Otc 2882 the Behaviour of Moored Ship in Waves Oortmerssen 1977

7/28/2019 Otc 2882 the Behaviour of Moored Ship in Waves Oortmerssen 1977

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O TC 2882T H E B E H A V IOU R OF MOOR E D S H l P S I N WAVES. y Gerard van Oortmerssen, Nether1 ands Ship Model Basin

Copyright 1977. Offshore Technology ConferenceThis paper was presented at the 9th Annual OTC in Houston, Tex., M ay 2- 5, 19 77 . The m aterial s subject to correction by the author. Permisslon o copy Is restricted o an abstract of not more &an 300 words

BSTRACTA method is described for the mathemati-cal simulation of the behaviour of a mooredship in waves. The method is based on theequations of motion i n the time domain accor-

ding to C ummins, while the hydrodynamic loadson the ship are obtained by means of the three,-fromdimensional source technique.of computations for a ship moored toa jetty are discussed and compared with ex-erimental results.

INTRODUCTION -UP t il l a few decades a go, the mooring ofships has been mainly a matter of practical

experience. Ships were moored in harbours orsheltered areas only , where t he external force:are in general limited to the rather steadycurrent and wind forces.With the development of the ocean indus-ry and the advent of very l arge ship s, whichcan only be accommodated i n a few harboursfficient water depth, the need aroseo moor ships in exposed areas. To this Pur-cial mooring facilities were aesigneaexerted by the environmentn vt he moored ship. Nowadays a variety ofngements is in operation.Because of the short history and fa stnt of mooring in exposed areas, theesign of terminals c an not be based on em-

sm. O n the other hand the problem is too

red,s hip by means of experi-small scale models. Although modelides an effective tool to deter-maximum motions o f

for design purposes, thisFirst, model tests are expensive and

essential that elasticity prop-mooring lines and fenders are simu-v e r y carefully, and sophisticated

facilities are needed to simulate th e relevantenvironmental conditions. For this reason testprograms are usuaZly restricted to final de-sign configurations and selected weather co ndi-tions which are assume d to be the most cri ti-cal' Further, the fundamental insight gainedmodel tests o n these complicated systemsis limited. Only the resulting output ismeasured without learning much of the mecha-nism which causes this output. As an examplethe low frequency motion of a moored shipobserved in tests in irregular waves may bementioned.On a basis of model test results it is notpossible to conclude whether this motion iscaused by second order ef fects in the waveloads, or by the fact that the elasticityproperties of the mooring system as- non-linear.It is, therefore, highly desirable todispose o l a mathematical simulation methodfor the behaviour of a moored object. Such amethod can help to increase the understandingof moored ship behaviour, and can be used asa tool in optimization studies in the earlystages of design, prior to model testing.The basis of each mathematical model for thesimulation of the behaviour of moored o bjectsis the law of dynamics of ~ e w t o n :

a(&)- = pdt

There-or,sinc e the inertia m of the ship may b eregarded as constant:

m; = FThe external force P is composed of- arbitrarily in time varying for ces due tothe waves;- hydrodynamic and hydrostatic restori ngforces, which are a function of the motiansof the ship;- restoring forces due to the mooring system,


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w hi ch 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 sp o s i t i o n of 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 mo ti on t h e o r y , it i scommon 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 sa s f o l l o w s :

a , b a nd c a r e c o e f f i c i e n t s wh ic h d e s c r i b e t h eh yd ro dy na mi c a nd 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 io n ofm ot io n , i n t h e s e n s e t h a t it r e l a t e s t h e i n -s t a n t an e o u s m ot io n v a r i a b l e s t o t h e i n s t a n t a -n eo us 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 no n l y b e us e d a s a d e s c r i p t i o n i n t h e f re q ue n cydomain of a s t e a d y o s c i l l a t o r y m ot io n, s i n c et h e hy dr od yn am ic c o e f f i c i e n t s a and b dependon 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 or k o n t h e m oo re d s h i p u r o-

I n t h i s p ap er 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 ic h . is b a s ed o n t h e e q u a t i o n s o fm o ti on i n t h e t i m e d o ma in a s t h e y h a v e f i r s tbe en fo rm ul at ed by Cummins [ l31 .T h e se 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 st r u e d i f f e r e n t i a l e q u at i on s ; t h e y g i v e t h ei 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 et w ee n t h e m ot io nv a r i a b l e s a nd t h e e x t e r n a l f o r c e s . I n t h e s ee q u a t io n s t h e v a r i o u s f a c t o r s g o ve rn i ng t h er e sp o ns 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 .T he o n l y a s s u m p t io n i n v o l v e d i s l i n e a r i t yo f t h e h y dr od yn am ic r e s t o r i n g f o r c e s . Non-l i n e a r a nd as ym me tr ic m oo ri ng c h a r a c t e r i s t i c sc a n b e d e a l t w i t h , a nd t h e e x c i t i n g f o r c e mayb e a r b i t r a r y , w hich 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 nd w in d- a nd c u r r e n t f o r c e s c a n b e i n -c lu d ed i n t h e f o r c i n g f u n c t i o n .

I le m p u b l i s h e d s o f a r h a s b e e n b a s e d o ne q u a t i o n ( 1 ) , w he re t h r e e c a t e g o r i e s c a n b e IDESCRIPT1O' OF THE -.d i s c e rn e d w i t h r e g a r d t o t h e s i m p l i f y i n ga s s u m p t i o n s m a d e .Some i n v e s t i g a t o r s , a s f o r i n s t a n c eKaplan and Pu tz Cl? , L e e n d e r t s e 1 2 1 , Muga131 an d ~ e i d l 41 l i n m i e e d t h e e l a s t i c i t yc h a r a c t e r i s t i c s o f t h e m o or in g s y s t e m . T her e s t o r i n g f o r c e s o f t h e m o or in g a i d s c a n t h e nbe i n c o r p o r ~ t e d n t h e h y d r o s t a t i c t e rm c x an dt h e e q u a t i on s ( 1 ) o f m o ti on i n t h e f re q ue n cydomain c a n b e s o l v e d e a s i l y , w i t h t h e r e s t s i c -t i o n t h a t o n l y ha rm on ic e x c i t a L i o n s c a n b eu s e d . O t h e r s , a s A b ra ms on a n d W i l s o n C51 ,Yang [6] a n d ~ i l n e r T J a d d n o n - l i n e a r t e r m st o e q u a t i on ( 1 ) t o a cc ou nt f o r t h e r e s t o r i n gf o r c e s o f t h e m o or in g sy s t e m , a h d s o l v e t h ee q u a t i o n s by me ans o f t h e me th od o f e q u i v a l e n tl i n e a r i z a t i o n , a ssu ming t h a t t h e e x c i t a t i o ni s p ur e s i n u s o i d a l an d t h a t , a s i n t h e e a r l i e rm e n ti o n ed m e th o d, t h e r e s p o n s e o f t h e s h i p i ss i m p l e h a rm o ni c t o o , w i t h a f r e q u en c y e q u a l t ot h a t o f t h e e x c i t a t i o n .T h i s i s n o t r e a l i s t i c , s i n c e o b se r v a t i on sb o t h i n m odel and f u l l S -c al e s i t u a t i o n s h av er e v e a l e d t h a t a l s o o t h e r modes o f m o t i on mayo c c u r .The work of W ilson and Awad al la C8, 91 ,Lean C101 , Wilson [ l 13 an d Bomze C121 be -l o n g s t o t h e t h i r d c a t e g o r y , wh ic h i s c h a r a c -t e r i z e d by t h e a ss u mp ti o n t h a t t h e h yd ro dy na -m ic c o e f f i c i e n t s a and h i n e qu a t i o n ( 1 ) a r ei nd e pe nd e nt o f t h e f r e q u en c y , s o t h a t t h i se q u a t i o ns r e g ar d e d as a* a c t u a l d i f f e r e n t i a le q u a t i o n . T he s o l u t i o n , w h ic h i s f ou nd e i t h e rb y a p p r o x i m at e a n a l y t i c a l m e th o ds ( r e f . C81,C101 ) o r by f i n i t e d i f f e r e n c e i n t e g r a t i o ni n t h e t i m e do mai n ( r e f . C91 , C111 an d t.123)~may C o n t a i n c om p o n en 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 rm o n i c ) o r h i g h e r ( s u p e r ha r m o n i c ) t h a nt 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 ss u m pt i on of c o n s t a n th yd ro dy na mi c c o e f f i c i e n t s c an 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 sh aL lo w w a t er t h e s ec o e f f i c i e n t s a pp ea r t o b e ve r y s e n s i t i v e t oc h a ng 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 om ain d e s c r i p t i o n o f t h e b e ha v i o u r o f t h emoored sh ip i s n e e de d wh ic h t a k e s i n t o a c c o u n tt h e f r e q u e nc y d ep en de nc y o f t h e f l u i d r e a c t i o nf o r c e s .

I n t h i s s e c t i o n , a g e n e ra l d e s c r i p t i o ni s g i v e n o f a m a t he m a ti c al t e c h n i q u e t o s im u-l a t e t h e b e h av i o ur o f a s h i p , moored t o aj e t t y . F or 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 .C o n s i d e r a s h i p , moored by means o f anumber o f m o o ri ng l i n e s w i t h n o n - l i n e a r e l a s -t i c c h a r a c t e r i s t i c s , t o a j e t t y e q ui p pe d w i t hf e n d e r s , a s s c h em a t iz e d i n F i g u r e 1 .

A s p a c e f i x e d , r i g h t h a nd e d s y s te m o fc o o r d i n a t e s G X I X 2 X i s u s e d , w i t h i t s o r i g i ni n t h e c e n t r e o f g ? a v i ty o f t h e s h i p .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 om ai n c a nb e w r i t t e n a s :

6 tZ { ( M .+m ) f + K k j ( t - r ) ; . ( r ) d r + C k j x j l =j = 1 k~ k j J' -m J

n1 n f= x k ( t ) + C ~ ~ ~ ( t )X ~ ~ ~ ( t )i = l i = l

k = 1 ' 2 , a . e . 6 -.............***-....--2 )

M k j i s an i n e r t i a m a t r i x . S i n ce t h e o r i g i n o ft h e s ys te m o f a x es c o i n c i d e s w i t h t h e c e n t r eo f g r a v i t y of t h e s h i p i n i t s r e s t p o s i t i o n ,i t i s f o un d t h a t


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m 0 0 0o m o oo o m o

0 00 0 0 00 0 0 - 1 4 6 O '6-where m = m as s o f t h e s h i p ,

t hIk = moment o f i n e r t i a in t h e kmode,I = p r od u c t o f i n e r t i a .k

Ck. i s t h e m a t r i x o f h y d r o s t a t i c r e s t o r i n gf o gc e c o e f f i c i e n t s .he h yd ro dy na mi c r e a c t i o n f o r c e s a c t i n g on t h es h i p a r e e x p r e s s e d by a c o n s t a n t " ad de d ma ss "c o e f f i c i e n t m . an d a r e t a r d a t i o n f u n c t i o nw h ic h t a % d s i n t o a c c o u n t t h e m emory$ J i c t o f t h e f r e e wa te r s u r f a ce .and K d e s c r i b e t h e h y dr od yn am ic f o r c ek j t h e k t hj mo de , du e t o m o ti on o f t h e s h i pn t h e j t h mode.i e [ l ? ] h a s shown t h e r e l a t i o n s h i p b e-e en t h e s e q u a n t i t i e s a n d t h e f r e qu e n cy -e p en d e nt a d d ed m as s a nd da mp in g c o e f ' ~ ~ c r @ n t s : -

m2K k j ( t ) = 7 1 -(W ) c o s cot du - m S . . . . ( 3 )0 k ~

m1 1 1m = ak j (W ) + 7 / K k j ( t ) S i n t d t . . . ( 4 )k W o= f r e q u e n c y - d e p e n d e n t a d d e d m a ssk j c o e f f i c i e n t ,b = f r e q u e n c y - d e p e n d e n t d a m p i n g

k j c o e f f i c i e n t .

1 i s a n a r b i t r a r i l y c ho se n value o f t h em i v e n b y ( 4 ) is i n d e p e n d e n t o fe v a l u e o f kj w $.s t h e c o m p o n e n tn h e kth mode o f th en i n t h e 5 t h mooringine, h i c h d e p e n d sn t h e i n s ta n t a n e o us l e n g t h o f t h e l i n e , t h eof t h e line nd t h e l o a d - e l on g a -o n c h a r a c t e r i s t i c s .

s t h e c o m p o n e n tn h e k t h m ode t h ein t h e i t h w h i ch is a functionof t h e f e n de r and t h e e l a s t i

i s t h e t o t a l number o f m oo ri ng l i n e s , n pi e number o f f e n d e r s .( t ) re p r es 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 lo a d i n t h e k-mode.t h e equationsf in h e t i m em ai n a n a r b i t r a r y m o t i o n is d e s c r i b ed a s as 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 ce m e n t s .e b a s i c a s s u m p t i o n is , t h a t a t any t i m e h et a l f l u i d r e a c t i v e fo r c e i s t h e sum o f t h et o t h e i n d i v i d u a l i m pu l s iv e d i s -n t s , e ac h r e a c t i o n b e i n g c a l c u l a t e dm e l a g f r om t h e i n s t a n t o fe c o r r e s p o n d in g i mp u l s i v e m o ti o n, o r i n

t h e r e m u st ,axis+ l i n e a r

b et we en t h e m o ti on s a n d t h e f l u i d r e a c t i v ef o r c e s .F o r t h e c o m p u t a t i on o f c o n s t a n t a d d e dmass 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 ndt i m e h i s t o r i e s o f wave l o a d s , t h e a dd ed massa nd da mp in g c o e f f i c i e n t s a n d wave l o a d s a r en e ed e d o ve r t h e c o m p le t e r a n g e o f f r e q u e n c i e so f i n t e r e s t .F o r t h e c o m p u t a t io n of d e ep w a t e r s h i pm o t io n s , t h e s o - ca l l e d s t r i p t h e o r y , h a sp 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 an ta ge 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 ng i -t u d i n a l d i m en s io n i s l a r g e r e l a t i v e t o t h el a t e r a l an d v e r t i c a l d im e ns i on s . F or s u c h as l e n d e r bod y t h e t h r e e - d i m e n s i o n a l p r o bl e mc a n b e r e du 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 pr o bl 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 me th od h a s b e e n r e f i n e d b y many a u t h o r sand 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 . Adrawback i s , h o we v er , t h a t no in f o rm a t 'o nC a n b e t h e su rgeU n l i k e t h e d e e p w a t e r p r o b le m , v e r y f ews t u d i e s h a ve b e en p r e s e n t e d on t h e m o t io n s 0 9a 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 pt e d t h e s t r i p t h e o r yf o r a r e s t r i c t e d w a te r d ep th . Fo 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 ac h y i e l d s u s e -fu l b u t it can b e usedl a t e r a l m o t io ns , e s p e c i a l l y i n t h e lo we r f r e -quency s i n c e t h e i s b a s i -c a l l y t w o- d im e ns i on a l, r e q u i r i n g t h a t t h e f lo wo f w a te r p as s e s e n t i r e l y u nd e r ne a th t h e k e e lo 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 ev er , 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 . T hew a t e r f l ow s p a r t l y u n d e r ne a t h t h e s h i p an dp a r t l y a r o un d t h e e n ds . I n t h e e x tr e me c a s e ,t h e s h i p s i t t i n g on bo t to m , w a t e r c a n moveo n l y a ro u nd t h e e n ds o f t h e s h i p .

A m et ho d o f s o l u t i o n , o f wh ic 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 on g as l i n e a r i t y i sa s c e r t a i n e d , i s p r o v i d e d b y t h e t h r e e- d i me n -s i o n a l s o u r c e t ec h n i q u e. T h i s t e c h n i q u e h a sb ee 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 ewy e a r s f o r t h e c o m p u ta t io n o f wave l o a d s o nl 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 dj u s t a s w e l l f o r s h i p s ha pe d b o di e s ( s e e r e f .[l71 ) . A s a n ex am pl e o f t h e r e s u l t s o b t a i n e dw i t h t h i s m et h od , F i g u r e 2 s ho ws a c o m p a r i s o nO f w av e f o r c e s o n a 2 0 0 ,0 0 0 t a wt a n k e r i n s h a l l ow w at e r ( k e e l c l e a r a n c e 20p e r c e n t o f t h e d r a f t ) w i t h v a lu e s o b t a i ne dfro m m od el t e s t s . F i g u r e 3 sh ow s a s i m i l a r

Of t h e a d d e d mass an d dampingc o e f f i c i e n t f o r m ot io n i n t h e sway mode. I tbecomes o b v io us f r om t h i s f i g u r e , t h a t t h ea d d e d m a ss i n s w a y , w h i c h i s q u i t e i m p or t a ntf o r t h e b e h a v io u r o f a m oo re d s h i p , i s s t r o n g -: l y d e p e n d e n t On t h e f r equency O fA s h a s b e en s t a t e d , t h e c o n s t a n t "a dd ed

m as s" c o e f f i c i e n t s a nd r e t a r d a t i o n f u n c t i o n sw hi ch a p pe a r i n t h e e q u a t i o n s o f m o t io n i n t h et i m e d o m a in , c a n b e f r o m t h e a d d e dm as s a n d d am pi ng c o e f f i c i e n t s i n t h e f r e q u e n cydomain . As an example , F ig u r e 4 s ho ws t h er 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 , s way a ndhe a ve m o t i o n s o f a 2 0 0 , 0 0 0 t a n k e r inw 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 ft h e sh ip ' s d r a f t .W it 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 ew i t k . t e c h n i u e i t i s a 19 0 p o s s i b l e t o t a k e t h e i n -f luencef a quay O n t h eo t h e : 6 c o e f f i c i e n t ~ n t o a cc o un t . F i gu r e 5 showsc o m p u te d a n d measured v a l u e s o f t h e a d d e d m a s s


4/12

L

a nd d am pin g c o e f f i c i e n t s i n t h e s w ay mode f o ra 2 00 ,0 0 0 t dw t a n k e r b e s i d e a q u a y .The n um e r i c a l s o l u t i o n of t h e 6 c o u p l e ds ec on d o r d e r d i f f e r e n t i a l e q u a t i on s ( 2 ) i sc a r r i e d t h r ou g h a c c o r di n g t o t h e f o l l o w in gp r o c e d ur e . S up po se t h e s i m u l a t i o n ha s a r r i v e dt o t h e moment t , A t i s t h e ki me i n c r e m e n ta p p l i e d , s o t h e e q u a t i o n s o f m o ti on ha ve t ob e s o l v e d f o r t h e m oment t + A t . F i r s t , t h ev e l o c i t i e s f o r t + a n d t + A t a r e p r e d i c t e d

2by e x t r a p o l a t i n g t h e o b t a i n e d t im e h i s t o r i e s .To t h i s e n d , t h e v e l o c i t y i s e x pa n de d i n aT a y lo r s e r i e s :~ t~( t A t ) = ( t ) A t % . ( t ) i7 i j ( t )J J J 2 .A t= ( t A t t . ( t ) + { f j ( t ) -J J

. . . . . . . . . . . . . . . . . .t . ( t - a t ) ) ( 5JA tI n a s i m i l a r way x . t + - -) is fou nd . Subs e-J 2q u e n t l y , t h e new p o s i t i o n a nd o r i e n t a t i o n a r e

p r e d i c t e d b y n u m er i ca l i n t e g r a t i o n o f t h ev e l o c i t i e s , a p p l y i ng S i mp s on 's r u l e :

A t A tx . ( t + A t ) = r . ( t ) + -g { x . ( t ) + 4 + -) +J J J J 2+ x . ( t + A t ) ) . . . . . . . . . . . . . . . . .( )J

The t im e h i s t o r y o f t h e v e l o c i t i e s i s nowknown u n t i l t h e mom e n t + A t , s o t h e c o n v o -l u t i o n i n t e g r a l s c an b e co m pu t ed . The numeri-c a l i n t e g r a t i o n o f t h e s e c o n vo l u t i o n i n t e g r a l si s c a r r i e d o u t by means o f S i m ps o n' s r u l e ,u s i n g a t i m e i n cr e me n t e q u a l t o t h e t i m e s t e pA t , a p p l i e d f o r t h e solutionf t h e e q u a t i o n so f m o t io n . T he u p p e r b ou nd o f t h e i n t e g r a l s ,i n t h e o ry i n f i n i t y , i s f i x e d a t 2 5 s e c on d s ,w h i c h a p p a r e n t l y i s s u f f i c i e n t .S u b s e q u en t l y , t h e v a l u e s o f t h e m oo ri ng l i n ea nd f e n d e r f o r c e s as w e l l as t h e h y d r o s t a t i cr e s t o r i n g f o r c e s c a n b e c a l c u l a t e d f o r t h enew c o o r d i n a t e s . A f t e r s u b s t i t u t i o n of t h e s ef o r c e s i n ( 2 ) , 6 l i n e a r e q u a t i o n s a r e ob-t a i n e d f r o m wh ic h t h e a c c e l e r a t i o n s f j ( t + A t )c an b e f o un d. F i n a l l y , t h e p r e d i c t e d v e l o c i -t i e s a r e c he ck ed by i n t e g r a t i o nf t h e a c c e -l e r a t i o n s . I n c as e t h e d i f f e r e n c e i s a c c e p -t a b l e , t h e c om pu ta ti un c o n t i n ue s f o r t h e n e x tt im e s t e p ; i f n o t , t h e t i m e in cr em e nt h a s t ob e d e c r e a s e d . When s t a r t i n g t h e c o m pu t a ti o np r o c e s s a t t = 0 , n o r e l i a b l e p r e d i c t i o n o fi . ( ~ t ) a n b e made. T h e r e f o r e , a n i t e r a t i o np g o c e d u r e i s u se d i n t h a t c a s e : t h e v a l u e s o ft h e velocitieso u n d a f t e r solvingh e equa-t i o n s o f m o ti o n a r e u s ed a s a new p r e d i c t i o na nd t h e p r o c e s s i s r e p ea t e d u n t i l t h e p r e d i c -t e d a n d co mpu te d v e l o c i t i e s a t t = A t a r e i ns a t i s f a c t o r y a gr ee me nt .The m a t h e ma t i c a l s i m u l a t i o n p r o c e s s d e s -c r i b e d b e f o r e h a s b e e n p r og r am m ed i n FORTRANf o r u s e o n a C o n t r o l D a t a 6 6 00 c o m pu t er .i n a u t B ~ ~ i ~ ~ some para:eters9 t h e

- i n e r t i a m a t r i x o f t h e s h i p ,- m a t r i x o f 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- a dd ed mass c o e f f i c i e n t s a nd r e t a r d a t i o nf u n c t i o n s o f t h e s h i p ,- c o o r d i n a t e s o f f e n d e r s a nd b o l l a r d s ,- 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 o f fe n d e r s a ndl i n e s ,- p r e t e n s i o n i n mo or in g l i n e s ,- t im 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 nd e l a s t i c i t yf u n c t io n s o f l i n e s a nd f e n d er s a r e r ea d i n a sa 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 of 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 , whenn e e de d , f o u nd b y m ean s 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 gr a m , t h e c om pu te dt i m e h i s t o r i e s o f mo ti o ns an d f o r c e s c a n b ep 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 ef o r f u r t h e r a n a l y s i s .T he 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 te -m a t i c c om p ut a ti on s w i t h v a r y i n g t i m e s t e ph 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 ss 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 nu m e ri c a ls o l u t i o n o f t h e e q u a t i o n s o f m ot io n i n c a s et h e s h i p i s m oored a g a i n s t s t i f f f e n d e r s . W it ht h i s t i m e s t e p , t h e r e q u i r e d c om pu ti ng t i m ea m o un t s t o 1 s ec on d f o r 10 s ec o n ds r e a l t i m e .

EXPERIMENTAL VERIFICATIONTo c h ec k t h e a d e q ua c y o f t h e m a t h e m a t i c a lm od el f o r m oor ed 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 pr og ra m h a s be e n c a r r i e d o u t t o a na -l y s e t h e m o ti on b e h a v i o ur o f a mo ored s h i p i nr e g u l a r an d i r r e g u l a r w av es . A f t e r w ar 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 rs i m u l a t i o n o n t h e c om pu te r t o s e e w he th e r t h ephenomena b e mean''o f t h e m a t h e m at i c al s i m u l a t i o n .F or a d e t a i l e d d e s c r i p t i o n a nd c o m pl et e r e-s u l t s r e f e r e n c e i s made t o c 1 4 1 . H e r e , o n l yt h e m ai 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 *T he s t u d y w as c o n d u c t e d f o r a l o a d e d2 0 0, 00 0 t dw t a n k e r , m oo re d t o a n o p e n j e t t yi n w a t e r w i t h a d e p t h a m ou nt in g t o 7 .2 t i m e st h e d r a f t a f t h e v e s s e l . The m a in p a r t i c u l a r so f t h e s h i p a r e g i ve n i n Ta bl e I .T h e 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 e6 . T he v e s s e l wa s mo o r ed b y m ea n s o f 4 l i n e s ,e a ch r e p r e s e n t i n g tw o o r t h r e e w i r e s w i t hn yl on t a i l s i n r e a l i t y . The l o ad - e l o n ga t io nc u rv e s o f t h e l i n e s a r e s hown i n F i g u r e 7 .I n e a c h l ine pre tens ion was of 20t o n s . The p a i r o f s t i f f f e n d e r s h ad a l i n e a r

e l a s t i c i t y a mo un ti ng t o 1 575 to n/ m.The mod el t e s t s w er e c a r r i e d o u t i n t h eS h a ll o w W a te r L a b o r a t o ry o f t h e N e t h e r l a n d sS h i p Mo del B a s i n o n a s c a l e o f 1 :82 . 5 ,The l o a d s i n m oo r in g l i n e s a nd f e n d e r sw e r e m e a su r e d b y m ea ns o f s t r a i n g a ug 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 re d b y m ea ns o fp o t e n t i o m e t e r s . Al l s i g n a l s w e r e r e c o r d e db o t h o n m a g ne t ic t a p e a nd p ap e r c h a r t .T he m od el t e s t s w er e b a s e d o n F r o u d e ' sl aw 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 iv e s e r i e s o f t e s t s i nregular beam paves ( a = 9 0 ~ ) as carriedutt o i n v e s t i g a t e t h e b e h av i o ur o f t h e m ooreds h i p o v e r a w id e r a ng e o f p e r i o d s , v a r y i n g fr on .


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9 t o 41 s e c o n d s . T he l o n g p e r i o d w a ve s a r e n o tm er el 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 ob 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 bo ur o s c i l -l a t i o n s , w hi c h so me t im e s c a u s e p r o bl 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 mo ti on , 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 s way m o ti o n was t h eo c c u r r e n c e o f a mean d i s p l a c e m e n t i n a d d i t i o nt o t h e o s c i l l a t o r y m o ti on . I n s h o r t waves t h i sm o t io n h ad t h e s am e f r e q u e n c y a s t h e w a v es ,w i t h b ou nc 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 ls t r e n g t h a t t i me i n t e r v a l s e q u a l t o t h e w avep e r i o d . I n c e r t a i n l o n g w av es , h o we ve r, as u b -h a r m on i c m o t i o n w as o b s e r v e d : t h e s h i pm o t io n was co mp o sed o f a m o t io n w i t h f r e q u e n cyw e q u al t o t h a t o f t h e e x c i t i n g wa ve s, onw h i ch a m o t i o n was s u p e r i m p os e d w i t h f r e q u e n c ye i t h e r w/3 o r w/2.I m p ac t s a g a i n s t t h e f e n d e r s o c c u r re d t h e n a tin tervalsf 3 O r wave periodsS u b ha r m on i c m o t i o n w as o b s e r v e d o n l y i n w a ve sw i t h s uc h a f r e q ue n c y , t h a t t h e f r e q u e nc y o ft h e s ub ha r mo ni 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 of t h e moored w h i c hi n t h e sway t o0 .0 7 r a d . s e c . - l ( d ue t o t h e n o n - l in 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 oo r in g s ys te mt 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 qu e nc y de p en ds o nt 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 eof m ot ion was decreases, t h e s u b h a r m on i c m o ti o l it h ed 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 fr e g u l a r m o t io n wa s f ou n d, f o r i n s t a n c e witha l t e r n a t i n g l i g h t a nd he av y b ou nc es a g a i n s tt h e f e n d e r s .The c o m p u t a ti o n s w er e c a r r i e d o u t w i t hp u r e h a r m on i c f o r c i n g T u n c t i o n s , w h ic h w e reo b t a i n e d f ro 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 et e c h n i q u e , t a k i n g i n t o a c c o un t a wave h e i g h te q ua l t o t h a t , m ea sure d i n t h e b a s i n w i t ho u tt h e s h i p m od el b e in g t h e r e .Time h i s t o r i e s w e re c om pu te d f o r a p e r i o do f 1 00 0 s e c o n d s , a l t h o u g h i t a p p ea r e d t h a ta f t e r 500 s ec o nd 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 ou nd , t h a t i n a l l c a s e s t h et y p i c a l m o ti on b e h av i ou r o b se r v ed i n t h ewas pred ic ted by t h e m a t h e -m a t i c a l m od el . B e s i d e s , t h e q u a n t i t a t i v eagreement b e t w e e n a n d ap-g e ar 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 sb e tw een m easu red an d co m p uted v a lu es o f m o t io na nd mo or in g f o r c e a m p l i t nd e s w e re i n g e n e r a ll e s s t h a n 20 p e r c e n t .As an ex am p le F i g u re 8 shows t h e r e s u l t s f o ra wave c o n d i t i o n i n wh i ch s u bh a r mo n i c r e s -p o n s e o c c u r r e d .

A l so l o n g c r e s t e d i r r e g u l a r w ave s h av eb e en 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 nd i -t i o n was u s e d , o f w hi ch t h e s p e c t r a l d e n s i t yi 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 fwave a t t a c k , a = 9 0 , 1 35 a n d 1 80 d e g r e e s - T hem ea su re me nt 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 o2 1 0 0 pro to type and b e g a n l o o Os ec on ds a f t e r s t a r t i n g t h e wave g e n e r a to r ,a v o i d i n g t h a t t r a n s i e n t phe nom ena wo ul d i n f l l r -e nc 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 ap 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 ds . The f i r s l .400 s e c on 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 ,t h u s l e a v i n g '2100 seconds fo r analysis.

Of a l l m e a s u r e d a n d co mp ut ed t i m e h i s t o -r i e s o f mo ti on s a nd f o r c e s a s p e c t r a l a n a l y s i swas c a r r i e d o u t . B e s i d es s p e c t r a , t h i s a na l y-s i s y i e l d e d t h e f o l lo w i n g s t a t i s t i c a l q u an t i -t i e s :- m ean v a lu e ,roo t mean s q u a r e v a lu e ,- s i g n i f i c a n t d ou bl e a m p li t u de ,- m a x ~ m u m and minimumalue,

T he wave sp e c t ru m w as s i m u la te d i n t h ec o m p u t a t i o n by m ea ns o f 1 5 s i n e w a v e s , r a n g i n gi n f r eq u e n c y f ro m 0 .4 25 t o 1 .1 25 r a d . s e e . - l ,T o t h a t end t h e m e a s u r e d wave spec t rum f s e eF i g u r e 9 ) was s u b d i v i d e d i n t o 1 5 b a n d s o fc o n s t a n t w i d t h . E ac h b a nd was r e p r e s e n t e d b ya com ponen t , havingh e c e n t r e f r e q u e n c yo f t h a t ba nd a nd a h e i g h t , f o l l o w i n g f r om t h eb a n d a r e a . T h e s e wave r o m p o n e n t s w e r e summeaw i t h a r b i t r a r y p h a se a n g l e s , and w i t h t h ea 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 om put ed w i t ht h e th r e e - d i me n s i o n a l s o u r c e t e c h n i qu e t h e

func t ion s in t h e 6 modes w ere d e t e r -m i n e d .M a t he m a ti c a l s i m u l a t i o n s showed t h a t i nbe am s e a s t h e r o l l m o t i o n a t t h ef 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 et h e o r y . T h e r e f o r e it ap p eared d e s i r a b l e t ot a k e s o m e e x t r a rol l a s d e t e r m i n e df ro m mod el t e s t s , i n t o a c c o u nt .I n w av es f ro m 9 0 and 1 3 5 d e g r e e s a gooda gr e em e nt was o b s e r v e d be t w e en t h e = e s u l t s ofexpe r i m en t s a nd t h e m at hem a t i ca l simula-t i o n . As an ex am p le , F i g u re 10 sho ws t h em e a s ur e d a n d c om p ut e d s p e c t r a o f m o t i o n s a n df o r c e s f o r 1 3 5 d e g r e e s w a v e s , whileh e mos ti 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 on f o r t h i scase i s tabula ted i nI t a pp e ar e d t h a t t h e s p e c t r a o f h or i z on -t a l m o t i o n s a nd m o or i n g f o r c e s s how p e a k si n t h e l ow 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 ad wa ve s ( a = l 8 0d 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 ad 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 gf u n c t io 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 yl o a d s i n s u r g e , h e av e a nd 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 oo ri ngf o r c e s a r o un d t h e p r e t e n s i o n a nd a s m a l l s u r g eo s c i l l a t i o n a ro und t h e z e r o p o s i t i o n . I nr e a l i t y , h ow ev er , t h e s u r g e c o n s i s t so f a l a r g e r a m p l i tu de o s c i l l a t i o n ar ou nd amean d i s p l a c em e n t , a nd t h e v a r i a t i o n s i n t h em oo ri ng l o a d s a r e a l s o l a r g e r . I t was assumedt 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 tf o r c e p l a y s an importanto l e . T o c h e c k thisa s s u m p t io n , t wo c a l c u l a t i o n s we r e m ad e: o new i t h a n add i t iona l c o n s t a n t and on e w i t h as lo w ly v a r y i n g d r i f t f o r c e , whi ch i s much morer e a l i s t i c i n i r r e g u l a r wa ve s. As an a p pr o xi -m a ti on o f t h e m ag ni tu de 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, , p e r f o r m e d ona d i f f e r e n t s h i p mode l in a similar s e at i o n w er e u s e d.W i th h e additionallowly varying r if t o r c ea g o o d a g r e e m e n t was f o u n d . Obviously,h ei n f l u e n c e of t h e l ow -f re qu en cy d r i f t f o r c e isthur8essential in t h i s condition,ANALYS I S OF RESULT S

The r e s u l t s o f t h e investigations d e s c r i bi n t h e p r e v io u s s e c t i o n ha ve r e v e a le d i n t e -


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r e s t i n g f e a t u r e s of t h e b eh a vi o ur 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 b eam s e a s , t h e s way m ot i onc 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 nd 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 av es a s ub h a rm o n ics wa y m o t i o n wa s f o u n d w i t h f r e q u e n c y w / 2 o rw/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 in g f o r c e s s ho w l ow -f re q ue n cy p e ak s , c l e a r l y d i s t i n c t f ro m t h er a n g e o f w av e f r e q u e n c i e s . From t h e c om g ut a-t i o n s i t f o l l o w e d t h a t i n h ea d waves t h e s e -c on 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 ei n e x c i t i n g t h i s low f re q u en c y b e h a v i o ur , b u ti 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 , l owf r e q u e n c y p e a k s w e r e f o u n d d ue t o f i r s t o r d e rwave e x c i t a t i o n o n l y . T h i s l ow f r e q u en c y be -h a v i o u r i n i r r e g u l a r w aves m us t a l s o be d i s -t i n g u i s h e d f r o m t h e s u b ha rm o ni c m ot 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 om po ne nt i nt 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 re a s s u b h a r m on i c m o t i o n s o n l y o c-c u r r e d i n w ave s w i t h f r e q u e n c i e s l o w e r t h a na 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 ho wn , t h i s t y p i c a l b e h a v i o u r o fa mo or ed 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 ic a l m od el d e s c r i b e d h e r e , b u tt 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 us e a s i m p l i f i e d a n a l y t i c a la p p r o a c h .The a bo ve m en t i on e d s p e c i a l f e a t u r e s o fm oo red s h i p m o t io n s o c c u r m a i n ly i n t h e h o r i -z o n t a l m o de s , s u r g e , s w ay an d y aw . When t h e s em o t io 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 ou p l ed , i ti s o b s e r v ed t h a t t h e r e s t o r i n g f o r c e a nd momen';i n s u r g e an 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 :

3f ( x ) = ax + @X . . . . . . . . . . . . . . . . . ( 7 )w he re 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 ed i s p l a c e m e n t a n d a a nd B a r e c o n s t a n t s . Whenm o re o ve r t h e f r e q u e n c y - d e pe n d e n c y o f t h ea 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 pi ng i s neg-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 of m ot io n b e-comes

2 + ux + 8x 3 = ~ ( t ) .. . . . . . . . . ( 8 )T h i s i s t h e w el l- kn ow n D u f f i n g e q u a t i o n , w h ic hh as be en 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 eo f n on - l i n e ar v i b r a t i o n s ( s e e f o r i n s t a n c eS t o k e r , 1 1 9 3 ) . The s o l u t i o n s of t h e D u f f i nge q u a t i on 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 motiondue t o a ha rm on ic e x c i t a t i o n F = F c o saw t i s : X = A c o s w t ,- t h e r e s p o n s e shows a I I ~ ~ ~ ~ ~ ~ ~ ~ ~ Ih a p e ,- w i t h F = Fa cos w t , s u bh a rm o ni c s o l u t i o n se x i s t w i t h f re q u en c y w / 3 ,- 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 ar mo ni ccomponents , F = F , c o s w l t + F 2 c o s w t ,2t h e m o t i on s c o n t a l n c om wo ne nt s w i t h f r e -

q u e n c i e s 2 b 1 2 w 2 and 2w2 + U , , t h e s o -c a l le c .c o m bi n a ti o n t-o ne s, 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 nd w 2 .

The elasticityf t h e mooringy s t e m int h e sway mode is e s s e n t i a l l y i f ferentn 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 pu s h in g a g a i n s t t h e f e n d e r s , t h ef o r c e i s d i f f e r e n t f r o m t h e c a se t h a t t h e s h i pp u l l s a t t h e l i n e s . The mo st s i m p l e way t os c he m a ti z e su c h a r e s t o r i n g f o r c e i s :

2 3f ( x ) = a x + Bx + yx . . . . . . . . . . . . . . ( 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 fa d d ed m as s a n d da m p in g , t h e e q u a t i o n o f m o t i o nbecomes :

22 + ux + fix + y x 3 = ~ ( t ). . . . . . . . . . 1 0 )S o l u t io n s o f t h i s e q u a t i o n a r e d i sc u s s e d i nr e f . C141 . I t a p p ea r s t h a t :- w i t h an e x c i t a t i o n F = F c o s w t , t h e f i r s to r d e r a p p r o xi m a ti o n of t a e r e s u l t i n g m o ti ons ho ws 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 + B

- s u b h a r m o n ic 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 ,- w i t h b i -f r eq u e nc y e x c i t a t i o n ( F = F lw , t + F2 c o s w t ) c o m b in a ti on t o n e s c a n b e2f o u nd w l t h f r e q u e n c i e s 2 wl 2 w 2 , 2 w 2 2 w land w , 2 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 pp ro ac hshows how a mean d i sp la ce me nt and low f r e -q ue nc y m o t io 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 rp a r t i c u l a r s o f t h e mo or in g sy st 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 fm o t io n i n t h e f r e q u e nc y d om ai n c a n o n l y b eu s ed 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 eb e h a v i o u r o f t h e mo or ed 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 oo ri ng s ys te 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 an d dam pingt e r m s b e i n g s t r o n g l y d e p e n de 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 ta t t h e s ub ha rm on ic f r eq u e nc y i s a r ou n d f o u rt 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 hi chc l e a r l y i l l u s t r a t e s t h e n e c es s a t y t o t a k et h e f re q ue n cy - de p en d en c y o f t h e c o e f f i c i e n t si n t o a c c o u nt .The lo w fr e qu e n cy r e s p o n s e i n i r r e g u l a rs e a s f r om 90 a n d 1 3 5 d e g r e e s i s o b v i o u s l y ar e s u l t o f t h e ph eno me no n o f c o m b i n a t i o n t o n e s ,s i n c e su b h ar m o ni c r e s p o n s e w as n o t f o u n d i nt h e s e p a r a t e c om po ne nt 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 low f re qu e nc y d r i f t f o r c e sad de 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 con-d i t i o n s . F o r a n o t h e r m oo ri ng c o n f i g u r a t i o nw i t h s o f t f e n d e r s , ho we ve r, t h e d r i f t f o r c ed i d have

FUTURE DEVELOPMENTS

The pr og r am pa c k ag 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 be ha -v i o u r o f s h i p s , moored t o op en o r s o l i dj e t t i e s , an d ca n a l s o b e a p p l i e d f o r t h e de-t e r m i n a t i o n o f i mp ac t l o a d s d u r i n g b e r t h i n g

J


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m a n o e u v r e s ( s e e r e f . L201). T he a n a l y s i s i st h e o r e t i c a l , e x ce p t f o r t h e wave d r i f t f o r c e s ,w hi ch s t i l l s h o u l d b e d e t e rm i n e d e x p er i m e n ta l -l y . I n v i e w o f t h e i m p or ta n c e of t h e s e ' f o r c e st o t h e b e h av i o u r o f m oo re d s h i p s , much e f f o r tw i l l b e n ee d ed f o r t h e d e ve lo p me n t o f r e l i a b l ep r e d i c t i o n me th o ds . I t i s e x p e c te d , t h a tw i t h i n t h e n e x t two y e a r s m eans w i l l becomea 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 elo w f r e q u en 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 ,

r o g r e s s i s m ad e b y a mo ng o t h e r s D a l z e l l c2 11a n d P i n k s t e r L181 .I t i s i n t e nd e d t o e x t e nd t h e a p p l i c a b i -l i t y o f t h e p ro gra m i n s uc h a way, t h a t s i n g l eo i n t m o o r i ng s c a n b e s i m u l a t e d . T h i s m ea 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 ha ng es 09e a d i n g o n t h e wav e l o a d s a n d t h e d y n a m ic s o ft h e b uo y mu st 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 lode l .

ONCLUSIONS --T he m a t h e m a t i c a l m o d el d e s c r i b e d i ss 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 oored s h i pi o ns i n s i x d e g r e e s o f f re e d om i n wa ve s.

he t y p i c a l m o t io n b e h a v i o u r as o b s e r ve d i nf u l l s c a l e an d i n model t e s t s i s r e p r o d u c e de m a t h e m a t i c a l m o de l.s h i p , m oo re d i n ra nd om 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 lo w f r e q u e n cy b e h a v i o u r :s ub h ar mo n ic r e s p o n s e t o c e r t a i n h a rm o ni cc o m po n e nt s , w i t h a f r e q u e n c y a m o un t in g t o1 12 o r 1 13 o f t h a t o f t h e e x c i t i n g wavec o m po n e nt . T h i s s u b h a r m o n i c r e s p o n s e i sa 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 oo ri n gs y s t e m ," c o m b i na t i o n t o n e s " , a lo w f r e q u e n c y m o t i o ni n d u c ed by t h e s i m u l t a n e o u s a c t i o n o f m oret h a n o ne h a r m o n i c w av e c o m p o n e n t .l~~h i sp hen om en on o r i g i n a t e s f r om t h e n o n - l i n e a re l a s t i c i t y c h a r a c t e r i s t i c s ,lo w f re 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 wf r e qu 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 .

k 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 ,g e n e r a l f o r c e o r mom ent,c e n t r e o f g r a v i t y ,

k moment o f i n e r t i a i n t h e k - th mode,k p r od u c t o f i n e r t i a ,

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 - th modek j d ue t o m o ti o n i n t h e j - t h mode,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- th mode duei k t o t h e i - t h m oo ri n g l i n e ,

k i n e r t i a m a t r i x ,i k f o r c e o r moment i n t h e k - t h mode d u et o t h e i - t h f e n d e r ,

S s p e c t r a l d e n s i t y o f l t h e wa v es ,

X k w av e f o r c e o r moment i n t h e k - t h m od e,a added mass coefficientn t h e k-thk j d ue t o m o t ia n i n t h e j - t h mode,b d am pi ng c o e f f i c i e n t i n t h e k - t h mode d u ek j t o m o ti on i n t h e j - t h m ode,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 ro 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 do m ,m m as 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 ed m a ss c o e f f i -k j c i e n t n t h e k - t h mode d u e t o m o t i o ni 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,ja 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 en c y .

REFERENCES

1 . K a p l a n , P , a n d P u t z , R . R . : " Th e m o t i o n s o fa 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 wa ve s a nd t h e i r i n f l u e n c e o n co n-s t r u c t i o n o p e r a t i o n " : N B Y - 32206 , M ar i neA d v i s o rs , I n c . L a J o l l a , 1 96 2.

2 . L e e n d e r t s e , J . J . : " A n a l ys i s o f t h e r e sp o n s eo f m oored 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 tcocea n w aves" : R and C o r p o r a t i o n MemorandumRM-3368 PR, 196 3.

3. Muga, B.J.: " E x pe r im e n ta l an d t h e o r e t i c a ls t u d y o f m o t i o n o f a b a r g e a s m o ore d i no c e an w av es ": 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 of m o t i on s o fs h i p s moored i n i r r e g u l a r s e a s " : P r oc .N.A .T.O. A d va n ce d S t u d y I n s t i t u t e o nA n a l y t i c a l T r e at m e nt o f P ro b le m s i n t h eB e r t h i n g a n d M o or in g o f S h i p s , W a l l in g -f o r d , 1 9 7 3 , p p . 2 2 1 -2 2 9.

5 . Abramson, H . N . a nd W i l s on , 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 lr e s p o ns e o f m oo red 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-M in: " ~ o t i o n s f m oored s h i p s i ns i x d e g r e e s o f f r ee d o m ": 9 t h S ym po siu m o nN a v a l H y d ro d y na m i cs , P a r i s , 1 9 7 2 ,

7 . K i l n e r , F.A .: "M ode l t e s t s o n t h e m o t i o no f m o or ed s h i p s p l a c e d o n l o n g wa v es t1 :P r o c . 7 t h C o nf . o n C o a s t a l E n g i n e e r i n g ,The Hague , 196 0, Volume 2 pp . 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 rm o ni 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 mo or ed s h i p s " : O f f s h o r e T ec hn 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 1 , p a p e r OTC1420, Volume 11 pp . 65- 80 .


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TABLE I

,9 . W i l s o n , J . F . a n d A w a d a l l a , N . G . : "Computers i m u l a t i o n o f n o n - l i n e a r m o t io n o f m oo re ds h i p s t 1 : P r o c . N .A .T .O . A d v an c ed 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 Tr ea tm en t o fp r ob l em s in h e ~ ~ ~ t h i ~ ~nd ~~~~i~~fS 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 98 .1 0 . L e a n , G . H . : " S u b h ar m o n ic m o t i o n s o f m o o r e ds h i p s s u b j e c t e d t o wave a c t i o n " : T r an s -

a c t i o n s o f t h e Ro ya l I n s t i t u t i o n ofN a v al A r c h i t e c t s , 1 9 71 , V o l. 1 1 3 , p p .387-399

1 1 . W i ls o n , B.W.: " P r o g r e s s i n t h e s t u d y o fs h i p s m o or e d i n w a v e s ": P r o c . N.A .T.O .A dv an ce d S t u d y I n s t i t u t e o n A n a - ly t ic a lT r e at m e n t o f P r o bl e ms i n t h e B e r t h i n g a ndM oo ri ng o f S h i p s , W a l l i n g f o r d , 1 9 73 ,pp . 143 - 213 .

12 . B ~ ~ ~ ~ ,, : " ~ ~ ~ l ~ t i ~ ~ le t e r E i n a t i o n o fs h i p m o ti o n a n d m o o r in g f o r c e s " : O f f s h o r eT e c h n ol 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 . -

13. Cummins , W.E.: "The im pu l s e r e sp on sef 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 . , 1962.

1 4 . O o r t m e r s s e n , G . v a n : " Th e m o t i o n s o f am o o r e d s h i p in 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 r o g r e s s to w ar 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 S ym po si um o n N a v a l B yd r od 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 ho n t h e h e a v i n g an d p i t c h i n g m o t i on s o f as h i p m oving i n l o n g i t u d i n a l r e g u l a r h e a dw a ve s ": S c h i f f s t e c h n i k , V o l. 1 5 , No. 7 9 ,1968 , pp . 127 - 132 .1 7 . O o r t m e r s s e n , G . v a n : " Th e 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 ce an E n g i n e e r i n g ,V o l . 3 , p p . 2 2 1- 2 55 , P e rg a m on P r e s s ,

1976 .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 do r d e r w ave f o r c e s o n v e s s e l s m oo re d a ts e a " : 11 th Symposium on Nava l Bydrodyna-m i c s , L o n d on , M a rc h , 1 9 7 6 .

1 9. S t o k e r , J . J . : " No n- li ne ar v i b r a t i o n s i nm e c h a ni c a l an d e l e c t r i c a l s y s t em 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 . , New Y o r k ,1957

2 0 . O o r t m e r s s e n , G . v a n : " Th e b e r t h i n g o f al a r g e t a n k e r t o a j e t t y " : O f f sh o r e Techno-l o g y C o n f e r e n c e , H o u s to n , 1 9 7 4 , p a p e rOTC 2100 .

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 waves" :J o u r n a l o f s h i p r e s e a r c h , V ol . 1 8 , N o . 1 ,M ar ch , 1974 .

. .

Main d im en s io n s 2 0 0 ,0 0 0 TDW T an k er

L e ng t h be t we e n p e r p e n d i c u l a r sB r e a d t hD r a f tV ol um e o f d i s p l a c e m e n tB loc k c o e f f i c i e n tM id sh ip s e c t i o n c o e f f i c i e n tP r i s m a t i c c o e f f i c i e n tD i s t an c e o f c e n t r e of g r a v i t y t o m id sh ip s e c t i o nH e i g h t o f cen t r e o f g r a v i t yM e t a c e n t r i c h e i g h tL o n gi t u d i n a l r a d i u s o f g y r a t i o nT r a n sv e r s e r a d i u s o f g y r a t i o n

310 .00 m47.20 m18 .90 m

235 ,000 m30 .850 .9950 .8556 .6 1 m

1 3 . 3 2 m5.78 m

7 7 .5 0 m17 .00 m


9/12

Pi 1 tPiPtPi31 FENDER /WAVE DIRECTION

F i g . 1 - D e f i n i t i o n s k et c h.


10/12

THEORETICALo EXPERIMENTAL

F ig . 2 - Trans fer fun ct io ns o f wave exc i ted forces and moments ; a = 225degrees ; wa te r dep t h l d ra f t = 1.2.

:P: - Non-d imens iona l added mass and damping co ef f i c i en ts i n sway; water d ep th ld ra f t = 1.2;sway amp1i ude.6

- HEORETLCALEXPERIMENTAL

o Sz : 0.825 mo L 2 : 1.650 mo S2 : 2.475 m

2 . -. - ~L

0+ * & A *OO 2 4 - "


11/12

. ---.-

~ ~ . . p~5 10 15 20 25secondsFIG, 4 - RETARDATION FUNCTIONS FOR SURGE, SWAY AND HEAVE ,

-

CHOCK FENDER DOLPHIN

10

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 L0 E X P E R I M E N T A L

--

158.40 m l 1m4Orn

20

RI. Ua

1 0 - - - -

RI.R Im

"

.,lP - -0 + + -0

0 0OO 2 4 6 '0

u q 2 w d $ 4 6-FIG, 5 - NON-DIMENSIONAL ADDED MASS AND DAMPING COEFFI CIE NCTS I N SWAY; DIST ANCE BETWEEN SH IP AND QUAY 16 , 5 M; WATER DEPTH/DRAFT 1.2


12/12

" C m

KILL

L I K 1

LME 1

LINE 3

L M

MOORIN5 LlNE I

FIG. 8 - COMPUTED AND MEASURED SHIP MOTIONS AND MOORING FORCES;REGULAR WAVES FROM 90 DEGREES; = 0.212 RAD. SEC-1;WAVE HEIGHT0.9 M,

t o m m

MOORlNO LlNE 2

SURGE

PITCHA

FENDER I

MOORING LINE 4

' \L'0 0 5 l 0 0 5 1 0

Documents

Otc 2882 the Behaviour of Moored Ship in Waves Oortmerssen 1977