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Sub-committee on S a f e t y o f P i s h i n g V e s s e l o , t h e 1 0 t h S e s s i o n

AN ANALÏSIS OP SHIP MOTIOHS AND HEELING I H THE POLLOWIHG SEA

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The p r o h l e m o f s h i p ' s s a f e t y i n f o l l o w i n g sea c o n d i t i o n s i s o f a complex n a t u r e . There have been two t r e n d s w i t h i n t h e framework o f t h i s p a r t i c u l a r s u b j e c t t h e f o r m e r o f w h i c h i s concerned w i t h t h e s t u d y o f t h e r e s t o r i n g moment.

The r e s t o r i n g moment v a r i e s a c c o r d i n g t o t h e s h i p ' s p o s i t i o n on t h e wave and i s a t i t s minimum w i t h t h e m a i n frame r i g h t a t t h e c r e s t o f t h e wave. T h i s , however, does n o t e x h a u s t t h e p r o b l e m o f s a f e s a i l i n g i n a f o l l o w i n g sea.

P r a c t i c a l s a i l i n g as w e l l as r e s e a r c h show t h a t t h e r e a r e two d i s t i n c t modes o f s h i p ' s d i r e c t i o n a l m o t i o n i n t h e f o l l o w i n g sea:

s h i p m o t i o n a t a mean speed a p p r o a c h i n g t h a t I n calm w a t e r ; s h i p m o t i o n w i t h t h e wave, when t h e f o l l o w i n g wave i m p a r t s t h e s h i p i t s own speed c a u s i n g i t , f o r a c e r t a i n p e r i o d o f t i m e , t o p r e s e r v e i t s p o s i t i o n unchanged w i t h r e g a r d t o t h e wave p r o f i l e . A s h i p d r i v e n by t h e f o l l o w i n g wave e x p e r i e n c e s , as a g e n e r a l r u l e , d i r e c t i o n a l i n s t a b i l i t y p o s s i b l y r e s u l t i n g i n b r o a c h i n g accompanied by s u b s t a n t i a l h e e l i n g . C o n s i d e r a b l e h e e l i n g may a l s o be e x p e r i e n c e d w h i l e man-e u v man-e r i n g i n t h man-e c o n d i t i o n s o f a f o l l o w i n g sman-ea. S t u d i e s concerned w i t h yawing o r s h i p ' s m a n e u v e r i n g i n a f o l l o w i n g sea and t h e h e e l i n g produced t h e r e b y make up t h e second t r e n d o f e f f o r t i n s e c u r i n g s a f e t y s a i l i n g i n f o l l o w i n g sea c o n d i t i o n s .

T h i s l a t t e r group o f problems was I n i t i a l l y s t u d i e d by K.Davidson (Ref. 1 ) and O.Grim ( R e f . 2 ) who c o n s i d e r e d d i r e c t i o n a l

: a b H i t y and d i r e c t i o n a l m o t i o n o f a s h i p . The c o m p l i c a t e d Lture o f t h e s e phenomena o b v i o u s l y a c c o u n t s f o r t h e f a c t l a t s i n c e t h e t i m e o f t h e s e e a r l y p u b l i c a t i o n s t h e p r o b l e m

ken as a whole has n o t been c o n c l u s i v e l y s o l v e d , w h i l e r t a i n d e d u c t i o n s f r o m t h e s t u d y o f i t a s e p a r a t e a s p e c t s t e n appear q u e s t i o n a b l e b e i n g b u i l t on u n j u s t i f i e d assumpt-n s .

The p r e s e n t paper p r o v i d e s a b r i e f s t a t e m e n t o f c e r t a i n s u i t s f r o m t h e s t u d y o f s h i p m o t i o n s i n a r e g u l a r f o l l o w i n g away, o b t a i n e d by D.M.Ananyev and Yu.L.Makov o f t h e K a l i n i n -ad T e c h n i c a l I n s t i t u t e f o r F i s h e r i e s .

ïor a t h e o r e t i c a l s t u d y o f t h e p r o b l e m o f d i r e c t i o n a l t i o n and y a w i n g a system o f d i f f e r e n t i a l e q u a t i o n s has been ed:

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X.y.Z - o r t h a g o u a l c o o r d i n a t e s o f a r i g h t - h a n d e d

stem o f body axes, moving w i t h t h e s h i p ,

^ 1 ' "^2 - speed o f s h i p a l o n g t h e axes x and y ( P i g . l ) s p e c t i v e l y ,

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- -V Sin p ^ -vp ,

- mass moment o f i n e r t i a about t h e v e r t i c a l c e n t r e a x i s ,

- added mass moment about t h e same a x i s .

The r i g h t - h a n d p a r t o f t h e e q u a t i o n g i v e s f o r c e s ' a c t i n g on t h e s h i p and moments about t h e v e r t i c a l a x i s p a s s i n g t h r o u g h the s h i p ' s c e n t r e o f g r a v i t y . TheYo and M „ s t a n d f o r com-ponents due t o a n g u l a r speed VJ^ and d r i f t a n g l e p . I n d e f a u l t of r e l i a b l e d a t a p e r t a i n i n g t o t h e e f f e c t o f a seaway on t h a t f o r c e and moment t h e y a r e assumed t o be t h e same as f o r c a l m w a t e r and a r e d e t e r m i n e d by t h e c o n v e n t i o n a l p r o c e d u r e adopted

f o r t h e t h e o r y o f s t e e r i n g .

I n c o n s i d e r i n g s h i p ' s maneuvering t h e Yo "^^^ a l s o i n c l u d e components caused by changes I n r u d d e r a n g l e . The f i r s t e q u a t i o n i n c l u d e s p r o p u l s i o n t h r u s t T and mean r e s i s t a n c e R b o t h dependant on t h e s h i p ' s speed and t h e e n g i n e o p e r a t i o n r e g i m e , t h e s e f o r c e s b e i n g l i k e w i s e assumed t h e same as f o r calm w a t e r . I n t h i s case t h e R r e p r e s e n t s t o w i n g r e s i s t a n c e .

The e q u a t i o n f i n a l l y i n c l u d e s d i s t u r b i n g f o r c e s made up of a component d e t e r m i n e d by t h e P r o u d e - K r y l o v h y p o t h e s i s and the second d i f f r a c t i o n component caused by t h e f a c t t h a t t h e wave i s r e f l e c t e d by t h o s h i p . The f o r c e d e t e r m i n e d by t h e P r o u d e - K r y l o v h y p o t h e s i s depends, a t a g i v e n wave, o n l y on t h e i n s t a n t a n e o u s p o s i t i o n o f t h e s h i p w i t h r e s p e c t t o t h e wave. For a f i r s t a p p r o x i m a t i o n the d i f f r a c t i o n component i s d e t e r -mined by t h e same f a c t o r s . I n t h i s case a m p l i t u d e s Jf^.T^.M^j^,

depend on s h i p ' s d i m e n s i o n s and t h e l e n g t h t o h e i g h t p r o p o r t -i o n s o f t h e wave, as w e l l as on t h e course a n g l e (p w -i t h r e s p e c t t o t h e d i r e c t i o n o f t h e waves (see P i g . l ) . The s h i p ' s p o s i t i o n w i t h r e s p e c t t o t h e wave p r o f i l e can be expressed by means o f phase q w h i c h I s : ? - ^ ( 3 ) where X, - w a v e l e n g t h , OC^ .- d i s t a n c a f r o m a c e r t a i n f i x e d wave t r o u g h {MM^) t o t h e s h i p ' s c e n t r e o f g r a v i t y ( P i g . l ) . S i n c e we a r e m o s t l y i n t e r e s t e d i n an a n a l y s i s o f s h i p m o t i o n s on a l o n g wave, when t h e l a t t e r u s u a l l y surpasses t h e s h i p , p o s i t i v e d i r e c t i o n o f t h e CC^ a x i s i s o p p o s i t e t o t h e d i r e c t -I o n o f t h e wave. An asymmetry o f t h e s h i p w i t h r e s p e c t t o t h e m a i n frame g i v e s t h e i n i t i a l phases 5^,6^ • C a l c u l a t i o n s have shown t h a t i n t h e case o f s t a n d a r d h u l l forms t h e 5^ and öj phases may be n e g l e c t e d , w h i l e 0^ s h o u l d be talcen i n t o a c c o u n t o n l y w i t h c o n s i d e r a b l e t r i m .

An a n a l y s i s o f e x p e r i m e n t a l d a t a has shown t h a t i n a

l o n g i t u d i n a l d i s t u r b i n g f o r c e t h e e f f e c t o f waves r e f l e c t e d f r o m t h e s h i p may be n e g l e c t e d and t h i s f o r c e can be f o u n d by t h e P r o u d e K r y l o v h y p o t h e s i s . I n t h e case o f a t r a n s v e r s e d i s t u r b -i n g f o r c e and a yaw moment ("^ and ) n e g l e c t o f t h e d i f f r a c t i o n component w i l l u n d e r e s t i m a t e t h e f o r c e and moment c o n s i d e r a b l y . T h e r e f o r e , w i t h a t r a n s v e r s e d i s t u r b i n g f o r c e and a yaw moment d i f f r a c t i o n components a r e t a k e n i n t o a c c o u n t .

I t s h o u l d be mentioned t h a t i n t h e p r a c t i c a l a p p l i c a t i o n 0 of m o t i o n e q u a t i o n s ( l ) t h e e f f e c t s o f p i t c h and h e a l were n e g l e c t e d . The system o f e q u a t i o n s ( l ) i s l a r g e l y n o n l i n e a r , w h i c h c o m p l i c a t e s i t e s o l u t i o n . The e a s i e s t f o r s o l u t i o n i s t h e p r o b l e m o f d i r e c t i o n a l m o t i o n w i t h t h e wave, i . e . w i t h <p = 0 and w i t h no yaw. I n t h i s case we o n l y have t o d e a l w i t h t h e f i r s t o f t h o • two e q u a t i o n s w h i c h becomes: ma-^kj^ - T-R^X^Sin^ . ( 5 ) The n o n l i n e a r e q u a t i o n ( 3 ) may be s o l v e d g r a p h i c a l l y or n u m e r i c a l l y ( R e f . 3 i 7 ) . F i g . 2 shows c o m p u t o r p r o c e s s e d r e s u l t s ( R e f . 7 ) o f d i r e c t -i o n a l m o t -i o n o f a s m a l l model w -i t h t h e f o l l o w -i n g e l e m e n t s : A = 5.25 k g . ; L = 0.80m; -g" =5-0; " ^ - 2 , 5 ; = 0.64; C„, = 0.80; = 0.84. The model m o t i o n s i n P i g . 2 a r e r e p r e s e n t e d by s o - c a l l e d phase t r a j e c t o r i e s , i . e . a dependence o f speed upon s h i f t .

To r e p r e s e n t s h i f t use has been made o f t h e model phase w i t h r e s p e c t t o t h e wave q ( i n r a d i a n s ) . On t h e v e r t i c a l ' a x i s i s g i v e n , i n a n o n - d i m e n s i o n a l f o r m , t h e r a t e o f s h i p ' s

l a g b e h i n d t h e wave:

where C - speed o f wave.

ïor a B M p w i t h a z e r o speed U = 1 ; f o r m o t i o n a t a speed e q u a l t o t h e speed o f t h e wave C/ = 0. T i n e a l o n g phase t r a j e o t o r l o s aoorues i n t h e d i r e c t i o n i n d i c a t e d by a r r o w s . F i g . 2 r e f e r s t o t h e oase o f model m o t i o n s on r e g u l a r f o l l o w ^ i n g wares w i t h a l e n g t h e q u a l t o t h e s h i p ' s l e n g t h ( A = i ) , w h i l e t h e h e i g h t i s 1/20 o f t h e wave l e n g t h ( = 2 0 ) .

P i g s . 2a, 2b, 2c show model m o t i o n s a t a p r o p u l s i o n t h r u s t s e c u r i n g , i n calm w a t e r , a l o w e r speed t h a n t h e speed o f t h e wave, t h e Proude number, i n calm w a t e r , b e i n g 0.25, 0.30, O.35 r e s p e c t l T e l y .

The diagrams l u P i g . 2 d ( P r . = O.4O) show m o t i o n s w i t h a p r o p u l s i o n t h r u s t e q u i v a l e n t , i n calm w a t e r , t o a speed e q u a l t o t h e speed o f t h e ware.

F i n a l l y , P i g s . 2e and 2 f a p p l y t o t h e case when t h e model speed I n calm w a t e r exceeds t h a t o f t h e ware ( P r . = 0.45, O.5O). The m o t i o n r e g i m e I n calm w a t e r i s shown by a d o t t e d l i n e i n P i g . 2 . The phase o f t h e c a p t u r e d ( e n t r a p p e d ) s h i p i s d e t e r m i n e d by t h e e q u a t i o n : S i n y " - - 2 1 ^ , ( 5 ) § w h i c h can be e a s i l y d e r i v e d f r o m . e q u a t i o n ( 3 ) . The l a t t e r e q u a t i o n c o r r e s p o n d s t o two e q u i l i b r i u m p o s i t -i o n s one o f w h -i c h ( cf -i n P -i g . 2 ) -i s s t a b l e , w h -i l e t h e o t h e r ( ^ * ) I s u n s t a b l e . I n t h i s case, when t h e model's main frame i s on t h e f o r w a r d s l o p e o f t h e wave and w i t h a Froude number

< 0.40 i n calm w a t e r we o b t a i n a p o s i t i o n o f e t a b l e e q u i l i b r l i v a and, c o n v e r s e l y , an u n s t a b l e p o s i t i o n i s o b t a i n e d w i t h t h e m a i n frame on t h e backward s l o p e . W i t h F r . > 0 , 4 0 we g e t an o p p o s i t e p i c t u r e . Two s t a b l e regimes o f s h i p m o t i o n s a r e p o s s i b l e i n f o l l o w -i n g sea c o n d -i t -i o n s :

1. M o t i o n a t a speed a p p r o a c h i n g t h a t f o r calm w a t e r cond i t i o n s ancond w i t h p e r i o cond i c a l speecond changes. T h i s r e g i m e o f s u r g -i n g m o t -i o n s -i s -i n d -i c a t e d -i n P -i g . 2 by a dot-and-dash l -i n e .

2. Ship m o t i o n a t a speed synchronous w i t h t h e wave ( e n -t r a p p e d c o n d i -t i o n ) i s r e p r e s e n -t e d by ( J * p o i n -t s i n -t h e d i a g r a m .

An a n a l y s i s o f t h e diagrams l e a d s t o t h e c o n c l u s i o n t h a t the c o n d i t i o n s f o r any o f t h o s e s t a b l e regimes a r e d e t e r m i n e d not m e r e l y by t h e f o r c e s b u t a l s o by t h e i n i t i a l m o t i o n c o n d i t -i o n s . W -i t h F r . = 0.25 ( P -i g . 2 a ) t h e c o r r e l a t -i o n o f f o r c e s -i a s u c h t h a t t h e model can n e v e r V c a p t u r e d by t h e wave. W i t h P r . = O.3O the model i s swung i n t o one o f t h e two s t a b l e m o t i o n r e g i m e s a c c o r d i n g t o t h e i n i t i a l c o n d i t i o n s ( i . e . t h e i n i t i a l <J and

U c o m b i n a t i o n ) . The g" and U c o m b i n a t i o n zone where t h e s h i p

i s i n v a r i a b l y e n t r a p p e d by t h e wave i s shaded i n P i g . 2 b . W i t h Fr.= 0.35 t h e p r o p u l s i o n t h r u s t i s s t r o n g enough t o r e n d e r t h e o h i p c a p t u r e d by t h e f o l l o w i n g wave no m a t t e r what t h e i n i t i a l c o n d i t i o n s may be ( P i g . 2 c ) . When t h e model has a t e n d e n c y t o move f a s t e r t h a n t h e wave t h e p i c t u r e i s q u a l i t a t i v e l y t h e same. The case o f F r . = 0.45 ( F i g . 2 e ) i s s i m i l a r t o t h a t o f

of F r . = 0.50

F r . = 0.35, w h i l e t h e m o t i o n regimes'^remind t h o s e o b t a i n i n g w i t h F r . =

The same model hao been used t o s t a g e an e x p e r i m e n t w i t h d i f f e r e n t w a v e l e n g t h s and w i t h = 1/25 ( E e f. 8 ) . P i g .3 shows an i n s t a n c e o f e x p e r i m e n t a l d a t a f o r 3^= 1-75 and a p r o p u l s i o n t h r u s t e q u i v a l e n t t o Pr.= 0.43 i n calm w a t e r . As i s seen i n Pig.3b t h e model under t e s t ( c o n t i n u o u s l i n e ) i s c a p t u r e d by

the wave; t h e d o t t e d l i n e here shows c a l c u l a t e d d a t a . The d i a -gram p r o v i d e s a c l e a r i n s t a n c e o f f a i r l y s a t i s f a c t o r y c o i n c i d e n c e ' o f b o t h e x p e r i m e n t a l and c a l c u l a t e d r e s u l t s .

Of m a j o r i m p o r t a n c e , w i t h i n t h e scope o f t h e p r o b l e m under s t u d y , I s t h e d e t e r m i n a t i o n o f b o u n d a r y v a l u e s o f wave elements and t h e engine o p e r a t i o n r e g i m e , w h i c h p r o v i d e a p o i n t of d i v i s i o n between t h e m o t i o n r e g i m e s o f a c a p t u r e d s h i p and t h o s e a t a mean speed b e i n g o t h e r t h a n t h e speed o f t h e wave. The engine o p e r a t i o n i s c h a r a c t e r i z e d by speed o r t h e Froude number i n calm w a t e r , and 3 0 t h e s e b o u n d a r i e s a r e e a s i e s t t o be shown g r a p h i c a l l y I n - c o o r d i n a t e s .

The l i g h t e r and d a r k e r s h a d i n g i n P i g s . 4 and 5 denote two entrapment zones b e i n g d i f f e r e n t i n n a t u r e and o b t a i n e d by c a l c u l a t i n g phase t r a j e c t o r i e s ( H e f . 7 ) :

1 . Zone One i n w h i c h entrapment o c c u r s o n l y under c e r t a i n i n i t i a l c o n d i t i o n s ( l i g h t e r s h a d i n g ) . The b o u n d a r i e s o f t h i s ^ zone a r e d e t e r m i n e d by t h e s t a t i c c o n d i t i o n (5) ofSin5r*= 1 .

2. Zone Two where e n t r a p m e n t o c c u r s i n v a r i a b l y , I . e . under any i n i t i a l c o n d i t i o n s ( d a r k e r s h a d i n g ) .

C i r c l e s i n F i g . 4 show e x p e r i m e n t a l d a t a , t h e b l a c k ones s i g n i f y i n g entrapment and t h e b l a n k ones an absence o f same. W i t h i n t h e zone where entrapment t a k e s p l a c e o n l y under c e r t a i n I n i t i a l c o n d i t i o n s b o t h k i n d s o f m o t i o n r e g i m e s a r e

1 0

p o s s i b l e , v i z . e n t r a p m e n t and s u r g i n g a t a speed a p p r o a c h i n g t h a t f o r calm w a t e r c o n d i t i o n s .

I n t h i e case, however, w i t h = 1.0 and F r . = O.JO, f o r i n s t a n c e , ( P i g . 2 b ) a s e l f - p r o p e l l e d s h i p c a n n o t be c a p t u r e d by t h e wave ( s i n c e i n p i c k i n g up speed t h e phase t r a j e c t o r i e s r e p r e s e n t i n g t h i s p r o c e s s a r e always above t h e d o t - a n d - d a s h l i n o d e n o t i n g s u r g i n g - P i g . 2 b ) and so cannot e n t e r t h e shaded zone b u t w i l l m a i n t a i n s e t s u r g i n g m o t i o n s .

As t h e s h i p swings o u t o f e n t r a p m e n t t h e p r o c e s s changes. I f a c a p t u r e d s h i p t h a t had been moving, b e f o r e e n t r a p m e n t , a t a Froude number F r . = 0.40, f o r i n s t a n c e , ( P i g . 2 d ) d r o p s i t s engine speed t o o b t a i n F r . = O.3O i n calm w a t e r , t h e n i t s m o t i o n s w i l l be d e s c r i b e d by t h e phase t r a j e c t o r i e s i n P i g . 2 b . A t t h e f i r s t moment t h e i n i t i a l m o t i o n c o n d i t i o n s w i l l r e m a i n t h o

same, i . e . { 7 = 0 and = q* = 0 ( P i g . 2 d ) . Tho phase ttrajeotory p r o c e e d i n g f r o m t h i s p o i n t i n P i g . 2 b i s shown by a d o t t e d l l i i e . I t always r e m a i n s w i t h i n t h e shaded zone and, hence, t h e s h i p w i l l c o n t i n u e b e i n g e n t r a p p e d , t h o u g h s h i f t e d t o a n o t h e r p o i n t on t h e same wave ( q*= 1.5, P i g . 2 b ) . A f t e r d r o p p i n g speed t h e s h i p w i l l r e m a i n e n t r a p p e d as l o n g as t h e P r . v a l u e i s i n t h e f i r s t zone ( F i g s . 4 , 5 ) .

Thus, i n e s t i m a t i n g t h e p o s s i b i l i t y o f a s h i p becoming c a p t u r e d by t h e wave we a r e c h i e f l y i n t e r e s t e d i n t h o b o u n d a r i e s o f t h e second zone, whoreas i n t h e case o f t h e s h i p b r e a k i n g l o o s e f r o m e n t r a p m e n t t h e b o u n d a r i e s o f t h e f i r s t zone appear the more i m p o r t a n t , s i n c e t h o s e a r e t h e b o u n d a r i e s t h a t g i v e an i d e a o f how f a r t h e s h i p ' s speed i s t o be r e d u c e d t o o b t a i n

f r e e d o m f r o m e n t r a p m e n t . F i g . 5 showa, f o r i n s t a n c e , t h a t w i t h

= 1.25 t h e s h i p cannot g e t f r e e f r o m e n t r a p m e n t even i f

the engine i s a l t o g e t h e r s t o p p e d .

The h o u n d a r i e s o f Zone Two where entrapment i s u n a v o i d a b l e

can a l s o be f o u n d w i t h o u t p l o t t i n g phase t r a j e c t o r i e s b u t f r o m

an a n a l y s i s o f s u r g i n g s t a b i l i t y . W i t h i n t h e zone o f i n e v i t a b l e

entrapment under any i n i t i a l m o t i o n c o n d i t i o n s t h e s u r g i n g i s

u n s t a b l e and s e t s u r g i n g m o t i o n s cannot t a k e p l a c e . T h e r e f o r e , t h e s t a b i l i t y boundary f o r s u r g i n g m o t i o n s i s t h e boundary o f t h e zone i n w h i c h e n t r a p m e n t c o n s t a n t l y o c c u r s . Set s u r g i n g m o t i o n s (shown by a d o t - a n d - d a s h l i n e i n P i g . 2 ) a r e expressed m a t h e m a t i c a l l y t h u s : ^ - 6 t + A g - , ( 6 ) where 0 - f r e q u e n c y o f e n c o u n t e r w i t h t h e wave w h i c h i s : ö - 6 „ - . ^ ^ „ . ( T ) 0 g - f r e q u e n c y o f t h e wave A g - a d d i t i o n a l phase e x p r e s s i n g s e t o s c i l l a t i o n s w i t h a p e r i o d ^ w i t h r e s p e c t t o movement a t a mean speed .

F o r a Judgment o f s e t s u r g i n g m o t i o n s we need t h e s o l u t i o n o f e q u a t i o n ( 5 ) i n t h e f o r m o f ( 6 ) . Ae i s seen f r o m Ref.5 t h e p r o b l e m o f e n t r a p m e n t by t h e f o l l o w i n g wave can be s a t i s f a c t o r i l y s o l v e d by l i n e a r i z i n g t h e d i f f e r e n c e o f r e s i s t a n c e and p r o -p u l s i o n t h r u s t . Here we a -p -p l y t h i s method t o e s t i m a t e s u r g i n g by assuming: 12 F o r s i m p l i f i c a t i o n l e t - u s f u r t h e r assume A^ t o be s m a l l and g e t an a p p r o x i m a t i o n : S i n ( ö t + A g ) - A g • C o s 0 t = 1 - S i n O t . Then t h e o r i g i n a l e q u a t i o n ( 3 ) w i l l become a M a t h i e u e q u a t i o n where A g i s a f u n c t i o n . The s t a b i l i t y o f s o l v i n g a M a t h i e u e q u a t i o n i s e x p l o r e d b y t h e a p p r o p r i a t e m a t h e m a t i c a l methods. Note t h a t t h e above e q u a t i o n ( 9 ) b e i n g o n l y an a p p r o x i m a t

-i o n t h e mean speed -i n a f o l l o w -i n g sea o b t a -i n e d by t h -i s . m e t h o d ^

does n o t d i f f e r f r o m t h e mean speed i n calm w a t e r .

The s t a b i l i t y o f s u r g i n g m o t i o n s i s d e t e r m i n e d b y t w o p a r a m e t e r s : „ X^ ( 1 0 )

^ ' p m ( i

-^kj

Uo'

p m ( l f / c j U,

a(R-T)

av

( 1 1 ) where t h e d e r i v a t i v e o f d i f f e r e n c e b e t w e e n a t o w i n g r e s i s t a n c e and p r o p u l s i o n t h r u s t c a n be f o u n d b y g r a p h i c d i f f e r e n t i a t i o n and by t a k i n g i n t o a c c o u n t t h e s h i p ' s speed i n c a l m w a t e r . C7

= I

1 - ( 1 2 ) 13

The r e s u l t s o f e x p l o r i n g s t a b i l i t y a r e r e p r e s e n t e d by a d i a g r a m I n P i g . 6 w h i c h , i n t h e f o r m o f H—f(Q), expresses the s t a b i l i t y b o u n d a r i e s , i . e . t h e b o u n d a r i e s o f t h e zone where entrapment always t a k e s p l a c e . Tho zone i n . w h i c h entrapment always accure i s shaded. The c o n c l u s i o n euggests i t s e l f t h a t t h e d i a g r a m I n P i g . 6 i s a p p l i c a b l e t o a l l k i n d o f s h i p s and so may s e r v e as a u n i v e r s a l diagram o f s u r g i n g s t a b i l i t y .

I n P i g . 5 t h e dot-and-dash l i n e shows t h e entrapment zone d e t e r m i n e d by t h i s u n i v e r s a l s t a b i l i t y d i a g r a m . The s t a b i l i t y d i a g r a m I n P i g . 6 i s o b t a i n e d by t h e use o f e x p r e s s i o n ( 9 ) . I f i n s t e a d o f e x p r e s s i o n ( 9 ) a more e x a c t e x p r e s s i o n i s t a k e n we o b t a i n a more e x a c t f o r m o f diagram and t h e b o u n d a r i e s ^ f en-trapment zone 'which a p p r o a c h t h o s e f o u n d f r o m computor-processed d a t a o f s h i p m o t i o n s .

Tho use o f t h e s t a b i l i t y d i a g r a m makes i t p o s s i b l e , a l t h o u g h n o t w i t h o u t a c e r t a i n degree o f e r r o r , t o d e t e r m i n e t h e b o u n d a r -i e s o f t h e entrapment zone t h r o u g h c o n v e n t -i o n a l c a l c u l a t -i o n methods w i t h o u t r e s o r t t o a computer. Entrapment o f t h e s h i p by a f o l l o w i n g wave I s t h e f i r s t s t a g e o f t h e p r o c e s s . Under t h e s e c i r c u m s t a n c e s t h e s h i p has g e n e r a l l y l i t t l e d i r e c t i o n a l s t a b i l i t y , w h i c h i s n a t u r a l l y f o l l o w e d by t h e second s t a g e , a t u r n on t h e wave accompanied by h e e l i n g .

I f t h o e f f e c t o f h e e l on t h e s h i p ' s m o t i o n i n a h o r i z o n t a l p l a n e be n e g l e c t e d , t h e n t o c a l c u l a t e t h e s h i p ' s t u r n use can be made o f a system o f e q u a t i o n s ( l ) , and t o f i n d t h e h e e l i n g a n g l e we can use t h e e q u a t i o n ( H e f . 6 ) :

h e e l i n g a n g l e ,

i n e r t i a moment o f t h e s h i p w i t h r e s p e c t t o t h e l o n g i t u d i n a l c e n t r e a x i s OX,

added i n e r t i a moment,

arm o f added mass w i t h r e s p e c t t o 0 * a x i s , a m p l i t u d e o f d i s t u r b i n g moment w i t h r e s p e c t t o

OX a x i s ,

r i g h t i n g l e v e r ,

arm o f l a t e r a l f o r c e Y „ caused b y d r i f t and s h i p ' s t u r n w i t h r e s p e c t t o OX a x i s ,

damping c o e f f i c i e n t p r o p o r t i o n a l t o t h o speed squaxed.

The f o l l o w i n g i s an i n s t a n c e o f some c o m p u t o r - p r o c e s s e d . e s u l t s ( K e f . 6 ) i n c a l c u l a t i n g t h e yaw and h e e l o f a cargo s h i p w i t h t h e s e e l e m e n t s :

^ = 1 5 1 0 t . ; X = 60.4m; B = X O m , c i = 3 . 3 . ; = 0.690;

C^= 0.815; C^- 0.965.

The O b j e c t o f c a l c u l a t i o n s was t o f i n d t h e yaw caused b y an i n i t i a l t r a n s v e r s e i m p u l s e w i t h t h o r u d d e r a m i d s h i p s . M t h e

. t h e s h i p was assumed c a p t u r e d by t h o f o l l o w i n g i n i t i a l moment t h e s h i p was as

. a v e . The s h i p ' s speed i n calm w a t e r was 15.5 k n o t s ( P r . = 0 . 3 2 7 ) .

where 0

K

M^^

QZ

-The d i s - t u r b i n g f o r c e s and moments were d e t e r m i n e d by t a k i n g i n t o account t h e d i f f r a c t i o n a l p a r t except t h e l o n g i -t u d i n a l f o r c e f o r w h i c h -t h i e member was n o -t i n -t r o d u c e d . The d i f f r a c t i o n component was c a l c u l a t e d a p p r o x i m a t e l y by t h e s t r i p t h e o r y methods.

The c a l c u l a t i o n f o r m u l a e f o r d i f f r a c t i o n components a r e r a t h e r b u l k y when s t r i p t h e o r y t e c h n i q u e s a r e used, so we suggest s i m p l e r dependences w h i c h y i e l d s i m i l a r , i f somewhat l o w e r v a l u e s i n comparison w i t h t h o s e o b t a i n e d t h r o u g h t h o a t r i p t h e o r y procedure ( R e f . 4 ) :

where Y„. and M^^ a r e a m p l i t u d e s a c c o r d i n g t o t h e F r o u d e -K r y l o v h y p o t h e s i s .

As can be seen f r o m t h e above f o r m u l a e ( 1A) wave d i f f r a c t -i o n a d d -i t -i o n s a r e a p p r o x -i m a t e d t h r o u g h t h e added mass and I n e r t i a moment. Since t h e s e a d d i t i o n s account f o r 50 - 80 p o r c e n t o f t h e a m p l i t u d e o b t a i n e d by t h e F r o u d e - K r y l o v method tho d i f f r a c t i o n components o f a t r a n s v e r s e f o r c e and t h e yaw-i n g moment s h o u l d n o t be n e g l e c t e d .

The o t h e r f o r c e s i n e q u a t i o n s ( l ) and ( I 3 ) were assumed the same as t h o s e f o r calm w a t e r . The r e s t o r i n g moment was i n t r o d u c e d by t h e m e t a c e n t r i c f o r m u l a .

16

The r e s u l t s o f c a l c u l a t i o n s i n t h e csiBe o f a wave w i t h t h e elements = 0.828, = l / 2 0 a r e r e p r e s e n t e d i n P i g s . 7 and 8 where t h e s h i p m o t i o n parameters a r e g i v e n w i t h r e s p e c t to t h e d i m e n s i o n l e s s p a t h .S" w h i c h i s r e l a t e d t o speed by t h e e x p r e s s i o n : - • . (15) P i g .7 shows t h e f o l l o w i n g s h i p m o t i o n p a r a m e t e r s : phase w i t h r e s p e c t t o a f i x e d wave t r o u g h {q), d i r e c t i o n a l a n g l e w i t h r e s p e c t t o t h e wave (Cp ) , d r i f t a n g l e ip),

JL - r e l a t i o n o f s h i p ' s speed t o t h e speed o f t h e wave;

c

Q - d i m e n s i o n l e s s a n g u l a r speed f o u n d by t h e f o r m u l a :

Q - W ^ ^ • ( 1 6 )

The d o t t e d l i n e i n P i g .7 shows t h e s h i p ' s speed i n c a l m w a t e r ( ) •

A l l a n g l e s a r e expressed i n r a d i a n s .

The i n i t i a l i m p u l s e i n t h i s p a r t i c u l a r i n s t a n c e was a p p l i e d t o t h e a f t p a r t and had g i v e n t h e s h i p t h e I n i t i a l d r i f t a n g l e

Jb^ = 1° and t h e i n i t i a l a n g u l a r speed = 0 . 1 3 7 5 .

As can be seen f r o m P i g .7 t h e s h i p swings o u t o f s y n c h r o -nism w i t h t h e wave, a t f i r s t s l o w l y and t h e n more and more r a p i d l y ( t h e change o f g and U ) and t u r n s b r o a d s i d e a l o n g

the wave, f i r s t on i t s f o r w a r d s l o p e ( u p t o ^ = 1.57) and t h e n on t h e backward s l o p e o f t h e f i r s t and t h e s u c c e s s i v e waves.

« . .... . . a .

... ^

h . . . . „ „ „ ^ „ ^ „ ^ ^ ^ ^^^^ ^^^^^^^^

. ... . M . _^ ^ ^ " i s s t a b l e .

^ven W i t h the a s s u m p t i o n s i n t r o d u c e d Tor s o l v i n g t h e p r o b l e m c a l c u l a t i o n r e s u l t s (Pi;.3 7 e ) n h ^ - •,

^ ^ i g s . ^ . e ; o b v i o u s l y p r o v i d e a t r u e p i c t u r e .

I n t h e above i n s t a n c e the r e s t o r i n g moment was f o u n d by t h e m e t a c e n t r i c f o r m u l a . For an a n a l y s i s o f t h e p o s s i b i l i t y o f c a p s i z i n g we o b v i o u s l y need i n t r o d u c i n g i n t o our c a l c u l a t i o n s t h e r e s t o r i n g moment as d e t e r m i n e d by Heed's d i a g r a m . A l s o t h e n a t u r a l c o n c l u s i o n i s t h a t we need e x t e n s i v e e x p e r i m e n t i n g r o r t h e s t u d y o f y a w i n g and h e e l i n g i n a f o l l o w i n g sea. I t would be w e l l t o m e n t i o n . i n c o n c l u s i o n , t h a t b o t h en-t r a p m e n en-t and y a w i n g o c c u r o n l y w i en-t h i n a m a en-t en-t e r o f one , en-two waves. T h e r e f o r e , wave i r r e g u l a r i t y seems t o be o f l i t t l e con sequence t o t h e problem d i s c u s s e d , and the r e s u l t s o b t a i n e d f o r r e g u l a r waves a r e amply r e p r e s e n t a t i v e o f the r e a l s t a t e o f a f f a i r s .

18

Beferences.

K.Davidson: "Note on t h e S t e e r i n g o f Ships i n F o l l o w i n g Seas". P r o c . o f t h e 7 t h I n t e r n . Congress f o r A p p l . Mechanics. V o l . 2 , p t 2. 1948.

O.orim: "Das S c h i f f i n von A c h t e r n A u f l a u f e n d e r See". J a h r b u c h der STG. B d. 4 5 , 1951.

D.M.Ananyev: "On Entrapment o f S h i p by t h e F o l l o w i n g Sea." P r o c . o f the 1 6 t h Conference on t h e Theory o. S h i p , i s s u e 73. R e s e a r c h S o c i e t y o f S h i p b u i l d

-i n g I n d u s t r y , C e n t r a l B o a r d , L e n -i n g r a d . 1966. D.M.Ananyev: " D i f f r a c t i o n F o r c e s and S t e e r i n g i n a Seaway".

T r a n s , o f T a l l i n P o l y t e c h n i c a l I n s t i t u t e . Ser.A, Mo.222, 1965.

D.M.Ananyev: " D i r e c t i o n a l S t a b i l i t y o f S h i p i n a P a r t i c u l a r i n s t a n c e o f F o l l o w i n g Seas." T r a n s , o f T a l l i n P o l y t e c h n i c a l I n s t i t u t e , Ser.A, No.222, 1965-D.M.Ananyev: ".awing and H e e l i n g i n a F o l l o w i n g Sea." From

" S e a w o r t h i n e s s and S t e e r i n g " , p u b l . 1 0 5 , Research S o c i e t y o f S h i p b u i l d i n g I n d u s t r y .

L e n i n g r a d .

1969-...L.Makov: " C e r t a i n R e s u l t s i n S t u d y i n g S h i p Entrapment t h e F o l l o w i n g Sea." From " S e a w o r t h i n e s s and S t e e r i n g " , p u b l . 1 2 6 . Research S o c i e t y o f S h i p -b u i l d i n g I n d u s t r y . L e n i n g r a d .

1969-.3 comparlaou o f e x p e r i m e n t a l and oomputenvproceaaed r e s u l t s :

1 _ e x p e r i m e n t a l r e s u l t a , 2 - oompater-prooeaad r e a u l t a .

2.0 2.5

—-F i g . 4 Two e n t r a p m e n t z o n e s a t — - 25. C i r c l e s show n e x p e r i m e n t a l r e s u l t s : • - e n t r a p m e n t , 0 - a b s e a o e o f e n t r a p m e n t .

O 0,5 1,0 i,5 2,0 2,S J.0 JI5 iO

? i g . 7 . S h i p motion parameters d u r i n g i t s t u m on the wave under the i n f l u e n c e of i n i t i a l broadside i m p u l s e .

8. Yu.L.Makov: "An E x p e r i m e n t a l Study o f S h i p Entrapment by t h e F o l l o w i n g Sea." P r o c . o f t h e 1 5 t h Conference on E x p e r i m e n t a l Naval H y d r o m e c h a n i c s y , publ.128, Research S o c i e t y o f S h i p b u i l d i n g I n d u s t r y , L e n i n g r a d , 1959-2 i o t c : R e f e r e n c e s 3 - 8 a r e p u b l i s h e d i n R u s s i a n .

_iijC-}

o f 0 0 0 1 ?