Mathematical modelling of catamaran hydromechnics and seakeeping

mmm

Maritime Research) Institute Netherlands

2, Haagsleeg, P.O. Box 28

6700 AA Wageningen, The Netherlands

Telephone + 31 8370 93911, Telex 45148 nsmb nl Telefax + 31 8370 93245 r n 1 6 0 9 1 8 - P a p e r CADMO'92 MATHEMATICAL MODELLING OF CATAMARAN HYDROMECHANICS AND SEAKEEPING B y : D a l l i n g a , H u b r e g t s e , S i r e g a r a n d R u s t e n b u r g J u n e 1 9 9 2 L J

MATHEMATICAL MODELLING OF CATAMARAN HYDROMECHANICS AND SEAKEEPING R.P. D a l l i n g a A.H. H u b r e g t s e F.R.T. S i r e g a r A. R u s t e n b u r g The M a r i t i m e Research I n s t i t u t e N e t h e r l a n d s The M a r i t i m e Research I n s t i t u t e N e t h e r l a n d s T e c h n i c a l U n i v e r s i t y D e l f t C o l l e g e o f Advanced T e c h n o l o g y , Haarlem, Department o f N a v a l A r c h i t e c t u r e ABSTRACT There i s a v i d e s p r e a d i n t e r e s t i n t h e use o f u n c o n v e n t i o n a l c r a f t f o r t r a n s p o r t and r e c r e a t i o n a l p u r p o s e s . V a r i o u s t y p e s , l i k e catamarans, SWATH, h y d r o f o i l s , s u r f a c e e f f e c t s h i p s and h y b r i d c o n c e p t s a r e c u r r e n t l y under s t u d y o r i n s e r v i c e around t h e v o r l d . Among t h e s e c r a f t t h e c o n v e n t i o n a l catamaran con-t i n u e s con-t o r e c e i v e s p e c i a l a con-t con-t e n con-t i o n because o f con-t h e r e l a con-t i v e l y l a r g e deck a r e a i n r e l a t i o n t o t h e main d i m e n s i o n s .

D e s p i t e t h e a d v a n t a g e s , t h e a p p l i c a t i o n o f t h e catamaran c o n c e p t i n a r e a ' s exposed t o vaves has sometimes l e a d t o a d i s a p p o i n t i n g p e r f o r m a n c e . I n o r d e r t o o b t a i n i n s i g h t i n t h e n a t u r e o f t h i s s e a k e e p i n g p r o b l e m t h e o r e t i c a l c a l c u l a t i o n s v e r e p e r f o r m e d v i t h t h r e e d i f f e r e n t programs. Tvo t v o - d i m e n s i o n a l and one t h r e e - d i m e n s i o n a l p o t e n t i a l t h e o r y method v e r e used t o e v a l u a t e t h e hydrodynamic c h a r a c t e r i s t i c s o f a medium speed c a t -amaran. The r e s u l t s v e r e compared m u t u a l l y and v i t h p u b l i s h e d e x p e r i m e n t a l d a t a . The d i f f e r e n c e s and agreements b e t v e e n t h e v a r i o u s r e s u l t s p r o v i d e i n s i g h t i n t h e n a t u r e o f catamaran h y d r o m e c h a n i c s . Based on t h e o b s e r v a t i o n s r e q u i r e m e n t s f o r an adequate m a t h e m a t i c a l model a r e f o r m u l a t e d . I n a d d i t i o n t o t h e m o d e l l i n g o f hydromechanics a method i s d e s c r i b e d t o g i v e t h e m o t i o n c h a r a c t e r i s t i c s a m e a n i n g f u l i n t e r -p r e t a t i o n . T h i s i s a c h i e v e d by a d o -p t i n g a s e t o f m o t i o n c r i t e r i a and a vave c l i m a t e . The m e r i t s o f t h i s t y p e o f " o p e r a b i l i t y " a n a l y s i s , i n c l u d i n g t h e s e l e c t i o n o f a p p r o p r i a t e c r i t e r i a and a vave c l i m a t e d e s c r i p t i o n a r e d i s c u s s e d .

1. INTRODUCTION

The o b j e c t i v e o f a s e a k e e p i n g a n a l y s i s i s t o q u a n t i f y t h e p e r formance i n v a v e s . F o r passenger v e s s e l s under n o r m a l o p e r a -t i o n a l c o n d i -t i o n s -t h e i m p o r -t a n -t a s p e c -t s o f -t h i s p e r f o r m a n c e a r e :

* passenger c o m f o r t governed by:

. sea s i c k n e s s , r e l a t e d t o v e r t i c a l a c c e l e r a t i o n l e v e l s . m o b i l i t y , r e l a t e d t o t r a n s v e r s e a c c e l e r a t i o n s

. p e r c e p t i o n o f t h e s h i p s b e h a v i o u r , r e l a t e d t o r o l l m o t i o n s and g l o b a l o r l o c a l vave i m p a c t phenomena

* t h e r e l i a b i l i t y o f t h e s e r v i c e governed by: . added r e s i s t a n c e i n v i n d and vaves

I n extreme c i r c u m s t a n c e s t h e s a f e t y o f t h e p a s s e n g e r s , r e l a t e d t o extreme a c c e l e r a t i o n l e v e l s and c a p s i z i n g , i s an i m -p o r t a n t i s s u e . Performance I n t h e o f f s h o r e i n d u s t r y i t i s q u i t e common t o e x p r e s s t h e p e r -formance o f a g i v e n v e s s e l i n terms o f t h e f r a c t i o n o f t i m e t h a t c r i t e r i a f o r v a r i o u s m o t i o n components a r e exceeded i n a d e s i g n " v a v e - c l i m a t e " . I t i s r e f e r r e d t o as " o p e r a b i l i t y " (nonexceedance o f c r i t e r i a ) o r "dovntime" (exceed(nonexceedance o f c r i t e r i a ) . A l -t h o u g h -t h i s q u a n -t i -t y i s n o -t -t h e o n l y and f i n a l measure f o r -t h e p e r f o r m a n c e o f a d e s i g n i t i s an i m p o r t a n t f i g u r e because i t r e -l a t e s t h e t h r e e most i m p o r t a n t i n g r e d i e n t s o f a p e r f o r m a n c e a n a l y s i s ; s h i p c h a r a c t e r i s t i c s , vave c l i m a t e and p e r f o r m a n c e c r i t e r i a . T h e o r e t i c a l frame v o r k The e v a l u a t i o n o f o p e r a b i l i t y f i g u r e s r e q u i r e s a p r e d i c t i o n o f t h e dynamic b e h a v i o u r f o r e v e r y p o s s i b l e vave c o n d i t i o n c o n s t i -t u -t i n g a vave c l i m a -t e . For -t h i s r e a s o n -t h e b a s i s o f mos-t me-thods i s l i n e a r ( o r l i n e a r i z e d ) s e a k e e p i n g t h e o r y . The a n a l y s i s con-s i con-s t con-s o f a c o n v o l u t i o n o f t h e m o t i o n c h a r a c t e r i con-s t i c con-s , i n t e r m con-s o f s o c a l l e d response f u n c t i o n s , and t h e v a r i o u s vave c o n d i -t i o n s , i n -t e r m s o f -t h e i r energy s p e c -t r a . I n -t h e f o l l o v i n g ve v i l l adhere t o t h i s a p p r o a c h , a l s o because i t y i e l d s v a l u a b l e i n s i g h t i n t h e n a t u r e o f t h e m o t i o n c h a r a c t e r i s t i c s o f catama-r a n s . 2. CATAMARAN HYDROMECHANICS 2.1. L i n e a r T h e o r y W i t h i n t h e f r a m e v o r k o f l i n e a r s e a k e e p i n g t h e o r y t h e m o t i o n r e s -ponse i n vaves may be t r e a t e d as t h e p r o d u c t o f t h e vave i n d u c e d f o r c e s a c t i n g on t h e c a p t i v e s h i p moving among vaves and t h e s t r u c t u r a l and hydrodynamic r e a c t i o n f o r c e s e x p e r i e n c e d by t h e o s c i l l a t i n g v e s s e l i n calm v a t e r . The l a t t e r a r e e x p r e s s e d i n t e r m s o f t h e h y d r o s t a t i c r e s t o r i n g f o r c e s , an i n e r t i a t e r m ("added mass") and a damping t e r m .

L i n e a r p o t e n t i a l t h e o r y i s used m o s t l y t o e v a l u a t e t h e com-ponents i n t h e e q u a t i o n s o f m o t i o n . I n g e n e r a l t h i s t h e o r y d e c r i b e s t h e f l u i d f l o v i n terms e i g h t v e l o c i t y p o t e n t i a l s , one f o r t h e i n c o m i n g u n d i s t u r b e d v a v e , one f o r t h e d i f f r a c t e d vave and s i x p o t e n t i a l s d e s c r i b i n g t h e f l o v due t o m o t i o n response o f

t h e s t r u c t u r e . The p r o b l e m o f q u a n t i f y i n g t h e v e l o c i t y p o t e n t i a l s c o r r e s p o n d s e s s e n t i a l l y t o t h e p r o b l e m o f f i n d i n g a p o t e n t i a l w h i c h , among o t h e r t h i n g s , a n n i h i l a t e s t h e p a r t i c l e v e l o c i t i e s p e r p e n d i c u l a r t o t h e h u l l . I n t h e case t h e i n c o m i n g wave p r o v i d e s t h e " d i s -t u r b a n c e s " -t h e r e s u l -t i s -t h e " d i f f r a c -t e d " wave, -t h e sum o f b o -t h y i e l d s t h e wave i n d u c e d f o r c e s . I n case t h e v a r i o u s m o t i o n com-ponents p r o v i d e t h e d i s t u r b a n c e t h e v e l o c i t y p o t e n t i a l s y i e l d t h e hydrodynamic r e a c t i o n f o r c e s ("added" mass and d a m p i n g ) .

The m o t i o n s f o l l o w from t h e wave i n d u c e d f o r c e s cuid h y d r o dynamic r e a c t i o n f o r c e s a f t e r a c c o u n t i n g f o r t h e mass d i s t r i b u -t i o n and -t h e h y d r o s -t a -t i c r e s -t o r i n g f o r c e s .

[m+a] + [ b ] + [ c ] = ^ = w i t h :

x^ = m o t i o n response i n 6 degrees o f freedom (see F i g u r e 1) F^ = wave i n d u c e d e x c i t a t i o n

[m]= s t r u c t u r a l mass d i s t r i b u t i o n (a] = added mass

(b] = hydrodynamic damping ( c ] = h y d r o s t a t i c r e s t o r i n g f o r c e s S t r i p t h e o r y U i t h i n " s t r i p " - t h e o r y t h e h u l l i s s c h e m a t i z e d i n t o two-dimen-s i o n a l c r o two-dimen-s two-dimen-s - two-dimen-s e c t i o n two-dimen-s . The hydrodynamic c h a r a c t e r i two-dimen-s t i c two-dimen-s o f t h e i n d i v i d u a l e l e m e n t s i n t e r m s o f t h e hydrodynamic r e a c t i o n f o r c e s i n sway, r o l l and heave a r e e v a l u a t e d w i t h o u t a c c o u n t f o r t h e t h r e e - d i m e n s i o n a l c h a r a c t e r o f t h e h u l l . The c h a r a c t e r i s t i c s o f t h e t o t a l h u l l ( a l s o i n p i t c h and yaw, s u r g e i s n e g l e c t e d ) a r e o b t a i n e d by i n t e g r a t i n g o v e r t h e l e n g t h o f t h e v e s s e l .

I n t h e p r e s e n t work two methods were used t o e v a l u a t e t h e c h a r a c t e r i s t i c s o f t h e t w o d i m e n s i o n a l s e c t i o n s : a boundary i n -t e g r a l and a c o n f o r m a b l e mapping -t e c h n i q u e . B o -t h me-thods a r e r e f e r r e d t o i n t e r m s o f t h e name o f t h e r e l a t e d computer code: S2D f o r t h e f i r s t and CATHO f o r t h e second approach.

I n t h e case o f nonzero f o r w a r d speed t h e r e s u l t s a r e c o r r e c t e d on b a s i s o f s l e n d e r body t h e o r y , t h e g r a d i e n t o f t h e r e -a c t i o n f o r c e s o v e r t h e l e n g t h o f t h e v e s s e l i s -a b -a s i c e l e m e n t i n t h e speed c o r r e c t i o n . V a r i o u s approaches e x i s t i n t h i s r e -s p e c t , b o t h -s t r i p t h e o r y program-s a r e ba-sed on " O r d i n a r y S t r i p Theory" ( F l o k s t r a , 1974, [ 2 ] ) . S2D

The program S2D i s based on a d i s c r e t i z a t i o n i n w h i c h t h e t w o -d i m e n s i o n a l h u l l c r o s s - s e c t i o n s a r e r e p r e s e n t e -d by a number o f n o d a l p o i n t s . The n a t u r e o f t h e s c h e m a t i z a t i o n a l l o w s a r b i t r a r y h u l l f o r m s , so t h e d o u b l e - h u l l c o n f i g u r a t i o n o f a catamaran can be a c c o u n t e d f o r .

The v e l o c i t y p o t e n t i a l s s a t i s f y t h e boundary c o n d i t i o n s e x a c t l y a t t h e n o d a l p o i n t s , t h e hydrodynamic r e a c t i o n f o r c e s a r e ob-t a i n e d a f ob-t e r i n ob-t e g r a ob-t i o n o f ob-t h e r e l a ob-t e d p r e s s u r e o v e r ob-t h e " p a n e l s " r e l a t e d t o each n o d a l p o i n t (Wu, 1987 [ 1 4 ] ) . 0 0 0 Boundary c o n d i t i o n s i A4>(x) (Vn)4'(x) ( - 0) 2 + g |-)<}>(x) The s o l u t i o n o f t h e t w o - d i m e n s i o n a l p o t e n t i a l i n t h e a f o r e m e n t i o n e d e q u a t i o n s can be d e r i v e d u s i n g f r o m Green's second i d e n t i t y : 1 1 r f x e V ())(x) = 4n * ( i ) ( V . n ) G(x,^) - G(x,^) (V.n)(}.(^)dS, x G T 0 J S X & V = 0 0 X i n f l u i d domain X on t h e body s u r f a c e X a t t h e f r e e s u r f a c e r a d i a t i o n c o n d i t i o n a t » i n w h i c h G d e s c r i b e s t h e Green's f u n c t i o n s a t i s f y i n g L a p l a c e and t h e boundary c o n d i t i o n s . The p o t e n t i a l a t t h e body s u r f a c e g i v e s t h e p r e s s u r e due t o u n i t m o t i o n s i n sway, heave and r o l l and t h e d i f f r a c t i o n f o r c e s : p = -p(}>-pgz S i n c e (})^ i s assumed t o be harmonic i n t i m e t h e f o r c e t r a n s -f e r -f u n c t i o n T ^ j t h e n becomes: 2 T.J = -ü) p ƒ (^.njdS S

The added mass and damping terms f o l l o w f r o m ; ^ i j ^ ^ <t>.njdS ; b^^ = -itop ƒ (j)^n dS

S S

The wave i n d u c e d f o r c e s a r e o b t a i n e d f r o m t h e i n c i d e n t wave and t h e hydromechanic r e a c t i o n f o r c e s by means o f an a p p r o x i m a -t i v e me-thod common -t o mos-t -t r a d i -t i o n a l s -t r i p -t h e o r y programs. I n t h e n e a r f u t u r e t h e e x a c t d i r e c t method w i l l be implemented. CATHO

The c a l c u l a t i o n o f t h e hydrodynamic p r o p e r t i e s o f t h e s h i p sec-t i o n s i n CATMO f o l l o w s sec-t r a d i sec-t i o n a l m o n o - h u l l s sec-t r i p - sec-t h e o r y . I sec-t i s based on t h e n o t i o n t h a t , by means o f a t r a n s f o r m a t i o n t e c h -n i q u e , a t y p i c a l s h i p s e c t i o -n ca-n be a p p r o x i m a t e d by a c i r c u l a r

shape, t h e c h a r a c t e r i s t i c s o f w h i c h can be c a l c u l a t e d a n a l y t i -c a l l y . The t r a n s f o r m a t i o n t e -c h n i q u e adopted i n t h e -c a l -c u l a t i o n s i s r e f e r r e d t o as t h e " F r a n k - c l o s e - f i t " method ( F l o k s t r a , 1974 [ 2 ] ) . The s c h e m a t i z a t i o n l i m i t s a p p l i c a t i o n t o s i m p l e symmetric m o n o - h u l l t y p e h u l l f o r m s , t h e t y p i c a l catamaran c o n f i g u r a t i o n can n o t be a c c o u n t e d f o r .

I n CATMO t h e catamaran c h a r a c t e r i s t i c s were e s t i m a t e d f r o m t h e s i n g l e - h u l l c h a r a c t e r i s t i c s by n e g l e c t i n g most i n t e r f e r e n c e s between b o t h h u l l s . The hydrodynamic r e a c t i o n f o r c e s i n t h e sway, yaw, heave and p i t c h modes a r e assumed t o be t w i c e t h e v a l u e s o f each h u l l . I n t h e r o l l mode a c o r r e c t i o n i s a p p l i e d f o r t h e l o c a l heave m o t i o n a s s o c i a t e d w i t h a r o l l m o t i o n around a p o i n t o u t s i d e t h e p l a n e o f symmetry o f an i n d i v i d u a l h u l l . For the added mass (and s i m i l a r f o r t h e damping) one o b t a i n s :

i ^ i ' " " •

'i\ir° ' •

The wave i n d u c e d f o r c e s a c t i n g on t h e p o r t and s t a r b o a r d s i d e h u l l a r e c a l c u l a t e d s e p a r a t e l y r e l a t i v e t o a h y p o t h e t i c a l o r i g i n on each h u l l and t r a n s f o r m e d t o t h e a c t u a l c e n t r e o f g r a v i t y t o o b t a i n t h e t o t a l wave f o r c e s (see F i g u r e 2 ) :

,CAT^,MONO ^ ,MONO . 2,3,5,6 ^SB ^PS

^CAT ^MONO ^ ^MONO ^ „MONO ^ ^ ^MONO F/ = F, + F, + r . F . + r.F

^SB ^PS ''SB ^PS

I n o r d e r t o f a c i l i t a t e t h e use o f CATMO t h e a s y m m e t r i c c r o s s - s e c t i o n s o f t h e p r e s e n t v e s s e l (see F i g u r e 3) were schema-t i z e d i n schema-t o symmeschema-tric s e c schema-t i o n s w i schema-t h "average" p r o p e r schema-t i e s .

Three d i m e n s i o n a l t h e o r y

The c a l c u l a t i o n u n d e r l y i n g t h e t h r e e d i m e n s i o n a l approach i s , l i k e S2D, based on a n o d a l r e p r e s e n t a t i o n o f t h e h u l l , b u t i n t h i s case d i v i d i n g t h e h u l l i n t o two d i m e n s i o n a l p a n e l s . The so-l u t i o n o f t h e p o t e n t i a so-l i s based on t h e boundary c o n d i t i o n s f o r the t h r e e d i m e n s i o n a l f l u i d domain. The p o t e n t i a l i s s o l v e d u s i n g a Green's f u n c t i o n , s a t i s f y i n g t h e t h r e e - d i m e n s i o n a l boun-d a r y c o n boun-d i t i o n s . I n t h i s p o t e n t i a l t h e i n f l u e n c e o f a l l o t h e r p a n e l s i s t a k e n i n t o a c c o u n t and a t h r e e d i m e n s i o n a l s o l u t i o n i s o b t a i n e d . C o n t r a r y t o t h e s t r i p t h e o r y w h i c h n e g l e c t s t h e i n t e r -a c t i o n s o f t h e d i f f e r e n t s e c t i o n s . T h i s s c h e m -a t i z -a t i o n -a l l o w s a r b i t r a r y h u l l f o r m s .

The t h e o r y u n d e r l y i n g t h e program was d e s c r i b e d by van Oortmerssen, 1976 [ 9 ] . Forward speed i s p r e s e n t l y n o t i n c l u d e d . The r e s u l t s a r e r e f e r r e d t o i n terms o f t h e name o f t h e computer code: DIFFRAC

2.2. C o r r e l a t i o n

The r e s u l t s o f t h e t h r e e c a l c u l a t i o n methods i n t e r m s o f m o t i o n s were compared m u t u a l l y and w i t h t h e r e s u l t s o f e x p e r i m e n t a l w o r k p u b l i s h e d by DTNSRDC (Wahab, 1971 [ 1 3 ] , Lee, 1973 [ 7 ] ) . The w o r k c o n c e r n e d seakeeping e x p e r i m e n t s w i t h a model o f a 64 m Sub-m a r i n e Rescue Ship o f t h e ASR 21 c l a s s . T e s t s were c a r r i e d o u t f o r a range o f r e g u l a r waves, speeds, h e a d i n g s and h u l l s e p a r a -t i o n s y i e l d i n g i n -t e r e s -t i n g d a -t a f o r v a l i d a -t i o n o f a m a -t h e m a -t i c a l model. The main p a r t i c u l a r s a r e i n d i c a t e d i n T a b l e 1 and F i g u r e 3.

2.2.1. Hydrodynamic r e a c t i o n f o r c e s

The hydrodynamic r e a c t i o n f o r c e s i n terms o f added mass and damping a t z e r o - f o r w a r d speed a r e i n d i c a t e d i n F i g u r e s 5 and 6. I n t e r a c t i o n e f f e c t s

Hydromechanic i n t e r a c t i o n o f t h e two h u l l s m a n i f e s t s i t s e l f i n t h e v e r t i c a l as w e l l as i n t h e h o r i z o n t a l p l a n e . I n t h e v e r t i c a l p l a n e dynamic a m p l i f i c a t i o n e f f e c t s a r e observed around 0.95 r a d / s . T h i s f r e q u e n c y i s c l o s e t o an a p p r o x i m a t i o n o f t h e n a t u -r a l f -r e q u e n c y i n t h e heave mode o f t h e w a t e -r column i n between t h e h u l l s g i v e n by:

"z " J(c^/Cm+a^)

w i t h c as t h e h y d r o s t a t i c " s p r i n g s t i f f n e s s " , m as t h e mass o f w a t e r t r a p p e d i n between t h e h u l l s and a t h e "added mass". Re-g a r d i n Re-g t h e l a t t e r t h i s q u a n t i t y can be a p p r o x i m a t e d by t h e volume o f a h a l f - c y l i n d e r below t h e h u l l s p a c i n g .

I n t h e t r a n s v e r s e mode i n t e r a c t i o n s a r e observed around 1.7 r a d / s . T h i s f r e q u e n c y c o r r e s p o n d s c l o s e l y w i t h a s t a n d i n g asym-m e t r i c wave i n between t h e h u l l s (see F i g u r e 4 ) .

The two programs a c c o u n t i n g f o r t h e i n t e r a c t i o n s between t h e h u l l s y i e l d v e r y d i f f e r e n t r e s u l t s , t h e t w o - d i m e n s i o n a l program p r e d i c t s t h e s t r o n g e s t i n t e r a c t i o n s , e s p e c i a l l y i n t h e p i t c h mode. C o n s i d e r i n g t h e d i f f e r e n c e between b o t h approaches o v e r e s t i m a t i o n o f e n d - e f f e c t s must be r e s p o n s i b l e f o r t h e ob-s e r v e d d i f f e r e n c e ob-s .

The f a c t t h a t v i s c o u s e f f e c t s a r e n e g l e c t e d i n t h e c a l c u l a -t i o n me-thods means -t h a -t -t h e dynamic a m p l i f i c a -t i o n e f f e c -t s i n be-tween t h e h u l l s a r e e x a g g e r a t e d by p o t e n t i a l t h e o r y , e s p e c i a l l y i n t h e t w o - d i m e n s i o n a l case. Comparing t h e p r e s e n t r e s u l t s w i t h r e s u l t s o f f o r c e d o s c i l l a t i o n t e s t s w i t h v a r i o u s two d i m e n s i o n a l t w i n - h u l l c o n f i g u r a t i o n s (Lee, 1971 [ 6 ] ) t h i s v i e w seems t o be c o n f i r m e d q u a l i t a t i v e l y . I n o r d e r t o o b t a i n a p r a c t i c a l d e s i g n t o o l t h e c a l c u l a t e d r e s u l t s s h o u l d be c o r r e c t e d t o some e x t e n t i f t h e i n t e r f e r e n c e f r e q u e n c y i s i n a r e l e v a n t wave f r e q u e n c y r a n g e .

V i s c o u s e f f e c t s

Vhen c o n s i d e r i n g c o n v e n t i o n a l m o n o h u l l s g e n e r a l p r a c t i c e i s t o a c c o u n t f o r t h e f a c t t h a t t h e r o l l damping due wave making e f -f e c t s i s s m a l l . T h i s i s a c h i e v e d by a d d i n g s e m i - e m p e r i c a l es-t i m a es-t e s f o r es-t h e damping f r o m o es-t h e r s o u r c e s . R e g a r d i n g caes-tamaran s t r u c t u r e s one m i g h t be i n c l i n e d t o assume t h a t t h e wave making damping i n t h e r o l l mode i s l a r g e r . The t a b l e below summarizes t h e c a l c u l a t e d r o l l c h a r a c t e r i s t i c s i n beam seas a t z e r o speed, t o g e t h e r w i t h t h e r e s u l t s o f m o t i o n decay t e s t s (Wahab, 1971 [ 1 3 ] ) . Program Nat. f r e q . [ r a d / s ] R o l l Peak r o l l response Program Nat. f r e q . [ r a d / s ] E x c i t . [MNm/m] Damping [MNms] Coupled [deg/m] Uncoupled [deg/m] CATHO 1.14 22.6 158 9.4 8.0 S2D 1.05 7.3 21 16.5 20.0 DIFFRAC 1.01 14.1 31 19.3 26.0 E x p e r i m e n t s 1.02

_-

60 11.5

-A f i r s t o b s e r v a t i o n i s t h e l a r g e d i f f e r e n c e s between t h e v a r i o u s t h e o r e t i c a l approaches, t h e second o b s e r v a t i o n i s t h e f a c t t h a t t h e most advanced c a l c u l a t i o n methods u n d e r e s t i m a t e t h e r o l l damping. T h i s i n d i c a t e s t h a t , l i k e i n t h e case f o r m o n o h u l l s , r o l l damping f r o m non-wavemaking s o u r c e s i s i m p o r t a n t as w e l l and s h o u l d be a c c o u n t e d f o r .

R e g a r d i n g t h e heave and p i t c h c h a r a c t e r i s t i c s i t i s n o t e -w o r t h y t h a t t h e a b s o l u t e magnitude o f t h e -wave making damping

(as a c c o u n t e d f o r by p o t e n t i a l t h e o r y ) i s r e l a t i v e l y s m a l l . T h i s i s t h e r e s u l t o f t h e l i m i t e d b r e a d t h t o d r a f t r a t i o o f t h e i n d i -v i d u a l h u l l s , see V u g t s , 1970 [ 1 2 ] .

2.2.2. Wave i n d u c e d f o r c e s

The r e s u l t s o f t h e c a l c u l a t i o n s r e g a r d i n g t h e wave i n d u c e d f o r c e s a r e shown i n F i g u r e 7 f o r t h e head and beam sea c a s e s .

The e f f e c t o f symmetric i n t e r a c t i o n s between t h e h u l l s a r o u n d 0.95 r a d / s i s c l e a r l y p r e s e n t i n t h e heave and p i t c h ex-c i t a t i o n . The f a ex-c t t h a t i n head seas t h e S2D p i t ex-c h e x ex-c i t a t i o n i s l a r g e s t a t t h e s e f r e q u e n c i e s i n d i c a t e s t h a t e x a g g e r a t i o n o f t h e i n t e r a c t i o n s , t o g e t h e r w i t h an e x a g g e r a t i o n o f t h e c o n t r i b u t i o n o f t h e v e s s e l s e x t r e m i t i e s , c o n t r i b u t e t o t h e d i f f e r e n c e s .

A l s o sway and r o l l e x c i t a t i o n l e v e l s as p r e d i c t e d by t h e t h r e e programs show c o n s i d e r a b l e d i f f e r e n c e s . I n t h e sway mode t h e two d i m e n s i o n a l approaches y i e l d r e l a t i v e l y h i g h e x c i t a t i o n l e v e l s . Around 0.8 r a d / s t h e wave l e n g t h i s n e a r l y t w i c e t h e l e n g t h o f t h e v e s s e l . Keeping i n mind t h a t i n t h i s a r e a t h r e e

d i m e n s i o n a l d i f f r a c t i o n e f f e c t s around a s i n g l e h u l l a r e l i k e l y t o m a n i f e s t themselves i t i s suggested t h a t t h e t w o - d i m e n s i o n a l s c h e m a t i z a t i o n i s one o f t h e main reasons f o r t h e d i f f e r e n c e s . R e g a r d i n g t h e r o l l t h e l a r g e s t d i f f e r e n c e s o c c u r a t h i g h e r f r e -q u e n c i e s where " s h a d i n g " i s l i k e l y t o o c c u r : t h e l e e w a r d h u l l i s l e s s exposed t h a n t h e windward h u l l . The f a c t t h a t S2D ( t w o d i -m e n s i o n a l so e x a g g e r a t e d s h a d i n g ) u n d e r e s t i -m a t e s t h e e x c i t a t i o n and CATHO (no s h a d i n g ) o v e r e s t i m a t e s t h e e x c i t a t i o n seems t o c o n f i r m t h i s h j ^ p o t h e s i s .

2.2.3. M o t i o n response

The m o t i o n c h a r a c t e r i s t i c s o f s h i p s and s t r u c t u r e s a r e m o s t l y p r e s e n t e d i n terms o f "response f u n c t i o n s " o r "response a m p l i -t u d e o p e r a -t o r " RAO r e p r e s e n -t i n g -t h e magni-tude o f a m o -t i o n com-ponent p e r meter wave h e i g h t as a f u n c t i o n o f t h e wave p e r i o d . The RAO'S a r e a f u n c t i o n o f speed and h e a d i n g .

The heave, r o l l and p i t c h response as p r e d i c t e d w i t h t h e t w o -d i m e n s i o n a l metho-ds a r e compare-d w i t h t h e Wahab's e x p e r i m e n t a l r e s u l t s i n F i g u r e s 8 and 9.

C o n s i d e r i n g t h e d i f f e r e n c e s i n t h e u n d e r l y i n g wave i n d u c e d e x c i t a t i o n and hydrodynamic r e a c t i o n f o r c e s t h e m u t u a l c o r r e l a -t i o n be-tween -t h e -t h e o r e -t i c a l r e s u l -t s and -t h e -t e s -t r e s u l -t s i s s u r p r i s i n g l y good. A c t u a l l y f o r most o f t h e d a t a p r e s e n t e d by Wahab, 1974 [ 1 3 ] f o r v a r i o u s headings and h u l l s p a c i n g s a v e r y r e a s o n a b l e c o r r e l a t i o n i s o b t a i n e d .

The asymmetric i n t e r a c t i o n i n - b e t w e e n t h e h u l l s can be r e c o g n i s e d i n t h e S-2D r e s u l t s , i t shows as a second peak i n t h e r o l l response i n bow q u a r t e r i n g waves. The t e s t r e s u l t s a t ( n o n -z e r o speed) do n o t r e f l e c t t h i s i n t e r a c t i o n . T h i s agrees w i t h i n - h o u s e e x p e r i e n c e , t h e c h a r a c t e r o f t h e i n t e r a c t i o n between two h u l l s i s a f f e c t e d s t r o n g l y by f o r w a r d speed i n a way w h i c h i s n o t a c c o u n t e d f o r by s t r i p t h e o r y .

The l o w damping o f t h e h u l l i n t h e v e r t i c a l p l a n e i s r e -f l e c t e d i n dynamic a m p l i -f i c a t i o n e -f -f e c t s i n t h e heave response (more t h a n 2 m/m). H e a v e - p i t c h c r o s s c o u p l i n g i n t r o d u c e s t h e d o u b l e peaked c h a r a c t e r o f t h e p i t c h response a t t h e n a t u r a l f r e q u e n c y o f heave ( r e m o v i n g t h e c o u p l i n g removes t h e d o u b l e peaked c h a r a c t e r ) .

The r e s u l t s i n F i g u r e 10 show t h a t , l i k e i n t h e case o f t h e m o n o h u l l , p o t e n t i a l t h e o r y u n d e r e s t i m a t e s t h e r o l l damping. The e x a g g e r a t e d r o l l damping i n CATMO (see S e c t i o n 2.2.1.) p r o v i d e s a v e r y r e a s o n a b l e e s t i m a t e o f t h e r o l l r e s p o n s e .

3. OPERABILITY

The b a s i c i n g r e d i e n t s o f any performance a n a l y s i s a r e t h e b a s i c c h a r a c t e r i s t i c s o f t h e v e s s e l t h e p r e v a i l i n g wave c l i m a t e and c r i t e r i a f o r "maximum" b e h a v i o u r . The r e s u l t s a r e a f f e c t e d by

t h e " m i s s i o n " o f t h e v e s s e l w h i c h governs q u a n t i t i e s l i k e speed and h e a d i n g .

3.1. M o t i o n c h a r a c t e r i s t i c s

The s i x degrees o f freedom m o t i o n s a t t h e c e n t r e o f g r a v i t y may be r e g a r d e d as t h e " b a s i c " c h a r a c t e r i s t i c o f a s t r u c t u r e . These r e s u l t s a l l o w d e r i v a t i o n o f i n t e r e s t i n g q u a n t i t i e s l i k e v e r t i c a l and h o r i z o n t a l a c c e l e r a t i o n l e v e l s and t h e r e l a t i v e wave e l e v a t i o n ( f o r i n s t a n c e a t t h e f r o n t o f t h e b r i d g e s t r u c t u r e c o n n e c t -i n g t h e h u l l s ) . The heave, r o l l and p -i t c h c h a r a c t e r -i s t -i c s o f t h e p r e s e n t v e s s e l as p r e d i c t e d by CATMO a r e i n d i c a t e d i n F i g u r e 11 as a f u n c t i o n o f wave f r e q u e n c y and h e a d i n g . They show t h a t i n low f r e q u e n c y , l o n g waves t h e v e s s e l f o l l o w s t h e wave p r o f i l e : the r o l l and p i t c h a n g l e s r e l a t e t o t h e e f f e c t i v e wave s l o p e , the heave m o t i o n t o t h e wave a m p l i t u d e . I n h i g h f r e q u e n c y , v e r y s h o r t waves t h e e x c i t a t i o n l e v e l s d r o p : t h e l o c a l f o r c e s c a n c e l o u t a f t e r i n t e g r a t i o n over t h e e n t i r e h u l l . I n a d d i t i o n t h e i n -e r t i a f o r c -e s i n c r -e a s -e b-ecaus-e o f t h -e h i g h -e r f r -e q u -e n c i -e s asso-c i a t e d w i t h s h o r t waves.

The maximum response i s f o u n d i n between t h e h i g h and l o w f r e -quency regime: t h e r e l a t i v e l y l a r g e peaks i n t h e response a r e r e l a t e d t o dynamic a m p l i f i c a t i o n e f f e c t s .

A c c e l e r a t i o n s

A c c e l e r a t i o n s a r e t h e p r o d u c t o f t h e m o t i o n a m p l i t u d e and t h e square 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 . As a r e s u l t r e l a t i v e l y h i g h v a l u e s o f t h e v e r t i c a l a c c e l e r a t i o n s a r e o b t a i n e d i n head and bow q u a r t e r i n g waves. The h o r i z o n t a l a c c e l e r a t i o n s a r e g e n e r a l l y e x p r e s s e d i n t e r m s o f e f f e c t i v e v a l u e s i n a s h i p bound system o f c o o r d i n a t e s ( a l o n g t h e d e c k ) . T h i s means t h a t t h e y i n c l u d e t h e g r a v i t y component due t o f i n i t e a n g l e s o f r o l l and p i t c h . T h i s e x p l a i n s p a r t l y t h e s i m i l a r c h a r a c t e r o f t h e r o l l response and t r a n s v e r s e a c c e l e r a t i o n s . 3.2. Vave c l i m a t e B e a u f o r t Numbers The s i m p l e s t way t o c h a r a c t e r i s e an o f f s h o r e e n v i r o n m e n t i s t o c h a r a c t e r i s e t h e w i n d c l i m a t e , f o r i n s t a n c e by t h e f r e q u e n c y o f o c c u r r e n c e o f t h e v a r i o u s B e a u f o r t numbers, and r e l a t i n g t h e s e w i n d c l a s s e s t o "average" wave c o n d i t i o n s . A l t h o u g h o f t e n used i n s h i p o p e r a t i o n s t h i s approach f a i l s t o a c c o u n t f o r t h e f a c t t h a t one wave h e i g h t can show d i f f e r e n t p e r i o d c h a r a c t e r i s t i c s . The f a c t t h a t t h e w i n d speed and d i r e c -t i o n a r e n o -t v e r y c o n s -t a n -t f a c -t o r s means -t h a -t -t h e waves a r e h a r d l y e v e r i n an e q u i l i b r i u m c o n d i t i o n . I f t h e waves a r e g r o w i n g most o f t h e e n e r g y i n p u t w i l l t a k e p l a c e a t t h e h i g h f r e q u e n c y t a i l o f t h e wave s p e c t r u m , r e s u l t i n g i n a s t e e p w i n d wave. T h i s i m p l i e s t h a t t h e average p e r i o d w i l l be r e l a t i v e l y s h o r t . I f t h e w i n d speed d r o p s t h e c o n t i n u i n g

non-l i n e a r t r a n s f e r o f wave energy from h i g h e r t o non-l o w e r f r e q u e n c i e s w i l l t u r n t h e w i n d wave i n t o a l o n g p e r i o d s w e l l .

The s i t u a t i o n i s a g g r a v a t e d by t h e f a c t t h a t t h e b e h a v i o u r o f s h i p s and o f f s h o r e s t r u c t u r e s i s q u i t e s e n s i t i v e t o wave h e i g h t and wave p e r i o d ( m o t i o n r e s p o n s e ) and t h e r e l a t e d w i n d speed (speed l o s s , dynamic p o s i t i o n i n g ) .

S c a t t e r d i a g r a m s

I n t h e o f f s h o r e i n d u s t r y t h e a v a i l a b i l i t y o f wave measurements has l e a d t o t h e i n t r o d u c t i o n and use o f s o c a l l e d wave s c a t t e r -d i a g r a m s , w h i c h r e f l e c t t h e j o i n t s t a t i s t i c s o f wave h e i g h t an-d average wave p e r i o d . The wave c o n d i t i o n a t a p a r t i c u l a r i n s t a n t i s c h a r a c t e r i s e d by two p a r a m e t e r s , t h e s i g n i f i c a n t wave h e i g h t and t h e average p e r i o d (most f r e q u e n t l y t h e average z e r o u p -c r o s s i n g p e r i o d ) . A " t y p i -c a l " s i n g l e - p e a k e d s p e -c t r a l shape ( f o r i n s t a n c e t h e JONSWAP o r ISSC f o r m u l a t i o n ) i s used t o model t h e wave s p e c t r u m . A l t h o u g h t h i s s c h e m a t i z a t i o n i s r a t h e r u n s a t i s

-f a c t o r y i n t h e -f r e q u e n t case o -f combined w i n d , wave and s w e l l s c a t t e r diagrams seem a good s t a r t i n g p o i n t f o r an o p e r a b i l i t y a n a l y s i s . '

The sample a n a l y s i s i n t h e p r e s e n t work i s based on wave s t a t i s t i c s p u b l i s h e d by BMT (Hogben, 1986 [ 4 ] ) . They r e p r e s e n t average a n n u a l c o n d i t i o n s i n t h e w e s t e r n M e d i t e r r a n e a n sea. 3.3. C r i t e r i a P r e s e n t l y t h e p e r f o r m a n c e o f passenger s h i p s , c r u i s e l i n e r s and f e r r i e s i s o f t e n r e l a t e d t o passenger c o m f o r t . A l t h o u g h o f t e n n e g l e c t e d t h e p e r f o r m a n c e o f t h e crew may a l s o y i e l d o p e r a t i o n a l l i m i t s . An i n c r e a s i n g volume o f p a s s e n g e r s , a t h i g h e r speeds i n more exposed a r e a ' s o f t h e w o r l d , t o g e t h e r w i t h h i g h e r demands on t h e p e r f o r m a n c e o f t h e crew o f modern s h i p s ( a s m a l l e r crew w i t h more c o m p l i c a t e d t a s k s ) may l e a d t o more s t r i n g e n t c r i t e r i a f o r t h e m o t i o n l e v e l s i n t h e near f u t u r e .

C o m f o r t c r i t e r i a a r e g e n e r a l l y r e l a t e d t o t h e m o b i l i t y o f and s e a s i c k n e s s . The l a t t e r i s m o s t l y r e l a t e d t o v e r t i c a l a c c e l e r -a t i o n l e v e l s . M o b i l i t y o f ( s t -a n d i n g ) p-assengers -and p e r s o n n e l i s governed by t h e combined o c c u r r e n c e o f v e r t i c a l and t r a n s v e r s e a c c e l e r a t i o n s , ( D a l l i n g a , 1992 [ 1 ] , Graham, 1990 [ 3 ] ) .

I n o r d e r t o d e m o n s t r a t e t h e c h a r a c t e r i s t i c s o f a catamaran v e s s e l i n t e r m s o f p e r f o r m a n c e as a passenger v e s s e l c r i t e r i a r e g a r d i n g s e a s i c k n e s s and m o b i l i t y were a d o p t e d . The v a l u e s s u g g e s t e d f o r t h e v e r t i c a l and t r a n s v e r s e a c c e l e r a t i o n l e v e l s i n t h e N o r d f o r s k s t u d y ( N o r d f o r s k , 1987 ( 8 ] ) were used. They a r e summarised on t h e n e x t page.

3.4. Performance

I n F i g u r e 12 sample o p e r a b i l i t y f i g u r e s a r e d e r i v e d f o r head and beam seas f o r v a r i o u s h u l l s p a c i n g s . As a c o m p a r i s o n t h e c h a r a c

-t e r i s -t i c s o f a -t y p i c a l 64 m m o n o h u l l ( w i -t h r o l l s -t a b i l i z e r s ) a r e i n c l u d e d . The r e s u l t s show t h a t t h e s m a l l e s t h u l l s p a c i n g p r o v i d e s t h e l a r g e s t r o l l a n g l e s , a l t h o u g h t h e e f f e c t o f h u l l s p a c i n g i s n o t v e r y l a r g e . The r e l a t e d o p e r a b i l i t y i s around 65X, w h i c h i s n o t p a r t i c u l a r l y h i g h compared t o t h e p e r f o r m a n c e o f t h e m o n o h u l l . U n i t s RMS S i g n i f i c a n t d o u b l e a m p l i t u d e A v . l / N t h s i n g l e a m p l i t u d e S i g n i f i c a n t d o u b l e a m p l i t u d e N = 1 N = 3 T r a n s i t passengers T r a n s v . a c c e l . [m/s^] 0.4 1.6 0.50 0.8 V e r t i c . a c c e l . R o l l a n g l e [m/s^] [deg] 0.5 2.5 2.0 10.0 0.63 3.13 1.0 5.0 I n terms o f t h e t r a n s v e r s e a c c e l e r a t i o n s t h e catamaran w i t h t h e s m a l l e s t h u l l s p a c i n g a l s o p r o v i d e s t h e l a r g e s t r e s p o n s e . C o n t r a r y t o t h e r o l l response t h e p e r f o r m a n c e i s s i m i l a r t o t h a t o f t h e m o n o h u l l . The r e a s o n f o r t h i s phenomenon may be r e l a t e d t o t h e r e l a t i v e p h a s i n g o f t h e r o l l and sway m o t i o n s . I t was a l s o o b s e r v e d on b o a r d m o n o h u l l s w i t h a r e l a t i v e l y h i g h t r a n s -v e r s e s t a b i l i t y ( H u t c h i s s o n , 1981 [ 5 ] ) .

The dynamic a m p l i f i c a t i o n e f f e c t s i n heave and p i t c h a r e , t h e main r e a s o n f o r t h e f a c t t h a t , d e s p i t e t h e same l e n g t h , t h e p e r f o r m a n c e o f t h i s catamaran w i t h r e l a t i v e l y n a r r o w h u l l s i s l e s s t h e n t h a t o f t h e m o n o - h u l l . 4. CONCLUSIONS The r e s u l t s o f t h e p r e s e n t i n v e s t i g a t i o n d e m o n s t r a t e t h e c o m p l i c a t e d n a t u r e o f catamaran hydromechanics. D e s p i t e t h i s c o m p l e x i -t y and -t h e f a c -t -t h a -t -t h e p r e d i c -t i o n programs do n o -t a c c o u n -t f o r some p h y s i c a l phenomena, l i k e speed dependent ' s h a d i n g e f f e c t s ' , i t may be c o n c l u d e d t h a t s t r i p t h e o r y p r o v i d e s a v e r y r e a s o n a b l e f i r s t e s t i m a t e o f t h e dynamics o f catamaran v e s s e l . The more advanced method, w h i c h a c c o u n t s f o r t w o - d i m e n s i o n a l i n t e r a c t i o n , r e q u i r e s a d d i t i o n a l r o l l damping and a way t o a c c o u n t f o r v i s -cous and speed e f f e c t s e f f e c t s i n t h e i n t e r a c t i o n modes.

The r e s u l t s o f t h e o p e r a b i l i t y a n a l y s i s show t h a t t h e mo-t i o n c h a r a c mo-t e r i s mo-t i c s can be g i v e n a m e a n i n g f u l i n mo-t e r p r e mo-t a mo-t i o n i n terms o f passenger c o m f o r t . Comparing t h e r e s u l t s w i t h t h o s e f o r a m o n o h u l l o f s i m i l a r l e n g t h i t was shown t h a t , d e s p i t e l a r g e r r o l l a n g l e s , t h e t r a n s v e r s e a c c e l e r a t i o n l e v e l s o f t h e catamaran are n o t e x c e s s i v e .

The r e s u l t s a l s o d e m o n s t r a t e a d i s a d v a n t a g e o f v e r y n a r r o w h u l l s ; t h e l a c k o f wave making damping y i e l d s r e l a t i v e l y h i g h heave and p i t c h m o t i o n s .

ACKNOWLEDGEMENT

The p r e s e n t p u b l i c a t i o n i s based on work by R u s t e n b u r g , 1991 [ 1 0 ] and S i r e g a r , 1991 [ 1 1 ] . T h e i r e n t h u s i a s m and commitment i s g r a t e f u l l y acknowledged.

BIBLIOGRAPHY

1. D a l l i n g a , R.P.; " S e l e c t i o n on Seakeeping", MARIN '60 Work-shop on Advanced V e s s e l s , Wageningen 1992.

2. F l o k s t r a , C.; "Comparison o f S h i p M o t i o n T h e o r i e s w i t h E x p e r i m e n t s f o r a C o n t a i n e r s h i p , P u b l . No. 462, MARIN, 1974. 3. Graham, R.; " M o t i o n - I n d u c e d I n t e r r u p t i o n s as O p e r a b i l i t y

C r i t e r i a " , N a v a l E n g i n e e r s J o u r n a l , March 1990.

4. Hogben, N., Dacuhna, N.M.C. and O l l i v e r , G.F.; " G l o b a l Wave S t a t i s t i c s " , BMT, London, 1986.

5. H u t c h i s s o n , B.L. " R i s k and O p e r a b i l i t y A n a l y s i s i n t h e M a r i n e E n v i r o n m e n t " , SNAME t r a n s a c t i o n s . V o l . '89, 1981. 6. Lee, CM., Jones, H.D. and B e d e l , J.W., "Added Mass and

Damping C o e f f i c i e n t s o f Heaving Twin C y l i n d e r s i n a Free S u r f a c e , R e p o r t 3695 NSRDC Bethesda, August 1971.

7. Lee, CM., Jones, H.D. and Curphey, R.M., " P r e d i c t i o n o f M o t i o n and Hydrodynamic Loads o f Catamarans, H a r i n e Tech-n o l o g y , October 1973.

8. N o r d f o r s k , "Assessment o f Ship Performance i n a Seaway", JSBN 87-982637-1-4.

9. Oortmerssen, G. van; "The M o t i o n s o f a Moored S h i p i n Waves, P u b l i c a t i o n No. 510 NSMB, 1976.

10. R u s t e n b u r g , A.; "Seakeeping Performance o f Catamarans", A f s t u d e e r s c r i p t i e , T.H. Haarlem, m e i 1991.

11. S i r e g a r , F.R.T.; Comparison o f Three C a l c u l a t i o n Methods f o r Hydrodynamic C h a r a c t e r i s t i c s and M o t i o n s o f Catamarans i n Waves".

12. V u g t s , J.H.; "The Hydrodynamic Forces and S h i p M o t i o n s i n O b l i q u e Waves", Rep. No. 150 s ( S 2 . 2 3 8 ) , TNO D e l f t , 1970. 13. Wahab, R., P r i t c h e t t , C. and R u t h , L . C ; "On t h e B e h a v i o u r

o f t h e ASR Catamaran i n Waves", M a r i n e T e c h n o l o g y , J u l y 1971.

14. Wu, X.J. and P r i c e , W.G.; "A M u l t i p l e Green's f u n c t i o n ex-p r e s s i o n f o r t h e Hydrodynamic a n a l y s i s o f M u l t i - h u l l S t r u c t u r e s " , A p p l i e d Ocean r e s e a r c h . V o l . 9, No. 2, 1987.

T a b l e 1 Main p a r t i c u l a r s o f t h e ASR catamaran D e n o m i n a t i o n Symbol U n i t Value Length between p e r p e n d i c u l a r s ^PP m 64.00 Beam o v e r a l l B m 26.20 Beam o f one h u l l ^H m 7.30 D r a f t D m 5.50 H u l l s e p a r a t i o n (between h u l l s ) S m 11.60 H u l l s e p a r a t i o n ( h e a r t - t o - h e a r t ) r m 18.90 D i s p l a c e m e n t t o n s 2750 L o n g i t u d i n a l c e n t r e o f g r a v i t y a f t o f F.P. m 32.16 V e r t i c a l c e n t r e o f g r a v i t y KG m 6.40 L o n g i t u d i n a l r a d i u s o f g y r a t i o n k y y GH^ m 16.00 T r a n s v e r s e m e t a c e n t r i c h e i g h t k y y GH^ m 17.98 N a t u r a l r o l l p e r i o d s 6.17 N a t u r a l p i t c h p e r i o d s 5.35 N a t u r a l heave p e r i o d T z s 5.35 F i g u r e 1. S i g n and h e a d i n g c o n v e n t i o n s F i g u r e 2. Wave i n d u c e d e x c i t a t i o n i n CATHO

F i g u r e 3. ASR catamaran (Wahab, 1971) Symmetric Asymmetric F i g u r e 4. H u l l i n t e r a c t i o n modes WAVE DIRECTION 90 D E a . S / B h = 158 a WAHAB (MODEL T E S T S ) CATMO S-2D DIFFRAC 1-8 0.0 0.6 1 0 Wave frequency (rad/s) Wave frequency (rad/s)

V3 = O kn . S / B h = 158 e - &-— CATMO — S-2D — O F F R A C SURGE E \ , CM « ft 01 Z < 7500-1 7500-1 R O a CM E 2 < 1 1 1 1 t / 1 ; ; / ''/^. SWAY E CM 2.5MCf- ^.5M</-0.5 1.0 HEAVE I . z < I 9

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AREA 2 6

All directions

ROLL RESPONSE OF 64 m LENGTH SHIPS IN BEAM SEAS MEDITERRANEAN SEA, SUMMER

DOWKTIME ANALYSIS - TOTAL DOWNTIME 404 THOUSANDTH l*CTVI>E OUANTITY MONOHULL C A T A U A R A N H M p « . « . 7 l C A T A U W M N H M P « . I . M C A T A U A R A N H M | I I S > I . 4 I C A T A U A R A N H u p A - l. T * C A T A U A A M K cnntnioN A V E R A Q E O O W M T I U E I R H I X A M P I . T M O U S A N O T H 10.00 4 M • 10.00 10.00 10.00 10.00 10.00 too T 5 10 ZERO-UP CROSSING PERIOD In <

15 F i g u r e 12. O p e r a b i l i t y a n a l y s i s i n becim seas T r a n s v e r s e a c c e l e r a t i o n s , 1 5 . 1 k n o t s , S / B H = 1- 5 B 1 5 +

AREA 2 6

All directions TT?—r

TRANSV. A C C E L RESPONSE OF 64 m LENGTH SHIPS IN BEAM SEAS MEDITERRANEAN SEA, SUMMER

DOWNTIME ANALYSIS - TOTAL DOWNTIME 526 THOUSANDTH I X O Y P E OUMfTITV CRITERION A V E R A M 0GMA<T»1E

8 I O N . D . AMP1_ 'mOU&ANDTH MONOHULL 1.W rrWZ C A T A M A R A N C A T A M A R A N H M i y S - l . 0 0 C A T A U A R A N H M| > « > 1 . 4 I C A T A M A R A N H M P « . 1 . 7 < C A T A M A R A N H M P « ^ 11 123 U 2 101 2« S I OH ' H - J ' ^ ^ '-r-0 5 1'-r-0 Z E R O - U P C R O S S I N G P E R I O D in s