The longitudinal distribution of low frequency hydrodynamic derivatives for lateral motions in shallow water

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

THE LONGITUDINAL DISTRIBUTION OF

LOW FREQUENCY HYDRODYNAMIC DERIVATIVES

FOR LATERAL MOTIONS IN SHALLOW WATER

ING. W. BEUKELMAN PROF.IR. J. GERRITSMA

• I

R e p o r t n r . 5 6 2 A S e p t e m b e r 1 9 8 3

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 " T h i s report is p r e p a r e d b y T H D / W L / N S P u n d e r c o n r r a c l n o . 8 2 / 1 5 2 8 / 7 . 3 . o [ the S l i c h t i n g C o ö r d i n a t i e M a r i t i e m O n d e r i o e k . " " © T H D / W L / N S P , 1 9 8 2 , " " R e p r o d u c t i o n i n w h o l e o r In port h y m e o n s of p r i n t , f o t o c o p y , m i c r o f i l m o r in a n y o t h e r w a y i s o n l y p e r m i l l e d a f t e r p r e c e d i n g w r i t t e n c o n s e n t o l T H D / W L / N S P . "

a s t e r i s k f o r v a l u e o f segment i n d i c a t i o n f o r d i m e n s i o n l e s s p r e s e n t a t i o n o f t h e hydrodynamic c o e f f i c i e n t s i n d i c a t i o n f o r d i m e n s i o n l e s s p r e s e n t a t i o n o f s e c t i o n a l v a l u e s o f hydrodynamic c o e f f i c i e n t s

t Ü V , V X O t i m e v e l o c i t y o r v e l o c i t y - c o m p o n e n t o f model a t h w a r t v e l o c i t y o r a c c e l e r a t i o n o f model c o - o r d i n a t e system f i x e d i n space X y z Y y a 3 6 e 6 1^ X P V

>

c o - o r d i n a t e system f i x e d t o model h o r i z o n t a l f o r c e e x e r t e d by o s c i l l a t o r c o e f f i c i e n t o f damping f o r c e f o r sway c o e f f i c i e n t o f added mass f o r sway c o u p l i n g c o e f f i c i e n t due t o t h e d i s t r i -b u t i o n o f damping f o r yaw

c o u p l i n g c o e f f i c i e n t due t o t h e d i s t r i -b u t i o n o f t h e added mass moment o f i n e r t i a

f o r yaw sway d i s p l a c e m e n t phase a n g l e between f o r c e o r moment and m o t i o n d r i f t a n g l e yaw a n g l e wave l e n g t h phase a n g l e between p e r i o d i c m o t i o n o f b o t h f a s t e n i n g p o i n t s d e n s i t y o f w a t e r c i r c u l a r firequency o f o s c i l l a t i o n volume o f d i s p l a c e m e n t o f model

i i i -N o t a t i o n A f a s t e n i n g p o i n t a f t on model L V a r e a o f w a t e r l i n e a, b, c, e h y d r o d y n a m i c c o e f f i c i e n t s r e l a t e d t o t h e e q u a t i o n f o r s h i p m o t i o n

a added mass o r added mass moment o f i n e r t i a a t speed ( r e l a t e d t o s h i p m o t i o n s ) , a m p l i t u d e ( i n d e x ) B b r e a d t h , f a s t e n i n g p o i n t f o r e on model b d a m p i n g - c o e f f i c i e n t a t speed ( r e l a t e d t o s h i p m o t i o n s ) , t a n k w i d t h C_, b l o c k c o e f f i c i e n t B d c o u p l i n g c o e f f i c i e n t f o r added mass ( r e l a t e d t o s h i p m o t i o n s ) Fn Froude number (Fn = ) gL g a c c e l e r a t i o n due t o g r a v i t y h d e p t h o f w a t e r I mass moment o f i n e r t i a o r - p r o d u c t k wave number L l e n g t h o f model 1 d i s t a n c e between o s c i l l a t o r l e g s (= 1 m) 1 ^ l e n g t h o f segment N h y d r o d y n a m i c y a w i n g moment w i t h r e s p e c t t o z - a x i s , damping a t z e r o speed N c o u p l i n g moment c o e f f i c i e n t due t o t h e d i s t r i b u t i o n o f damping f o r sway N. c o u p l i n g moment c o e f f i c i e n t due t o t h e ^ d i s t r i b u t i o n o f added mass f o r sway

c o e f f i c i e n t o f t h e moment o f damping f o r yaw N. c o e f f i c i e n t o f t h e added mass moment o f

•'^ i n e r t i a f o r yaw

r = {jj yaw a n g u l a r v e l o c i t y

f = yaw a n g u l a r a c c e l e r a t i o n T d r a u g h t o f model

I n t r o d u c t i o n . The l o n g i t u d i n a l d i s t r i b u t i o n o f t h e h o r i z o n t a l h y d r o -dynamic f o r c e s a c t i n g on a s l o w l y o s c i l l a t i n g s h i p i n s h a l l o w w a t e r i s o f i n t e r e s t f o r t h e d e t e r m i n a t i o n and a n a l y s i s o f t h e s t e e r i n g and m a n o e u v r i n g c h a r a c t e r i s t i c s o f s h i p s i n c o n f i n e d w a t e r s . As a f i r s t e x p l o r a t i o n o f t h e p r o b l e m o f t h i s l o n g i t u d i n a l d i s t r i b u t i o n f o r c e d o s c i l l a t i n g e x p e r i m e n t s have been c a r r i e d o u t w i t h a segmented model i n s h a l l o w w a t e r , p e r -f o r m i n g v e r y low -f r e q u e n c y yaw- and sway o s c i l l a t i o n s . A l s o some s t a t i c yaw t e s t s were i n c l u d e d f o r comparison w i t h t h e c o r r e s p o n d i n g l o w f r e q u e n c y v a l u e s . These e x p e r i m e n t a l d a t a a r e p r e s e n t e d f o r comparison w i t h c a l c u l a t e d r e s u l t s , f o r i n s t a n c e t h o s e o b t a i n e d f r o m s t r i p t h e o r y methods, f i n i t e element t e c h n i q u e s o r s o c a l l e d d i f -f r a c t i o n methods. A l l o f t h e s e c a l c u l a t i o n s a r e based on t h e a s s u m p t i o n o f an i d e a l f l u i d and t h u s n e g l e c t i n g t h e e f f e c t s o f v i s c o s i t y . A comparison o f t h e e x p e r i m e n t a l v a l u e s and c a l c u l a t i o n s c o u l d show t h e i n f l u e n c e o f v i s c o s i t y on t h e magnitude and d i s t r i b u t i o n o f t h e hydrodynamic d e r i v a t i v e s i n p a r t i c u l a r a f t o f t h e m i d s h i p s e c t i o n .

Because o f s e p a r a t i o n e f f e c t s i n t h i s r e g i o n , t h e a p p l i c a -t i o n o f s -t r i p -t h e o r y me-thods i s q u e s -t i o n a b l e , a l -t h o u g h i -t i s n o t known t o what e x t e n t s e p e r a t i o n e f f e c t s i n f l u e n c e the sway and yaw f o r c e s .

I n comparison w i t h 3 - d i m e n s i o n a l f l o w c a l c u l a t i o n s f o r v e r y low f o r w a r d speeds and f r e q u e n c i e s , i t i s perhaps p o s s i b l e t o show t h e p r o g r e s s b e i n g made w i t h r e s p e c t t o s i m p l e r two d i m e n s i o n a l s t r i p methods.

I n t h i s r e p o r t t h e r e s u l t s o f e x p e r i m e n t s w i t h a model con-s i con-s t i n g o f con-seven con-segmentcon-s c a r r y i n g o u t low f r e q u e n c y yaw and sway m o t i o n s i n s h a l l o w w a t e r a r e g i v e n . Two f o r w a r d speeds and a range o f w a t e r depths have been c o n s i d e r e d . The measured t o t a l hydrodynamic f o r c e s , as w e l l as t h e i r

2

-d i s t r i b u t i o n a l o n g t h e l e n g t h o f t h e mo-del a r e compare-d w i t h c a l c u l a t i o n s a c c o r d i n g t o t h e s t r i p t h e o r y , t a k i n g i n t o a c c o u n t t h e e f f e c t o f s h a l l o w w a t e r .

The model.

The e x p e r i m e n t s have been c a r r i e d o u t w i t h a 2.3 m model o f t h e S e r i e s S i x t y . T h i s model had been used e a r l i e r f o r a s i m i l a r i n v e s t i g a t i o n f o r h i g h e r o s c i l l a t i o n f r e -q u e n c i e s , o f i n t e r e s t f o r t h e a n a l y s i s o f s h i p m o t i o n s i n waves, i n deep w a t e r as w e l l as s h a l l o w w a t e r 1, 2, 3 . The main p a r t i c u l a r s o f t h e model a r e summarized i n T a b l e 1.

Table 1 ; Dimensions o f s h i p model.

L e n g t h between p e r p e n d i c u l a r s L 2. 258 m W a t e r l i n e l e n g t h L^^ 2. 296 m Beam B 0.32 2 m D r a u g h t T 0.129 m Volume o f d i s p l a c e m e n t V 0.0657 m^ B l o c k c o e f f i c i e n t C_, 0.700 W a t e r p l a n e a r e a 0.572 m^ L o n g i t u d i n a l moment o f i n e r t i a o f w a t e r p l a n e I.^ 0.1685 m"* L o n g i t u d i n a l p o s i t i o n o f t h e c e n t r e o f b u o y a n c y ( f o r w a r d o f L^_/2) LCB 0.011 m LL C e n t r o i d o f w a t e r p l a n e ( f o r w a r d o f L,,/2) 0.038 m M e a s u r i n g p r o g r a m .

The h y d r o d y n a m i c d e r i v a t i v e s have been d e t e r m i n e d by means o f f o r c e d o s c i l l a t i o n e x p e r i m e n t s f o r p u r e sway and yaw m o t i o n s o f t h e s h i p model.

The r e l a t i o n between t h e measured o s c i l l a t i o n f o r c e s and t h e hydrodynamic d e r i v a t i v e s i s g i v e n i n A p p e n d i x 1 , f o r t h e whole model, as w e l l as f o r each o f t h e seven segments o f t h e model. I n T a b l e 2 t h e model speeds (Fn, U ) , t h e

o s c i l l a t i o n f r e q u e n c i e s co and t h e o s c i l l a t i o n a m p l i t u d e s y o f t h e sway- and yaw m o t i o n s are g i v e n . I t s h o u l d be

a

remarked t h a t t h e a m p l i t u d e s o f t h e h o r i z o n t a l d i s p l a c e m e n t o f t h e p o i n t s A and B i n t h e yaw mode are e q u a l , b u t t h e i r m o t i o n s d i f f e r by t h e a n g l e c)) (see F i g u r e A l . 1 Appendix 1) . The l e n g t h between p e r p e n d i c u l a r s i s t a k e n as t h e r e f e r e n c e l e n g t h . Table 2: T e s t c o n d i t i o n s . Fn U m/s r=il) r a d / s r L

^ ^

IT

CO r a d / s (f,/2 degrees 0 r l a . 0 cüsm-j meter , (jüL c . = — 0 .0675 0.318 0 .0376 0 .267 0.26 22.2 0 .191 1 . 85 0 .0675 0.318 0 .0376 0 .267 0.50 38 .2 0.061 3.55 0 .0675 0.318 0 .0376 0 .267 0.75 49 .7 0 .033 5.33 0 .103 0 .485 0.0335 0 .156 0.26 15 .0 0.249 1.21 0 .103 0 .485 0.0335 0 .156 0.50 27.3 0.073 2.33 0 .103 0 .485 0.0335 0 .156 0 .75 37.7 0 .037 3.49

The s t a t i c d r i f t a n g l e e x p e r i m e n t s have been c a r r i e d o u t f o r

a range - 10° < 3 < 10° i n s t e p s o f 1 degree.

For t h i s c o n d i t i o n v = r = f = 0.

For t h e f o r c e d o s c i l l a t i o n t e s t s and t h e s t a t i c d r i f t a n g l e t e s t s t h e f o l l o w i n g w a t e r d e p t h / d r a u g h t ratio'r:i have been con-s i d e r e d :

h/T = 1.15; 1.2; 1.5; 1.8; 2.4

A scheme o f t h e m e c h a n i c a l o s c i l l a t o r and t h e e l e c t r o n i c

m e a s u r i n g system i s g i v e n i n F i g u r e 1. T h i s measuring system determines: o n l y t h e i n - p h a s e and t h e q u a d r a t u r e component o f the o s c i l l a t i o n f o r c e s , u s i n g as a r e f e r e n c e t h e phase o f t h e a t h w a r t s h i p ' s d i s p l a c e m e n t y o f t h e model, r e s p e c t i v e l y t h e yaw a n g l e ^ . F i g u r e 2 i s a p i c t u r e o f t h e h o r i z o n t a l

4

-By u s i n g a segmented model an e s t i m a t e o f t h e 2 - d i m e n s i o n a l v a l u e s o f added mass and damping and t h e i r d i s t r i b u t i o n a l o n g t h e l e n g t h o f t h e s h i p m o d e l can be made, based on t h e measured f o r c e s , see 1 and Appendix 1 .

R e s u l t s o f t h e model e x p e r i m e n t s and t h e c a l c u l a t i o n s .

The e x p e r i m e n t a l d i s t r i b u t i o n s and YV. a r e g i v e n i n t h e F i g u r e s 3 and 4 as s o l i d l i n e s , f o r t h e two f o r w a r d speeds o f t h e model, t h e o s c i l l a t i o n f r e q u e n c i e s and t h e w a t e r d e p t h -d r a u g h t r a t i o ' s c o n s i -d e r e -d . F o r c o n s t a n t w a t e r -d e p t h v a r i a t i o n o f t h e l a w f o r w a r d speed and t h e l o w o s c i l l a t i o n f r e q u e n c i e s have o n l y a s m a l l i n f l u e n c e on t h e d i s t r i b u t i o n s Y" and

V YV. . T h i s i n f l u e n c e i s c e r t a i n l y s m a l l i n comparison w i t h t h e i n f l u e n c e o f w a t e r d e p t h . I n F i g u r e 5 t h e d i s t r i b u t i o n s o f Y" f o r t h e case o f <-0 = 0 V ( s t a t i c t e s t s ) a r e g i v e n f o r two d r i f t a n g l e s ( 3 = 5 and

3 = 10°) and f o r two modelspeeds and f i v e w a t e r d e p t h / d r a u g h t r a t i o ' s . For t h e w a t e r d e p t h / d r a u g h t r a t i o ' 3 h/T = 1.15, 1.5 and 2.4 t h e measured Y^ and Y^ d i s t r i b u t i o n s f o r t h r e e o s c i l l a t i o n f r e q u e n c i e s and f o r t h e s t a t i c t e s t s a r e p l o t t e d i n F i g u r e s 6 and 7. Of i n t e r e s t a r e t h e n e g a t i v e s i d e f o r c e s o c c u r r i n g f o r t h e segments 1 , 2 and 3 i n t h e a f t p a r t o f t h e s h i p m o d e l . The e x p e r i m e n t a l v a l u e s o f t h e s i d e f o r c e s as f o u n d w i t h t h e s t a -t i c -t e s -t s agree s a -t i s f a c -t o r y w i -t h -t h e v a l u e s f o u n d w i -t h -t h e l o w e s t o s c i l l a t i o n f r e q u e n c y ( 03 = 0.26 r a d / s ) . The d i s t r i b u t i o n o f Y^ i n t h e a f t p a r t o f t h e s h i p m o d e l i s s l i g h t l y dependent on t h e o s c i l l a t i o n f r e q u e n c y . To a l e s s e r degree t h i s i s a l s o o b s e r v e d i n t h e f o r w a r d p a r t o f t h e s h i p model. I n F i g u r e 6 a l s o t h e v a l u e s o f Y^ based on s t r i p t h e o r y c a l c u l a t i o n s a r e d e p i c t e d . I t i s emphasized t h a t no v i s -c o s i t y e f f e -c t s a r e i n -c l u d e d i n t h e s e v a l u e s . I n c o m p a r i s o n w i t h t h e e x p e r i m e n t a l v a l u e s i t i s shown t h a t v i s c o s i t y has an i m p o r t a n t i n f l u e n c e on t h e s i d e f o r c e s i n t h e

a f t e r p a r t o f t h e s h i p m o d e l , p r o b a b l y as a r e s u l t o f s e p a r a t i o n , b u t t h e i n f l u e n c e i n t h e forv/ard p a r t i s much s m a l l e r .

For t h e s m a l l e s t w a t e r d e p t h r a t i o (h/T = 1.15) t h e c o r r e

-l a t i o n between c a -l c u -l a t i o n and e x p e r i m e n t i s n o t s a t i s f a c t o r y .

The d i s t r i b u t i o n o f t h e hydrodynamic mass (Y'^) o v e r t h e l e n g t h o f t h e s h i p m o d e l (see F i g u r e 7) agrees f a i r l y good w i t h t h e c a l c u l a t i o n , e x c e p t f o r t h e s m a l l e s t w a t e r d e p t h -r a t i o .

I n g e n e r a l t h e i n f l u e n c e o f f o r w a r d speed i n t h e c o n s i d e r e d range i s n o t i m p o r t a n t and t h e same a p p l i e s t o t h e i n f l u e n c e o f t h e o s c i l l a t i o n - f r e q u e n c y f o r w a t e r d e p t h - d r a u g h t r a t i o ' s e x c e e d i n g 1.5.

For s m a l l e r w a t e r d e p t h r a t i o ' s Y'.' i n c r e a s e s w i t h t h e o s c i l l a t i o n f r e q u e n c y .

The v a l u e s f o r Y^ and Y^ f o r t h e whole model and t h e c r o s s c o u p l i n g d e r i v a t i v e s Y^ and Y^ a r e g i v e n i n t h e F i g u r e s 8 and 9. A l s o t h e c o r r e s p o n d i n g c a l c u l a t e d v a l u e s a r e

d e p i c t e d i n t h e s e F i g u r e s . Reference i s made t o Appendix 2 w i t h a v i e w on t h e s i g n c o n v e n t i o n s as used i n t h i s work.

To some e x t e n t t h e t e n d e n c i e s as a l r e a d y m e n t i o n e d f o l l o w f r o m t h e s e f i g u r e s . The i m p o r t a n t i n c r e a s e o f t h e hydrodynamic

mass f o r v e r y s m a l l w a t e r d e p t h r a t i o ' s i s known a l s o f r o m o t h e r s o u r c e s .

The measured and c a l c u l a t e d v a l u e s o f Y" and Y'.' a r e g i v e n

V V

i n t h e T a b l e s 4 - 9 f o r comparison w i t h o t h e r and f u t u r e c a l c u l a t i o n s .

The s t a t i c d r i f t a n g l e r e s u l t s a r e g i v e n i n T a b l e 10.

I n g e n e r a l i t may be s t a t e d t h a t t h e agreement between e x p e r i m e n t and c a l c u l a t i o n i s b e t t e r when t h e w a t e r d e p t h r a t i o as w e l l as 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 i n c r e a s e .

Remarks.

1. The d i m e n s i o n l e s s o s c i l l a t i o n f r e q u e n c i e s to' = — are g i v e n i n Table 2. An e x t r a p o l a t i o n o f t h e measured f o r c e s t o 0 3 = 0 i s n o t p o s s i b l e because o f i n a c c u r a c i e s i n t h e measured f o r c e s . 2. The f r e q u e n c i e s o f 2 d i m e n s i o n a l s t a n d i n g waves i n t h e t a n k a r e h i g h e r t h a n t h e f r e q u e n c i e s o f o s c i l l a t i o n as used i n t h e e x p e r i m e n t s . The s t a n d i n g wave f r e q u e n c i e s f o l l o w f r o m : (JO^ = k g t a n k h kh i n c o m b i n a t i o n w i t h where b = t a n k w i d t h = 4.21 m.

The s t a n d i n g wave f r e q u e n c i e s a r e summarized i n T a b l e 3 .

T a b l e 3; S t a n d i n g wave f r e q u e n c i e s ( r a d / s ) . (JÜ h/T A/b h/T 2/1 2/2 2/3 2/4 1 .15 0.89 1 .81 2.74 3.71 1.20 0.92 1.85 2 .81 3.82 1 .50 1.03 2.08 3.18 4 . 37 1.80 1.13 2 . 30 3.53 4.92 2.40 1.31 2 . 70 4 .25 6 .22 3. W i t h r e s p e c t t o t h e i n f l u e n c e o f waves r e f l e c t e d f r o m t h e t a n k w a l l s i t may be remarked t h a t f o r t h e model e x p e r i m e n t t h e v a l u e o f ^ v a r i e s f r o m 0.0084 t o 0.037. T h e r e f o r e t h e o s c i l l a t i n g model i s always h i t by r e f l e c t e d r a d i a t i o n waves. The m a g n i t u d e o f t h i s e f f e c t c o u l d n o t be d e t e r m i n e d because t h e d i s t o r t i o n o f t h e s u r f a c e by t h e s e r a d i a t e d and r e f l e c -t e d waves c o u l d n o -t be o b s e r v e d .

Appendix 1 1. D e t e r m i n a t i o n o f t h e hydrodynamic d e r i v a t i v e s by f o r c e d o s c i l l a t i o n e x p e r i m e n t s . Sway For t h i s m o t i o n t h e f o l l o w i n g e q u a t i o n s o f m o t i o n a r e used: 1/2 < > A ^ 1/2 'A V s i n (Ü t a F i g u r e A l . 1 Sway (Y. - m)v + Y V = Y ^ s i n ( o j t + e ) = Y^ + Y^ V V a A ts N.v + N V = N sin(cot + 6) = (Y^ - Y ^ ) l / 2 V V a a A ( A l . l ) W i t h t h e a t h w a r t s h i p ' 3 d i s p l a c e m e n t y V s i n w t i t f o l l o w s t h a t : ^ a Y. = V Y -V N. V N V -Y cose a Y s i n e a y^ ü 3 -N cos6 3. N sinö a y^a3 + m (A1.2)

8

-For each o f t h e seven segments a s i m i l a r e x p r e s s i o n h o l d s :

( y t - m*)v + Y*v = Y* s i n ( o j t + e*) (A1.3)

V V a and: ' V Y = V where: 5 + m * .

*

Y s i n e a y^oj : A I . 4 ) m - t h e mass o f t h e c o n s i d e r e d segment Y - t h e a m p l i t u d e o f t h e o s c i l l a t i o n f o r c e on t h e a segment Pure y a w i n g . I n t h e case o f p u r e y a w i n g : v = v = 3 = l3 = 0

where v i s t h e a t h w a r t s h i p ' s v e l o c i t y component. I n o t h e r words t h e v e l o c i t y v e c t o r o f t h e c e n t r e o f g r a v i t y (G) l i e s i n t h e l o n g i t u d i n a l p l a n e o f symmetry. Than: y ^ = y^sin(Lot - ^) and y^ = y ^ s i n ( c o t + ^) :

i

o Ó lüü ^ 5 2 ^ 2 Ü F i g u r e A 1.2 Yawing,

For each c o m b i n a t i o n o f t h e d i s t a n c e 1 , t h e f r e q u e n c y ui and t h e f o r w a r d speed o f t h e s h i p U, de phase a n g l e <t> i s a d j u s t e d

t o o b t a i n v = v = 3 = é = 0.

The e q u a t i o n s o f m o t i o n o f t h e p u r e y a w i n g m o t i o n a r e :

where;

y.r- + (Y - mU) r = Y cos (cot + a) r r a (N. - I ) f + N r = N cos (tot + 3)

r z z r a

I = t h e mass moment o f i n e r t i a and

ZZ ^a = (^B-I t f o l l o w s t h a t : -Y .1 cosa Y. = _{f 2y co2sin(J)/2} a y 1 . s i n a Y_, = ,^ + mU -N c o s 3 N. = ,^ 2 + I r ^) Lii'' zz a N s i n g N = ^ (A1.5) r 2y^a3sin(j)/2 ^^^^^^ r CO

For each o f t h e segments s i m i l a r e x p r e s s i o n s a r e v a l i d :

*

^ - y ^ l c o s a ^ r 2y co2sincf)/2 x y 1 s x n a Y = TT^^ r^TTö- + m U (A1.7) r 2y üJsxn(j)/2 a where: x. - t h e h o r i z o n t a l d i s t a n c e between t h e c e n t r e o f X g r a v i t y o f t h e segment and G.

10 -2. D e t e r m i n a t i o n o f t h e s t a t i c d e r i v a t i v e s From t h e measured s i d e f o r c e s Y = Y + Y i t f o l l o w s t h a t : 3. r\ Jj V V Y - Y W i t h : V = - U tgB ^ - U -^-^—-we f i n d : Y 1 Y Tpr (A1.8) V (Yg - y ^ ) u

and f o r each o f t h e segments: Y * l Y ^ (A1.9) V (Yg - Y^)U 3. D i m e n s i o n l e s s p r e s e n t a t i o n o f t h e m e a s u r i n g r e s u l t s . The d i m e n s i o n l e s s hydrodynamic d e r i v a t i v e s a r e d e f i n e d as f o l l o w s : For t h e segments s i m i l a r e x p r e s s i o n s a r e v a l i d .

The mean v a l u e s o f t h e hydrodynamic d e r i v a t i v e s have been d e t e r m i n e d by: Y.' Y ' Y^ = and Y" = -j— ( A l . l l ) s ^ s where: 1 ^ i s t h e l e n g t h o f t h e c o n s i d e r e d segment. I t i s assumed t h a t t h e d i s t r i b u t i o n s o f t h e hydrodynamic d e r i v a t i v e s o v e r t h e l e n g t h o f t h e shipmodel can be r e p r e -s e n t e d by c o n t i n u o u -s c u r v e -s . The c u r v e -s can be d e t e r m i n e d f r o m t h e mean v a l u e s as g i v e n by A l . l l and t h e known t o t a l v a l u e s , b e i n g t h e sum o f t h e segment v a l u e s .

I n v i e w o f t h e r e s u l t s as f o u n d f o r Y' and Y l t h e d i s t r i -r -r

Appendix 2.

T r a n s f o r m a t i o n o f hydrodynamic d e r i v a t i v e s t o t h e manoeuvring s i g n c o n v e n t i o n .

Sway

The e q u a t i o n o f m o t i o n f o r swaying m o t i o n s i n waves can be w r i t t e n as f o l l o w s : (pV + a ) y + b y = Y^sin(cot + e) (A2.1) YY YY a S u b s t i t u t i n g : y = y ^ s i n t o t g i v e s f o r t h e q u a d r a t u r e component o f t h e s i d e f o r c e : b coy = Y s i n e yy -'a a I n Appendix 1 we f o u n d (see A 1 . 2 ) : -Y s i n e Si

I n v i e w o f t h e s i g n c o n v e n t i o n s a d o p t e d (see F i g u r e A2.1 and A2.2) i t f o l l o w s t h a t : Y = -b V y y S i m i l a r l y : (A2.3) ^v = -a yy (A2.4)

12

-Yawing.

Pure y a w i n g i s d e f i n e d by t h e absence o f a d r i f t a n g l e , t h u s t h e v e l o c i t y v e c t o r o f G i s t a n g e n t t o t h e p a t h o f G.

F i g u r e A2.1 Pure y a w i n g .

T h i s m o t i o n can be s p l i t up i n a t r a n s l a t i o n o f G (sway) and a r o t a t i o n (yaw)

The e q u a t i o n s o f m o t i o n o f sway and yaw o f a s h i p i n waves can be w r i t t e n as f o l l o w s (see (1) i n 3 ) : (pV + a ) y + b y + d > + e , 4 j = Y cos (wt + a ) ^ YY yy Y^ a By s u b s t i t u t i o n o f ; (A2.5) y = y^sincot 2y. \l) = Ip cosojt = —=j-^ s i n cosoot (A2.6:) we f i n d w i t h Appendix 1 (A1.6) ÜJ (m + a ) 1 Y = - e — r Y^ + mu 2 s i n and; (A2.7) b 1 2a3sin|

The r e s u l t s c a l c u l a t e d w i t h (A2.6) and (A2.7) a r e p r e s e n t e d i n F i g u r e 9.

1 4

-References.

G e r r i t s m a , J . and W. Beukelman,

The d i s t r i b u t i o n o f t h e hydrodynamic f o r c e s on a h e a v i n g and p i t c h i n g s h i p model i n s t i l l w a t e r ,

5 t h O f f i c e o f N a v a l Research Symposium 1964, Bergen, Norway.

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.

I n t e r n a t i o n a l S h i p b u i l d i n g P r o g r e s s , 1967.

The d i s t r i b u t i o n o f hydrodynamic mass and damping o f an o s c i l l a t i n g s h i p f o r m i n s h a l l o w w a t e r ,

I n t e r n a t i o n a l S h i p b u i l d i n g P r o g r e s s , 1982.

4] K e i l , H. ,

Die Hydromechanische K r a f t e by d e r p e r i o d i s c h e n Bewegung zwei d i m e n s i o n a l e r Körper an d e r O b e r f l a c h e f l a c h e r Gewasser, B e r i c h t n r . 305, I n s t i t u t für S c h i f f b a u d e r U n i v e r s i t a t Hamburg, 1974. 2 G e r r i t s m a , J . and W. Beukelman, 3^ Beukelman, W. and J . G e r r i t s m a ,

h/T = 2.4 Y" * 1 0 ^ V Y'J * 1 0 ^ V S e c t i o n Nr: Fn = 0 . 0 6 7 5 Fn = 0 . 1 0 3 Fn = 0 . 0 6 7 5 Fn = 0 . 1 0 3 S e c t i o n Nr: CO CO Oü S e c t i o n Nr: 0 . 2 6 0 . 5 0 0 . 7 5 0 . 2 6 0 . 5 0 0 . 7 5 0 . 2 6 0 . 5 0 0 . 7 5 0 . 2 6 0 . 5 0 0 . 7 5 1 - 6.0 - 3 . 1 + 1 . 1 - 5.5 - 3.5 - 0.5 - 1.0 - 0.8 - 1.8 - 0.8 - 0.6 - 0.8 2 + 3.7 + 6.9 + 9.3 + 5.6 + 8.6 + 9.9 - 4.7 - 5.3 - 5.9 - 4.8 - 5.3 - 5.7 3 + 3.3 + 5.7 + 6.0 + 4.2 + 6.4 + 6.9 - 7.0 - 7.5 - 8.0 - 7 . 1 - 7.6 - 8.0 4 - 4.3 - 3.0 - 4 . 1 - 3 . 1 - 2.5 - 2.8 - 7.7 - 8 . 1 - 8.7 - 7.8 - 8 . 1 - 8.5 5 - 1 0 . 6 - 1 1 . 8 - 1 2 . 8 - 9.2 - 9.8 - 1 1 . 3 - 7.7 - 7.8 - 8.0 - 7.9 - 8.0 - 8 . 1 6 - 1 6 . 3 - 1 6 . 4 - 1 7 . 6 - 1 5 . 5 - 1 6 . 1 - 1 7 . 2 - 6 . 1 - 6.3 - 6.7 - 6.4 - 6.4 - 6.6 7 - 4 2 . 2 - 4 0 . 4 - 4 1 .5 - 4 1 . 6 - 4 1 . 6 - 4 2 . 0 - 3.3 - 3.7 - 3.8 - 3.3 - 3.5 - 3.7 Whole model Y' * 1 0 ^ V Y: * 1 0 ^ _V Whole model - 2 3 . 6 - 2 0 . 1 - 1 9 . 2 - 2 1 . 2 - 1 9 . 0 - 1 8 . 4 - 1 2 . 1 - 1 2 . 7 - 1 3 . 8 - 1 2 . 3 - 1 2 . 8 - 1 3 . 4

T a b l e 5: Sway ( E x p e r i m e n t . ) h/T = 1 . 5 Y" * 1 0 ^ V Y'.' * 1 0 ^ V Fn = 0 . 0 6 7 5 Fn = 0 . 1 0 3 Fn = 0 . 0 6 7 5 Fn = 0 . 1 0 3 S e c t i o n Nr: CO CO OJ 0 3 S e c t i o n Nr: 0. 2 6 0 . 5 0 0 . 7 5 0 . 2 6 0 . 5 0 0 . 7 5 0 . 2 6 0 . 5 0 0 . 7 5 0 . 2 6 0 . 5 0 0 . 7 5 1 - 4.0 + 2.4 + 6.5 - 4.0 + 0.0 + 7.3 - 0.9 - 0.9 - 2.8 - 1 . 1 - 0.0 - 1 . 3 2 + 7 . 1 + 1 7 . 1 + 2 4 . 4 + 9.9 + 1 8 .2 + 2 5 . 2 - 6.2 - 7.7 - 1 1 . 3 - 6.8 - 7 . 9 - 1 0 . 1 3 + 2 . 3 + 9.0 + 1 1 . 1 + 3.9 + 1 0 . 8 + 1 2 . 4 - 9.6 - 1 2 . 0 - 1 3 . 7 - 1 0 . 8 - 1 2 . 3 - 1 5 . 3 4 - 9 . 6 - 6 . 4 - 7 . 4 - 9 . 0 - 5 . 1 - 7 . 1 - 1 0 . 6 - 1 2 . 9 - 1 4 . 9 - 1 1 . 5 - 1 3 .0 - 1 6 . 0 5 - 1 9 . 7 - 2 1 . 5 - 2 5 . 6 - 1 9 . 5 - 1 9 . 2 - 2 6 . 9 - 1 1 . 5 - 1 2 . 9 - 1 4 . 4 - 1 2 . 3 - 1 3 .6 - 1 5 . 8 6 - 3 1 . 5 - 3 4 . 1 - 3 8 . 0 - 3 2 . 3 - 3 3 .2 - 4 1 . 2 - 8.2 - 1 0 . 0 - 1 1 . 0 - 8.9 - 1 0 . 2 - 1 1 . 4 7 - 5 8 . 3 - 6 0 . 7 - 6 4 . 8 - 5 9 . 3 - 5 9 . 1 - 6 5 . 7 - 2.3 - 4.8 - 5.3 - 2.8 - 4.3 - 4.1 VJhole model Y ' * 1 0 ^ V Y: * 1 0 ^ V VJhole model - 3 6 . 8 - 3 0 . 2 - 3 0 . 0 - 3 5 . 7 - 2 8 . 2 - 3 0 . 7 - 1 6. 0 - 1 9 . 8 - 2 3 .8 - 1 7 . 5 - 1 9 .8 - 2 3 .9

h/T = 1 . 1 5 Y" * 10^ V Y'.' * 10^ V Fn = 0 . 0 6 7 5 Fn = 0 . 1 0 3 Fn = 0.0675 Fn = 0.103 S e c t i o n Nr: CÜ Ü 3 S e c t i o n Nr: 0.26 0.50 0.75 0. 26 0.50 0.75 0.26 0.50 0.75 0.26 0.50 0.75 1 - 3.5 + 1.0 - 8.1 - 3.8 + 6.2 -18.8 - 2.4 - 3.2 - 8.6 - 1.3 - 1.5 -13 .8 2 + 5.2 + 19.3 + 10.7 + 6.3 + 29.4 -32-4 - 9.5 -13.6 -25.2 -10.8 -13 . 9 -43 .9 3 - 8.1 + 3.9 + 31.7 - 8.4 + 8.8 -127.4 -15.5 -22.1 -35.4 -17 .4 -23 .4 -55.1 4 -32.7 -27 .8 -88.9 -35.5 -26.5 -223 .8 -16.9 -25.3 -38.3 -16.7 -24.8 -45.6 5 -52.1 -48.7 -127 . 9 -55.5 -52.1 -264.4 -15.4 -23 .1 -33.9 -15.3 -23 .7 -25.6 6 -76.3 -80.3 -159.4 -83 .7 -84.1 -267 .3 - 9.4 -17.4 -23 .2 - 6.2 -14 . 9 - 4.9 7 -111.6 -10 9. 6 -168.3 -119.1 -109.4 -200.1 - 3.4 - 5.3 - 3.9 - 8.6 - 0.4 - 3.9 Whole model Y' * 10^ V Y'. * 10^ V Whole model -93.5 -78.1 -185.3 -96.8 -73 . 2 -366.5 -21.3 -35.6 -54.7 -19.1 -33.1 -45.3

T a b l e 7: Sway ( C a l c u l a t i o n s . ) h/T = 2 . 4 Y" * 10^ V Y!' * 10^ V Fn = 0.0675 Fn = 0.103 Fn = 0.0675 Fn = 0.103 Ord. Nr: OJ CO Ord. Nr: 0. 26 0.50 0.75 0. 26 0.50 0.75 0 . 26 0.50 0.75 0 . 26 0.50 0.75 0 +82 . 0 +8 2.1 +82.2 +82.0 +82 .1 +82.2 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2 2 + 8.4 + 8.0 + 7.3 + 8.4 + 8.1 + 7.6 - 4.4 - 4.4 - 4.4 - 4.3 - 4.4 - 4.4 4 + 21 . 9 + 20.9 + 19 . 2 +22.1 + 21 .4 + 20. 2 - 6.2 - 6.2 - 6.2 - 6.2 - 6.2 - 6.2 6 +19.5 + 17.8 + 14.9 + 19.7 +18.6 + 16.6 - 8.6 - 8.5 - 8.5 - 8.6 - 8.5 - 8.5 8 + 5.4 + 3.4 - 0.1 + 5.6 + 4.3 + 2.0 - 9.9 - 9.9 - 9.8 - 9.9 - 9.9 - 9.8 10 - 0.8 - 2.8 - 6.3 - 0.5 - 1.8 - 4.1 -10.0 -10 . 0 -10.0 -10.0 -10.0 -10.0 12 - 3.5 - 5.5 - 9.0 - 3.3 - 4.6 - 6.8 -10 . 0 -10.0 -10.0 -10.0 -10.0 -10.0 14 -14.0 -15.7 -18.6 -13.8 -14.9 -16.7 - 9.3 - 9.2 - 9.2 - 9.3 - 9.2 - 9.2 16 -18.3 -19.3 - 2 1 . 0 -18.1 -18 .7 -19.7 - 7.6 - 7.6 - 7.5 - 7.6 - 7.6 - 7.5 18 -13 . 0 -13.4 -14.2 -13 . 0 -13.2 -13.6 - 5.9 - 5.9 - 5.9 - 5.9 - 5.9 - 5.9 20 -107 .2 -107.3 -107.5 -107 .2 -107 .3 -107 .5 0 . 0 0 . 0 0 . 0 0 . 0 0.0 0.0 Whole model Y' * 10^ V Y; + 10^ V Whole model - 1 . 6 - 4.4 - 9.2 - 1 . 2 - 3.1 - 6.2 -17 . 0 -16.9 -16.9 -17 . 0 -16.9 -16.9

h/T = 1 . 5 Y" * 1 0 ^ V Y'.' * 1 0 ^ V Fn = 0 . 0 6 7 5 Fn = 0 . 1 0 3 Fn = 0 . 0 6 7 5 Fn = 0 . 1 0 3 Ord. Nr: CO CO Ord. Nr: 0 . 2 6 0 . 5 0 0 . 7 5 0 . 2 6 0 . 5 0 0 . 7 5 0 . 2 6 0 . 5 0 0 . 7 5 0 . 2 6 0 . 5 0 0 . 7 5 0 + 1 0 1 . 1 + 1 0 1 . 0 + 1 0 0 . 9 + 1 0 1 . 1 + 1 0 1 . 0 + 1 0 0 . 9 - 0.2 - 0.2 - ,0.2 - 0.2 - 0.2 - 0.2 2 + 2 6 . 0 + 2 4 . 5 + 2 2 . 0 + 2 6 . 2 + 2 5 . 0 + 23 . 0 - 6.0 - 6.0 - 6 . 0 - 6.0 - 6.0 - 6 . 0 4 + 5 0 . 0 + 4 5 . 2 + 3 7 . 2 + 5 0 . 4 + 4 6 . 8 + 4 0 . 7 - 1 0 . 3 - 1 0 . 2 - 1 0 . 0 - 1 0 . 3 - 1 0 . 2 - 1 0 . 0 6 + 4 2 . 4 + 3 3 . 5 + 1 9 . 3 + 4 3 . 2 + 3 6 . 7 + 2 6 . 3 - 1 5 . 6 - 1 5 . 3 - 1 4 . 8 - 1 5 . 6 - 1 5 . 3 - 1 4 . 8 8 + 1 0 . 3 + 0.7 - 1 4 . 8 + 1 1 . 5 + 4.9 - 5.6 - 1 8 .6 - 1 8 . 1 - 1 7 .3 - 1 8 . 6 - 1 8 . 1 - 1 7 .3 1 0 - 3.5 - 1 2 . 7 - 2 7 . 4 - 2.3 - 8.3 - 1 8 . 0 - 1 8 . 9 - 1 8 . 4 - 1 7 .7 - 1 8 . 9. - 1 8 . 4 - 1 7 . 7 1 2 - 9.8 - 1 8 . 6 - 3 2 . 8 - 8.6 - 1 4 . 2 - 2 3 . 3 - 1 8 . 9 - 1 8 . 4 - 1 7 . 7 - 1 8 . 9 - 1 8 . 4 - 1 7 . 7 1 4 - 3 4 . 3 - 4 0 . 4 - 5 0 . 3 - 3 3 .2 - 3 6 . 6 - 4 2 . 2 - 1 7 . 2 - 1 6 . 8 - 1 6 . 1 - 1 7 . 2 - 1 6 . 8 - 1 6 . 1 1 6 - 4 7 . 3 - 5 0 . 0 - 5 4 . 5 - 4 6 . 7 - 4 7 . 8 - 4 9 . 7 - 1 3 . 0 - 1 2 . 8 - 1 2 . 5 - 1 3 . 0 - 1 2 . 8 - 1 2 . 5 1 8 - 3 7 . 9 - 3 8 . 8 - 4 0 . 4 - 3 7 . 7 - 3 8 . 0 - 3 8 . 6 - 8.6 - 8.5 - 8.5 - 8.6 - 8.5 - 8.5 2 0 - 1 3 7 .8 - 1 3 7 .7 - 1 3 7 .5 - 1 3 7 .8 - 1 3 7 .7 - 1 3 7 . 5 - 0.0 - 0.0 - 0.0 - 0.0 - 0.0 - 0.0 Whole model Y' * 1 0 ^ V Y: + 1 0 ^ V Whole model - 5 . 1 - 1 7 . 0 - 3 6 . 3 - 3 5 . 3 - 1 1 . 3 - 2 4 . 0 - 2 9 . 6 - 2 9 . 0 - 2 8 . 1 - 2 9 . 6 - 2 9 . 0 - 2 8 . 1

T a b l e 9; Sway ( C a l c u l a t i o n s . ) h/T = 1 . 1 5 Y" * ] 0 ^ V YV * 10^ V Fn = 0.0675 Fn = 0.103 Fn = 0.0675 Fn = 0.103 Ord., Nr: CO CO to Ord., Nr: 0.26 0.50 0.75 0.26 0.50 0.75 0.26 0.50 0.75 0.26 0.50 0.75 0 +146.3 +145.7 +144.6 + 146 .3 +145.7 +144.6 - 0.3 - 0.3 - 0.3 - 0.3 - 0.3 - 0.3 2 + 79.0 + 71.7 + 59.8 + 79.5 + 73.2 + 63.3 -10.2 -10.1 - 9.9 -10.2 -10.1 - 9.9 4 + 137 .3 +102.8 + 54.4 +139.3 +109.8 + 68.9 -22.2 -21 .1 -19.4 -22 .2 -21 .1 -19.4 6 +125.9 + 51.4 - 34.5 +131.2 + 68.9 - 1.6 -37 .6 -33 .3 -27 .8 -37.6 -33.3 -27 .8 8 - 0.2 - 5 8 . 1 -122.2 + 7.8 - 33.2 - 77.9 -46.3 -39.1 -30.9 -46.3 -39.1 -30.9 10 - 23.2 - 72.4 -128.7 - 15.2 - 47.5 - 84.4 -46.3 -39.1 -30.9 -46 .3 -39.1 -30.9 12 - 5.9 - 61.2 -122.8 + 2.1 - 36.3 - 78.5 -46.3 -39.1 -30.9 -46.3 -39.1 -30 . 9 14 -137.4 -141.2 -155.2 -130.6 -119.5 -115.7 -42.7 -36.8 -29.8 -42.7 -36.8 -29.8 16 -145. 2 -140.3 -138.8 -141.9 -128 . 9 -116.0 -29.8 -27 .5 -24 . 4 -29.8 -27 .5 -24 .4 18 -118.6 -116.5 -114.2 -117.6 -112.9 -106.5 -16.5 -16.1 -15 . 5 -16.5 -16.1 -15.5 20 -223 . 6 -221 . 9 -218.9 -223.6 -221.9 -218.9 0.0 0.0 0.0 0.0 0.0 0.0 Whole model Y' * 10^ V Y l + 10^ V Whole model - 29.0 - 91.9 -168.9 - 68.8 - 60.6 -110.8 -68.8 -60 . 6 -50.8 -68.8 -60.6 -50.8

* • Y V" - V V 1. s Fn S e c t i o n Nr. Y' V Whole model h/T Fn 1 2 3 4 5 6 7 Y' V Whole model h/T . 0675 .103 -3.9 -3.4 -10.8 -12.3 - 4.6 - 5.9 - 9.8 - 8.2 -20.4 -18.3 -33.5 -31.3 -60.6 -58. 0 -39.5 -32.7 1.5 1.5 . 0675 .103 -4.3 -1.8 - 5.5 - 7.5 - 2.8 - 4.0 - 3.8 - 3.1 - 9.3 - 9.5 -16.5 -16.1 - 4 1 . 0 -41.7 -21 .6 -19.6 2.4 2.4 . 0675 . 103 -8.0 -8.3 -13 . 8 - 3.3 -28 .2 -14.2 -56 .1 -41 . 5 -80.6 -63 . 2 -110.3 -93 .5 -151.5 -127 . 9 -144 . 9 -113.8 1.15 1.15

- 22

M O D U L A T E D C A R R I E R G E A R E D M O T O R S C O T C H Y O K E P H A S E S H I F T E R S T E E L pox G I R D E R D E M O D U L A T O R T = I N T E G R A T O R 3 S T R A I N G A U G E D Y N A M O M E T E R IN P H A S E C O M P O N E N T Q U A D R A T U R E C O M P O N E N T

F i g u r e 1. PRINCIPLE OF MECHANICAL OSCILLATOR AND ELECTRONIC CIRCUIT

SWAY -/.Ol 0 I 0 0 0 -40 Fn=0.0675 U)=0.26 Fn=0.103 W = Q26 Fn = 0.0675 CJ = Q50 1

I

2

I

3

U I ^ r e

--2 A = 1.2 ^ = 1.15 Experiment Fn = 0.103 (JO = 0.50 Vv "10 -/.Ol 0 .3 0 0 0 T T 2 _{I S} Fn=0.0675 (iJ=0,75 Fn=0.103 (0=0.75 1

I

2

I

3

I

^5

I

6

I

7 ^ ü = 2.4 ü=1.5 --1.15 F i g u r e 3: E x p e r i m e n t a l d i s t r i b u t i o n o f t h e damping c o e f f i c i e n t Y V

Experiment Fn = 0.103 ÜJ = 0.50 ) — ^ — )-) s — -- -\ -\ -\ .—. \ .—. \ / \ / 1 2 _{3 1 U} 5 6 7 Fn=0.0675 to = 0.75

M

2

I

3 Fn=0.103 U)=0.75 =3 ^ — - -- -- -•X /

\

-1 5 6 7 F i g u r e 4: E x p e r i m e n t a l d i s t r i b u t i o n o f t h e added mass c o e f f i c i e n t Y! V

STATIC TESTS Fn =0.0675 13=5° Fn = 0103 (3 = 5° -40 0 -40 0 -40, y^.10 3 0 -40 0 -40 1 ^ 1 I 2 I 3 I 4 ^-2.4 ^=1.5 ü=1.2 i h _{= 1.15} Fn=Q0675 (3=10° Fn =0.103 (3=10° -40 0 -40 0 -40 -40 0 -40 2 I 3 I 4 I

s K e

1 1 2 1 3 ^=2.4 5 P 6 R 1=1.5 ^=1.15 F i g u r e 5: D i s t r i b u t i o n o f t h e damping c o e f f i c i e n t f r o m s t a t i c t e s t s .

C a l c u l a t i o n 0 0) = ₂₆ A to = 50 • to = 0, 75 O OJ ' 0, 26 A to = _{0, 50} U to 0. 75 - + - s t a t i c B = 5

F i g u r e 6: Measured and c a l c u l a t e d d i s t r i b u t i o n o f Y" f o r

V

— — — C a l c u l a t i o n / • Q CO ~ 0, 26 A CO = 0. 50 • CO = 0. 75 0 CO = 0. 26 A CO = 0. 50 • CO = 0. 75

F i g u r e 7: Measured and c a l c u l a t e d d i s t r i b u t i o n o f YÏ. h/T = 2.4, 1.5 and 1.15.

F i g u r e 9: Comparison o f measured and c a l c u l a t e d c o e f f i c i e n t s f o r yawing as f u n c t i o n o f t h e w a t e r d e p t h - d r a u g h t r a t i o .