Some notes on roll stabilization by means of free surface tanks

T E C H N I S C H E H O G E S C H O O L DELFT AFDELING DER MARITIEME TECHNIEK

L A B O R A T O R I U M V O O R S C H E E P S H Y D R O M E C H A N I C A Some n o t e s on r o l l s t a b i l i z a t i o n by means o f f r e e s u r f a c e t a n k s . I r . J . J . v a n d e n B o s c h R e p o r t n r : 1 2 3 - P L e c t u r e h e l d b e f o r e N o r g e s T e k n i s k e H^Jgskole, I n s t i t u t t f o r S k i p s b y g g i n g , T r o n d h e i m , Norway, 14-15 S e p t e m b e r '64

Delft University of Technology Ship Hydromechanics Laboratory Mekelweg 2

2628 CD D E L F T The Netherlands Phone 0 1 5 - 7 8 6 8 8 2

ROLL STABILIZATION Guest L e c t u r e s by I r . J . J . v a n den Bosch T r o n d h e i m September 14.- 15. 196A N O R G E S T E K N I S K E H 0 G S K O L E INSTBTUTT FOR S K I P S B Y G G I N G II TRONDHEIM — N.T.H.

SOME NOTES ON ROLL STABILIZATION BY MEANS OF FREE SURFACE TANKS.

By I r JoJ« v a n den Bosch<

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

S i n c e a l m o s t a , c e n t u r y t h e r e have been c o u n t l e s s e f f o r t s t o r e d u c e t h e r o l l i n g m o t i o n o f s h i p s a t sea. From t h e o u t s e t o f t h e steam e r a t h e need was f e l t t o

r e p l a c e t h e s t a b i l i z i n g a c t i o n o f t h e s a i l s by some o t h e r d e v i c e w h i c h w o u l d r e l i e v e t h e s h i p o f t h e e x c e s s i v e r o l l i n g m o t i o n .

The b i l g e k e e l was a d o p t e d and d i d a good d e a l t o r e a c h t h i s g o a l , b u t p r o v e d i n a d e q u a t e f o r v e r y s t i f f s h i p s w i t h l a r g e i n e r t i a . So o t h e r ways were t r i e d w i t h more o r l e s s s u c c e s s . D u r i n g t h e l a s t decades i n t e r e s t i n t h i s s u b j e c t has a g a i n

f l a r e d up and a r e m a r k a b l e p o i n t i s t h a t most o f t h e knowledge t h a t has been g a t h e r e d i n t h e p a s t seems t o be a t t h e d i s p o s a l o f o n l y a few p e r s o n s o r f i r m s . Whereas t h i s i s u n d e r s t a n d a b l e when i t c o n c e r n s i n t r i c a t e s e r v o mechanisms, i t

seems t o be a b i t odd t h a t a d e s i g n e r has t o f a l l back upon a c o n s u l t i n g o f f i c e f o r t h e d e s i g n o f a r e l a t i v e l y s i m p l e p a s s i v e t a n k system w h i c h w i l l g i v e s a t i s f a c t o r y r e s u l t s i n n i n e t y p e r c e n t o f t h e c a s e s .

The p u r p o s e o f t h i s l e c t u r e i s t o g i v e some i n s i g h t i n t h e g e n e r a l problem, o f t h e r o l l i n g and t o s t i m u l a t e t h e i n t e r e s t i n r o l l r e d u c t i o n as a p a r t o f t h e d e s i g n o f a s h i p . The methods used f o r t h e s t a b i l i a a t i o n o f s h i p s can be s e p a r a t e d i n t o :

A. p a s s i v e systems and B. a c t i v e systems.

W i t h t h e p a s s i v e system t h e s t a b i l i z i n g a c t i o n i s d e v e l o p e d w i t h o u t i n t e r m e d i a ^ o f any e x t e r n a l power s o u r c e . W i t h t h e a c t i v e s y s t e m , s e n s i n g e l e m e n t s a r e used t o d e t e c t t h e m o t i o n and t h i s i n p u t s i g n a l i s used t o c o n t r o l t h e mechanism w h i c h p r o d u c e s t h e s t a b i l i z i n g moments. W i t h a c t i v e systems a h i g h e r d e g r e e o f s t a b i l i -z a t i o n c a n be a t t a i n e d ; t h e drawbacks a r e t h e g r e a t e r c o s t s and m a i n t e n a n c e . The d e v i c e s w h i c h have been used c o n s i s t o f f o u r g e n e r a l t y p e s ;

a. t h e m o v i n g s o l i d w e i g h t ,

b. t h e u s e o f t h e g y r o s c o p i c e f f e c t , c. e x t e r n a l f i n s ,

A t t h e end o f t h e 1 9 t h c e n t u r y t h e r e have been s e v e r a l a t t e m p s t o use t h e s o l i d w e i g h t s y s t e m , b u t p r o b a b l y because t h e s c i e n c e o f s e r v o mechanisms was n o t so w e l l u n d e r s t o o d and most systems were u n d e r d i m e n s i o n e d , none o f t h e s e e x p e r i m e n t s p r o v e d v e r y s u c c e s s f u l . The f i r s t a p p l i c a t i o n o f t h e g y r o s c o p i c e f f e c t came f r o m S c h l i k i n 1907. T h i s was a p a s s i v e s y s t e m w i t h a v e r y l a r g e w e i g h t and poor c o n t r o l o f t h e damping o f t h e s y s t e m i t s e l f .

Some y e a r s l a t e r S p e r r y d e s i g n e d t h e a c t i v a t e d c o u n t e r p a r t o f t h i s system. The s e n s i n g e l e m e n t c o n s i s t e d o f a s r a a l l s e n s i t i v e g y r o w h i c h c o n t r o l l e d by means o f an e l e c t r i c m o t o r t h e l a r g e m a i n g y r o s c o p e w h i c h d e v e l o p e d t h e s t a b i l i z i n g moment.

The o l d e s t known and most used d e v i c e i s t h e p a s s i v e e x t e r n a l f i n : t h e b i l g e k e e l . B i l g e k e e l s have a c o n s i d e r a b l e e f f e c t on t h e r o l l i n g a m p l i t u d e s o f a s h i p and I s h o u l d n o t a d v o c a t e t h e o m i t t a n c e o f b i l g e k e e l s when a n o t h e r d e v i c e i s i n s t a l l e d , a l t h o u g h i n some i n s t a n c e s t h e use o f b i l g e k e e l s i s p r o h i b i t e d by o t h e r demands ( i n t h e case o f i c e b r e a k e r s f o r e x a m p l e ) . M o v i n g f i n s a r e e n j o y i n g much i n t e r e s t d u r i n g t h e l a s t d e c e n n i a and r i g h t l y so because f o r f a s t s h i p s m o v i n g f i n s a r e u n e x c e l l e d as t h e s t a b i l i z i n g moments i n c r e a s e w i t h t h e s q u a r e o f t h e speed. W i t h low speeds, however, and many s h i p s t r a v e l w i t h

speeds b e l o w 15 k n o t s , t h e e f f e c t o f t h e f i n s d i m i n i s h r a p i d l y . A n o t h e r o b j e c t i o n i s t h e v u l n e r a b i l i t y .

R o l l s t a b i l i z a t i o n by means o f w a t e r t r a n s f e r was f i r s t m e n t i o n e d by Wa^ts i n 1883. A l t h o u g h h i s s y s t e m , t h e f r e e s u r f a c e t a n k , showed some p r o m i s e , i n t e r e s t d i e d away t i l l i n a b o u t 1960 a somewhat a l t e r e d s y s t e m o f f r e e s u r f a c e t a n k t u r n e d up, t h e s o - c a l l e d f l u m e t a n k c o v e r e d w i t h numerous A m e r i c a n p a t e n t s .

Q u i t e a n o t h e r t y p e o f a n t i - r o l l i n g t a n k was i n v e n t e d by Frahm, who used t h e p r i n c i p l e o f t h e U-tube. A b o u t 18 y e a r s l a t e r M i n o r s k y c o n c e i v e d t h e a c t i v e

c o u n t e r p a r t . The U-tube and i t s m o d i f i c a t i o n s p r o v e d s u c c e s s f u l i n many c a s e s , b u t appeared t o g i v e t r o u b l e s i n o t h e r cases under s p e c i a l c i r c u n s t a n c e s .

B e f o r e e n t e r i n g i n t o t h e s p e c i f i c b e h a v i o u r o f t h e a n t i r o l l i n g t a n k s , I w i l l e n l a r g e somewhat upon t h e d e s c r i p t i o n o f t h e r o l l i n g m o t i o n o f t h e s h i p a t sea and t o t h a t p u r p o s e we a r e l o o k i n g a t :

The s h i p as a l i n e a r damped m a s s - s p r i n g system.

Assume i n t h e f i r s t i n s t a n c e t h e s h i p i n s t i l l w a t e r . When we g i v e t h i s s h i p a l i s t and s e t i t f r e e , i t w i l l p e r f o r m o s c i l l a t i n g m o t i o n s w i t h g r a d u a l l y

d e c r e a s i n g amplitude» We o n l y c o n s i d e r t h e r o l l i n g m o t i o n , w h i c h can be d e s c r i b e d m a t h e m a t i c a l l y as:

-3-^ i s t h e r o l l i n g a n g l e , (j) i t s f i r s t t i m e d e r i v a t i v e , t h e a n g u l a r v e l o c i t y , and (Jl* t h e r o l l i n g a c c e l e r a t i o n . I i s t h e mass moment o f i n e r t i a w i t h r e s p e c t t o t h e f o r e and a f t a x i s t h r o u g h t h e c e n t e r o f g r a v i t y . T h i s moment o f i n e r t i a c o m p r i s e s n o t o n l y t h a t o f t h e s h i p p r o p e r , b u t a l s o a component due t o t h e w a t e r w h i c h has been s e t i n m o t i o n .

N i s a damping c o e f f i c i e n t . The damping r e s u l t s f r o m wave m a k i n g and v i s c o u s f o r c e s . R i s a s p r i n g c o n s t a n t , t h e moment needed t o g i v e t h e s h i p a l i s t o f one r a d i a n , so R = GM A • GM i s m e t a c e n t r i c h e i g h t . A i s d i s p l a c e m e n t . The s o l u t i o n o f t h e e q u a t i o n , w i t h t h e c o n d i t i o n s t h a t a t t = 0 ^ ^ and 41 = 0 i s f o u n d by t h e s u b s t i t u t i o n o f ({> = e*^ , where a i s a complex number. The r e s u l t i s ! 21 N ij) = $ e ( c o s OJ t + r r — s i n u t ) , H S Z X OJ^ S w h e r e i n : T h i s i s t h e e q u a t i o n o f an o s c i l l a t o r y m o t i o n w i t h d e c r e a s i n g a m p l i t u d e . The p e r i o d o f o s c i l l a t i o n i s : •J, = 2n . 2 j t w h i c h i s c a l l e d t h e n a t u r a l p e r i o d o f t h e r o l l i n g m o t i o n . As N i s m o s t l y v e r y s m a l l i n r e l a t i o n t o R and I , t h e t e r m can be n e g l e c t e d and so i t can be s t a t e d t h a t f o r p r a c t i c a l p u r p o s e s t h e n a t u r a l p e r i o d i s i n d e p e n d e n t o f t h e damping. W i t h :

R = GM • A and

I = ^ k ,

where g i s t h e a c c e l e r a t i o n o f g r a v i t y and k iö r a d i u s o f i n e r t i a f o r r o l l . The e x p r e s s i o n f o r t h e n a t u r a l p e r i o d becomes!

T = 2 n k

\'

F o r o r d i n a r y c a r g o - s h i p s k = 0,35 - 0,41 B. The r e s u l t i n g m o t i o n i s s k e t c h e d i n f i g u r e 1. The r a t i o o f two v a l u e s o f 41, say dij^ and .^^ , s e p a r a t e d by a t i m e i n t e r v a l e q u a l t o t h e n a t u r a l p e r i o d T , amounts tos

0

11 >2

e 21 T 3

T h i s r a t i o i s a measure f o r t h e damping o f t h e system.

L e t us suppose nov t h a t t h e s h i p i s s u b j e c t t o a s i n u s o i d a l v a r y i n g r o l l i n g moment: M = M s i n (tot + e ) . w wa w^ The d i f f e r e n t i a l e q u a t i o n o f m o t i o n becomes: + N4. + R* = M s i n (ojt + € ) . ^ ^ wa w The s o l u t i o n o f t h i s e q u a t i o n c o n s i s t s o f a t r a n s i e n t p a r t s i m i l a r t o t h e s o l u t i o n o f t h e f r e e m o t i o n and a p a r t i c u l a r s o l u t i o n w h i c h we assume t o be o f a s i n u s i o d a l f o r m . A f t e r some t i m e t h e t r a n s i e n t m o t i o n has d i e d b u t and t h e s t e a d y s t a t e s o l u t i o n , i n w h i c h we a r e i n t e r e s t e d r ^ j a a i n s . From t h i s s o l u t i o n t h e a m p l i t u d e and t h e phase o f t h e m o t i o n a r e deduced:

M wa . T 2,2 ^ ^2 2' Toj ) + N w w a r c t g N to R - I t o When to = 0 i . e . t h e s t a t i c c o n d i t i o n , t h e a m p l i t u d e becomes e q u a l t o t h e h e e l i n g a n g l e o f t h e s h i p s u b j e c t e d t o t h e s t a t i c h e e l i n g moment M^^ : M " s t wa R The r a t i o : £ = "st R T 2 2 — Z (R - I t o ) + to i s c a l l e d t h e dynamic m a g n i f i c a t i o n f a c t o r .

.5-I f we i n t r o d u c e : 2 R and t h e f r e q u e n c y r a t i o s ^ = A we f i n d : and: f = - A ) + V A V A € = a r c t g -2 " A - L

I n f i g u r e 2 t h e m a g n i f i c a t i o n f a c t o r and t h e phase a n g l e a r e shown as a f u n c t i o n o f A f o r d i f f e r e n t v a l u e s o f v . F o r o r d i n a r y s h i p s v i s a p p r o x i m a t e l y 0 , 1 . The

s h a r p n e s s o f t h e peaks f o r t h i s v a l u e a t A = 1 t s i m p o r t a n t . F o r A = 1 i t

1 o f o l l o w s t h a t f = ~ and t h e phase a n g l e becomes 90 .

Because o f t h e l a r g e m a g n i f i c a t i o n due t o t h e s m a l l damping i n r o l l , r o l l a m p l i t u d e s w i l l be l a r g e when t h e s h i p i s e x c i t e d a t i t a own n a t u r a l f r e q u e n c y . I t i s more p r a c t i c a l t o m o d i f y t h e f o r m e r p r e s e n t a t i o n t o a s l i g h t e x t e n t .

Suppose t h e s h i p i s h e l d f i x e d i n a r e g u l a r l o n g c r e s t e d sea w i t h t h e waves coming i n o v e r t h e beam. The wave l e n g t h i s assumed l a r g e i n r e l a t i o n t o t h e s h i p s beam and d r a u g h t . The s h i p i s s u b j e c t f e d t o e x c i t i n g moments w h i c h a r e a p p r o x i m a t e l y p r o p o r t i o n a l t o t h e wave s l o p e and t o t h e s t i f f n e s s o f t h e s h i p . M = a R = R wa a A = wave s l o p e a m p l i t u d e . r = wave a m p l i t u d e . ^ = wave l e n g t h .

The r o l l a m p l i t u d e i s now p r e s e n t e d as i t s r a t i o t o t h e wave s l o p e a m p l i t u d e :

^ • R

f

Because o f t h e assumed p r o p o r t l n n a l i t y o f t h e s t i f f n e s s and t h e e x c i t i n g moment i n waves t h e v a l u e o f becomes e q u a l t o t h e wave s l o p e .

L e t us assume two s h i p s , s i m i l a r i n f o r m , d i s p l a c e m e n t and r a d i u s o f g y r a t i o n , b u t d i f f e r i n g i n t h e m e t a c e n t r i c h e i g t h GM. Suppose t h a t s h i p A i s two t i m e s as s t i f f as s h i p B. Then t h e r a t i o o f t h e n a t u r a l f r e q u e n c i e s o f t h e two s h i p s i s s

And t h e r a t i o between t h e non d i m e n s i o n a l damping c o e f f i c i e n t s becomes;

A t t h e r e s p e c t i v e n a t u r a l f r e q u e n c i e s w i t h A = 1 t h e r a t i o o f t h e m a g n i -^A / f i c a t i o n f a c t o r becomes — = /2 . So t h e s t i f f e r s h i p A has b o t h a l a r g e r ^B \' m a g n i f i c a t i o n f a c t o r and a l a r g e r v a l u e f o r t h e n a t u r a l f r e q u e n c y . The a n g u l a r a c c e l e r a t i o n becomes , as a r e s u l t o f t h e l a r g e r s t i f f n e s s o f t h e s h i p A^ 2,8 as l a r g e as t h a t o f t h e e a s i e r s h i p B. T h i s i s m e n t i o n e d t o emphasize t h e i m p o r t a n c e o f r o l l s t a b i l i z a t i o n f o r s h i p s w i t h r e l a t i v e l y l a r g e m e t a c e n t r i c h e i g h t s . Large m e t a c e n t r i c h e i g h t s a r e a

f e a t u r e o f many s m a l l s h i p s w i t h low i f r e e b o a r d , where t h e s e l a r g e v a l u e s a r e needed on b e h a l f o f t h e r a n g e óf s t a t i c s t a b i l i t y . Examples a r e t r a w l e r s , t u g s and

s u p p l y - v e s s e l s f o r o f f - s h o r e d r i l l i n g i n s t a l l a t i o n s . Work on deck o f t h e s e v e s s e l s can be e x c e p t i o n a l l y t i r i n g and a t some t i m e s even d a n g e r o u s .

On c a r g o - s h i p s l a r g e a c c e l e r a t i o n s t e n d t o p u t h i g h l o a d s on t h e s h i p s t r a n s v e r s e s t r e n g t h members and sometimes i t p r o v e s .rtécessary t o t a k e s p e c i a l measures t o p r o t e c t t h e c a r g o .

I n many d e s i g n s o f f a s t c a r g o - s h i p s t h e f e a r f o r e x c e s s i v e s t i f f n e s s s e t s a l i m i t t o t h e beam. An i n c r e a s e o f t h e beam, however, w o u l d have made i t p o s s i b l e t o d e c r e a s e t h e power as a s m a l l e r b l o c k c o e f f i c i e n t c o u l d have been a d o p t e d .

The e q u a t i o n s w h i c h have been shown, d e s c r i b e i n a v e r y s i m p l i f i e d way t h e r o l l i n g m o t i o n o f a s h i p i n a r e g u l a r beam sea. B u t r e a l i s t i c seas a r e n o t r e g u l a r and so we have t o c o n s i d e r f u r t h e r methods t o p r e s e n t a more g e n e r a l t r e a t m e n t o f t h e r o l l i n g m o t i o n , d e p e n d i n g on t h e p r e s e n t a t i o n o f t h e i r r e g u l a r seaway.

-7-We assume an i r r e g u l a r l o n g - c r e s t e d s e a , i . e . a sea where t h e waves corae m a i n l y f r o m one d i r e c t i o n . Such a sea can be t h o u g h t t o be composed o f an

i n f i n i t e ( v e r y l a r g e ) number o f r e g u l a r waves w i t h d i f f e r e n t f r e q u e n c i e s , v e r y s m a l l h e i g h t and w i t h a random phase. F o r e v e r y wave component t h e e n e r g y p e r u n i t area i s :

p i s mass d e n s i t y . The e n e r g y i s i n d e p e n d e n t o f t h e wave p e r i o d . We can w r i t e t h e e q u a t i o n o f t h e s u r f a c e e l e v a t i o n o f one component a t a g i v e n l o c a t i o n : i q " ^ q The t o t a l e l e v a t i o n i s g i v e n by: n 1 The mean s q u a r e d v a l u e o v e r a l o n g t i m e i s : T O w h i c h e q u a l s : 2 2 o r 2 pg

We now assume t h e e x i s t a n c e o f a f u n c t i o n S ( u ) d e f i n e d by: w + d(jo

S(w) i s c a l l e d t h e energy s p e c t r u m . I n f i g u r e 3 t h e g e n e r a l shape o f a seawave s p e c t r u m i s shown.

From t h e s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n o f t h e m o t i o n s i t f o l l o w s t h a t f o r any component q t h e m o t i o n a m p l i t u d e i s p r o p o r t i o n a l t o t h e a m p l i t u d e o f t h e wave i n p u t : say

f = cos (toqt + e^)

The f u n c t i o n Y i s c a l l e d t h e r e s p o n s e a m p l i t u d e o p e r a t o r , € i s t h e phase q q a n g l e w i t h r e g a r d t o t h e wave, Y^ and d e f i n e t h e f r e q u e n c y r e s p o n s e o f t h e r o l l i n g m o t i o n . The t o t a l r e s p o n s e o f t h e s h i p i s f o u n d by s u p e r p o s i t i o n o f t h e s e p a r a t e r e s p o n s e s t o t h e wave components: n Y * r cos (o) t + e ) q q q q 1

The mean s q u a r e d v a l u e o f t h e r o l l a n g l e amounts t o : n

2 q q r

The s p e c t r u m o f t h e m o t i o n i s d e f i n e d by:

OJ

The e x c i t i n g moment f o r r o l l i s d i r e c t l y p r o p o r t i o n a l t o t h e wave slope' and n o t t o t h e wave h e i g h t as a l s o t h e wave l e n g t h p l a y s a r o l e . T h i s o b s c u r e s t h e d i r e c t r e l a t i o n o f t h e r e s p o n s e a m p l i t u d e o p e r a t o r t o t h e a m p l i t u d e m a g n i f i c a t i o n f a c t o r . F o r t h a t r e a s o n i t i s b e t t e r t o t r a n s f o r m t h e wave h e i g h t s p e c t r u m t o a wave s l o p e s p e c t r u m a c c o r d i n g t o t h e f o l l o w i n g r e l a t i o n : 2 0J_ r . 2

The wave s l o p e s p e c t r u m can be d e f i n e d a s :

(j+dcj cj+d(jo ^ 2

The f r e q u e n c y r e s p o n s e has t o be b r o u g h t i n a c c o r d a n c e w i t h t h i s n e w l y d e f i n e d s p e c t r u m . The new a m p l i t u d e r e s p o n s e o p e r a t o r shows a c l o s e r r e s e m b l a n c e t o t h e a m p l i t u d e m a g n i f i c a t i o n f a c t o r o f t h e l i n e a r m a s s - s p r i n g system. I n f i g u r e 4 t h e d e t e r m i n a t i o n o f t h e r o l l s p e c t r u m w i t h t h e a i d o f t h e wave s l o p e s p e t t r u m and the c o r r e s p o n d i n g f r e q u e n c y r e s p o n s e f u n c t i o n i s shown s c h e m a t i c a l l y .

-9-Because o f t h e r e l a t i v e f l a t n e s s o f t h e wave s l o p e s p e c t r u m and t h e s h a r p n e s s o f t h e a m p l i t u d e o p e r a t o r , t h e r o l l s p e c t r u m i s a l s o s h a r p , w h i c h means t h a t l a r g e r o l l a m p l i t u d e s a r e always a s s o c i a t e d w i t h f r e q u e n c i e s i n t h e n e i g h b o u r h o o d o f t h e n a t u r a l r o l l f r e q u e n c y o f t h e s h i p .

Because o f t h e t r a n s f o r m a t i o n o f t h e wave s p e c t r u m f r o m wave h e i g h t t o wave t l o p e , t h e t o p o f t h e wave s p e c t r u m has s h i f t e d t o much l a r g e r v a l u e s o f w. T h i s

means t o s m a l l e r wave l e n g t h s . Here a g a i n i t i s e v i d e n t t h a t as r e g a r d s r o l l i n g t h e c o n d i t i o n s a r e most u n f a v o u r a b l e f o r s m a l l s h i p s . Resuming we can s t a t e t h a t ? a) N a t u r a l damping o f t h e r o l l i n g m o t i o n i s s m a l l , r e s u l t i n g i n l a r g e a m p l i t u d e s under c e r t a i n c i r c u m s t a n c e s . b) Large r o l l a m p l i t u d e s a r e a s s o c i a t e d w i t h f r e q u e n c i e s i n t h e n e i g h b o u r h o o d o f t h e n a t u r a l f r e q u e n c y o f t h e s h i p . c) S m a l l and s t i f f v e s s e l s w o u l d b e n e f i t most f r o m r o l l s t a b i l i = z a t i o n . I n t h e p r e v i o u s p a r t o f t h i s l e c t u r e I have t r i e d t o make c l e a r t h a t s m a l l s t i f f v e s s e l s w o u l d b e n e f i t most f r o m an e f f e c t i v e means o f r o l l s t a b i l i z a t i o n . The consequence o f t h i s i s t h a t t h e s t a b i l i z i n g d e v i c e must be r e l a t i v e l y cheap t o produce and m a i n t a i n . I t s b e h a v i o u r i n use s h o u l d n o t be c r i t i c s 1 so t h a t a l t e r i n g c o n d i t i o n s s h o u l d n o t a f f e c t t o a l a r g e degree t h e a c t i o n o f t h e d e v i c e . The use s h o u l d n e v e r be dangerous w h a t e v e r t h e e n v i r o n m e n t a l c o n d i t i o n s w o u l d be. The s t a b i l i z i n g d e v i c e s h o u l d be e f f e c t i v e w i t h low o r z«ro speed. To o u r knowledge t h e f r e e s u r f a c e t a n k f u l f i l s b e s t t h e above c o n d i t i o n s .

The f i r s t t i m e t h a t t h i s t y p e o f t a n k i s m e n t i o n e d i n littérature i s i n 1883 when P. W a t t s d e s c r i b e s t h e i n s t a l l a t i o n o f such a t a n k on b o a r d t h e armoured s h i p I n f l e x i b l e . D u r i n g t h e d e s i g n o f t h i s v e s s e l about 1875 i t was f o u n d n e c e s s a r y t o a c c e p t a g r e a t m e t a c e n t r i c h e i g h t i n t h e n o r m a l c o n d i t i o n and t h e t h o u g h t a r o s e i f i t would n o t be p o s s i b l e t o have some d e v i c e w h i c h w o u l d r e d u c e t h e t e n d e n c y t o r o l l . From t h e s e c o n s i d e r a t i o n s t h e "waterchamber" as i t was c a l l e d , was b o r n , a r e c t a n g u l a r compartment e x t e n d i n g o v e r t h e f u l l b r e a d t h o f t h e s h i p and f i l l e d w i t h w a t e r t o a c e r t a i n s p e c i f i c l e v e l . T h i s d e v i c e o f w h i c h t h e e f f e c t was a s c e r

-t a i n e d i n model d i d n o -t show -t o f u l l advan-tage i n p r a c -t i c e , becaose o f -t h e -two chambers, o r i g i n a l l y i n t e n d e d f o r r o l l r e d u c t i o n , t h e one w h i c h was by f a r t h e l a r g e s t was a p p r o p r i a t e d f o r s t o w a g e , as t h e I n f l e x i b l e was a n a v a l v e s s e l and r o o m was s c a r c e .

W i t h t h e r e m a i n i n g s m a l l e r t a n k s e v e r a l t e s t s were i n t e n d e d j b u t t h e I n f l e x i b l b e i n g a n a v a l v e s s e l was r e q u i r e d f o r s e r v i c e a t t h e bombardment o f A l e x a n d r i a and so t h e most i m p o r t a n t t e s t s were never c a r r i e d o u t .

A l t h o u g h t h e t a n k showed some promise t h e d i s c u s s i o n f o l l o w i n g t h e r e a d i n g o f t h e paper, r e v e a l e d an a l m o s t g e n e r a l m i s g i v i n g a b o u t t h e i d e a o f a f r e e f l u i d s u r f a c e i n a p a r t o f t h e s h i p . To d e m o n s t r a t e t h i s , I q u o t e ; Mr. W. J o h n ;

"We a l l know t h a t f r e e w a t e r i n a s h i p does r e d u c e t h e ^ s t a t i c a l s t a b i l i t y . We a l s o know t h a t f r e e w a t e r i n a s h i p i s a dangerous t h i n g , u n l e s s i t i s w e l l under c o n t r o l i t i s l i k e d e a l i n g w i t h f i r e or gunpowder o r any o t h e r dangerous t h i n g , "

S i r Edward Reed:

"... b u t i t does appear t o me t h a t t h e paper does nöt t o u c h , s u f f i c i e n t l y i n any r a t e , t h e dangerous e l e m e n t s i n v o l v e d i n t h e use o f f r e e w a t e r i n t h e s h i p , "

Mr. J , D ' A g u i l a r Samuda;

"We have c o n s i d e r e d and have f o u n d t h e g r e a t e s t p o s s i b l e m i s c h i e f

a r i s e f r o m h a v i n g w a t e r f r e e i n t a n k s when used as b a l l a s t and I have v e r y g r e a t d i f f i c u l t y i n s p e a k i n g a t a l l e x c e p t t o say t h i s

I f a i l t o see f r o m t h e d e s c r i p t i o n g i v e n tc u s , t h e good e f f e c t w h i c h i t i s d e s i r e d t o i m p r e s s upon us w o u l d r e s u l t f r o m c h a n g i n g

t h e o l d s t y l e and a d o p t i n g t h e f r e e w a t e r i n s t e a d o f t h e c o n f i n e d w a t e r . "

I have e n l a r g e d somewhat on t h i s p i e c e o f h i s t o r y because as f a r as I can t r a c e one o f t h e m a i n r e a s o n s f o r t h e d y i n g o u t o f t h e i n t e r e s t i s t h i s a t t i t u d e and even nowadays many s h i p s o f f i c e r s d i s t r u s t any a n t i - r o l l i n g t a n k .

I n 1885 W a t t s r e a d a second paper on t h e same s u b j e c t b e f o r e t h e I.N.A. i n w h i c h he r e p o r t e d t h e r e s u l t s o f e x p e r i m e n t s c a r r i e d o u t w i t h s i m i l a r t a n k s

a b o a r d t h e E d i n b u r g h , b o t h i n model and on f u l l s c a l e . The r e s u l t s showed c l e a r l y t h e g r e a t s t e a d y i n g power o f t h e waterchambers and a l s o a r e m a r k a b l e agreement between t h e f u l l s c a l e and model v a l u e s .

The m e e t i n g r e c e i v e d t h e paper w i t h much more b e n e v o l e n c e and a p p r e c i a t i o n t h a n t h e p r e v i o u s one. N e v e r t h e l e s s t h e r e l u c t a n c e t o a d o p t t h e a c t i o n o f f r e e w a t e r as a means o f s t a b i l i z a t i o n , was c l e a r l y n o t i c e a b l e .

1 1

-Some t i m e ago t h e D e l f t S h i p b u i l d i n g L a b o r a t o r y t o o k up t h e r e s e a r c h o f r o l l s t a b i l i z a t i o n by means o f t a n k s .

The work w h i c h i s done by t h e sea i n r o l l i n g t h e s h i p hgs t o be a b s o r b e d as much as p o s s i b l e by t h e s t a b i l i z e r . The w o r k done by a h a r m o n i c v a r y i n g moment a c t i n g on a body s u b j e c t e d t o a h a r m o n i c r o t a t i o n o f t h e same f r e q u e n c y i s done o n l y by t h a t component o f t h e moment w h i c h i s i n phase w i t h t h e v e l o c i t y . From t h i s we c o n c l u d e t h a t t h e s t a b i l i z i n g moment s h o u l d l a g t h e m o t i o n by a phase a n g l e o f 90 d e g r e e s .

As l a r g e r o l l a n g l e s a r e a s s o c i a t e d w i t h t h e n a t u r a l f r e q u e n c y o f t h e s h i p o r r a t h e r w i t h a band o f f r e q u e n c i e s i n . t h i s neighbourhoods, t h e n a t u r a l f r e q u e n c y o f t h e s t a b i l i z e r s h o u l d i n t h e f i r s t i n s t a n c e be e q u a l t o t h e n a t u r a l f r e q u e n c y o f t h e s h i p . The s t a b i l i z i n g moment s h o u l d be i n c o u n t e r p h a s e t o t h e wave moemant.

The e x p e r i m e n t a l i n s t a l l a t i o n as s h o r n i n f i g u r e 5 has been assembled w i t h a v i e w t o t h e above m e n t i o n e d c o n s i d e r a t i o n s . An o s c i l l a t o r w h i c h produces a s i n u s -o i d a l r -o t a t i -o n i s d r i v e n by a f r e q u e n c y c -o n t r -o l l e d e l e c t r -o m -o t -o r . The m-odel t a n k i s mounted on a c r a d l e w h i c h i s suspended f r o m t h e a x i s o f r o t a t i o n i n such a way t h a t i n t h e p o i n t o f s u s p e n s i o n o n l y t h e f o r c e s p e r p e n d i c u l a r t o t h e a x i s a r e a b s o r b e d , t h e moment a r o u n d t h a t a x i s i s t a k e n up by a s t r a i n - g a g e dynamometer.

By an e l e c t r o n i c m e a s u r i n g s y s t e m t h e components o f t h e moment i n - p h a s e and 90 degrees o u t - o f phase a r e d e t e r m i n e d . From t h e s e two d a t a t h e a m p l i t u d e and t h e phase a n g l e o f t h e moment a r e c a l c u l a t e d . A m p l i t u d e and f r e q u e n c y o f t h e m o t i o n , t h e w a t e r l e v e l i n t h e t a n k and t h e p o s i t i o n o f t h e t a n k w i t h r e g a r d t o t h e r o l l i n g a x i s , can be v a r i e d . A r e c t a n g u l a r t a n k o f t h e f o l l o w i n g - d i m e n s i o n s was o s c i l l a t e d s l e n g t h I (measured f o r w a r d and a f t ) 150 mm b r e a d t h b (measured a c r o s s t h e s h i p ) 983 mm I n most o f t h e e x p e r i m e n t s t h e d i s t a n c e s f r o m t h e b o t t o m o f t h e t a n k t o t h e a x i s o f r o t a t i o n amounted t o 200 mm, t h i s d i s t a n c e was v a r i e d i n a l a t e r s t a g e o f t h e i n v e s t i g a t i o n .

When t h e o s c i l l a t o r was s e t i n a c t i o n f o r t h e f i r s t t i m e i t became c l e a r t h a t t h e b a s i c p h y s i c a l phenomenon r e s p o n s i b l e f o r t h e a t a b l l i ^ i n g e f f e c t i s t h e appearance o f a b o r e : a h y d r a u l i c jump t r a v e l l i n g t o and f r o a l o n g t h e b r e a d t h o f t h e t a n k . Because o f t h e " s t e p " i n t h e w a t e r s u r f a c e , t h e w a t e r was p i l e d up a l t e r n a t e l y on b o t h s i d e s w i t h such a phase l a g t h a t a moment c o u n t e r a c t i n g t h e m o t i o n was c r e a t e d .

The f u n d a m e n t a l n a t u r a l p e r i o d o f t h e h y d r a u l i c jump can be c a l c u l a t e d f o r s m a l l a m p l i t u d e s , as t h e v e l o c i t y o f t h e b o r e depends o n l y on t h e w a t e r d e p t h as a f i r s t a p p r o K i m a t i o n j '

c = ^ g h and: w h i c h r e s u l t s i n s as t h e n a t u r a l c i r c u l a r f r e q u e n c y . I n t h e n e x t s e r i e s o f f i g u r e s t h e b o r e i s shown i n f o u r c o n s e c u t i v e s t a g e s . The f r e q u e n c y o f t h e m o t i o n i s n e a r l y ei^ual t o t h e n a t u r a l f r e q u e n c y . I n f i g u r e 6 a t h e h y d r a u l i c jump i s shown w h i c h has j u s t r e v e r s e d i t s m o t i o n ; t h e t a n k i s a p p r o x i m a t e l y i n i t s extreme p o s i t i o n and t h e w a t e r i s s t i l l r u n n i n g d o w n h i l l t o f e e d t h e b o r e . I n t h e n e x t f i g u r e s , 6b, 6c, 6d t h e b o r e i s t o be seen on i t s way t o t h e o t h e r s i d e óf t h e t a n k , t h e h e i g h t has decreased somewhat on a c c o u n t o f t h e c o n t i n u i t y c o n d i t i o n . When t h e t a n k ( i . e , t h e s h i p ) i s r i g h t i n g i t s e l f i t has t o l i f t t h e w a t e r on t h e same t i m e t h a t t h e s h i p s t a b i l i t y t r i e s t o r i g h t e n t h e s h i p . So t h e w a t e r m o t i o n i n t h e t a n k c r e a t e s a c o u n t e r a c t i n g moment. I n t h e neKt s e r i e s o f f i g u r e s t h e p h y s i c a l phenomena o v e r a l a r g e r a n g e o f f r e q u e n c i e s a r e shown. I n t h e f i r s t f i g u r e ( 7 a ) , w h i c h d e m o n s t r a t e s t h e s i t u a t i o n f o r v e r y low f r e q u e n c i e s i t i s c l e a r t h a t t h e w a t e r s u r f a c e r e m a i n s h o r i z o n t a l . T h e r e i s no damping component, t h e o n l y e f f e c t i s t h e d e c r e a s e o f t h e GM v a l u e as a r e a u l t o f t h e f r e e s u r f a c e . I n t h e n e x t f i g u r e (7b) , r e p r e s e n t i n g t h e s i t u a t i o n f o r somewhat h i g h e r , b u t s t i l l low v a l u e s o f co , a s m a l l wave t r a i n appears as a

f o r e b o d e o f t h e b o r e . The t h i r d f i g u r e ( 7 c ) shows t h e h y d r a u l i c jump c l e a r l y . As soon as t h e b o r e a p p e a r e s , t h e q u a d r a t u r e component o f t h e moment becomes

i m p o r t a n t . T h i s happens a t a f r e q u e n c y much l o w e r t h a n t h e n a t u r a l f r e q u e n c y , a l t h o u g h t h i s depends on t h e a m p l i t u d e o f t h e m o t i o n . The n e x t f i g u r e ( 7 d )

shows t h e b o r e w i t h a f r e q u e n c y h i g h e r t h a n t h e n a t u r a l f r e q u e n c y . The q u a d r a t u r e component i s s t i l l c o n s i d e r a b l e . Then ( f i g u r e 7e) a t a r e l a t i v e l y s h a r p l y d e f i n e d p e r i o d t h e b o r e t r a n s f o r m s i n t o a s i n g l e s t e e p c r e s t e d wave -which i s h u r l e d f r o m s i d e t o s i d e j t h e damping moment d r o p s f a s t w i t h t h e i n c r e a s e o f t h e f r e q u e n c y . I n t h e l a s t f i g u r e o f t h i s s e r i e s ( 7 f ) t h e " f r o z e n s t a t e " o f t h e f l u i d i s shown; the l i q u i d a c t s as a s o l i d raass because t h e r e s u l t a n t o f t h e a c c e l e r a t i o n o f g r a v i t y and t h e c e n t r i f u g a l a c c e l e r a t i o n i n t e r s e c t s t h e a x i s o f r o t a t i o n .

The raain r e s u l t s o f t h e s e r i e s o f t e s t s a r e p r e s e n t e d i n a s e r i e s o f f i g u r e s * The m o t i o n i s d e t e r r a i n e d by 5 v a r i a b l e s when a t a n k o f u n i t l e n g t h i s c o n s i d e r e d :

-13-t h e a m p l i -13-t u d e 4» t h e f r e q u e n c y u t h e t a n k b r e a d t h b t h e d i s t a n c e t o t h e a x i s o f r o t a t i o n s t h e w a t e r d e p t h h .

I n f i g u r e 8 an example o f t h e measured i n - p h a s e and q u a d r a t u r e components i s

g i v e n . The a m p l i t u d e o f o s c i l l a t i o n i n t h i s case was <t>^ = 0,10 r a d i a n s (5,7 d e g r e e s ) . A m p l i t u d e and phase o f t h e moment c a l c u l a t e d f r o m t h e s e measurements a r e sho\m i t

f i g u r e 9. The e x p e r i m e n t a l p o i n t o f r e s o n a n c e i . e . t h e f r e q u e n c y f o r w h i c h t h e phase a n g l e e becomes 90 degrees does n o t c o i n c i d e w i t h t h e t h e o r e t i c a l n a t u r a l fr«auency as d e f i n e d e a r l i e r e x c e p t f o r s m a l l a m p l i t u d e s .

A c o m p l e t e p i c t u r e o f t h e dependancy o f t h e measured moment on t h e a m p l i t u d e and f r e q u e n c y o f m o t i o n i s g i v e n i n t h e f i g u r e 10.

The measurements were c a r r i e d o u t f o r one t a n k b r e a d t h , one p o s i t i o n o f t h e c e n t r e o f r o t a t i o n and one w a t e r l e v e l . From f i g u r e 12 i t appears t h a t t h e moment a m p l i t u d e i n c r e a s e s a p p r o x i m a t e l y a c c o r d i n g t o t h e square r o o t o f t h e a m p l i t u d e o f m o t i o n .

From t h e e x p r e s s i o n f o r t h e n a t u r a l f r e q u e n c y f o r s m a l l a m p l i t u d e s , i t a p p e a r s t h a t t h e o n l y p o s s i b i l i t y t o change t h e resonance p o i n t w i t h a g i v e n t a n k , i s a change i n t h e w a t e r d e p t h . As shown i n f i g u r e 8, t h e component w h i c h i s 90 d e g r e e s o u t o f phase i s l a r g e s t i n t h e n e i g h b o u r h o o d o f t h e r e s o n a n c e p o i n t . S i n c e t h i s component i s r e s p o n s i b l e f o r t h e r o l l damping i t i s e v i d e n t t h a t i n p r i n c i p l e t h e n a t u r a l f r e q u e n c i e s o f t h e s h i p and t h e t a n k s h o u l d be e q u a l .

But i n p r a c t i c e i t p r o v e s advantageous i n some cases o f v e r y l o w f r e q u e n c i e s and c o n s e q u e n t l y low w a t e r l e v e l s t o choose a h i g h e r n a t u r a l f r e q u e n c y f o r t h e t a n k , because t h e accompanying g r e a t e r w a t e r d e p t h i n c r e a s e s t h e amount o f w a t e r and so t h e moment a m p l i t u d e t o suet an e x t e n t t h a t t h e n e t e f f e c t i s a l a r g e r damping, i n s p i t e o f t h e u n f a v o u r a b l e phase a n g l e . The moment a m p l i t u d e a p p e a r s t o be^ a p p r o x i m a t e l y p r o p o r t i o n a l t o t h e square r o o t o f t h e r a t i o ^, M = c o n s t a n t / — , see f i g u r e 13.

The i n f l u e n c e o f t h e t a n k b r e a d t h i s most i m p o r t a n t . Assume two t a n k s , one w i t h a l a r g e r b r e a d t h t h a n t h e o t h e r , r o t a t i n g a b o u t t h e c e n t e r l i n e o f t h e b o t t o m .

h Jt

Assume a l s o t h e same ^ r a t i o and t h e same ^ r a t i o . Then one o f t h e t a n k s can be c o n s i d e r e d as a s c a l e model o f t h e o t h e r . T h i s means t h a t w i t h t h e r e s p e c t i v e n a t u r a l f r e q u e n c i e s t h e r a t i o o f t h e moment a m p l i t u d e s i s :

^ 1

o r expressed o t h e r w i s e s t h e moment a m p l i t u d e oer u n i t t a n k l e n g t h v a r i e s as t h e t h i r d power o f t h e t a n k b r e a d t h , p r o v i d e d t h e — r a t i o i s t h e same.

To v e r i f y t h i s i n an e x i s t i n g t a n k m o d e l two s i d e w a l l s were b u i l t , w h i c h 3

r e d u c e d t h e t a n k b r e a d t h t o o f i t s f o r m e r v a l u e . The d i s t a n c e o f t h e t a n k b o t t o m t o t h e a x i s o f r o t a t i o n was n o t b r o u g h t i n t h e same r a t i o t o t h e b r e a d t h because o f l a c k o f t i m e . The r e s u l t s o f t h e measurements a r e g i v e n i n f i g u r e 14. The t e s t s were c a r r i e d o u t f o r two w a t e r d e p t h s . The moment a m p l i t u d e s f o r t h e n a r r o w t a n k a r e f a r l e s s t h a n f o r t h e f u l l w i d t h t a n k , b u t when t h e moments a r e e x p r e s s e d as n o n - d i m e n s i o n a l c o e f f i c i e n t s s M _a ^^a " 3 ^ pg b-' i and p u t on a b a s i s o f t h e f r e q u e n c y r a t i o a c c o r d i n g t o t h e i r r e s p e c t i v e v a l u e s o f ( f i g u r e 15) t h e agreement i s good, e s p e c i a l l y i n v i e w o f t h e f a c t t h a t t h e i n f l u e n c e o f t h e arm s has n o t been e l i m i n a t e d .

E x p e r i m e n t s as t o t h e i n f l u e n c e o f t h e p o s i t i o n o f t h e t a n k i n h e i g h t i n d i c a t e t h a t f o r i n c r e a s i n g h e i g h t w i t h r e s p e c t t o t h e a x i s o f r o t a t i o n t h e moment a m p l i t u d e i n c r e a s e s and t h e phase a n g l e becomes s l i g h t l y s m a l l e r . The r e s u l t s a r e shown

i n t h e f i g u r e s 16 and 17. I n f i g u r e 16 an example o f t h i s i n f l u e n c e on t h e

q u a d r a t u r e component i s g i v e n f o r one v a l u e o f t h e m o t i o n a m p l i t u d e . I n f i g u r e 17 t h e r e s u l t s a r e c o m p i l e d f o r d i f f e r e n t a m p l i t u d e s .

R e s u m i n g _ t h e _ r e s u l . t s _ s o _ f a r s _

1. A f r e e s u r f a c e t a n k b r o u g h t i n t o o s c i l l a t i o n c r e a t e s a c o u n t e r a c t i n g moment.' M a g n i t u d e and phase depend on a nxmiber o f p a r a m e t e r s ,

2. The e s s e n t i a l p h y s i c a l phenomenon i s t h e appearance o f a b o r e , t r a v e l l i n g t o and f r o . The c o u n t e r a c t i n g moment i s m a i n l y due t o t h e d i s p l a c e m e n t o f t h e c e n t r e o f g r a v i t y o f t h e m o v i n g f l u i d w i t h a c o r r e c t p h a s e l a g w i t h t h e m o t i o n o f t h e t a n k . T h i s i s p r e s e n t e d s c h e m a t i c a l l y i n f i g u r e 18. 3. As l o n g as p u r e r o l l i n g i s considered,, t h e a c t i v e s t a b i l i z i n g moment n e v e r changes i n t o an e x c i t i n g moment. I n w h i c h d e g r e e o t h e r m o t i o n s , e s p e c i a l l y t r a n s v e r s o s c i l l a t i o n s , i n f l u e n c e t h e b e h a v i o u r o f t h e t a n k i s n o t y e t known. 4. E x p e r i m e n t s show t h a t t h e moment a m p l i t u d e p e r u n i t t a n k l e n g t h i s a p p r o x i -m a t e l y p r o p o r t i n a l t o t h e p r o d u c t s f o r a s p e c i f i e d p o s i t i o n o f t h e a x i s o f r o t a t i o n and f o r OJ = o)^^..

-15-5. The h i g h e r t h e a x i s o f r o t a t i o n o f t h e t a n k t h e h i g h e r t h e moment a m p l i t u d e . As f a r as can be c o n s i d e r e d f r o m t h e e x p e r i m e n t s t h e b e s t p o s i t i o n o f t h e t a n k i s as h i g h i n t h e s h i p as p o s s i b l e ; t h e t a n k s h o u l d e x t e n d o v e r t h e f u l l b r e a d t h o f t h e s h i p . E s t i m a t i o n _ o f _ t a n k ^ e f f e c U

We have seen t h a t t h e r e c t a n g u l a r t a n k , p a r t l y f i l l e d w i t h w a t e r , can i n c r e a s e t h e r o l l damping o f t h e s h i p . Now we have t o c o n s i d e r t h e c o m b i n a t i o n o f t h e s h i p and t a n k c h a r a c t e r i s t i c s .

We know t h a t t h e i n - p h a s e and q u a d r a t u r e components o f t h e t a n k moment depend on t h e f r e q u e n c y o f o s c i l l a t i o n . We a l s o know t h a t t h e v i r t u a l moment o f i n e r t i a and t h e damping o f t h e s h i p depend on t h e f r e q u e n c y . However, o u r knowledge o f the s h i p c h a r a c t e r i s t i c s i n t h i s r e s p e c t i s s t i l l v e r y s r a a l l , so f o r t h e t i m e we have t o r e s o r t t o i n d i v i d u a l raodel t e s t s .

When a shipraodel i s o s c i l l a t e d i n r o l l , t h e r o l l i n g raoraent, t h e raotion a m p l i t u d e and t h e phase o f t h e m o t i o n can be measured. From t h e s e raeasurements the v i r t u a l raoraent o f i n e r t i a and t h e daraping can be d e t e r r a i n e d as a f u n c t i o n o f f o r w a r d speed and f r e q u e n c y o f o s c i l l a t i o n .

L e t us assurae f o r t h e raoraent t h a t we know f o r a c e r t a i n speed t h t v i r t u a l moment o f i n e r t i a , t h e s t i f f n e s s o f t h e s h i p and t h e damping as r e g a r d s t h e r o l l i n g m o t i o n . A l l v a l u e s on a b a s i s o f f r e q u e n c y . Then we can c o r r e c t t h e s h i p v a l u e s by t h e c o r r e s p o n d i n g t a n k v a l u e s . The r e s u l t s can be v e r i f i e d by f o r c e d o s c i l l a t i o n t e s t s o f t h e combined system: t a n k and s h i p .

I t i s n o t p o s s i b l e t o s e p a r a t e t h e mass- and s t i f f n e s s p a r t s o f t h e i n - p h a s e moment o f t h e t a n k . I n most cases t h e mass o f t h e t a n k w a t e r i s s m a l l i n compar-i s o n w compar-i t h t h e mass o f t h e s h compar-i p . However, t h e r e d u c t compar-i o n o f GM , due t o t h e f r e e s u r f a c e e f f e c t o f t h e t a n k can be r e l a t i v e l y l a r g e . C o n s e q u e n t l y t h e i n - p h a s e component o f t h e t a n k moment i s c o n s i d e r e d e n t i r e l y as a r e d u c t i o n o f t h e s p r i n g c o n s t a n t R. I n t h i s way t h e f r e q u e n c y r e s p o n s e o f t h e s h i p + t a n k s y s t e m can be d e t e r m i n e d a p p r o x i m a t e l y . F o r a q u i c k e s t i m a t i o n o f t h e t a n k d i m e n s i o n s i n a c e r t a i n case we have t o e s t i m a t e t h e s h i p ' s r a d i u s o f i n e r t i a , t h e v a l u e o f t h e n o n - d i m e n s i o n a l damping c o e f f i c i e n t 6nd t h e m e t a c e n t r i c h e i g h t . From t h e e q u a t i o n o f m o t i o n o f t h e s h i p w i t h o u t t a n k : l i * + N* + R* = M^^ s i n (ojt + e^) ,

we have d e r i v e d t h e n a t u r a l r o l l f r e q u e n c y : 8 R I ' and t h e r o l l a m p l i t u d e a t r e s o n a n c e (e = 90°): w M (J. . - B 8 t a Nto V s w i t h N s V

The d i f f e r e n t i a l e q u a t i o n o f t h e s h i p + t a n k sysjtem can be w r i t t e n as

t

IA + NA + R(J. = M* s i n (tot + e ) + M s i n (tot + e) wa w a

where M s i n (tot + e) i s t h e s u p e r i m p o s e d t a n k moment, a

The two components:

M s i n ojt cos e and M cos tot s i n e

a a a r e r e s p e c t i v e l y c o n s i d e r e d a s : ^ • (J) and - ^ ' (S> M a /ïR = - — cos 6 and ^ m = - s i n e So t h e d i f f e r e n t i a l e q u a t i o n becomes: I4,* + (N + /^N)4> + (R + .^)(1) = M^^ s i n (cjt + e^)

-17-From t h i s e q u a t i o n we f i n d : ^* ^ / R + ^ s < a _ w a ^ ^ wa. a (N + ^ ) c o : N + ^ M R + Z J l s

The wave moment M^^ a l t e r s a p p r o x i m a t e l y p r o p o r t i o n a l t o t h e s p r i n g c o n s t a n t . So we can w r i t e : * ^wa

J

I N

\l

R R + ^ N + M R When th« r e s o n a n c e f r e q u e n c i e s o f t h e t a n k and t h e s h i p c o i n c i d e i t f o l l o w s t h a t a t r e s o n a n c e : . = 90 ° w € = - 90 ° and: t h u s :

*

4) V + A V GL The q u a n t i t y Av •= — c a n be c a l c u l a t e d p e r u n i t t a n k l e n g t h . I n t h i s way t h e t a n k l e n g t h , n e c e s s a r y^ ï^ a c h i e v e t h e d e s i r e d r e d u c t i o n p e r c e n t a g e - w h i c h i s a r b i t r a r y - can be d e t e r m i n d e d . The i n f l u e n c e o f t h e f r e e s u r f a c e on t h e s t a t i c s t a b i l i t y must be t a k e n i n t o a c c o u n t . A l t h o u g h f o r s m a l l h e e l i n g a n g l e s , t h e r e d u c t i o n o f t h e m e t a c e n t r i c h e i g h t can be c o n s i d e r a b l e t h i s i n f l u e n c e d e c r e a s e s i n most cases when t h e h e e l i n g a n g l e i n c r e a s e s .

I have m e n t i o n e d t h e e x i s t e n c e o f a t o t a l l y d i f f e r e n t t y p e o f p a s s i v e a n t i ¬ r o l l i n g t a n k ; t h a t based on t h e p r i n c i p l e o f t h e U - t y b e . About t h i s t a n k t h e r e has appeared i n l i t e r a t u r e much more t h a n o f t h e f r e e s u r f a c e t a n k . The d i f f e r e n c e i n

t h e ^ a r a c t e r i s t i c s o f t h e two t y p e s a r e c o n s i d e r a b l e . For an i n d i v i d u a l case we made a c o m p a r i s o n between t h e U-tube and t h e f r e e s u r f a c e t a n k . Such a c o m p a r i s o n depends t o a l a r g e d e g r e e on many r e s t r i c t i o n s , f o r i n s t a n c e , t h e w e i g h t o f t a n k w a t e r can be l i m i t e d or t h e space w h i c h i s a v a i l a b l e . N e v e r t h e l e s s t h e d i f f e r e n c e s i n

c h a r a c t e r a r e c l e a r l y shown i n t h e f i g u r e s 19, 20 and 2 1 , The U-tube i s r e l a t i v e l y e f f e c t i v e i n a n a r r o w f r e q u e n c y band. T h i s t y p e i s d i f f i c u l t t o tune p r o p e r l y . The f r e e s u r f a c e t a n k , a l t h o u g h i n t h i s case somewhat l e s s e f f e c t i v e a t t h e r e s o n a n c e f r e q u e n c y , i s much k i n d e r t o d i f f e r e n c e s i n t u n i n g . T h i s i s e s p e c i a l l y i m p o r t a n t f o r s m a l l s h i p s ( f o r i n s t a n c e a f r e s h h a u l o f f i s h on deck o f a t r a w l e r can a l t e r i t s n a t u r a l p e r i o d c o n s i d e r a b l y ) . Because o f t h e i n h e r e n t l a r g e damping o f t h e f r e e s u r f a c e t a n k i t s e l f , s e c o n d a r y peaks i n t h e a m p l i t u d e r e s p o n s e o p e r a t o r a r e o f no i m p o r t a n c e . A p a r t f r o m t h e r e c t a n g u l a r t a n k , we have t e s t e d s e v e r a l m o d i f i c a t i o n s o f t h e f r e e s u r f a c e t a n k . Tanks w i t h c o n t r a c t i o n s w i t h rounded o r s h a r p e n t r i e s , t a n k s w i t h w i r e gauges as homogeneous damping, t a n k s w i t h p a r t i a l l y r a i s e d b o t t o m e t c . A l l o f t h e s e m o d i f i c a t i o n s showed a s h i f t o f t h e phase a n g l e s t o l o w e r f r e q u e n c i e s

and a d e c r e a s e o f t h e moment a m p l i t u d e was o b s e r v e d . I n g e n e r a l t h e e f f e c t was a s m a l l g a i n a t low f r e q u e n c i e s w h i c h may be o f use i n t h e case o f l o n g p e r i o d s h i p s -and a c o n s i d e r a b l e l o s s a t t h e h i g h e r f r e q u e n c i e s .

A l t h o u g h we a r e o n l y a t t h e b e g i n n i n g o f our r e s e a r c h program c o n c e r n i n g r o l l s t a b i l i z a t i o n , we e x p e c t t h a t i n a r e l a t i v e l y s h o r t t i m e we s h a l l have a t our d i s p o s a l s u f f i c i e n t d a t a t o e n a b l e e v e r y s h i p d e s i g n e r t o j u d g e t h e f e a s i b i l i t y and p o s s i b i l i t y of t h e a p p l i c a t i o n o f f r e e s u r f a c e t a n k s as a means o f r o l l s t a b i l i z a t i o n .

Y(U))

0(0))

-0)

Electranic strain indRatir Carrier o n v U f icr Resotvcr

h phase corrponent Quadrature ccmpcnent

I 1 1 1 1 1 1 jrr:. -K s M

Principle of experimental set-up

n o t e : s = a = - 0 . 5 2 2 m b=1.00 m h/b=0.06 ip=a=0.0667 Some s u c c e s s i v e p o s i t i o n s of t h e b o r e f o r W=2.00

FIG.6

ID = 1.00 w a t e r h o r i z o n t a l . W=1.25 s m a l l w a v e l e t s . s o l i t a r y w a v e . „ f r o z e n ' s l a t e . n o t e : s = a = - 0 5 2 2 m b = 1.00 m ^P = a = 0.0667 I l l u s t r a t i o n of p h e n o m e n a f o r d i f f e r e n t f r e q u e n c i e s .

FIG. 7

M',co«e

E x a m p l e of m e a s u r e d I n - p h a s e and quadrature comporrents of the moment.

Ü) ..T.-<

E x a m p l e of moment amplitude a s a function of frequency.

dtgree, -180

e - 9 0

E x a m p l e of phase angles a s a function of frequency.

o l<^-010 M I ^ . U O

1.0 2J0 3.0 4.0

Ü) ««-«

Moment amplitudes for d i f f e r e n t a m p l i t u d e s of motion.

P h a s e a n g l e s for different amplitude* of motion.

P h a s t a n g l e s a s a function of the amplitude of motion f o r ( ü = U ) ,

10 ^^^^^ ^

/

/

/

/ / 1 0 . 0 4 0 . 0 8 0.12

Moment amplitudes a s a f u n c t i o n of the r a t i o w a t e r depth / t a n k breadth forWsCJ^jt

degrees - 1 6 0 E - 9 0 ss-020m b=a9e3m O 0- • — 0 " 0 0 4 , 0J08 0.12

P h a s e a n g l e s a s a function of the ratio w a t e r depth/tank breadth forCOzCJot'

U f " - 1 8 0 e -90 I 0 b a U I 3 m Il/tl.a0(1 X a b . O S D m »jb>0.061

////

X • 1

^

10 2 f l 3.0 4.0 0) i.c.-'

P h a s s angles for two tank w i d t h s .

' 2 0 . • • - O J O r n b . : ^ a 0 j a 3 V b . a o d b . a 9 * 3 m Yi^tJiStta IB

F I G . U

e - 9 0 O b . 0 9 f 3 m r V b . a o d • b . V ( « 0 J i 3 0.0(1 0 b . a s i J m I ^ . Q O S J • b . ^ a O M ) i ^ . o . o n 1' J • w^n^*^^ 0.5 1.0 •1.5 2.0

P h a s e angles for two tank widths on a nondimensional frequency b a s i s .

0.020 0J015 OJOIO OJOOS S B - O i O m iPj«ft10 O b . a l l 3 m iVb.ODd • b . ^ i 0 5 l 3 ijb.0.0(1 a b . l U I 3 m t v b . a O M • b . ^ i i a s i 3 h)b.0D62

4^

- T J - O ^ -r— •

\ ^

\ o ^ \ \

\ V

\ \

05 1.0 1.5 'ot 2.0

Moment amplitude coefficients for two tank widths on a nondimensional frsQuefKy basis.

F I G . 1 5

- 1 5 - 1 0 - 5 O b . U I 3 • Sa+Q20m (p^«I110 • ! • a m tj'0\tt 0 | p- 0 ] 0 m >Pj>aio • t> - a 4 0 m i(i,.0.10 \ 0 \ \ \ \ ' 1 ^ « ^ n no 1.0 2X1 3.0 iD (1) Jtc->

Quadrature components a s a function of tank distance to the a x i s of rotation forU) = U),,^

+ 0.20

Moment a m p l i t u d e s a s a function of tank d i s t a n c e to the a x i s of rotation for

i t t g r t t i - 1 9 0 I E - 9 0 -P,.00333 < ( l , . 00667 ipj-O.lO f , . 0 1 5 i(i,«OJO -0.40 -0.20 . _{• 0 2 0}

P h a s e angles a s a function of tank distance to the a x i s of rotation for(i)=W

mofn. max.Mg. mom neo. mom. l e r e

monvpo.. mommo».pt.5 mompot.

7 o.l.3ni^ 8 0)1.7V' 9

P o s i t i o n of bore during one period of rolling at tani< r e s o n a n c e ( e - 9 0 ' ' ) ; v i e w in positive x - d i r e c t i o n .

kgcm 120

60

^ ^ ^ ^ + .—.^^^^^^

ube

Free surface

\

+ — -

_\

\ \ 1 ,1

0.5 1.0 2.0 3.0 ^.0

0) sec-''

FIG.20