) I1 / 2 1/ 2 for (mK)

I1/2 < 0.25

=2(K1/2 for Wm) 1

/ 2 > 0.25

k3 = I + 2 (mK)1/2 - [1+4(mK) /21/2

k4 = 1 + 2 (mK) 1/2 + [l+4(mK) 1/2 1/2

(122)

6 = 4(mK)1 2 -1

N3 (k) =K

- - 1/2 1/2

(k) (-k) (3k+l) sign [6k+1+2(mK)4 2

2

U 2

m - g

2g

As in special cases, when K 0, G 3 becomes (see Appendix D)

I3 ((LI (2 sinx) (123)"3 4 x Mo

and when m 0,

34

kA.,

G3 [ (-H (Klxl) - Y (Kjxj) + 2i J (Klx!)] (124)

where H is the Struve function, and J and Y 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:

'WATiI OA, SWATH 6C, and SWATH 6D. Experimental tests for motion have been carried

out for all three models. All models have the samp 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*

Dimension SWATH 6A SWATH 6C SWATH 6D

Displacement, long tons 2579 2602 2815

Length at waterline, m 52.5 60.2 68.0

Length of main hull, m 73.2 73.2 73.2

Beam of each hull at waterline, m 2.2 2.6 2.2

Hull spacing between centerline, m 22.9 22.9 22.9

Droft at midship, m 8.1 8.1 8.1

Maximum diameter of main hull, m 4.6 4.6 4.6

Longitudinal center of gravity 35.5 34.7 36.1aft of main hull nose, m

Vertical center of gravity, m 10.4 10.4 9.1

Longitudinal GM, m 6.1 13.7 26.4

Radius of gyration for pitch, m 16.9 17.7 19.0

Waterplane area, m 193.9 181.4 253.9

Length of strut, m 52.4 25.8 (per 25.8 (per

strit) strut)

Strut gap, m 0 8.6 16.4

Maximum strut thickness, m 2.2 "2.6 3.1

*Dimensions are full-scale.

35

Figures 6, 7, and 8 show the added mass and dampirg 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 Lee. The two-dimenoional 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.

2

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 diLcrepancy may result from the different methods used to solve

the strip theory. In the derivation of the unified slender-body theory, Newman3

solved the strip theory with the multipole expansion method (Equation (50)). In

Equation (50) a3 .s a source located at the origin and is usbd to solve the three-

dimensional source strength (Equation (62)), from which the correction factor C. is

computed as in Equation (61). The only part of the source located at the origin is

a The other part of the source is a m from Equation (50), which is included in thz

computation 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) a is distributed

on the contour of the section, and the corresponding 5oorce a , which is located0

at the origin, is computed by the relation in Enuption (105). To solve the three-

dimensional source strength, a3 was replaced with u , but a includes the effect of

o3 and am. It is difficult to decompose ao into a3 and am in the Frank close-fit

method. If ao is not computed accurately, the three-dimensional source qj ini

Equation (62) cannot be solved correctly, even though the kernal function t(x) has

been evaluated properly. Therefore, the difference between o3 and ao could possibly

caujse Lhe large discrepancy In added mass coefficients.

36

.I

3 3

STRIP THEORY

- UNIFIED SLENDER-BODY THEORY

2 2

A 3 33I-- 1 1/2j)- .5~*

0 0

wU 1-i

3 1 10.6

2 0.4

A55 B5 5

MI-2 (g) /

1 0.2

0 0.5 1 00 0.1

1/2

F igulre 6 Added > ~~and I)amip ing Coneff ic ient.,; of SWATH 6A in Foll owingScas at 20 Knots

37

[3

I 3

-STRIP THEORY

- -- -UNIFIED SLENDER-BODY THEORY

A33 B33

1/2M

-(-

II

I I .0.5 I 1 0.5oU 1

g-4

0.6

2 - 0.4 -

A B55A5 5 ML(gL 1/2

MI2

1 0.2

0 ~0.010.5 1 0 0.5

41/2

Figure 7 - Added Mass and Damping Coefficients of SWATH 6C in FollowingSeas at 20 Knots

38

3

STRIP THEORY

UNIFIED SLENDER-BODY THEORY

2 \2

A3 3 B3 3

1- 1 - ------..

0 0

0 0.5 1 0 0.5WoU 1

3 0.6

I

I

2 0.4-

A55 B55

ML 2 ML(gI\ 1/2

1 0.2 _

\ /

\ /\ /

o I 0.0 "0 0.5 0 0.5

-(!g)1/2

Figure 8 - Added Mass and Damping Coefficients of SWATH 6D in Following

Seas at 20 Knots

39

When (inK) 1/2 = w~U/g =0.25, there is a discontinuity in the computations. This

singular behavior can be explained by the Green function. If (inK) 12approaches

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 X/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 X/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 X/L values. Pitch amplitude, computed by the mixed method, agrees well with

experiment for all X/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

3

STRIP THEORY---

2

F 3 _

1.4 I

.0! j

F 5,gVA

0.5

0.0 I0 1 2 3 4 5

L

lite 9- Exc it ing Force and Moment of SWATH OA in F-ollowingSeas at 20 Knots

41

r

3A

/0////

2,,,/ o0

F3

pg VA/L

0 A

0

0 00.75 -1I\

S\ 0 / \~

I \ ,

0.5

F5pgVA 10

STRIP THEORY

0.25 -- UNIFIED SLENDER-

BODY THEORY

/ 0 EXPERIMENT

CU 1

0 .0 _0 2 3 I 4

LFigure 10 -Exciting Force and Moment of SWATH1 6D in Following

Seas at 20 Knots

42

1.0

'314 0A+

0.5 -00

+ +. 0

+ 0

0.0 ~ )0 ~ 0 ~~STRIP THEORY1

3 -BODY THEORY0 EXPERIMENT

/ \ + MIXED METHOD

++

I 0 0

0 +

0++-

ol +0 1 2 3 4 5

L

V, it II - ('t iont )t S WATHi 6A in Fol lowing icas at 20) Knots

43

STRIP THEORY

--- UNIFIED SLENDER-

l~I 0.5

0 EXP RI EN

3 / ~ + MIXED METHOD

2

010 12 3 4 5

L

Figure 12 -Motion oif 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 /L values.

'Thu pitcl, 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

dot"s not compute the pitch amplitude properly. This indicates that the added mass

ond damping coefficients in Figure 8 by the slender-body method may be i!. significant

error. This error might be caused by solving the strip theory with the Frank close-

fit method Ind by replacing c 3 with o 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 Newman 3 is applied

to predict the motion of SWATH ships in following seas. Only for a limited range of

/1. vlues is there an improvement for heave motion. For pitch motion, except small

,moumit or frtquencits,, the re-ults are worse than those of strip theory. When the

encounter frelquencV is 1are, the pitch motion becomes extremely large. The reason

for thi:s discrepancv seems to lie in the fact that the strip theory is solved by the

Irank close-fit method, and not hy the multipole expansion. From the present study,

tht- followin g -oncl i;ions nay be drawn:

1. In the unified slender-body approach, the largest contribution to the

v,.,i\,,Thn~iTh co.-f i i tnt; rem its from the solution of the strip theory itself.

I ,r"t ,'TIJMU( ;it ion of tile s;ouir'e strength at the origin is necessary to compute the

t ,rtt-d iun-; iona I -our(, ;trengt h exactly. Therefore, for the outer approximation

,,i lr ii t hcorv, th, m1 it i porhe expansion method should be applied instead of the

!rink cl .,t- f it method.

.'. A mre, careful analvsis should be made in computing the excitation forces

i,v the strip theory. Application of the Helmholtz equation is mathematically more

c, rrut . However, there is a singularity in this solution when the heading angle

')t t hi iflcoullming waves is z,ro or 180 degrees. The two-dimensional Laplace equation

! !, t ie heave excit;it ion force correctly, but the pitch excitation moment is

Murq, edl inc'orrect Iy.

45

o EXPERIMENT- -- .--

+- MIXED METHOD

00

0.0- 1

1.5 -STRIP THEORY +

- -- UNIFIED SLENDER-BODY THEORY

00

00

0.0 1 2 345

L-

Figure 13 -Motion of SWATH 61) in Follow~ing Seas at 20 Knots

46

I. According to the unified slender-body theory it should be valid for all

trquencies. 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.

Bl'cauL of the above mentioned discrepancies, until further research

mentioned in items I and 2 are carried out, the application of the unified slender-

hodv thtorv to improve the prediction of motion of SWATH ships is not recommended.

ACKNOWLEDGMENTS

Ihe alithor expresses his thanks to Dr. C.M. Lee for his useful advice and

gritefully acknowledges the support of Ms. M.D. Ochi.

47

APPENDIX A

DERIVATION OF G 2D (y,z) WITH EXPONENTIAL FUNCTION

Equation (30) can be rewritten as

C - LZ f e kz(eik Y+el ky ) dk (A.1)

0

For the first term of Equation (A.1), we change the variable as

w = -i(k-K) (y-iz) (A.2)

ad

d -dw(A3d i(y-iz)(A3

The contour of the integral path will be different depending upon y <0. When y > 0,

we take the inte gral path in the following figure

k-PLANE w-PLANE

Pr; ubsituingEquations (A.2) and (A.3) into Equation (A.1), the first term

U kz+iky dkeiK(y-iz) e-w d A4

J k-K f wo iK(y-iz)

---- c-F49

If we consider the following integral along the closed path in the w-plane

f edw + + = i

then, the integral over F = Re vanishes as R - o and Equation (A.4) becomes

iK(y-iz)

ekz+iky dk eiK(y- iz ) e -W

f K d e [2T i-dw]0 k-K

eiK(yiz) [27i+E (Kz+iKy)] (A.5)

When y < 0, the integral path is given in the following figure

K

iKly-iz)k-PLANE w-PLANE

and the first term of Equation (A.1) becomes

J e kz+iky e iK(y-iz) e-W

e- dk f e dw (A. 6)

o iK(y-iz)

Tf we consider that the following integral along the closed path is in the w-plane,

50

p-T

e_ dw= {+ +f=o0w

e~ dw

then the integral over T' Re vanishes as R - o, and Equation (A.6) becomes

kz+iky iK(y-iz) e - w

e- dk = - ei - dwk-K fw

iK(y-iz)= e El (Kz+iKy) (A.7)

For the second term of Equation (A.l), we apply the following transformation of the

vd riao~le

w = i(k-K) (y+iz) (A.8)

in d

dk dw (A.9)i(y+iz)

,ith the -ane procedure for derivation of Equations (A.5) and (A.7), the second term

i,,i (A. I) is given as

f .... dk = e El(Kz-iKy) for y > 0 (A.10)

0

Kz-iKy [27i+EI(Kz-iKy)] for y < 0 (A.11)

B" uiIlt itut ion of Equations (A.5), (A.7), (A.10), and (A.11) into Equation (A.1),

51

G(y~z) = ~ieKiY [2l7i+E (Kz+iKy)] + ez +iY E1 (Kz +iKy)) for y > 0

-- 1. Re leKz+iKy E (Kz+iKy)) -ie Kz+iKlYl (A.12)

52

APPENDIX B

DERIVATION OF G (y,z) FOR SMALL ARGUMENT

2D

We expand the exponential function and the exponential integral in Equation (31)

for small Kr where y = r sin 0 and z = -r cos 6

Kz+iKye i = 1 + Kz + iKy(i+Kz) (B.l)

eKi - 1 + Kz + iIjy (i+Kz) (B.2)

E (Kz+iKy) = E I(Krei(T-0))

= -y - 2n(Kr) -i(7-0) + Kz + iKy (B.3)

By substitution of Equations (B.1), (B.2), and (B.3) into Equation (31), G is2D

given as

G2D(Y,Z) = - - Re ([(1+Kz+iKy(l+Kz)] [-y-n(Kr)-i(7-e)+Kz+iKy]j

- i [(l+Kz)+iKlyl(l+Kz)]

- (1+Kz) [y+Qn(Kr)-Kz+Kv,-i-] (B.4)

53

APPENDIX C

FOURIER TRANSFORM OF THE THREE-DIMENSIONAL GREEN FUNCTION

Equation (36) can be expressed as

22

1k 1 +Z xpizk 2+ZC(x,y,z) - 4~ e 1 dk ~ k±.212 K)dQ (C.1)

47 (k 2 )Z (k 2+Z) 2 ~

where

K = (C.2)g

The first double integral of Equation (C.1) is known. Therefore, we rewrite Equation

(C.I) as

1G(x,yz) GB(X,y,z) (C.3)

where R +2 x2 + 2 + z2 and G3 is given as

i(Xyz) -ikx idk I f K exp z(k2+9 2 ) -iyk] (C.4)

3' 2727 2 2 1/2 2 2 1/2_

3_- (k +) f[(k +Z ) -K]

If we assume y and z become very small, Equation (C.3) is expressed as

G(x,O,O) G= - 1/- G(x'OO) (C.5)221/2 32 2

27T(x +r )

2 2 2where r v + z

55:lL:I.llG '. l bi ll q i

The Fourier transform of Equation (C.5) is

G*(kK) =- Ko(Ikjr) - GB*(k,K) (C.6)

and from the definition given in Equation (38), G3* is given

- K d

G3 *(k,K) = IJ 1/2 1/2 (C.7)-M (k2 +k2 ) [(k2 +Z2) -K]

In Equation (C.5), K(X) is the modified Bessel function. For small r, Ko(kjr) can

be expanded

K 0(jkjr) -n ( -2r y (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

G*(k,K) = (jk n ) +Y] - G3*Tr2 3

= G f*(k,K,<) (C.9)2D 7T

where G2D is Equation (33), G 3* is Equation (C.7), and f* is given by

f*(k,K,K) = Zn-- -i + G 3* (C.10)

56

APPENDIX D

DERIVATION OF G3 (x,O,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.

1. x> 0

a. First Integral of Equation (115)

Evaluate the following integral in the complex plane of w

K exp(-iwx cos 6) 2 dw = += - 2 i (residue) (D.1)w 1 ()/2+(1/2I/2f

- [K +() WCos e] c c2 c3

12 3

WV+J

C2

w-PLANE

w [(K)1/2 + (m) /2u cos 2 m Cos (W-ul) (w-u 2 )

the residue becomes

K exp(-iu 2x cos 0) -iK exp(-iu2x cos 0)

2 (c 1/2-m Cos 2 (u2-ul [4(mK) 1/2C

57

. . .

where u and u2 are given in Equation (110). The first integral in Equation (D.1)

is the same as the first in Equation (115). The second integral along the contour

c2 becomes zero

-Tr/2

dw = lim - K exp(-iRe x cos 6) R e i 6 i dfRR-f2 Re id 20

c 2 o

The last integral of Equation (D.1) becomes

C °. K exp(-iivx cos 6) i dv f iK exp(-vx cos 6) dvJ )dw 1/2 1/2 2 f 1/2 1/2 2

S f_ iv - [(K) +(m) iv cos 6] o iv+[(K) -i(m) v cos 6]c3-0

Therefore, the first integral of Equation (115) is given by

00

K exp(-iux cos @) exp(-iu 2x cos 0)2du =- 2K 1/2

o U- [(K) /2+(m) /2u cos 02 [4(mK)I/2 cos 6 -1]

+ iK exp(-vx cos 6) dv (D.2)

f1/2 1/2 2d D2

0 iv +[(K) -i(m) V Cos 6]

b. Second Integral of Equation (115)

Evaluate the following integral in the complex plane of w

K exp(-i cos 0) 2 dw = + f + f = - 2ri (residue)

w - [(K) /2+(m) /2w cos 0] cI c2 c3

58

VUU C1 =L 1

C3 [C

w-PLANE

The residue is

K exp(-iu 2 x cos 0) -K exp(-iu 2x cos

-m cos2 O(u2-u) [l-4(mK) 1/2 Cos 1/2

The integral along c2 becomes zero as shown in (a) and the integral along c3 becomes

J = ) 0d K exp(-iivx cos e) Jd 1/2 ex( /v ose 2 dv2 f11 12

J 1/2 1/2 0 (K) I - cos 6]c-3 -- iv-[(K) +i(m) v cos 0o iv+[K i(m)i/v 2

Therefore, the second integral of Equation (115) is given by

f K exp(-iu,,x cos iK exp(-vx cos

Sdu - 27Ki 1/2 - 2 dv (D.3)S[ 1-4(mK)I/2 Cos o iv+[(K)1/2i(m)i/2 C

59

c. Third Integral of Equation (115)

Evaluate the following integral in the complex plane of w

K exp(iux cos 0) dw = + F + f =27i (residue)1/2 1/2 2 f

W-[ -( w cos 6] Cl c 2 c3

V

c2

C3

u 3 u 4 U

c I = L2

The residue is w-PLANE

K exp(iu4 x Cos C) K ex(iu 3x cos 0)

-m cos "(u 4-u3) -m Cos 2 "-(u 3 -u 4 )

-K exp(iu 4x cos ') K exp(iu 3 x Cos 6)1/ /2 1/2

[l+4(mK) /2 cos 1 [l+4(mK) / 2 cos 0]

The integral along c, becomes zero as shown in previous cases and the integral along

c3 becomes

30

dw =K exp(iivx os ) idvf o / 1/2 ]

c 3 iv-[(K) /2 (m) iv Cos

- - iK exp (-vx cos ) dv

f 1/2_ 1/2 2o iv-[(K) -L(m) v cos

60

Therefore, the third integral of Equation (115) is given by

exp(iu_4_ x Cos _) exp(iu 3 x cos ')( ) du = - 27Ki 4cs + 2Ki

L2 [l+4(mK) /2 cos 0] [1+4(mK) o/2 ; ,7

iK exp(-ux cos - dv (D.4)f 1/? 1/2 2o iv-(K) --i(m) v cos 0]

By adding Equations (D.2) through (D.4), g3 is expressed

iT/2

93 f dO J K exp(-vx cos 0) dv2 TT f 1/2_ 1/"o o iv+[(K) -i(m) V Cos

7TOS"/2 /

f d0f K exSp(-vx cos ") dv I K exp(-i 2xos27 2 f 1/2_ 1/2 f d1/

Siv-[(K) I/ i(m) v cos o [4(mK) cos

7/2 K exp(-iu2x cos /) K exp(iu 4x cos

f [[-4(mK) Cos [l+4(mK) I/ 2 cosT 0

T

2 K exp(iu 3X cos

-11 1/2 d' (D.5)

o [l+4(mK) cos "]

2. x< 0

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 eviluate the complex integrals given in Case I in the first quadrant, and for the

61

third integral, we evaluate the complex integral in the fourth quadrant. The ex-

pression for g3 similar to Equation (D.5) is given by

i 9Jd:; .....K exp(vx Cos 0) dv3 2 12

0 0 tv-[(K) /+i(m) -v cos 0]

o+. -K.-(-vx cos 0) dv 1' -1 K exp(-iu x cos 0)

2 dis 1K2 df2

o o iv+[(K) /+i(m) v cos 0 ] o [4(mK) cos 0-11

ir/2 K exp(-iUIx Cos 0)

+ 1/2 - /2 dO (D.6)

0 [ 1-4 (inK) Cos 0 1

.o r g4 ' the same procedure Can he applied. But, the numerator, K, should be

replaced by -i[mu cos 0+2(inK) I2 or i[mu cos 0-2(inK) 1/2 ] with appropriate variables

,11nd poles.

If we let

A = K

u/2 (D.7)A4 i[mu cos 0+2(inK)

B 4 = [mu cos 0-2(mK) 11

wt, can express g for n = 3 and 4 as follows

72 A (-i,)) exp(-ux cos 0)

dO --dduf f 1 1/'2 112

il+ [(K) -i(m) 1 Cos 0]

+ (2 i u' )exp(-ux cos 0) du f A ( 2 0) exp(-iux cos 0)+ 2 f n ... .. . .-- - -I1 __- _ _ 4 -i

f ill-[(K) - i (1) Cos 0 ]( 1/2o 0 0 14 (inK) cos 0-1](D.8)

(cont.)

62

7T/2 /A n (u 2) exp(-iu2 x cos ) B n/(U 4, exp(iu4x Co.-;+/ / d (I -- -- .. . . . .. .. . .

-f 1/2 112 d/1120 [1-4(mK) cos 0] o [1+4(mK) 1 Cos

T

B/ B(U.3,0) exp(iu x cos 0)

dO for x 0 (D.8)

o [1+4(mK) Cos cO

i/2 AnA(iu,O) exp(ux cos 0)gn = k dOf ____

n 2 2 f 1/2 1/2 du

o o iu-[(K) /+i(m) u cos

T /2 B n (-iuQ) exp(ux cos 0)

+ 2 J dO j du

2r2 f1/2 1120 o iu+[(K) '/+i(m) /2 cos 0]

p co )I T/2 An (u ,) exp(-iu x cos 7)n( Ul0 exp(-iu-Ix Cos 0) Ai ud-'(.9-Ii-- dO + ~ ~__ ------ '-----dO, (D.9)

- 11/2 f /2 112o [4(nK) cos 0-1] 0 (1-4(mK)l cos O)

for x < 0

The first two double-integral terms in Equations (D.8) and (D.9) can be reduced

with substitution of u cos 0 = v (Reference 17) to:

1. n= 3 and x > 0 for A3 =B 3 =K

63

n/2

(1) i O f K exp(-vx) sec e dv3 2 1/2 1/2 2

o o iv sec e+[(K) -i(m) vi

TT/2 M+ dO f K exp(-vx) sec 6 dv

2 2 1/2_ 1/2]2

o o iv :ec O-[(K) -(m) vi

K f e-vxI dO K f e-VX dv

7T 2 4 2 2 1/2_ 1/2v4}i/20 0 +[(K)I i(m)i Cos ev2+[

The denominator can be written as

+[(K) 11-i(m)/2v] = m I i m 2 (D.10)

where

2 2 2

m (K-mv) - (4mK-1) v

m = 4(mK) 1/2 v (K-mv2)

The last integral becomes

2 11/2K e e-vx K ( (ml +Th 2 )2-v

(mi)m1-/2 d = f J 1/2 e dv (D.II)

o (m +M2)

With the change of the variables

mi = r cos 0

64

i :4

m 2 = (sign m 2 ) r sin 0

-m I

C2o2 1/2

(m 2+M)12

sin 0 2 2 1/2

(m +m 2 )

L1

the numerator becomes

(m+im) 1/2 1 2 [( 2 11/2 /2+i(sign in) 2 22 1/2 1/21-221/2 1(l 2 ) +1 +i g 2 ) 1 (mm 2 ) 1l

(1)Finally, g3 can be expressed

g 2(2)1/27. [(m+m)+mliI+i(sign m) +m2e) -1 dv2() 7' 2 2

0 (m2 +m 2 )

(D.12)

2. n = 3 and x< 0

Following the same method as that of Case 1, we can derive

S() , K [22 21 2in 2 ) e(m2+m2) .1]/2 22

=(m 1 +ml+m2) -mliJ L11/2dvg3 2(2)1/2 T ( M2 + isg 2 e2 d

(m0 m2)

(D.13)

where

m2 = - 4(mK) I / 2 v (K-mv ) (D.14)

65

With Equations (D.12) and (D.14), gil) can be written for all values of x as

follows

(1) K 2 2 1/2 1/222 /_]1 eV I3 2 112 (Ml+m2 +mI + i (sign mv2 ) L ml+m2) - 2 2 1/2 dv

0 (2))(m 1 +m2 )

(D.15)

where

m = 4(mK) I/2 v (K-mv 2 ) (sign x) (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

g 1/2 J R4 (v) m{[ +m2 ) +m 1J +i(sign m 2 ) ml+m 2 )

4 2(2) 1/ o 2

0

11/2 e-v 'xj

-m 2 21/2 dv (D.17)

(m+m2 )

where

R4 (v) = -i 2(mK) 1/2 - 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) bp

n/2 An(Ul,) exp(-iu 1 x cos 0)(2 J 1/2 1/2 dO (D.19)

gn f [l-4(mK) / 2 cos 0]

T

66

where

A 3 (ule) = K and A4 (U1,0) = -i[mu 1 cos 0+2(mK) I/2

With the change of the variable

2 mu1 cos 0 = kIk (D.20)

where u1 is given in Equation (108) and

12121/2 1m~/2

k = I - 2(mK)1/2 -[1-4(mK) / 2] for (mK) < 0.25

= 2(mK)1/ 2 for (mK)I / 2 > 0.25 (D.21)

(2)gn is transformed asn

(2) 2ik I K or -1F--- +2 (nK)l/ klkn 7 f 4 1/2 exp i m X) dk (D.22)

o [klk+2(mK) 1/2 -4klk 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 mu2 cos 6 = k 2k (D.23)

2 mu4 cos 0 = k4k (D.24)

2 mu3 cos 0 = k 3k (D.25)

67

where k2, k and k3 are given by

k2 = I- 2(mK) /2 + [1-4(mK) / 2] for (inK) < 0.25

1/2 1/2= 2(mK) for (mK) > 0.25 (D.26)

1/2 1/21/2

k 4 1 1 + 2(mK)I /2 + [1+4(mK) / 2] (D.27)

k = 1 + 2(nK) 1/2 - [1+4(mK)1/ 2 1 (D.28)(2)

With these changes of variables, g for x > 0 becomes

i [~k+(m)i/2 1k22 2 1/2 x i xd

(2) - 2 Kor 2+2(mK) 1/2 exp dk

1 {[~k+(mK)1/2 -4k~k29exp _xdk

2iT f 1/2 4_k2 21/2 exp

1 [jk 2k2(mK) + 4k2 k21k 4f K or , -2(mK) 1 7x kk4 xd

Wit th chng of + ox 3~~~( 2 -4xk dk (D.29)

Wit th chngeofthe variable

Cos 0= 11/2 (D.30)k+2 (iK)

68

The third term of Equation (D.9) for x < 0 becomes

(3) _1 Ae n (uO) exp(-iux cos 0)gn =T 1/2 1/2 de

o [4(mK) cos 8-1]

2(mK)1/2 )i 1/2 .k 211or - (4mK-k2) -ijj +2(mK)l dk

2 12

l-2(mK) 1/2 ([2(mK) 1/2-k][k+2(mK) 1/2] [k+2(mK) 1/2] 2 ) 1/2

exp + [4(mK)1/2-k 2 ] (D.31)

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)

k 2(mK) /2-1 (D.32)4(mK) 112-i 4(mK) 12_l

S(3)can be finally expressed as follows for all values of x

(3) (k) exp -2 (mK)l11x I exp -2akx _

gn - 7 [Ek(l-k) (k+l) (k+2) 1/2 dk

x exp {_[S(lk)(k+1)] /2 (D.33)

69

where

6=4(mK) -1

N = K (D.34)

1 -- 1/211/

N4 = 2 [6(l-k)(6k+l)] (sign x) - 2 [6k+l+2(mK) I / 2

Then, g3 and g4 given in Equations (115) and (116) can be expressed with n = 3

and 4

g g(1) (2) + (D.35)gn = gn gn gn ( 5

where g() is Equation (D.15) for n = 3 and Equation (D.17) for n = 4; g(2 ) is

Equation (D.22) when x < 0 and Equation (D.29) when x > 0; and g(3) is Equationn

(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 = 0 or K = 0, we can integrate these equations

explicitly.

1. Steady Forward Motion, K = 0

Because of K in the numerator,

(1) (2) (3) = 0g3 =g 3 93 =

From Equations (D.1O), (D.16), and (D.18),

m l = v 2 (m2 v2 +l)

m = 0

R 4(v) = - mv sign x

70

By substitution of these relations into Equation (D.17), gl becomes

g(1) 1 [-mv sign x] (2m l 12 e-vlx dv1/2 [ sign x2 m I

0

m e-Vx

= - ign x f-~ dv2rT gn X 2 2 1/2

o (m v +I)

4 sign x o-,I -Yo kmj (D.36)

In Equation (D.36), Ho(x) is the Struve functionand Y (x) is the Bessel function of

the second kind.

When K = 0, kI = k = 0 and k = k = 2. Therefore, for x < 0, 92) =.3 2 44

For x > 0, from Equation (D.29), g(2) is given by

(2) -2i-2 f _ -ik exp) d

g4 2i-2 1/2 exp / dkI [(2k)4-4"4k 2]

1 !(2k)4-44k 2]

kxS-) OS ---

2J o i/2 dk'T 1/21 (k'-I)

= - " ( M..) (D.37)

71

The integral limit of g(3) in Equation (D.31) is transformed from the variable 0that is between 0 and 0 = cos 1i/[4(1K)i/2]). If 1/[4(mK) 1/2 ] is larger than 1,

T (3)0 becomes zero. Therefore, when K = 0, g4 becomes zero. By adding EquationsT(D.36) and (D.37), g4 is given by

g - ( H ) - [2+ sign x] Y (D.38)

Finally, from Equation (114), G3 (x,0,0) is given

g3CxOO 9 [ Ho (2k) +(2+ sign x) Y ( jCD 9

2. Pure Oscillation, m = 0(1)),i erg becomes zero. When m=0 1 k

Because R4 , Equation (D.18), is zero, g (1) , k = k

0 and k= = 0. Therefore, 4 , Equation (D.22), is zero when x < 0. When

x > 0, as the complex argument of the exponential function becomes infinitive in the(2)beoezeo Teuprlmtf

first and second terms in Equation (D.29), g4 becomes zero. The upper limit of

the integral for g43 ) in Equation (D.33) origined from 0T which becomes zero in this

case. Therefore, there is no contribution from 94 to g3 when m = 0.WihR2 +2 ad 0 (1). g

3 = K, m1 = v , n m2 = O g3 s

(1) K 1/2 e dv

3 3/2 2mTl m

0

= K f _-v_ xl dvo (K2+v 2 )

= K C f 11/2 exp(-Ktlxl) dt

2[ (t 2 +l)0

= K4 [Ho(Kjxj) - Y o (K jx j) ] (D.40)

72

()is evaluated

With A Kand u 1 = K from Equation (115), as m becomes zero, 83 sevlae

from Equation (D.9) for x < 0

iT/2

(2) =i oK exp(-iK cos 0) dO for x < 0 (D.41)g3 7= f

Because u -K, and u2 and u4 disappear, (2) for x > 0 is given from Equation (D.8)

2/2

(2) f K exp(iKx cos 0) dO for x > 0 (D.42)g3 =

For all values of x, g is given by

(2) = exp(iK1 x , cos e) deg3 =~ j

= [-H (Kjx )+iJ (Kx!)] (D.43)2 o 0

Because of n = 0,g is zero. Finally, is given

G K [-1o(K~x')-Y (K x)+2iJo(KIxi)] (D.44)

3 93 4 o 0

73

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," DTNSRDC 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.O., "The Steady Motion of a Slender Ship," Ph.D. thesis, University

of Cambridge, Cambridge, England (1963).

5. Tuck, E.O., "The Application of Slender Body Theory to Steady Ship Motion,"

DTNSRDC Report 2008 (Jun 1965).

h Tuck, E.O., "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," DTNSRDC 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

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II

AD-A107 723 DAVID W TAYLOR NAVAL SHIP RESEARCH AND DEVELOPMENT CE-ETC F/6 13/10PREDICTION OF MOTIONS OF SWATH SHIPS IN FOLLOWING SEAS. (U)NOV 81 Y S HONG

UNCLASSIFIED DTNSRDC81/039 N

SUPPLEMENTA

*INFO R,.MATION'

KERRATA SHEET

>1 4UNCLASSIFIED report DTNSRDC-81/039, November 1981

1. Page 20, Table 1: Equation for A at bottom of table, replace fourth term

53

by - g xn3 R dS

- PU n3 RdS

2. Page 21, Table 1 (continued): Equation for A55 at middle of table, replace

eighth term

f wJJ2 n 4 dS

by

2--2- f xn C5I dS

3. Page 35, Table 2: Value given for "Waterplane area" for "SWATH 6D" as

"253.9," fourth line from bottom of table, should be 211.2.

4. Page 51, top of page: Insert integral sign in front of quantity on left-

hand side of equation so that

e dww

becomes

w dw