The distribution of the hydrodynamic forces on a heaving and pitching shipmodel in still water

D i s c u s s i o n o f t h e p a p e r s :

"A NEW APPRAISAL OF STRIP THEORY",

By: V a s s i l o p o u l o s and Mandei,

and

"THE DISTRIBUTION OF THK HYDRODYNAMIC FORGES QN A

HEAVING AND PITCHING SHIPMODEL I N STILL WATER" ,

By: Q e r r i t a m a and Beukelsian.

Both p r e s e n t e d a t t h e F i f t h Symposium on

Naval Hydrodynamics, Bergen 1964„

By: T.R. Dyer»

1

-10 I n t r o d u c t i o n .

The paper by V a s s i l o p o u l o s and Handel ^1 ] r i g o r o u s l y examined much o f p r e s e n t seakeeping t h e o r y , and i s e s p e c i a l l y v a l u a b l e f o r

emphasis on developing a b a s i s f o r p r a c t i c a l s h i p design a p p l i c a t i o n . The paper by G e r r i t s m a and Beukelman [ 2 ] c o n t a i n s s i g n i f i c a n t e x p e r i -m e n t a l r e s u l t s , and a c l e a r c o n c i s e p r e s e n t a t i o n o f s t r i p t h e o r y . I t I t i s a m e a n i n g f u l b r i d g e between t h e o r y and p h y s i c a l phenomena. However, these two papers have d i s c r e p a n c i e s between them, and t h e paper by V a s s i l o p o u l o s and Mandei [1] d i s a g r e e s w i t h t h e r e s u l t s o f K o r v i n - K r o u k o v s k y [3]. The d i f f e r e n c e s have been examined below, w i t h r e s p e c t t o : f i r s t , t h e s t r i p t h e o r y ; second, t h e choice o f axes; and t h i r d , t h e e x p e r i m e n t a l r e a u l t a i n [2]»

The two p a p e r s , [1] and [2] d i s a g r e e i n t h e e v a l u a t i o n o f some m o t i o n d e r i v a t i v e s . Let i t be emphasized t h a t no disagreement e x i s t s as t o t h e form o f t h e c o e f f i c i e n t s o f t h e e q u a t i o n s o f m o t i o n . The d i s t i n c t i o n between t h e c o e f f i c i e n t s o f t h e e q u a t i o n s o f m o t i o n , and t h e m o t i o n d e r i v a t i v e s , i s i m p o r t a n t . The c o e f f i c i e n t s , a, b, c, . . A, B, C . . ., c o n t a i n m o t i o n d e r i v a t i v e s . Which m o t i o n d e r i v a t i v e s appear i n each c o e f f i c i e n t i s independent o f t h e method o f e v a l u a t i o n o f t h e hydrodynamic f o r c e s . Since b o t h papers p r e s e n t f i n a l r e s u l t s f o r f i x e d axes, i t i a g r a t i f y i n g . , t h a t agreement e x i s t s on the m o t i o n d e r i v a t i v e s c o n t a i n e d i n each c o e f f i c i e n t . M o t i o n d e r i v a t i v e s a r e ex-p r e s s e d as Z. , Z , M., M e t c . , i n t h e n o t a t i o n o f [l]<. The disagreement

" " 8 K

i n e v a l u a t i o n o f m o t i o n d e r i v a t i v e s i s due t o d i f f e r e n c e s i n t h e a p p l i -c a t i o n o f s t r i p t h e o r y t o t h e d e t e r m i n a t i o n o f hydrodynami-c for-ces»

These d i f f e r e n c e s i n a p p l i c a t i o n are due s o l e l y t o one d i f f e r i n g a s s u m p t i o n . T h i s i n v o l v e s what G e r r i t s m a and Beukelman {,2] r e f e r t o aa

" t h e e f f e c t o f f o r w a r d speed", however, " t h e e f f e c t o f f o r w a r d apeed on s t r i p t h e o r y " i s more p r e c i s e . [2] c l e a r l y shows the n e c e s s i t y o f a f o r w a r d speed c o n s i d e r a t i o n , but two t y p e s o f c o n s i d e r a t i o n s a r e i n e e l u d e d : those a r i s i n g from s t r i p t h e o r y e v a l u a t i o n o f the m o t i o n d e r i ~ v a t i v e a , and those a r i s i n g o n l y from t h e f i x e d a x i s mechanics o f a r i g i d b o d y o

-2O S t r i p t h e o r y .

D i f f e r e n c e s i n a t r i p t h e o r y a p p l i c a t i o n e x p l a i n a l l d i s c r e p a n c i e s between f l ] and V a s s i l o p o u l o s and Handel [ l ] s t a t e t h a t : "Each o f t h e s t r i p s i s assumed t o b e l o n g t o a s p e c i f i c i n f i n i t e c y l i n d e r o s c i l l a t i n g a t z e r o f o r w a r d speed and i t a b e h a v i o u r i s assumed independent and i s o -l a t e d from t h e n e i g h b o u r i n g s t r i p " .

Since a system moving w i t h f o r w a r d speed i s c o n s i d e r e d , t h e r a t e o f change o f added mass comes i n t o t h e f o r m u l a f o r t h e hydrodynamic f o r -c e , as i s p u t f o r w a r d i n 2 , w h i l e t h e f l o w i n ea-ch s t r i p i s independent o f t h e f l o w i n t h e n e i g h b o u r i n g s t r i p s . T h i s i s n o t c o n s i d e r e d i n [ l l , w h i c h s t a t e s : "The i n t r o d u c t i o n o f terms dependent on t h e r a t e o f change o f added mass over t h e s h i p l e n g t h i a i n c o n s i s t e n t w i t h t h e use o f two-d i m e n s i o n a l t h e o r y " . T h e r e f o r e , i t seems t h a t [ l ] assiunes'the e f f e c t o f f o r w a r d speed on s t r i p t h e o r y " t o be n e g l i g i b l e .

Table I compares (columns 3 and k) t h e f i n a l r e s u l t s o f [ l ] and [21 , each e x p r e s s i o n o f a m o t i o n d e r i v a t i v e i s e n c l o s e d i n b r a c k e t s

Disagreement e x i s t s i n c o e f f i c i e n t s B, C and E, and t h e a p p a r e n t a g r e e ment i n some o t h e r c o e f f i c i e n t s i s due o n l y t o c a n c e l l a t i o n o f speed e f -f e c t s o The speed e -f -f e c t s w i l l be u n t a n g l e d -from t h e m a t h e m a t i c s , showing t h e i r exact r o l e s , and u n c o v e r i n g no e r r o r s i n e i t h e r tlJ o r [23 .

The " e f f e c t o f f o r w a r d speed on s t r i p t h e o r y " may be a n a l y s e d by s u b d i v i s i o n i n t o a " t h r e e d i m e n s i o n a l c o r r e c t i o n " and a "speed c o r r e c t i o n " o Consider f i r a t t h e " t h r e e d i m e n s i o n a l c o r r e c t i o n " , w h i c h e x p r e s s -e s chang-e o f add-ed mass a l o n g t h -e s h i p s l -e n g t h y I t can b-e s-e-en i n F i g o 7 and 8 o f [ 2 l , t h a t f o r w a r d speed haa l i t t l e e f f e c t on a' and b', f o r t h e t w o - d i m e n s i o n a l c y l i n d r i c a l form o f t h e m i d s h i p s e c t i o n ( s e c t i o n No. ^)» C o n v e r s e l y , t h e t h r e e - d i m e n s i o n a l f o r w a r d and a f t e r s e c t i o n s show marked speed dependence i n b', t h e s e c t i o n a l damping c o e f f i c i e n t . K o r v i n K r o u -k o v s -k y , (3j page 123, t a -k e s s e c t i o n a l a r e a t o be a f u n c t i o n o f time» T h i s a r e a , o f t h e s p e c i f i c s e c t i o n i n s t a n t a n e o u s l y i n c o n t a c t w i t h t h e h y p o t h e t i c a l aheet o f w a t e r , must o b v i o u s l y change as t h e s h i p p r o g r e s s -es t h r o u g h t h e s h e e t , m' and N' a r e f u n c t i o n s o f s e c t i o n a l a r e a , and so must a l s o be f u n c t i o n s o f t i m e . The " t h r e e - d i m e n s i o n a l c o r r e c t i o n " thus i a expressed by c o n s i d e r i n g m' a f u n c t i o n o f t i m e , and / 0.

The "apeed c o r r e c t i o n " i s c o n s i d e r e d s e p a r a t e l y from t h e " t h r e e -d i m e n s i o n a l " c o r r e c t i o n , i n o r -d e r t o c l e a r l y show i t s r e l a t i o n t o o t h e r v e l o c i t y t e r m s . The "speed c o r r e c t i o n " i s found by c o n s i d e r i n g X, t h e d i s t a n c e from t h e b o d y a x i s o r i g i n t o t h e sheet o f w a t e r , t o be a f u n c -t i o n o f -t i m e , as -t h e s h i p p r o g r e s s e s -t h r o u g h -t h e s h e e -t .

-1 C o e f f i -c i e n t s 3 -TABL I o C o e f f i c i e n t s o f e q u a t i o n s o f m i o n . d e r i v a t i v e s H e s u l t s o f [ 2 ] , w i t h s t r i p t h e o r y c o r r e c t e d f o r f o r w a r d speed. R e s u l t s o f [1] R e s u l t s o f [2], w i t h s t r i p t h e o r y n o t c o r r e c t e d f o r f o r w a r d speed 3 E -Z -z. -z, . ( Z ^ . u ^ Z ^ ) q -(M + u M.) q o w -M f m' dx L' ' L J ( N' dx L ^ L J - J^N' X dx + V [• - gS 1 + V j N' dx m' X dx .''L J M'x^ dx +v f n . x d x - V r f ffl.xdx J •'L . m' X dx ^ L J N' X dx + V m J|A ( x ) d x JN ( x ) d x ^ g J B ( x ) dx - j ( x ) X dx - | N ( X ) X dx + J j ^ ( x ) dx . p g B( x ) X dx + U j N ( x ) d x j •) J o *- -J r 2 J|l. ( x ) x dx r 2 7 r r T N( x ) x . - u ( x ) x d x ^ 3 g j B ( x ) x 2 d x -U^ j N ( x ) x d x - J } ^ ( x ) dx - N ( x ) X dx - ƒ) g ]B ( X ) X dx • ' dx f N» dx L ^ L • • X dx HL - f N' x d x | + V [ffi] J L J - '/3 g S ^ + J j ^ ^' "^^J «' x^ dx 2 1 " f N' X dx - V m'x dl L ^ L J V^L p g I ] - v ' f N'xdx - j ^ . ' x d x - f N' X dx . J L J

N e t e s : I ) For c o n v e n i e n c e , t h e n o t a t i o n o f [ 1 ] and [2] i s m i x e d , i n ^.11 cases i n t e n t s h o u l d be c l e a r . 2 ) For c o n s i s t e n c y , c o e f f i c i e n t s i n column 3 have been r e a r r a n g e d r e l a t i v e t o t h e i r f o r . i«

Suoh a c o r r e c t i o n i s independent o f t h e t h i r d dimension and would a l s o appear i n t h e c o n s i d e r a t i o n o f t w o - d i m e n s i o n a l cylinders« T h i s c o r r e c t i o n i s , however, c o n f u s e d by s i m i l a r i t y t o terms a r i s i n g from t h e mechanics o f movable a x i s systems. Care must be t a k e n t o d i s t i n g u i s h between these a i m i l a r t e r m s .

The r o l e o f " t h e e f f e c t o f f o r w a r d speed on s t r i p t h e o r y " i s most e a s i l y seen by c a r r y i n g o u t t h e t h e o r e t i c a l d e r i v a t i o n o f [ 2 ] , b u t e l i -m i n a t i n g a l l " e f f e c t o f f o r w a r d speed" c o r r e c t i o n s . A l l such c o r r e c t i o n s w i l l be i d e n t i f i e d by braces ^ ^. Since a l l assumptions s h o u l d now agree, t h e r e s u l t i n g m o t i o n d e r i v a t i v e s s h o u l d agree w i t h those o f

R e f e r i n g now t o a r t i c l e 4.1, S t r i p Theory, o f [2] , f o r pure heave

we have, w i t h t h e n o t a t i o n o f [ 2 ] : i = - è ( » ' ^ ) - N' - a / ) g y Z ^ . D i f f e r e n t i a t i n g , we o b t a i n : = - ^ " ' ^ ^ 1 ^ ] ^o^ - ^ ' ^ o - ^f^y^o" N o t i n g t h a t t h e " t h r e e - d i m e n s i o n a l c o r r e c t i o n " : » » » • I / dm dx dm dx (" „ dm 7 t j ~ dx d t • • • " dx d t " " [ ^ dx j ' we o b t a i n , as i n [2] :

But n e g l e c t i n g t h e " t h e e f f e c t o f f o r w a r d speed" we have: dm

d t

where 2 D i n d i c a t e s n e g l e c t o f t h e " t h r e e d i m e n s i o n a l " and "speed" c o r r e c -t i o n s . I n -t e g r a -t i n g we o b -t a i n :

( 6 ) 2D

N o t i c e t h a t c o e f f i c i e n t b = J'^^ N dx i s t h e same w i t h and w i t h o u t c o r r e c -t i o n s , becauae:

f o r t h e case o f m' = 0 a t x = -L/2 and x = + L/2, see [ 2 ] and [5]»

5

-Now c o n s i d e r i n g the moment we a««

K ' = (x m' ) 2^ + ( N ^ - X ^ ^ ) + 2 /o g X y Z . . . ( ? ) H • • . and: (M*) = (Xm') 2^ + (N* x ) Z^ + 2 / ) g y . . . n 2 D o r o I n t e g r a t i n g we o b t a i n : (M„) = ( f m * x d x ) 2 + ( f N ' x d x ) Z + ftgS Z . . , a 2 D L ° L o r w o N e t i c e t h a t , i n [ 2 ] , t h e c o e f f i c i e n t E = , N ' X dx + Vm , L because: - Sv f X ^ dx? = + Vm

whan i n t e g r a t e d by p a r t s , f o r m = 0 a t the ends, see [ 2 and [5 Next we must c o n s i d e r t h e s h i p i n pute p i t c h :

2 D ( 8 ) 2D I I P

An e x p r e s s i o n f o r Z must be f o u n d . I f we c o n s i d e r the p o i n t where t h e o

mevable x a x i s p i e r c e s the h y p o t h e t i c a l sheet o f w a t e r , l e t t i n g t h e d i s -placement o f t h a p o i n t be Z^, we have Z^ = ( _ x 0 ) , and d i f f e r e n t i a t i n g we have:

2^ = (_xö - X e) = (-xé + v o ) ,

w h i c h agrees w i t h [2]. Bear i n mind t h a t t h i s i s an e x p r e s s i o n o f the velo-= c i t y o f a p o i n t on t h e s h i p , r e l a t i v e t o f i x e d axes, t h u s x must be t a k e n aa a f u n c t i o n o f t i m e (eee Fay [ 6 ] )o However, i f x i s n o t a f u n c t i o n o f t i m e , then 2 = - x Ö . T h i s w i l l l e a d t o r e s u l t s d i s a g r e e i n g w i t h those o f

o

[ l ] o Thus i t appears t h a t V a s s i l o p o u l o s and Mandei [1] do, i n t h i s case, o s n s i d e r x a f u n c t i o n o f t i m e . T h i s i s n o t done e x p l i c i t l y , b u t i s a con-sequence o f t h e c o n v e r s i o n from moving t o f i x e d - a x i s systems. [1] develops the form o f the c o e f f i c i e n t s i n d e p e n d e n t l y o f t h e m e t i o n d e r i v a t i v e s , and i s t h u s a b l e t o c o n s i d e r t h e e f f e c t o f f o r w a r d speed on the r i g i d body mechanics, w h i l e n o t c o n s i d e r i n g i t s e f f e c t on t h e s t r i p t h e o r y . T h i s i s because [ l ] c o n s i d e r s o n l y u^, the c o n s t a n t v e l o c i t y o f the s h i p and not x.

-The paraiB«t»r x appears o n l y a s a r e s u l t o f s t r i p t h e o r y ; and i e t h u s c o n s i d e r e d , i n [ l ] , o n l y i n t h e f i n a l e v a l u a t i o n o f t h e c o e f f i c i e n t s o Now c o n t i n u i n g , b u t n o t n e g l e c t i n g V © , we h a v e : ^p ^ " "it + v e ) - N * ( - x ö + v e ) + 2 ^ 3 g y x e . N o t i n g t h a t i n ]L2] t h e s i g n o f 2 ƒ> g y x 0 must be p o s i t i v e . We t h e n h a v e : = -m' ( - X Ö - [ i © ^ + V ê ) - [ f f ' ^ (-x© + V © ) ^ - N ' ( - X © + V © ) + 2 / ) g y x O , a n d : F p = m' X Ö + (N' x + [ x m ' ^ - V m * - ^ x V - | | - ^ ) © + + ( 2 / , g y x -H [ V ^ _ N' V ) © . . . ( 9 ) At t h i s s t a g e t h e " s p e e d c o r r e c t i o n " i s n e g l e c t e d by and ^ x m'^ d i s -a p p e -a r s , y i e l d i n g : ( F p ) ^ ^ = m' x 8 + ( N ' X - Vm') 9 + ( 2 ^ g y X - N' V) © . . . ^^^2 D I n t e g r a t i o n g i v e s u s : ( F ) = ( f m' X d x ) 8 + ( f N ' X dx - Vm) © + (/)g S - v f N' dx)© o .o ( 1 0 ) P 2 D J L X ' Note t h a t t h e c o e f f i c i e n t e, i n [ 2 , i s a l s o : I r ' r N x d x - Vm, b e c a u a e x m dx and V c a n c e l , and t h a t g, i n [ 2 ] , i s : pe - v j N' dx, b e c a u s e ƒ ^ dx = 0. L L Now c o n s i d e r i n g moments : M' = - m ' x ^ S -(N* x^-H [ x m ' x ] - V m' x - [ x ^ V ) © = - ( 2 ^ g y x ^ +

b^^'Trl

-«Ad I n t e g r a t i n g I (M^) = = ( fm' d x ) Ö- ( f N 'X ^ dx - v f m' X dx) Ö -- ( / ) g l -- v f N * x d x ) 0 0 . . ( 1 2 ) * t 2D Comparing t h i s w i t h [ 2 ] we s e e t h a t i n c o e f f i c i e n t B: * ƒ m ' x - v y m ' x - | x ^ V ^ d x = 0 , L L when t h e l a s t i n t e g r a l i s e v a l u a t e d by p a r t s f o r m =0 a t the e n d s . T h i s l e a d s to a c o e f f i c i e n t w i t h s e e m i n g l y no u M. t e r m . However, we now s e e t h a t u Mo = + V l m x d x , w h i c h i s c a n c e l e d by a - V/ m x d x term i n o w J L ^ L A c o m p a r i s o n o f t h e s e r e s u l t s (column 5 ) w i t h t h o s e o f [ 1 ] (column k) i n T a b l e I shows t h e r e s u l t s to be i d e n t i c a l . T h i s shows t h a t the d i f f e -r e n c e s between [ l ] and [zJ do, i n d e e d , -r e s u l t o n l y f-rom a d i f f e -r i n g assump-t i o n r e g a r d i n g assump-the e f f e c assump-t o f f o r w a r d s p e e d on s assump-t r i p assump-t h e o r y e v a l u a assump-t i o n o f t h e hydrodynamic t e r m s . I f the i n t e g r a l s i n [ 2 ] and [ 5 ] , mentioned above, a r e a p p l i e d to K o r v i n - K r o u k o v s k y ' s 3 c o e f f i c i e n t s e, B, C and E , t h e y a r e s e e n t o a g r e e w i t h t h o s e o f G e r r i t s m a and Beukelman [ 2 ] , w i t h " t h e e f f e c t o f f o r w a r d s p e e d on s t r i p t h e o r y " c o r r e c t i o n s i n c l u d e d . The c o e f ¬ f i c i e n t s i n [ 3 ] a r e more g e n e r a l , not r e q u i r i n g m = 0 a t the e n d s . T h u s , K o r v i n - K r o u k o v s k y ' s d i s a g r e e m e n t w i t h [ 1 ] does not r e s u l t from " e r r o n e o u s t i a e - d i f f e r e n t i a t i o n " , but o n l y from a d i f f e r i n g a s s u m p t i o n r e g a r d i n g t h e a p p l i c a t i o n o f s t r i p t h e o r y .

The key to the t h e o r y , i n [z] , i s i n t h e ^ (m' Z^) t e r m s , w h i c h g i v e r i s e to the " t h r e e - d i m e n s i o n a l c o r r e c t i o n " and l e a d to use o f t h e "speed c o r r e c t i o n " . Tiiose s p e e d terms found i n a l l p a p e r s , due t o r i g i d body m e c h a n i c s , a r i s e from the e x p r e s s i o n :

z^ = -

+ v e ,

w h i c h a p p e a r s i n [ l ] , a s :

3» C h o i c e o f axeso

The work i n [ l l i s b a s e d on t h e work o f A b k o w i t z [kJ. A b k o w i t z ' d e r i v a t i o n o f t h e e q u a t i o n s o f motion i s p e r f o r m e d f i r s t on a s y s t e m o f movable a x e s , t h e n c o n v e r t e d t o f i x e d a x e s . G e r r i t s m a and B e u k e l = man [ 2 ] , and K o r v i n - K r o u k o v s k y [ 5 j work w i t h f i x e d a x e s . It? i s sometimes s u g g e s t e d t h a t d i f f e r e n c e s a r i s e from t h e s e two a p p r o a c h e s . O b v i o u s l y t h e p h y s i c a l phenomena, and t h u s the motion d e r i v a t i v e s , do not change a s man changes i m a g i n a r y a x i s s y s t e m s . However, s i n c e t h e c o e f f i c i e n t s a r e a s s o c i a t e d w i t h d i f f e r e n t p a r a m e t e r s i n the d i f f e r e n t s y s t e m s (ego Z and Z ^ ) , t h e i r forms «ust change. C o e f f i c i e n t s e, g, C and D a l l c o n -t a i n a -term o f -t h e p r o d u c -t o f V and a mo-tion d e r i v a -t i v e , when w r i -t -t e n f o r f i x e d a x e s . B u t , i n a movable a x i s s y s t e m t h e s e v e l o c i t y dependent terms d i s a p p e a r o These v e l o c i t y terms a r e q u i t e s i m i l a r t o some o f t h e " e f f e c t o f f o r w a r d s p e e d on s t r i p t h e o r y " c o r r e c t i o n s o I n o r d e r to dem o n s t r a t e the e f f e c t o f a x i s s y s t e dem s , and t o ahow t h a t t h e s p e e d c o r r e c t i o n s f o r s t r i p t h e o r y a r e i n d e p e n d e n t o f a x i s s y s t e m s , t h e f i x e d -a x i s r e s u l t s o f [ 2 ] w i l l be c o n v e r t e d to movable a x e s . These r e s u l t s can t h e n be compared w i t h t h e m o v a b l e - a x i s r e s u l t s o f [l]»

To c l e a r l y s e e what happens to the "speed e f f e c t on s t r i p t h e o r y "

main i n expanded form.

C o n s i d e r i n g f i r s t t h e e q u a t i o n s p r e s e n t e d f o r f i x e d a x e s , i n the t e r m s , t h e y v / i l l r e m a i n i n s i d e b r a c e s and the e q u a t i o n s w i l l r e -form o f [ l ] , the n o t a t i o n o f [ 2 ] : F = ^ V ( 2 ^ ) , and: M = k ^ ^ V ö , a n d : F = ( j m ' d x ) 2 ^ + (ƒ N' d x ) Z ^ + (/)gA^)Z^ - (ƒ m' x d x ) 8 ) © -• 2 m X d x ) 8 + (ƒ ' 2 N X dx + m X dx D X dx -- ( j T N ' x d x -- [ v / ^ x ^ d x ^ ) Z ^ - / > g S ^ Z ^ 9

9

-Now t h e s e e q u a t i o n s may be t r a n s f o r m e d t o t h o s e f o r movable a x e s by e x p r e s s i n g o f t h e c e n t r e o f g r a v i t y i n t e r m s o f Z , a s f o l l o w s : Z = Z c o s <è ^ 7i o Z = Z - V 0 O

2 = 2

Ó

V (2

a 7

The " e f f e c t o f s p e e d on s t r i p t h e o r y " terms r e m a i n ; and i f motion d e r i -v a t i -v e s a r e s u b s t i t u t e d f o r t h e -v a r i o u s t e r m s , t h e r e s u l t s a r e i d e n t i c a l to t h o s e o f f o r movable a x i s s y s t e m s . E v i d e n t l y t h e n , t h e a r b i t r a r y c h o i c e o f a x i s - s y s t e m has no e f f e c t on s t r i p t h e o r y , and no e r r o r s have been made i n [ l ] o r [ 2 ] . A l l v e l o c i t y dependent t e r m s now p r e s e n t a r e t h e r e s u l t o n l y o f s t r i p t h e o r y .

-ko G o a p a r i s o n w i t h e x p e r i n i e n t a l resuits»

A s s e e n i n T a b l e I , t h r e e c o e f f i c i e n t s show d i s c r e p a n c i e s

between [ l ] and [ 2 ] , t h e s e a r e B, C and E . The t h e o r e t i c a l v a l u e s o f th«ae p a r a m e t e r s , due to b o t h [ l ] and [ 2 ] s h o u l d be compared w i t h t h e e x p e r i m e n t a l r e s u l t s o f [ 2 ] . I n F i g . 1 3 , [z] i n d i c a t e s t h a t s t r i p t h e o r y s h o u l d be c o r r e c t e d , f o r the d e t e r m i n a t i o n o f E , I n F i g . I 5 , [ 2 ] g i v e s v a l u e s o f A and Bo C o e f f i c i e n t A i n c l u d e s t h e s p e e d terms o f C, w h i c h were moved by d i v i s i o n by to o

These speed terms a r e e q u a l to V E , i n b o t h £lj and [ 2 ] . S i n c e E seems t o r e q u i r e s p e e d t e r m s i n t h e s t r i p t h e o r y , so a l s o does c o e f -f i c i e n t A , or C. The e x p e r i m e n t a l agreement w i t h t h e t h e o r e t i c a l A i s q u i t e good i n [ 2 ] . Thus o n l y B r e m a i n s to be c o n s i d e r e d . T h e o r y and e x p e r i m e n t do not a g r e e w e l l i n [ 2 ] . However, a s s e e n i n T a b l e I I , the t h e o r e t i c a l v a l u e s from [ 1 ] show even worse a g r e e m e n t , f o r a? = 6 r a d / s e c o TABLE I I . V a l u e s o f c o e f f i c i e n t B CO = 6 r a d / s e c . Fn 0 . 1 5 0 . 2 0 0 . 2 5 0 o 3 0 E x p e r i m e n t 6 o 3 6 . 1 6 . 0 _{5 . 7} T h e o r y i n 2 7 . 7 . 7 . 7 7 . 7 7 . 7 T h e o r y i n 1 k.O 2 . 8 1 . 6 Ook I t must be c o n c l u d e d t h a t t h e s t r i p t h e o r y does r e q u i r e a c o r r e c -t i o n f o r f o r w a r d s p e e d , and -t h a -t w i -t h s u c h a c o r r e c -t i o n s a -t i s f a c -t o r y e v a l u a t i o n i s o b t a i n e d f o r a l l c o e f f i c i e n t s , e x c e p t Bo - 1 1

r i

-5O Summary and c o n c l u s i e n s .

•

The d e r i v a t i o n o f the e q u a t i o n s o f motion a p p e a r i n g i n [ 1 ] , and due t o A b k o w i t s [ 4 ] , seems a more r i g o r o u s and s a t i s f y i n g methodo However, i t does not a t t e m p t to e v a l u a t e t h e motion d e r i v a t i v e s . When s u c h e v a l u a t i o n was made, i n [ l ] , i t was assumed t h a t f o r w a r d s p e e d d i d not a f f e c t t h e a t r i p t h e o r y . The e x p e r i m e n t a l r e s u l t s o f [ 2 ] do n o t appear to j u s t i f y t h i s a s s u m p t i o n . The method i n [ 2 ] , due i n p a r t t o K o r v i n - K r o u k o v a k y [ 3 ] > d e r i v e s t h e e q u a t i o n s o f motion and e v a l u a -t e s -the m e -t i o n d e r i v a -t i v e s i n one p r o c e s s . E x p e r i m e n -t a l r e s u l -t s a g r e e w i t h t h e r e s u l t s o f t h i s method. However, the method does not seem a s e l e g a n t or f l e x i b l e a s t h a t due to A b k o w i t z .

The a s s u m p t i o n r e g a r d i n g the e f f e c t o f f o r w a r d s p e e d on s t r i p t h e o r y i s the o n l y d i f f e r e n c e between t h e s e p a p e r s . No e r r o r s have been made by e i t h e r a u t h o r s , or by K o r v i n K r o u k o v s k y . I t seems, to t h i s d i s c u s s e r , most p r a c t i c a l t o use the d e r i v a t i o n o f A b k o w i t z , and V a s s i l o -p o u l o s and Mandei, to s t u d y t h e e q u a t i o n s o f m o t i o n ; and to use the method o f K o r v i n - K r o u k o v s k y , IVatanabe, and G e r r i t s m a and Beukelman,

t o e v a l u a t e the m o t i o n d e r i v a t i v e s . I t i s , t h e r e f o r e hoped t h a t t h i s d i s c u s s i o n w i l l c o n t r i b u t e to the u n d e r s t a n d i n g o f t h e s e two a p p r o a c h

-l o V a s s i l o p o u l o s and Mandei- T h i s Symposiunio

2 o G e r r i t s m a and Beukelman » T h i s Symposiumo

3 . K o r v i n - K r o u k o v s k y , "Theory o f seakeeping"» SNAME, New York I 9 6 I 0

k. A b k o w i t z ,

"The l i n e a r i z e d e q u a t i o n s o f m o t i o n f o r p i t c h i n g and heaving s h i p s ' Symposium on t h e Behaviour o f Ships i n a Seaway,

NoS.MoB. iVageningen 1 9 5 7 . 5 . Watanabe,

"On t h e t h e o r y o f p i t c h and heave o f a a h i p " . Technology r e p o r t s o f Kyushu U n i v e r s i t y , 1 9 5 8 .

6 0 Fay,

"The motions and i n t e r n a l r e a c t i o n s o f a v e s s e l i n r e g u l a r waves". J o u r n a l o f Ship Research 1 9 5 8 o

G r a t e f u l acknowledgement must a l s o be made t o P r o f . G e r r i t s m a , I r . J . B c van den Brug and W. Beukelman a t D e l f t , f o r v a l u a b l e a s s i s -tance and d i s c u s s i o n ; and t o P r o f . A b k o w i t z and P r o f . Mandei, f o r t h e i i n s t r u c t i o n and guidance a t M.I.To