Prediction of relative maotion of ships in waves

Scheepshydromechardcft ^ ^ T ) Archief

MekcKveg 2, 2628 C D Defft ^ Tel.: GIB - 7S5373 • F?« 015 • 781S38

Prediction of Relative Motion of Ships in

Waves

Chouag M. Lee

O f f i c e of Naval Research, A r l i n g t p n , VA 22207, U . S . A . John F . 0'Dea and WilliaLm G. Meyers

Davtd W. Taylor Navai Ship R&D Center Bethesda, MD 20084

ABSTRACT

An a n a l y t i c a l method i s develpped f o r p r e d i c t i n g the v e r t i c a l motipn pf a point pn a s h i p r e l a t i y e to the f r e e s u r f a c e . The method accounts f c r the deformation of tbe f r e e s u r f a c e caused by d i f f r a c t i o n and by the waves generated by the motion pf the s h i p . Gpmputed r e s u l t s are compared to experimental r e s u l t s f o r two h u l l forms. The phase r e l a t i o n s among the i n c i d e n t , d i f f r a c t e d and radiated wave cpmponents are found to p l a y a s i g n i f i c a n t r o l e In determining the t p t a l f r e e * surface motion. The s t r i p theory used i n the p r e s a i t work appears to be inaccurate i n p r e d i c t i n g correct phase r e l a t i o n s h i p s f p r these comppn^ts.

INTRODUCnmi

In an assessment of the seakeeping q u a l i t i e s of a ship the deck wetness, bottom slamming, and rudder or p r o p e l l e r emergence are some of the important f a c t o r s to be taken i n t o account. The occurrence of these events i s d i r e c t l y governed by the s o - c a l l e d " r e l a t i v e mptipn." The r e l a t i v e motipn i s the measure of the v e r t i c a l motion of a ship w i t h respect tP the undulating f r e e - s u r f a c e mption.

For instance, i f a ship i s s a i l i n g i n a long s w e l l , the v e r t i c a l motion pf any p p l n t s pn the ship would be i n unison w i t h the v e r t i c a l motion of the f r e e s u r f a c e d i r e c t l y below or above the h u l l p o i n t s ; hence, the r e l a t i v e motion would be z e r o , (to the other hand, when a large ship i s moving i n small waves and, t h e r e f o r e , there i s p r a c t i c a l l y no motion of the ship but steady forward motion, the r e l a t i v e motion of the ship would be the negative pf the wave mption.

Current p r a c t i c e i n cpmputing ship r e l a t i v e motion u s u a l l y neglects i n t e r f e r e n c e e f f e c t s caused by ship-generated waves on the incoming waves. The main reason f o r n e g l e c t i n g the defcrmatlpn e f f e c t on the • oncoming waves has been due to the d i f f i c u l t i e s involved i n i t s computa-t i o n . A ship moving i n waves creacomputa-tes varipus wave componencomputa-ts which

would i n t e r f e r e w i t h the oncoming waves. These i n t e r f e r i n g waves are generated by the forward motion, the wave-excited o s c i l l a t o r y motions

In s i x degrees of freedom, and the d i f f r a c t i o n by the ship h u l l . The Importance of c a l c u l a t i n g accurate f r e e - s u r f a c e motion alongside a ship h u i l i n the r e l a t i v e motion c a l c u l a t i o n has been w e l l recognized by ship motion researchers because of the c l e a r experimental evidence p r e -sented by Cox and Gerzina (1974), Gerzina and Woo (1975), and Bales et a l . (1975), showing the discrepancies i n the e x i s t i n g t h e o r e t i c a l methods. The purpose of the present i n v e s t i g a t i o n i s to compute the aforementioned i n d i v i d u a l components of the ship-generated waves and

incorporate them i n t o the c a l c u l a t i o n of r e l a t i v e motions of s h i p s . To check the v a l i d i t y of the v a r i o u s assumptions made i n the theor e t i c a l a n a l y s i s , the c a l c u l a t e d theor e s u l t s atheore c o theor theor e l a t e d w i t h model e x -perTimental r e s u l t s obtained at the Maneuvering and Seakeeping Basin of DTNSRDC.

The a n a l y t i c a l method developed here i s based on a two-dimensional approximation w i t h i n the context of s t r i p theory, which was described In d e t a i l hy Lee (1982). The main reasons f o r employing the

two-dimensional approximation are f i r s t , f o r i t s s i m p l i c i t y i n i n c o r p o r a t i n g i n t o an e x i s t i n g ship motion computer program which i s a l s o based on s t r i p theory given by Salvesen et a l . (1970) and second» f o r checking the v a l i d i t y of the r e l a t i v e motion p r e d i c t i o n based e n t i r e l y on a s t r i p theory. Since there has been no c o n c l u s i v e evidence to demonstrate that ship motion i s b e t t e r predicted by three-dimensional theories than by s t r i p theory, the present i n v e s t i g a t i o n , u n t i l a r e l i a b l e other method i s developed, i s deemed as the necessary f i r s t step toward improving the p r e d i c t i o n of the r e l a t i v e motions of ships w i t h i n the present s t a t e o f -t h e - a r -t i n ship mo-tion -theory.

THEORETICAL ANALYSIS

Formulation of the problem i s made under the assumption pf an i d e a l f l u i d , the v e l o c i t y vector f i e l d of which can be represented by the gradient of the v e l p c i t y p o t e n t i a l f u n c t i o n $. I t i s assumed that the depth of the water i s i n f i n i t e and that no current and wind e x i s t . I t

i s a l s o assumed that the response of a ship to the wave e x c i t a t i o n i s l i n e a r and that the i r r e g u l a r ocean waves can be represented by a l i n e a r s u p e r p o s i t i o n of various harmonic wave components. Thus, the s h i p response to the i r r e g u l a r ocean waves can be obtained by determining the frequency response f u n c t i o n of the ship to harmonic wave e x c i t a -t i o n s .

The coordinate system to be used i n t h é a n a l y s i s i s a right-handed Cartesian coordinate system which t r a n s l a c e s on the calm-water plane w i t h the mean speed of the s h i p . The o r i g i n i s located on the calm-water plane d i r e c t l y above or below the center of the g r a v i t y of the ship at i t s mean p o s i t i o n . The x - a x i s i s d i r e c t e d toward the mean course of the s h i p , and the z - a x i s i s d i r e c t e d v e r t i c a l l y upward, as shown i n Figure 1.

WAVE DIRECTION

H « 180 Degrees Conesponds to Head Waves

M » 90 Degrees Correapcmds to Staiboard Beam Waves

H = 0 Degrees Correqitonde to Foltowing Waves

AP 4, =• SURGE « J = SWAY FP {, ° HEAVE = ROLL tj = PtTCH = YAW

Figure 1 - Description of Coardinate System

T o t a l f l u i d disturbances generated by the progressive s i n u s o i d a l waves of length X, heading angle V, and amplitude Ç^, w i t h a ship of slender geometry undergoing o s c i l l a t o r y laotion at a mean speed U , can be d e s c r i b e d , w i t h i n the l i n e a r a n a l y s i s j by

iü) t

$ ( x . y . , z , t ) = -Ux + (t)g(x.y.z) + Re[<i)j)(x,y,z)e ® ] (1) where represents the disturbance of the f l u i d by the ship at the steady speed Ü i n calm water; 0^ represents the o s c i l l a t i n g f l u i d disturbance generated by the i n c i d e n t wave and the motion of the s h i p ;

i , t and OJ^, r e s p e c t i v e l y , are the Imaginary u n i t , time, and the wave-encounter frequency which i s r e l a t e d to the incident wave frequency Ü) by Wg = tü-2ïïUcosij/X, and Re means the r e a l part of what f o l l o w s .

The o s c i l l a t o r y v e l o c i t y p o t e n t i a l (J>oi which Is given i n the form of complex amplitude, can be f u r t h e r decomposed i n t o

6

k=i

k^k (2)

where $ j represents the incident-wave p o t e n t i a l ; <(>j) the d i f f r a c t i o n p o t e n t i a l ; 4»^ the r a d i a t i o n Wave_potential associated w i t h the k t h mode of motion of the s h i p ; and the complex amplitude of the

displacement of the ship from i t s mean p o s i t i o n i n the d i r e c t i o n of the kth mode of motion.

The incident-wave p o t e n t i a l <|>i i s e x p l i c i t l y given by

* j ( x , y . z ) = 3-iK(xcosu + yslnu) + Kz

where g i s the g r a v i t a t i o n a l a c c e l e r a t i o n and K = üj^/g - 2Tr/X i s the wave riumber i n deep water. The d i f f r a c t i o n p o t e n t i a l should s a t i s f y the d i f f r a c t i o n p r i n c i p l e of water waves, i . e . ,

dn on

on the ship h u l l s u r f a c e SQ at i t s mean p o s i t i o n where 3/9n means the

normal d e r i v a t i v e on and the normal vector n i s i n t o the h u l l . The f r e e - s u r f a c e e l e v a t i o n Ç ( x , t ) can be obtained i n terms of * from the B e r n o u l l i equation by

Ç ( x , y , , t ) = - i - - U ^ ) * ( x , y , 0 , t ) + 0 ( * ' )

iüi t - Y *sx^^»y'^^ *(,^<x,y,0)e

Ü) iw t

- 1 ^ (j)oe ^ (5)

where the subscript x means the p a r t i a l d e r i v a t i v e w i t h respect to x and Re i s omitted w i t h the understanding t h a t , h e r e a f t e r , whenever a product i n v o l v i n g e^^^ i s present, only the r e a l part of i t w i l l be r e a l i z e d .

I f Ç i s decomposed i n t o the steady and o s c i l l a t o r y p a r t s , we can d e f i n e where id) t + Coe (6) C g U . y ) = J ^^^^ Co(x,y) = W ( x , y , 0 ) - ±ui^^^] (6b)

If we denote the displacement of the ship from i t s mean p o s i t i o n by ^\^it) where k = l , 2 , . . . 6 , i n d i c a t e s the surge, sway, heave, r o l l , p i t c h and yaw, r e s p e c t i v e l y , we can express the time dependent v e r t i c a l displacement of a point x = ( x , y , z ) on the s h i p , Çy, which i s o f t e n r e f e r r e d to as "absolute motion," by

Ç y ( x . t ) = C3(t) - xC5(t) + yC^(t) (7)

f o r a given i n i t i a l v e r t i c a l p o s i t i o n z . Thus, the r e l a t i v e motion at the point x i s obtained by

5R(iL>t) = ?v(2^,t) - Ç ( x . y . t ) (8)

To determine i f a chosei point on the deck w i l l bê immersed under the f r e e surface or a p o i n t on the ship bottom w i l l be r a i s e d above the f r e e surface we can examine whether | Cr | IS greater than z+Çg-Çg

where z i s the v e r t i c a l coordinate of the point and Çg i s the v e r t i c a l displacement c f the point due to sinkage and t r i m of the s h i p .

As described i n the f o r e g o i n g , i n order to determine the r e l a t i v e motion and the chance of immersion of deck or emergence of ship bottom, we need to know the absolute motion Ç^, and the o s c i l l a t o r y and steady f r e e surface e l e v a t i o n , ÇQ and Çg, and the sinkage and t r i m of the

s h i p . These q u a n t i t i e s can be obtained i f we can determine the v e l o -c i t y p o t e n t i a l s , (î>s» <|>D» and f o r k = l , 2 , . . . , 6 .

STEADY EFFECTS

When a ship i s advancing i n waves, the mean freeboard may be changed by s e v e r a l e f f e c t s . These include sinkage, t r i m and wave pro-f i l e due to pro-forward speed i n calm water, plus a p o s s i b l e a d d i t i o n a l mean s h i f t i n these q u a n t i t i e s caused by the o s c i l l a t o r y motions of

the ship and waves, and i t s forward speed. Various t h e o r e t i c a l methods are a v a i l a b l e to c a l c u l a t e the steady e f f e c t s associated w i t h a ship at a constant speed i n calm water, ranging from simple t h i n - s h i p theory to three-dimensional source panel d i s t r i b u t i o n methods as shown by B a i and McCarthy (1979). From the model experiments conducted i n the past, i t i s w e l l known that the steady wave p r o f i l e Çg can be s i g n i f i c a n t l y i n f l u e n c e d by the sinkage and t r i m of a s h i p ; however, due to the ex-treme complexities i n the mathematical modelling of the bow-wave phenomenon no e x i s t i n g computational methods have succeeded i n c o r -r e c t l y p -r e d i c t i n g the e f f e c t of sinkage and t -r i m on Cg. Since the main focus of the present study i s on the p r e d i c t i o n of the r e l a t i v e motion Cr, no attempts w i l l be made to improve the p r e d i c t i o n of Çg w i t h more r i g o r o u s a n a l y s i s .

In the present study, an e m p i r i c a l method derived by Bishop and Bales (1978) i s used to p r e d i c t calm water sinkage and t r i m , a i ^ a t h i n - s h i p assumption i s made to c a l c u l a t e the bow wave p r o f i l e , the e m p i r i c a l formulas f o r sinkage z ^ , and t r i m 6 Q , are given as quadratic and cubic equations, r e s p e c t i v e l y , which are based on a regression

a n a l y s i s f o r a number of destroyer type h u l l s . The equations may be given i n nondimensional form as

z J L = a . F + a, n

3 • <^n)

(9a)

(9b)

where L i s the l e n g t h o f the ship and F^^ i s the Froude number based on s h i p l e n g t h . The c o e f f i c i e n t s have been derived separately f o r h u l l s w i t h and without bow domes or bulbous bows, and are given i n Table 1.

Table 1 - C o e f f i c i e n t s f o r P r e d i c t i n g Sinkage and Trim i n Calm Water

For Ships With Bulbous Bows -0.00120 -0.01492 -1.1-37 11.793 -23.779 For Ships Without Bulbs 0.00081 -0.02095 ^0.682 8.507 -17.129 OTADY-WAVE POTENTIAL

Assuming t h a t t h é beam B , of the s h i p i s much l e s s than the length L , we use the well-known t h i n - s h i p theory f i r s t introduced by M i c h e l l

( f o r a concise d e s c r i p t i o n see Wehausen (1973)) to obtain (|)g which i s given by * g ( x , y , z ) sC") f ç ( Ç . O - dCdC /(x-Ç)Z+y^+Cz-Ç)^ Tn f f G o ( x - C . y , z + O f ç ( C , O d Ç d Ç s ( 0 (10)

where S denotes the l o n g i t u d i n a l center plane of s h i p ; f i s the h a l f l o c a l beam; and the Green f u n c t i o n representing a source of u n i t strength t r a n s l a t i n g i n the d i r e c t i o n of the p o s i t i v e x - a x i s w i t h a constant v e l o c i t y U at the depth of Ç from the calm water surface i s given by d e / dke»^^^^^> Go(x-Ç,y.z+Ç) = TTÜ' , cos[k(x-g)cose]cos(ky sin0.) kcos^e - g/U-. / • 2t j ^ ( z + ç ) s e c 2 e + - | f - / dOsec^e e _{s i n - ^ ( x - C > s e c e} cos ysinOsec^Ö

)

i n which ƒ means the p r i n c i p a l - v a l u e i n t e g r a l .

Within the f i r s t - o r d e r of B / L , the wave p r o f i l e along the side of the h u l l can be obtained from Equations (6a), (10), and (U> by

Çg(x,0) = ^ *g^(x,0,0) = Reï- j ^ f f fçdÇdÇ sC»)

( / . 2Tr

( V q *^®sece e - dk + ko ƒ sec^edO / " sec^edO ƒ

t7» / - ^ i i / • » kZ dSsecS ƒ e dk + k, ' — S o ^ n ƒ e dk 0 k - kgsec^e • ) ^ f ç d Ç d C ƒ dBsec^e e^asec^ez-,(0) 0 J (12)

where

ko = g/U^ and Z* = C + i(JE-C)cose RADIATION AND DIFFRACTION POTENTIALS

In the f o r e g o i n g s e c t i o n we assumed that the beam of the ship i s much smaller than the l e n g t h . We f u r t h e r assume that the d r a f t T i s a l s o much smaller than the l e n g t h , and that the l o n g i t u d i n a l slope of the ship i s much smaller than the transverse s l o p e . In mathematical expressions, the above assumptions correspond to B / L , D / L , n j - 0(e) f o r small p o s i t i v e number ë where n j i s the x-component of the u n i t normal vector n - ( n j , n2, ng) on the s h i p s u r f a c e . We o f t e n c a l l the.body which f i t s the geometric property described above a slender body. In p a r a l l e l w i t h t h i s slender body geometry, i f we assume that the disturbances of the f l u i d due to the wave d i f f r a c t i o n and the o s c i l l a t o r y body motion are 0(e) i n the x - d i r e c t i o n compared to those i n the y - and z - d i r e c t i o n , we can approximate {t)D and f o r k F l , 2 , . . . , 6 by the s o - c a l l e d s t r i p theory, that i s , f o r g i v ö i x these p o t e n t i a l s can be treated as f u n c t i o n s of y and z o n l y .

For the r a d i a t i o n p o t e n t i a l s 4>jj^ f o r k=2, 3 and 4, the twodimens i o n a l twodimens o l u t i o n f o r i n f i n i t e l y long h o r i z o n t a l c y l i n d e r twodimens having twodimens h i p -l i k e c r o s s s e c t i o n s i s we-l-l-known [Tasai (1959), P o r t e r (1960), and Frank (1967)]. Then invoking the slender-body assumption, we can determine [Salvesen et a l . (1970)1 that

(j)-(y,z;x) C13)

(14)

The d i f f r a c t i o n p o t e n t i a l i s obtained i n the same manner as f o r the r a d i a t i o n p o t e n t i a l s 0? and <^3 [Lee (1982)] except f o r imposing the kinematic body-boundary coriditions as f o l l o w s :

9(j) jr

— L « - û)Ç. e tN2Slnucos(Kysiriy) 9N A 2

+ N , a i n ( K y s i n u ) ] = the odd p a r t of

9*3

I n i u ç ^ e- [N2Sinîisin(Kysiny)

- NjCOsXKysinli) ] « the even part of

(- ^ *x)

where the t i l d a sign i s üsed to d i f f e r e n t i a t e the p o t e n t i a l s from and <|)3 and N « (ng, n^) i s the u n i t normal vector i n the y - z plane. Then we obtain the s o l u t i o n f o r *j) by

(17) S t r i c t l y speaking, *d obtained by Equation (17) may only be v a l i d f o r p - ±n/2 (beam waves) or r e l a t i v e l y long waves of order of ship l e n g t h . However, the approximation of Equation (17) i s maintained i n the present work w i t h the a n t i c i p a t i o n that It w i l l not s i g n i f i c a n t l y degrade our s o l u t i o n f o r the r e l a t i v e motion.

MOTION OF A SHIP

The displacement of a ship from i t s mean e q u i l i b r i u m p o s i t i o n i n s i x degrees of freedom i s obtained by s o l v i n g two sets of l i n e a r i z e d coupled equatipns of motion which are shown below

( A i i + M)Çi + B i i C i + MzqCs =• F i e

1(0 t e

<A33 + M)i3 + B33C3 + C33Ç3 + A3,t*5 + B 3 J , + C33Ç3 « F.e

iü)_t

(18a)

(18b)

M^o^i + A53Ç3 + B „ t 3 + C53C3 + (A3, + 1^)1^

iw t + B55C5 + GssCs - Fge _(18c) (A22 + M)?2 + B22Ç2 + ( A j ^ - Mzo)ë^ + B j ^ i i u t e (19a)

(A^2 - MZ(,)C2 + \2^2 + (A,^ + I^)Ç^ + B^,Ç^ iü) t e + Cj^^Ç^ + (A^6 - 1^6)^6 + B^gÇg = F^e A62^2 + S - ^ ' + ^^««^ - ^•^'^^^•^ + ^'^»'^ (19b) 6 it + (Açg+ Ig)Ce + Bg,Ç, - F , e itl) t e (19c) In the foregoing equations M i s the mass of the s h i p ; I]^, I 5 and Ig are the mass moment of I n e r t i a about the x , y , and z a x i s , r e -s p e c t i v e l y , and I^g = f I j pjj^xz dv where I^S dv i -s the -ship volume i n t e g r a l , the point mass d e n s i t y ; C^r^'s are the h y d r o s t a t i c r e s t o r -ing c o e f f i c i e n t s which are given by

Cas »

^^Jf ^^^^

where p i s the i n t e g r a l ever the waterplane area and GM and GM^ are, r e s p e c t i v e l y , the transverse and I p n g i t u d i n a l metacentric heights. The hydrodynamic c o e f f i c i e n t s A ^ j , B y and represent the added masses, damping c o e f f i c i e n t s , aria wave e x c i t a t i o n f o r c e s .

According to the s t r i p theory, the expressions f o r the hydro-dynamic c o e f f i c i e n t s i n terms of the r a d i a t i o n p o t e n t i a l s can be given by: (see e . g . , Lee (1976))*

(21)

The d i f f e r e n c e i n the sign from Lee (1976) i s due to the change i n the harmonic time dependence from e~^*^ct giüJet ^ present work.

^ i k '

ji^

v |

H^i^ = jf^'(<i)^j,(y,z;x) - 2UN36^j. + ^ÜNjó^g)*^ ds (23) So

So

and i s the Kronecker d e l t a .

s i n c e the equations o f mption g i v e n by (18) and (19) are l i n e a r , we can r e a d i l y set

iü) t

where f ^ i s the complex amplitude of the i t h mode o f motion indep^dent of time. S u b s t i t u t i o n o f Equation (25) i n t o Equations (18) and (19) y i e l d s two a l g e b r a i c equations, which can be e a s i l y inverted to f i n d the s o l u t i o n s f o r ^± f o r 1 = 1 , 2 , . - . , 6 . The amplitude and phase of each mode of motion then can be obtained by

= arctan ( I m X i / R e f i ) (26b)

where i s the phase lead r e f e r r e d to the i n c i d e n t wave c r e s t at the coordinate o r i g i n , and Im means the imaginary part of what f o l l o w s . EXPERIMENTAL PROCEDURE

The r e l a t i y e motion experiments were conducted i n the Maneuvering ^ d Seakeeping (MASK) Basin at D'ntSRDC. This basin i s U O m long by 73 m wide. Pneumatic wavemakers are mounted ori two adjacent sides of the b a s i n , and there are s l o p i n g beaches on the opposite sides to absorb the waves. The towirig c a r r i a g e i s supported from a r o t a t a b l e

b r i d g e . By proper alignment of the bridge angle and appropriate choice of wavemaker bank, any desired heading of a s h i p model r e l a t i v e t o wave d i r e c t i o n can be obtained.

Two ship h u l l fprms were selected f o r the purppse of v a l i d a t i o n . The f i r s t , designated Ship A, i s a modem, h i g h speed c o n t a i n e r s h i p . The second, designated Ship B , i s a t y p i c a l Naval combatant h u l l f o r m . Body plans of the two h u l l s are shown i n F i g u r e 2, and t h e i r p r i n c i p a l c h a r a c t e r i s t i c s are shown i n Table 2. Ship A has a r e l a t i v e l y l a r g e

bulbous bow and c r u i s e r s t e r n , while Ship B has no bulb but does have a wide transom s t e m .

Body Plan fpr Ship A Body Plan for Ship B

Hgure 2 - Body Plans for Ship A and Ship B

Table 2 - P r i n c i p a l C h a r a c t e r i s t i c s pf Ship A arid Ship B

Ship A Ship B LWL 274.3 m 124.4 m B 32.2 m 13.7 m T 10.4 m 4.5 m ^B 0.5,3 0,46 c 0.94 0.75 X 0.56 0.61 1.28 m 1.33 m

The models were attached to the c a r r i a g e by means of a heave s t a f f . A r o l l - p i t c h gimbal was attached to the bottom of the s t a f f , so that the models had three degrees of freedom: heave, r p l l , and p i t c h . The mpdels were cpnstralned from surging, swaying pr yawing. In a d d i t i o n , experiments on one model (Ship A) were done w i t h the h u l l r i g i d l y r e s t r a i n e d , i n order to measure d i f f r a c t i o n e f f e c t s i n waves.

Because of varying requirements regarding the placement of i n s t r u -mentation on each model they had d i f f e r e n t r a d i i of g y r a t i o n f o r p i t c h

and r o l l . Ship A had a p i t c h radius of 0.25 times the l e n g t h , but because of c e r t a i n heavy instruments on c e n t e r l i n e the r o l l radius was only 0.24 times the beam. Chi the other harid. Ship B had a more r e a l i s -t i c r o l l radius of 0,38 -times -the beam, bu-t because of ins-trumen-ts located f a r forward the p i t c h radius was f o r c e d to a value of 0.27 times length.

A l l experiments were performed i n regular head (U - 180 ) and bow (Vi = 2^5 ) waves. A l l measurements were made e l e c t r o n i c a l l y and f e d to a carriage-mounted d i g i t a l computer. R i g i d body motions were mea-sured by potentiometers mounted to the heave s t a f f . Incident wave e l e v a t i o n was measured by an u l t r a s o n i c transducer mounted a p p r o x i -mately one-half model length i n f r o n t of the bow. R e l a t i v e motions alongside the h u l l s were measured using r e s i s t a n c e - t y p e wave probes. These probes were f lush-iaounted i n the s i d e of the h u l l i n the case of Ship A, but were mounted s l i g h t l y o f f the side on short Outriggers f o r Ship B. U l t r a s o n i c transducers could not be used f o r r e l a t i v e motipn measurements near a h u l l , since the h u l l side would r e f l e c t the sonic pulse and cause spurious measurements.

A l l data were harmonically analyzed and the f i r s t harmonic of the incident wave and a l l responses were used to c a l c u l a t e l i n e a r trarisfer f u n c t i o n s . Mean values were c a l c u l a t e d by averaging s i g n a l s over an integer number of c y c l e s of the f i r s t harmonic.

RESULTS

The presentation and d i s c u s s i o n of r e s u l t s may be separated i n t o the kinematic and nonkinematic components of r e l a t i v e motion. The kinematic components are simply the r i g i d body motions and the incident wave e l e v a t i o n . The vector combination of these components, taking proper account of phase angles, r e s u l t s i n the kinematic estimate of r e l a t i v e motion. The a d d i t i o n a l dyriamic components are those due to the d i f f r a c t i o n of the i n c i d e n t wave by the presence of the s h i p plus the r a d i a t i o n of waves due to the o s c i l l a t i o n of the s h i p . F u r t h e r -more, there are the mean s h i f t s caused by the steady forward motion of

the s h i p .

The mean sinkage and trim measured i n calm water are compared to c a l c u l a t i o n s using Equations (93) and (9b) i n Figure 3. The agreement f o r sinkage i s g e n e r a l l y s a t i s f a c t o r y f o r both s h i p s , but i s l e s s so f o r t r i m . Part of t h i s l a t t e r discrepancy i s caused by the mean mea-sured t r i m angles being of the order of magnitude of one-tenth degree, which i s near the l i m i t which the instruments can r e s o l v e . However,

f ó r the case of Ship A at Fjj «= 0.30, an absolute v e r t i c a l motion t r a n s -ducer was mounted at the s t e m , which confirmed that there was a s l i g h t bow up p i t c h a t t i t u d e .

Steady wave p r o f i l e s are compared i n Figure 4 . The measured values have been corrected f o r sinkage and t r i m . The r e s u l t s are i n

q u a l i t a t i v e agreement w i t h Equation (12) f o r both s h i p s . However, the shape of the wave p r o f i l e f o r Ship A i s not w e l l predicted at Fn = 0.30, p o s s i b l y because of i t s bulbous bow (see Figure 2 ) , and the wave p r o -f i l e -f o r Ship B does not r i s e near the bow at Fn = 0.15, as p r e d i c t e d by Equation (12).

Predicted and measured r i g i d body t r a n s f e r f u n c t i o n s are p r e s ö i t e d i n Figures 5 and 6 f o r Ship A and Ship h, respectivelyw In these

f i g u r e s , heave has been nondimensionalized by the i n c i d e n t wave a m p l i -tude, w h i l e p i t c h and r o l l have been nondiinerisionalized by wave s l o p e . P o s i t i v e heave i s d e f i n e d upward (see Figure 1 ) , while p o s i t i v e p i t c h i s bow down and p o s i t i v e r o l l i s starboard s i d e down. Phase angles are defined as phase leads w i t h respect to maximum wave e l e v a t i o n at the l o n g i t u d i n a l center of g r a v i t y . Absolute v e r t i c a l motdLon near the bpw ( S t a t i o n 2 f o r Ship A and S t a t i o n 2.5 f o r Ship B) has been c a l c u -l a t e d according to Equation (7) using experimenta-l and t h e o r e t i c a -l heave and p i t c h t r a n s f e r f u n c t i o n s , and i s shown i n Figure 7.

The c o r r e l a t i o n between theory and experiment i n head waves f o r these motions i s g e n e r a l l y s a t i s f a c t o r y f o r Ship A at the lower speed

(Fjj = 0.10). However, at the higher speed (Fn = 0.30) the measured heave does not show the strong resonant peak predicted by s t r i p theory, and p i t c h magnitudes are a l s o somewhat l e s s than p r e d i c t e d . As a r e -s u l t , the predicted ab-solute motion i n Figure 7 i -s -s u b -s t a n t i a l l y higher than measured at Fn - 0.30. In the case of Ship B , the c o r r e l a t i o n f o r heave i s e x c e l l e n t , but measured p i t c h i s larger than predicted a t both speeds. Consequently, the measured absolute motion shown f o r Ship B

i n Figure 7 i s s l i g h t l y greater than p r e d i c t e d .

The trends i n bow waves (u = 225°) are s i m i l a r f o r p i t c h and heave. In the case of r o l l , the c o r r e l a t i o n between theory and e x p e r i -ment i s poor. The magnitude and frequency of the peak are poorly

pre-d i c t e pre-d . I t i s suspectepre-d that these r e s u l t s are causepre-d by a combination of inaccurate r o l l damping estimates, together w i t h the f a c t that the model t e s t s were conducted w i t h sway and yaw r e s t r a i n e d . In any case,

as w i l l be shown below, r o l l has only a minor e f f e c t on v e r t i c a l r e l a -t i v e mo-tion a-t -the bow i n bow waves. I -t w i l l have a more impor-tan-t e f f e c t i n the v i c i n i t y of the midship arid f o r beam or stern waves.

The v a r i o u s components of r e l a t i v e motion are discussed i n d e t a i l below- At low speeds, the previous r e l a t i v e motiori theory which only coinputes kinematic terms appears to give adequate p r e d i c t i o n s . The consequences of l a r g e r e l a t i v e motions are more severe at high speeds, and the dynamic components associated w i t h d i f f r a c t i o n and r a d i a t i o n are a l s o l a r g e r . Therefore, i n the f o l l o w i n g discussions emphasis i s placed on the higher speed case, Fj^ «= 0.30.

For completeriess, a l l the components of absolute and r e l a t i v e motion f o r Ship A at one speed and heading are tabulated i n the Appen-d i x .

-xm is -.003 < -1 I S _-.002 (O -.001 — i — I — r — CALCULATED O MEASURED J L .3 UJ 2 g BOW a DOWN 9 •

1

Hgure 3 - Comparison of Predicted and Measured Sinkage and Trim in Calm Water

.010 .005 .010 .005 1 1 r THEORY EXPERIMENT Fn = 0.30 • T Fn = 0.10 SHIP A THEORY EXPERIMENT F„ = 0.15 — o F „ = 0 . 3 0 - - - ^ • « SHIP B 1 STAO

Rgure 4 - Comparispn pf Theoretical and Experimental Bow Wave Prbfiles in Cbim Water

< S! < UI X 1.5 1.0 0.5 180° S* O u = 225 = Î80° THEORY MEASURED Fn •= 0-1 F_ = 0 . 3 — - — < -180° li) X Ü I*--Q. 0-5 -180' " ^ T ^ n g c j g a a ft Q u = 225" u = 180 u = 225

Figure 5 - Thepretical and Experimental Resülts fpr Rigid Body MPtions for Ship A

I 1 1 1 u = 225° I— 1 1 THEOTY MEASURED F- « 0.15 ^ _ 0 u s 180<* -180° u = 225° as 1.0 2J) 2.5 < •w*? 3 -O flC -180' H = 225° OS l i ) 1.S 2.0 2,5 3.0

Rgure 6 — Tiieoretical and Experimental Results f w Rigid Body Motions for Ship B

~ 2 1 1 — — ^ n f THEORY r MEASURED SHIP A _ - F„ = 0.10 O HEAO WAVES fl . Fn - 0'30

•

-^•^-^

Rgure 7 — Theoretical and Experirriehtal R^ults for Absolute Vertical Motion Near the Bow for Ship A and Ship B in

Head and Bow Waves

The components of wave e l e v a t i o n due to motion ( r a d i a t i o n compo-nent) and d i f f r a c t i o n are shown i n Figures 8 and 9 f o r Ship A. In Figure 8, the i r i d i v i d u a l component magnitudes are shown, together w i t h t h e i r phase angles while iri Figure 9 the p r e d i c t e d components are shown combined, r e s u l t i n g i n the t o t a l m o d i f i e d wave e l e v a t i o n . The combination of i n c i d e n t plus d i f f r a c t e d wave i s shpwn i n Figure 8 f o r comparison to experimental r e s u l t s f o r Sl:iip A, sirice the d i f f r a c t f f i i component can only be measured i n combination w i t h the i n c i d e n t wave i n an experiment. As showri, the predicted d i f f r a c t e d wave i s only of s i g n i f i c a n t magnitude f o r short wavelengths i n bow waves, and even i n t h i s wavelength region i n head waves., the d i f f r a c t i o n e f f e c t i s q u i t e anall.. On the c o n t r a r y , the measured d i f f r a c t i o n e f f e c t on Ship A was s i g n i f i c a n t over the e n t i r e wavelength range i n both head and bow waves. Regarding the phase a n g l e s , i t i s important to note t h a t the phase angles are changing r a p i d l y i n the r e g i o n of X / L = 1.0, and i t

i s i n t h i s regiori that the r e l a t i v e motion t r a n s f e r fiinctioris reach t h e i r peak v a l u e s . It i s a l s o s i g n i f i c a n t that the r a d i a t i o n and d i f f r a c t i o n components at long wavelengths have approximately the same niagriitude, while t h e i r phases are approximately 180 degrees a p a r t . This i n d i c a t e s that these e f f e c t s w i l l tend to c a n c e l out to make the iricident wave remain unmodified at long wavelengths, even though the

i n d i v i d u a l components may be s i g n i f i c a r i t . F i n a l l y , i t should be noted i n Figure 8 that the p r e d i c t e d d i f f r a c t i o n component i s approximately 90 degrees out of phase from the i n c i d e n t wave at a l l wavelengths. However, the phase of the measured i n c i d e n t plus d i f f r a c t e d wave was

SP close to that of the i n c i d e n t wave alone that the d i f f r a c t i o n

compo-nent appears to be i n phase w i t h the i n c i d e n t wave, w i t h i n exper^imental accuracy. The f a c t that the measured amplitudes of the i n c i d e n t plus the d i f f r a c t e d waves are greater than those of the i n c i d e n t wave alone implies the closeness of the phases f o r these two wave cœnponents. On t h i s basis of r e a s o n i n g , the discrepancy between the p r e d i c t e d and measured r e s u l t s of the amplitudes of the i n c i d e n t plus the d i f f r a c t e d waves at long wayelengths shown i n Figure 9 on the weather s i d e could be r e s u l t i n g from the erroneous p r e d i c t i o n of the phase angles o f the d i f f r a c t e d waves by the s t r i p theory employed i n t h i s work. As w i l l be shown below, there i s a l s o reason t o b e l i e v e that the phase angle of the r a d i a t e d waves, as p r e d i c t e d by s t r i p theory, a l s o d i f f e r s from the a c t u a l r a d i a t e d phase angle by approximately 90 degtees.

tNOOENT ft DIFFRACTB) WAVE OIFFRACTB} WAVE

INCIDENT WAVE MOflbN (^ERATEO WAVE

MEASURED INOOENT PUJS OimiACm}

;?

-HEAD yVAVES ' ' BOW WAVES

180° g 90

<

Rgure 8 — Wave Anrplitiules and Phases Due to Motion and Diffraction at Station 2 on Ship A at Fn = 0.30 for

< UÎ a Q. < Ul â i

<

INCIMNT PLUS DEFFRACTED WAVE MOTION GENERATED WAVE _ - - TOTAL WAVE

• MEASURED INDDENT PLUS DIFFRACTED 1 1 - WEATHER^ SIDE - WEATHER^ SIDE

f

Rgure 9 — Amplitudes of Incident Plus Diffracted Waves, Motipn Generated Waves and Total Waves at the Weather and Lee Sides

at Station 2 of Ship A at F„ ^ 0.30 for Bow Waves

The r e l a t i v e motion t r a n s f e r f u n c t i o n s of both Ship A and Ship B are shown i n Figure 10 f o r head waves and Figures 11 and 12 f o r bow waves at two d i f f e r e n t speeds* The agreement between theory and e x periment i s somewhat v a r i a b l e , and i n g e n e r a l the i n c l u s i o n of d i f f r a c -t i o n and r a d i a -t i o n e f f e c -t s does no-t provide a s i g n i f i c a n -t improvemen-t. In f a c t , i n spme cases the p r e d i c t i o n i s s l i g h t l y worse when these e f f e c t s are added. The d i f f e r e n c e between the weather and l e e s i d e s i a bow waves i s s m a l l , except i n short waves where the d i f f r a c t i o n component i n the new theory p r e d i c t s a s h e l t e r i n g e f f e c t . The e f f e c t of r o l l i s only n o t i c e d at long wavelengths, where the absolute motion as p r e d i c t e d by Equation (7) shoWs a s l i g h t d i f f e r e n c e between weather and l e e s i d e s .

C a r e f u l examination of a l l the components of r e l a t i v e motion ( e i t h e r p r e d i c t e d or measured) shows that r e l a t i v e motion i s s t r o n g l y a f f e c t e d by both magnitudes and phases of these components. In order to mpre c l e a r l y i l l u s t r a t e these e f f e c t s , Figures 13-15 are presented i n the form o f v e c t o r diagrams. In these i l l u s t r a t i o n s , the ccmplex amplitude of the v a r i o u s components (motions and waves) are represented as v e c t o r s wliose magnitude i s the absolute v a l u e of the q u a n t i t y and whose phase i s as d e f i n e d i n Equation (26b). Phase angles are measured

O MEASURED PRIENT THEORY PREVIOUS THEORV SHIPB STATION 2.5 SHIP A STATION Z Fn = aio •iin»jr SHIP A STATION 2 Fn » lUO ' 1 —1 0 SHIPB O 0 STATIPN 2.6 " Fn - 03,0 " o - / -1 _{..1 —} m.

Rgure 10 — Theoretical and Experimental Results for Relative IMotion Near the Bow for Ship A at Fp^O.IO and 0.30 and

for Ship B at F^^O.IS and 0,30 ih Head Waves

T SHIP A WEATHER SIDE Fn = 0.10 O MEASURED PRESENT THEORY PREVIOUS THEORY SHIP A LEE SIDE SHIP A WEATHER SIDE Fn = 0.30 SHIP A LEE SIDE

Figure 11 — Theoretical and Experimental Results for Relative Motion on the Weather and Lse Sides at Station 2 fpr

SHIPS WEATHER SIDE F_« 0.15 O MEASURED PMSÈNT THEORY — — PREVIOUS THEORY T SHIP B L^SIDE F^ S ai8 H SHIPB LEE SIDE SHIPB W»THER SIDE F.« 0.30 Œ F „ = a 3 0 i 2 3 1 2 A/L . Rgure 12 - Thepretical and Experimental Results for Relative Motion

on the Weather and Lee Sides at Station 2.5 for Ship B at Fn = 0.15 and 0 JO in Bow Wayes

MAGNITUDE MOOIFIEO . (DIFFRACTED AND \ ^ ^ BA04ATED SHlFTEaWJ ± 1 8 0

I

Figure 13 - Vector Diagrain of Calculated Wave Components at Statioin 2,5 On Ship B at J/L = 1.0. Fn=

>

I

l i t

§1

51

LU Q 3 Z O

<

s

as p o s i t i v e counterclockwise from the r e a l a x i s , and a l l amplitudes havé been nondimensionalized by the Incident wave amplitude.

The various components of c a l c u l a t e d o s c i l l a t o r y wave motion at S t a t i o n 2.5 on Ship B are i l l u s t r a t e d as vectors i n Figure 13 f o r one speed and one wavelength. The sum of the i n c i d e n t , r a d i a t e d (from both heave and p i t c h ) and d i f f r a c t e d waves i s defined as the " m o d i f i e d " wave. In a d d i t i o n , the e f f e c t on the m o d i f i e d wave of s h i f t i n g tlie phases of the radiated and d i f f r a c t e d waves by -9Ö degrees i s shown.

The complete vector construction of r e l a t i v e motion f o r the same c o n d i t i o n i s shown i n Figure 14. The absolute v e r t i c a l motion i s simply a v e c t o r combination of heave and p i t c h according to Equation

(7), The r e l a t i v e motion denoted " p l d " i s the absolute motion w i t h the undisturbed i n c i d e n t wave subtracted. The '*new" r e l a t i v e motion includes the p r e d i c t e d r a d i a t i o n and d i f f r a c t i o n e f f e c t s . In other words, i t i s constructed by s u b t r a c t i n g the modified wave from the absolute v e r t i c a l motion. Âs can be seen i n Figures 10 and 14, the new method shows no improvement over the p l d , except f o r the phase angle, when compared t o experimentally measured r e l a t i v e motion. How-ever, when the radiated and d i f f r a c t e d phases are s h i f t e d by -90 degrees, the agreement i s s i g n i f i c a n t l y improved. Although not i l l u s -t r a -t e d , a s i m i l a r r e s u l -t i s found a-t o-ther waveleng-ths i n head seas.

The various components of r e l a t i v e motion i l l u s t r a t e d f o r Ship B i n Figures 13 and 14 are based on s t r i p theory c a l c i i l a t i o n s of absolute motions, r a d i a t e d and d i f f r a c t e d waves, since i n t h i s p a r t i c u l a r case the experimentally measured absolute motions agreed c l o s e l y w i t h the predicted v a l u e s , and np measurements of the r a d i a t e d or d i f f r a c t e d components were a v a i l a b l e . In the case of Ship A, there was c o n s i d e r -able discrepancy i n the absolute motion at high speed, p a r t i c u l a r l y i n the heave motion. Furthermore, experimental measurem^ts of d i f -f r a c t i o n e -f -f e c t s (see Figure 8) were a v a i l a b l e -f o r t h i s h u l l -form and a l s o d i f f e r e d s i g n i f i c a n t l y from the p r e d i c t e d v a l u e s . Therefore, the complete r e l a t i v e motion was constructed i n Figure 15 using measured values of the v a r i o u s components, where a v a i l a b l e . Since no forced o s c i l l a t i o n experiments were done to oieasure the r a d i a t e d wave compo-n r ä t , t h i s compocompo-necompo-nt was determicompo-ned by combicompo-nicompo-ng the r a d i a t e d wave p o t e n t i a l s obtained by the s t r i p theory w i t h experimentally d e t e r -mined complex motion amplitudes The absolute v e r t i c a l motion at S t a t i o n 2 was a l s o c a l c u l a t e d from the measured motions, r a t h e r than being d i r e c t l y measured w i t h a displacement transducer at that s t a t i o n . The c a l c u l a t i o n of the r e l a t i v e motion by the new method shows s l i g h t l y better agreement w i t h the measurement than the o l d method. However, when the estimated radiated vave component i s phase s h i f t e d -90 de-grees, there i s e x c e l l e n t agreement In both maghitude and phase.

The r e s u l t s presented above have been f o r a s t a t i o n near the bow f o r both ships i n head and bPW waves. R e l a t i v e mptions near the bow u s u a l l y a f f e c t s h i p operations the most because of t h e i r i n f l u e n c e on such phenomena as slamming or deck wetness. However, there may be cases where r e l a t i v e motions f u r t h e r a f t along the h u l l are of concem, p a r t i c u l a r i y where freeboard i s s m a l l or operations such as r e p l e n i s h -ment have to be c a r r i e d out over the s i d e . Figiire 16 i s presented to

S < «w. «ut e < «ut —1 1 r — ! PRESENT THEORY 1 A/L = 2.0 _ PREVIOUS THEORY

-

-

--

_•

•

•

Stations for Ship B at F„ e 0, 3 0 in Head Waves for X/L==Z0,1.0and 0.5

show the r e l a t i v e motion along the f u l l l e n g t h of Ship B f o r several values of X / L . The agreement between p r e d i c t i o n and measurement i s g e n e r a l l y s a t i s f a c t o r y except at S t a t i o n 2.5 at X / L = 1.0 and 0 . 3 . I n t e r e s t i n g l y , the agreement at the bow ( S t a t i o n 0) i s b e t t e r than at S t a t i o n 2 . 5 . This i s apparently due to the f a c t that there i s l i t t l e or no r a d i a t i o n or d i f f r a c t i o n at t h i s s t a t i o n . The bow of Ship B has a raked stem l i n e and no bulb or dome.

Qae unexpected phencœaenon discovered In the experiments was a

s h i f t i n the mean values of v a r i o u s q u a n t i t i e s w h ^ running i n waves, as compared to t h e i r values i n calm water at the s ^ e speed. Some of these r e s u l t s are presented i n Figures 17 and 18, f o r Ship A. Mean s h i f t s were detected i n heave, p i t c h and r e l a t i v e motion near the bow. The s i g n a l from a v e r t i c a l absolute motion transducer at the s t e m was a l s o a v a i l a b l e and served to i n d e p e n d ^ t l y c o n f i r m the heave and p i t c h mean s h i f t s . As shown i n Figure 17, the heave showed a s l i g h t r i s e (pr

decrease i n sinkage) i n the wavelength region near X / L = 1.0, compared to the l e v e l i n calm water. S i m i l a r l y , p i t c h had a s m a l l bow up ten-dency at the same wavelengths. V e r t i c a l motion at the s t e r n became more negative (increased s i n k a g e ) , which i s consistent w i t h the bow up p i t c h s h i f t . The net r e s u l t of these s h i f t s i s that the absolute v e r t i c a l motion of the forward part of the ship increases i n waves,

compared to the t r i m i n calm water at the 3am:e speed.

The mean value of r e l a t i v e mption at S t a t i o n 2 becoiaes l e s s nega-t i v e i n waves (bow i s r i s i n g r e l a nega-t i v e nega-to l o c a l f r e e surface e l e v a nega-t i o n ) which i s consistent with the d i r e c t i o n of mean s h i f t i n absolute

motion. However, the magnitude of r i s e i n r e l a t i v e motion i s less than would be expected from the r i s e i n absolute v e r t i c a l motion at t h i s s t a t i o n . This implies that there i s an absolute r i s e i n the mean f r e e s u r f a c e near the h u l l . At S t a t i o n 0, on the other hand, the mean s h i f t i n r e l a t i v e motion i s negative at a l l wavelengths. This means that there i s a mean r i s e i n the f r e e surface which i s even greater than the mean r i s e of the bow.

The data i n Figures 17 and 18 are a sample of the mean s h i f t s observed. S i m i l a r r e s u l t s were fpimd i n bPw waves, and i n the data f o r Ship B at F^ ° 0.30. Mean s h i f t s were not c l e a r l y detectable f o r e i t h e r ship at the lower speeds. At present, we know of no a n a l y t i c a l p r e d i c t i o n method f o r these mean s h i f t s . I t i s suspected that they are caused by quadratic i n t e r a c t i o n s between various components of the f i r s t order o s c i l l a t o r y p o t e n t i a l . In other words, i t could be analo-gous to added r e s i s t a n c e or d r i f t f o r c e s , except i n t h i s case the mean f o r c e and moment of i n t e r e s t are the heave and p i t c h e x c i t a t i o n . Since t h i s i s s t i l l an open question, and we do not have s u f f i c i e n t e x p e r i -mental data to determine whether the s h i f t s vary l i n e a r l y ,

qiiadrat-i c a l l y or qiiadrat-i n some other f a s h qiiadrat-i o n w qiiadrat-i t h wave amplqiiadrat-itude, the data of Figures 17 and 18 have simply been made nondimensional w i t h respect to shtp length.

-.003

3=1

-.001

1 1 CALM WATER MEAN

O < 0 0 o ° ° 0 o — — I I z o. .

h

w

-.002 o CALM VVATER MEAN

--.001 r-1 r-1 « _u

I.

2. CALM WATER MEAN

À/L

Rgure 17 ~ Change öf Mean Values of Rigid Body Motions in Waves for Ship A In Head Seas, Fn=0.30

-.015 -.010 O 0 ' STATION 0 O 0 c O O O O O 1

GALM WATER MEAN

1

0 1 2 3

GALM WATER MEAN ' STATION 2 0 O O O O ^ 0 0 0

0 O

1 •

A/L

Figure 18 — Change of Mean Vahie of Relative Motion at Stations 0 and 2 for Ship A

in Head Waves, Fn = 0.3p

The mean s h i f t s i n waves are q u i t e s m a l l compared to the s t a t i c f r e e b o a r d , as are the s h i f t s due to sinkage, t r i m and bow wave p r o f i l e i n calm water. However, accurate p r e d i c t i o n of these components i s u l t i m a t e l y as important as the p r e d i c t i o n of t r a n s f e r f u n c t i o n s of the o s c i l l a t o r y components, because the frequency of occurrence of e v ö i t s such as slamming and deck wetness i n random seas i s a s e n s i t i v e f u n c -ti£»i of the mean d r a f t or f r e e b o a r d . For i n s t a n c e , imder the assump-t i o n of R a y l e i g h ' s law of p r o b a b i l i assump-t y d i s assump-t r i b u assump-t i o n assump-the average number of occurrences per hour that a given l e v e l F i s Rxreefled. i s given by:

2 ÎT - 3600 n ^ _27T

ft)

Where and 0^ are the standard d e v i a t i o n of r e l a t i v e motion and r e l a t i v e v e l o c i t y ^ r e s p e c t i v e l y , and the r e l a t i v e v e l o c i t y standard d e v i a t i o n i s obtained from the second moment of the r e l a t i v e motion spectrum. Thus, the frequency of occurrence i s an e x p o n e n t i a l f u n c t i o n of the s q ù a r e of the l e v e l to be exceeded (which could be freeboard i n

the case of deck wetness or p r o p e l l e r depth i n the case of p r o p e l l e r emergence), and even a sinall change i n the mean can have a r e l a t i v e l y l a r g e e f f e c t on the frequency of occurrence.

CONCLUDING REMARKS

The comput:ed and measured r e s u l t s r e v e a l no c o n c l u s i v e evidence that i n c l u s i o n o f the d i f f r a c t e d and motion-generated waves» as com-puted by s t r i p theory, provide much improvement f o r the computation of r e l a t i v e motions, compared to the o l d method f o r which only tbe k i n e -matic terms are i n c l u d e d . However, there i s strong evidence from the experimental r e s u l t s that the magnitudes of these terms are s i g n i f i -cant and t h a t inproved p r e d i c t i o n of th& associated phase angles w i l l n o t i c e a b l y improve the c o r r e l a t i o n between p r e d i c t i o n s and experiments. I t i s a i s o f e l t that improvements i n p r e d i c t i o n of the magnitude o f the d i f f r a c t i o n component must be made.

The d e f i c i e n c i e s of s t r i p theory i n p r e d i c t i n g r e l a t i v e motion may came from several sources. The f r e e surface e l e v a t i o n from B e r n o u l l i ' s equation (Equation (5)) Includes a term p r o p o r t i o n a l to the product of forward speed and the « V I A T d e r i v a t i v e of the o s c i l l a t o r y p o t e n t i a l *

Since i n s t r i p theory t h i s term i s considered to be of higher prder than the time d e r i v a t i v e term, i t has npt been included i n the present c a l c u l a t i o n s . However, f o r ain a c t u a l h u l l form which has s u b s t a n t i a l l o n g i t u d i n a l curvatures i n the bow and s t e m regions^ t h i s term may be c o m p t a b l e i n magnitude to the time d e r i v a t i v e term. The nature o f the term could also expain why phase angle computations appear to be l e s s s a t i s f a c t o r y at higher speeds and i n head waves.

I n order to accomplish a s i g n i f i c a n t Improvement i n the p r e d i c t i o n of r e l a t i v e motion, i t i s s t r o n g l y f e l t that improved t h e o r e t i c a l

methods beyond the s t r i p theory and t h i n - s h i p theory f o r f r e e - s u r f a c e motion as w e l l as body motion I n c l u d i n g the n o n l i n e a r e f f e c t s should be developed i n a s s o c i a t i o n w i t h c a r e f u l l y conducted systematic e x p e r i -ments.

ACKNO\a.EDG£MmS

The a n a l y t i c a l work on s h i p r e l a t i v e mptions was sponsored by the Naval Sea Systems Command under the General Hydromechanics Research

(GHR) Program administered by the David W. Taylor Naval Ship Research and Development Center. Experimental work was sponsored under the GBR program, the U . S . Navy S h i p s , Subs and Boats E x p l o r a t o r y Development Program, and the U . S . Coast Guard Commercial Vessel Safety Program i n the O f f i c e of Merchant Marine S a f e t y . Much of the experimental work was i n i t i a t e d by the l a t e H . K . Bales and c a r r i e d out by Harry D. Jones and Richard C. Bishop. C a l c u l a t i o n s of calm water wave p r o f i l e s were provided by D r . Y . S . Hong.

References

B a i , K . J . , and J . H, McCarthy (Eds.) (Nov. 1979). Proceedings of the Workshop on. Ship Waver-Res i s tance. Computations. DTNSRDC.

Bales, N. K. e t a l . (Nov, 1975). V a l i d i t y of a . S t r i p Theory-Linear Superposition Approach.to P r e d i c t i n g - P r o b a b i l i t y of,Peck,Wetness f o r a F i s h i n g . V e s s e l . DTNSRDC Report SPD-e43-01.

Bishop, R. C , and N. K. Bales (Jan. 1978). A Synthesis. of Bow Wave P r o f i l e and Change of L e v e l Data f o r Destroyer^Type. H u l l s with A p p l i c a t i o n to. Computing ; Minimum. Required Freeboards. DTNSRDC Report SPD-81i-01.

Cox, G. G., and D. M. Gerzina (1975)^ ; A Comparison, of P r e d i c t e d

Experimental Seakeeping (^laracterlsticB. f o r Ships With and Without Large Bow Bulbs, Report of the Seakeeping Comnlttee, Appendix 2, Proceedings of the ISth I n t e m a t i o n a l Towing Tank Conference, V o l . Frank, W. (1967). O s c i l l a t i o n of Cylinders, i n or Below the Free

gur-: face o f Deep F l u i d s , NSRDC Report 2375.

Gerzina, D. M., and E. L. Woo (Dec, 1975). ; CVA 68 R e l a t i v e Motion I n v e s t i g a t i o n . DTNSRDC Report SPD-656-01.

Hong, Y. S. (June 1977). Numerical C a l c u l a t i o n o f Second-Order Wave Resistance, Journal of Ship : Resea:rch 20(2), 94-106.

Lee, C. M. (1976). T h e o r e t i c a l Prédiction, of Motion of SmallrWater-.plane-Area. Twin H u l l (SWATH) Ships ; in.Waves, DTNSRDC Report

76-0046 (see Appendix A).

Lee, C. M. (1982). Computation of. R e l a t i v e Motion , of Ships to Waves. DTNSRDC Report 82/019. Also presented at the T h i r d I n t e m a t i o n a l Conference on Numerical Ship Hydrodynamics, P a r i s , 1981.

Porter, W. R. (1960), Pressure D i s t r i b u t i o n s , Added Mass, Damping C o e f f i c i e n t s f o r C y l i n d e r s O s c i l l a t i n g i n a Free Surface, Inst. Engr... Res.. Unly, of C a l i f . , Berkeley. Series 82, Issue No. 16, Salvesen, N., 0, F a l t i n s e n , and E. 0. Tiick (1970). Ship Motion and

Sea Loads, SNAME Trans.. V o l . 78.

T a s a i , F. (1959). On the Damping Force and Added Mass of Ships Heaving and P i t c h i n g , J . Zosen K i o k a i 005, 47-56,

Wehausen, J , V, (1973). The Wave Resistance of Ships, Advances i n Applied Mechanics 03.

APPENDIX-^AMPLITÜDES AND PHASES OF MOTIONS, WAVE COMPONENTS GENERATED BÏ

EWaa

MDDIFIKD WAVE, ABSOLUTE MOTION AND RELATIVE MOTION AT STATION 2 FOR SHIP A AT F » 0.30 IN BOW WAVES

n

Anplitudes and phases o f the r i g i d body motion, together with the wave components generated by the corresponding degree o f freedom, are shown. Magnitude and phase of the d i f f r a c t i o n and modified waves are also shown. Magnitudes of the wave components and t r a n s l a t i o n a l mo-tions are nondimensionalized by Incident wave amplitude, while those of r o t a t i o n a l motions are nondimensionalized by incident wave slope. A l l phase angles are degrees of lead with respect to maximum wave e l e v a t i o n at the o r i g i n *

T a b l e 3 - A m p l i t u d e s and Phases o f M o t i o n s , Wave Components G e n e r a t e d by E a c h M o t i o n , I n c i d e n t Wave, D i f f r a c t e d Wave,

T o t a l M o d i f i e d Wave »..Absolute M o t i o n and R e l a t i v e M o t i o n a t S t a t i o n 2 f o r S h i p A a t i n Bow Waves Fn = 0.30 SUAT HEAVE HOTIOH WAVE C0M?OHENT MOTIOK WAVE COKPOliiQlT

U E AHP PHASE ÀHP PHASE AHF PHASE AMP PHASE ^ 5 5 1.44 1.19 1.00 .76 .50 .23 .40 .58 .66 .74 .87 1.18 2.11 .418 82.5 .297 89.0 .240 88.3 .179 86.8 .080 82.6 .018 -57.4 .004 64.2 .094 -94.5 .177 -114.5 .198 -125.6 .187 -136.6 .101 -159.8 .028 34.9 .008 139.7 1.014 0 1.092 -.6 1.222 -4.6 1.442 -19.0 .978 -88.5 .081 17.6 .010 139.7 .077 -68.3 .132 -82.3 .169 -91.6 .223 -111.3 .174 -170.5 .016 -93.2 .003 54.0 II ROLL Pl TCH WAVE WAVE

MOnOK HOTIOH COHPOHERT X/L AHF PRASE AMP PHASE AMP PHASE AKP PHASE 2.55 .40 3.096 -60.5 .025 122.4 .786 -98.3 .078 46.4 1.44 .58 .859 -129.9 .033 22.8 .850 -113,2 .219 9.2 1.19 .66 .631 -136.4 .041 4.B .870 -124.6 .305 - l O . l 1.00 .74 .473 -140.3 .046 -9.6 .868 -141.8 ,400 -34.7 .76 .89 .260 -141.6 .040 -32.4 .562 164.3 .382 -100.4 .50 1.18 .019 -122.2 .005 -43.4 .031 123.7 .034 -154.8 .23 2.11 .005 6B.2 .004 118.7 .002 67.1 .006 168.3 YAM MOTION WAVE COMPONENT TOTAL RACIATED WAVE

X/L " E AKP PRASE AHF PHASE AHP PHASE

2 . 5 5 1 . 4 4 1 . 1 9 1 . 0 0 . 7 6 . 5 0 . 2 3 . 4 0 . S B . 6 6 . 7 4 . 8 9 1 . 1 8 Z . U . 2 2 5 1 6 9 . 3 . 1 7 1 - 1 7 9 . 1 . 1 4 9 - 1 7 5 . 1 . 1 2 6 - 1 7 1 . 5 . 0 6 3 - 1 6 4 . 7 . 0 2 1 - 1 4 0 . 6 . 0 0 1 - 1 7 7 . 3 . 0 7 6 2 5 . 7 . 2 1 7 1-6 . 3 1 2 - 7 . 5 . 3 9 0 - 1 5 . 5 . 4 0 7 - 3 0 . 9 . 1 8 8 - 3 5 . 8 . 0 2 8 - 9 4 . 7 . 1 4 2 - 2 3 . 0 . 4 7 2 - 3 0 . 3 . 6 7 7 - 3 8 . 5 . 8 6 1 - 5 1 . B . 6 1 4 - 8 5 . 5 . 1 9 6 - 4 0 . 9 . 0 2 2 - 1 3 2 . 0 INCIDEKT WAVE DIFFRACTION WAVE INCIIŒST PLÜS DIFFRACTION WAVE' TOTAL HODIFIED VAVE

X/L " E AMP PHASE AMP PHASE AMP PHASE AHP PHASE

2 . 5 5 1.44 1 . 1 9 1 . 0 0 . 7 6 . 5 0 . 2 3 . 4 0 . 5 8 . 6 6 . 7 4 . 8 9 I . I B 2 . 1 1 1 . 0 0 0 4 5 . 3 8 0 . 6 " 9 7 . 5 1 1 6 . 0 1 5 2 . 3 - 1 3 0 . 6 1 4 3 . 6 . 1 2 3 1 4 8 . 3 . 2 1 9 1 7 7 . 7 .'263 - 1 6 9 . 7 . 3 1 1 - 1 5 6 . 5 . 4 1 0 - 1 3 1 . 3 , 6 0 6 - 7 2 . 3 . 7 1 4 1 6 5 . 0 . 9 8 0 5 2 . 3 . 9 9 9 9 3 . 2 1 . 0 2 1 1 1 2 . 4 1 . 0 6 0 1 3 3 . 1 1 , 1 6 6 1 7 2 . 3 1 . 4 1 6 - 1 0 9 . 2 1 . 6 8 5 1 5 2 . S 1 . 0 2 5 4 4 . 7 . 8 3 5 6 5 . 0 .541 7 5 . 0 . 2 1 5 1 5 3 . Ó 1 . 2 0 0 - 1 5 7 . 7 1 . 5 0 0 - 1 0 2 . 3 1 . 6 9 0 1 5 3 . 2 ABSOLUTE MOTtON PRESENT RELATIVE HOTIOH PREVIOUS RELATIVE MOTIOK X/L AHF PHASE AMP PHASE AHP PHASE

2 . 5 5 1 . 4 4 1 . 1 9 1 . 0 0 . 7 6 . 5 0 . 2 3 . 4 0 . 5 8 . 6 6 . 7 4 . 8 9 1 . Î B 2 . 1 1 1 . 1 4 3 3 3 . 7 2 . 2 6 8 4 0 . 7 2 . 8 3 0 3 3 . 6 1 3 . 4 0 6 1 7 . 3 1 2 . 5 2 8 - 3 7 . 8 1 . 2 1 1 -35.'3 1 . 0 2 3 - 1 3 8 . 4 . 4 5 1 8 , 0 1 . 5 4 6 2 7 . 8 2 . 4 4 9 2 5 . 2 3 . 5 6 3 1 4 . 9 3 , 2 9 2 - 1 9 . 4 1.43Ö 6 9 . 9 1.682 - 2 7 . 5 . 4 7 8 8 . 7 1 . 6 3 2 1 7 . 5 2 . 5 5 3 1 3 . Ö 3 . 6 9 2 1.8 3 . 5 1 7 - 3 5 . 0 1 . 0 4 1 3 7 . 8 . 9 9 6 - ? 7 . 7

Dîiscussion

J.H. P a t t i s o n (Naval Sea Systems Command)

In recent years, various research programs i n ship hydrodynamics have provided h u l l form designers with valuable t o o l s f o r p r e d i c t i n g seakeeping performance^* Results of the p r e d i c t i o n s have allowed the designers to choose h u l l form and appendage parameter values f o r best seakeeping performance. Even so, the t o o l s are most accurate f o r p r e d i c t i n g absolute motions of the center of g r a v i t y of a ship and l e a s t accurate f o r p r e d i c t i n g r e l a t i v e motions between p o i n t s on the ship and the water surface. T h i s l i m i t s the accuracy with which deck wetness and danqping are p r e d i c t e d . The authors are t o be complimented f o r developing an analyticcü. method £oc p r e d i c t i n g r e l a t i v e motion that includes the e f f e c t s of the moving ship on the wave f i e l d . I t i s hoped t h a t t h i s methodology can be included i n Improved t o o l s f o r the h u l l form designer i n the near future.

The work presented represents a good step toward accurately rep-resenting r e l a t i v e motions i n that they include both dynamic and steady e f f e c t s not previously incorporated i n t o the p r e d i c t i v e t o o l s * The dynamic e f f e c t s include wave d i f f r a c t i o n and r e f l e c t i o n by the ship as w e l l as wave generation by the mpying ship. The steady e f f e c t s include sinkage, trim, and the steady-wave p r o f i l e caused by the forward move-ment of the ship. However, to make the problem t r a c t a b l e , the authors chose the l i n e a r equations of motion, which l i m i t s the a p p l i c a b i l i t y

to l i n e a r ranges of ocean waves and ship responses. For p r e d i c t i o n s of absolute motion, the l i m i t occurs at Sea State 6, or lower, depend-ing on the s i z e of the ship. I t occurs to the discuasor that nonlinear e f f e c t s on r e l a t i v e motion may occur a t lower sea s t a t e s .

Another l i m i t a t i o n i n the work reported i s that only the under-water h u l l form i s considered. Aisove-under-water features, such as knuckles and flare« are expected to have s i g n i f i c a n t e f f e c t s on the predicted r e l a t i v e motion. Also, i t i s not c l e a r that r e s u l t s obtained i n regu-l a r waves are d i r e c t regu-l y a p p regu-l i c a b regu-l e to i r r e g u regu-l a r waves.

The discusser agrees with the authors that a systematic s e r i e s of ship model t e s t s are needed to v e r i f y and improve the methods f o r pre-d i c t i n g r e l a t i v e motion. Furthermore, the s e r i e s shoulpre-d inclupre-de both the underwater h u l l form and above-water h u l l form and features. To be u s e f u l to the h u l l form, parameter and feature v a r i a t i o n s should be kept w i t h i n reasonable, p r a c t i c a l bounds.

*Meyers, W.G., T.R. Applebee, and A.E. H a i t i s , "User's Manual f o r the Standard Ship Motion Program, SMP," DTiHSBDC Ship Performance Department Report OTNSRDC/SPO-Q936-01 (Sept. 1981).

E.N. Comstock (Naval Sea Systems Command)

The r e s u l t s of t h i s inproved p r e d i c t i v e method, as presented i n Figure 16, r a i s e a question r e l a t i v e to experience we have had i n using e x i s t i n g r e l a t i v e motion p r e d i c t i v e techniques without radiated or d i f f r a c t e d wave p b t e n t i a l s . S p e c i f i c a l l y , we have observed an appreciable increase i n p r e d i c t e d r e l a t i v e motion between s t a t i o n s 10 and 12 f o r s e v e r a l ships. T h i s phenomenon, which y i e l d s a characteris-t i c "W" shape characteris-t o characteris-the r e l a characteris-t i v e mocharacteris-tion p r o f i l e , has been observed wicharacteris-th regular and Irregular Waves at v a r i o u s speeds i n head and bow seas. T h i s c h a r a c t e r i s t i c increase i n r e a c t i v e motion has been observed and confirmed by model t e s t s . Would the authors please comment on whether they have experienced t h i s same e f f e c t and t h e i r thoughts r e l a t i v e to our experience.

L.J. Doctors (University of New South Wales)

Equation (12) g i v e s the wave p r o f i l e along the side of the h u l l on the assumption that the ship i s " t h i n . " The two t e s t cases, whose body plans are reproduced i n Figure 2 would probably be better de-s c r i b e d ade-s e i t h e r " f l a t " de-shipde-s Or "de-slender" bodiede-s. Do the authorde-s f e e l that there would be much e r r o r i n the r e s u l t s because of t h e i r choice o f these t e s t cases?

H.T. Wang (Naval Research Laboratory)

I wish to commend the authors f o r taking a f i r s t step toward com- | puting the t o t a l wave p r o f i l e alongside a ahip c o n s i s t i n g of the i n c i

-dent wave> the wave due to steady forward motion, r a d i a t i o n waves, and : the d i f f r a c t i o n wave. The authors use t h i n - s h i p theory to c a l c u l a t e

tite steculy-wave and slender-body theory to c a l c u l a t e the r a d i a t i o n and d i f f r a c t i o n waves. These t h e o r i e s are reasonably accurate over most

of the ship. However, i n the bow area, where the authors c a l c u l a t e j the r e l a t i v e motion, the ship carinot be considered e i t h e r t h i n or i slender. For exanple, the term 3 f / 3 ^ the l o n g i t u d i n a l d e r i v a t i v e of

the ship half beam, » at the forward perpendicular i n the authors' Equation (12) f o r the steady-wave p r o f i l e .

Z would l i k e to ask how the bow geometry was modeled by the au-thors and i f they made any attempts to make c o r r e c t i o n s to t h e i r the-o r i e s a t the bthe-ow area. Zn p a r t i c u l a r , d i d they attempt tthe-o make use the-of the many s t u d i e s on the bow-wave p r o f i l e conducted by Professor ^ i l v i e and h i s students a t the U n i v e r s i t y of Hichigan? Or d i d they t r y the s i n g u l a r i t y gap technique, i n i t i a t e d by Professor Lahdwebec, where the s i n g u l a r i t y d i s t r i b u t i o n does not extend a l l the way to the blunt edge? The extent of the gap i s a f u n c t i o n of the geometric p r o p e r t i e s of the leading edge, p r i n c i p a l l y the radius of curvature.