The behaviour of moored ships in waves

OTC

2882

THE BEHAVIOUR OF MOORED SH l

PS

I N WAVES

.

by Gerard

van

Oortmerssen, Nether1 ands Ship

Model

Basin

Copyright 1977. Offshore Technology Conference

This paper was presented at the 9th Annual OTC in Houston, Tex., May 2-5,1977. The material is subject to correction by the author. Permisslon to copy Is restricted to an abstract of not more &an 300 words

ABSTRACT

A

method is described for the mathemati-

cal simulation of t h e behaviour of a moored

ship in waves. The method is based on the equations of motion in the time domain accor- ding to Cummins, while the hydrodynamic loads

on the ship are obtained by means of the three,-from

dimensional source technique.

of computations

for a ship moored

to

a jetty are discussed and compared with ex- perimental results.

INTRODUCTION -

UP t i l l a few decades ago, the mooring of ships has been mainly a matter of practical experience. Ships were moored in harbours or sheltered areas only, where the external force: are in general limited to the rather steady current and wind forces.

With the development of the ocean indus- try and the advent of very large ships, which can only be accommodated i n a few harbours with sufficient water depth, the need arose to moor ships in exposed areas. To this Pur- pose special mooring facilities were aesignea to absorb the loads exerted by the environment

o n v t h e moored ship. Nowadays a variety of

mooring arrangements is in operation. Because of the short history and fast development of mooring in exposed areas, the design of terminals can not be based on em- pirism. On the other hand the problem is too complicated for an analytical treatment. fore it is common practice to study the be- haviour of a moored,ship by means of experi- ments with small scale models. Although model testing provides an effective tool to deter- mine mooring forces and maximum motions of the moored ship for design purposes, this method inheres a few drawbacks.

First, model tests are expensive and time consuming. The test set-up is compli- cated, it is essential that elasticity prop- erties of mooring lines and fenders are simu-

laCed

v e r y carefully,

and sophisticated

facilities are needed to simulate the relevant environmental conditions. For this reason test programs are usuaZly restricted to final de- sign configurations and selected weather condi- tions which are assumed to be the most criti- cal'

Further, the fundamental insight gained model tests on these complicated systems is limited. Only the resulting output is measured without learning much of the mecha- nism which causes this output. As an example the low frequency motion of a moored ship observed in tests in irregular waves may be mentioned.

On a basis of model test results it is not possible t o conclude whether this motion is caused by second order effects in the wave loads, or by the fact that the elasticity properties of the mooring system as- non- linear.

It is, therefore, highly desirable to dispose o l a mathematical simulation method for the behaviour of a moored object. Such a method can help to increase the understanding of moored ship behaviour, and can be used as a tool in optimization studies in the early stages of design, prior to model testing.

The basis of each mathematical model for the

simulation of the behaviour of moored objects is the law of dynamics of ~ e w t o n :

a(&)

- = p

dt

There-or,since the inertia m of the ship may b e

regarded as constant:

m; =

F

The external force P is composed of

-

arbitrarily in time varying forces due to

the waves;

-

hydrodynamic and hydrostatic restoring

forces, which are a function of the motians of the ship;

w h i c h a r e a f u n c t i o n o f t h e i n s t a n t a n e o u s p o s i t i o n o f t h e s h i p . I n t h e c l a s s i c a l s h i p m o t i o n t h e o r y , i t i s common p r a c t i c e t o f o r m u l a t e t h e e q u a t i o n s a s f o l l o w s : a , b a n d c a r e c o e f f i c i e n t s w h i c h d e s c r i b e t h e h y d r o d y n a m i c a n d 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 . I n f a c t , ( 1 ) i s n o t a r e a l e q u a t i o n o f m o t i o n , i n t h e s e n s e t h a t i t r e l a t e s t h e i n - s t a n t a n e o u s m o t i o n v a r i a b l e s t o t h e i n s t a n t a - n e o u s v a l u e o f t h e e x c i t i n g f o r c e s . I t c a n o n l y b e u s e d a s a d e s c r i p t i o n i n t h e f r e q u e n c y domain o f a s t e a d y o s c i l l a t o r y m o t i o n , s i n c e t h e h y d r o d y n a m i c c o e f f i c i e n t s a a n d b d e p e n d on t h e f r e q u e n c y o f m o t i o n . A n a l y t i c a l w o r k o n t h e moored s h i p u r o - I n t h i s p a p e r a m a t h e m a t i c a l m o d e l i s d e s - c r i b e d w h i c h .is b a s e d on t h e e q u a t i o n s o f m o t i o n i n t h e t i m e d o m a i n a s t h e y h a v e f i r s t b e e n f o r m u l a t e d b y Cummins [l31

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T h e s e e q u a t i o n s o f m o t i o n may b e c o n s i d e r e d a s t r u e d i f f e r e n t i a l e q u a t i o n s ; t h e y g i v e t h e i n s t a n t a n e o u s r e l a t i o n s h i p b e t w e e n t h e m o t i o n v a r i a b l e s a n d t h e e x t e r n a l f o r c e s . I n t h e s e e q u a t i o n s t h e v a r i o u s f a c t o r s g o v e r n i n g t h e r e s p o n s e o f t h e s h i p a r e s e p a r a t e d i n t o c l e a r - l y i d e n t i f i a b l e u n i t s . The o n l y a s s u m p t i o n i n v o l v e d i s l i n e a r i t y o f t h e h y d r o d y n a m i c r e s t o r i n g f o r c e s . Non- l i n e a r a n d a s y m m e t r i c m o o r i n g c h a r a c t e r i s t i c s c a n b e d e a l t w i t h , a n d t h e e x c i t i n g f o r c e may b e a r b i t r a r y , w h i c h means t h a t b e s i d e s f i r s t . o r d e r wave f o r c e s a l s o s l o w l y v a r y i n g d r i f t . f o r c e s a n d wind- a n d c u r r e n t f o r c e s c a n b e i n - c l u d e d i n t h e f o r c i n g f u n c t i o n .

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C111 a n d t.123)~ may C o n t a i n c o m p o n e n t s w i t h f r e q u e n c i e s l o w e r ( s u b h a r m o n i c ) o r h i g h e r ( s u p e r h a r m o n i c ) t h a n t h a t o f t h e f o r c i n g f u n c t i o n . U n f o r t u n a t e l y , t h e a s s u m p t i o n o f c o n s t a n t h y d r o d y n a m i c c o e f f i c i e n t s c a n n o t b e j u s t i - f i e d : e s p e c i a l l y i n s h a L l o w w a t e r t h e s e c o e f f i c i e n t s a p p e a r t o b e v e r y s e n s i t i v e t o c h a n g e s i n f r e q u e n c y . C o n s e q u e n t l y , a t i m e - d o m a i n d e s c r i p t i o n o f t h e b e h a v i o u r o f t h e moored s h i p i s n e e d e d w h i c h t a k e s i n t o a c c o u n t t h e f r e q u e n c y d e p e n d e n c y o f t h e f l u i d r e a c t i o n f o r c e s . I n t h i s s e c t i o n , a g e n e r a l d e s c r i p t i o n i s g i v e n o f a m a t h e m a t i c a l t e c h n i q u e t o s i m u - l a t e t h e b e h a v i o u r o f a s h i p , moored t o a j e t t y . F o r a more d e t a i l e d d e s c r i p t i o n r e f e - r e n c e is made t o [l41

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in t h e i t h

w h i c h i s a f u n c t i o n o f t h e d e f l e c t i o n of t h e f e n d e r a n d t h e e l a s t i p r o p e r t i e s . n i s t h e t o t a l number o f m o o r i n g l i n e s , n p t i e number o f f e n d e r s . x k ( t ) r e p r e s e n t s t h e t i m e h i s t o r y of t h e t o t a l e n v i r o n m e n t a l l o a d i n t h e k-mode. t h e e q u a t i o n s o f i n t h e t i m e d o m a i n a n a r b i t r a r y m o t i o n i s d e s c r i b e d a s a s u c c e s s i o n o f s m a l l i m p u l s i v e d i s p l a c e m e n t s . The b a s i c a s s u m p t i o n i s , t h a t at any t i m e t h e t o t a l f l u i d r e a c t i v e f o r c e i s t h e sum o f t h e r e a c t i o n s t o t h e i n d i v i d u a l i m p u l s i v e d i s - p l a c e m e n t s , e a c h r e a c t i o n b e i n g c a l c u l a t e d a n a p p r o p r i a t e t i m e l a g f r o m t h e i n s t a n t o f t h e c o r r e s p o n d i n g i m p u l s i v e m o t i o n , o r i n

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a l i n e a r b e t w e e n t h e m o t i o n s a n d t h e f l u i d r e a c t i v e f o r c e s . F o r t h e c o m p u t a t i o n o f c o n s t a n t a d d e d mass c o e f f i c i e n t s , r e t a r d a t i o n f u n c t i o n s a n d t i m e h i s t o r i e s o f wave l o a d s , t h e a d d e d mass a n d d a m p i n g c o e f f i c i e n t s a n d wave l o a d s a r e n e e d e d o v e r t h e c o m p l e t e r a n g e o f f r e q u e n c i e s o f i n t e r e s t . F o r t h e c o m p u t a t i o n o f d e e p w a t e r s h i p m o t i o n s , t h e s o - c a l l e d s t r i p t h e o r y , h a s p r o v e n i t s u s e f u l n e s s . T h i s t h e o r y t a k e s a d - v a n t a g e o f t h e f a c t t h a t f o r s h i p s t h e l o n g i - t u d i n a l d i m e n s i o n i s l a r g e r e l a t i v e t o t h e l a t e r a l a n d v e r t i c a l d i m e n s i o n s . F o r s u c h a s l e n d e r body t h e t h r e e - d i m e n s i o n a l p r o b l e m c a n b e r e d u c e d s u c c e s s f u l l y t o a l o c a l two- d i m e n s i o n a l p r o b l e m . A f t e r i t s p r e s e n t a t i o n , t h e m e t h o d h a s b e e n r e f i n e d by many a u t h o r s a n d t h e r e s u l t s a r e i n g e n e r a l r e l i a b l e . A d r a w b a c k i s , h o w e v e r , t h a t no i n f o r m a t ' o n C a n b e t h e s u r g e U n l i k e t h e d e e p w a t e r p r o b l e m , v e r y f e w s t u d i e s h a v e b e e n p r e s e n t e d on t h e m o t i o n s 0 9 a s h i p i n s h a l l o w w a t e r . K i m C161 h a s a d a p t e d t h e s t r i p t h e o r y f o r a r e s t r i c t e d w a t e r d e p t h . F o r t h e v e r t i - c a l modes o f m o t i o n t h i s a p p r o a c h y i e l d s u s e - f u l but it c a n b e u s e d l a t e r a l m o t i o n s , e s p e c i a l l y i n t h e l o w e r f r e - q u e n c y s i n c e t h e i s b a s i - c a l l y t w o - d i m e n s i o n a l , r e q u i r i n g t h a t t h e f l o w o f w a t e r p a s s e s e n t i r e l y u n d e r n e a t h t h e k e e l o f t h e s h i p . I n s h a l l o w w a t e r , h o w e v e r , t h r e e - d i m e n s i o n a l e f f e c t s become i m p o r t a n t . The w a t e r f l o w s p a r t l y u n d e r n e a t h t h e s h i p a n d p a r t l y a r o u n d t h e e n d s . I n t h e e x t r e m e c a s e , t h e s h i p s i t t i n g on b o t t o m , w a t e r c a n move o n l y a r o u n d t h e e n d s o f t h e s h i p . A m e t h o d o f s o l u t i o n , o f w h i c h t h e v a l i - d i t y i s u n r e s t r i c t e d a s l o n g a s l i n e a r i t y i s a s c e r t a i n e d , i s p r o v i d e d by t h e t h r e e - d i m e n - s i o n a l s o u r c e t e c h n i q u e . T h i s t e c h n i q u e h a s b e e n a p p l i e d s u c c e s s f u l l y d u r i n g t h e l a s t f e w y e a r s f o r t h e c o m p u t a t i o n o f wave l o a d s on l a r g e o f f s h o r e s t r u c t u r e s , b u t i t c a n b e u s e d j u s t a s w e l l f o r s h i p s h a p e d b o d i e s ( s e e r e f . [l71 ) . A s a n e x a m p l e o f t h e r e s u l t s o b t a i n e d w i t h t h i s m e t h o d , F i g u r e 2 shows a c o m p a r i s o n O f wave f o r c e s o n a 2 0 0 , 0 0 0

t a w

t a n k e r i n s h a l l o w w a t e r ( k e e l c l e a r a n c e 2 0 p e r c e n t o f t h e d r a f t ) w i t h v a l u e s o b t a i n e d f r o m m o d e l t e s t s . F i g u r e 3 shows a s i m i l a r O f t h e a d d e d m a s s a n d damping c o e f f i c i e n t f o r m o t i o n i n t h e sway mode. I t becomes o b v i o u s f r o m t h i s f i g u r e , t h a t t h e a d d e d m a s s i n s w a y , w h i c h i s q u i t e i m p o r t a n t f o r t h e b e h a v i o u r o f a m o o r e d s h i p , i s s t r o n g - : l y d e p e n d e n t O n t h e f r e q u e n c y O f A s h a s b e e n s t a t e d , t h e c o n s t a n t " a d d e d m a s s " c o e f f i c i e n t s a n d r e t a r d a t i o n f u n c t i o n s w h i c h a p p e a r i n t h e e q u a t i o n s o f m o t i o n i n t h e t i m e d o m a i n , c a n b e f r o m t h e a d d e d m a s s a n d d a m p i n g c o e f f i c i e n t s i n t h e f r e q u e n c y d o m a i n . As a n e x a m p l e , F i g u r e

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shows t h e r e t a r d a t i o n f u n c t i o n s f o r s u r g e , sway a n d h e a v e m o t i o n s o f a 2 0 0 , 0 0 0 t a n k e r in w a t e r w i t h a d e p t h , 20 p e r c e n t i n e x c e s s o f t h e s h i p ' s d r a f t . W i t h t h e t h r e e - d i m e n s i o n a l s o u r c e w i t k . t e c h n i ¶ u e i t i s a 1 9 0 p o s s i b l e t o t a k e t h e i n - f l u e n c e O f a q u a y O n t h e o t h e : 6 c o e f f i c i e n t ~ i n t o a c c o u n t . F i g u r e

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t i m e h i s t o r y o f t h e e x t e r n a l l o a d s . The r e t a r d a t i o n f u n c t i o n s a n d e l a s t i c i t y f u n c t i o n s o f l i n e s a n d f e n d e r s a r e r e a d i n a s a number o f d i s c r e t e p o i n t s , s u f f i c i e n t t o f i x t h e c u r v e s . I n t e r m e d i a t e v a l u e s a r e , when n e e d e d , f o u n d b y means o f i n t e r p o l a t i o n s u b - r o u t i n e s . As o u t p u t o f t h e p r o g r a m , t h e c o m p u t e d t i m e h i s t o r i e s o f m o t i o n s a n d f o r c e s c a n b e p r i n t e d , p l o t t e d o r dumped on d i g i t a l t a p e f o r f u r t h e r a n a l y s i s . The c o m p u t i n g t i m e i s l i n e a r l y p r o p o r t i o - n a l t o t h e i n v e r s e o f t h e t i m e s t e p . S y s t e - m a t i c c o m p u t a t i o n s w i t h v a r y i n g t i m e s t e p h a v e shown t h a t a s t e p o f 0 . 2 s e c o n d s i s s u f f i c i e n t l y s m a l l f o r a n a c c u r a t e n u m e r i c a l s o l u t i o n o f t h e e q u a t i o n s o f m o t i o n i n c a s e t h e s h i p i s moored a g a i n s t s t i f f f e n d e r s . W i t h t h i s t i m e s t e p , t h e r e q u i r e d c o m p u t i n g t i m e a m o u n t s t o 1 s e c o n d f o r 10 s e c o n d s r e a l t i m e . EXPERIMENTAL VERIFICATION To c h e c k t h e a d e q u a c y o f t h e m a t h e m a t i c a l m o d e l f o r moored s h i p s , a n e x t e n s i v e e x p e r i - m e n t a l p r o g r a m h a s b e e n c a r r i e d o u t t o a n a - l y s e t h e m o t i o n b e h a v i o u r o f a moored s h i p i n r e g u l a r a n d i r r e g u l a r w a v e s . A f t e r w a r d s , t y p i c a l t e s t s i t u a t i o n s w e r e s e l e c t e d f o r s i m u l a t i o n on t h e c o m p u t e r t o s e e w h e t h e r t h e p h e n o m e n a b e mean'' o f t h e m a t h e m a t i c a l s i m u l a t i o n . F o r a d e t a i l e d d e s c r i p t i o n a n d c o m p l e t e r e - s u l t s r e f e r e n c e i s made t o c141

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H e r e , o n l y t h e m a i n f i n d i n g s w i l l b e s u m m a r i z e d * The s t u d y was c o n d u c t e d f o r a l o a d e d 2 0 0 , 0 0 0 tdw t a n k e r , m o o r e d t o a n o p e n j e t t y i n w a t e r w i t h a d e p t h a m o u n t i n g t o 7 . 2 t i m e s t h e d r a f t a f t h e v e s s e l . The m a i n p a r t i c u l a r s o f t h e s h i p a r e g i v e n i n T a b l e I . The m o o r i n g l a y - o u t i s d e p i c t e d i n F i g u r e

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l i n e s , e a c h r e p r e s e n t i n g two o r t h r e e w i r e s w i t h n y l o n t a i l s i n r e a l i t y . The l o a d - e l o n g a t i o n c u r v e s o f t h e l i n e s a r e shown i n F i g u r e 7. I n e a c h l i n e a p r e t e n s i o n was o f 20 t o n s . The p a i r o f s t i f f f e n d e r s h a d a l i n e a r e l a s t i c i t y a m o u n t i n g t o 1 5 7 5 t o n / m . The m o d e l t e s t s w e r e c a r r i e d o u t i n t h e S h a l l o w W a t e r L a b o r a t o r y o f t h e N e t h e r l a n d s S h i p Model B a s i n on a s c a l e o f 1 : 8 2 . 5 , The l o a d s i n m o o r i n g l i n e s a n d f e n d e r s w e r e m e a s u r e d b y means o f s t r a i n g a u g e t r a n s - d u c e r s . T h e m o t i o n s w e r e m e a s u r e d b y means o f p o t e n t i o m e t e r s . All s i g n a l s w e r e r e c o r d e d b o t h on m a g n e t i c t a p e a n d p a p e r c h a r t . The m o d e l t e s t s w e r e b a s e d o n F r o u d e ' s l a w o f s i m i l i t u d e . F i r s t a n e x t e n s i v e s e r i e s o f t e s t s i n

r e g u l a r beam p a v e s ( a = 9 0 ~ ) was c a r r i e d out

t o i n v e s t i g a t e t h e b e h a v i o u r o f t h e moored s h i p o v e r a w i d e r a n g e o f p e r i o d s , v a r y i n g f r o n .

9 t o 41 s e c o n d s . The l o n g p e r i o d waves a r e n o t m e r e l y o f t h e o r e t i c a l i n t e r e s t , b u t may a l s o b e a s s o c i a t e d w i t h s e i c h e s o r h a r b o u r o s c i l - l a t i o n s , w h i c h s o m e t i m e s c a u s e p r o b l e m s i n - . p r a c t i c e . - I n a l l c o n d i t i o n s t e s t e d , t h e s h i p a t - t a i n e d a p e r i o d i c m o t i o n , a f t e r a p e r i o d o f t r a n s i e n c e . C h a r a c t e r i s t i c o f t h e sway m o t i o n was t h e o c c u r r e n c e of a mean d i s p l a c e m e n t i n a d d i t i o n t o t h e o s c i l l a t o r y m o t i o n . I n s h o r t waves t h i s m o t i o n h a d t h e same f r e q u e n c y a s t h e w a v e s , w i t h b o u n c e s a g a i n s t t h e f e n d e r s o f e q u a l s t r e n g t h a t t i m e i n t e r v a l s e q u a l t o t h e wave p e r i o d . I n c e r t a i n l o n g w a v e s , h o w e v e r , a s u b - h a r m o n i c m o t i o n was o b s e r v e d : t h e s h i p m o t i o n was composed o f a m o t i o n w i t h f r e q u e n c y

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e q u a l t o t h a t o f t h e e x c i t i n g w a v e s , on w h i c h a m o t i o n was s u p e r i m p o s e d w i t h f r e q u e n c y e i t h e r w/3 o r w/2. I m p a c t s a g a i n s t t h e f e n d e r s o c c u r r e d t h e n a t i n t e r v a l s O f 3 O r wave periods S u b h a r m o n i c m o t i o n was o b s e r v e d o n l y i n w a v e s w i t h s u c h a f r e q u e n c y , t h a t t h e f r e q u e n c y o f t h e s u b h a r m o n i c ( w / 3 o r w / 2 ) was c l o s e t o t h e " n a t u r a l " f r e q u e n c y o f t h e m o o r e d w h i c h i n t h e sway t o 0 . 0 7 r a d . s e c . - l ( d u e t o t h e n o n - l i n e a r e l a s - t i c i t y c h a r a c t e r i s t i c o f t h e m o o r i n g s y s t e m t h e r e i s no w e l l - d e f i n e d - n a t u r a l f r e q u e n c y ; i n f a c t t h e r e s o n a n c e F r e q u e n c y d e p e n d s on t h e a m p l i t u d e of m o t i o n ) . When t h e a m p l i t u d e o f m o t i o n was decreases, t h e s u b h a r m o n i c m o t i o l i t h e d i s a p p e a r e d . I n some c a s e s a n o t h e r mode o f r e g u l a r m o t i o n was f o u n d , f o r i n s t a n c e with a l t e r n a t i n g l i g h t a n d h e a v y b o u n c e s a g a i n s t t h e f e n d e r s . The c o m p u t a t i o n s w e r e c a r r i e d o u t w i t h p u r e h a r m o n i c f o r c i n g T u n c t i o n s , w h i c h w e r e o b t a i n e d f r o m t h e t h r e e - d i m e n s i o n a l s o u r c e t e c h n i q u e , t a k i n g i n t o a c c o u n t a wave h e i g h t e q u a l t o t h a t , m e a s u r e d i n t h e b a s i n w i t h o u t t h e s h i p m o d e l b e i n g t h e r e . Time h i s t o r i e s w e r e c o m p u t e d f o r a p e r i o d o f 1000 s e c o n d s , a l t h o u g h i t a p p e a r e d t h a t a f t e r 500 s e c o n d s t h e r e s u l t s became s t a - t i o n a r y . I t was f o u n d , t h a t i n a l l c a s e s t h e t y p i c a l m o t i o n b e h a v i o u r o b s e r v e d i n t h e was p r e d i c t e d b y t h e m a t h e - m a t i c a l m o d e l . B e s i d e s , t h e q u a n t i t a t i v e agreement b e t w e e n a n d ap- g e a r e d t o b e s a t i s f a c t o r y : t h e d i f f e r e n c e s b e t w e e n m e a s u r e d a n d c o m p u t e d v a l u e s o f m o t i o n a n d m o o r i n g f o r c e a m p l i t n d e s w e r e i n g e n e r a l l e s s t h a n 20 p e r c e n t . As a n e x a m p l e F i g u r e 8 shows t h e r e s u l t s f o r a wave c o n d i t i o n i n w h i c h s u b h a r m o n i c r e s - p o n s e o c c u r r e d . A l s o l o n g c r e s t e d i r r e g u l a r waves h a v e b e e n t a k e n i n t o c o n s i d e r a t i t i n . One s e a c o n d i - t i o n was u s e d , o f w h i c h t h e s p e c t r a l d e n s i t y i s shown i n F i g u r e 9 , w i t h t h r e e a n g l e s o f wave a t t a c k , a = 9 0 , 135 a n d 180 d e g r e e s - The m e a s u r e m e n t s l a s t e d a p e r i o d c o r r e s p o n d i n g t o 2 1 0 0 p r o t o t y p e and b e g a n l o o O s e c o n d s a f t e r s t a r t i n g t h e wave g e n e r a t o r , a v o i d i n g t h a t t r a n s i e n t phenomena w o u l d i n f l l r - e n c e t h e r e s u l t s . The c o m p u t a t i o n s w e r e p e r f o r m e d f o r a p e r i o d c o r r e s p o n d i n g t o 2500 s e c o n d s . The f i r s l . 400 s e c o n d s r e p r e s e n t a p e r i o d O F t r a n s i e n c e ,

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b a n d s o f c o n s t a n t w i d t h . Each b a n d was r e p r e s e n t e d b y a c o m p o n e n t , h a v i n g t h e c e n t r e f r e q u e n c y o f t h a t b a n d a n d a h e i g h t , f o l l o w i n g f r o m t h e b a n d a r e a . T h e s e w a v e r o m p o n e n t s w e r e summea w i t h a r b i t r a r y p h a s e a n g l e s , a n d w i t h t h e a i d o f t h e t r a n s f e r f u n c t i o n s c o m p u t e d w i t h t h e t h r e e - d i m e n s i o n a l s o u r c e t e c h n i q u e t h e f u n c t i o n s i n t h e

6

m o d e s w e r e d e t e r - m i n e d . M a t h e m a t i c a l s i m u l a t i o n s showed t h a t i n beam s e a s t h e r o l l m o t i o n at t h e f r e q u e n c y i s o v e r e s t i m a t e d b y t h e t h e o r y . T h e r e f o r e i t a p p e a r e d d e s i r a b l e t o t a k e s o m e e x t r a roll a s d e t e r m i n e d f r o m m o d e l t e s t s , i n t o a c c o u n t . I n w a v e s f r o m 9 0 a n d 1 3 5 d e g r e e s a good a g r e e m e n t was o b s e r v e d b e t w e e n t h e = e s u l t s o f e x p e r i m e n t s a n d t h e m a t h e m a t i c a l simula- t i o n . As a n e x a m p l e , F i g u r e 10 shows t h e m e a s u r e d a n d c o m p u t e d s p e c t r a o f m o t i o n s a n d f o r c e s f o r 1 3 5 d e g r e e s w a v e s , while t h e most i n t e r e s t i n g d i g i t a l i n f o r m a t i o n f o r t h i s c a s e i s tabulated i n I t a p p e a r e d t h a t t h e s p e c t r a o f h o r i z o n - t a l m o t i o n s a n d m o o r i n g f o r c e s show p e a k s i n t h e low f r e q u e n c y r a n g e . F o r t h e c a s e o f h e a d waves ( a = l 8 0 d e g r e e s ) , i n f i r s t i n s t a n c e a v e r y b a d t o r r e - l a t i o n *as f o u n d : t h e t h e o r e t i c a l f o r c i n g f u n c t i o n s c o n s i s t o n l y o f s m a l l o s c i l l a t o r y l o a d s i n s u r g e , h e a v e a n d p i t c h mode, r e s u l - t i n g i n s m a l l v a r i a t i o n s o f t h e m o o r i n g f o r c e s a r o u n d t h e p r e t e n s i o n a n d a s m a l l s u r g e o s c i l l a t i o n a r o u n d t h e z e r o p o s i t i o n . I n r e a l i t y , h o w e v e r , t h e s u r g e c o n s i s t s o f a l a r g e r a m p l i t u d e o s c i l l a t i o n a r o u n d a mean d i s p l a c e m e n t , a n d t h e v a r i a t i o n s i n t h e m o o r i n g l o a d s a r e a l s o l a r g e r . I t was a s s u m e d t h a t i n t h i s c a s e t h e s e c o n d o r d e r wave d r i f t f o r c e p l a y s a n i m p o r t a n t r o l e . T o c h e c k t h i s a s s u m p t i o n , two c a l c u l a t i o n s w e r e made: o n e w i t h a n a d d i t i o n a l c o n s t a n t and o n e w i t h a s l o w l y v a r y i n g d r i f t f o r c e , w h i c h i s much more r e a l i s t i c i n i r r e g u l a r w a v e s . As a n a p p r o x i - m a t i o n o f t h e m a g n i t u d e o f t h e d r i f t f o r c e , m e a s u r e m e n t s by P i n k s t e r

t18,

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p e r f o r m e d o n a d i f f e r e n t s h i p m o d e l in a similar s e a t i o n w e r e u s e d . W i t h L h e a d d i t i o n a l slowly v a r y i n g d r i f t f o r c e a g o o d a g r e e m e n t was f o u n d . O b v i o u s l y , t h e i n f l u e n c e o f t h e l o w - f r e q u e n c y d r i f t f o r c e is thur8essential i n t h i s condition, A N A L Y S I S O F RESULTS The

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r e s t i n g f e a t u r e s of t h e b e h a v i o u r o f a s h i p , moored t o a j e t t y . I n r e g u l a r beam s e a s , t h e sway m o t i o n c o n s i s t s 09 a n o s c i l l a t i o n a r o u n d a mean d i s - p l a c e m e n t . I n c e r t a i n l o n g w a v e s a s u b h a r m o n i c sway m o t i o n was f o u n d w i t h f r e q u e n c y w/2 o r w / 3 . I n i r r e g u l a r s e a s , t h e s p e c t r a o f h o r i - z o n t a l m o t i o n s a n d m o o r i n g f o r c e s show low- f r e q u e n c y p e a k s , c l e a r l y d i s t i n c t f r o m t h e r a n g e of wave f r e q u e n c i e s . From t h e c o m g u t a - t i o n s i t f o l l o w e d t h a t i n h e a d waves t h e s e - c o n d o r d e r wave f o r c e p l a y s a n i m p o r t a n t r o l e i n e x c i t i n g t h i s low f r e q u e n c y b e h a v i o u r , b u t i n t h e o t h e r wave d i r e c t i o n s c o n s i d e r e d , low f r e q u e n c y p e a k s w e r e f o u n d d u e t o f i r s t o r d e r wave e x c i t a t i o n o n l y . T h i s l o w f r e q u e n c y b e - h a v i o u r i n i r r e g u l a r waves m u s t a l s o b e d i s - t i n g u i s h e d f r o m t h e s u b h a r m o n i c m o t i o n i n r e - g u l a r w a v e s : t h e l o n g e s t wave c o m p o n e n t i n t h e s p e c t r u m h a d a f r e q u e n c y o f 0 . 4 2 5 r a d . s e c . - l w h e r e a s s u b h a r m o n i c m o t i o n s o n l y o c - c u r r e d i n waves w i t h f r e q u e n c i e s l o w e r t h a n a p p r o x i m a t e l y 0 . 2 1 r a d . s e c . - l . As was s h o w n , t h i s t y p i c a l b e h a v i o u r o f a moored s h i p c a n b e p r e d i c t e d b y t h e c o m p l i - c a t e d m a t h e m a t i c a l m o d e l d e s c r i b e d h e r e , b u t t o u n d e r s t a n d why t h e s e modes o f m o t i o n o c c u r , i t i s h e l p f u l t o u s e a s i m p l i f i e d a n a l y t i c a l a p p r o a c h . The a b o v e m e n t i o n e d s p e c i a l f e a t u r e s o f moored s h i p m o t i o n s o c c u r m a i n l y i n t h e h o r i - z o n t a l modes, s u r g e , sway a n d yaw. When t h e s e m o t i o n s a r e c o n s i d e r e d a s b e i n g u n c o u p l e d , i t i s o b s e r v e d t h a t t h e r e s t o r i n g f o r c e a n d momen'; i n s u r g e a n d yaw a r e n o n - l i n e a r , b u t s y m m e t r i c , a n d h e n c e t h e y c a n b e s c h e m a t i z e d a s : 3 f ( x ) =

ax

+

@ X

. . . ( 7 )

w h e r e f ( x ) i s t h e r e s t o r i n g f o r c e , X i s t h e d i s p l a c e m e n t a n d a a n d

B

a r e c o n s t a n t s . When m o r e o v e r t h e f r e q u e n c y - d e p e n d e n c y o f t h e a d d e d m a s s i s i g n o r e d a n d t h e d a m p i n g i s n e g - l e c t e d , t h e s i m p l i f i e d e q u a t i o n o f m o t i o n b e - comes 2 + u x + 8 x 3 = ~ ( t ) .

. . .

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( 8 ) T h i s i s t h e w e l l - k n o w n D u f f i n g e q u a t i o n , w h i c h h a s b e e n t r e a t e d e x t e n s i v e l y i n t h e l i t e r a t u r e o f n o n - l i n e a r v i b r a t i o n s ( s e e f o r i n s t a n c e S t o k e r , 1193 ) . The s o l u t i o n s o f t h e D u f f i n g e q u a t i o n show t h e f o l l o v i n g p a r t i c u l a r s :

-

t h e f i r s t o r d e r o f t h e m o t i o n due t o a h a r m o n i c e x c i t a t i o n F = F c o s a w t i s : X = A c o s

w t ,

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t h e r e s p o n s e s h o w s a I I ~ s h a p e , ~ ~ ~ ~ ~ ~ ~ ~ I

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w i t h F = Fa c o s w t , s u b h a r m o n i c s o l u t i o n s e x i s t w i t h f r e q u e n c y w / 3 ,

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w i t h e x c i t a t i o n s c o n s i s t i n g o f two h a r m o n i c c o m p o n e n t s , F = F , c o s w l t

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F 2 c o s w t , 2 t h e m o t i o n s c o n t a l n comwonents w i t h f r e - q u e n c i e s 2 b 1

2

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U , , t h e s o - c a l l e c . c o m b i n a t i o n t-ones, b e s i d e s t h e b a s i c f r e - q u e n c i e s w , a n d w 2 . T h e e l a s t i c i t y o f t h e m o o r i n g s y s t e m i n t h e sway mode i s e s s e n t i a l l y d i f f e r e n t i n t h a t s e n s e , t h a t t h e r e s t o r i n g f o r c e i s a s y m m e t r i - c a l : when p u s h i n g a g a i n s t t h e f e n d e r s , t h e f o r c e i s d i f f e r e n t f r o m t h e c a s e t h a t t h e s h i p p u l l s a t t h e l i n e s . The m o s t s i m p l e way t o s c h e m a t i z e s u c h a r e s t o r i n g f o r c e i s : 2 3 f ( x ) = a x

+

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+

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. . . ( 9 )

When a g a i n i g n o r i n g f r e q u e n c y - d e p e n d e n c y o f a d d e d mass a n d d a m p i n g , t h e e q u a t i o n o f m o t i o n b e c o m e s : 2

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t h e f i r s t o r d e r a p p r o x i m a t i o n o f t a e r e s u l t i n g m o t i o n shows a n o s c i l l a t i o n a r o u n d a mean v a l u e : X = A c o s

a t

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s u b h a r m o n i c m o t i o n may o c c u r w i t h f r e q u e n - c i e s w/2 a s w e l l a s w / 3 ,

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w i t h b i - f r e q u e n c y e x c i t a t i o n ( F = F l w , t

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F2 c o s w t ) c o m b i n a t i o n t o n e s c a n b e 2 f o u n d w l t h f r e q u e n c i e s 2wl

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w 2 . Thus t h i s s i m p l i f i e d a n a l y t i c a l a p p r o a c h shows how a mean d i s p l a c e m e n t a n d low f r e - q u e n c y m o t i o n s o r i g i n a t e f r o m t h e n o n - l i n e a r p a r t i c u l a r s o f t h e m o o r i n g s y s t e m . I t w i l l b e c l e a r , t h a t t h e e q u a t i o n s o f m o t i o n i n t h e f r e q u e n c y d o m a i n c a n o n l y b e u s e d Lo e x p l a i n c e r t a i n p a r t i c u l a r s o f t h e b e h a v i o u r o f t h e moored s h i p , b r o u g h t a b o u t by t h e p e c u l i a r i t i e s o f t h e m o o r i n g s y s t e m , t h e c o e f f i c i e n t s o f t h e i n e r t i a a n d d a m p i n g t e r m s b e i n g s t r o n g l y d e p e n d e n t on f r e q u e n c y . I n t h e p r e s e n t c a s e , t h e i n e r t i a c o e f f i c i e n t a t t h e s u b h a r m o n i c f r e q u e n c y i s a r o u n d f o u r t i m e s l a r g e r t h a n a t t h e wave f r e q u e n c y , w h i c h c l e a r l y i l l u s t r a t e s t h e n e c e s s a t y t o t a k e t h e f r e q u e n c y - d e p e n d e n c y o f t h e c o e f f i c i e n t s i n t o a c c o u n t . The l o w f r e q u e n c y r e s p o n s e i n i r r e g u l a r s e a s f r o m 9 0 a n d 1 3 5 d e g r e e s i s o b v i o u s l y a r e s u l t of t h e phenomenon o f c o m b i n a t i o n t o n e s , s i n c e s u b h a r m o n i c r e s p o n s e was n o t f o u n d i n t h e s e p a r a t e c o m p o n e n t s o f t h e s p e c t r u m . Com- p u t a t i o n s w i t h l o w f r e q u e n c y d r i f t f o r c e s a d d e d t o t h e f o r c e i n p u t d i d n o t y i e l d s i g n i - f i c a n t l y d i f f e r e n t r e s u l t s f o r t h e s e wave c o n - d i t i o n s . F o r a n o t h e r m o o r i n g c o n f i g u r a t i o n w i t h s o f t f e n d e r s , h o w e v e r , t h e d r i f t f o r c e d i d h a v e FUTURE DEVELOPMENTS The p r o g r a m p a c k a g e a v a i l a b l e a t N.S.M.B. i s o p e r a t i o n a l f o r t h e a n a l y s i s o f t h e b e h a - v i o u r o f s h i p s , moored t o o p e n o r s o l i d j e t t i e s , a n d c a n a l s o b e a p p l i e d f o r t h e d e - t e r m i n a t i o n o f i m p a c t l o a d s d u r i n g b e r t h i n g J

m a n o e u v r e s ( s e e r e f . L201). The a n a l y s i s

i s

t h e o r e t i c a l , e x c e p t f o r t h e wave d r i f t f o r c e s , w h i c h s t i l l s h o u l d b e d e t e r m i n e d e x p e r i m e n t a l - l y . I n v i e w o f t h e i m p o r t a n c e o f t h e s e ' f o r c e s t o t h e b e h a v i o u r o f m o o r e d s h i p s , much e f f o r t w i l l b e n e e d e d f o r t h e d e v e l o p m e n t o f r e l i a b l e p r e d i c t i o n m e t h o d s . I t i s e x p e c t e d , t h a t

w i t h i n t h e n e x t two y e a r s means w i l l become a v a i l a b l e t o c a l c u l a t e t h e t i m e h i s t o r y of t h e l o w f r e q u e n c y d r i f t f o r c e s . On t h i s s u b j e c t , P r o g r e s s i s made b y among o t h e r s D a l z e l l c211 a n d P i n k s t e r L181

.

I t i s i n t e n d e d t o e x t e n d t h e a p p l i c a b i - l i t y o f t h e p r o g r a m i n s u c h a way, t h a t s i n g l e p o i n t m o o r i n g s c a n b e s i m u l a t e d . T h i s m e a n s , t h a t t h e i n f l u e n c e o f s i g n i f i c a n t c h a n g e s 0 9 h e a d i n g on t h e wave l o a d s a n d t h e d y n a m i c s o f t h e buoy m u s t b e i n c l u d e d i n t h e m a t h e m a t i c a l m o d e l . CONCLUSIONS - - The m a t h e m a t i c a l m o d e l d e s c r i b e d i s s u i t a b l e f o r t h e s i m u l a t i o n o f m o o r e d s h i p m o t i o n s i n s i x d e g r e e s o f f r e e d o m i n w a v e s . The t y p i c a l m o t i o n b e h a v i o u r as o b s e r v e d i n f u l l s c a l e a n d i n m o d e l t e s t s i s r e p r o d u c e d b y t h e m a t h e m a t i c a l m o d e l . A s h i p , moored i n random s e a s , c a n e x p e r i e n c e t h r e e t y p e s o f l o w f r e q u e n c y b e h a v i o u r :

-

s u b h a r m o n i c r e s p o n s e t o c e r t a i n h a r m o n i c c o m p o n e n t s , w i t h a f r e q u e n c y a m o u n t i n g t o 1 1 2 o r 1 1 3 o f t h a t o f t h e e x c i t i n g wave c o m p o n e n t . T h i s s u b h a r m o n i c r e s p o n s e i s a r e s u l t o f t h e n o n - l i n e a r i t y o f t h e m o o r i n g s y s t e m ,

-

" c o m b i n a t i o n t o n e s " , a l o w f r e q u e n c y m o t i o n i n d u c e d b y t h e s i m u l t a n e o u s a c t i o n o f more t h a n o n e h a r m o n i c w a v e c o m p o n e n t . ~l~~ t h i s phenomenon o r i g i n a t e s f r o m t h e n o n - l i n e a r e l a s t i c i t y c h a r a c t e r i s t i c s ,

-

low f r e q u e n c y m o t i o n s e x c i t e d b y t h e l o w f r e q u e n c y s e c o n d o r d e r wave d r i f t f o r c e . NOMENCLATURE C k j m a t r i x o f r e s t o r i n g f o r c e c o e f f i c i e n t s , F g e n e r a l f o r c e o r moment, G c e n t r e o f g r a v i t y , I k moment o f i n e r t i a i n t h e k - t h mode, I k j p r o d u c t o f i n e r t i a , K r e t a r d a t i o n f u n c t i o n i n t h e k - t h mode k j _{d u e t o m o t i o n i n t h e j - t h mode,} L l e n g t h o f t h e s h i p , f o r c e o r moment i n t h e k - t h mode d u e L i k t o t h e i - t h m o o r i n g l i n e , M k j i n e r t i a m a t r i x , N i k f o r c e o r moment i n t h e k - t h mode d u e t o t h e i - t h f e n d e r , S S s p e c t r a l d e n s i t y o f l t h e w a v e s ,

X k wave f o r c e o r moment i n t h e k - t h mode, a a d d e d mass c o e f f i c i e n t i n t h e k-th k j due t o m o t i a n i n t h e j - t h mode, b d a m p i n g c o e f f i c i e n t i n t h e k - t h mode d u e k j t o motion i n t h e j - t h mode, g a c c e l e r a t i o n o f g r a v i t y , j , k s u b s c r i p t s r a n g i n g f r o m 1 t o

6

u s e d f o r a d i r e c t i o n o r a d e g r e e o f f r e e d o m , m m a s s o f t h e s h i p , m f r e q u e n c y - i n d e p e n d e n t a d d e d m a s s c o e f f i - k j _{c i e n t i n t h e k - t h mode d u e t o m o t i o n} i n t h e j - t h mode, X d i s p l a c e m e n t i n t h e j - t h mode, j

a

a n g l e o f wave i n c i d e n c e , w c i r c u l a r f r e q u e n c y . R E F E R E N C E S 1 . K a p l a n , P , a n d P u t z , R . R . : "The m o t i o n s o f a moored c o n s t r u c t i o n - t y p e b a r g e i n i r r e - g u l a r w a v e s a n d t h e i r i n f l u e n c e on con- s t r u c t i o n o p e r a t i o n " : N B Y - 3 2 2 0 6 , M a r i n e A d v i s o r s , I n c . L a J o l l a , 1 9 6 2 . 2 . L e e n d e r t s e , J . J . : " A n a l y s i s o f t h e r e s p o n s e o f moored s u r f a c e a n d s u b s u r f a c e v e s s e l s tc o c e a n w a v e s " : Rand C o r p o r a t i o n Memorandum RM-3368 PR, 1 9 6 3 . 3 . Muga, B.J.: " E x p e r i m e n t a l a n d t h e o r e t i c a l s t u d y o f m o t i o n o f a b a r g e a s m o o r e d i n o c e a n w a v e s " : U n i v e r s i t y o f I l l i n o i s , H y d r a u l i c E n g i n e e r i n g S e r i e s No. 1 3 , 1967.

4 .

S e i d l , L.H.: " P r e d i c t i o n o f m o t i o n s o f s h i p s moored i n i r r e g u l a r s e a s " : P r o c . N.A.T.O. Advanced S t u d y I n s t i t u t e o n A n a l y t i c a l T r e a t m e n t o f P r o b l e m s i n t h e B e r t h i n g a n d M o o r i n g o f S h i p s , W a l l i n g - f o r d , 1 9 7 3 , p p . 221-229.

5 .

Abramson, H . N . a n d W i l s o n , B a s i l W . : "A f u r t h e r a n a l y s i s o f t h e l o n g i t u d i n a l r e s p o n s e o f m o o r e d v e s s e l s t o s e a o s c i l - l a t i o n s " : P r o c . A.S.C.E. 8 5 , 1 9 5 9 , W W ~ , p . 1 7 3 .

6 . Yang, I-Min: " ~ o t i o n s o f moored s h i p s i n s i x d e g r e e s o f f r e e d o m " : 9 t h Symposium o n N a v a l H y d r o d y n a m i c s , P a r i s , 1 9 7 2 , 7 . K i l n e r , F.A.: "Model t e s t s on t h e m o t i o n o f moored s h i p s p l a c e d on l o n g w a v e s t 1 : P r o c . 7 t h C o n f . on C o a s t a l E n g i n e e r i n g , The H a g u e , 1 9 6 0 , Volume 2 p p . 723-745. 8 . W i l s o n , J . F . a n d A w a d a l l a , N . G . : "Sub- h a r m o n i c r e s p o n s e i n t h e n o n - l i n e a r o s c i l - l a t i o n s o f m o o r e d s h i p s " : O f f s h o r e Techno- l o g y C o n f e r e n c e , H o u s t o n , 1 9 7 1 , p a p e r OTC 1 4 2 0 , Volume 11

pp.

6 5 - 8 0 .

TABLE I

,

9 . W i l s o n , J . F . a n d A w a d a l l a , N . G . : " C o m p u t e r s i m u l a t i o n o f n o n - l i n e a r m o t i o n o f m o o r e d s h i p s t 1 : P r o c . N.A.T.O. A d v a n c e d S t u d y I n s t i t u t e o n A n a l y t i c a l T r e a t m e n t o f p r o b l e m s i n t h e ~ ~ ~a n d t~~~~i~~ h i o f ~ ~ S h i p s , W a l l i n g f o r d , 1 9 7 3 , p p . 2 7 7 - 2 9 8 . 1 0 . L e a n , G . H . : " S u b h a r m o n i c m o t i o n s o f m o o r e d s h i p s s u b j e c t e d t o wave a c t i o n " : T r a n s - a c t i o n s o f t h e R o y a l I n s t i t u t i o n o f N a v a l A r c h i t e c t s , 1 9 7 1 , V o l . 1 1 3 , p p . 387-399 1 1 . W i l s o n , B.W.: " P r o g r e s s i n t h e s t u d y o f s h i p s m o o r e d i n w a v e s " : P r o c . N.A.T.O. A d v a n c e d S t u d y I n s t i t u t e o n A n a - l y t i c a l T r e a t m e n t o f P r o b l e m s i n t h e B e r t h i n g a n d M o o r i n g o f S h i p s , W a l l i n g f o r d , 1 9 7 3 , p p . 1 4 3 - 2 1 3 . 1 2 . B ~ H , : ~" ~~ ~ ~~ d e t e r E i n a t i o n l, ~ t i ~ o f ~ l s h i p m o t i o n a n d m o o r i n g f o r c e s " : O f f s h o r e T e c h n o l o g y C o n f e r e n c e , H o u s t o n , 1 9 7 4 , p a p e r OTC 2 0 7 2 . -

1 3 . Cummins, W.E.: "The i m p u l s e r e s p o n s e f u n c t i o n a n d s h i p m o t i o n s " , D.T.M.B. R e p o r t 1 6 6 1 , W a s h i n g t o n , D . C . , 1 9 6 2 . 1 4 . O o r t m e r s s e n , G . v a n : "The m o t i o n s o f a m o o r e d s h i p i n w a v e s " , N . S . M . B . p u b l i c a - t i o n No. 5 1 0 , 1 9 7 6 . 1 5 . O g i l v i e , T . F . : " ~ e c e n t p r o g r e s s t o w a r d t h e u n d e r s t a n d i n g a n d p r e d i c t i o n o f s h l p mo- t i o n s " : 5 t h Symposium o n N a v a l B y d r o d y - n a m i c s , B e r g e n , 1 9 6 4 . 1 6 . K i m , C . H . :

h he

i n f l u e n c e o f w a t e r d e p t h o n t h e h e a v i n g a n d p i t c h i n g m o t i o n s o f a s h i p m o v i n g i n l o n g i t u d i n a l r e g u l a r h e a d w a v e s " : S c h i f f s t e c h n i k , V o l . 1 5 , No. 7 9 , 1 9 6 8 , p p . 1 2 7 - 1 3 2 . 1 7 . O o r t m e r s s e n , G . v a n : "The m o t i o n s o f a s h i : i n s h a l l o w w a t e r " : O c e a n E n g i n e e r i n g , V o l . 3 , p p . 2 2 1 - 2 5 5 , P e r g a m o n P r e s s , 1 9 7 6

.

1 8 . P i n k s t e r , J . A . : "Low f r e q u e n c y s e c o n d o r d e r wave f o r c e s o n v e s s e l s m o o r e d a t s e a " : 1 1 t h Symposium on N a v a l B y d r o d y n a - m i c s , L o n d o n , M a r c h , 1 9 7 6 . 1 9 . S t o k e r , J . J . : " N o n - l i n e a r v i b r a t i o n s i n m e c h a n i c a l a n d e l e c t r i c a l s y s t e m s " : I n t e r s c i e n c e P u b l i s h e r s , I n c . , N e w Y o r k , 1957 20. O o r t m e r s s e n , G . v a n : "The b e r t h i n g o f a l a r g e t a n k e r t o a j e t t y " : O f f s h o r e T e c h n o - l o g y C o n f e r e n c e , H o u s t o n , 1 9 7 4 , p a p e r OTC 2 1 0 0 . 2 1 . D a l z e l l , J . F . : " C r o s s - b i s p e c t r a l a n a l y s i s : a p p l i c a t i o n t o s h i p r e s i s t a n c e i n w a v e s " : J o u r n a l o f s h i p r e s e a r c h , V o l . 1 8 , N o . 1 , M a r c h , 1 9 7 4 . . . Main d i m e n s i o n s 2 0 0 , 0 0 0 TDW T a n k e r L e n g t h b e t w e e n p e r p e n d i c u l a r s B r e a d t h D r a f t Volume o f d i s p l a c e m e n t B l o c k 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 P r i s m a t i c c o e f f i c i e n t D i s t a n c e o f c e n t r e

of

g r a v i t y t o m i d s h i p s e c t i o n H e i g h t o f

c e n t r e

o f g r a v i t y M e t a c e n t r i c h e i g h t 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 T r a n s v e r s e r a d i u s o f g y r a t i o n 3 1 0 . 0 0 m 4 7 . 2 0 m 1 8 . 9 0 m 2 3 5 , 0 0 0 m 3 0 . 8 5 0 . 9 9 5 0 . 8 5 5 6 . 6 1 m 1 3 . 3 2 m 5 . 7 8

m

7 7 . 5 0 m 1 7 . 0 0 m

Pi1 tPiPtPi3

1

FENDER

/

WAVE DIRECTION

THEORETICAL

o EXPERIMENTAL

Fig. 2

-

Transfer f u n c t i o n s o f wave e x c i t e d forces and moments; a = 225

degrees; water d e p t h l d r a f t = 1.2.

:P:

_{sway amp1}

-

Non-dimensional added mass and damping c o e f f i c i e n t s i n sway; water d e p t h l d r a f t _{i tude.} = 1.2;

6

-

THEORETLCAL EXPERIMENTAL o Sz : 0.825 m o L 2 : 1.650 m o S2 : 2.475 m 2 . -. - ~L 0 + * & A * OO 2 4 - "

. - --.- ~ ~ . . p~ 5 10 15 20 ₂₅ seconds

FIG, 4

-

RETARDATION FUNCTIONS FOR SURGE, SWAY AND HEAVE,

-

CHOCK FENDER DOLPHIN 1 0 0 P. 1. 200.000 TDW TANKER . - - :

.

.. . -. . : 139.45 m

1

13S.45 m 4 -. . -- _{T H E O R E T I C A L} 0 E X P E R I M E N T A L -- 158.40 m l 1m4Orn 20 RI

.

EU a 1 0 - - - - RI .RI m "

.,l

-P -

-

"

0

+ + -

0 0 0 OO 2 4 6 '0

u q

2 w d $ 4 6 -

FIG, 5

-

NON-DIMENSIONAL ADDED MASS AND DAMPING COEFFICIENCTS I N SWAY; DISTANCE BETWEEN S H I P AND QUAY 16,5 M; WATER DEPTH/ DRAFT

1.2

B

" C m KILL L I K 1 LME 1 LINE 3 L M 4 MOORIN5 LlNE I

FIG. 8

-

COMPUTED AND MEASURED SHIP MOTIONS AND MOORING FORCES; REGULAR WAVES FROM 90 DEGREES; = 0.212 RAD.

SEC-1;

WAVE HEIGHT

0.9 M, t o m m MOORlNO LlNE 2 SURGE PITCH

A

FENDER I MOORING LINE 4 ' \ L' 0 0 5 l 0 0 5 1 0