Wave effects on deformable bodies

E L S E V I E R

Printed in Great Britain. A l l rights reserved 0141-1187/94/$07.00

J . N. Newman

Department of Ocean Engineermg, MIT, Cambridge, Massacliusetts 02139, USA

The linearized frequency-domain analysis o f wave radiation and diffraction by a three-dimensional body i n a fixed mean position is extended to a variety of deformable body motions. These include continuous structural deflections, and also discontinuous motions which can be used to represent multiple interacting bodies. A general methodology is adopted with the body deflection defined by an expansion i n arbitrary modal shape functions, and the response i n each mode is obtained as a logical extension o f the usual analysis f o r rigid-body modes. Illustrative computations are presented f o r the bending of a freely-floating barge, and of a vertical column with cantilever support at the bottom. For these structural deflections the use of orthogonal polynomials is emphasized, as an alternative to the more conventional use o f natural modes. Also presented are computations for the motions of two rigid barges connected by a hinge joint, and for a finite array o f images used to approximate wall effects on a cylinder i n a channel. Results f r o m the latter problem are compared with more analytical solutions, and it is shown that practical results can be obtained f o r the first-order hydrodynamic force coefficients using only a few images.

1 I N T R O D U C T I O N

W a v e - i n d u c e d m o t i o n s o f ships a n d o f f s h o r e p l a t f o r m s are u s u a l l y described b y six r i g i d - b o d y modes i n c l u d i n g t r a n s l a t i o n s (surge, sway, heave) p a r a l l e l t o the Cartesian axes, a n d r o t a d o n s ( r o l l , p i t c h , y a w ) a b o u t these axes. A d d i t i o n a l 'generalized' modes o f m o t i o n are s i g n i f i c a n t i n v a r i o u s a p p l i c a t i o n s i n c l u d i n g c o n t i n u o u s s t r u c t u r a l deflections, d i s c o n t i n u o u s deflections o f m u l t i - c o m p o n e n t bodies w i t h m e c h a n i c a l j o i n t s , a n d the i n t e r a c t i o n s o f separate bodies. These a p p l i c a t i o n s are q u i t e d i f f e r e n t i n a p r a c t i c a l context, b u t s i m i l a r m a t h e m a t i c a l l y . T h u s i t is convenient t o analyze t h e m f r o m a single generalized a p p r o a c h . W a v e - i n d u c e d s t r u c t u r a l loads are i m p o r t a n t f o r m a n y p r a c t i c a l a p p l i c a t i o n s , b u t i n m o s t cases the s t r u c t u r a l a n d h y d r o d y n a m i c analyses are p e r f o r m e d separately. T h i s is a p p r o p r i a t e f o r s t i f f structures, where the eigenfrequencies o f elastic deflections are substan-t i a l l y h i g h e r substan-t h a n substan-the frequencies o f substan-the firssubstan-t-order w a v e loads. I n these circumstances the analysis o f w a v e r a d i a t i o n a n d d i f f r a c t i o n can be p e r f o r m e d neglecting the s t r u c t u r a l modes, a n d the h y d r o d y n a m i c analysis o f these m o d e s can be restricted t o the e v a l u a t i o n o f the c o r r e s p o n d i n g added-mass coefficients u s i n g a s i m p l i f i e d h i g h - f r e q u e n c y a p p r o x i m a t i o n o f the free-surface effect. O n the o t h e r h a n d , i n a p p l i c a t i o n s where the eigen-frequencies o f elastic deflections f a l l w i t h i n the s p e c t r u m o f first-order wave loads, the t w o p r o b l e m s m u s t be

c o u p l e d t o account f o r wave r a d i a t i o n i n the analysis o f the s t r u c t u r a l modes.

I n s t r u c t u r a l analysis i t is c o m m o n t o define a special set o f natural m o d e shapes, w h i c h c o r r e s p o n d t o the a c t u a l elastic deflections o f the b o d y i n a specified p h y s i c a l c o n t e x t . T h i s is d i f f i c u l t f o r h y d r o e l a s t i c p r o b l e m s , w h e r e the m o d e shapes are affected b y the h y d r o d y n a m i c pressure field a n d c a n n o t be specified i n advance. T h i s d i f f i c u l t y can be a v o i d e d i f the s t r u c t u r a l d e f l e c d o n is represented instead b y a s u p e r p o s i t i o n o f simpler mathematical m o d e shapes w h i c h are s u f f i c i e n t l y general a n d c o m p l e t e t o represent the p h y s i c a l m o t i o n . F o r example, the s t r u c t u r a l deflections o f a floating slender ship can be expressed b y the free u n d a m p e d ' w e t ' b e n d i n g m o d e s o f the h u l l i n water, b y the ' d r y ' modes o f the same structure i n air, as r e c o m m e n d e d b y B i s h o p a n d P r i c e , ' o r m o r e s i m p l y the o r t h o g o n a l modes o f a u n i f o r m beam, as s h o w n b y G r a n . ^ A n even s i m p l e r representation can be developed i n terms o f o r t h o g o n a l p o l y n o m i a l s , despite the f a c t t h a t these l a c k a p h y s i c a l basis a n d d o n o t satisfy the a p p r o p r i a t e free-end b o u n d a r y c o n d i t i o n s .

A n a l o g o u s o p t i o n s exist i n d e f i n i n g the modes f o r d i s c o n t i n u o u s d e f l e c t i o n o f m u l t i p l e bodies. T h e usual d e f i n i t i o n o f each m o d e is w i t h one b o d y p e r f o r m i n g a u n i t a m p l i t u d e o f m o t i o n a n d the others fixed. I f N separate bodies osciflate i n d e p e n d e n t l y , w i t h six degrees o f m o t i o n f o r each b o d y , the ensemble has a t o t a l o f 6N degrees o f f r e e d o m . T h e i n d i v i d u a l bodies, a n d modes o f

48 _{J. N. Newman}

m o t i o n , can be denoted b y a p a i r o f c o r r e s p o n d i n g indices, d e n o t i n g f o r example the a m p l i t u d e o f m o t i o n f o r b o d y n i n m o d e j b y the s y m b o l . H o w e v e r this is an a w k w a r d n o t a t i o n w h i c h obscures the a n a l o g y w i t h other a p p l i c a t i o n s . I t is m o r e convenient t o consider a single ' g l o b a l b o d y ' , d e f i n e d g e o m e t r i c a l l y b y the ensemble o f N separate bodies. F o r this example there are a t o t a l o f 6N modes o f the g l o b a l b o d y , o r m o r e i f the i n d i v i d u a l bodies are d e f o r m a b l e .

A n y specified d i s t r i b u d o n o f n o r m a l v e l o c i t y o n the ( g l o b a l ) b o d y surface is considered t o be a ' m o d e ' i n the present w o r k . T h e d e c o m p o s i d o n o f c o m p l i c a t e d b o d y m o t i o n s i n t o separate modes is j u s t i f i e d b y linear s u p e r p o s i t i o n , p r o v i d e d suitable m o d a l f u n c t i o n s are i n c l u d e d . The a p p r o p r i a t e d e f i n i t i o n o f each m o d e is n o n - u n i q u e , a n d subject t o choice. I n specific applica-t i o n s applica-the choice o f modes c a n be based o n physical considerations, a n d c o m p u t a t i o n a l efficiency.

T h e case o f t w o rigid bodies j o i n e d b y a hinge o r p i n j o i n t is a simple example, applicable t o a p a i r o f floating barges j o i n e d together b y an e q u i v a l e n t m o o r i n g arrangement. T h e hinge c o n s t r a i n t reduces the t o t a l n u m b e r o f degrees o f f r e e d o m f r o m t w e l v e t o seven, i n c l u d i n g six r i g i d modes o f the g l o b a l b o d y plus one a n g u l a r d e f l e c t i o n o f the hinge. S o l v i n g the l a t t e r p r o b l e m d i r e c t l y avoids the unnecessary s o l u t i o n f o r five e x t r a r a d i a t i o n potentials, a n d p e r m i t s the equations o f m o t i o n t o be solved i n a consistent m a n n e r w i t h o u t i m p o s i n g ad hoc constraints.

S i m i l a r considerations a p p l y i f the m e t h o d o f images is used t o analyse the m o t i o n s o f a single b o d y near a v e r t i c a l w a l l , or i f a finite n u m b e r o f images are used to a p p r o x i m a t e the effects o f t w o p a r a l l e l w a l l s i n a wave c h a n n e l . I f the t o t a l n u m b e r o f images is / , the general p r o b l e m f o r = ƒ + 1 separate bodies i n v o l v e s 6N r i g i d - b o d y m o t i o n s , b u t i n the u s u a l a p p l i c a t i o n the m o t i o n o f each image is either the same o r opposite t o t h a t o f the a c t u a l b o d y . S o l v i n g the l a t t e r p r o b l e m d i r e c t l y , w i t h a p p r o p r i a t e d e f i n i t i o n s o f the n o r m a l v e l o c i t y o n each image, reduces the t o t a l n u m b e r o f r a d i a t i o n potentials b y the f a c t o r \ / N .

Section 2 f o r m u l a t e s the linear h y d r o d y n a m i c analysis f o r a g l o b a l b o d y w i t h a r b i t r a r y specified modes o f m o t i o n . T h i s analysis leads t o s t r a i g h t f o r w a r d extensions o f the u s u a l addedmass, d a m p i n g , a n d e x c i t i n g -f o r c e coe-f-ficients -f o r a rigid b o d y . Special a t t e n t i o n is r e q u i r e d f o r the h y d r o s t a t i c coefficients, f o l l o w i n g the analysis i n Ref. 3. I l l u s t r a t i v e a p p l i c a t i o n s are then described i n Sections 3 - 6 .

F o r each o f the above applications the h y d r o d y n a m i c analysis is p e r f o r m e d using the three-dimensional r a d i a t i o n / d i f f r a c t i o n code W A M I T , based o n the b o u n d a r y - e l e m e n t ' p a n e l ' m e t h o d . F u r t h e r detafls r e g a r d i n g this code are given i n R e f s 4 - 7 . T o p r o v i d e a c o m p u t a t i o n a l basis f o r a n a l y z i n g the general p r o b l e m o f a b o d y w i t h a r b i t r a r y specified modes, t h e p r o g r a m has been extended so t h a t the user can specify the

f u n c t i o n a l f o r m o f generalized modes i n a d d i t i o n t o the six c o n v e n t i o n a l rigid-body m o t i o n s . P r o v i s i o n is m a d e t o specify e x t e r n a l mass, d a m p i n g , a n d stiffness matrices w i t h coefficients c o r r e s p o n d i n g t o each o f the modes o f m o t i o n . F o r each o f the applications described b e l o w , discretizations w i t h increasing numbers o f panels have been used t o v e r i f y numerical convergence. F o r example, i n the analysis o f the vertical c o l u m n i n Section 4, t w o discretizations ( w i t h N = 512,2048) t o t a l panels o n the c o l u m n are employed, a n d the results compared; the m a x i m u m relative difference observed i n the m o d a l amplitudes is 2 % . A s s u m i n g the numerical errors are p r o p o r t i o n a l t o \ / N , as i n Ref. 7, i t f o l l o w s that the results presented, based o n the finer discretization, are accurate w i t h i n 0-5%. Similar convergence tests a n d e r r o r estimates have been made f o r all o f the results presented.

2 F O R M U L A T I O N A N D F I R S T - O R D E R P R E S S U R E F O R C E

T i m e h a r m o n i c m o t i o n s o f s m a l l a m p l i t u d e are c o n -sidered, w i t h the c o m p l e x f a c t o r e'"' a p p l i e d t o a l l first-order o s c i l l a t o r y quantities. T h e b o u n d a r y c o n d i t i o n s o n the b o d y a n d free surface are linearized, a n d p o t e n t i a l flow is assumed. Cartesian coordinates are d e f i n e d w i t h z = 0 as the plane o f the u n d i s t u r b e d f r e e surface a n d z = -h as the b o t t o m . W i t h i n the fluid d o m a i n is a b o d y ( o r ensemble o f bodies) w i t h w e t t e d surface Sf,. T h e b o d y m a y be floating, submerged, o r b o t t o m - m o u n t e d .

Six c o n v e n t i o n a l r i g i d - b o d y m o t i o n s (surge, sway, heave, r o l l , p i t c h , y a w ) are d e n o t e d b y t h e indices 7 = 1 , 2 , 3 , 4 , 5 , 6 , respectively. T h e same n o t a t i o n is extended t o other modes o f b o d y d e f o r m a t i o n (7 = 7 , 8 , . . . , / ) . E a c h m o d e m a y be d e f i n e d b y a vector 'shape f u n c t i o n ' S j ( x ) w i t h Cartesian c o m p o -nents uj,vj,wj. F o r r i g i d - b o d y t r a n s l a t i o n ( 7 = 1 , 2 , 3 ) the shape f u n c t i o n Sj(x) is a u n i t vector i n the c o r r e s p o n d i n g d i r e c t i o n , a n d f o r rigid-body r o t a t i o n s (7 = 4 , 5 , 6 ) a b o u t the o r i g i n x = 0, S^(x) = Sy_3 x x . T h e displacement o f an a r b i t r a r y p o i n t w i t h i n the b o d y , due t o m o t i o n i n the 7 t h m o d e w i t h c o m p l e x a m p l i t u d e ( j , is represented b y the p r o d u c t ^jSj{x). T h e vector S^- is assumed t o be c o n t i n u o u s a n d d i f f e r e n t i a b l e near the b o d y surface S/,, w i t h divergence Dj = V • Sj. T h e divergence is zero f o r each r i g i d - b o d y m o d e . T h e n o r m a l c o m p o n e n t o f Sj o n S/, is expressed i n the f o r m

nj = Sj-n = Ujn:, + Vjny + Wjn, (2.1) T h e u n i t n o r m a l vector n p o i n t s o u t o f the fluid d o m a i n

a n d i n t o the b o d y .

T h e fluid v e l o c i t y field is represented b y the g r a d i e n t o f the v e l o c i t y p o t e n t i a l ^ , governed b y Laplace's e q u a t i o n i n the fluid d o m a i n

A p p r o p r i a t e b o u n d a r y c o n d i t i o n s o n the free surface a n d b o t t o m are dz K(j) = 0, o n z = 0 o n = -h (2.3) (2.4) H e r e , K = to^/g, were g is tlie acceleration o f g r a v i t y . T h e v e l o c i t y p o t e n t i a l o f the i n c i d e n t wave is d e f i n e d b y

igA cosh [/c(z + /;)] ^_ik(^,o,0+y _sin/3)

(2.5) w cosh kh

where the w a v e n u m b e r k is the positive real r o o t o f the dispersion r e l a t i o n

.2 U> g

k t a n h kh (2.6)

a n d f3 is the angle between the d i r e c t i o n o f p r o p a g a t i o n o f the i n c i d e n t wave a n d the positive x axis.

T h e v e l o c i t y p o t e n t i a l <f> can be expressed i n the f o r m

4> = <I>R + 4>D (2-7)

't>D = 4>i + <f>s (2-8) is the d i f f r a c t i o n p o t e n t i a l , (ps is the scattering

c o m p o n e n t representing the disturbance o f the i n c i d e n t wave b y the fixed b o d y , a n d

J

^R = Y.^j't>j (2-9) is the r a d i a t i o n p o t e n t i a l due t o the b o d y ' s m o t i o n s . I n

each m o d e , (pj is the c o r r e s p o n d i n g u n i t - a m p l i t u d e r a d i a t i o n p o t e n t i a l .

O n the u n d i s t u r b e d p o s i t i o n o f the b o d y b o u n d a r y , the r a d i a t i o n and d i f f r a c t i o n potentials are subject t o the c o n d i t i o n s 0 ^ = '""^-d<pD dn 0 (2.10) (2.11) T h e b o u n d a r y v a l u e p r o b l e m is c o m p l e t e d b y i m p o s -i n g a r a d -i a t -i o n c o n d -i t -i o n o f o u t g o -i n g waves, f o r the p o t e n t i a l s (p^ a n d (pj. C o r r e s p o n d i n g t o each m o d e o f m o t i o n , generaUzed first-order pressure forces are defined i n the f o r m

pni dS 'iu)(p + gz)nidS (2.12)

H e r e p is the fluid pressure, w h i c h is evaluated i n the last f o r m o f (2.12) f r o m the Hnearized B e r n o u f l i e q u a t i o n .

T h e c o n t r i b u t i o n t o (2.12) f r o m the t e r m /w0 involves a t r i v i a l extension o f the c o r r e s p o n d i n g r i g i d - b o d y analysis. A f t e r s u b s t i t u t i n g (2.10) f o r t h e c o m p o n e n t s o f the r a d i a t i o n p o t e n t i a l , added-mass and d a m p i n g

matrices are defined i n the f o r m ÜJ aij - iujbij -lup

Pjn^dS-(2.13) S i m i l a r l y , the generalized wave-exciting f o r c e is

Xi = —iujp So

hriidS: , - dS

So dn

(2.14) T h e indices i a n d j can take o n any values w i t h i n the ranges o f the r i g i d - b o d y modes ( 1 - 6 ) a n d extended modes ( 7 , 8 , . . . ) . Green's t h e o r e m c a n be a p p l i e d t o (2.13) t o establish r e c i p r o c i t y , a n d t o (2.14) t o derive the H a s k i n d relations between the generalized e x c i t i n g f o r c e a n d the c o r r e s p o n d i n g r a d i a t i o n p o t e n t i a l .

T h e d e f o r m a t i o n o f the b o d y geometry m u s t be considered i n d e r i v i n g the c o n t r i b u t i o n t o the general-ized f o r c e (2.12) f r o m the h y d r o s t a t i c pressure -pgz, since this pressure is o f order one. Three surfaces o f i n t e g r a t i o n are defined as f o f l o w s : the i n i t i a l w e t t e d surface o f the b o d y p r i o r t o the m o d a l displacement Sj is denoted as Sg; the c o r r e s p o n d i n g d e f o r m e d surface a f t e r the n o r m a l displacement i n the m o d e j is denoted as Sg; a n d the closed surface E is defined i n c l u d i n g Sq, Sg, a n d the p o r t i o n o f the plane z = 0 l y i n g between these t w o open surfaces i f they intersect the free surface. O n S the n o r m a l vector is d e f i n e d i n a consistent m a n n e r , t o p o i n t o u t o f the enclosed v o l u m e p.

W i t h these d e f i n i t i o n s , the change i n the h y d r o s t a t i c generalized f o r c e c o m p o n e n t Fj due t o a u n i t displace-m e n t o f the b o d y i n displace-m o d e j is defined b y the displace-m a t r i x

ztZj dS - pg znt dS = pg znj dS (2.15) U s i n g (2.1) a n d the divergence t h e o r e m

pg zSj -ndS = pg V - ( z S , ) d F (2.16) F o r s m a l l d e f o r m a t i o n s the v o l u m e f is t h i n , a n d the last i n t e g r a l c a n be a p p r o x i m a t e d t o first o r d e r as the surface i n t e g r a l o f the p r o d u c t o f the i n t e g r a n d a n d the distance Hj between the t w o b o u n d a r y surfaces a n d S^. T h u s

pg njW • (zS,) dS = pg nj{Wi + zDi) dS (2.17) H e r e the generalized f o r c e is d e f i n e d i n a fixed reference f r a m e , a n d o n l y the h y d r o s t a t i c pressure is considered. A s a result, (2.17) includes some c o n t r i b u t i o n s w h i c h n o r m a l l y are n o t considered, such as the r o l l (or p i t c h ) m o m e n t due t o a sway ( o r surge) displacement, b o t h o f w h i c h are equal t o the displaced b o d y v o l u m e times the m o m e n t a r m associated w i t h the c o r r e s p o n d i n g dis-placement. N o r m a l l y , f o r a f r e e l y - f l o a t i n g b o d y , these

30 _{J. N. Newman}

c o n t r i b u t i o n s are balanced b y the g r a v i t a d o n a l f o r c e due t o the b o d y mass. I n the generalized analysis o f a d e f o r m a b l e b o d y the c o r r e s p o n d i n g mass f o r c e m u s t be evaluated separately, f o r each m o d e , depending o n the m o d e shape a n d mass d i s t r i b u d o n . T h e coefficients (2.17) depend n o t o n l y o n the n o r m a l displacement (2.1), b u t also o n the divergence D ; a n d the v e r t i c a l c o m p o n e n t o f S,-. I n general the h y d r o s t a d c m a t r i x (2.17) is n o t symmetric.^

3 B E N D I N G O F A S L E N D E R B A R G E

A s the first example o f s t r u c t u r a l deflections, we consider a ship freely floating i n head seas. T h e ship is assumed to be a slender beam, w i t h v e r t i c a l displace-m e n t R e ( ^ ( x ) e ' " ' ) along the l e n g t h . T h e s t r u c t u r a l d e f l e c t i o n is governed b y the beam e q u a t i o n '

-o?m£, + {Eia' = Z[x) (3.1)

where m{x) is the d i s t r i b u t i o n o f mass, a n d primes denote d i f f e r e n t i a t i o n w i t h respect t o x. T h e p r o d u c t o f the m o d u l u s o f elasticity E a n d m o m e n t o f i n e r t i a I is the stiffness f a c t o r EI. Z{x) is the l o c a l pressure f o r c e a c t i n g o n a v e r t i c a l section o f the ship. Since b o t h ends o f the ship are free, the a p p r o p r i a t e b o u n d a r y c o n d i -tions are

0, ( £ / O ' = 0, f o r A- = ± L / 2 (3.2) where L is the ship's length, a n d the o r i g i n is at the m i d s h i p section.

T h e d e f l e c t i o n i m a y be expanded i n an a p p r o p r i a t e set o f modes, i n the f o r m

(3.3)

where the (complex) a m p l i t u d e o f each m o d e is u n k n o w n . A d o p t i n g the m e t h o d o f w e i g h t e d residuals (3.1) is m u l t i p l i e d b y / ( x ) a n d i n t e g r a t e d a l o n g the l e n g t h , t o give the system o f equations

f L / 2 j J —L/2 Mx)Z{x)dx (L/2 -L/2 •L/2 . -L/2

These can be r e w r i t t e n i n the f o r m L/2 -L/2

fi{x)Z{x)dx

(3.4)

(3.5)

where the coefficients o n the l e f t - h a n d side are the mass m a t r i x L/2 -L/2 mfi{x)fj{x)dx (3.6) a n d the stiffness m a t r i x (L/2 Cij = tL/2 EIf/'{x)fj"{x)dx J-L/2 (3.7) T h e stiffness m a t r i x (3.7) is derived b y i n t e g r a t i n g the c o r r e s p o n d i n g t e r m i n (3.4) twice b y parts, a n d i n v o k i n g the b o u n d a r y c o n d i t i o n s (3.2). B o t h (3.6) a n d (3.7) are symmetric matrices.

T h e i n t e g r a l o n the r i g h t - h a n d side o f (3.4) includes c o n t r i b u t i o n s t o the linearized pressure f o r c e Z ( x ) f r o m the d i f f r a c t i o n a n d r a d i a t i o n p r o b l e m s , as described i n Section 2. S u b s t i t u t i n g ( 2 . 1 3 - 2 . 1 5 ) , a n d m o v i n g the r a d i a t i o n coefficients t o the l e f t side o f (3.4), gives a c o n v e n t i o n a l set o f equations o f m o t i o n

E + ^ i j ) + i'^bij + C,j + Cy] = Xi (3.8) J

Pertinent modes o f m o t i o n i n head seas include surge, heave, p i t c h , a n d the s t r u c t u r a l deffection.

T w o d i f f e r e n t sets o f m o d e f u n c t i o n s w i l l be used f o r this case. T h e first are the n a t u r a l modes f o r b e n d i n g o f a u n i f o r m beam w i t h free ends, given by^

ƒ 1 / c o s / t 2 y 9 ^ c o s h / t 2 j ^ ^•^ 2 \ cos K2J cosh K2j (3.9) _ 1 fsinK2j+ig ^ sinhK2y+ig'^ 2 V s i n K 2 y + i smhK2j+iy Here the n o r m a l i z e d c o o r d i n a t e is q = 2x/L, a n d J = 1 , 2 , . . . . T h e degenerate f u n c t i o n s fg = 1 a n d f \ = q

correspond t o the heave a n d p i t c h modes respectively. T h e f a c t o r s KJ are the positive real roots o f the e q u a t i o n

( - 1 ) - ' ' t a n K y - f t a n h K ^ - = 0 (3.10) T h e first f o u r r o o t s are K2 = 2-3650, Ks = 7-0686 « 3 = 3-9266, « 4 = 5-4978, N o t e t h a t f j { x ) is an even o r o d d f u n c t i o n o f x a c c o r d i n g as j is even or o d d . T h e first f o u r b e n d i n g modes are s h o w n as the s o l i d lines i n F i g . 1.

These f u n c t i o n s are o r t h o g o n a l , w i t h the n o r m a h z e d values f j { \ ) = \, a n d the i n t e g r a t e d values

r l

fi{q)fM)^^ = \^ii for 0 ' > 2 ) o r ( 7 > 2 )

(3.11) where the K r o e n e c k e r delta is equal t o 1 w h e n ; = j, otherwise zero. Assuming a u n i f o r m mass distribution, m is a constant and the mass m a t r i x is diagonal, w i t h the values

(L/2 fi{2x/L)fj{2x/L)dx Mij _m L/2 1 - 1 fi{q)fj{q)M = \M6ij (3.12)

1.00 0.50 0.00 -0.50 -1.00 -1.00

Fig. 1. Natural modes (3.9) (solid lines) and Legendre polynomial modes (dashed lines). The normalized coordinate

q is equal to ± 1 at the ends o f the ship.

T h e stiffness m a t r i x is s i m p l i f i e d i n this case b y n o t i n g t h a t i f EI is constant, a n d the i n d i v i d u a l modes (3.9) s a t i s f y the b o u n d a r y c o n d i t i o n s (3.2), the i n t e g r a n d o f (3.7) can be w r i t t e n instead i n the o r i g i n a l f o r m o f (3.4) as the p r o d u c t fifj-""^ where the second f a c t o r is the f o u r t h derivative. F r o m the d i f f e r e n t i a l e q u a t i o n

a n d t h e o r t h o g o n a l i t y r e l a t i o n (3.11) i t f o l l o w s t h a t

Cij = 4{EI/L')KUi, (3.13)

T h e alternative m o d a l f u n c t i o n s a p p l i e d to this p r o b l e m are the Legendre p o l y n o m i a l s Pi{q), w h i c h are p l o t t e d f o r c o m p a r i s o n i n F i g . 1. These are simpler m a t h e m a t i c a l l y t h a n the n a t u r a l modes (3.9), b u t the o f f d i a g o n a l elements o f t h e stiffness m a t r i x are n o n -zero f o r even values o f m + n.

A s a specific example, a rectangular barge is considered w i t h l e n g t h L = 80 m , b e a m B = 10 m , a n d d r a f t T = 5m. T h e t o t a l mass M is d i s t r i b u t e d u n i f o r m l y t h r o u g h o u t the v o l u m e , u p t o a 'deck' 5 m above the free surface. A constant sdffness f a c t o r S E / / / = 0 - l M s ~ ^ is assumed. T h e water d e p t h is i n f i n i t e .

T a b l e 1 compares the response i n each o f eight b e n d i n g modes at a wave p e r i o d o f 8 s, u s i n g the t w o a l t e r n a t i v e m o d e shapes. T h e effectiveness o f each is i n d i c a t e d b y the m a g n i t u d e s o f the c o r r e s p o n d i n g a m p l i t u d e s a n d b y t h e convergence o f the sums w h i c h represent the t o t a l m a g n i t u d e o f the b e n d i n g m o t i o n at the b o w {q = 1). T h e ' c o n v e r g e d ' values o f these t w o sums agree t o f o u r s i g n i f i c a n t figures, c o n f i r m i n g t h a t the Legendre p o l y n o m i a l s are a p p l i c -able t o this p r o b l e m even t h o u g h they d o n o t satisfy the b o u n d a r y c o n d i t i o n s (3.1). H o w e v e r the n a t u r a l modes

Table 1. Amplitude of each mode and sum (total deflection at the bow) based on the natural free-free beam modes (3.9) and on the

Legendre polynomials, for the wave period 8 s

Mode, j Natural modes Legendre modes

Sum Sum 2 0-173821 0-173821 0-193646 0-193646 3 0-010693 0-173545 0-013896 0-193324 4 0000051 0-173494 0-021882 0-171505 5 0-000133 0-173517 0-003925 0-171512 6 0-000011 0-173528 0-002072 0-173581 7 0-000069 0-173534 0-000541 0-173568 8 0-000003 0-173537 0-000051 0-173517 9 0-000028 0-173540 0-000002 0-173517

(3.9) p r o v i d e a m o r e effective expansion. F o r example, the results u s i n g one o r t w o n a t u r a l modes give three o r f o u r decimals accuracy, respectively, i n the deflected a m p l i t u d e at the b o w . B y c o m p a r i s o n , five Legendre modes are r e q u i r e d t o achieve the same p r e c i s i o n .

A d d i t i o n a l c o m p u t a t i o n s have been p e r f o r m e d using Chebyshev p o l y n o m i a l s f o r the m o d e f u n c d o n s . T h e r e is n o s i g n i f i c a n t change i n the rate o f convergence relative t o the Legendre p o l y n o m i a l s . A n advantage o f the Legendre modes, f o r a u n i f o r m mass d i s t r i b u t i o n , is t h a t the mass m a t r i x (3.12) is d i a g o n a l . F o r the Chebyshev modes there is c o u p l i n g i n the mass m a t r i x between a l l even o r o d d modes, i n c l u d i n g c o u p l i n g o f heave o r p i t c h w i t h t h e c o r r e s p o n d i n g s y m m e t r i c o r a n t i s y m m e t r i c b e n d i n g modes.

C o m p u t e d values o f the response-amphtude o p e r a t o r s i n heave, p i t c h , a n d the first t w o b e n d i n g m o d e s are s h o w n i n F i g . 2 f o r the range o f wave periods between 5 a n d 12 s. T h e r i g i d - b o d y modes o f heave a n d p i t c h d o m i n a t e the t o t a l v e r t i c a l m o t i o n s . F o r other values o f the stiffness f a c t o r the b e n d i n g - m o d e a m p h t u d e s are m o d i f i e d , a p p r o x i m a t e l y i n inverse p r o p o r t i o n t o EI. 2.00 PITCH HEAVE "5.00" "6.00''''V.'ÓÖ ••8,00 '''9\oÖ'''ïó'.Oo''ïi.oÖ''Ï2.bo PERIOD

Fig. 2. Vertical amplitude at the bow due to heave, pitch, and the first two bending modes.

52 / . N. Newman

These results are based o n a d i s c r e t i z a t i o n w i t h 2304 panels o n the submerged p o r t i o n o f the barge. A t o t a l o f 64 l o n g i t u d i n a l , 16 transverse, a n d 8 v e r t i c a l s u b d i v i -sions are used, w i t h cosine w e i g h t - f a c t o r s t o p r o v i d e finer spacing near the corners a n d the free surface. Convergence tests c o m p a r i n g these results w i t h a relatively coarse d i s c r e t i z a t i o n (576 panels) i n d i c a t e a n accuracy o n the o r d e r o f 1 % f o r the results i n T a b l e 1 a n d F i g . 2.

4 B E N D I N G O F A V E R T I C A L C O L U M N

T h e second example used t o i l l u s t r a t e the analysis o f s t r u c t u r a l deflections is a slender v e r t i c a l c o l u m n w i t h its free u p p e r end at the plane o f the free surface z = 0, a n d w i t h c l a m p e d s u p p o r t at the b o t t o m z = -h. F o r s i m p h c i t y i t is assumed t h a t the c o l u m n is s t r u c t u r a l l y u n i f o r m a l o n g its l e n g t h , w i t h a c o n s t a n t stiffness f a c t o r EI. A u n i f o r m d i s t r i b u t i o n o f mass m is assumed a l o n g the c o l u m n , a n d a c o n c e n t r a t e d mass nio is placed at the free surface z = 0 t o account f o r the 'superstructure'.

R e g u l a r i n c i d e n t waves o f f r e q u e n c y to are assumed, causing a h o r i z o n t a l d e f l e c t i o n R e ( ^ ( z ) e ' " ' ) o f the c o l u m n . T h e b e a m e q u a t i o n (3.1) is applicable, w i t h the c o o r d i n a t e x replaced b y z a n d the h o r i z o n t a l pressure f o r c e X'{z) o n the r i g h t - h a n d side. H o w e v e r the b o u n d a r y c o n d i t i o n s are d i f f e r e n t i n this case.

A p p r o p r i a t e b o u n d a r y c o n d i t i o n s at the b o t t o m are

C = 0, ^' = 0 at -h (4.1)

A t the upper end, where there is no b e n d i n g m o m e n t a n d a concentrated i n e r t i a l l o a d

e" = 0, EI^" -o?mo^ at 0 (4.2) A p p l y i n g the m e t h o d o f w e i g h t e d residuals, as i n (3.3¬ 3.8), gives the Hnear system

E</-[-'^'(«'7 + ^u) + '^bij + Cij] = Xi (4.3) where the mass m a t r i x and stiffness m a t r i x are defined b y

Mij = m " fi{z)fj{z)dz + m^mfj{Q) (4.4) J -h

EI f l \ z ) f ! \ z ) A z

H e r e (4.5) is derived i n a m a n n e r s i m i l a r t o (3.7) b y p a r t i a l i n t e g r a t i o n , a n d the second b o u n d a r y c o n d i t i o n (4.2) results i n an extra c o n t r i b u t i o n t o the mass m a t r i x (4.4). Since the d e f l e c t i o n is h o r i z o n t a l , a n d depends o n l y o n the v e r t i c a l c o o r d i n a t e , there is n o c o n t r i b u t i o n f r o m the h y d r o s t a t i c m a t r i x (2.17).

A n a p p r o p r i a t e set o f o r t h o g o n a l p o l y n o m i a l s y;(z) (y = 1 , 2 , 3 , . . . ) can be d e f i n e d to represent the

d e f l e c t i o n , i n the f o r m f j { z ) = q^P;.M, {q=l+z/h) (4.6) 1.00 0.50 0.00 -0.50 H -1.00 0.00 1.00

Fig. 3. Mode functions (4.6) f o r ( = 1,2,3,4. The normalized coordinate q is zero at the bottom and 1 at the free surface. Here the n o r m a l i z e d v e r t i c a l c o o r d i n a t e q increases f r o m zero at the b o t t o m to 1 at the free surface, Pj is a p o l y n o m i a l o f degree n, a n d the f a c t o r q^ is i n t r o d u c e d t o satisfy the b o u n d a r y c o n d i t i o n s (4.1). T h e p o l y n o -mials Pf{q) can be derived f r o m the Jacobi p o l y n o m i a l s Pj"'^\2q- 1).^ T h e o r t h o g o n a l i z a t i o n r e l a t i o n implies t h a t a = 0 a n d /? = 4. T h e first f o u r members are

Po = 1 Pt =6q-5 Pz = 2 8 ^ ^ - 4 2 ^ + 1 5 P3 = nOq^ - 252q^ + 168^ - 35 M o r e generally h m ! ( « - m ) ! ( 4 + « - m ) ! ^ (4.7) These p o l y n o m i a l s are n o r m a l i z e d w i t h P * ( l ) = 1. T h e c o r r e s p o n d i n g o r t h o g o n a l i t y r e l a t i o n is (4.5) ^^Pnq)P;iq)q'dq 2i + 5 (4.8) T h e first f o u r f u n c t i o n s (4.6) are p l o t t e d i n F i g . 3. E a c h m o d e shape is p r o p o r t i o n a l to q^ = {l+z/h)^ near the b o t t o m , a n d equal to 1 at the free surface. F o r intermediate depths they oscillate w i t h {J - 1) zeros, w i t h i n a range o f a p p r o x i m a t e l y ± 1 / 2 except close t o the free surface where the a m p l i t u d e increases t o 1. I n the n o t a t i o n o f M e i r o v i t c h , ' the m o d a l f u n c t i o n s (4.6) are admissible since they satisfy the geometric b o u n d a r y c o n d i t i o n s (4.1).

W i t h the m o d a l f u n c t i o n s defined b y (4.6) the mass m a t r i x (4.4) c a n be evaluated. A f t e r u s i n g the

1 able 2. Coefficients Qj of the stiffness matrix (4.5); each entry Table 3. Amplitude of each mode in this table should be multiplied by the factor El/h^

Period Mode 1 Mode 2 Mode 3 Mode 4

/' _{J=l 7 = 2 7 = 3} ; = 4 5-0 0-04813 0-00423 0-00010 0-00002 1 4 16 32 52 6 0 0-29934 0-02551 0-00041 0-00007 2 16 172 380 640 6-5 1-14505 0-09698 0-00135 0-00024 7356 12496 7-0 0-44875 0-03795 0-00046 0-00008 3 32 380 _{5 5} 8-0 0-22035 0-01875 0-00019 0-00004 52 640 12496 220652 9-0 0-16106 0-01391 0-00014 0-00004 4 52 640 5 35 100 0-13055 0-01151 0-00013 0-00004 (4.9) o r t h o g o n a l i t y r e l a t i o n (4.8)

A l l elements o f the stiffness m a t r i x (4.5) are non-zero r a t i o n a l f r a c t i o n s . T a b l e 2 gives the elements o f this m a t r i x f o r (iJ) < 4.

A s a n i l l u s t r a t i o n , consider a vertical c i r c u l a r c y l i n d e r o f r a d i u s 10 m , i n water o f d e p t h 200 m . T h e d i s t r i b u t e d mass a l o n g the l e n g t h o f the c y l i n d e r is assumed t o be h a l f o f its displaced mass. A concentrated mass mo e q u a l t o the t o t a l displaced mass (twice the d i s t r i b u t e d mass) is located at the free surface. T h e stiffness o f the cylinder is chosen such t h a t the r a t i o El/h^ = 0-41 mo s~'^; this results i n a resonance o f the first b e n d i n g m o d e at a wave p e r i o d o f 6-5 s.

C o m p u t a t i o n s have been p e r f o r m e d i n c l u d i n g the first eight modes (4.6), at a sequence o f wave periods between 5 a n d 10 s. T h e ampUtudes o f the first a n d second modes are p l o t t e d i n F i g . 4. T h e a m p l i t u d e s o f the h i g h e r modes are t o o small t o indicate i n this figure. A l s o s h o w n i n F i g . 4 is the t o t a l response at the u p p e r end o f the c o l u m n . T h e first a n d second modes generally are o u t o f phase. A s a result, the t o t a l response is a b o u t 1 0 % less t h a n the a m p h t u d e o f the first m o d e .

T h e m o s t i m p o r t a n t feature i n F i g . 4 is the h i g h l y

-Fig. 4. Amplitudes of the first bending mode (short dashed curve), second mode (long dashed curve), and the total response o f the vertical column at the upper end (solid curve).

t u n e d resonance at 6-5 s, w i t h a m a x i m u m h o r i z o n t a l d e f i e c t i o n at the free surface s l i g h t l y greater t h a n the i n c i d e n t wave a m p l i t u d e . T h e o n l y d a m p i n g w h i c h is i n c l u d e d here is t h a t due t o linear wave r a d i a t i o n , w h i c h at resonance is 2-6% o f the c r i t i c a l d a m p i n g . T h e occurrence o f this resonance can be a t t r i b u t e d t o the f a c t t h a t the wave e x c i t a t i o n , a n d d a m p i n g are c o n f i n e d t o a relatively s m a l l p a r t o f the c o l u m n near the free surface. T h i s is i n c o n t r a s t t o the floating barge i n Section 3, where the entire vessel is near the free surface. T a b l e 3 shows the c o m p u t e d amphtudes o f the first f o u r modes at representative wave periods, c o n f i r m i n g the r a p i d convergence o f the m o d a l expansion.

I t s h o u l d be n o t e d t h a t i n the n u m e r i c a l a p p r o a c h f o l l o w e d , the b o u n d a r y c o n d i t i o n s (4.2) are n o t e x p l i c i t l y i m p o s e d as constraints o n the s o l u t i o n . Instead t h e y are u t i l i z e d o n l y i n the weaker c o n t e x t o f the end c o n d i t i o n s i n the p a r t i a l i n t e g r a t i o n w h i c h yields the mass a n d stiffness matrices ( 4 . 4 - 4 . 5 ) . One can v e r i f y n u m e r i c a l l y t h a t the b o u n d a r y c o n d i t i o n s (4.2) are i n f a c t satisfied b y the n u m e r i c a l s o l u t i o n , b u t the relatively slow convergence o f higher derivatives hinders this c o n f i r m a t i o n . F o r example, c o m p u t e d values o f |C"(0)| decrease f r o m a m a x i m u m o f 0-005 to 0-0004 w h e n the n u m b e r o f modes is increased f r o m f o u r t o eight. E v e n w i t h eight modes i n c l u d e d , the t w o sides o f the second b o u n d a r y c o n d i t i o n (4.2) d i f f e r i n the t h i r d s i g n i f i c a n t figure ( i n the second s i g n i f i c a n t figure f o r the lowest wave p e r i o d ) .

I t is interesting t o c o m p a r e the use o f n a t u r a l modes w i t h the m e t h o d a d o p t e d here, based o n the o r t h o g o n a l p o l y n o m i a l modes (4.6). N a t u r a l modes c o u l d be used, as i n Section 3, a c c o u n t i n g f o r the b o u n d a r y c o n d i t i o n s (4.2) at the u p p e r end o f the c o l u m n . Since the second c o n d i t i o n (4.2) w o u l d o n l y be satisfied at the corre-s p o n d i n g eigenfrequency f o r each m o d e , thicorre-s a p p r o a c h w o u l d n o t o f f e r any c o m p u t a t i o n a l advantage relative t o the simpler p o l y n o m i a l modes used here.

T h e c o m p u t a t i o n s s h o w n here are based o n a d i s c r e t i z a t i o n o f the c o l u m n w i t h a t o t a l o f 2048 panels, c o r r e s p o n d i n g t o 128 panels u n i f o r m l y spaced a r o u n d the circumference a n d 64 v e r t i c a l subdivisions between the free surface a n d b o t t o m . T h e v e r t i c a l spacing uses a cosine w e i g h t f a c t o r t o give a finer s u b d i v i s i o n near the free surface a n d a relatively coarse spacing near the b o t t o m . A s n o t e d i n Section 1, a c o m p a r i s o n w i t h c o m p u t a t i o n s using A ' ^ = 5 1 2 panels

54 _{/ . N. Newman} 40m 10m 40m 10m 5ni 5m

Fig. 5. Configuration o f tlie hinged barges. shows differences o n the order o f 2 % , i n d i c a t i n g t h a t the

accuracy o f the resuhs presented here w i t h N = 2048 is a p p r o x i m a t e l y 0-5%.

F o r this special case, where the b o d y is a v e r t i c a l c i r c u l a r c y l i n d e r o f constant r a d i u s e x t e n d i n g f r o m the b o t t o m t o the free surface, the h y d r o d y n a m i c analysis c o u l d be p e r f o r m e d m o r e d i r e c t l y u s i n g H a v e l o c k ' s ' ° o r t h o g o n a l e i g e n f u n c t i o n expansion. T h e p a n e l m e t h o d used here has the advantage t h a t i t is applicable t o any p r a c t i c a l b o d y o f a r b i t r a r y three-dimensional f o r m .

I n a p p l i c a t i o n s where the stiffness f a c t o r EI and mass d i s t r i b u t i o n m are n o n - u n i f o r m , t h e i r f u n c t i o n a l dependence o n z m u s t be i n c l u d e d i n the integrals ( 4 . 4 - 4 . 5 ) . T h i s m a y require n u m e r i c a l e v a l u a t i o n o f these integrals, a n d w i l l result i n non-zero o f f - d i a g o n a l elements o f the mass m a t r i x , b u t no f u n d a m e n t a l d i f f i c u l t i e s are envisaged.

5 M O T I O N S O F A H I N G E D B A R G E

I f separate r i g i d vessels are connected mechanically, the r e s u l t i n g modes o f m o t i o n o f the g l o b a l b o d y i n c l u d e b o t h r i g i d modes o f the ensemble a n d d i s c o n t i n u o u s relative m o t i o n s between the separate vessels. A s a n example we consider t w o r e c t a n g u l a r barges, each o f w h i c h is 40 m l o n g b y 10 m beam a n d 5 m d r a f t , connected end-to-end b y a simple hinge w i t h a gap o f 10 m separating the t w o barges ( F i g . 5). T h e hinge axis is located a t the o r i g i n ( x = 0, z = 0 ) , m i d w a y between the t w o barges i n the plane o f the f r e e surface. E a c h barge is assumed t o be i n static e q u i l i b r i u m , w i t h u n i f o r m mass d i s t r i b u t i o n t h r o u g h o u t its length, beam, a n d vertically i n the range ( - 5 < z < 5 m ) . T h e w a t e r d e p t h is assumed t o be i n f i n i t e .

I n c i d e n t head waves are considered, w i t h resultant m o t i o n s i n surge ( ^ i ) , heave (^3), p i t c h (,^5), a n d i n the a d d i t i o n a l h i n g e d m o d e (^7). T h e first three modes are d e f i n e d c o n v e n t i o n a l l y , w i t h heave the v e r t i c a l m o t i o n at the hinge p o i n t a n d p i t c h the average a n g u l a r p o s i t i o n o f the t w o barges. I n the h i n g e d m o d e the r i g h t - h a n d a n d l e f t - h a n d barges are p i t c h e d a b o u t the y axis to angles ± ^ 7 , respectively. Since the g e o m e t r y is s y m m e t r i c a b o u t the hinge axis, a n d the displacement i n this m o d e is s y m m e t r i c , there is no c o u p l i n g w i t h surge o r p i t c h , a n d the l a t t e r modes are n o t affected b y the hinge d e f l e c t i o n .

T h e response-amplitude operators f o r heave and the hinge d e f l e c t i o n are s h o w n i n F i g . 6. These t w o modes are d y n a m i c a l l y similar, a n d t h e i r phase angles ( n o t s h o w n ) are p r a c t i c a l l y i d e n t i c a l . Since a p o s i t i v e hinge d e f l e c t i o n i m p l i e s negative v e r t i c a l m o t i o n at the outer

ends o f the barges, there is substantial c a n c e f l a t i o n resulting f r o m these t w o modes. T o illustrate this p o i n t , the heave a m p h t u d e at the m i d s h i p p o s i t i o n o f each barge (x = ± 2 5 m ) is also s h o w n i n F i g . 6.

A l s o s h o w n b y the dashed line i n F i g . 6 is the heave a m p l i t u d e o f the same p a i r o f barges w h e n the hinge is r i g i d (^7 = 0 ) . Except f o r a s m a f l s h i f t o f the peak w a v e p e r i o d a n d a m p l i t u d e , this response is p r a c t i c a l l y i d e n t i c a l to the heave response o f the h i n g e d barges as measured at t h e i r m i d s h i p sections. A t a wave p e r i o d o f 8 s, c o r r e s p o n d i n g to a w a v e l e n g t h o f 100 m , there is p r a c t i c a l l y n o heave m o t i o n o f the r i g i d c o n f i g u r a t i o n , or o f the h i n g e d c o n f i g u r a t i o n at the m i d s h i p sections. T h e results s h o w n i n F i g . 6 are based o n a descretization w i t h 2460 panels o n b o t h barges, u s i n g 32 l o n g i t u d i n a l subdivisions o n each barge, 16 trans-verse subdivisions a n d 8 v e r t i c a l subdivisions. S y m m e t r y a b o u t the planes x = 0 a n d y = 0 p e r m i t the s o l u t i o n s f o r c a n o n i c a l s y m m e t r i c a n d a n t i s y m m e t r i c p o t e n t i a l s , i n one q u a d r a n t consisting o f one side o f one barge. T h u s the t o t a l n u m b e r o f u n k n o w n s i n each c a n o n i c a l s u b - p r o b l e m is reduced to 640. Cosine spacing is used t o give smaller panels near the corners o f the barge a n d near the free surface. C o m p a r i s o n w i t h the resuhs f o r a

3.00 n

5.00 6.00 7.00 8.00 9,00 10.00 11.00 12.00

PERIOD

Fig. 6. Amplitudes of the symmetric modes of the hinged barge. Heave ( ^ 3 ) is normalized by the incident wave amplitude A and angular deflection i n the hinged mode ( ^ 7 ) is normalized by the wave slope KA. The lower solid curve shows the heave amplitude at the midship section o f each separate barge, and the dashed line shows the heave amplitude o f the same pair o f barges with a rigid connection instead o f a

coarse discretization u s i n g a t o t a l o f 640 panels shows m a x i m u m differences i n the range 1-5%. T h i s implies a n accuracy o f 1 % o r better f o r the results p l o t t e d i n F i g . 6.

I n s u m m a r y , the v e r t i c a l m o t i o n s o f the h i n g e d a n d r i g i d c o n f i g u r a t i o n s are p r a c t i c a l l y i d e n t i c a l at the m i d s h i p sections o f each barge, b u t substantially greater f o r the h i n g e d c o n f i g u r a t i o n at the hinge axis ( b y a f a c t o r o f a b o u t f o u r at resonance).

6 W A L L E F F E C T S I N A C H A N N E L

W a v e interactions w i t h a b o d y i n a channel o f r e c t a n g u l a r cross-section m a y be analyzed b y the m e t h o d o f images. T o i l l u s t r a t e this type o f p r o b l e m w e consider the case o f a v e r t i c a l c i r c u l a r c y l i n d e r o f finite d r a f t , centered i n a rectangular channel. T h i s p r o b l e m has been studied b y Y e u n g a n d S p h a i e r ' ' a n d also b y L i n t o n a n d E v a n s ; i n b o t h o f these w o r k s the g e o m e t r y o f the b o d y is restricted, b u t the images are accounted f o r i n a m o r e a n a l y t i c m a n n e r . H e r e we e m p l o y a finite n u m b e r o f image bodies, a n d study the convergence as this n u m b e r is increased. A s s u m i n g this p r o c e d u r e can be carried o u t successfully, i t offers the possibiHty o f p e r f o r m i n g c o m p u t a t i o n s f o r m o r e general b o d y shapes i n a channel.

T o p e r m i t c o m p a r i s o n w i t h the results i n the above references, the dimensions are chosen such t h a t

T = 2a,d = 2a, a n d h= 10a, where a is the c y l i n d e r

r a d i u s , T its d r a f t , d the channel h a l f - w i d t h , and h the channel d e p t h .

A single array o f image c y l i n d e r s is r e q u i r e d , w i t h the axes aX y = ±2nd {n= 1,2,2,... ,N). T h u s there is a t o t a l o f 27V^ + 1 cylinders i n c l u d i n g the c e n t r a l b o d y a n d its images o n b o t h sides. T h e c o m p u t a t i o n s are p e r f o r m e d w i t h A ' ' = 0 , 1 , 2 , 4 , 8 , c o r r e s p o n d i n g to arrays w i t h 1,3,5,9,17 cyhnders. E a c h c y l i n d e r is discretized i n a n i d e n t i c a l m a n n e r , w i t h 32 a z i m u t h a l s u b d i v i s i o n s , 8 v e r t i c a l subdivisions o n the side, a n d 8 r a d i a l subdivisions o n the b o t t o m , g i v i n g a t o t a l o f 512 panels o n each c y l i n d e r a n d u p t o 8704 panels f o r the largest a r r a y w i t h 17 separate bodies. Since the geometry is s y m m e t r i c a b o u t x = 0 a n d y = 0, the n u m b e r o f u n k n o w n s is reduced b y a f a c t o r o f f o u r . T h e c o n f i g u r a t i o n w i t h A'^ = 2 is s h o w n i n F i g . 7.

S i n g u l a r features m u s t be expected as a result o f the c h a n n e l w a l l s . T h e simplest o f these are the c u t - o f f frequencies, w h i c h f o r s y m m e t r i c p r o p a g a t i n g waves occur at kd = n, 2-K, 2>-K,..., c o r r e s p o n d i n g t o A : a = 1 - 5 7 , 3 ' 1 4 , 4 - 7 1 , . . . . F o r a n t i s y m m e t r i c waves,

kd = 7 r / 2 , 3 7 r / 2 , . . . , c o r r e s p o n d i n g t o ka = = 0 - 7 9 ,

2 - 3 6 , . . . . H e r e the relevant plane o f s y m m e t r y is the c h a n n e l centerplane, hence f o r a b o d y w i t h geometric s y m m e t r y a b o u t this plane the modes o f surge, heave a n d p i t c h a n d the d i f f r a c t i o n p r o b l e m are s y m m e t r i c whereas the modes o f sway, r o l l a n d y a w are a n t i

-Fig. 7. Perspective o f the vertical cylinder and f o u r images showing the discretization with 512 panels on each body. The channel walls are indicated by the straight lines, i n the plane o f

the free surface.

s y m m e t r i c . I n a d d i t i o n t o the c u t - o f f frequencies, L i n t o n a n d Evans'•^ have s h o w n t h a t a n a n t i s y m m e t r i c t r a p p e d wave exists at kd= 1-41, c o r r e s p o n d i n g t o ka = 0-70.

I r r e g u l a r frequencies also m u s t be a n t i c i p a t e d i n the present c o m p u t a t i o n s , w h i c h are based o n the s o l u t i o n o f a b o u n d a r y - i n t e g r a l e q u a t i o n using the free-surface Green's f u n c t i o n . T h e i r r e g u l a r frequencies w h i c h a f f e c t the heave f o r c e c o r r e s p o n d t o the zeros o f the Bessel f u n c t i o n Jo{ka), o c c u r i n g a t fca = 2 - 4 0 , 5 - 5 2 , . . . . F o r surge a n d sway, the zeros o f Ji{ka) are r e l e v a n t , o c c u r r i n g at A:fl = 3 ' 8 3 , 7 - 0 2 , . . . . M e t h o d s exist t o r e m o v e the effects o f the i r r e g u l a r frequencies, as described b y Lee a n d Sclavounos,'^ b u t t h e present results are s h o w n w i t h o u t this c o r r e c t i o n t o i l l u s t r a t e the r e s u l t i n g n u m e r i c a l errors.

F i g u r e 8 shows the surge exciting f o r c e ( p i t c h is s i m i l a r ) . T h e c o m p u t a t i o n s f o r the five d i f f e r e n t arrays are i n d i c a t e d b y dashed lines o f increasing l e n g t h , a n d the results o f Y e u n g a n d Sphaier are s h o w n b y the s o h d line. T h e w a l l effects are relatively s m a l l , a n d t h e results depend o n l y w e a k l y o n the n u m b e r o f images.

4.00 n

0,00 I J I I I I I I I I I I I I I I I I I I I 11 I I M I M M

0.00 1.00 2.00 3.00 4.00 5.00

1 a

Fig. 8. Surge exciting force f o r the cylinder in a channel. The dashed lines o f increasing length denote arrays w i t h 1,3,5,9, arid 17 bodies. The solid line is reproduced f r o m Yeung and Sphaier."

56 / . T V . Newman

0.00 1.00 2,00 3,00

Fig. 9. Heave exciting force f o r tiie cylinder in a channel. (See Fig. 8 f o r details.)

I n c r e a s i n g the n u m b e r o f images tends t o i m p r o v e the c o m p a r i s o n w i t h the results o f Y e u n g & Sphaier, a n d the m o s t apparent difference occurs i n the v i c i n i t y o f ka = TT, c o r r e s p o n d i n g t o the second c u t - o f f f r e q u e n c y f o r the channel. A weaker singular f e a t u r e is apparent i n the v i c i n i t y o f ka = 4-7, near the t h i r d s y m m e t r i c c u t - o f f f r e q u e n c y . The resuhs near the f i r s t c u t - o f f f r e q u e n c y are r e m a r k a b l y u n i f o r m . F i g u r e 9 shows the c o r r e s p o n d i n g results f o r the heave e x c i t i n g f o r c e , where the w a l l effects are m i n o r . A v e r y small ' h u m p ' is apparent at the first i r r e g u l a r f r e q u e n c y {ka = 2-40). ( B o t h e x c i d n g forces are c o m p u t e d u s i n g the H a s k i n d reladons, w i t h the i n t e g r a d o n carried o u t over the b o d y surface.)

T h e added-mass a n d d a m p i n g coefficients i n surge, sway, a n d heave are s h o w n i n Figs 1 0 - 1 5 . F o r surge

5.00 -| 4.00 H 3.00 i 2.00 H 1.00 d 0.00 1 I I I I I M M 0.00 1.00 2.00 3.00 4.00 5.00 t o

Fig. 11. Surge damping coefficient for the cylinder i n a channel. (See Fig. 8 f o r details.)

(Figs 10 a n d 11) the p r i n c i p a l feature occurs near the first s y m m e t r i c cut-off" f r e q u e n c y ka= 1-57. T h e r e is no c o r r e s p o n d i n g feature apparent at /ca = 4 . 7 1 . F o r sway a m u c h stronger feature is present near the t r a p p e d - w a v e f r e q u e n c y /cfl = 0-70, a n d first a n t i s y m m e t r i c c u t - o f f f r e q u e n c y ka = 079. W h e n the channel walls are accounted f o r exactly, the sway d a m p i n g is zero b e l o w the first c u t - o f f frequency. T h e finite arrays d e m o n s t r a t e Stokes' p h e n o m e n o n , o s c i l l a t i n g a b o u t zero i n an a t t e m p t t o m i m i c the sharp c u t - o f f . A weaker f e a t u r e is apparent at the second and t h i r d c u t - o f f frequencies ka = 2-36,3-92. T h e heave added mass a n d d a m p i n g d o n o t e x h i b i t such d i s t i n c t i v e features. T h e r e is a substantial increase i n the h i g h - f r e q u e n c y a d d e d mass due t o the c h a n n e l walls, b u t even the a r r a y w i t h t w o

7.00 6.00 ^ 5.00 H 4.00 3.00 2.00 1.00 h i

Dk

/ \ \

;:\.'

Fig. 10. Surge added-mass coefficient f o r the cylinder i n a channel. (See Fig. 8 f o r details.)

Fig. 12. Sway added-mass coefficient for the cylinder i n a channel (See Fig. 8 for details.)

20.00 15.00 10.00 H 5.00 0.00 1

il

Fig. 13. Sway damping coefficient for tlie cylinder i n a channel. (See Fig. 8 f o r details.)

images gives a close a p p r o x i m a t i o n t o the added mass i n this regime. F o r l o w frequencies the added mass a n d d a m p i n g are i n f l u e n c e d s i g n i f i c a n t l y b y the walls, a n d the d i f f e r e n t arrays show substantial differences. T h e l o w - f r e q u e n c y l i m i t o f the added mass is u n b o u n d e d due t o the w a l l s , as i n the case o f a t w o - d i m e n s i o n a l b o d y . E x c e p t i n the l o w - f r e q u e n c y regime there is g o o d agreement between the present results a n d the m o r e exact c o m p u t a t i o n s o f L i n t o n a n d Evans.

These results indicate t h a t the role o f the c u t - o f f frequencies varies f o r d i f f e r e n t f o r c e coefficients. T h i s is an o b v i o u s c o n c l u s i o n f o r d i f f e r e n t modes, f o r example between surge a n d sway, b u t a c o m p a r i s o n o f d i f f e r e n t coefficients f o r the surge m o d e reveals a m o r e subtle

2.50

2.00

1.50

1.00

0.00 1.00 2.00 3.00 4.00 5.00

Fig. 14. Heave added-mass coefficient for the cylinder i n a channel. The dashed lines of increasing length denote arrays with 1,3,5,9, and 17 bodies. The solid line is reproduced f r o m

Linton and Evans.

Fig. 15. Heave damping coefficient for the cylinder in a channel. (See Fig. 14 f o r details.)

d i s t i n c t i o n . A s n o t e d above, the e x c i t i n g f o r c e s h o w n i n F i g . 8 is r e m a r k a b l y u n i f o r m at the first c u t - o f f f r e q u e n c y , w i t h a substantial i r r e g u l a r f e a t u r e at the second c u t - o f f f r e q u e n c y , whereas the added mass a n d d a m p i n g (Figs 10 a n d 11) are precisely the opposite. T o a t t e m p t t o e x p l a i n this difference, i t m a y be n o t e d t h a t the pressure f o r c e due t o a free wave w h i c h is s y m m e t r i c a l a b o u t the centerplane j = 0, a n d subject t o c u t - o f f i n a channel w i t h walls at y = ±d, is p r o p o r t i o n a l t o the i n t e g r a l

tl-IT

cos 6 sin {k^a cos 9) cos {kyü sin 9) d9

H e r e kl + kj :k\ky=n7:/d{n= 1,2,3,...), and 9 denotes the p o l a r angle a r o u n d the c y l i n d e r . W i t h respect t o the added mass a n d d a m p i n g , one w o u l d expect t h a t the d i s c o n t i n u i t y near the n t h c u t - o f f f r e q u e n c y m i g h t be q u a l i t a t i v e l y p r o p o r t i o n a l t o this i n t e g r a l , whereas the H a s k i n d e x c i t i n g f o r c e w o u l d c o n t a i n an e x t r a w e i g h t f a c t o r cos(/cacos6') associated w i t h the i n c i d e n t - w a v e p o t e n t i a l . T h u s one m i g h t anticipate t h a t the differences i n i r r e g u l a r i t y near the first t w o c u t - o f f frequencies m i g h be associated w i t h d i f f e r e n t relative m a g n i t u d e s o f these integrals. T h i s is b o r n e o u t b y c o m p u t a t i o n s , i n s o f a r as the i n t e g r a l f o r the added mass a n d d a m p i n g is a b o u t f o u r times greater t h a n f o r the H a s k i n d e x c i t i n g f o r c e at the first i r r e g u l a r f r e q u e n c y , a n d a b o u t f o u r times smaller at the second. N o t e h o w e v e r t h a t a l l o f these integrals v a n i s h at the c u t - o f f frequencies, i n p r o p o r t i o n t o the f a c t o r kxO. T h u s the p o s s i b i l i t y remains t h a t this is n o t a complete e x p l a n a t i o n o f the above differences.

One c o m p u t a t i o n a l p r o b l e m w h i c h s h o u l d be recog-nized is t h a t the t o t a l n u m b e r o f panels, a n d u n k n o w n s , is p r o p o r t i o n a l t o the n u m b e r o f image bodies. F o r the largest array, w i t h 17 separate bodies a n d 512 panels o n

58 / . N. Newman

each b o d y , a t o t a l o f 8704 panels are used. E x p l o i d n g the geometric s y m m e t r y o f this c o n f i g u r a t i o n reduces the t o t a l n u m b e r o f u n k n o w n s t o 2176. One p o s s i b i l i t y w h i c h has n o t been explored is to decrease the n u m b e r o f panels o n the images as the distance f r o m the central b o d y increases. I n this m a n n e r i t m a y be possible t o reduce substantially the t o t a l n u m b e r o f panels, w i t h o u t d e g r a d i n g the accuracy o f the c o m p u t e d forces o n the b o d y itself.

T h e use o f a t r u n c a t e d image a r r a y w i t h a finite n u m b e r o f image bodies has been c r i t i c i z e d i n the m o r e a n a l y t i c w o r k o f L i n t o n a n d Evans, due to the i n c o r r e c t f o r m o f the resulting s o l u t i o n i n the f a r field. H o w e v e r f o r c o m p u t a t i o n s o f the first-order f o r c e coefficients a c t i n g o n a b o d y i n a channel, a n d f o r other l o c a l flow parameters, the e r r o r i n the f a r field seems i r r e l e v a n t . T h e present resuhs suggest t h a t p r a c t i c a l c o m p u t a t i o n s c a n i n f a c t be p e r f o r m e d w i t h a r e l a t i v e l y s m a l l n u m b e r o f images, except f o r the singular features w h i c h occur near c u t - o f f frequencies. T h e significance o f these singular features is reduced i n practice b y i m p e r f e c t w a f l refiections, a n d b y transient effects w h i c h attenuate the resonant reflections.

T w o issues w h i c h have n o t been addressed here are the effects o f channel wafls o n the l o c a l pressure field a n d o n the second-order m e a n d r i f t f o r c e . M c l v e r ' ' * ' ' ^ has n o t e d t h a t these quantities are affected s t r o n g l y by w a l l effects. T h e requirement o f large n u m b e r s o f panels m a y be a significant b a r r i e r f o r c o m p u t a t i o n s o f the m e a n d r i f t f o r c e at the relatively large wavenumbers s h o w n i n Figs 1 0 - 1 5 .

7 D I S C U S S I O N

T h i s paper extends the m e t h o d o l o g y o f three-dimens-i o n a l r a d three-dimens-i a t three-dimens-i o n a n d d three-dimens-i f l f r a c t three-dimens-i o n based o n the panel m e t h o d o f c o m p u t a t i o n t o the analysis o f various d e f o r m a b l e b o d y m o t i o n s . T h e c o n t i n u o u s deflections w h i c h are i l l u s t r a t e d f o r the v e r t i c a l c o l u m n a n d floating barge are t y p i c a l o f s t r u c t u r a l deflections. T h e discon-t i n u o u s m o discon-t i o n s o f discon-the h i n g e d barge a n d a r r a y o f cylinders are examples where the p h y s i c a l a p p h c a t i o n is q u i t e d i f f e r e n t . T h e specific examples used f o r i l l u s t r a -tion are g e o m e t r i c a l l y simple, b u t the c o m p u t a t i o n a l a p p r o a c h is applicable to m o r e general bodies.

F o r s t r u c t u r a l deflections the p r a c t i c a l i m p o r t a n c e o f a c o m p l e t e h y d r o d y n a m i c analysis depends o n the p a r t i c u l a r a p p l i c a t i o n . F o r relatively stiff" f r e e l y - f l o a t i n g bodies, i n c l u d i n g c o n v e n t i o n a l ships, the frequencies o f h u h v i b r a t i o n s are substantiafly higher t h a n those o f first-order wave e x c i t a t i o n , a n d the a m p h t u d e s o f these m o t i o n s are m u c h smaller t h a n the r i g i d - b o d y modes. T h u s the h y d r o d y n a m i c a n d s t r u c t u r a l analyses can be c a r r i e d o u t separately. H o w e v e r this s i m p l i f i e d proce-d u r e is n o t a p p r o p r i a t e i f the b o proce-d y is m o r e flexible. T h e slender barge analysed i n Section 3 illustrates the

m e t h o d o l o g y w h i c h c a n be used i n such cases. F o r the slender v e r t i c a l c o l u m n treated i n Section 4, a h i g h l y -t u n e d resonan-t b e n d i n g deflec-tion occurs w i -t h i n -the range o f s i g n i f i c a n t wave energy, increasing the i m p o r t a n c e o f the s t r u c t u r a l d e f l e c t i o n . A s the develop-m e n t o f o f f s h o r e p l a t f o r develop-m s extends i n t o deeper waters this type o f p r o b l e m m a y become particularly i m p o r t a n t .

One p o i n t emphasized i n b o t h o f these examples is the f e a s i b i f l t y o f using m a t h e m a t i c a l m o d e shapes w h i c h d i f f e r f r o m the m o r e c o n v e n t i o n a l p h y s i c a l modes o f the b o d y . The v e r t i c a l c o l u m n is an example where the use o f o r t h o g o n a l p o l y n o m i a l s is p a r t i c u l a r l y effective, a n d where the c o r r e s p o n d i n g n a t u r a l modes w o u l d be relatively c o m p l i c a t e d due to the concentrated mass at the free surface. Conversely, the floating barge fllustrates the relative efficiency o f using n a t u r a l modes o f a b e a m w i t h free ends, a l t h o u g h the use o f o r t h o g o n a l p o l y n o m i a l s also is feasible.

I t is c u s t o m a r y i n engineering applications t o p e r f o r m a d y n a m i c analysis o f the specific structure, i g n o r i n g the effects o f the s u r r o u n d i n g fluid to determine the con-esponding n a t u r a l ' d r y ' modes. I n the case o f a ship, f o r example, these w o u l d be the actual physical modes o f free v i b r a t i o n o f the huU, at different eigenfrequencies, accounting f o r the distributions o f structural stiffness and mass t h r o u g h o u t the ship a n d neglecting a l l h y d r o -dynamic a n d hydrostatic pressure forces. F o r cases where the stiffness a n d mass are substantially continuous, such an approach seems unnecessarily comphcated b y c o m -parison w i t h the alternatives o f representing the wave-induced deflections i n terms o f either the n a t u r a l modes (3.9) f o r a simple u n i f o r m beam, o r appropriate o r t h o g o n a l p o l y n o m i a l s .

T h e h i n g e d barge, w h i c h is considered i n Section 5, illustrates the m e t h o d o l o g y f o r s t u d y i n g m u l t i p l e bodies w i t h mechanical constraints. T h i s type o f p r o b l e m c o u l d be analysed i n a m o r e general manner b y c o n s i d e r i n g each separate b o d y to have six independent degrees o f f r e e d o m , and i m p o s i n g the c o n s t r a i n t i n a special post-processor. T h u s the f o u r relevant modes o f surge, heave, p i t c h , a n d hinge d e f l e c t i o n f o r the h i n g e d barge w o u l d be represented b y six modes (surge, heave a n d p h c h f o r each separate b o d y ) , a n d t w o constraints w o u l d be i m p o s e d stating t h a t the h o r i z o n t a l a n d v e r t i c a l m o t i o n at the hinge p o i n t is the same f o r b o t h barges. Such a procedure is a w k w a r d f r o m the c o m p u t a t i o n a l stand-p o i n t , since the r a d i a t i o n stand-p o t e n t i a l o f the extra modes must be evaluated, a n d the constraints are i m p o s e d i n an ad hoc m a n n e r . T h e present a p p r o a c h avoids these c o m p l i c a t i o n s .

T h e p r o b l e m o f w a l l effects o n a c y l i n d e r , treated i n Section 6, is another a p p l i c a t i o n o f d i s c o n t i n u o u s modes. I n this case generalized modes are used t o represent the a p p r o p r i a t e m o t i o n s o f each image b o d y , so as t o satisfy the b o u n d a r y c o n d i t i o n s o n the w a l l s . T h i s p r o b l e m also c o u l d be treated w i t h i n d e p e n d e n t modes o f each image b o d y , b u t the r e l a t i v e l y large

n u m b e r o f images and c o r r e s p o n d i n g independent modes w o u l d substantially increase the c o m p u t a t i o n a l b u r d e n . A C K N O W L E D G E M E N T S T h i s w o r k was p e r f o r m e d as p a r t o f the J o i n t I n d u s t r y P r o j e c t ' W a v e effects o n o f f s h o r e structures', s u p p o r t e d b y the C h e v r o n O i l F i e l d Research C o m p a n y , C o n o c o N o r w a y , M o b i l O i l C o m p a n y , the N a t i o n a l Research C o u n c i l o f Canada, N o r s k H y d r o , O f f s h o r e T e c h n o l o g y Research Center, Petrobras, Saga P e t r o l e u m , the Shell D e v e l o p m e n t C o m p a n y , S t a t o i l , a n d D e t N o r s k e V e r i t a s Research. T a b u l a r d a t a files f o r the solid curves s h o w n i n Figs 9 a n d 10 were k i n d l y p r o v i d e d b y the a u t h o r s o f Refs 11 a n d 12.

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