Some notes on the transvers stability of ships in irregular longitudinal waves

R e p o r t I o . 303

LABORATORIUM VOOR

SCHEEPSBOUWKUNDE

T E C H N I S C H E H O G E S C H O O L D E L F T Some n o t e s on t h e t r a n s v e r s e s t a b i l i t y o f s h i p s i n i r r e g u l a r l o n g i t u d i n a l waves by B, de Jons March 1971•

Slimmary

T h i s paper c o n t a i n s t h e e s s e n t i a l f e a t u r e s o f c h a p t e r s i x o f t h e a u t h o r ' s d o c t o r ' s t h e s i s 1.

We c o n s i d e r two phenomena w h i c h a r e i m p o r t a n t f o r t h e t r a n s v e r s e s t a b i l i t y i n i r r e g u l a r l o n g i t u d i n a l waves, p a r t i c u l a r l y , i n f o l l o w i n g wa.ves, namely, l o s s o f d y n a m i c a l s t a b i l i t y and spontaneous r o l l i n g .

W i t h r e s p e c t t o t h e f i r s t m e n t i o n e d phenomenon, we expound a method t o

d e t e r m i n e t h e average numher o f t i m e s p e r second t h a t t h e r e s i d u a l d y n a m i c a l s t a b i l i t y i s n e g a t i v e d u r i n g a c e r t a i n t i m e i n t e r v a l i f a g i v e n e x t e r n a l moment i s a p p l i e d on t h e v e s s e l . R e g a r d i n g t h e o t h e r phenomenon, we g i v e a method t o c a l c u l a t e t h e p r o b a b i l i t y t h a t a s h i p m i g h t d e v e l o p a spontaneous r o l l i n g m o t i o n . F i n a l l y , we a p p l y t h e g i v e n methods t o d e t e r m i n e t h e s e s t a t i s t i c a l q u a n t i t i e s f o r t h e m.s. "S.A. van der S t e l " .

I n t r o d u c t i o n

For a l o n g t i m e t h e s t a h i l i t y o f a s h i p was c h a r a c t e r i z e d hy t h e v a l u e s o f i t s r i g h t i n g arms o f t h e s t a h i l i t y moment and i t s d y n a m i c a l s t a b i l i t y . B o t h

q u a n t i t i e s r e f e r t o s t i l l w a t e r .

A l r e a d y W. Froude and S i r Reed r e c o g n i z e d t h a t under extreme c o n d i t i o n s t h e problem o f t h e s t a h i l i t y o f a s h i p i n a seaway s h o u l d be c o n s i d e r e d f r o m t h e p o i n t o f vie\j o f dynamics r a t h e r t h a n s t a t i c s . T h i s i s c l e a r f r o m t h e f a c t t h a t f o r a s n i p i n waves t h e r i g h t i n g arms w i l l v a r y c o n t i n u o u s l y . T h e r e f o r e , f o r a s h i p i n an i r r e g u l a r sea, t h e l e n g t h s o f t h e s e arms a r e random f u n c t i o n s ,

t h e s t a t i s t i c a l p r o p e r t i e s o f t h e s e f u n c t i o n s b e i n g d e t e r m i n e d by t h e s t a t i s t i c a l c h a r a c t e r i s t i c s o f t h e seaway and t h e geometry o f t h e s h i p . I n o r d e r t o he a b l e t o p r e d i c t t h e s a f e t y o f a s h i p w i t h r e s p e c t t o i t s t r a n s v e r s e s t a b i l i t y i n an a n a l y t i c a l way, we s h o u l d have s u f f i c i e n t knowledge about b o t h t h e b e h a v i o u r o f a s h i p i n an i r r e g u l a r sea and t h e e x t e r n a l moments w h i c h m i g h t work on a s h i p , e.g. due t o waves, w i n d , e t c . However, s i n c e up t o t h e p r e s e n t h o t h t h e b e h a v i o u r o f a s h i p under extreme c o n d i t i o n s and s u f f i c i e n t knowledge about t h e n a t u r e o f t h e sea on v a r i o u s s h i p r o u t e s a r e s t i l l l a c k i n g , t h e p r e s e n t s t a n d a r d s o f s t a b i l i t y a r e t o a h i g h degree based on a s t a t i s t i c a l i n v e s t i g a t i o n o f t h e

e x p e r i e n c e s w h i c h have been e n c o u n t e r e d by s h i p s i n t h e p a s t . The i n v e s t i g a t i o n has been i n i t i a t e d by J. Rahola ("The J u d g i n g o f t h e S t a h i l i t y o f Ships and t h e D e t e r m i n a t i o n o f t h e Minimiun Amount o f S t a b i l i t y " , H e l s i n k i , 1 9 3 9 ) .

V/ith r e s p e c t t o t h e d y n a m i c a l phenomena c o n c e r n i n g t h e t r a n s v e r s e s t a h i l i t y o f a s h i p i n a f o l l o w i n g sea, i t i s r e p o r t e d t h a t s h i p s e x h i b i t e d sometimes s i g n s o f a t e m p o r a r y l o s s o f t r a n s v e r s e s t a b i l i t y when t h e s h i p was about t h e same as t h e wave c e l e r i t y . These phenomena are i n p a r t i c u l a r observed f o r f a s t c o n t a i n e r s h i p s w i t h a low p r i s m a t i c c o e f f i c i e n t .

I t happens t h a t , i n case t h e s h i p has a t some moment an a n g l e o f i n c l i n a t i o n due t o an e x t e r n a l moment, i t remains f o r a l o n g t i m e i n t h a t p o s i t i o n and appears n o t t o be i n a h u r r y t o r e s t o r e t h e u p r i g h t p o s i t i o n . O n l y , hy making a d r a s t i c course change, a q u i c k r e s t o r a t i o n o f t h e u p r i g h t p o s i t i o n can be e f f e c t e d .

I t IS easy t o u n d e r s t a n d t h a t t h e s e phenomena a r e e s s e n t i a l l y due t o a change m t h e t r a n s v e r s e s t a h i l i t y o f t h e s h i p . I t has been shown by F.B. A r n d t and S. Roden [ 2 ] t h a t i n a harmonic f o l l o w i n g wave t h e curve o f t h e r i g h t i n g B.TTPB v a r i e s p e r i o d i c a l l y between two extreme p o s i t i o n s w i t h a f r e q u e n c y e q u a l t o t h e /~ These extreme p o s i t i o n s a r e d e t e r m i n e d by t h e geometry o f t h e s h i p , t h e wave l e n g t h and t h e wave h e i g h t . F u r t h e r , f o r e v e r y wave h e i g h t , t h e a m p l i t u d e o f o s c i l l a t i o n o f t h e s t a h i l i t y c u r v e has i t s l a r g e s t v a l u e when t h e wave l e n g t h i s about e q u a l t o t h e s h i p l e n g t h , a t t a i n i n g i t s l o w e s t v a l u e when t h e c r e s t i s i n t h e m i d s h i p r e g i o n and i t s h i g h e s t v a l u e f o r a t r o u g h i n t h i s r e g i o n .

For i n c r e a s i n g o r d e c r e a s i n g v a l u e s o f t h e wave l e n g t h , t h e i n f l u e n c e o f t h e waves on t h e t r a n s v e r s e s t a b i l i t y w i l l d i m i n i s h . I f now t h e l o w e r l i m i t o f t h e c u r v e o f r i g h t i n g arms i s below t h e minimum p r e s c r i b e d s t a b i l i t y c u r v e , t h e n , d u r i n g each p e r i o d o f e n c o u n t e r , t h e r e i s a t i m e i n t e r v a l i n vrhich t h e ' s t a b i l i t ; y o f t h e s h i p does n o t f u l f i l t h e minimum r e q u i r e m e n t s o f t r a n s v e r s e s t a b i l i t y . / f r e q u e n c y o f e n c o u n t e r .

. 3 .

T h i s t i m e i n t e r v a l w i l l become l o n g e r when t h e s h i p speed i s v a r i e d i n such a vray t h a t t h e f r e q u e n c y o f e n c o u n t e r has a s m a l l e r v a l u e .

I n c o n n e c t i o n w i t h t h i s i s s u e two papers o f Grim [ 3 , H ] s h o u l d he m e n t i o n e d . S t a r t i n g from t h e f a c t t h a t t h e l a r g e s t change i n t h e t r a n s v e r s e

s t a h i l i t y occurs when t h e wave l e n g t h i s e q u a l t o t h e s h i p l e n g t h , he

i n v e s t i g a t e s i n w h i c h measure t h e random f o l l o w i n g sea on t h e s i d e s o f t h e s h i p can he a p p r o x i m a t e d b y a r e g u l a r wave w i t h a wave l e n g t h e q u a l t o t h e s h i p l e n g t h 3 . Then t h e a m p l i t u d e o f t h i s s o - c a l l e d e f f e c t i v e wave i s a

random t i m e f u n c t i o n .

The power spectrum o f t h i s f u n c t i o n i s used f o r t h e d e t e r m i n a t i o n o f s t a t i s t i c a l q u a n t i t i e s w i t h r e s p e c t t o t h e a m p l i t u d e o f t h i s wave which g i v e an i n s i g h t i n t o t h e changes o f t h e t r a n s v e r s e s t a b i l i t y i n t h a t p a r t i c u l a r random sea.

I n h i s o t h e r paper h Grim p o i n t s o u t t h a t t h e p e r i o d i c changing o f the t r a n s v e r s e s t a b i l i t y i n a r e g u l a r l o n g i t u d i n a l wave can g i v e r i s e t o a spontaneous r o l l i n g m o t i o n . The i n i t i a l m e t a c e n t e r h e i g h t i s

assumed t o v a r y h a r m o n i c a l l y about some average v a l u e w i t h a f r e q u e n c y e q u a l t o t h e f r e q u e n c y o f e n c o u n t e r . Then t h e w e l l - k n o w n d i f f e r e n t i a l e q u a t i o n o f M a t h i e u i s o b t a i n e d w h i c h c h a r a c t e r i z e s t h e r o l l i n g m o t i o n i n l o n g i t u d i n a l waves. Depending on t h e v a l u e s o f t h e c o e f f i c i e n t s , t h i s e q u a t i o n y i e l d s a s t a b l e o r u n s t a b l e s o l u t i o n . I f t h e s o l u t i o n i s u n s t a b l e t h e v e s s e l w i l l d e v e l o p a spontaneous r o l l i n g m o t i o n . I n t h i s r e p o r t we d e r i v e some s t a t i s t i c a l q u a n t i t i e s w h i c h c h a r a c t e r i z e t h e s e n s i t i v e n e s s o f a v e s s e l i n an i r r e g u l a r l o n g i t u d i n a l sea f o r t h e above m e n t i o n e d phenomena.

• f

The d y n a m i c a l s t a b i l i t y o f a s h i p i n an i r r e g u l a r f o l l o w i n g sea

The s t a h i l i t y moment M^^ o f a s h i p i n s t i l l w a t e r w h i c h has a h e e l i n g angle ((> i s g i v e n hy

M ^ - P 1 ( 4 , ) = y V l (tf)) ( 1 )

s t

P r e p r e s e n t s t h e w e i g h t o f t h e s h i p w h i c h i s e q u a l t o t h e w e i g h t Y V o f t h e volume V o f d i s p l a c e d f l u i d w i t h s p e c i f i c

g r a v i t y y, and !(<})) i s t h e r i g h t i n g arm o f t h e moment when t h e v e s s e l has a h e e l i n g angletj) . The d y n a m i c a l s t a h i l i t y D^^ o f a s h i p a t a g i v e n h e e l i n g a n g l e (f) i s g i v e n by

Y T

0

I t i s a w e l l - k n o w n f a c t t h a t t h e s t a b i l i t y moment can change c o n s i d e r a b l y i n comparison w i t h t h e s t i l l w a t e r v a l u e when t h e s h i p i s s a i l i n g i n f o l l o w i n g waves, [ 2 , 3 , ! + ] . I t has been e s t a b l i s h e d t h a t t h i s f a c t i s v e r y o b v i o u s e s p e c i a l l y f o r a wave l e n g t h w h i c h i s e q u a l t o t h e s h i p l e n g t h . For a s h i p i n

a r e g u l a r f o l l o w i n g wave t h e r i g h t i n g arms o f t h e s t a b i l i t y moment a r e c o n s i d e r a b l y r e d u c e d when t h e s h i p i s on a wave

c r e s t , w h i l e , i n a t h r o u g h o f t h e wave, t h e s e arms can be much l a r g e r t h a n t h e c o r r e s p o n d i n g s t i l l w a t e r v a l u e s . F i g . 1 . I t may he assujiied t h a t , a p p r o x i m a t e l y , t h e r i g h t i n g arms and, c o n s e q u e n t l y , a l s o t h e d y n a m i c a l s t a h i l i t y have i t s most f a v o u r a b l e and u n f a v o u r a b l e v a l u e s when t h e v e s s e l i s i n a t r o u g h o r on a c r e s t o f t h e wave, r e s p e c t i v e l y . C o n s e q u e n t l y , t h e d y n a m i c a l s t a b i l i t y a t a g i v e n h e e l i n g a n g l e f o r a s h i p i n a r e g u l a r l o n g i t u d i n a l wave i s a p e r i o d i c f u n c t i o n o f t h e t i m e w i t h a p e r i o d T w h i c h i s e q u a l t o t h e p e r i o d o f e n c o u n t e r . The shape o f t ^ i s f u n c t i o n depends among o t h e r t h i n g s on t h e m o t i o n s o f t h e s h i p i n t h e waves. I n t h i s c o n n e c t i o n we s h o u l d m e n t i o n a paper o f Grim [ 5 ] . He showed t h a t , i n case t h e wave c e l e r i t y i s l a r g e r t h a n t h e s h i p s p e e d , t h e r e l a t i v e speed

between wave and s h i p i s l a r g e r m a viave t h r o u g h t h a n on a c r e s t o f t h e wave. C o n s e q u e n t l y , t h e s t a p e o f D , c a l c u l a t e d on t h e h a s i s o f a c o n s t a n t r e l a t i v e speed, w i l l he more f a v o u r a b l e t h a n t h e one, based on a v a r y i n g r e l a t i v e speed.

Assume t h a t t h e s h i p e x p e r i e n c e s i n t h e u p r i g h t p o s i t i o n a c o n s t a n t e x t e r n a l moment M w h i c h may v a r y f o r d i f f e r e n t v a l u e s o f t h e h e e l i n g a n g l e . I n o r d e r t o compare t h i s moment w i t h t h e s t a b i l i t y moment, we r e p r e s e n t i t i n t h e curve o f t h e r i g h t i n g arms h y a graph w i t h o r d i n a t e s , g i v e n h y M /yV, ( F i g . 2 ) . I t i s e a s i l y seen t h a t i n case t h e a r e a o f t h e h a t c h e d r e g i o n I i n F i g . 2 i s s m a l l e r t h a n t h e area A-j.-^. o f r e g i o n I I , t h e s h i p w i l l o b t a i n a s t a b l e e q u i l i b r i u m .

For A-j- = t h e e q u i l i b r i u m i s n e u t r a l w h i l e no s t a b l e e q u i l i b r i u m e x i s t s when A^ ^"^n*

We now d e f i n e t h e f u n c t i o n D as t h e d i f f e r e n c e between t h e areas o f t h e r e g i o n s I I and I :

°r = ^ 1 1 - ^ I

So f o r a s h i p s a i l i n g i n a r e g u l a r l o n g i t u d i n a l wave D i s a p e r i o d i c f u n c t i o n and may have a shape as r e p r e s e n t e d "Ey F i g . 3 . We assume t h a t t h e s h i p i s s a i l i n g i n a r e g u l a r l o n g i t u d i n a l wave w i t h such a wave l e n g t h and wave h e i g h t t h a t D has a n e g a t i v e v a l u e on a wave c r e s t and a p o s i t i v e one i n a t r o u g h . Then i n each p e r i o d o f e n c o u n t e r t h e r e i s an i n t e r v a l T d u r i n g w h i c h D <o. So, i n t h i s i n t e r v a l t h e s h i p i s i n an u n s t a b l e p o s i t i o n . Whether o r n o t c a p s i z i n g w i l l o c c u r depends on t h e l e n e t h o f t h e t i m e i n t e r v a l T t h e v a l u e s o f D w i t h i n t h i s • . o . r i n t e r v a l and t h e d y n a m i c a l p r o p e r t i e s o f t h e v e s s e l . We assurae_the s t i l l w a t e r v a l u e o f t h e f u n t i o n D ( t ) t o be g i v e n b y D . I t i s obvious t h a t f o r v e r y s m a l l v a l u e s o f t h e wave heighï f;^ t h e f u n t i o n D_^(t) o s c i l l a t e s p e r i o d i c a l l y about D_^. However, i f i s i n c r e a s e d _ t h e average v a l u e o f D ^ ( t ) w i l l i n g e n e r a l n o t remain e q u a l t o D^.

We can now i n v e s t i g a t e f o r each wave l e n g t h X w h e t h e r t h e r e e x i s t s a wave h e i g h t ? <; (X) such t h a t t h e v e s s e l i s i n an u n s t a b l e p o s i t i o n f o r alS°wave h e i g h t s C > C f o r a t l e a s t a p e r i o d j i n each p e r i o d o f e n c o u n t e r T^, ( F i g . 3 ) , where T^ i s d e t e r m i n e d by t h e wave l e n g t h A and t h e s h i p speed. Of c o u r s e , we can r e s t r i c t o u r i n v e s t i g a t i o n s t o t h e wave l e n g t h s w h i c h

c o r r e s p o n d w i t h t h e p e r i o d s o f e n c o u n t e r , s a t i s f y i n g t h e i n e q u a l i t y T > x . For t h e v a l u e s ofA where no v a l u e f o r C i s f o u n d , we d e f i n e C ( A ) = °°.

C o n s i d e r i n g t h e f u n c t i o n D ( t ) f o r s e v e r a l r e g u l a r waves w i t h d i f f e r e n t wave l e n g t h s b u t t h e same h e i g h t , we may e x p e c t , on b a s i s o f t h e b e h a v i o u r o f t h e d y n a m i c a l s t a b i l i t y , t h a t t h e

f u n c t i o n D ( t ) has i t s l a r g e s t a m p l i t u d e f o r waves w h i c h have about t h e same wave l e n g t h as t h e l e n g t h L o f t h e s h i p .

For t h i s r e a s o n we may expect t h a t t h e f u n c t i o n ^ ( A ) a t t a i n s

• Pi O

l t s s m a l l e s t v a l u e i n t h e n e i g h b o u r h o o d o f A=L.

So, f i n a l l y , we o b t a i n e d a f u n c t i o n <; (A ) w h i c h has a v a l u e f o r e v e r y wave l e n g t h x- The shape o f this°function i s d e t e r m i n e d b y t h e geometry o f t h e s h i p , i t s speed, and f u r t h e r by t h e v a l u e s o f t h e p a r a m e t e r s and t ^ . S i n c e , i t appears t o be i m p o r t a n t , i n t h e f u t u r e , t o be aware o f t h e dependence o f t; on t h e p a r a m e t e r s T and M we s h a l l denote r (A ) by r ( A ; IS ° , ' ' ) .

O . G 3,0 SO S O

C o n s i d e r now a s h i p s a i l i n g m i r r e g u l a r l o n g i t u d i n a l waves. The s u r f a c e o f t h e waves i s assumed t o have a s t a t i o n a r y and e r g o d i c n o r m a l d i s t r i b u t i o n w i t h a r e l a t i v e l y narrow power

s p e c t r u m . Then t h e waves a r e q u a s i - h a r m o n i c , (see a p p e n d i x ) , and can be r e p r e s e n t e d by t h e f u n c t i o n .

^ ( t ) cos (w t + e ( t ) )

ih)

The i n s t a n t a n e o u s a m p l i t u d e C ( t ) and i n s t a n t a n e o u s f r e q u e n c y 0) ( t ) = 0) + Ö(t) a r e random f u n c t i o n s w h i c h v a r y s l o w l y as compared w i t h cos w t . F u r t h e r , i t i s c l e a r t h a t t h e a p p a r e n t a m p l i t u d e and f r e q u e n c y have f o r such a s p e c t r u m t h e same p r o h a b i l i t y d e n s i t y as t h e c o r r e s p o n d i n g i n s t a n t a n e o u s q u a n t i t i e s w h i c h have a d e n s i t y g i v e n hy

I n v i r t u e o f t h e q u a s i - h a r m o n i c c h a r a c t e r o f t h e waves, we d e t e r m i n e t h e p o s i t i o n o f t h e s h i p i n t h e waves i n a q u a s i -s t a t i o n a r y way, i . e . , i f t h e wave-s have on t h e l o c a t i o n o f t h e v e s s e l an i n s t a n t a n e o u s a m p l i t u d e and f r e q u e n c y g i v e n h y x,^ and Ü). , t h e n t h e i n s t a n t a n e o u s w a t e r l i n e o f t h e v e s s e l i s d e t e r m i n e d as i f i t i s i n a r e g u l a r wave w i t h t h i s f r e q u e n c y and a m p l i t u d e . I t i s c l e a r t h a t t h e f u n c t i o n D ^ ( t ) i s a q u a s i - p e r i o d i c f u n c t i o n , i . e . , t h e a p p a r e n t p e r i o d T w h i c h i s measured f r o m t r o u g h t o t r o u g h v a r i e s o n l y v e r y s l o w l y . F o r t h e same r e a s o n , t h e g u a n t i t y T w h i c h i s t h e i n t e r v a l o f t i m e i n each p e r i o d T , d u r i n g w h i c h t h e v e s s e l has no s t a b l e e q u i l i b r i i m , i s a q u a s i - p e r i o d i c f u n c t i o n . I n v i r t u e o f t h e d e f i n i t i o n o f t h e f u n c t i o n ? ( A ) t h e ^ p r o b a b i l i t y P {.T> T ; M } t h a t i n some apparen? p e r i o d T t h e r e s i d u a l d y n a m i c a l s t a b i l i t y has n e g a t i v e v a l u e s f o r a t l e a s t a t i m e i n t e r v a l i s g i v e n b y : P { T > T ;M }=ƒ du)7 dC p (?^,a)) ( 5 ) r o' e o 0 of a'

where, s i n c e A = •^^,t:he f u n c t i o n ? ( A ; M ,T ) can a l s o be

w r i t t e n as a f u n c t i o n o f t h e f r e q u e n c y o f ?he wave. T h i s f u n c t i o n has t o be d e t e r m i n e d n u m e r i c a l l y .

I t i s o b v i o u s t h a t t h e average l e n g t h o f t h e a p p a r e n t p e r i o d s i s t w i c e t h e average d i s t a n c e between two s u c c e s s i v e p a s s i n g s o f t h e wave s u r f a c e t h r o u g h t h e s t i l l w a t e r l e v e l .

I n correspondence w i t h t h e t e r m i n o l o g y o f random n o i s e t h e o r y we c a l l such a p a s s i n g a z e r o . D e n o t i n g t h e random d i s t a n c e between two s u c c e s s i v e zeros byx we o b t a i n :

E{ T }= 2 E{ T } ( 6 )

e

F u r t h e r E { x } i s e q u a l t o t h e r e c i p r o c a l o f t h e e x p e c t e d number o f zeros p e r u n i t t i m e [ 6 ] . T h i s i s c l e a r from t h e f o l l o w i n g argument. I f we r e p r e s e n t t h e e x p e c t e d number o f zeros p e r u n i t t i m e b y 3. Then i n a v e r y l o n g i n t e r v a l T t h e r e a r e 3T zeros and t h e r e f o r e a l s 3T i n t e r v a l s .

C o n s e q u e n t l y , t h e average l e n g t h o f such an i n t e r v a l i s i n d e e d g i v e n h y 1 . , 2

So : 3 E { T ^ } = - ( 7 ) The average number o f apparent p e r i o d s i n a t i m e i n t e r v a l

o f l e n g t h T i s g i v e n b y T

— ( 8 ) E T

Then t h e average number o f t i m e s t h a t i n t h e t i m e i n t e r v a l T t h e r e s i d u a l d y n a m i c a l s t a b i l i t y D i s n e g a t i v e f o r a t l e a s t a t i m e i n t e r v a l T i s g i v e n by : 0 T e Pr { T > T ; M } 0 ' e .9) D i v i d i n g t h i s e x p r e s s i o n by T, we o b t a i n t h e average numher o f t i m e s p e r u n i t t i m e t h a t D < o f o r a t l e a s t a t i m e i n t e r v a l x . D e n o t i n g t h i s q u a n t i t y by n(D <o;x ,M ) , we f i n d af-Ber s u b s t i t u t i o n o f (T) : . r o e n(D <o;x ,M ) ^ ^ ^ " ^ ^ ^ ^ o ' ^ ^ (10) r o e 2

For 3 we s u b s t i t u t e e x p r e s s i o n ( 5 1 ) , m w h i c h t h e second moments b ^ and b are r e p l a c e d by t h e i r o r i g i n a l e x p r e s s i o n s . TÊen,finally . ƒ V w (V ) dv| n(D <oix ,M ) =Pr{x>x ,M } r ' 0 ' e 0 e ƒ w(V) dv 0 1/2 (11) where t h e p r o b a b i l i t y i n t h e r i g h t hand s i d e i s g i v e n by t h e i n t e g r a l ( 5 ) w h i c h has t o be e v a l u a t e d n u m e r i c a l l y . By e v a l u a t i n g t h e s t a t i s t i c a l q u a n t i t y n(D <o;x ,M ) w h i c h depends on t h e p a r a m e t e r s M andx^ i n f o r m a t i o n i s " ' ^ o b t a i n e i w i t h r e s p e c t t o t h e t r a n s v e r s e s t a b i l i t y ol' t h e s h i p . For example, i f x^ has t h e same o r d e r o f magnitude as t h e n a t u r a l r o l l i n g p e r i o d o f t h e s h i p , t h e n a dangerous s i t u a t i o n a r i s e s . We assumed i n t h i s c h a p t e r t h e f u n c t i o n M ((j)) t o be independent o f t h e t i m e . I n a c t u a l c i r c u m s t a n c e s however t S i s f u n c t i o n w i l l be random t i m e dependent w i t h a component due t o t h e wave a c t i o n and a n o t h e r as a r e s u l t o f t h e w i n d .

We observe t h a t f o r a s h i p s a i l i n g i n an o b l i q u e i r r e g u l a r f o l l o w i n g sea t h e wave component o f t h i s d e n s i t y i s c o r r e l a t e d w i t h t h e d e n s i t y p ( c ,cü) f o r t h e p a r a m e t e r s o f t h e l o n g i t u d i n a l component o f t h i s s p e c i f i c seaway. T h e r e f o r e , t h e i n f o r m a t i o n g i v e n by e x p r e s s i o n ( I I ) i s not

complete and enables us o n l y t o g i v e a r a t h e r s u p e r f i c i a l judgement about t h e t r a n s v e r s e s t a b i l i t y o f t h e s h i p . I n o r d e r t o o b t a i n more t a n g i b l e d a t a we need more e x t e n s i v e i n f o r m a t i o n r e g a r d i n g t h e s t a t i s t i c a l p r o p e r t i e s o f t h e e x t e r n a l r o l l i n g moment f u n c t i o n M w h i c h i s n o t c o n s i d e r e d a t p r e s e n t .

I t i s c l e a r t h a t i n head waves w h i c h have about t h e same l e n g t h as t h e l e n g t h o f t h e s h i p t h e p e r i o d o f e n c o u n t e r i s v e r y s m a l l as

compared w i t h t h e r o l l i n g p e r i o d . For t h i s r e a s o n t h e t r a n s v e r s e s t a b i l i t y o f t h e s h i p w i l l be d e t e r m i n e d by t h e average v a l u e o f t h e r e s i d u a l

d y n a m i c a l s t a b i l i t y D o f t h e s h i p . I f t h i s average v a l u e i s p o s i t i v e t h e n we may expect a s a f e o p e r a t i o n o f t h e s h i p under t h e s e c i r c x i m s t a n c e s . Consequently,we can r e s t r i c t our d i s c u s s i o n s t o f o l l o w i n g seas.

We a l r e a d y n o t i c e d above t h a t i n t h i s case, f o r such v a l u e s o f x ^ and M w h i c h m i g h t g i v e dangerous s i t u a t i o n s w i t h r e s p e c t t o t h e t r a n s v e r s e

9 .

Spontaneous r o l l i n g i n i r r e g u l a r l o n g i t u d i n a l waves.

A l t h o u g h , t h e r e i s no e x t e r n a l r o l l i n g moment w o r k i n g on a s h i p i n

l o n g i t u d i n a l waves, i t has been e s t a b l i s h e d hy Grim b o t h t h e o r e t i c a l l y and e x p e r i m e n t a l l y t h a t r o l l i n g m o t i o n s can be e x c i t e d under t h e s e

c i r c u m s t a n c e s .

I n view o f our d i s c u s s i o n s i n t h e p r e c e d i n g s e c t i o n o f t h i s c h a p t e r , i t i s o b v i o u s t h a t f o r a s h i p , s a i l i n g i n a r e g u l a r l o n g i t u d i n a l wave, t h e i n i t i a l m e t a c e n t r i c h e i g h t GM v a r i e s p e r i o d i c a l l y w i t h a p e r i o d e q u a l t o t h e p e r i o d o f e n c o u n t e r . Analogous t o Grim's argument we assume t h e v a r i a t i o n o f GM t o be h a r r a o n i c a l . Consequently,we may w r i t e :

GM = GM + A GM s i n 0) t (12)

o e

where GM i s t h e average v a l u e o f GM and AGM t h e a m p l i t u d e o f i t s v a r i a t i o n . B o t h GM and AGM depend on t h e f r e q u e n c y o f e n c o u n t e r oi^, the wave l e n g t h an§ t h e wave a m p l i t u d e .

The i n i t i a l s t a b i l i t y moment w h i c h i s t h e r e s t o r i n g t e r m i n t h e e q u a t i o n o f m o t i o n f o r r o l l i n g becomes : M ^ = Y V { CM + AGM s i n o) t } (f) (13) s t ' o e ^ So we f i n d f o r t h e e q u a t i o n o f m o t i o n f o r r o l l i n g w i t h s m a l l a m p l i t u d e s i n l o n g i t u d i n a l waves : I (t>* + Wè + Y V { G M + A GM s i n 0) t}(j) = 0 (1^+) cj) O e

I n o r d e r t o reduce t h e amount o f m a t h e m a t i c a l work w h i c h , as w i l l be seen a t t h e end o f t h i s s e c t i o n , does n o t a f f e c t t h e g e n e r a l i t y o f t h e method expounded i n t h i s s e c t i o n we d e l e t e t h e damping t e r m i n t h i s e q u a t i o n . Then t h e w e l l - k n o w n d i f f e r e n t i a l e q u a t i o n o f M a t h i e u i s o b t a i n e d : I V + Y V { GM + AGM s i n oi t } ()> = 0 ( 1 5 ) (j) ^ o e I n t h i s e q u a t i o n we i n t r o d u c e t h e change o f v a r i a b l e s ; 00 t = 2v (16) e Then e q u a t i o n (15) can be w r i t t e n i n t h e f o l l o w i n g s o - c a l l e d c a n o n i c a l f o r m : 2 ^ + (a-2q cos 2v)^ = 0 • ( I T ) d t where b (JO and _{1 =^} (18)

GM ( ] 9 ) and yVGM (20) OJQ r e p r e s e n t s t h e n a t u r a l f r e q u e n c y o f t h e r o l l i n g m o t i o n . T h e ^ q u a n t i t y i n t h e l a s t f o r m u l a i s t h e t o t a l i n e r t i a o f s h i p and w a t e r . I t i s observed t h a t i n view o f t h e dependence o f a and q on GM^ and AQM

t h e parameters a and q i n e q u a t i o n ( I 7 ) depend on t h e f r e q u e n c y o f e n c o u n t e r , t h e wave l e n g t h and t h e wave h e i g h t .

I t i s w e l l - k n o w n t h a t t h e s o l u t i o n s o f e q u a t i o n ( I 7 ) can remain f i n i t e o r t e n d t o i n f i n i t y , once (j) has a v a l u e u n e q u a l z e r o .

For t h i s r e a s o n we c a l l t h e s e s o l u t i o n s s t a b l e o r u n s t a b l e , r e s p e c t i v e l y . The t y p e o f s o l u t i o n w h i c h i s o b t a i n e d depends on t h e v a l u e s o f a and q. A d e t a i l e d d i s c u s s i o n o f t h i s i s s u e i s f o u n d i n [t] . The r e s u l t s w h i c h a r e d e r i v e d i n t h a t r e f e r e n c e w i l l be used h e r e w i t h o u t f u r t h e r d i s c u s s i o n . The ( a , q ) - p l a n e can be s u b d i v i d e d , i n s o - c a l l e d s t a b l e and u n s t a b l e r e g i o n s , (see F i g . k), w h i c h c o r r e s p o n d w i t h s t a b l e and u n s t a b l e s o l u t i o n s o f e q u a t i o n ( 1 5 ) . Since i n o u r case t h e c o e f f i c i e n t s a and q have o n l y p o s i t i v e v a l u e s , we can r e s t r i c t o u r d i s c u s s i o n s t o t h e f i r s t quadrant o f t h e ( a , q ) - p l a n e . I n F i g .

h t h e s t a b l e r e g i o n s a r e bounded by t h e curves a. and b . , where i = 1,2, ....

C o n s e q u e n t l y , when we, A and Ca have such values'^'that t A e c o r r e s p o n d i n g p o i n t ( a , q ) i s s i t u a t e d i n an u n s t a b l e r e g i o n t h e n t h e v a l u e o f <j) w i l l i n c r e a s e unboundedly, once i t has a v a l u e u n e q u a l t o z e r o , e.g., due t o an e x t e r n a l moment. I n p r a c t i c a l cases t h e s e moments a r e always p r e s e n t , e.g., moments e x c i t e d b y w i n d , i r r e g u l a r i t i e s o f t h e waves, asymmetry o f t h e s h i p , e t c . T h e r e f o r e , we may expect t h a t t h e s e v a l u e s o f oig, A and w i l l always i n d u c e a r o l l i n g m o t i o n .

We^consider t h e v a l u e s o f a and q when f o r some c o n s t a n t and A t h e wave h e i g h t Ca. i s i n c r e a s e d f r o m Ca = 0. I t i s observed t h a t f o r Ca = 0 w h i c h c o l r e s p o n d s w i t h t h e s t i l l w a t e r case t h e p a r a m e t e r s have t h e v a l u e s : r 2 q = 0 and where : Ü) = o YV GM s t i n w h i c h GM^^^ r e p r e s e n t s t h e m e t a c e n t r i c h e i g h t f o r s t i l l w a t e r . A c c o r d i n g t o K e r w i n [ 8 ] t h e average v a l u e o f t h e m e t a c e n t r i c h e i g h t i n waves GM^ i n c r e a s e s when t h e wave a m p l i t u d e i s i n c r e a s e d .

C o n s e q u e n t l y j t h e v a l u e o f a i n c r e a s e s when f;^ i ^ i n c r e a s e d and, t h e r e f o r e , t h e v a l u e s o f a and q f o r c o n s t a n t v a l u e s o f Wg and \ and i n c r e a s i n g v a l u e s o f

a r e r e p r e s e n t e d i n F i g , 1| as a monotonic i n c r e a s i n g l i n e , F o r each s e t o f v a l u e s (tOg, X ) o f t h e s e p a r a m e t e r s such a c u r y e can be o b t a i n e d .

For e v e r y c o n s t a n t v a l u e o f Wg we can d e t e r m i n e i n t h e ( R , X ) - p l a n e a s e t o f r e g i o n s S-] , Sp, , S^, , where t h e r e g i o n S-[ c o n t a i n s t h e

( R , X ) - v a l u e s w h i c h c o r r e s p o n d w i t h t h e ( a , q ) — v a l u e s i n t h e u n s t a b l e r e g i o n between t h e boundary curves a£ and b j ^ .

C o n s i d e r i n g now a s h i p s a i l i n g i n an i r r e g u l a r l o n g i t u d i n a l sea w i t h a r e l a t i v e l y narrow power s p e c t r i i m , t h e n , a p p l y i n g a g a i n t h e q u a s i - s t a t i o n a r y a p p r o x i m a t i o n method, we f i n d f o r t h e p r o b a b i l i t y t h a t t h e s h i p i s i n a p o s i t i o n t h a t an u n s t a b l e r o l l i n g m o t i o n m i g h t be d e v e l o p e d : P = ƒƒ p(c^,tü. ) dRdüj (21 ) . 1 1 where t h e j o i n t p r o b a b i l i t y d e n s i t y p(Cg^,Wj_) i s g i v e n b y (i+ij) . The r e g i o n s S;[, i = 1,2, , have t o be d e t e r m i n e d n u m e r i c a l l y .

C o r o l l a r y

I t i s observed t h a t f o r t h e d e t e r m i n a t i o n o f t h e p r o b a b i l i t y ( 2 1 ) o n l y t h e b o u n d a r i e s i n t h e ( a , q ) - p l a n e between t h e s t a b l e and u n s t a b l e r e g i o n s o f t h e s o l u t i o n o f ( 1 5 ) a r e needed. I n [j] t h e b o u n d a r i e s between t h e s t a b l e and .unstable r e g i o n s o f t h e s o l u t i o n s o f ( l ^ l ) a r e d e t e r m i n e d . I t appears t h a t t h e

u n s t a b l e r e g i o n s o f ( l U ) a r e s u b r e g i o n s o f t h o s e o f ( 1 5 ) , t h e areas o f t h e s e r e g i o n s depending on t h e v a l u e o f W. C o n s e q u e n t l y , when t h e r o l l i n g m o t i o n f o r a s h i p i n l o n g i t u d i n a l waves i s r e p r e s e n t e d by e q u a t i o n {^h) t h e n e v e r y i n t e g r a t i o n r e g i o n Sj_ i n _expression ( 2 1 ) f o r t h e p r o b a b i l i t y P has t o be r e p l a c e d by a new r e g i o n S i , w h i c h i s based on t h e new b o u n d a r i e s between _the s t a b l e and u n s t a b l e r e g i o n s f o r t h e e q u a t i o n {^k). I t i s o b v i o u s t h a t

J l

N u m e r i c a l example

The s t a t i s t i c a l q u a n t i t i e s , i n t r o d u c e d above i n o r d e r t o d e s c r i b e t h e t r a n s v e r s e s t a b i l i t y o f a s h i p i n i r r e g u l a r l o n g i t u d i n a l waves, a r e d e t e r m i n e d i n t h i s s e c t i o n f o r t h e m.s. "S.A. van der S t e l " w h i c h i s a f a s t cargo s h i p w i t h a low p r i s m a t i c c o e f f i c i e n t . The v e s s e l i s assumed t o have a speed o f 25 k n o t s i n an i r r e g u l a r f o l l o w i n g sea w i t h a u n i - d i r e c t i o n a l power s p e c t r u m g i v e n b y : I ^ gg^ -3 ( a , ^ / 2 ^ v ) ^ ( 2 2 ) w ( v ) = q e o T h i s i s t h e s o - c a l l e d P i e r s o n - M o s k o v i t c h s p e c t r u m . The c o e f f i c i e n t s i n t h i s e x p r e s s i o n a r e g i v e n by a= 8.1 x 10~ , 3- 0.7^+ and = g/u i n w h i c h u i s t h e w i n d v e l o c i t y a t a . h e i g h t o f 19-5 m above t h e sea s u r f a c e . I t i s o b s e r v e d t h a t t h e t r a n s v e r s e s t a b i l i t y as c o n s i d e r e d above i s c o m p l e t e l y d e t e r m i n e d b y t h e b e h a v i o u r as a f u n c t i o n o f t h e t i m e o f b o t h t h e c u r v e o f r i g h t i n g arms and t h e m e t a c e n t r i c h e i g h t GM. These f u n c t i o n s a r e a t each i n s t a n t o f t i m e d e t e r m i n e d by t h e i n s t a n t a n e o u s w a t e r - l i n e o f t h e v e s s e l . I t i s assumed t h a t t h i s w a t e r - l i n e can be d e t e r m i n e d i n a s u f f i c i e n t l y

a c c u r a t e way by t a k i n g o n l y t h e h e a v i n g and p i t c h i n g m o t i o n s o f t h e v e s s e l i n t o a c c o u n t . I t has been o b s e r v e d t h a t o n l y waves w i t h l e n g t h s w h i c h a r e o f t h e same o r d e r o f m a g n i t u d e as t h e s h i p l e n g t h can s u b s t a n t i a l l y a f f e c t t h e

t r a n s v e r s e s t a b i l i t y . F o r t h i s v e s s e l w h i c h has a l e n g t h L = 152.50 m t h e c i r c u l a r f r e q u e n c i e s o f e n c o u n t e r have v a l u e s i n t h e range ( O . O . I 6 ) when t h e s h i p i s i n r e g u l a r waves w i t h l e n g t h s A s a t i s f y i n g O.5O < A/L < 1.25.

C o n s e q u e n t l y , we f i n d t h a t f o r t h i s waves t h e r e l a t i v e v e l o c i t y 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 waves i s s m a l l . S i n c e , i n a d d i t i o n , t h e f r e q u e n c y and

a m p l i t u d e o f an i r r e g u l a r sea w i t h a r e l a t i v e l y n a r r o w s p e c t r u m v a r y v e r y s l o w l y , we a p p l y t h e q u a s i - s t a t i o n a r y a p p r o x i m a t i o n method t o d e t e r m i n e t h e i n s t a n t a n e o u s r e l a t i v e p o s i t i o n o f t h e v e s s e l w i t h r e s p e c t t o t h e waves. U s i n g t h e s t r i p method as d e s c r i b e d by G e r r i t s m a and Beukelman [ 9 ] , t h e h e a v i n g and p i t c h i n g m o t i o n s a r e d e t e r m i n e d when t h e s h i p i s s a i l i n g i n r e g u l a r waves, f o r w h i c h O.5O L ^ A ^ 1.25 L.

I t t u r n s o u t t h a t t h e p i t c h i n g m o t i o n s a r e n e g l i g i b l e w i t h r e s p e c t t o t h e h e a v i n g o f t h e v e s s e l . The a m p l i t u d e and phase c h a r a c t e r i s t i c s f o r t h e h e a v i n g m o t i o n s a r e g i v e n i n F i g .

5-The b e h a v i o u r o f GM and o f t h e c u r v e o f r i g h t i n g arms i s d e t e r m i n e d n u m e r i c a l l y on b a s i s o f t h e geometry o f t h e h u l l o f t h e s h i p and o f t h e i n s t a n t a n e o u s

shapes o f t h e w a t e r - l i n e . S i n c e i t i s a r a t h e r t i m e consuming p r o c e d u r e t o f i n d t h e a c c u r a t e t i m e - d e p e n d e n t b e h a v i o u r o f GM and o f l((J)) f o r a sequence o f v a l u e s o f (Ca^.^), i t i s assumed t h a t t h e s e q u a n t i t i e s v a r y _ h a r m o n i c a l l y and a c h i e v e t h e i r maximum and minimum v a l u e s when t h e s h i p i s i n a t r o u g h and on a c r e s t o f t h e wave, r e s p e c t i v e l y . So, i t s u f f i c e s t o c a l c u l a t e f o r each c o u p l e o f v a l u e s (Ca,A) o n l y t h e v a l u e s o f GM and t h e c u r v e s o f r i g h t i n g arms when t h e s h i p i s on a wave c r e s t o r i n a wave t r o u g h .

These c a l c u l a t i o n s have heen c a r r i e d o u t f o r a sequence o f wave l e n g t h s and a sequence o f wave h e i g h t s . The v a l u e s o f t h e c o e f f i c i e n t s a and q i n e q u a t i o n ( I T ) a r e d e t e r m i n e d f r o m ( l 8 ) . The h e h a v i o u r o f t h e f u n c t i o n D^, as d e f i n e d by ( 3 ) , i s e a s i l y d e t e r m i n e d by i n t e g r a t i n g t h e r e g i o n s I and I I w h i c h a r e

i n d i c a t e d i n F i g . 2.

The procedure t o d e t e r m i n e t h e s e n s i t i v i n e s s o f t h e v e s s e l f o r spontaneous r o l l i n g has been c a r r i e d o u t f o r GM = 0.05m. F o r a sequence o f v a l u e s f o r A, t h e r e l a t i o n between t h e c o e f f i c i e n t s a and q and t h e wave a m p l i t u d e s have been d e t e r m i n e d and p l o t t e d m F i g . 6.

The r e g i o n Sp o f ( C a , ^ ) - v a l u e s w h i c h c o r r e s p o n d w i t h t h e ( a , q ) - v a l u e s i n t h e u n s t a b l e r e g i o n between t h e boundary curves ap and bp i s g i v e n i n F i g . T . The v a l u e o f t h e p r o b a b i l i t y (21) has been e v a l u a t e d n i i m e r i c a l l y and i t i s f o u n d t h a i

P = 0.00521 f o r B e a u f o r t 8

P = 0.0033T f o r B e a u f o r t 6

I n o r d e r t o i n d i c a t e t h e d i s t r i b u t i o n o f P over t h e wave l e n g t h s , we have

p l o t t e d i n F i g . T a l s o t h e i n t e g r a l /p(Ca,'^) ^Cg, over t h e r e g i o n Sp as a f u n c t i o n o f t h e wave l e n g t h .

The a c t u a l v a l u e s o f GM f o r t h e c o n s i d e r e d s h i p range f r o m O.5O t o 1.37 m. However, t h e c o e f f i c i e n t a i s f o r t h e s e v a l u e s o f GM so l a r g e and q so s m a l l t h a t no s u b s t a n t i a l v a l u e s f o r P c o u l d be o b t a i n e d . T h i s r e s u l t t h e n suggests t h a t t h e c o n s i d e r e d v e s s e l i s n o t s e n s i t i v e f o r spontaneous r o l l i n g .

The c a l c u l a t i o n s have a l s o been c a r r i e d o u t f o r t h e s h i p s a i l i n g i n head waves w i t h a speed o f 25 k n o t s . I t t u r n s o u t t h a t f o r a w i n d v e l o c i t y o f B e a u f o r t 8 t h e c o e f f i c i e n t q has always v a l u e s s m a l l e r t h a n 0.0075 and t h e v a l u e o f P i s t h e r e f o r e even f o r GM = 0.05 m n e g l i g i b l y s m a l l .

The s t a t i s t i c a l q u a n t i t y n (Dp<0; TQ, M^) has been d e t e r m i n e d f o r a c o n s t a n t e x t e r n a l moment, g i v e n b y M^ = 1 5Y'^ and f o r GM = O.8O m. The r e l a t i o n

between t h e f u n c t i o n C and XQ has been e v a l u a t e d f o r a sequence o f v a l u e s f o r A and p l o t t e d i n F i g . 8. From t h i s we d e r i v e t h e r e l a t i o n s h i p between t h e f u n c t i o n ^ao and t h e wave f r e q u e n c i e s cofor x^ = 5 , 1 0 and 20 s e c , ( t h e n a t u r a l r o l l i n g p e r i o d o f t h e v e s s e l i s about 20 sec. ( P i g . 9 ) - The i n t e g r a l ( 5 ) i s e v a l u a t e d n u m e r i c a l l y and i t s v a l u e i s s u b s t i t u t e d i n (11 ) . The d i s t r i b u t i o n o f t h e c o n t r i b u t i o n t o t h e p r o b a b i l i t y V^{x > x^jMg) over t h e f r e q u e n c i e s i s g i v e n i n F i g . 10. The r e l a t i o n between t h e q u a n t i t y n, g i v e n i n 11 , and x^ i s g i v e n i n F i g . 11.

.///-Appendix

Random phase model o f a s t r i c t l y s t a t i o n a r y Ganssian p r o c e s s ; i n s t a n t a n e o u s a m p l i t u d e and f r e q u e n c y .

We assume t h a t I ( t ) i s a s t r i c t l y s t a t i o n a r y Ganssian p r o c e s s w h i c h extends f r o m t = -°° t o t =«> . We f u r t h e r assume t h a t t h e s p e c t r a l d e n s i t y w ( v ) o f t h i s process i s c o n t i n u o u s . Then i t can he p r o v e d , [ 1 0 , s e c t . T-ll]» t h a t t h e p r o c e s s i s a l s o e r g o d i c , i . e . , t i m e and ensemble avarages y i e l d t h e same r e s u l t . W i t h o u t r e s t r i c t i n g t h e g e n e r a l i t y o f t h e d i s c u s s i o n we may assume t h a t

t h e avarage v a l u e o f t h e s i g n a l has zero v a l u e

E I ( t ) = 0 ( 2 3 ) The v a r i a n c e o f t h e s i g n a l i s e q u a l t o i t s t o t a l avarage power

E I ^ ( t ) = ƒ w ( v ) d v {2k)

o

We p r e s e n t t h e p r o c e s s by t h e random phase model N I ( t ) = Z c cos (OJ t -(j) ) (25) n=1 " The a n g l e s ^ tj) ,(j) a r e i n d e p e n d e n t s t o c h a s t i c v a r i a b l e s w i t h a u n i f o r m d i s t r i b u t i o n o v e r t h e range ( 0 , 2 i r J . E u r t h e r c = 2 w ( v ) A v j ^ , 00 = 2TT V and v = n A v (26) n L J ' n n n The c o n t i n u o u s s p e c t r a l d e n s i t y w ( v ) i s i n t h e r e p r e s e n t a t i o n ( 2 3 ) r e p l a c e d by a d i s c r e t e s p e c t r u m w'(v) w h i c h has o n l y v a l u e s f o r v-| , Vp, v ^ . ^ 2 w''(v)= Z I c 6 ( v - v ) (27) . n n

By a p p l y i n g t h e c e n t r a l l i m i t theorem i t can be shown t h a t t h e r e p r e s e n t a t i o n (23) converges t o a n o r m a l d i s t r i b u t e d p r o c e s s w i t h avarage v a l u e z e r o and v a r i a n c e ^ w ( v ) d v , when t h e d i s c r e t e s p e c t r u m (25) a p p r o x i m a t e s t h e

o

Analogous t o Rice's method we choose a f r e q u e n c y m^, w h i c h i s a r e p r e s e n t a t i v e midhand f r e q u e n c y o f t h e power s p e c t r u m , and w r i t e (25) i n t h e f o r m :

: t ) N = E n=1 c cos n , (JÜn m t -0) t -d CO t ) m I cos c m s m CO t - I s i n CO t ( 2 8 ) where N I = E c cos (co t - CO t - d) \ c , n n m ^n) n=1 N I = E c s i n (co t - 00 t - d) ) s . n n m n n=1 We d e f i n e 2 . .t) - a r c t g ^ I c

Combining (28 and ( 3 0 ) we can w r i t e ( 2 5 ) i n t h e f o r m

'29)

( 3 0 )

I ( t ) = C ( t ) cos (w t + e ( t ) ) ( 3 1 )

We assume t h e power spectrum t o be r e l a t i v e l y n a r r o w , i . e . , t h e c h a r a c t e r i s t i band w i d t h i s v e r y s m a l l as compared w i t h a r e p r e s e n t a t i v e midband f r e q u e n c y ,

e.g., t h e median o f t h e spectrum. Then t h e v a l u e s o f c^ w i l l decrease v e r y f a s t t o zero when | n-m ) i n c r e a s e s i n v a l u e . Thus I c ("t'l and I s ( t ) a r e v e r y s l o w l y v a r y i n g f u n c t i o n s as compared w i t h cos coj^t and f o r t h i s r e a s o n t h e envelope f u n c t i o n i ; ^ ( t ) and phase f u n c t i o n 6 ( 1 ) have a l s o t h i s p r o p e r t y . A s i g n a l l ( t ) w i t h a r e l a t i v e l y n a r r o w band w i d t h has a shape as r e p r e s e n t e d by F i g . 12. The f u n c t i o n l,^ ( t ) and tüi(t) - coj^ + ê(t) can be i n t e r p r e t e d as t h e i n s t a n t a n e o u s a m p l i t u d e and f r e q u e n c y o f t h e s i g n a l . From t h i s p h y s i c a l concept o f t h e f u n c t i o n s Ca("t) and ojj_(t) i t i s n o t c l e a r w h e t h e r o r n o t a d i f f e r e n t c h o i c e o f coj^ w i l l l e r d t o t h e same t.r^{t) and co£(t). However, i t has been shown by D u g u n d j i [ l 2 ] t h a t t h e s e f u n c t i o n s a r e i n d e p e n d e n t o f t h e c h o i c e o f 1%. He s t a r t e d f r o m t n e complex v a l u e d f u n c t i o n

Z ( t ) = I ( t ) + i I ( t )

w h i c h he c a l l e d t h e p r e - e n v e l o p e o f t h e wave f o r m I ( t ) .

The f u n c t i o n I ( t ) r e p r e s e n t s t h e H i l h e r t t r a n s f o r m J B J o f l ( t ) , d e f i n e d h y

I ( t ) = P.v. ^ ^ d 5 ( 3 3 )

— co

W r i t i n g Z ( t ) i n t h e f o r m Z ( t ) = | Z( t ) | e^**^^* ^, t h e f u n c t i o n s Ca("t) and 0)^(1) a r e d e f i n e d by

C ( t ) = I Z ( t )I

( 3 ^ ) 0). ( t ) = 4» ( t )

These d e f i n i t i o n s a r e independent o f w . Since s i n (üj^t- i>.^) i s t h e H i l h e r t t r a n s f o r m o f cos ( u ^ t - (Jj^), "the p r e - envelope o f t h e r e p r e s e n t a t i o n ( 2 5 ) i s g i v e n h y : Z ( t ) = ? c^e ^ ^ ^ n t - <^n) ( 3 5 ) n=1 By w r i t i n g ( 3 5 ) i n t h e shape Z ( t ) = iWjjjt N L n=1 c e n i { m ( 3 6 )

i t i s easy t o see t h a t [3h) g i v e s t h e same e x p r e s s i o n s f o r C a ( t ) and 0)^(1) as t h e d e f i n i t i o n s o f R i c e . T h i s makes c l e a r t h a t , i n d e e d , t h e

f u n c t i o n s Ca^"t) and 01^(1) = tOj^j + 9 ( t ) as d e r i v e d f r o m t h e R i c e - d e f i n i t i o n s ( 3 0 ) a r e i n d e p e n d e n t o f t h e c h o i c e o f Wjj,.

/

J o i n t p r o b a b i l i t y d e n s i t y o f t h e i n s t a n t a n e o u s a m p l i t u d e and f r e q u e n c y . I t i s seen f r o m ( 3 0 ) t h a t t h e f u n c t i q n s I c ( " t ) and I g ( t ) a r e r e l a t e d t o C a ( t ) and e ( t ) by : I ( t ) = C ( t ) cos e ( t ) (3T) I ( t ) = t,At) s i n e ( t ) s a I n v i e u w o f o u r d i s c u s s i o n s w i t h r e s p e c t t o t h e r e p r e s e n t a t i o n ( 2 5 ) f o r I ( t ) i t i s o b v i o u s t h a t t h e ^ p r o c e s s e s 1 ^ ( t ) and I ^ ( t ) and a l s o t h e i r t i m e -d e r i v a t i v e s I ( t ) an-d 1 ^ ( t ) have a n o r m a l -d i s t r i b u t i o n s w i t h a c o n t i n u o u s power s p e c t r i m and a r e t h e r e f o r e e r g o d i c . C o n s e q u e n t l y j f o r some f i x e d v a l u e o f t , t h e j o i n t d i s t r i b u t i o n o f t h e v a r i a b l e s 1 ^ ( t ) , I s ( t ) , I g ( t ) and I c ( t ) , w h i c h we denote by x-] , x g , X3 and x l | i s n o r m a l i n f o u r d i m e n s i o n s , g i v e n b y :

h k

p ( x . , x_, x„, X,) = ( 2 . ) - 2 | M

1

-^/2

^ x p { - ^ Z

2 T

| 4

X ^ X

} ( 3 8 ; r=1 s=1 ' ' X } r s Where | M [ i s t h e d e t e r m i n a n t o f t h e m a t r i x M = [ p ^ g » i n w h i c h p i s t h e c o r r e l a t i o n c o e f f i c i e n t o f t h e v a r i a b l e s x^ and Xg. F u r t h e r , M r e p r e s e n t s t h e c o f a c t o r o f p^^g i n M. U s i n g t h e n o t a t i o n r s ( 2 ^ )

"ƒ

.(v) (• V J d V m E X X r s .39, t h e m a t r i x i s f o u n d t o be M

Then ( 3 8 ) can be reduced t o p ( x ^ , x ^ , x ^ , x ^ ) exp b - b. o 1 -b.

iko)

( b . x ? - b . x . x . ^ 2 "*"^24 ^ 2^l''3''H^^o4 2B 2 1 1 1 2 )'

J 9 .

i n w h i c h

By u s i n g ( 3 7 ) we r e p l a c e t h e v a r i a b l e s x , x , x and x, hy c; , ^ , 9 and T h i s y i e l d s : _ 1 2 3 4 a a

p ( ; ,C ,9,9) = exp (U2)

I n t e g r a t i n g w i t h r e s p e c t t o 6 and i^g^over t h e ranges (0.2TT) and (-«>,=«), r e s p e c t i v e l y , we o b t a i n :

P(C^,0,

( 2 T b B ) \

2

^ ' { ^ 2 - +b^ 9^ } ( i i 3 )

I t has been shown above t h a t t h e b e h a v i o u r o f t h e f u n c t i o n s C ( t ) and ü).(t) are i n d e p e n d e n t o f t h e c h o i c e o f Wm» C o n s e q u e n t l y by choosing^cOjj^ = 0 we"^ o b t a i n f r o m (!|3) t h e d e s i r e d j o i n t d e n s i t y : p ( c ^ , CO.: 2 (2irb B ) ' 1 exp 2B { b' - 2b; . b^.. } The v a l u e s f o r t h e c o e f f i c i e n t s b ' i n t h i s f o r m u l a a r e o b t a i n e d f r o m t h e , c o r r e s p o n d i n g e x p r e s s i o n s ( 3 9 ) f o r b by s u b s t i t u t i n g v = 0 . n m

Average niimber o f zeros p e r u n i t t i m e o f t h e f u n c t i o n I ( t ) .

We assume t h a t t h e random f u n c t i o n l ( " t ) has a zero v a l u e i n some p o i n t t + T i n t h e t i m e i n t e r v a l ( t , t + d t ) . We f u r t h e r assume t h a t t h e l e n g t h d t o f t h e i n t e r v a l i s so s m a l l , t h a t i n t h i s i n t e r v a l l ( t ) can be c o n s i d e r e d as l i n e a r w i t h p r o b a b i l i t y a l m o s t one. I n t h e p o i n t o f i n t e r s e c t i o n we have I ( t ) + I T = o (^5) where 0<T< d t (ll6) Combining {k3) and {k6) y i e l d s - I d t < I ( t ) < 0 when I > 0 (hi) 0 < l ( t ) < - I d t when I < 0 Assume t h e j o i n t d e n s i t y o f t h e v a r i a b l e s I and I t o be r e p r e s e n t e d b y p ( l , i ) . T h i s p r o c e s s i s i n d e p e n d e n t o f t s i n c e i t i s s t a t i o n a r y , . I t i s c l e a r f r o m t h e r e p r e s e n t a t i o n (25) t h a t t h d e n s i t y i s n o r m a l i n two d i m e n s i o n s , w h i c h i s r e a d i l y f o u n d t o be 2 ' 2 p ( l , I )

=

1 T - exp { - - - - , }

(HQ)

2 T ( b ^ b 2 ' ) ' ^ o 2 b 2 where bp i s o b t a i n e d f r o m b ^ b y s u b s t i t u t i n g v^^ = 0 i n ( 3 9 ) . Then t h e p r o b a b i l i t y t h a t l ( t ) has a z e r o i n ( t , t + d t ) i s g i v e n b y : ^ O 0 - I d t • • • ƒ d l ;^ d l p ( I , I ) + ƒ d l ƒ d l p ( l , I ) (1+9) 0 - I d t - 0 ° 0 S i n c e d t i s i n f i n i t e s i m a l s m a l l we may r e p l a c e (il9) by OO d t ƒ I p ( l = o, I ) d l (50)

The average number o f zeros p e r u n i t t i m e , w h i c h we denote by S , i s g i v e n by 00

6 = ƒ I I I p ( l = o , I ) d l (5T)

O M

1

References 1 . Jong, B de

"Some aspects o f s h i p motions i n i r r e g u l a r heam and l o n g i t u d i n a l waves" T h e s i s , T e c h n i s c h e Hogeschool D e l f t , I97O,

2. A r n d t , B und Roden, S.

S t a b i l i t a t b e i v o r - und a c h t e r l i c h e m Seegang S c h i f f s t e c h n i k Bd. 5 - 1 9 5 8 - H e f t 2 9

3. Grim, 0.

B e i t r a g zu dem Problem der S i c h e r h e i t des S c h i f f e s im Seegang W i s s e n s c h a f t l i c h e Z e i t s c h r i f t d e r U n i v e r s i t a t Rostock 1 0 . J a h r g a n g I 9 6 I . Grim, 0. R o l l s c h w i n g u n g e n , S t a b i l i t a t und S i c h e r h e i t im Seegang 2 6 9 . M i t t e i l u n g d e r Hamhurgischen S c h i f f b a u v e r s u c h a n s t a l t 5. Grim, 0.

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 d e r S c h i f f b a u t e c h n i s c h e n G e s e l l s c h a f t 1+5. Band 1 9 5 1 6. McFadden, J.A. The A x i s - C r o s s i n g I n t e r v a l s o f Random F u n c t i o n s I.R.E. T r a n s a c t i o n s on I n f o r m a t i o n T h e o r y , December I 9 5 6 T . McLachlan, N.W. Theory and A p p l i c a t i o n s o f M a t h i e u F u n c t i o n s O x f o r d a t t h e C l a r e n d o n Press 8. K e r w i n , J.E. Notes on R o l l i n g i n L o n g i t u d i n a l Waves

Lab. v o o r Scheepsbouwkunde, T e c h n i s c h e Hogeschool D e l f t June 1 9 5 5

9. G e r r i t s m a , J . and Beukelman, V/.

A n a l y s i s o f t h e m o d i f i e d s t r i p t h e o r y f o r t h e c a l c u l a t i o n o f s h i p m o t i o n s and wave b e n d i n g moments

R e p o r t no. 9 6 S, Nederlands S c h e e p s s t u d i e c e n t r i m TNO 1 0 . Cramer, H a r a l d and L e a d b e t t e r , M.R.

S t a t i o n a r y and R e l a t e d S t o c h a s t i c Processes W i l e y & Sons, I n c .

11 . R i c e , S.O.

M a t h e m a t i c a l a n a l y s i s o f random n o i s e

B e l l System T e c h n i c a l J o u r n a l , v o l s 23 and 2h 12. D u g u n d j i , J .

Envelopes and p r e - e n v e l o p e s o f r e a l wave forms I.R.E. T r a n s a c t i o n s on I n f o r m a t i o n Theory, I958 13. T i t c h m a r s h , E.C.

I n t r o d u c t i o n t o t h e Theory o f F o u r i e r I n t e g r a l s O x f o r d U n i v e r s i t y P r e s s , New York.

- 1 8 0

Fig,5. A m p l i t u d e - a n d phosecharacteristics for heave, M s, " S , A , von der S t e l ". Fn = -3325.

-Fig.10a. D i s t r i b u t i o n of the c o n t r i b u t i o n to P r ( T > T o ; Me) over t h e w a v e l e n g t h s .

( M e / Y V = -15 : GM = -8 ; F n = 3 3 2 5 : B e a u f o r t 6 )

1 H

Fig.6. Behaviour of a_and q as a f u n c t i o n of for various

w a v e l e n g h t s

( G M = 0 5 ;

Fn=

3 3 2 5 )

CO

Fig.9. Relation between

andU)

for To= 5,10 and 20.