Prediction of wave-induced motions and loads for catamarans

DET NORSICE VERITAS

ClaHsilïcatioiuiiKl

Registry of Shipping

PREDICTION OF WAVE-INDUCED MOTIONS AND

LOADS FOR CATAMARANS

By Nils Nordenstrom, Odd Faltinsen and Bjorn Pedersen.

FFSHORE TECHNOLOGY CONFERENCE 200 North Central Expressv/ay

g l l a s , Texas 75206

NUMBER O T C 1 4 1 8

THIS IS A PREPRINT SUBJECT TO CORRECTION

P r e d i c t i o n o f W a v e - ! n d u c e d M o t i o n s a n d L o a d s F o r C a t a m a r a n s

By

Nils Hordenstry^m, Odd Faltinsen ajid Bjj^rn Pedersen, Det norske Veritas

© C o p y r i g h t 1 9 7 1

ffshore Technology Conference on behalf of American I n s t i t u t e of Mining, Metallurgical, and etroleum Engineers, Inc., The American Association of Petroleum Geologists, American I n s t i t u t e of hemical Engineers, American Society of C i v i l Engineers, The American Society of Mechanical

ngineers. The I n s t i t u t e of E l e c t r i c a l and Electronics Engineers, Inc., Marine Technology Society, oclety of Exploration Geophysicists, and Society of Naval Architects & Marine Engineers.

This paper was prepared for presentation at the Third Annual Offshore Technology Conference o be held i n Houston, Tex., A p r i l 19-21, 1971. Permission to copy i s restricted to an abstract f not more than 300 words. I l l u s t r a t i o n s may not be copied. Such use of an abstract should ontain conspicuous acknowledgment of v;here and by whom the paper i s presented.

I . A b s t r a c t

The paper d e s c r i b e s a method t o :;alculate motions and loads f o r a c a t a -M r a n a t zero speed i n r e g u l a r waves o f a r b i t r a r y f r e q u e n c y and d i r e c t i o n . The nydrodynamical i n t e r a c t i o n between t h e two h u l l s has been taken i n t o c o n s i d e -r a t i o n . I t has been found t h a t t h i s i n t e r a c t i o n has a s i g n i f i c a n t i n f l u e n c e Dn t h e c o e f f i c i e n t s f o r added mass and damping. Comparisons w i t h model t e s t s have g i v e n good agreement. Short and long t e r m response i n i r r e g u l a r waves have been p r e d i c t e d u s i n g wave c o n d i -t i o n s f o r -t h e N o r -t h Sea. Forces, mo-ments , a c c e l e r a t i o n s and motions have been c a l c u l a t e d . The methods developed are c o n s i d e r e d t o be u s e f u l t o o l s f o r p r e d i c t i n g motions and loads f o r c a t a -marans o r s i m i l a r v e s s e l s .

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

One o f t h e g r e a t e s t problems i n c o n n e c t i o n w i t h t h e d e s i g n o f o f f s h o r e s t r u c t u r e s , i s t h e l a c k o f adequate knowledge about t h e loads imposed by the sea. As a r e s u l t t h e f r e q u e n c y o f s e r i o u s damages t o o f f s h o r e s t r u c t u r e s i s v e r y h i g h . T h i s a p p l i e s p a r t i c u l a r -l y t o the u n c o n v e n t i o n a -l types o f

References and i l l u s t r a t i o n s a t end of paper.

s t r u c t u r e s l i k e f l o a t i n g d r i l l i n g p l a t forms f o r which t h e e x p e r i e n c e i s l i m . i -t e d , w h i l e f o r c o n v e n -t i o n a l s h i p s -t h e t r i a l and e r r o r procedure g r a d u a l l y has developed an adequate s t r e n g t h s t a n d a r d over a l o n g p e r i o d o f t i m e . The m e r i t s o f t h e t r i a l and e r r o r procedure a r e obvious s u c c e s s f u l s e r -v i c e e x p e r i e n c e i s the b e s t t e s t o f a d e s i g n . However, t h e l i m i t a t i o n s o f the t r i a l and e r r o r procedure a r e equal-l y obvious - i t does n o t work f o r unc o n v e n t i o n a l designs and f o r t h i s r e a -son i t becomes d i f f i c u l t t o s o l v e new problems i n c o n n e c t i o n w i t h new de-mands l i k e b i g s h i p s , f a s t t r a n s p o r t o r o f f s h o r e o i l d r i l l i n g . The demands a r e changing f a s t nowadays and c o n s e q u e n t l y the a t t e n t i o n has l a t e l y been more and more f o c u s e d on t h e o r e t i c a l p r e d i c t i o n s of wave induced motions and l o a d s f o r o f f s h o r e s t r u c t u r e s .

One may ask why n a v a l a r c h i t e c t s who have been w o r k i n g on t h e s e problems f o r c e n t u r i e s , d i d n o t a r r i v e a t suc-c e s s f u l s o l u t i o n s a l o n g t i m e ago. The answer i s s i m p l y t h a t methods t o des c r i b e t h e confudesed desea were n o t a v a i l -a b l e u n t i l Rice / I / i n 19U5 developed

Il-lk

PREDICTION OF WAVE-INDUCED MOTIONS AND LOADS FOR CATAMARANS OTC 1418,

h i s m a t h e m a t i c a l t h e o r i e s f o r random n o i s e . The i n t r o d u c t i o n o f s t a t i s t i c a l methods was t h e b e g i n n i n g o f a r e v o l u -t i o n V7hich has a l r e a d y c o m p l e -t e l y chan-ged the s i t u a t i o n . I n 1953 St. Denis and P i e r s o n /2/ a p p l i e d t h e new s t a t i s t i c a l t h e o r i e s t o s h i p problems and i n t r o d u c e d t h e l i n e a r s u p e r p o s i t i o n p r i n c i p l e f o r c a l c u l a t i o n o f response s p e c t r a f r o m wave s p e c t r a and t r a n s f e r f u n c t i o n s . Another impor-t a n impor-t simpor-tep f o r w a r d was impor-t h e s impor-t r i p - impor-t h e o r y development by Korvin-Kroukovsky and Jacobs 73/ f o r c a l c u l a t i o n o f t r a n s f e r f u n c t i o n s o f heave, p i t c h and v e r t i c a l bending moments and shear f d r e e s . L a t e -l y Sa-lvesen, Tuck and F a -l t i n s e n /4/ have developed methods t o c a l c u l a t e t r a n s f e r f u n c t i o n s a l s o f o r l a t e r a l motions (sway, yaw and r o l l ) as wéll as the l a t e r a l bending moments and shear f o r c e s ' a n d t h e t o r s i o n a l moments.

Among t h e l a r g e number o f model t e s t s i n waves one may mention t h e Vossers e t a l . 75/ s y s t e m a t i c runs w i t h v a r i e d models o f thé S e r i e s 60 form.

S t a t i s t i c a l methods were a l s o used t o analyse b o t h s h o r t and l o n g t e r m f u l l s c a l e measurements and f r o m the works o f Jasper /6/, Bennet e t a l . / 7 / , Nordenstrom /8/ and Lewis /9/ among o t h e r s , methods t o p r e d i c t l o n g t e r m s t a t i s t i c a l " l i f e t i m e " d i s t r i b u t i o n s of response were developed.

The sea environment was d e s c r i b e d by means o f wave s p e c t r a a c c o r d i n g t o Neumann 710/ and P i e r s o n and

Moskowitz 711/ and t h e d e s c r i p t i o n o f l o n g t e r m wave s t a t i s t i c s by W e i b u l l d i s t r i b u t i o n s vjas i n t r o d u c e d by Nordenstr^zSm 7127. T h i s l i s t o f a u t h o r s and events g i v e o f course o n l y a glimpse o f t h e l a r g e amount o f work c a r r i e d out w i t h i n t h i s f i e l d by a l a r g e number o f p e r -sons. We have p i c k e d t h e works we have found t o be u s e f u l f o r our pur-poses. However, t h i s l i s t i l l u s t r a t e s the i m p o r t a n t f a c t t h a t many d i s i p l i n e s are i n v o l v e d and have t o be i n v o l v e d i n o r d e r t o make p r e d i c t i o n s o f wave loads p o s s i b l e .

Our g o a l i n Det norske V e r i t a s when we i n t e n s i f i e d our e f f o r t s w i t h i n t h i s f i e l d i n t h e b e g i n n i n g o f t h e 60's was t w o f o l d . On one hand we c o n c e n t r a

-ted on s y n t h e s i s r a t h e r t h a n a n a l y s i s

i . e . we t r i e d t o put t o g e t h e r a l l t h e p i e c e s mentioned above and shown i n Fig. I I . I i n o r d e r t o see t h e e n t i r e p i c t u r e and_use a l l p a r t s o f i t t o pro-duce an answer. On t h e o t h e r hand we c o n c e n t r a t e d on c o m p u t e r i z a t i o n f r o m the v e r y b e g i n n i n g i n o r d e r t o develop p r a c t i c a l t o o l s f o r t h e d e s i g n e r . We s t i l l t h i n k t h a t t h e key words "synthe-s i "synthe-s " and "computer" r e a l l y g i v e t h e key, t o a p r a c t i c a l s o l u t i o n . " S y n t h e s i s " because o f t h e obvious f a c t t h a t f o r example a p e r f e c t t r a n s f e r f u n c t i o n i s of no v a l u e i f you do n o t know how t o use i t . "Computer" because t h i s i s t h e o n l y way t o make c o m p l i c a t e d t h e o r i e s u s e f u l i n p r a c t i c e .

Consequently we l o o k e d a t F i g . I I . I f i l l e d t h e gaps by d e v e l o p i n g "wave s t a t i s t i c s " and " i n t e g r a t i o n " and im-proved " t r a n s f e r f u n c t i o n s " . We t h e n o b t a i n e d t h e system o f computer p r o -grammes shown i n F i g . I I . 2 . To-day we c o n c e n t r a t e our e f f o r t s on t h e development o f t r a n s f e r f u n c t i -ons because we t h i n k t h a t f o r t h e t i m e b e i n g our methods t o use t r a n s f e r f u n c -t i o n s are adequa-te. The f i r s -t s -t e p was t h e p a r t i c i p a t i o n i n t h e d e v e l o p -ment o f t h e s i x - d e g r e e s - o f - f r e e d o m — computer programme f o r a s i n g l e h u l l based on t h e Salvesen, Tuck, F a l t i n s e n method as mentioned above. The second s t e p was t h e development o f t h e t h e o r y and computer programme f o r a catamaran p r e s e n t e d i n d e t a i l i n t h i s paper. T h i s f i r s t v e r s i o n o f t h e catamaran programme does n o t i n c l u d e f o r w a r d speed l a r g e r than z e r o , but we t h i n k t h a t a catamaran a t zero speed i s o f c o n s i d e r a b l e i n t e r e s t because o f i t s use i n c o n n e c t i o n w i t h rescue and d r i l l i n g o p e r a t i o n s e t c . I t i s our i n t e n -t i o n -t o i n c l u d e f o r w a r d speed i n -t h e f u t u r e . The n e x t b i g s t e p w i l l be t h e development o f a programme f o r s t i l l more c o m p l i c a t e d geometry l i k e d r i l -l i n g r i g s 5 and t h e f i r s t v e r s i o n o f such a programme has a l r e a d y been completed.

One o f t h e main s t r u c t u r a l ' p r o b l e m s f o r a catamaran i s t o d e s i g n t h e c r o s s -s t r u c t u r e which connect-s t h e two l i u l l -s and i t i s t h e n necessary t o p r e d i c t the l o a d s on t h i s s t r u c t u r e . One t y p e of l o a d i s slamming when t h e b o t t o m of t h e c r o s s - s t r u c t u r e h i t s t h e w a t e r . We w i l l not handle t h i s problem i n t h i s paper, but t h e motions c a l c u l a t e d by means o f t h e methods d e s c r i b e d here

r c 1 4 1 8 N. NORDENSTR0M, 0. FALTINSEN, B. PEDERSEN 11-15

may be used f o r t h i s purpose.

We w i l l here c o n c e n t r a t e on some of t h e motions and c o n n e c t i n g f o r c e s and moments between t h e two h u l l s . For s i m p l i c i t y we w i l l here o n l y c u t the catamaran a t t h e symmetry plane and s t u d y the c o n n e c t i n g f o r c e s and moments t h e r e .

The loads handled h e r e are shown i n F i g s . I I . 3 - I I . 7 .

I I I . T h e o r y

I I . I Motions i n r e g u l a r waves.

We w i l l c o n s i d e r a catamaran w i t h no f o r w a r d speed i n r e g u l a r s i n u s o i d a l waves. The waves can have an a r b i t r a r y d i r e c t i o n w i t h r e s p e c t t o t h e catama-ran. We are g o i n g t o assume t h a t t h e catamaran i s symmetric about t h e cent r e p l a n e , cent h a cent cent h e h u l l s o f cent h e c a cent a -maran are l o n g and s l e n d e r , and t h a t the responses a r e l i n e a r and harmonic. Under these assumptions i t can be shown t h a t -the d i f f e r e n t i a l e q u a t i o n s o f motions can be w r i t t e n as two s e t s o f e q u a t i o n s where surge i s n e g l e c t e d . The e q u a t i o n s a r e (M+A33)fi3 + B33n3 + C33n3 ^ ^ 3 5 ^ B " ^ 3 5 ^ 5 ^S5 ^ V I W t (1) ^ 3 ^ ^ B 5 3 ^ 3 " S 3 ^ 3 - ^ ^ ^ 5 5 ^ ^ 5 ) ^ 5 i w t _{( 2 )} (M+A22)n2+B22n2+C-^3^+A2^)n^ + B24n4+A2gng+B2gng = F2e i u t ( 3 ) ( - M V \ 2 ) S ^ B ^ 2 ' ^ 2 ^ " 4 4 ^ \ 4 ) ^ xwt ( 4 ) A g 2 n 2 + B g 2 n 2 + ( - i ^ g + A g ^ ) n ^ lü3t ( 5 )

Here A j j ^ and Bj]^ are t h e added-mass and damping c o e f f i c i e n t s ,Cj]^ are t h e h y d r o s t a t i c r e s t o r i n g c o e f f i c i -e n t s and Fg ar-e t h -e compl-ex a m p l i t u d -e s of t h e e x c i t i n g f o r c e and moment, w i t h the f o r c e and moment g i v e n by t h e r e a l p a r t _ o f Fje^""*^. F2 and F3 are t h e

a m p l i t u d e s o f sway and heave e x c i t i n g f o r c e s , w h i l e F 4 , F5 and Fg a r e t h e a m p l i t u d e s o f the r o l l , p i t c h and yaw e x c i t i n g moments. F^ i s t h e moment w i t h r e s p e c t t o t h e x - a x i s o f t h e c o o r d i n a t e system shown i n F i g . I I I . l . F5 and Fg are t h e moments w i t h r e s p e c t t o y and z a x i s r e s p e c t i v e l y , u i s t h e f r e -quency o f encounter and t h e f r e q u e n c y of response. When t h e catamaran has no f o r w a r d speed the f r e q u e n c y o f enc o u n t e r i s the same as t h e wave f r e -quency. n j ( j = 2 , 3 , 4 , 5 , 6 ) r e f e r t o sway, h e a v e , r o l l , p i t c h and yaw d i s p l a c e m e n t s r e s p e c t i v e l y . The t r a n s -l a t o r y d i s p -l a c e m e n t s a r e f o r a p o i n t which has the o r i g i n o f t h e c o o r d i n a t e system shown i n F i g . I I I . l , as mean p o s i t i o n . The dots i n t h e e q u a t i o n s of m o t i o n s t a n d f o r t i m e d e r i v a t i v e s so t h a t n j and fij are v e l o c i t y and a c c e l e r a t i o n terms. M i s the mass o f the catamaran, and 1144, I 5 5 and Igg a r e the moments o f i n e r t i a w i t h r e s p e c t t o x,y and z - a x i s r e s p e c t i v e l y . The o n l y p r o d u c t o f i n e r t i a which appears i s I ^ g , the r o l l - y a w p r o d u c t , which v a n i s h e s i f the catamaran has f o r e - and a f t - s y m m e t r y The c e n t r e o f g r a v i t y o f t h e catamaran i s l o c a t e d a t ( 0 , 0 ,z ) .

G

Since t h e h u l l s o f t h e catamaran are assumed t o be l o n g and s l e n d e r , i t can be shown t h a t

PREDICTION OF WAVE-INDUCED MOTIONS AND LOADS FOR CATAMARANS OTC 1418 A 55 ^55 22 ^22 4 4 ^24 'S6 ^65 (6) '35 • 53 '33 ^35 = ^53 - -^-^33^'''^^-/x^b^^ dx ( 8 ) 33 r ( 2 D ) , / b 2 2 ^ - ) d x = \ 2 - ^^24^"^>^ ^2 = ^1=24^^^^^^ (9) ^26 ^ \ 2 /xa^^^^D)^^ : /xb2 2^^°''dx ( 1 1 ) 26 62 (12) 46 64 A^„ = ;xè^,/2^^dx , '24 ^6 = ^64 = /-b2,^2D)^^ ^^3^ / x ^ a . . ( 2 D ) ^ ^ 2 2 B.. = /x^b ^2D) '22 dx (14) The i n t e g r a t i o n s a r e over t h e l e n g t h o f t h e catamaran. a j k ( 2 D ) g^^d bj]< are t w o d i m e n s i o n a l c r o s s s e c t i o n a l added mass and damping c o e f f i -c i e n t s , r e s p e -c t i v e l y .

a„„^^'^'* and b„„^^^'^ are t h e c o e f f i c i -: ^ (2D) , , (2D) ents f o r sway, a '44 and b t h e c o e f f i c i e n t s f o r r o l l (2D) 4 4 ^24 (2D) are and '24 t h e c o e f f i c i e n t s due t o cross c o u p l i n g between sway and r o l l , and a and b are t h e c o e f f i c i -(2D) are computed f o r 19 c r o s s s e c -'33 33 ents f o r heave. U s u a l l y a (2D)

t i o n s a l o n g t h e catamaran, and A., and B., a r e o b t a i n e d by summation.

:k

The t w o - d i m e n s i o n a l added mass and damping c o e f f i c i e n t s a r e d i s c u s s e d i n more d e t a i l i n Appendix 1. The h y d r o s t a t i c r e s t o r i n g c o e f f i -c i e n t s i n t h e e q u a t i o n s o f motions f o l l o w d i r e c t l y f r o m h y d r o s t a t i c con-s i d e r a t i o n con-s . They a r e '33 '35 '55 '44 = p g /2b dx = - p g /2bx dx = p g /2bx^ dx = p g V GM (15 y (16) (17) (18) Here b i s t h e s e c t i o n a l beam o f one o f t h e h u l l s o f t h e catamaran, p i s t h e mass d e n s i t y o f t h e w a t e r , g i s t h e g r a v i t a t i o n a l a c c e l e r a t i o n , V i s t h e d i s p l a c e d volume o f t h e catamaran and GM i s t h e m e t a c e n t r i c h e i g h t . The i n t e -g r a t i o n s a r e over t h e l e n -g t h o f t h e catamaran.

The e x c i t i n g f o r c e and moment com-ponents have been d e r i v e d i n a s i m i l a r way as d e s c r i b e d by Salvesen, Tuck and F a l t i n s e n 74/ f o r a s i n g l e h u l l s h i p . They can be w r i t t e n as

ÏC 1418 N. NORDENSTR0M, O FALTINSEN, B. PEDERSEN 11-17 Fj = ph fif. 3 = 2,3,4 h.)dx,

:

where t h e i n t e g r a t i o n s are over t h e l e n g t h o f the catamaran and

v j h e r e ^ - i k x cos e , i k y s i n e f . = ge ƒ N. e C 3 X e'^'^ds, j = 2,3,4 (20) and h j = we - i k x cos e , , f ( i H - - N „ s i n £)• C ^ ^ X i k y s i n e kz , , e e i|i j ds, j = 2,3,4 (21) The p o t e n t i a l o f t h e i n c i d e n t wave used i n the d e r i v a t i o n o f Fj i s i g h - i k ( x cos e-y s i n e) kz I (jj (22)

Here h i s the i n c i d e n t wave a m p l i t u d e , k i s the wave number and e i s the angle between i n c i d e n t wave and c a t a -maran heading (e = 180 deg. f o r head s e a s ) , ds i s an element o f arc a l o n g the submerged c r o s s - s e c t i o n Cj, a t x. N2 and Nj are components i n t h e y and z - d i r e c t x o n s o f t h e t w o - d i m e n s i o n a l outward u n i t normal v e c t o r i n t h e y-z p l a n e , and N4 = y N3-ZN2. Furthermore 11^2 5 '1^3 5 4*4 are t h e v e l o c i t y p o t e n t i a l s f o r the same problems as f i n d i n g two-d i m e n s i o n a l atwo-dtwo-detwo-d mass antwo-d two-damping

c o e f f i c i e n t s o f t h e catamaran f o r sway, heave, r o l l r e s p e c t i v e l y .

The e x c i t i n g moments F^ and Fg can be w r i t t e n as

Fg =-ph ƒ [ x ( f 3 + h 3 ) ] dx ( 2 3 )

F = ph ƒ [ x ( f +h ) ] dx ( 2 4 ) I t i s t o be n o t e d t h a t

ph ƒ f j dx j = 2 ,3

are t h e components i n y. and z - d i r e c t i o n o f t h e wellknown F r o u d e K r i l o f f f o r c e -components. The o t h e r p a r t o f t h e

ex-p r e s s i o n o f the e x c i t i n g f o r c e i s due tc( d i f f r a c t i o n o f t h e wace. I I I . 2 Loads i n r e g u l a r V7aves . We are g o i n g t o d e s c r i b e a method t o f i n d t h e dynamic loads on t h e h a l f p a r t s which are o b t a i n e d by i n t e r s e c -t i n g a l o n g -the c e n -t r e p l a n e o f -t h e ca-ta- cata-maran. These loads are c o n s i d e r e d t o be o f main i n t e r e s t i n t h e d e s i g n o f a catamaran. Let us w r i t e the f o r c e due t o these loads as V = i + i + V3 k and t h e moment as (25) M (26) The f o r c e i s t h e d i f f e r e n c e between the i n e r t i a f o r c e and t h e sum o f e x t e r -n a l f o r c e s a c t i -n g o-n t h e aboveme-ntio-ned h a l f p a r t o f t h e catamaran. I f t h e ex-t e r n a l f o r c e componenex-ts are s e p a r a ex-t e d i n t o 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 components R j , t h e e x c i t i n g f o r c e com-ponents X j , and t h e hydrodynamic f o r c e components Dj due t o t h e body m o t i o n we can w r i t e

V. R, _{( 2 7 )}

where I . are t h e i n e r t i a f o r c e compo-n e compo-n t s . S i m i l a r l y , t h e momecompo-nt i s e q u a l t o t h e d i f f e r e n c e between t h e moment due t o the i n e r t i a f o r c e and t h e moment due t o t h e sum o f t h e e x t e r n a l f o r c e s , so the e x p r e s s i o n f o r Vj a l s o a p p l i e s f o r j = 4,5,6.

11-18 _{PREDICTION OF WAVE-INDUCED MOTIONS AND LOADS FOR CATAMARANS OTC m i S H i ;}

VJe c o n s i d e r V^, V^, V^^ and V5 t o be o f main i n t e r e s t and w i l l o n l y s t u d y these f o r c e - and moment-components

i n t h i s paper. I n t h e f o l l o w i n g t e x t ^ 2 ' ^3' ^4 ^5 ^^^•^ ^'^ named r e s p e c -t i v e l y h o r i s o n -t a l a -t h w a r -t s h i p s f o r c e , v e r t i c a l shear f o r c e , v e r t i c a l bending moment and p i t c h c o n n e c t i n g moment be-tween t h e two h u l l s . The i n e r t i a f o r c e i s t h e mass t i -mes t h e a c c e l e r a t i o n . I j , j = 2,3,4,5 can be w r i t t e n I 2 = | ( n 2 " ^c ^28) ^3

- ¥^ '

^4 = 2^-^c^2'-yA^3^ "44^ _{n,,- [/ xy dM]n,} A •46, 2 " \ (30) Ip = -[ƒ xy dM]Ti^^ + ^ A ^ 2 [ƒ zy dMlri A ' 55:, (31) Here dM i s an i n f i n i t e s i m a l mass e l e -ment l o c a t e d a t a p o i n t ( x , y , z ) i n t h e c o o r d i n a t e system shown i n F i g . I I I . l . ƒ means t h a t t h e i n t e g r a t i o n i s A o v e r the abovementioned h a l f p a r t o f t h e catamaran, yp^ i s t h e y - c o o r d i n a t e o f t h e c e n t r e o f g r a v i t y o f the h a l f p a r t o f the catamaran. The o t h e r v a r i -a b l e s h-ave been e x p l -a i n e d e -a r l i e r i n t h e t e x t . The 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 and moment c o n t r i b u t i o n s w i l l be (32) "^33 -^35 2 "3 2 ^^5 2 "4 c c 33 35 V y B - T - ^ 3 " y B — ^ - P S 2 '33 c c ^53 ^55 _{2 "3 2 ^5 2 "4} '53 ( ? 3 ) (34) ( 3 5 ) Here yg i s t h e y - c o o r d i n a t e o f t h e ^ c e n t r e o f buoyancy o f t h e h a l f p a r t o f t h e catamaran. The o t h e r v a r i a b l e s have been d e f i n e d i n s e c t i o n I I I . l . Let us now c o n s i d e r t h e c o n t r i b u t i o i due t o t h e incoming waves. We do o n l y know t h e t o t a l d i f f r a c t e d wave f o r c e s and moments. But we are g o i n g t o s e t each component o f t h e d i f f r a c t e d wave f o r c e s and moments on the h a l f p a r t s o f t h e catamaran e q u a l t o one h a l f o f t h e c o r r e s p o n d i n g t o t a l d i f f r a c t e d wave f o r c e - and moment-components. We w i l l denote t h e d i f f r a c t e d f o r c e - and moment | components by F.^^^e^"^,

and t h e " F r o u d e - K r i l o f f " f o r c e and

mo-^ ( I ) iu)t < ment components by F^ e — We can w r i t e P ( D p (D) ^ 2 2 iüjt , , r - i k x cos £ X2 = — ~—e + pgh/e [ ƒ N„cos(ky s i n e ) e ^ ^ d s ] d x e c ^x2 lü)t (36) P ( D p (D) Y - 3 3 ,iüJt ^ - i k x cos e X_ = e + pgh/e [ ^7 Ng i s i n ( k y s i n G)e^^ds]dxe i w t Cx2 (37)

N. NORDENSTR0M, 0. FALTINSEN, B. PEDERSEN 11-19 p ( I ) ^ P (D) 4 4 „iü)t , , - i k x cos e X = e + p g h j e ^x2 ^ (38) F ( I ) + F . ƒ N„ i s i n ( k y s i n e)e'^^ds] dxe''"^"'' x2 ^ (39)

Here ds i s an element o f are a l o n g t h e submerged c r o s s - s e c t i o n C „ o f one o f the h a l f p a r t s a t x.

The hydrodynamic f o r c e and moment components due t o t h e m o t i o n o f t h e catamaran can be w r i t t e n as '22. 2 '"^2 ~ ^23^3 2 "4 "25"5 B. '2 4.. -n, - a^^n, ^26;. ^22; , ; '24' 2 "6 2 "2 "23"3 2 ""^4 B. '26 25"5 2 "^6 (40) "33.. '35.. 32"2 2 "3 "34"4 2 "5 ' 3 3" * ^36^6" ^32^2" — ^ 3 " ^ 3 4 ^ 35 • 2 "5 '•'se^'e (41) A 42.. ^•^2 " ^43*^3 2 "4 ""45"5 44.. '46.. N5'^5 aeon ^42 B Ho- b,,„n. 44' 2 "2 ''43"3 2 '"^4 B, '46' — ^ 6 (42) '5 3.. 52"2 M"^3"^54'^4 2 "5 '^56"6 5 5.. B b c o H 5 3 B no" bp.n, 55' 52"2 2 '3 ''54"4 2 "5 ^ 6 \ ( 4 3 )

Here A., and B., a r e t h e added mass and I k _:k

damping c o e f f i c i e n t s as used e a r l i e r i n t h e e q u a t i o n s o f m o t i o n s . The o t h e r added mass and damping c o e f f i c i e n t s a.,

-I and b.j^ a r e c a l c u l a t e d i n a s i m i l a r way a i A., and B., . I t i s t o be n o t e d

^ ] k :k t h a t c o r r e s p o n d i n g terms f o r t h e t o t a l catamaran a r e z e r o . By u s i n g t h e e x p r e s s i o n f o r I . , R. X. and D. t o g e t h e r w i t h t h e e q u a t i o n s ^ ^ . ( I ) (D) o f m o t i o n and w i t h F^ =F^ ^ ^ + F j '^ , t h e l o a d components can be w r i t t e n as:

The h o r i z o n t a l a t h w a r t s h i p f o r c e betweer t h e two h u l l s : - pgh/e • i k x cos e ' ƒ N c o s ( ky s i n E ) e ^ ^ d s ] d x e ^ x 2 ^ i w t (44)

11-20

PREDICTION OF WAVE-INDUCED MOTIONS AND LOADS FOR CATAMARANS OTC J^.

The v e r t i c a l shear f o r c e between t h e two h u l l s :

a32n2 +

X^f N i s i n ( k y s i n e ) e ^ ^ d s ] dxe""^^

x2 ^

(45) The v e r t i c a l bending moment between t h e two h u l l s : ^f^A + ^43^^3 ^S5 - ^^^^5 ' N 3 ^ 3 ' \ 5 ^ ^ y B - F ^ 3 + y g ^l^-Tig - p g h / e - i ' ^ ^ ^ [ p / N c o s ( k y s i n e)e'^^ds] dxe^'^^ x2 ^ (46)

The p i t c h c o n n e c t i n g moment between t h e two h u l l s : ^5 = ^52^2 + ^ " { ^ y + ^54>n4 + ( - / z y dM + a p p ) n ^ + b,^n A 56'"6 '^52"2 C, ' ' ^56^6 '

+ pgh/xe •ikx cos e

N x2 _iü)t , .-o i s i n (ky s i n e ) e ^ ^ d s l d x x2 (47)

I I I . 3 Short term d e s c r i p t i o n o f sea W i t h s h o r t term we here mean a pe-r i o d o f t i m e which i s s h o pe-r t enough t o make i t p o s s i b l e t o d e s c r i b e t h e sea as a s t a t i o n a r y random process.

A random process i s s t a t i o n a r y whe: i t s s t a t i s t i c a l p r o p e r t i e s r e m a i n un-changed i n t i m e . T h i s does n o t mean t h a t t h e process r e p e a t s i t s e l f . A par t i c u l a r p a t t e r n o f the sea w i t h i n a p a r t i c u l a r area a t a g i v e n i n s t a n t i s unique and never occurs a g a i n , and i t i s n o t p o s s i b l e t o p r e d i c t t h e appe arance o f t h e sea s u r f a c e a t a c e r t a i n t i m e i n t h e f u t u r e . The s t a t i s t i c a l p r o p e r t i e s o f t h e sea may, however, be p r e d i c t e d and t h i s i s a l l we need i n o r d e r t o make p r e d i c t i o n s r e g a r d i n g t h e performance o f a s h i p a t sea.

A s h o r t t e r m s t a t i o n a r y sea s t a t e may be d e s c r i b e d by means o f t h e wave spectrum. S e v e r a l d i f f e r e n t d e f i n i t i o n ' and t y p e s o f s p e c t r a may be found i n t h e l i t e r a t u r e . Here we w i l l use a var: ance spectrum which shows how t h e v a r i ance o f t h e s u r f a c e e l e v a t i o n i s d i s t r , buted over d i f f e r e n t f r e q u e n c i e s and d i r e c t i o n s . The s u r f a c e may be r e g a r -ded as a sum o f an i n f i n i t e number o f elementary s i n e waves o f d i f f e r e n t ^ f r e q u e n c i e s and d i r e c t i o n s and w i t h random phases. The spectrum i s u s u a l l y w r i t t e n as f o l l o w s : [ a ( w , a ) ] 2 = [ a ( w ) j ^ f ( a ) ( 4 8 ) where: . a ( u , a ] ^ i s t h e s h o r t c r e s t e d ( d i r e c -t i o n a l ) spec-trum [a(üj)p i s t h e spectrum as f u n c t i o n o f f r e q u e n c y f ( a ) i s t h e d i r e c t i o n a l i t y f u n c t i o i w i s t h e wave f r e q u e n c y ot i s t h e wave d i r e c t i o n ( o f t h e elementary wave r e l a ¬ t i v e t o t h e wave system) _a(ü3, a ) ] ^ dwda i s t h e v a r i a n c e o f t h e waves w i t h i n t h e f r e -quency i n t e r v a l w ± dü)/2 and w i t h i n t h e a n g u l a r i n t e r v a l a ± da/2.

TC 1 4 1 8 N. NORDENSTR0M, 0. FALTINSEN, B. PEDERSEN 11-21

Some i m p o r t a n t p r o p e r t i e s o f t h e 3ea a r e r e l a t e d t o t h e moments o f t h e ;vave spectrum. The e t h moment m^ i s de-f i n e d as

m = /ü)^[a(ü))] ^ dü) ( 4 9 ) rhe c r e s t - t o - t r o u g h wave h e i g h t H and the z e r o - c r o s s i n g p e r i o d T a r e d e f i n e d i n F i g . I I I . 2 . The s t a t i s t i c a l d i s t r i b u t i o n o f H may f o r most p r a c t i c a l purposes be d e s c r i -bed by means o f t h e f o l l o w i j ^ i g R a y l e i g h d i s t r i b u t i o n f u n c t i o n . P(H) = 1 - exp (-2(H/H^/3)M, (50) where ^-^/^ s i g n i f i c a n t wave h e i g h t d e f i n e d as t h e mean o f t h e one t h i r d h i g h e s t waves. The R a y l e i g h d i s -t r i b u -t i o n i s shown i n F i g . I I I . 3 . '1/3 i s r e l a t e d t o t h e area under t h e spectrum by «1/3 - ^ ^ ( 5 1 )

The average apparent p e r i o d T, w h i c h i s t h e mean o f T d e f i n e d above, i s r e l a t e d t o t h e zero and second mo-ments o f t h e spectrum.

2Tr / m^/mz (52)

I t may be shown t h a t t h e spectrum f o r m u l a e proposed by s e v e r a l i n v e s t i -g a t o r s -g e n e r a l l y may be d e f i n e d by t h e two parameters i^-^/S ^^^^ have adop t e d t h e Pierson-Moskowitz spectrum / l l / w h i c h we w r i t e on t h e f o l l o w i n g non-d i m e n s i o n a l f o r m /12/. T h i s spectrum i s shown i n F i g . I I I . 4 We use t h e f o l l o w i n g d i r e c t i o n a l i t y f u n c t i o n . See / 1 2 / . f ( a ) = 2 2 I T < < I T - cos a; - 2 - a - 2 0; elsewhere (54) The d i r e c t i o n a l i t y f u n c t i o n i s shown i n F i g u r e I I I . 5 . The l o g s l o p e f o r m o f t h e spectrum i n t r o -duced by E.V.Lewis / 1 3 / i s v e r y handy f o r p r a c t i c a l purposes.

^ a U n A ) j 2 ^ ^ ^ ^ j = [ a ( a 3 ) ] ^ [dJ| ( 5 5 )

g i v e s

8 ( T T g ) '

a(w) ( 5 6 )

^a(£nA)j 2 £g ^Y\e l o g - s l o p e spectrum

2 T r g

2 i s t h e wave l e n g t h . CO

The l o g - s l o p e Pierson-Moskowitz spectrum may t h u s be w r i t t e n as f o l l o w s : ; a U n X l ] 2 , 5 ( ^ ^ ) ^ e x p (- 4 T r ( - ^ ) ^ ) gT' ( 5 7 ) [ a ( c o ) ] ^ "'1/3 T STT 2 217 . 1 ,wT - \ exp (- - ( 2 ^ ) ) ( 5 3 ) T h i s spectrum i s shown i n F i g . I I I . 6 j ' I I I . 4 . Long t e r m d e s c r i p t i o n o f sea As mentioned i n t h e p r e v i o u s sec-t i o n a s h o r sec-t sec-t e r m s sec-t a sec-t i o n a r y sea s sec-t a sec-t e may be c h a r a c t e r i z e d by t h e two p a r a -meters s i g n i f i c a n t wave h e i g h t H-^^^^ and average apparent wave p e r i o d T. The l o n g t e r m d e s c r i p t i o n o f t h e sea may t h u s be o b t a i n e d by means o f t h e l o n g t e r m

11-22

PREDICTION OF WAVE-INDUCEp-MOTIONS AND LOADS FOR CATAMARANS OTC m i i

s t a t i s t i c a l d i s t r i b u t i o n s o f H^^^ and T. Most wave s t a t i s t i c s a r e g i v e n , however, i n terms o f t h e v i s u a l l y e s t i m a t e d

h e i g h t and p e r i o d T^. I n o r d e r t o use t h i s l a r g e amount o f d a t a we have developed a method t o d e s c r i b e t h e l o n g t e r m d i s t r i b u t i o n s o f and T 712/ and t h e necessary r e l a t i o n s h i p s between v i s u a l and t h e o r e t i c a l wave h e i g h t s and p e r i o d s 712/. We have found t h a t t h e l o n g t e r m c o n d i t i o n a l d i s t r i b u t i o n o f H when T V y l i e s w i t h i n s m a l l i n t e r v a l s may be w e l l d e s c r i b e d by t h e W e i b u l l d i s t r i b u t i o n f u n c t i o n which i s g i v e n by t h e f o l -l o w i n g e q u a t i o n : exp (-( H H .-H . C l o i (58) ^ i ^ ^ v ^ •the p r o b a b i l i t y t h a t t h e v i s u a l wave h e i g h t does n o t exceed when Ty l i e s w i t h -i n -i n t e r v a l No. -i . H Q ^ J H ^ ^ and a r e parameters o f t h e d i s t r i b u t i o n . Values o f t h e parameters f o r t h e N o r t h A t l a n t i c and t h e N o r t h Sea a r e g i v e n i n Tables I I I . l and I I I . 2 where P i denotes t h e p r o b a b i l i t y t h a t T^ f a l l s w i t h i n i n t e r v a l No. i . F i g . I I I . 7 shows t h e W e i b u l l d i s t r i b u t i o n s o f

v i s u a l wave h e i g h t on t h e N o r t h

A t l a n t i c . W e i b u l l d i s t r i b u t i o n s o f wave h e i g h t f o r many areas around t h e w o r l d may be found i n r e f e r e n c e s 7127 and 714/ The average v a l u e o f T f o r a g i v e n v a l u e o f T^ i s 712/. T = 2.83 T ^'^^ + _{s ( T ) seconds} (59) s ( T ) i s t h e s t a n d a r d d e v i a t i o n o f T when T^ l i e s w i t h i n t h e i n t e r v a l s de f i n e d i n Tables I I I . l and I I I . 2 . s ( T ) 1.0 second ₍₆₀₎ The d i s t r i b u t i o n o f T f o r a g i v e n valu, o f T^ may be d e s c r i b e d by means o f nor-T mal o r l o g - n o r m a l d i s t r i b u t i o n f u n c - ^ t i o n s . We have t r i e d b o t h assumptions ^ and found /15/ t h a t t h e c a l c u l a t e d response w i l l be p r a c t i c a l l y t h e same f o r these two t y p e s o f d i s t r i b u t i o n " f u n c t i o n s . For s i m p l i c i t y , t h e r e f o r e , we have chosen t h e normal d i s t r i b u t i o n f o r t h i s purpose. (The computer p r o

-gramme can handle b o t h ) . : i This normal d i s t r i b u t i o n may a c c o r d i n g

t o Eqs. ( 5 9 ) and ( 6 0 ) be w r i t t e n

p ( T ) = (2Tr)~i e x p ( - (T - 2.83.

(61)

The v a l u e o f H^^^ which i s exceeded

w i t h t h e same p r o b a b i l i t y as i s /12/ H-, ,„ = 1.68 H 1/3 V 0.75 ( 6 2 ) H . = 0.50 H 1.33 1/3

As an example we may choose t h e i n t e r -v a l 9.5 <T-v ' 1 1 . 5 seconds on t h e North A t l a n t i c . _ T h e p r o b a b i l i t y t h a t T^ f a l l s w i t h i n t h i s i n t e r v a l i s a c c o r d i n g t o Table I I I . l , p = 0.1385.

The average apparent p e r i o d i s .0. 44 2.83'10,5 ±1.0 = 7.95±1.0 secondi The c o r r e s p o n d i n g c u m u l a t i v e d i s t r i b u -t i o n o f H^yg IS P(H^/3): exp (H^,3/1.68)l/°-^5-0.75 ^ 4.05-0 . 75 .

The p r o b a b i l i t y t h a t H^^3 exceeds say 10 metres i s t h u s 0.0035 i f T l i e s w i t h i n t h e chosen i n t e r v a l .

TC 1418 N. NORDENSTR0M, 0. FALTINSEN, B. PEDERSEN 11-23

The p r o b a b i l i t y t h a t Ty l i e s w i t h i n

uhe chosen i n t e r v a l and H , exceeds LO metres i g 0.1385 • 0.0035 = 0.00048 .vhich corresponds t o a p p r o x i m a t e l y 0.00048 • 8760 = 4.1 h o u r s / y e a r . [1.5. S h o r t t e r m response The s h o r t t e r m i r r e g u l a r response nay be e s t i m a t e d f r o m wave spectrum and t r a n s f e r f u n c t i o n by means o f t h e l i n e a r s u p e r p o s i t i o n t e c h n i q u e i n t r o -duced by S t . Denis and P i e r s o n / 2 / .

The wave spectrum on t h e l o g - s l o p e form was d e f i n e d i n s e c t i o n I I I . 3 above

The t r a n s f e r f u n c t i o n d e s c r i b e s the response o f a s h i p t o a r e g u l a r s i n u s o i d a l wave o f any f r e q u e n c y , h e i g h t and d i r e c t i o n p r o v i d e d t h e ampli' tude o f t h e response i s a l i n e a r f u n c -t i o n o f -t h e a m p l i -t u d e o f -t h e wave. The t r a n s f e r f u n c t i o n s may be o b t a i n e d from model t e s t s o r t h e o r e t i c a l c a l c u -l a t i o n s . We w i -l -l here use t h e o r e t i c a -l r e s u l t s a c c o r d i n g t o s e c t i o n s I I I . l and I I I . 2 above. The t r a n s f e r f u n c t i o n , also c a l l e d a m p l i t u d e o p e r a t o r , i s usu-a l l y d e f i n e d usu-as t h e r usu-a t i o between re-* sponse and wave a m p l i t u d e as a f u n c t i o n

Df wave f r e q u e n c y . I n c o n n e c t i o n w i t h

the l o g - s l o p e spectrum used h e r e , how-ever, t h e t r a n s f e r f u n c t i o n i s t h e

r a t i o between a non-dimensional response]

K and t h e wave s l o p e h/X as a f u n c t i o n Df An(X/L). TR h/X (63) "/here h i s t h e wave a m p l i t u d e , X i s t h e wave l e n g t h and L i s t h e s h i p l e n g t h . X i s n o n - d i m e n s i o n a l as f o l l o w s : C r a n s l a t i o n s (sway and heave)

x = X j /L •^here X j i s t h e a m p l i t u d e o f nj f o r - 2,3. R o t a t i o n s ( r o l l , p i t c h and yaw) X = X j where x j i s t h e a m p l i t u d e o f n j f o r j = 4,5,6. H o r i z o n t a l a t h w a r t s h i p s f o r c e X = D^/yBL^ where D^^ i s t h e a m p l i t u d e o f V2. V e r t i c a l shear f o r c e X = D ^ / Y B L ^ where D^ i s t h e a m p l i t u d e o f V^. V e r t i c a l bending moment X = M ^ / Y B L ^ where i s t h e a m p l i t u d e o f V^^. P i t c h c o n n e c t i n g moment X = Mp/yBL^ where Mp i s t h e a m p l i t u d e o f V^. B i s t h e beam a t w a t e r l i n e amidships Y = Pg as d e f i n e d i n s e c t i o n I I I . l .

I n our n o t a t i o n s t h e parameter Ey-of t h e s h o r t t e r m R a y l e i g h d i s t r i b u t i o n of t h e v a r i a b l e x may be e s t i m a t e d by means o f Eq. ( 6 4 ) . E^(e, L, V, T, H^/3) T 7 / 2 " 2 ƒ ƒ [TR^ ( e , X/L, V ) ] - T T / 2 ^ a(£nX, a, T, H ,„) ; ^ d(JlnX)da ( 6 4 ) E .. i s t h u s t w i c e t h e v a r i a n c e o f t h e x c o n t i n u o u s v a r i a b l e c o r r e s p o n d i n g t o x.

11-24

PREDICTION OF WAVE-INDUCED MOTTONS AMH LOADS FOR PATAMARANS OTC 1

4j~-The R a y l e i g h d i s t r i b u t i o n i s g i v e n by Eq. ( 6 5 ) P ( x ) = 1 - exp ( - | - ) X (65) e = V P(x) i s t h e a n g l e between t h e s h i p ' s course and t h e d i r e c t i o n o f t r a v e l o f t h e wave system i'^ wind d i r e c -t i o n ) a + B i s t h e angle between s h i p and r e g u l a r wave i s t h e speëd o f t h e s h i p i s t h e p r o b a b i l i t y t h a t t h e depar-t u r e s o f " c r e s depar-t s " and " depar-t r o u g h s " o f a response v a r i a b l e f r o m i t s z e r o l e v e l (mean) do n o t exceed x. The s t a t i s t i c a l s h o r t t e r m d i s t r i -b u t i o n o f t h e response x i s t h u s com-p l e t e l y d e f i n e d by one s i n g l e com-parameter Ej, f o r g i v e n heading ( B ) , s i z e o f s h i p ( L ) , speed o f s h i p (V) and sea s t a t e

(T, H^ / 3 ) .

I t can be shown by s t u d y i n g Eq. (64) t h a t t h e s h o r t t e r m response may be p r e s e n t e d n o n - d i m e n s i o n a l l y as R.^ as f u n c t i o n o f f o r c o n s t a n t s h i p speed (V) and heading ( 3 ) .

T /gJL

(66)

(67)

The f o r m o f Eq. ( 6 6 ) i s chosen so t h a t t h e denominator and n o m i n a t o r a r e on t h e same p r o b a b i l i t y l e v e l i n t h e s h o r t t e r m d i s t r i b u t i o n s o f wave and response r e s p e c t i v e l y . Rx as a f u n c t i o n o f fi t h u s d e f i n e s c o m p l e t e l y t h e s h o r t t e r m response f o r 'a g i v e n geometry, speed and heading f o r any s i z e ( L ) and weather (H _1/3 T ) .

I I I . 6 . Long t e r m response

As d e s c r i b e d i n t h e p r e v i o u s sect i o n sect h e paramesecter E^^ which c h a r a c sect e r i -zes t h e s h o r t t e r m response may be foundj f o r any weather d e f i n e d by s i g n i f i c a n t wave h e i g h t H^^^ and average a p p a r e n t p e r i o d T. We have a l s o e s t a b l i s h e d i n s e c t i o n I I I . 4 a d e s c r i p t i o n o f t h e l o n g t e r m s t a t i s t i c a l d i s t r i b u t i o n o f H-^^^^ and T. I t i s t h u s c l e a r t h a t t h e l o n g t e r m s t a t i s t i c a l d i s t r i b u t i o n o f E^ ma be found by combining s h o r t t e r m r e -sponse c h a r a c t e r i s t i c s and l o n g t e r m weather s t a t i s t i c s . We have found t h a t t h i s c o m b i n a t i o n produces a VJeibull d i s t r i b u t i o n o f E (see e.g. /16/) which we w r i t e in'^the f o l l o w i n g way.

P ( / ^ ) = 1 - exp (-( 4 ^ ) " ^ )

(68) T h i s wès a l s o found f r o m l o n g t e r m f u i . s c a l e measurements /8/. The parameters ^: ; a and m a r e , as d e s c r i b e d below, ob¬

t a i n e d f r o m t h e s h o r t t e r m response c h a r a c t e r i s t i c s o f t h e v a r i a b l e s i n q u e s t i o n and t h e l o n g t e r m vjeather s t a t i s t i c s f o r t h e a c t u a l sea zone. As mentioned above each v a l u e o f E-j r e p r e s e n t s a s h o r t t e r m R a y l e i g h d i s t r i b u t i o n o f x and t h e l o n g t e r m d i s t r i b u t i o n o f x may t h u s be found by a summation o f a l l t h e s e s h o r t t e r m d i s t r i b u t i o n s w i t h due r e g a r d t o t h e i r d i f f e r e n t p r o b a b i l i t i e s o f occurence as g i v e n by Eq. ( 6 8 ) . 01 t< T l i : dl P ( x ) p ( / r ^ ) P ( x ) f exp( o dP(/E") d/Ë^ X | - ) p(/E") d/Ë^ E^ X X (69) ( 7 0 ) w i s t h e l o n g t e r m p r o b a b i l i t y t h a t t h e v a l u e o f t h e n o n - d i ¬ mensional v a r i a b l e i n question s does n o t exceed x i s t h e p r o b a b i l i t y d e n s i t y • f u n c t i o n f o r /E ' X

Eq. ( 6 9 ) was s t u d i e d i n /8/ and 1121. I t was shown t h a t P ( x ) may be a p p r o x i mated by t h e f o l l o w i n g W e i b u l l d i s t r i -b u t i o n f u n c t i o n

1418 N. NORDENSTR0M, 0. FALTINSEN, B. PEDERSEN 11-25 here A = ( f ) ^ ( 7 2 ) l d where b and k a r e f u n c t i o n s o f m iven i n T a b l e , I I I . 3 . I t i s thus p o s s i b l e t o f i n d d i r e c t -/ t h e parameters o f t h e l o n g t e r m s i b u l l d i s t r i b u t i o n o f x from t h e para-3 t e r s o f t h e l o n g t e r m W e i b u l l d i s t r i -j t i o n o f v'ËT'. The p r o b a b i l i t y l e v e l Q = 1 - P ( x ) i d t h e number o f response c y c l e s N a r e ïlated as f o l l o w s , and by assuming

1 average response p e r i o d i t i s t h u s D s s i b l e t o i n t e r p r e t t h e p r o b a b i l i t y 2vel as a p e r i o d o f t i m e . Q = 1/N ( 7 3 ) As shown i n F i g . I I I . 8 a s h i p l i f e f 20 years corresponds a p p r o x i m a t e l y 3 t h e p r o b a b i l i t y l e v e l Q = 10-8. le c o r r e s p o n d i n g v a l u e o f t h e v a r i a b l e 3 t h u s t h e most p r o b a b l e l a r g e s t v a l u e i r i n g t h e l i f e t i m e o f t h e s h i p . The p r a c t i c a l procedure f o r t h e a l c u l a t i o n o f l o n g t e r m d i s t r i b u t i o n s 3 g i v e n i n Appendix 2.

/. Data f o r s h i p and sea

Our c a l c u l a t i o n s are based on a arent catamaran f o r m w i t h p a r t i c u l a r s 3 g i v e n i n Table I V . 1 .

The model t e s t s were performed Lth a model i n s c a l e 1:30.

D u r i n g t h e model t e s t s t h e t r a n s -acers were p o s i t i o n e d as d e s c r i b e d ïlow. A l l d i s t a n c e s r e f e r t o f u l l : a l e , L - 119 m.

The r o d used f o r measuring heave as f a s t e n e d a t a p o i n t 6.5m a f t o f midships 4.2 m f r o m t h e c e n t r e p l a n e ( s t a r b o a r d s i d e ) 13.2 m above t h e k e e l . The t r a n s d u c e r s f o r measuring "lear f o r c e s and bending moments were

D c a t e d i n t h e c e n t r e p l a n e o f t h e s h i p ,

13.95 m above t h e k e e l and 32.75 m f o r e and a f t o f amidships.

P i t c h and r o l l were measured v ; i t h a gyroscope.

The l o n g t e r m responses have been c a l c u l a t e d employing wave s t a t i s t i c s f o r t h e N o r t h Sea. Parameters o f t h e wave h e i g h t d i s t r i b u t i o n s a r e g i v e n i n Table I I I . 2 . I n f o r m a t i o n on wave spec-t r u m and d i r e c spec-t i o n a l i spec-t y f u n c spec-t i o n used, i s g i v e n i n s e c t i o n I I I . 3 .

The c a l c u l a t e d r e s u l t s have been o b t a i n e d f r o m our computer programmes NVCAT and NV403 , r u n on our UNIVAC 1108 computer. M e t r i c u n i t s a r e used t h r o u g h o u t t h i s r e p o r t . R e s u l t s which a r e g i v e n f o r s h i p l e n g t h s d i f f e r e n t f r o m t h a t o f t h e p a r e n t catamaran a r e v a l i d f o r geome-t r i c a l l y s i m i l a r s h i p s , w i geome-t h a l l dimen-s i o n dimen-s dimen-s c a l e d w i t h t h e dimen-same f a c t o r . V. R e s u l t s and comparisons

V.1 Motions and loads i n r e g u l a r waves. Comparisons w i t h model t e s t r e s u l t s . The t e s t e d catamaran has been de-s c r i b e d i n c h a p t e r I V .

I n F i g . V.1 a r e shown comparisons f o r r o l l . The r o l l angles a r e g i v e n n o n - d i m e n s i o n a l l y w i t h r e s p e c t t o h/X where h i s t h e wave a m p l i t u d e and X i s t h e wave l e n g t h . Computations have been performed f o r two cases. The d i f f e r e n c e between t h e tV70 cases i s due t o d i f f e -r e n t -r o l l moments o f i n e -r t i a . The -r o l l moment o f i n e r t i a about t h e a x i s t h r o u g h t h e c e n t r e o f g r a v i t y was e x p e r i m e n t a l l y d e t e r m i n e d . As one case c a l c u l a t i o n s were performed w i t h t h i s v a l u e , and about t h e a x i s f o r which i t was d e t e r -mined. As t h e o t h e r case we used t h e same v a l u e , b u t about t h e a x i s i n t h e i n t e r s e c t i o n between c e n t r e p l a n e and w a t e r p l a n e . The l a t t e r case i s r e f e r r e d t o as t h e o r y w i t h c o r r e c t e d r o l l mo-ment o f i n e r t i a . The c o r r e s p o n d i n g r o l l moment o f i n e r t i a about t h e a x i s t h r o u g h c e n t r e o f g r a v i t y i n t h i s l a t t e r case i s a p p r o x i m a t e l y 2 0% lower t h a n t h e e x p e r i -m e n t a l l y deter-mined one. I t i s seen t h a t t h e o r y w i t h c o r r e c -t e d r o l l momen-t o f i n e r -t i a agrees v e r y

11-26

PREDICTION OF WAVE-INDUCED MOTIONS AND LOADS FOR CATAMARANS OTC 14]

w e l l w i t h e x p e r i m e n t s , w h i l e f o r t h e case w i t h g i v e n r o l l moment o f i n e r t i a t h e r e i s a d i f f e r e n c e i n t h e t h e o r e t i -c a l l y and e x p e r i m e n t a l l y d e t e r m i n e d r o l l resonance f r e q u e n c y . The same d i f f e r e n c e o c c u r r e d when t h e wave d i r e c -t i o n was 90°. T h i s case i s n o -t shown here. Some o f t h e d i f f e r e n c e i n r o l l resonance f r e q u e n c y between e x p e r i m e n t s and t h e o r y c o u l d be due t o t h e e x p e r i -m e n t a l i n a c c u r a c y i n d e t e r -m i n i n g t h e r o l l moment o f i n e r t i a . However, t h e most r e a s o n a b l e e x p l a n a t i o n i s non-l i n e a r e f f e c t s . I t i s t o be n o t e d t h a t i n some cases we have observed s i m i l a r d i f f e r e n c e s between e x p e r i m e n t a l l y and t h e o r e t i c a l l y determined r o l l resonance f r e q u e n c y f o r a s i n g l e - h u l l s h i p .

I t i s i n t e r e s t i n g t o note t h a t f o r a catamaran i n c o n t r a s t t o a s i n g l e -h u l l s -h i p , i t does n o t seem t o be necessary t o t a k e i n t o account v i s c o u s e f f e c t s when c a l c u l a t i n g t h e r o l l a m p l i -tudes a t t h e r o l l resonance f r e q u e n c y . I n F i g . V.2 a r e shown comparisons f o r heave. I t i s t o be noted t h a t t h e f a s t e n i n g p o i n t o f t h e r o d which

measured heave was n o t i n t h e c e n t r e -l i n e . See s e c t i o n I V . T h i s means t h a t r o l l a t , and i n the v i c i n i t y o f , t h e r o l l resonance f r e q u e n c y w i l l c o n t r i b u t e s i g n i f i c a n t l y t o t h e measured heave a m p l i t u d e . We have c a l c u l a t e d t h e r e f o r e heave f o r

the two cases: 1. Theory w i t h c o r r e c t e d r o l l moment o f i n e r t i a . 2. Theory w i t h g i v e n r o l l moment o f i n e r t i a . I t i s seen t h a t t h e agreement between t h e o r y and experiments i s a c c e p t a b l e , and as expected t h e case w i t h c o r r e c t e d r o l l ' moments o f i n e r t i a g i v e s t h e b e s t agree-ment.

X

We note t h e resonance o f heave a t /L = 0.35. R o l l has s m a l l i n f l u e n c e a t /L < 0.45 so t h e computed heave v a l u e s are a p p r o x i m a t e l y t h e same as t h e heave v a l u e s o f t h e c e n t r e o f g r a v i t y of t h e catamaran. S i m i l a r b e h a v i o u r o f heave f o r a s i n g l e - h u l l s h i p i s n o t • known. The reason f o r t h i s d i f f e r e n c e between t h e heave m o t i o n o f t h e catama-r a n , and t h a t o f a s i n g l e - h u l l s h i p i s most l i k e l y due t o hydrodynamic i n t e r -a c t i o n between t h e h u l l s o f t h e c -a t -a - ' maran.

.^^^^^ F i g - V,3 a r e shown comparisons between e x p e r i m e n t s and t h e o r y f o r p i t c h . The agreement i s good. We note the resonance o f p i t c h f o r ^/L = 0.35 and i n t h e same way as f o r heave t h i s ' resonance must be e x p l a i n e d by hydro-dynamic i n t e r a c t i o n between t h e h u l l s o f t h e catamaran.

I n F i g . V.4 a r e shown comparisons between e x p e r i m e n t s and t h e o r y f o r v e r t i c a l shear f o r c e between t h e two h u l l s . The v e r t i c a l shear f o r c e are g i v e n n o n - d i m e n s i o n a l l y w i t h r e s p e c t t (PgBLh. As i s seen f r o m t h e e x p r e s s i o n of V 3 , t h e v e r t i c a l shear f o r c e depend on t h e r o l l i n g m o t i o n and we have c a l -c u l a t e d t h e r e f o r e Dy w i t h -c o r r e -c t e d ro moment o f i n e r t i a and w i t h g i v e n r o l l moment o f i n e r t i a . I t i s seen t h a t the agreement between t h e o r y w i t h c o r r e c t e r o l l moment o f i n e r t i a and experiments are v e r y good.

I n F i g . V.5 a r e shown comparisons between experiments and t h e o r y f o r p i t c h c o n n e c t i n g moment. The p i t c h c o n n e c t i n g moment depends on t h e r o l l -m o t i o n . However, t h e -maxi-mu-m o f co-mpu- compu-t e d p i compu-t c h c o n n e c compu-t i n g momencompu-t does n o compu-t occur a t , o r i n t h e v i c i n i t y o f , t h e r o l l resonance f r e q u e n c y , so t h e o r y w i t h c o r r e c t e d r o l l moment o f i n e r t i a and t h e o r y w i t h g i v e n r o l l moment o f i n e r t i a do n o t d i f f e r s i g n i f i c a n t l y . I

I S seen t h a t t h e agreement between the

ory and experiments i s r e a s o n a b l y good I n F i g . V.6 i s p l o t t e d t h e o r e t i c a l and e x p e r i m e n t a l v a l u e s f o r v e r t i c a l wave bending moment between t h e two h u l l s . V4 does n o t depend on r o l l i n g m o t i o n , b u t i t seems t h a t t h e e x p e r i -ments do. Except f o r t h e e x p e r i m e n t a l V v a l u e a t r o l l - r e s o n a n c e t h e agreement between t h e o r y and e x p e r i m e n t s i s accel t a b l e .

V. 2 Short t e r m response a Only a few examples w i l l be g i v e n >^

of c a l c u l a t e d s h o r t t e r m response i n i r r e g u l a r waves. We have chosen t o p r e s e n t t h e v e r t i c a l shear f o r c e . The c a l c u l a t i o n s a r e performed w i t h correc- f

J^. NORDENSTROM-. O. FAT.TTM'^KN, R. PKnPRc;ir|^ 11-27

I n F i g s . V.7 - V.9 are shown response s p e c t r a f o r two v a l u e s o f t h e non-dimensional wave p e r i o d

n = T/gTT = 1.02 and 2.18. R e s u l t s

are g i v e n f o r t h e t h r e e headings: head, beam and f o l l o w i n g seas. The o r d i n a t e of these f i g u r e s i s t h e i n t e g r a n d o f Eq. ( 6 4 ) . The a b s c i s s a i s l i n e a r i n t h e loga r i t h m o f t h e wave l e n g t h t o s h i p l e n g t h r a t i o . I t should be noted t h a t the o r d i n a t e s c a l e o f F i g . V.8 d i f f e r s from t h a t o f F i g s . V.7 and V.9.

I t i s seen t h a t the response s p e c t r a i n head and f o l l o w i n g seas d i f f e r v e r y l i t t l e f o r t h e s m a l l e r va -lues. T h i s i s p a r t i c u l a r l y v a l i d f o r l a r g e s h i p s when these s m a l l e r v a l u e s are most i m p o r t a n t .

The response i n beam sea i s con-s i d e r a b l y l a r g e r t h a n i n t h e two other headings.

I n F i g . V.10 i s shown t h e s h o r t term parameter R as a f u n c t i o n o f fi. Results are g i v e n f o r f i v e d i f f e r e n t heading a n g l e s . Here i t i s a g a i n seen t h a t t h e responses i n head and f o l l o w i n g seas d i f f e r v e r y l i t t l e .

As an example o f how t o use these curves, we w i l l c a l c u l a t e t h e most probable l a r g e s t v e r t i c a l shear f o r c e during 200 l o a d c y c l e s i n beam sea f o r a s h i p w i t h l e n g t h L = 12 0 m and beam B = 30.2 m. The weather i s c h a r a c t e r i z e d 3y H 1/3 3 m and T 6 sec. T h i s g i v e s fi = 1.72. From F i g , /.lO we f i n d R = 8.6-10 -2 md BE = 8.6-10-2 • ^ 2.15-10 -3

The most p r o b a b l e l a r g e s t shear o r c e , i s found by employing t h e below e l a t i o n

V ,max

The f a c t o r Y B L ^ i s i n t r o d u c e d as the s h o r t term parameter i s made non-d i m e n s i o n a l as shown i n s e c t i o n I I I . 5 . We t h e n o b t a i n D V ,max 2.15-10 30.2 / I n 200' • 1.025 1 2 0 n o n s 780 t o n s

V.3 Long term response. A l s o t h e l o n g t e r m been c a l c u l a t e d f o r t h e t i o n e d i n s e c t i o n V.1, r o l l moment o f i n e r t i a r o l l moment o f i n e r t i a , curve i s g i v e n r o l l has or t h e d i f f e r e n c e s were n e g l i g i b l e . response,has two cases men-w i t h c o r r e c t e d and w i t h g i v e n Where o n l y one no i n f l u e n c e p r a c t i c a l l y A l l r e s u l t s are g i v e n on t h e proba b i l i t y l e v e l Q = IQ^, i . e . t h e y r e p r e s e n t t h e most p r o b a b l e l a r g e s t r e -sponse i n 108 l o a d c y c l e s . The f i g u r e s showing response as a f u n c t i o n o f s h i p l e n g t h are prepared assuming a l l heading angles t o be e q u a l l y p r o b a b l e . These f i g u r e s are marked " A l l headings".

The s h o r t - c r e s t e d n e s s o f t h e sea has been t a k e n i n t o account as e x p l a i n e d i n s e c t i o n I I I . 5 .

I n F i g . V.11 t h e r o l l i n g angle i s shown as a f u n c t i o n o f heading angle f o r a s h i p w i t h L = 120 m. I t has been found t h a t the same t r e n d w i t h r e s p e c t t o heading angle occurs f o r o t h e r s h i p l e n g t h s .

The reason why a s u b s t a n t i a l r o l -l i n g ang-le i s found a -l s o i n head and f o l l o w i n g seas, 180° and 0° r e s p e c t i v e -l y , i s t h a t t h e s h o r t c r e s t e d n e s s i s t a k e n i n t o c o n s i d e r a t i o n . F i g . V.12 g i v e s r o l l i n g a n g l e as a f u n c t i o n o f s h i p l e n g t h . I t i s seen t h a t v e r y l a r g e r o l l i n g angles are p r e -d i c t e -d . They may even be c h a r a c t e r i z e -d as u n b e l i e v a b l e . However, model t e s t s viere performed a l s o i n i r r e g u l a r waves c o r r e s p o n d i n g t o B e a u f o r t 6 and 7 f o r a

11-28

PREDICTION OF WAVE-INDUCED MOTIONS AND T.nAns FOR CATAMAKANR nrr i^p

s h i p o f l e n g t h 119 m. I t was concluded t h a t B e a u f o r t 6 would be the maximum o p e r a t i o n a l c o n d i t i o n . The s e t t i n g o f t h i s l i m i t was due n o t on^y t o r o l l i n g , but i t i l l u s t r a t e s t h e f a c t t h a t t h e p r e s e n t catamaran seems t o have an u n f o r t u n a t e d e s i g n . I n view o f t h i s a r o l l i n g angle o f 40° s i n g l e a m p l i t u d e f o r L = 120 m seems t o be r e a l i s t i c . The t r e n d w i t h s h i p l e n g t h i s t y p i c a l and i s c o n s i d e r e d t o be r e a s o n a b l e . From F i g . V.13 i t i s seen t h a t p i t c h and p a r t i c u l a r l y yaw i s v e r y l i t t l e i n f l u e n c e d by the heading a n g l e . This i s v a l i d a l s o f o r d i f f e r e n t s h i p l e n g t h s . The f r o m F i g l i s t i c , e s h i p l e n g p r e s s i o n s l i g h t l y f o r s h i p m a t e l y 50 1/3 t h e p l e n g t h s .

p i t c h and yaw angles found

V.14 are c o n s i d e r e d t o be r e a x c e p t may be f o r t h e s m a l l e r t h s . I t i s o u r g e n e r a l im-t h a im-t o u r compuim-ter programmes o v e r e s t i m a t e t h e response l e n g t h s s m a l l e r t h a n a p p r o x i

-m. The yaw a n g l e i s about i t c h angle f o r a l l s h i p

The heave and sway motions have v e r y s i m i l a r t r e n d s w i t h heading a n g l e , as i s seen from F i g . V.15. The same comment a p p l i e s t o F i g . V.16 showing heave and sway as a f u n c t i o n o f s h i p l e n g t h . The maximum response i s seen t o occur f o r a s h i p l e n g t h o f a p p r o x i m a t e l y 100 m. The downward t r e n d f o r l a r g e r s h i p s i s c o n s i d e r e d t o be r e a s o n a b l e . This r e s u l t may c a l l f o r a few comments. F i g . V.10 shows t h a t t h e s h o r t t e r m parameter /ST/E.^^^ may de-crease f o r a g i v e n wave p e r i o d when the s h i p _ l e n g t h i n c r e a s e s . I n o r d e r then t o o b t a i n a l a r g e response l a r g e wave p e r i o d s and wave h e i g h t s are necessary. The wave h e i g h t d i s t r i b u t i o n f o r each wave p e r i o d as w e l l as the p r o b a b i l i t y o f the d i f f e r e n t wave p e r i o d s a r e

t h e r e f o r e d e c i s i v e f o r t h e response. Consequently t h e s h i p l e n g t h f o r which t h e maximum response occurs i s depen-dent b o t h on the response t o a g i v e n wave and t h e wave s t a t i s t i c s f o r t h e ocean area i n q u e s t i o n . For the N o r t h Sea t h e most f r e q u e n t average wave p e r i -od i s s m a l l and t h e r e f o r e the maximum occurs a t t h e c o m p a r a t i v e l y s m a l l s h i p l e n g t h . I f wave s t a t i s t i c s f o r t h e N o r t h A t l a n t i c were used the maximum would move t o a l a r g e r s h i p l e n g t h .

V e r t i c a l a c c e l e r a t i o n has been ca c u l a t e d by combining heave and p i t c h w i t h due r e g a r d t o t h e i r phase d i f - h f e r e n c e , i . e . i t i s the v e r t i c a l acce-i-i l e r a t i o n a t c e n t r e l i n e . I n F i g . V.17 the v e r t i c a l acceler^^^ t i o n a t f o r w a r d and a f t p e r p e n d i c u l a r n as w e l l as a m i d s h i p s i s shown as a ^.' f u n c t i o n o f heading a n g l e . I t i s seeni' t h a t a t zero speed the a c c e l e r a t i o n s at f o r w a r d and a f t p e r p e n d i c u l a r s dif¬ f e r v e r y l i t t l e .

The h o r i z o n t a l a c c e l e r a t i o n shown"^"' i n F i g . V.18 i s found i n the-same way*" by combining sway and yaw. As o n l y ' these two v a r i a b l e s are c o n s i d e r e d t h e g i v e n r e s u l t s r e p r e s e n t t h e h o r i -z o n t a l a c c e l e r a t i o n i n the waterplaneiV

I n F i g . V.19 i s shown the d i s t r i - l . b u t i o n o f v e r t i c a l and h o r i z o n t a l aca l e r a t i o n over the l e n g t h o f the s h i p . I t must be emphasized t h a t these d i s t : b u t i o n s are not p i c t u r e s o f instanten-i o u s d instanten-i s t r instanten-i b u t instanten-i o n s , but express the . envelope o f a l l p o s s i b l e d i s t r i b u t i o n ; I t i s seen t h a t t h e h o r i z o n t a l accele¬ r a t i o n i s p r a c t i c a l l y c o n s t a n t over t h e l e n g t h o f the s h i p , w h i l e t h e v e r t i c a l a c c e l e r a t i o n a t f o r w a r d perpen-d i c u l a r i s twice'^the a c c e l e r a t i o n amit s h i p s . By c o n s i d e r i n g the r e l a t i v e i magnitude between heave and sway on oi hand and p i t c h and yaw on t h e o t h e r , one might expect t h a t t h e h o r i z o n t a l a c c e l e r a t i o n would be more s t r o n g l y dependent on p o s i t i o n i n the s h i p . '* The reason why t h i s i s n o t so i s founi i n t h e phase d i f f e r e n c e between sway and yaw. I t should be mentioned t h a t i p i c t u r e , v e r y s i m i l a r t o t h i s i s found f o r o r d i n a r y s h i p s . The dependence o f a c c e l e r a t i o n amidships on s h i p l e n g t h shown i n F i g . V.20 i s c o n s i s t e n t w i t h what i s found f o r o r d i n a r y s h i p s . ^ The v e r t i c a l r e l a t i v e m o t i o n

be-tween s h i p and wave a t c e n t r e l i n e has been c a l c u l a t e d f o r d i f f e r e n t l o c a -t i o n s i n -t h e s h i p from heave, p i -t c h ar wave c o n s i d e r i n g t h e phase r e l a t i o n s . The r e s u l t i s shown i n F i g s . V.21-22. When c a l c u l a t i n g t h i s response no

account has been t a k e n o f the p r o b a b i v l i t y t h a t the b r i d g e between t h e two h u l l s may submerge.

N. NORDENSTR0M, 0. FALTINSEN. B. PEDERSEN

11-29

I n F i g s . V.23 and 24 t h e v e r t i c a l hear f o r c e and t h e h o r i z o n t a l a t h w a r t -hips f o r c e i s shown versus heading ngle and s h i p l e n g t h r e s p e c t i v e l y .

The v e r t i c a l bending moment and he p i t c h - c o n n e c t i n g moment are shown n F i g s . V.25 - 26. I t i s c l e a r l y seen hat t h e d i f f e r e n c e between t h e c a l c u l a ions f o r t h e two d i f f e r e n t r o l l moments] f i n e r t i a i s p r a c t i c a l l y n e g l i g i b l e .

I n F i g s . V.2 3 and 2 5 t h e a b s o l u t e agnitudes o f shear f o r c e s and moments re g i v e n . These f i g u r e s a r e c a l c u l a -ed f o r a catamaran w i t h L - 120 m and

= 30.2.

Conclusions

The Fr^nk C l o s e - F i t method may be used t o c a l c u l a t e added mass and damping c o e f f i c i e n t s f o r s e c t i o n s w i t h more t h a n one h u l l .

The h y d r o d y n a m i c a l i n t e r a c t i o n between t h e two h u l l s has a l a r g e i n f l u e n c e on added mass and damp-i n g c o e f f damp-i c damp-i e n t s . T h e r e f o r e , t h damp-i s i n t e r a c t i o n has t o be t a k e n i n t o cons i d e r a t i o n . Viscous e f f e c t s seem t o be l e s s i m p o r t a n t a t r o l l resonance f o r a catamaran as compared t o an o r d i n a r y s i n g l e h u l l s h i p .

The agreement between c a l c u l a t i o n s and model t e s t s i n r e g u l a r waves i s g e n e r a l l y good except f o r v e r t i c a l bending moment a t r o l l r e -sonance. The model t e s t s seem t o show t h a t t h e r e i s a c o u p l i n g between r o l l i n g and v e r t i c a l ben-d i n g moment which i s n o t i n c l u ben-d e ben-d i n t h e t h e o r y . This i s p r o b a b l y due t o n o n - l i n e a r e f f e c t s .

Both model t e s t s and c a l c u l a t i o n s g i v e an a d d i t i o n a l heave and p i t c h resonance f o r s h o r t wave l e n g t h s which has n o t been observed f o r s i n g l e h u l l s h i p s . We c o n s i d e r t h i s t o be due t o t h e hydrodynami-c a l i n t e r a hydrodynami-c t i o n between t h e two h u l l s .

The value o f comparisons w i t h model t e s t s and d i s c r e p a n c i e s i n t h i s c o n n e c t i o n , s h o u l d n o t be overemphasized. This i s due t o t h e f a c t t h a t i r r e l e v a n t f a c t o r s l i k e

10,

t a n k vjall' e f f e c t s , e t c . may i n t r o duce e r r o r s i n t h e model t e s t r e -s u l t -s .

The i n f l u e n c e o f heading angle and s h i p s i z e on t h e c a l c u l a t e d l o n g term responses f o r t h e catamaran are e s s e n t i a l l y t h e same as f o r s i n g l e h u l l s h i p s . I t should be emphasized t h a t t h e t r a n s f e r f u n c t i o n s , i . e . t h e r e -sponse c h a r a c t e r i s t i c s i n r e g u l a r waves,are o f l i t t l e v a l u e f o r de-s i g n purpode-sede-s unlede-sde-s t h e y a r e combined w i t h wave s p e c t r a and l o n g t e r m wave s t a t i s t i c s i n o r d e r t o p r e d i c t long term s t a t i s t i c a l d i s t r i b u t i o n s . Short and l o n g t e r m s t a t i s t i c a l d i s t r i b u t i o n s o f t h e v a r i o u s r e ^ s p o n s e . v a r i a b l e s , as c a l c u l a t e d by means o f t h e methods o u t l i n e d here may be used f o r d e s i g n purposes. Years o f e x p e r i e n c e w i t h s i n g l e h u l l s h i p s have shown t h i s , and we f i n d no reason t o b e l i e v e t h a t t h i s would n o t be t r u e f o r c a t a -marans .

We c o n s i d e r t h a t p r i n c i p a l l y t h e same methods as o u t l i n e d i n t h i s paper may be used n o t o n l y f o r s i n g l e - h u l l s h i p s and catamarans, but a l s o f o r o t h e r f l o a t i n g ves-s e l ves-s l i k e d r i l l i n g p l a t f o r m ves-s . Nomenclature added mass ( j , k = 2 , 3 c o e f f i c i e n t s , . . .6) F. 3 ( I ) (D) damping c o e f f i c i e n t s ( j , k - 2 , 3 , 6) h y d r o s t a t i c r e s t o r i n g co-e f f i c i co-e n t s ( j , k = 2 , 3 ,. ..6) hydrodynamic f o r c e and moment components due t o body m o t i o n on t h e con-s i d e r e d h a l f p a r t o f t h e catamaran ( j = 1 , ....6) e x c i t i n g f o r c e and moment components ( j = 2 , 3 , . . . 6 ) d i f f r a c t e d wave f o r c e and moment ( j = 2 , 3 , . . . 6 ) " F r o u d e - K r i l o f f " f o r c e and moment ( j = 2 , 3 , . . . 6 ) i n e r t i a f o r c e and moment components on t h e c o n s i -dered h a l f p a r t o f t h e catamaran ( j =1,....6)

11-30

PREDICTION OF WAVE-INDUCED MOTIONS AND LOADS FOR CATAMARANS OTC • I . . 11 (2D) 3 k b. (2D) j k :k ^ j k D D, M M 'x2 ds x,y,z - moment o f i n e r t i a ( j = 4 , 5 , 6 ) r o l l y a w p r o d u c t o f i n e r -t i a - t w o - d i m e n s i o n a l s e c t i o n a l g e n e r a l i z e d normal compo-nents ( j = 2 , 3 , 4 ) h y d r o s t a t i c r e s t o r i n g f o r -ce and moment components on t h e c o n s i d e r e d h a l f p a r t o f t h e catamaran ( j = l , . . . . 6 ) - dynamic l o a d component's on t h e c o n s i d e r e d h a l f p a r t o f t h e catamaran ( j = l , 6) - e x c i t i n g f o r c e and moment components on t h e c o n s i -dered h a l f p a r t o f t h e catamaran ( j =1,....6) - t w o - d i m e n s i o n a l v e l o c i t y p o t e n t i a l s ( j = 2 , 3 , 4 ) - t w o - d i m e n s i o n a l s e c t i o n a l added mass c o e f f i c i e n t ( j ,k = 2,3,4) - t w o - d i m e n s i o n a l s e c t i o n a l damping c o e f f i c i e n t ( j ,k=2,3,4) - added mass c o e f f i c i e n t s o f t h e c o n s i d e r e d h a l f p a r t o f t h e catamaran - damping c o e f f i c i e n t s o f t h e c o n s i d e r e d h a l f p a r t o f t h e catamaran - d i s p l a c e m e n t s ( j = l , 2 . . . 6 r e f e r t o surge , sway , heave ,| r o l l , p i t c h and yaw r e s -p e c t i v e l y ) - c o r r e s p o n d i n g a m p l i t u d e s o f d i s p l a c e m e n t s - a m p l i t u d e o f v e r t i c a l shear f o r c e • a m p l i t u d e o f h o r i z o n t a l a t h w a r t s h i p s f o r c e • a m p l i t u d e o f v e r t i c a l bending moment • a m p l i t u d e o f p i t c h con-n e c t i con-n g momecon-nt • c o n t o u r a t x o f submer-ged c r o s s - s e c t i o n o f t h e catamaran • c o n t o u r a t x o f submerged c r o s s - s e c t i o n o f t h e h a l f p a r t o f t h e catamaran element o f a r c a l o n g t h e c o n t o u r o f a c r o s s -s e c t i o n c o o r d i n a t e system as de-f i n e d i n F i g . I I I . l u n i t v e c t o r s i n t h e x-,y-and z - d i r e c t i o n s P g Y = Pg L B b 2p V M V _ GM GK 0) A k h e ! = e + a [ a ( t o ) ] 2 [a(a) ,a)] •a(lnX)' L A -I f ( a ) m - r a d i u s o f t h e c i r c u l a r • c r o s s - s e c t i o n o f t h e c l i n d e r s c o n s i d e r e d i n t w o - d i m e n s i o n a l case - mass d e n s i t y o f water - a c c e l e r a t i o n o f g r a v i t • s p e c i f i c g r a v i t y o f wa - l e n g t h betv/een perpend c u l a r s • beam a t w a t e r l i n e amid s h i p s • s e c t i o n a l beam o f one tamaran h u l l • d i s t a n c e between the c t r e p l a n e s o f t h e two h • d i s p l a c e d volume o f tb catamaran • mass o f t h e catamaran • speed o f t h e catamaran ' m e t a c e n t r i c h e i g h t • v e r t i c a l d i s t a n c e t o c e n t r e o f g r a v i t y from k e e l b l o c k c o e f f i c i e n t w a t e r p l a n e c o e f f i c i e n t m i d s h i p s e c t i o n c o e f f i c i e n t y - c o o r d i n a t e o f t h e cei o f g r a v i t y o f a h a l f pi o f t h e catamaran y - c o o r d i n a t e o f t h e cei o f buoyancy o f a T i a l f p a r t o f t h e catamaran z - c o o r d i n a t e o f c e n t r e g r a v i t y f r e q u e n c y o f encounter wave l e n g t h wave number a m p l i t u d e o f i n c i d e n t ' angle between i n c i d e n t r e g u l a r wave and catam r a n heading (£:=180° i n head sea) a n g l e between d i r e c t i e , o f t r a v e l o f elementar r e g u l a r wave and s h o r t c r e s t e d wave system angle between s h i p ' s course and d i r e c t i o n o t r a v e l o f s h o r t c r e s t e wave system wave spectrum as a f u n t i o n o f f r e q u e n c y d i r e c t i o n a l wave spec-t r u m wave spectrum on l o g -s l o p e f o r m spectrum d i r e c t i o n a l i t , f u n c t i o n c ' t h moment o f t h e spe' t r u m c r e s t - t o - t r o u g h wave h e i g h t