DYNAMIC INSTABILITY AND CHAOTIC MOTIONS
OF A SINGLE-POINT-MOORED TANKER
S.D. Sharma, T . Jiang, T.E. Schellin
Federal Republic of Germany
A s y m p t o t i c motions o f an exemplary s u p e r t a n k e r vhen s i n g l e - p o i n t - m o o r e d , i n steady c u r r e n t , w i t h o r w i t h o u t c o n s t a n t w i n d o r r e g u l a r waves, were i n v e s t i g a t e d . N o n l i n e a r time-domain s i m u l a t i o n s o f phasespace t r a j e c t o r i e s and l o c a l l y l i n e a r i z e d s t a b i l i t y analyses o f e q u i l i b r i u m s t a t e s were p e r -formed. F i v e s e t s o f f o r c e s were modeled: l ) n o n l i n e a r q u a s i - s t e a d y hydrodynamic response and con-t r o l f o r c e s based on a f o u r - q u a d r a n con-t maneuvering model, 2 ) l i n e a r memory e f f e c con-t s due con-t o r a d i a con-t e d waves, 3) n o n l i n e a r mooring l i n e c h a r a c t e r i s t i c s , h) w i n d a c t i o n i n s t a n d a r d f o r m , and 5) f i r s t -and second-order f o r c e s due t o i n c i d e n t r e g u l a r waves. F i v e parameters were s y s t e m a t i c a l l y v a r i e d : c u r r e n t speed, f a i r l e a d l o c a t i o n , mooring l i n e l e n g t h , p r o p e l l e r r a t e , and r u d d e r d e f l e c t i o n . The system was found t o e x h i b i t i n t r i g u i n g phenomena such as m u l t i p l e e q u i l i b r i a and dynamic i n s t a b i l i t y , l e a d i n g t o s e l f - s u s t a i n e d o s c i l l a t i o n s o r even d e t e r m i n i s t i c chaos. Scale e f f e c t s were shown t o be s i g n i f i c a n t ; memory e f f e c t s , i n c o n s e q u e n t i a l . R e s u l t s are g r a p h i c a l l y p r e s e n t e d as s t a b i l i t y domains and b i f u r c a t i o n l o c i i n parameter space, and as motion h i s t o r i e s and t r a j e c t o r i e s i n phase space.
NOMENCLATURE Note: Symbols not l i s t e d conform t o ITTC s t a n
-d a r -d nomenclature. V e c t o r s an-d m a t r i c e s a r e i n b o l d p r i n t i components 1,2,3 r e p r e s e n t s u r g e , sway, and yaw; s u p e r s c r i p t T denotes t r a n s p o s e . S u b s c r i p t s A,E on c o o r d i n a t e s i n d i c a t e mooring l i n e (attachment p o i n t ) and s h i p e q u i l i b r i v m i s t a t e . S u b s c r i p t s A,Q,M,S,W on f o r c e s and mo-ments s t a n d f o r mooring l i n e , q u a s i - s t e a d y , me-mory, waves, and w i n d . Overbar means t i m e - a v e r a g e .
L A ^AU 0 p p Q
C i r c u l a r wave niimber o f component j I n s t a n t a n e o u s l e n g t h o f mooring l i n e Unloaded l e n g t h o f mooring l i n e N u l l m a t r i x o f s u i t a b l e dimension E f f e c t i v e p o s i t i o n o f mooring p o i n t 18 X 1 8 C o e f f . m a t r i x i n s t a b i l i t y a n a l . 6 x 6 C o e f f . m a t r i x i n s t a b i l i t y a n a l y s i s A A m p l i t u d e o f i n c i d e n t wave
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3 x 1 S t a t e v e c t o r f o r k = 0 , l , . . . , n A Mooring l i n e attachment p o i n t ( f a i r l e a d ) Mean wave p e r i o d 2iT/tüQA L , T L o n g i t u d i n a l , t r a n s v e r s e w i n d a t t a c k a r e a " ' ^ r e l Ship v e l o c i t y components r e l a t i v e t o w a t e r A 3 x 3 I n e r t i a m a t r i x i n s t a b i l i t y a n a l y s i s Speed o f c u r r e n t , w i n d
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3 x 3 Parameter m a t r i x f o r k = 0,1,. . . ,n V 3 x 1 V e l o c i t y v e c t o r (Uj.g-L,Vj.g-i,r)^ a 3 x 3 Hydrodyn. i n e r t i a m a t r i x f u n c t i o n X 3 x 1 P e r t u r b a t i o n v e c t o r (5Q,T)Q»X)''' B 3 x 3 Damping m a t r i x i n s t a b i l i t y a n a l y s i s y 18 X 1 S t a t e v e c t o r i n s t a b i l i t y a n a l y s i s 3 x 3 Parameter m a t r i x f o r k = 0,1,.. . ,n z 6 x 1 S t a t e v e c t o r i n s t a b i l i t y a n a l y s i s b 3 x 3 Hydrodyn. damping m a t r i x f u n c t i o n A(ij Frequency d i f f e r e n c e (u-j^-Ug)C 3 x 3 S t i f f n e s s m a t r i x i n s t a b i l i t y a n a l . C n Wave e l e v a t i o n , i t s H i l b e r t t r a n s f o r m D 3 x 3 A u x i l . m a t r i x i n s t a b i l i t y a n a l y s i s SOIQX Earthbound c o o r d i n a t e s c e n t e r e d a t E E E q u i l i b r i u m p o s i t i o n o f m i d s h i p p o i n t 0 PA Mass d e n s i t y o f a i r F 3 x 1 Force-couple v e c t o r (X,Y,N)''^ a E i g e n v a l u e i n s t a b i l i t y a n a l y s i s G 3 x 1 Wave d r i f t - f o r c e v e c t o r f u n c t i o n <'C,S,W D i r e c t i o n o f c u r r e n t , waves, w i n d H 3 x 1 Frequency-response v e c t o r f u n c t i o n Center f r e q u e n c y (u^+U2)/2
I I d e n t i t y m a t r i x o f s u i t a b l e dimension C i r c u l a r f r e q u e n c y o f wave component j Som D. Sharma, I n s t i t u t für S c h i f f b a u d e r U n i v e r s i t a t Hamburg, Lammersieth $0, 2000 Hamburg 60, FRG Tao J i a n g and Thomas E. S c h e l l i n , Germanischer L l o y d , V o r s e t z e n 32, 2000 Hamburg 1 1 , F. R. Germany
1 INTRODUCTION
I t has r e c e n t l y been f o u n d t h a t dynamic i n -s t a b i l i t y o f anchored, moored o r towed -ship-s can, under c e r t a i n r e a l i s t i c c o n d i t i o n s , l e a d not o n l y t o l a r g e - a m p l i t u d e s e l f - s u s t a i n e d os-c i l l a t i o n s b u t a l s o t o i n h e r e n t l y i r r e g u l a r mo-t i o n s , even i n mo-t h e absence o f random e x c i mo-t a mo-t i o n . This s o - c a l l e d d e t e r m i n i s t i c chaos can be endo-genous, i . e . o r i g i n a t e i n an autonomous system c o m p r i s i n g t h e moored s h i p and a steady c u r r e n t , o r i t can be f o r c e d by a simple p e r i o d i c e x c i t -a t i o n due t o -a r e g u l -a r h-armonic w-ave, f o r example. T h i s phenomenon i s , t h e r e f o r e , q u a l i t a t -i v e l y d -i f f e r e n t from t h e f a m -i l -i a r s t o c h a s t -i c response o f a s h i p t o s t o c h a s t i c e x c i t a t i o n i n i r r e g u l a r waves. The c o u n t e r i n t u i t i v e o c c u r rence o f c h a o t i c b e h a v i o r i n a f u l l y d e t e r m i n i s t i c mechanical system w i t h o u t any random i n put s t i l l evokes some s u r p r i s e and m i l d e x c i t e -ment , even though i t has a l r e a d y been demon-s t r a t e d i n a v a r i e t y o f comparable dynamic demon-sydemon-s- sys-tems, see f o r i n s t a n c e t h e a n t h o l o g y by Holden
(1986). I n t h i s paper we a t t e m p t a p a r a m e t r i c study o f t h e a s y m p t o t i c motions o f a t y p i c a l s u p e r t a n k e r when s i n g l e - p o i n t - m o o r e d (SPM) i n a u n i f o r m c u r r e n t , w i t h o r w i t h o u t a d d i t i o n a l ex-c i t a t i o n from ex-c o n s t a n t wind or r e g u l a r waves.
We focused on f i v e s e l e c t e d p a r a m e t e r s , namely c u r r e n t speed as t h e f o r e m o s t e n v i r o n -m e n t a l para-meter, f a i r l e a d l o c a t i o n and - moori n g l moori n e l e n g t h as p r moori n c moori p a l o p e r a t moori o n a l p a r a -m e t e r s , and p r o p e l l e r r a t e and r u d d e r angle s e t t i n g s as r e a d i l y a v a i l a b l e c o n t r o l p a r a m e t e r s . The e f f e c t s o f wind and waves were o n l y s p o t -checked, c h i e f l y i n search o f c h a o t i c m o t i o n s . The s h i p was t r e a t e d as a r i g i d body moving w i t h t h r e e degrees o f freedom: s u r g e , sway, and yaw. F i v e s e t s o f f o r c e s were c o n s i d e r e d : 1) q u a s i -steady n o n l i n e a r hydrodynamic response and con-t r o l f o r c e s , i n c l u d i n g h u l l - p r o p e l l e r - r u d d e r i n t e r a c t i o n s and c e r t a i n s c a l e e f f e c t s , 2) a d d i -t i o n a l l i n e a r hydrodynamic f o r c e s i n response t o t h e s h i p ' s motion h i s t o r y (memory e f f e c t ) , 3) r e s t o r i n g f o r c e g e n e r a t e d by t h e n o n l i n e a r l o a d - e l o n g a t i o n c h a r a c t e r i s t i c o f t h e mooring system, h) s t a n d a r d i z e d w i n d f o r c e s on above-water s h i p , and 5) s i m p l i f i e d f i r s t and second o r d e r f o r c e s due t o i n c i d e n t r e g u l a r waves. The r e q u i r e d dynamic a n a l y s i s was p e r f o r m e d by a j u d i c i o u s employment o f two complementary t e c h -n i q u e s : 1) g l o b a l -n o -n l i -n e a r s i m u l a t i o -n o f phase-space t r a j e c t o r i e s o r i g i n a t i n g f r o m a r b i t r a r y i n i t i a l c o n d i t i o n s and 2) l o c a l l y l i n e a r i z e d s t a b i l i t y analyses o f p o s s i b l e s t a t e s o f system e q u i l i b r i u m . R e s u l t s are s y s t e m a t i c a l l y p r e s e n t e d as s t a b i l i t y domains i n a f i v e - d i m e n s i o n a l para-meter space and supplemented by i l l u s t r a t i v e m o t i o n h i s t o r i e s and t r a j e c t o r i e s i n phase space. I n many ways t h i s paper i s an e x t e n s i o n o f our p r e v i o u s work, see J i a n g e t a l . (198?) and J i a n g and S c h e l l i n ( 1 9 8 8 ) , and we t a k e t h e o p p o r t u n i t y t o make two i m p o r t a n t c o r r e c t i o n s . Owing t o u n f o r t u n a t e programming e r r o r s , our e a r l i e r computed r e s u l t s d i s p l a y e d an a r t i f i c i a l e f f e c t o f memory on s t a b i l i t y and a s p u r i o u s h i g h e r frequency o s c i l l a t i o n i n t h e a s y m p t o t i c t r a j e c t o r i e s .
Beyond t h e t h e o r e t i c a l i n t e r e s t o u t l i n e d a¬ bove, motions o f an SPM s h i p possess, o f c o u r s e , p r a c t i c a l s i g n i f i c a n c e . Nowadays, s u p e r t a n k e r s are i n c r e a s i n g l y l o a d e d or d i s c h a r g e d a t u n p r o -t e c -t e d o f f s h o r e l o c a -t i o n s u s i n g SPM sys-tems. The moored s h i p i s i n t e n t i o n a l l y l e f t f r e e t o weathervane around t h e SPM t e r m i n a l . Hence, i n -s t a b i l i t y o f e q u i l i b r i u m a-s -such doe-s not con-t r a v e n e o p e r a b i l i con-t y . However, i m p e r c e p con-t i b l y slow s e l f - s u s t a i n e d h o r i z o n t a l o s c i l l a t i o n s o f t h e s h i p , e n t a i l i n g p o s s i b l y h i g h peak t e n s i o n s i n t h e m o o r i n g l i n e , c o n s t i t u t e a p o t e n t i a l h a z a r d t h a t needs t o be c a r e f u l l y m o n i t o r e d d u r i n g o p e r a t i o n o r , p r e f e r a b l y , c o n t r o l l e d by a c c u r a t e a n a l y s i s a t t h e design s t a g e . A c c o r d -i n g l y , a p l e t h o r a o f papers has been devoted t o t h i s problem i n r e c e n t y e a r s . For l a c k o f space, we c i t e o n l y two o u t s t a n d i n g monographs. F i r s t , t h e r e i s t h e d o c t o r a l d i s s e r t a t i o n o f P a p o u l i a s ( 1 9 8 7 ) , o b v i o u s l y c r o w n i n g an i m p r e s s i v e s e r i e s o f papers by B e r n i t s a s and h i s a s s o c i a t e s a t t h e U n i v e r s i t y o f M i c h i g a n . I t comprises a r i g -orous m a t h e m a t i c a l a n a l y s i s o f t h e dynamics o f mooring systems, w h i c h we f o u n d t r u l y i l l u m i -n a t i -n g . Seco-nd, t h e r e i s t h e v e r y r e c e -n t t h e s i s o f Vfichers ( I 9 8 8 ) , p u t t i n g t o g e t h e r numerous e x p e r i m e n t a l and t h e o r e t i c a l i n v e s t i g a t i o n s w h i l e f o c u s i n g on n o n l i n e a r wave e f f e c t s , i n -c l u d i n g i n t e r a -c t i o n between i n -c i d e n t waves and p r e v a i l i n g c u r r e n t . Moreover, t h e l o n g l i s t s o f r e f e r e n c e s c o n t a i n e d i n t h e s e two t h e s e s , t a k e n t o g e t h e r , make up a f a i r l y complete b i b -l i o g r a p h y on t h e s u b j e c t . 2 SIMULATIOM MODEL 2.1 C o o r d i n a t e s and K i n e m a t i c s
The f o u r c o o r d i n a t e systems used t o de-s c r i b e t h e m o t i o n de-s o f t h e moored de-s h i p are a l l i n d i c a t e d i n F i g . 1 . F i r s t , f o l l o w i n g s t a n d a r d c o n v e n t i o n , h o r i z o n t a l p o s i t i o n and o r i e n t a t i o n o f t h e s h i p a r e g i v e n by c o o r d i n a t e s x ^ j y ^ o f m i d s h i p p o i n t 0 and by heading angle i> i n an earthbound c o o r d i n a t e system, c o n v e n i e n t l y cen-t e r e d a cen-t cen-t h e e f f e c cen-t i v e mooring p o i n cen-t P and a l i g n e d t o some f i x e d g e o g r a p h i c a l d i r e c t i o n ( p r e f e r a b l y o p p o s i t e t o t h e p r e v a i l i n g c u r r e n t , waves or w i n d ) . I n t h i s system l e t t h e s t a t i c e q u i l i b r i u m c o o r d i n a t e s o f t h e s h i p (under t h e average a c t i o n o f c u r r e n t , waves and w i n d ) be
XoE)yoE>"''E • Then p e r t u r b a t i o n s from e q u i l i b
-r i u m a-re mo-re e x p e d i e n t l y exp-ressed i n a sec-ond e a r t h b o u n d c o o r d i n a t e system E^QnoXj cen-t e r e d a cen-t e q u i l i b r i u m p o i n cen-t E and a l i g n e d cen-t o e q u i l i b r i u m heading i/ig. These two systems are connected by t h e t r a n s f o r m a t i o n r u l e s :
^o = ( ^ O - ^ O E 5 ' ^ ° S > I ' E (yo-yoE' =^i"'J'E
% = (yo-yoE^ °°='''E - (^o-^oE^ s i n i l i j j ( 2 )
X = - ( 3 )
T h i r d , a shipbound c o o r d i n a t e system Oxy i s em-p l o y e d as u s u a l t o s i m em-p l i f y t h e deem-pendence o f hydrodynamic c o n t r o l and response f o r c e s on s h i p m o t i o n s . The r e l a t i o n between h o r i z o n t a l v e l o c i t y components u,v and Xo,yo r e s o l v e d
a l o n g moving c o o r d i n a t e s x,y and s t a t i o n a r y co-o r d i n a t e s Xco-o,yco-o, r e s p e c t i v e l y , i s c co-o n t a i n e d i n t h e t r a j e c t o r y e q u a t i o n s :
XQ = u cos i|) - V s i n i> = y cos i|i + u s i n i(i
ih) (5) (6) w h i c h a l s o d e f i n e yaw r a t e r . Center o f g r a v i t y G o f t h e s h i p and attachment p o i n t A o f t h e mooring l i n e have, o f c o u r s e , f i x e d c o o r d i n a t e s
XQ,yQ and xx>yk' r e s p e c t i v e l y , i n t h e shipbound system Oxy. F o u r t h , i t i s sometimes advantageous t o r e p l a c e t h e c o o r d i n a t e s XQ,y|| w i t h an e q u i v -a l e n t p -a i r I't^i'i'A r e p r e s e n t i n g t h e h o r i z o n t a l l i n e l e n g t h and d i r e c t i o n , r e s p e c t i v e l y , u s i n g t h e r e l a t i o n s : ^o ^ "•'"A ~ ^A * ^A * ^o ^ *A " ^A 'J' ~ ^A (7) and t h e i r i n v e r s e . The k i n e m a t i c e f f e c t o f a steady u n i f o r m c u r r e n t o f magnitude VQ and d i r e c t i o n i>Q s i m p l y shows up i n t h e d i s t i n c t i o n between s h i p v e l o c -i t y components u,v over ground and u^g-i,Vpg]^ r e l a t i v e t o ambient w a t e r : ^ r e l = ^ - s i n ( i J ^ 2- 4 ' ) (9) (10) W i t h o u t l o s s o f g e n e r a l i t y , angle can be t a k e n t o be 180°.
A b s o l u t e wind speed and d i r e c t i o n are i n -d i c a t e -d by V^^ an-d iji^, r e s p e c t i v e l y ; d i r e c t i o n o f p r o p a g a t i o n o f i n c i d e n t waves i s denoted by a n g l e ^q, see F i g . 1.
2.2 Equations o f Motion
The s h i p was t r e a t e d as a t r a n s v e r s a l l y symmetric r i g i d body h a v i n g t h r e e degrees o f freedom: s u r g e , sway,andyaw. P o s s i b l e minor e f f e c t s o f heave, p i t c h , and r o l l on t h e h o r i -z o n t a l motions were n e g l e c t e d . The e q u a t i o n s o f motion can t h e n be w r i t t e n i n t h e f o l l o w i n g s t a n d a r d form: (Ü - v r - r^Xg)m = X { v + u r + r x p ) m = Y i-Izz + ( v + u r ) X ( , (11) ( 1 2 ) (13) where m i s s h i p mass and I^z i t s moment o f i n e r -t i a abou-t a v e r -t i c a l a x i s -t h r o u g h 0. The n e -t e x t e r n a l t i m e - v a r y i n g h o r i z o n t a l f o r c e compo-n e compo-n t s X,ï r e s o l v e d a l o compo-n g axes x,y acompo-nd t h e i r mo-ment N about 0 r e s u l t , i n g e n e r a l , f r o m a com-p l e x i n t e r a c t i o n o f v a r i o u s com-p h y s i c a l com-phenomena. We s i m p l y c o n s i d e r e d a l i n e a r s u p e r p o s i t i o n o f f i v e e f f e c t s :
F = F n + F „ + F . + F , , + Fo ( l i t ;
where F r e p r e s e n t s f o r c e - c o u p l e (X,Y,N) and s u b s c r i p t s Q,M,A,W,S s t a n d f o r q u a s i - s t e a d y , memory, mooring l i n e , w i n d , and waves, r e s p e c t -i v e l y ; s u p e r s c r -i p t T denotes t r a n s p o s e . 2.3 S p e c i f i c a t i o n o f Forces
2.3.1 Quasi-Steady Forces
The q u a s i - s t e a d y hydrodynamic response and c o n t r o l f o r c e - c o u p l e FQ was c a l c u l a t e d a c c o r d i n g t o t h e f o u r - q u a d r a n t model o f Sharma ( I 9 8 2 ) as f u l l y documented by Oltmann and Sharma ( l 9 8 1 t ) . B a s i c a l l y , t h e f o r c e - c o u p l e elements are syn-t h e s i z e d as f o l o w s : - X j j j + Xjjj^ = y HI ÏHL + % C + Yp ^ H Q = N HI + N HL % C ^ Np + Nj^ ( 1 5 ) (16) ( 1 7 ) where s u b s c r i p t s H,P,R s t a n d f o r system elements h u l l , p r o p e l l e r , rudder and I,L,C f o r p h y s i c a l mechanisms i d e a l f l u i d , l i f t i n g , c r o s s f l o w e f f e c t s , r e s p e c t i v e l y ; t h e odd t e r m s i m p l y denotes o r d i n a r y r e s i s t a n c e t o pure l o n g i t u d i -n a l m o t i o -n . T h i s f o r c e - c o u p l e depe-nds l i -n e a r l y on a c c e l e r a t i o n s Uj,g]_,Vj.g2!^ ^-f^d i n a h i g h l y n o n l i n e a r way on v e l o c i t i e s Uj.gjs'^rel»^ w e l l as on c o n t r o l v a r i a b l e s p r o p e l l e r r a t e n and r u d d e r angle 6. The e x p l i c i t f o r m u l a t i o n s , de-r i v e d fde-rom e x t e n s i v e c a p t i v e model t e s t s and v a l i d a t e d by comparison w i t h f r e e model t e s t s i n a t o w i n g t a n k , a r e t o o l e n g t h y t o be r e p r o -duced here. However, t h r e e f e a t u r e s are w o r t h m e n t i o n i n g . F i r s t , t h e f o u r - q u a d r a n t model, un-l i k e many o t h e r s i n common use, does n o t break down near speed r e v e r s a l s (u,n=0) and i s , t h e r e -f o r e , s p e c i a l l y s u i t a b l e -f o r s i m u l a t i n g slow s h i p m o t i o n s . Second, t h e appearance o f Equa-t i o n s (15-17) n o Equa-t w i Equa-t h s Equa-t a n d i n g , Equa-t h i s model p a i n s t a k i n g l y accounts f o r t h r e e way h u l l p r o p e l l e r r u d d e r i n t e r a c t i o n s . T h i r d , i t i n c o r -p o r a t e s sim-ple e m -p i r i c a l c o r r e c t i o n s f o r t h e main Reynolds-number a s s o c i a t e d s c a l e e f f e c t s on h u l l r e s i s t a n c e and wake w i t h i m p o r t a n t r a m i f i c a t i o n s f o r p r o p e l l e r and r u d d e r f o r c e s . 2.3. F o r c e s A s s o c i a t e d w i t h Memory A l i n e a r response f o r c e a s s o c i a t e d w i t h hydrodynamic memory was c a l c u l a t e d by means o f a f i n i t e s t a t e space model f u l l y d e s c r i b e d i n a p r e v i o u s paper, see J i a n g e t a l . ( 1 9 8 7 ) . F i n a l r e s u l t can be summarized as f o l l o w s : F M ( t ) = Ca(o) - a(<»)]v(t) + S o ( t ) ( 1 8 ) V k < ^ ) = V i - k ( * ) - V o( t ) - B j ^ v ( t ) w i t h k = 0,1,...,n and S j j ^ ^ ( t ) = 0 ( 1 9 ) .)T. Here V i s t h e v e l o c i t y v e c t o r (uj.gi,Vpg3^,r; v { t ) i s t h e a c c e l e r a t i o n v e c t o r (uj.gi,Vpg2,f i ' ^ , Sj^ a r e f o r m a l r e c u r s i v e 3 x 1 s t a t e v e c t o r s t h a t e f f e c t i v e l y s t o r e motion h i s t o r y d u r i n g t i m e domain s i m u l a t i o n ; Aj^jBj^ a r e 3 x 3 parameter ma-t r i c e s o f an n ma-t h - o r d e r s ma-t a ma-t e space a p p r o x i m a ma-t i o n .
i d e n t i f i e d by a l e a s t squares f i t t o t h e o r e t i -c a l l y -c a l -c u l a t e d 3 x 3 m a t r i x f u n -c t i o n s a ( u ) , b(w) c o m p r i s i n g frequency dependent hydrodynamic i n e r t i a and damping c o e f f i c i e n t s , r e s p e c t i v e l y , i n coupled surge, sway, and yaw.
Note t h a t t h i s f o r m u l a t i o n accounts f o r memory e f f e c t s due o n l y t o waves r a d i a t e d by an o s c i l l a t i n g h u l l w i t h o u t mean f o r w a r d speed, and even t h a t o n l y up t o f i r s t o r d e r . Other p o s s i b l y i m p o r t a n t memory e f f e c t s , such as those due t o v o r t e x shedding a t l o w f r e q u e n c i e s , were d i s r e g a r d e d .
2.3.3 Mooring L i n e Force
The h i g h l y n o n l i n e a r l o a d e l o n g a t i o n p r o -p e r t y o f t h e mooring system was a-p-proximated i n t h e range o f i n t e r e s t by t h e e m p i r i c a l f o r m u l a : F^ = | ( l + s g n A L j ^ ) C ^ i L ^ , ALj^ = L^ - L 'AU (20) w i t h c o e f f i c i e n t Cj^ = 0.0113 kNm"^. Here F^ i s h o r i z o n t a l component o f l i n e t e n s i o n , L;^ i s i n -stantaneous h o r i z o n t a l d i s t a n c e o f attachment p o i n t A from e f f e c t i v e mooring p o i n t P, and L^^y i t s r e f e r e n c e v a l u e c o r r e s p o n d i n g t o t h e un-s t r e t c h e d no-load c o n d i t i o n , un-see F i g . 1 . Note t h a t Fy^ r e p r e s e n t s t o t a l s t a t i c r e s t o r i n g f o r c e a r i s i n g f r o m t h e e l a s t i c i t y o f t h e mooring l i n e and t h e c a t e n a r y a c t i o n o f an anchored buoy o r t h e r i g h t i n g moment o f an a r t i c u l a t e d t o w e r , e t c . The i n e r t i a o f t h e mooring system was con-s i d e r e d t o be n e g l i g i b l y con-s m a l l compared t o t h a t o f t h e s h i p .
2.3.h Wind Forces
The wind-generated f o r c e - c o u p l e , due t o a steady wind a c t i n g on t h e above-water p a r t s o f t h e h u l l and s u p e r s t r u c t u r e , was c a l c u l a t e d as u s u a l by means o f t h e e m p i r i c a l f o r m u l a s :
% =
Y,., = ^1
PA "^w % 2 PA % A L '^YW 2 PA VW A L V„ AT L C NW (21) (22) (23) where p;^ i s mass d e n s i t y o f a i r , V;^ i s mean e f -f e c t i v e wind speed ( a c c o u n t i n g i n s t a n d a r d i z e d ways f o r v e r t i c a l p r o f i l e and f o r t u r b u l e n c e l e v e l ) , Aj, and Acj a r e l o n g i t u d i n a l ( b r o a d s i d e ) and t r a n s v e r s e (head-on) p r o j e c t e d above-water areas, L i s s h i p l e n g t h between p e r p e n d i c u l a r s , and Cxy,CYy,Cuy a r e s h i p - t y p e dependent f o r c e and moment c o e f f i c i e n t s , o b t a i n e d from wind t u n n e l model t e s t s , as f x i n c t l o n s o f wind angle o f a t t a c k (IT - i>\] + ^). Required wind areas f o r s u b j e c t t a n k e r a r e g i v e n i n Table I ; a p p r o p r i -a t e n u m e r i c -a l v-alues o f f o r c e c o e f f i c i e n t s were t a k e n from OCIMF (1977, pp. 23-27). I n f l u e n c e o f s h i p motions u , v , r on w i n d f o r c e was i g n o r e d . 2.3.5 Wave Forces Wave f o r c e generated on t h e s h i p by t h e a c t i o n o f ambient sea waves was approximated as t h e sum:where Fg(^) i s t h e u s u a l f i r s t - o r d e r f o r c e and Fg'^) a second-order s l o w l y v a r y i n g d r i f t f o r c e . The o n l y wave e x c i t a t i o n c o n s i d e r e d was t h e s u p e r p o s i t i o n o f two l o n g - c r e s t e d r e g u l a r waves o f e q u a l a m p l i t u d e A and s l i g h t l y d i f f e r e n t f r e -quencies p r o p a g a t i n g i n a g i v e n d i r e c t i o n ijig, t h e r e b y c r e a t i n g wave groups o f l o w f r e q u e n c y c l o s e t o a n a t u r a l frequency o f t h e moored s h i p . The f i r s t - o r d e r f o r c e was c o n s t r u c t e d by superimposing responses t o i n d i v i d u a l wave com-ponents :
2
FJI ) = Re I H(u)j,i(.s-i(.) A J—1
• exp - i k j ( x o cosifig + yo sini|is) + i njj t ( 2 5 ) where i s t h e wave number c o r r e s p o n d i n g t o frequency u j and H t h e h u l l - f o r m dependent com-p l e x frequency-rescom-ponse v e c t o r , here c a l c u l a t e d by means o f an a v a i l a b l e computer program based on 3-D p o t e n t i a l t h e o r y as d e s c r i b e d by D s t e r g a a r d et a l . ( 1 9 7 9 ) . F o l l o w i n g M a r t h i n s e n ( 1 9 8 3 ) , t h e second-o r d e r f second-o r c e was e s t i m a t e d as f second-o l l second-o w s . L e t C be wave e l e v a t i o n a t 0 and ri i t s H i l b e r t t r a n s f o r m , i . e . 2
C + iTi = J A e x p - i k j ( x o cosijis + y,-, sini^^g) J = l j t ] ( 2 6 ) Then t h e s l o w l y v a r y i n g d r i f t f o r c e i s p r o p o r -t i o n a l -t o l o c a l envelope-ampli-tude squared: F ( 2 ) ( C ^ + n^) G(U(,,ij)g-ij<) ( 2 7 ) where G i s t h e h u l l - f o r m dependent d r i f t - f o r c e c o e f f i c i e n t v e c t o r c a l c u l a t e d f o r a r e g u l a r wave o f frequency equal t o t h e c e n t e r f r e q u e n c y üJc=(u]^+ü)2)/2 o f t h e wave group. F o l l o w i n g Cox
( 1 9 8 2 ) , t h e dependence o f G on r e l a t i v e d i r e c -t i o n o f wave p r o p a g a -t i o n can be approxima-ted as
CXS PS'''2 c o s ^ { H i s - < p ) Cjs P g ' V sin3(i))3-i(,) CNS Pg'^'^ sin2(ii.g-i;)) (28) (1) (2) (21.) where p i s mass d e n s i t y o f w a t e r , g a c c e l e r -a t i o n due t o g r -a v i t y , -and V d i s p l -a c e d volume o f t h e s h i p . The nondimensional c o e f f i c i e n t s Cxg, CYg,Cug were here e s t i m a t e d by means o f a n o t h e r a v a i l a b l e computer program based on 3D p o t e n -t i a l -t h e o r y as documen-ted by Clauss e -t a l . ( 1 9 8 2 ) . Numerical v a l u e s f o r s u b j e c t t a n k e r a t 2ir/uc=10 s, f o r example, were found t o be Cxg = 0.25,
Cyg = 3.50, and Ciig = -1.25.
I n f l u e n c e o f s h i p motions u,v,r on wave f o r c e and i n t e r a c t i o n o f i n c i d e n t waves w i t h ambient c u r r e n t were i g n o r e d .
2.1) Canonized S i m u l a t i o n Equations
We expressed t o t a l system dynamics i n c a -n o -n i c a l form by a s e t o f 6 + 3 { -n + l ) f i r s t - o r d e r coupled n o n l i n e a r o r d i n a r y DE's c o n s i s t i n g o f t h r e e k i n e m a t i c t r a j e c t o r y Equations {h-6),
t h r e e dynamic e q u a t i o n s d e r i v e d by e x p l i c i t l y
s o l v i n g Equations (1113) f o r u,v,r t h a t o t h e r -wise occiir l i n e a r l y on b o t h s i d e s , and 3 ( n + l ) Equations ( 1 9 ) f o r r a t e s o f change o f n+1 s t a t e v e c t o r s S]^ o f dimension t h r e e . The l a s t 3 ( n + l ) e q u a t i o n s were dropped when memory e f f e c t s were o p t i o n a l l y d i s r e g a r d e d . For t h e s u b j e c t t a n k e r , n=3 t u r n e d o u t t o be a s a t i s f a c t o r y approxima-t i o n as demonsapproxima-traapproxima-ted i n our p r e v i o u s work. Hence, t h e system comprised e i t h e r 18 ( w i t h memory) o r 6 ( w i t h o u t memory) s i m u l a t i o n e q u a t i o n s t h a t were n u m e r i c a l l y i n t e g r a t e d w i t h c o n t r o l l e d a c -curacy u s i n g computer s u b r o u t i n e LSODA, s u p p l i e d by I n t e r n a t i o n a l M a t h e m a t i c a l and S t a t i s t i c a l L i b r a r i e s , I n c . , and s p e c i f i c a l l y recommended f o r s o - c a l l e d s t i f f systems.
For c h e c k i n g purposes, we d e r i v e d and p r o -grammed an a l t e r n a t i v e s e t o f s i m u l a t i o n equa-t i o n s u s i n g s equa-t a equa-t e v a r i a b l e s l'js^,i>j^,^ ,IjA'Ï'A'^ i n -s t e a d o f XQ,y|-,,i|i,u,v,r. There wa-s n o t h i n g unique about e i t h e r c h o i c e . I n p r i n c i p l e , an i n f i n i t e v a r i e t y o f a l t e r n a t i v e c o o r d i n a t e systems must l e a d t o e q u i v a l e n t t r a j e c t o r y s i m u l a t i o n s (and i d e n t i c a l s t a b i l i t y assessments). Indeed, t h i s e x e r c i s e h e l p e d us i n c a t c h i n g two s u b t l e p r o gramming e r r o r s t h a t had l o n g evaded our a t t e n -t i o n .
3 STABILITY ANALYSIS 3.1 E q u i l i b r i u m S t a t e s
For any g i v e n steady e x c i t a t i o n ( c u r r e n t , w i n d , and waves) and f i x e d c o n t r o l s e t t i n g s ( p r o -p e l l e r r a t e and r u d d e r a n g l e ) t h e moored s h i -p can be expected t o have one o r more e q u i l i b r i u m s t a t e s . We d e t e r m i n e d these s t a t e s f r o m Equat i o n s (911) by s u b s Equat i Equat u Equat i n g e x p l i c i Equat f o r m u l a -t i o n s f o r a l l f o r c e s and e n f o r c i n g -t h e zero motion c o n d i t i o n by s e t t i n g u , v , r , u , v , r = 0 (29) The r e s u l t i n g s e t o f t h r e e n o n l i n e a r a l g e b r a i c e q u a t i o n s was s o l v e d i t e r a t i v e l y t o y i e l d (some-t i m e s m u l (some-t i p l e ) e q u i l i b r i a p o s i (some-t i o n s o f (some-t h e s h i p i d e n t i f i e d by c o o r d i n a t e s XQg,yQE,i))g. Aux-i l Aux-i a r y e q u Aux-i l Aux-i b r Aux-i u m v a l u e s such as l Aux-i n e l e n g t h L;^, d i r e c t i o n ^j^, and t e n s i l e f o r c e F;^ t h e n f o l l o w e d f r o m Equations ( 7 , 8 , 2 0 ) . Note t h a t t h e memory dependent v e c t o r SQ i s a u t o m a t i c a l l y zero i n any e q u i l i b r i u m s t a t e .
3.2 L o c a l L i n e a r i z a t i o n
We chose t o express p e r t u r b a t i o n s from any e q u i l i b r i u m s t a t e E by a v e c t o r X, c o m p r i s i n g d i s p l a c e m e n t s Co,1o,X, and b y i t s t i m e d e r i v a -t i v e s X,5(. I -t was found -t h a -t -t o f i r s -t o r d e r = ( u , v , r ) T , ( u , v , r ) T ( 3 0 ) L o c a l l i n e a r i z a t i o n a t E w i t h r e s p e c t t o p e r t u r -b a t i o n s t h e n t r a n s f o r m e d motion Equations ( 1 1 ¬ 13) i n t o t h e f o l l o w i n g second-order v e c t o r DE: A x + B x + C x = So w i t h A = A J + A Q + A M , B = BQ+BM, C = CQ+CA+CW+CS (31)
Here c o e f f i c i e n t m a t r i c e s A,B,C comprise e i t h e r g l o b a l system parameters such as body i n e r t i a :
0, m , mx, 0, mXQ, I ^ and hydrodynamic i n e r t i a A^ = - 0 / 3 i , 3 / 3 v , 8 / 3 J ) ^ (XQ,YQ,NQ) Ca(o) - a(<»>)] (32) (33) (3lt) or l o c a l g r a d i e n t s o f f o r c e s w i t h r e s p e c t t o p e r t u r b a t i o n s c a l c u l a t e d anew a t each E: BT = - 0 / 8 u , 3 / 3 v , 3 / 3 r ) T ( X n , Y Q , N o ) 0 , 0 , { a i i ( 0 ) - a i i ( - ) } V c s i n ( ^ E - i ; - c ) 0,0,{a22(0)-a22(")}Vccos(i|'E-'J'c), L0,0,{a32(0)-a32(°)}Vccos('l'E->l'c)J ^A -(3/3C^,3/3no,3/3x)^(XQ,Yg,,NQ) -0/3C„,8/3^0,3/3x)'^(XA,Yji,NA) -(3/3f ,3/3n„,3/3x)'^(X„,Y - 0/ 3 ?O, 3 / 3T1O, 3 / 3X ) ^ W'^W'^w' (Xg,?s,5s) (35) (36) (37) (38) (39) (1*0) Note t h a t wave f o r c e s were reduced t o t h e i r time-averaged components i n o r d e r t o enable a h e u r i s t i c assessment o f t h e e f f e c t o f waves on s t a b i l i t y by d i s r e g a r d i n g t h e p u r e l y o s c i l l a t o r y components.
A l l d i f f e r e n t i a t i o n s necessary f o r o b t a i n -i n g t h e above J a c o b -i a n m a t r -i c e s were performed n m e r i c a l l y b y means o f t h e f i v e - p o i n t Lagrange f o r m u l a w i t h a s u i t a b l e s t e p s i z e and v a l i d a t e d , where p o s s i b l e , by comparison w i t h a n a l y t i c a l d e r i v a t i o n s . One sample s e t i s reproduced i n Table I I .
T o t a l system dynamics i s t h e n d e s c r i b e d by l i n e a r i z e d E q u a t i o n ( 3 1 ) a l o n g w i t h a l r e a d y l i n e a r Equations ( 1 9 ) . For f u r t h e r a n a l y s i s we found i t e x p e d i e n t t o combine them i n t o a s i n g l e set o f f i r s t - o r d e r DE's. For t h i s we i n t r o d u c e d ( r e c a l l n=3) an extended s t a t e v e c t o r : = [ j T ^ s T 3T 3T ,T ^ T ] and a composite c o e f f i c i e n t m a t r i x : (1*1) - A - 4 , A - 1 , 0 , 0 , 0 , - A- l C -B3 . -A3, I , 0 , 0 , -B3D p - -B2 , -A2, 0 , I , 0 , -B2D r — - B l , - A l , 0 , 0 . I , - B i D -BQ , -AQ, 0 , 0 , 0 , -BQD 1 , 0 , 0 , 0 , 0 , 0 (1+2) c o n t a i n i n g an a u x i l i a r y m a t r i x 0, 0, Vcsin(.)<E-'('c) 0, 0, Vccos{i^E-i|;c) LO, 0, 0 (lt3) The l i n e a r i z e d system i s t h e n d e s c r i b e d by a s i n g l e f i r s t - o r d e r v e c t o r DE o f dimension 6+3(n+1):
iUh)
These e q u a t i o n s s i m p l i f i e d s u b s t a n t i a l l y when memory was o p t i o n a l l y d i s r e g a r d e d . V e c t o r s Sj^ became d i s p e n s a b l e and m a t r i c e s A ^ j B f ^ dropped o u t o f E q u a t i o n ( 3 l ) . Consequently, Equations {kl,k2,hk) c o u l d be r e p l a c e d by Q i - ( A J + A Q ) - 1 B Q , - (A J + A Q ) - 1 C I , 0 (^5) ikG) ( l i 7 ) r e s p e c t i v e l y , t h u s r e d u c i n g system dynamics w i t h o u t memory t o a f i r s t - o r d e r v e c t o r DE o f dimension 6 ( i n s t e a d o f l 8 w i t h memory). I n p a s s i n g , we n o t e t h a t when t h e a l t e r -n a t i v e c o o r d i -n a t e system me-ntio-ned i -n S e c t i o -n
2.h was used, l o c a l l i n e a r i z a t i o n was done nu-m e r i c a l l y w i t h r e s p e c t t o p e r t u r b a t i o n s "('A-'f'AE.X i n s t e a d o f Co.flo.X above. 3.3 S t a b i l i t y Assessment
L o c a l s t a b i l i t y o f any examined e q u i l i b r i u m s t a t e was assessed i n t h e sense o f Lyapunov by s o l v i n g a c l a s s i c a l e i g e n v a l u e p r o b l e m , i . e .
IP
I I = 0 IQ - a l j = 0 ( i t 8 ) depending on whether memory was t o be r e t a i n e d o r n o t . E i t h e r l 8 ( w i t h memory) o r 6 ( w i t h o u t memory) e i g e n v a l u e s were o b t a i n e d .Consider f i r s t t h e g e n e r i c case where a l l e i g n v a l u e s have nonzero r e a l p a r t s (sometimes c a l l e d h y p e r b o l i c c a s e ) . Then t h e i n t e r p r e t a -t i o n i s s -t r a i g h -t f o r w a r d . I f a l l r e a l p a r -t s a r e n e g a t i v e d e f i n i t e , t h e e q u i l i b r i u m s t a t e i s s t a b l e , and t h e autonomous system s h o u l d asymp-t o asymp-t i c a l l y r e asymp-t u r n asymp-t o i asymp-t a f asymp-t e r a s u f f i c i e n asymp-t l y s m a l l , a r b i t r a r y i n i t i a l d i s t u r b a n c e . I f one or more r e a l p a r t s are p o s i t i v e d e f i n i t e , t h e e q u i -l i b r i u m s t a t e i s u n s t a b -l e . Even when t h e i n i t i a -l d i s t u r b a n c e i s a r b i t r a r i l y s m a l l , t h e system w i l l almost never r e t u r n t o such an u n s t a b l e e q u i l i b r i u m . I t may a s y m p t o t i c a l l y wander away t o a n e i g h b o r i n g s t a b l e e q u i l i b r i u m , e n t e r a p e r i o d i c o r b i t ( l i m i t c y c l e ) , g e t t r a p p e d i n a q u a s i - p e r i o d i c o r b i t on a t o r u s , or execute c h a o t i c motions on a m a n i f o l d o f f r a c t a l d i -mension ( s t r a n g e a t t r a c t o r ) i n d e f i n i t e l y , see e.g. Holden and Muhamad ( I 9 8 6 ) .
I n t e r p r e t a t i o n o f t h e d e g e n e r a t e , nonhyp e r b o l i c case i s n o t so easy. The s i m nonhyp l e s t a b i l -i t y c r -i t e r -i o n j u s t s t a t e d f a -i l s . I n g e n e r a l , t h i s s i t u a t i o n can o n l y a r i s e when a c o n t i n u o u s v a r i a t i o n o f one or more system parameters i s c o n s i d e r e d . Then i t r e p r e s e n t s a c r i t i c a l p o i n t , where l o n g - t e r m system b e h a v i o r undergoes a q u a l i t a t i v e change ( b i f u r c a t i o n ) . Complete c l a s -s i f i c a t i o n o f b i f u r c a t i o n -s i n m u l t i p a r a m e t e r systems i s t o o complex t o be a t t e m p t e d h e r e . L e t us c o n c e n t r a t e on two t y p i c a l s i t u a t i o n s . I f a pure r e a l e i g e n v a l u e passes t h r o u g h z e r o , a s t a t i c (saddle-node o r p i t c h f o r k ) b i f u r c a t i o n o c c u r s , i m p l y i n g a t r a n s i t i o n between s i n g l e and m u l t i p l e e q u i l i b r i a . I f t h e r e a l p a r t o f a p a i r o f complex-conjugate e i g e n v a l u e s passes t h r o u g h z e r o , a dynamic ( H o p f ) b i f u r c a t i o n o c c u r s , i m p l y i n g a t r a n s i t i o n between s t a b l e e q u i l i b r i u m and l i m i t c y c l e . For f u r t h e r de-t a i l s , see e.g. Guokenheimer and Holmes ( 1 9 8 6 ) .
k RESULTS AHD DISCUSSION
h.1 Scope o f C a l c u l a t i o n s
C a l c u l a t i o n s were done f o r a f u l l y loaded 150 000 t o n deadweight s u p e r t a n k e r i n deep w a t e r I t s p r i n c i p a l p a r t i c u l a r s a r e reproduced i n Table I . T h i s exemplary s h i p was chosen m a i n l y because i t s complete s e t o f c o e f f i c i e n t s f o r q u a s i - s t e a d y hydrodynamic f o r c e s , based on ex-t e n s i v e ex-t e s ex-t s w i ex-t h a 1 : 35 s c a l e model a ex-t ex-t h e Hamburg Ship Model B a s i n , was a v a i l a b l e from Oltmann and Sharma (198I*, Table 3 ) . A d d i t i o n a l c o e f f i c i e n t s r e q u i r e d t o r e p r e s e n t hydrodynamic memory e f f e c t s were computed and documented by us i n a p r e v i o u s paper, c f . J i a n g e t a l . (1987, Table I I ) .
E q u i l i b r i u m s t a t e s were determined and c o r r e s p o n d i n g s t a b i l i t y analyses c a r r i e d out s y s t e m a t i c a l l y , i n t e n d i n g t o cover t h e e n t i r e p r a c t i c a l range o f t h e f i v e parameters i n t r o -duced i n S e c t i o n 1. To r e a s o n a b l y l i m i t t h e amount o f c o m p u t a t i o n , o n l y two parameters were v a r i e d a t a t i m e w h i l e k e e p i n g t h e o t h e r s con-s t a n t a t con-s e l e c t e d r e f e r e n c e v a l u e con-s acon-s l i con-s t e d i n Table I I I . T h i s generated e x c a c t l y t e n t w o d i -mensional s e c t i o n s t h r o u g h a f i v e - d i m e n s i o n a l parameter space. Two c l a r i f y i n g remarks a r e i n o r d e r . F i r s t , a l t h o u g h our a l g o r i t h m a l l o w s i n dependent v a r i a t i o n o f t h e two f a i r l e a d c o o r d i -nates x;^,y^, o n l y a o n e - d i m e n s i o n a l v a r i a t i o n n e a r l y f o l l o w i n g t h e r e l e v a n t deck c o n t o u r (see F i g . 2) was i n v e s t i g a t e d i n o r d e r t o r u l e o u t u n r e a l i s t i c mooring c o n f i g u r a t i o n s . Second, t h e r e f e r e n c e v a l u e o f p r o p e l l e r r a t e was not a f i x e d c o n s t a n t ( s a y , n=0) b u t v a r i e d somewhat from p o i n t t o p o i n t , always c o n f o r m i n g t o t h e p h y s i c a l c o n d i t i o n o f zero t o r q u e Q ( f r e e w h e e l -i n g p r o p e l l e r ) . We have shown elsewhere t h a t t h e a l t e r n a t i v e c h o i c e o f zero t u r n i n g r a t e ( p r o -p e l l e r l o c k e d or h e l d f i x e d by engine f r i c t i o n ) does n o t a p p r e c i a b l y a l t e r r e s u l t s .
E s s e n t i a l l y a l l s t a b i l i t y c a l c u l a t i o n s were conducted f o r each o f f o u r d i s t i n c t a l t e r -n a t i v e s : - model c o n d i t i o n w i t h memory e f f e c t - model c o n d i t i o n w i t h o u t memory e f f e c t - s h i p c o n d i t i o n w i t h memory e f f e c t - s h i p c o n d i t i o n w i t h o u t memory e f f e c t The t e r m "model c o n d i t i o n " i m p l i e s s i m p l y s c a l -i n g model hydrodynam-ics t o s h -i p s -i z e a c c o r d -i n g t o Froude's l a w , d i s r e g a r d i n g a l l v i s c o u s s c a l e e f f e c t s , and hence r e f l e c t s r e s u l t s as expected from 1 : 35 s c a l e model t e s t s i n a t a n k . By con-t r a s con-t , con-t h e l a b e l " s h i p c o n d i con-t i o n " i m p l i e s ITTC s t a n d a r d c o r r e c t i o n s t o s h i p r e s i s t a n c e and wake, a c c o u n t i n g f o r t h e d i f f e r e n c e i n Reynolds number between model and f u l l s c a l e , w i t h s u b s t a n t i a l consequences n o t a b l y f o r p r o p e l l e r and r u d d e r a c t i o n . The concepts " w i t h and w i t h o u t memory" have been d e f i n e d e a r l i e r i n S e c t i o n 2.3.2.
Numerous t r a j e c t o r y s i m u l a t i o n s were p e r -form.ed f o r s e l e c t e d s e t s o f parameter v a l u e s .
m a i n l y t o i l l u s t r a t e t h e d i f f e r e n c e i n system b e h a v i o r i n t h e v i c i n i t y o f s t a b l e and u n s t a b l e e q u i l i b r i a . Except f o r a few t e s t cases, these t i m e consuming computations were p e r f o r m e d o n l y f o r t h e s h i p c o n d i t i o n w i t h o u t memory. S e v e r a l values o f wind and waves ( t a k e n a g a i n s t t h e c u r r e n t ) were a l s o t r i e d i n search o f more i n t e r -e s t i n g t r a j -e c t o r i -e s , s p -e c i a l l y t h o s -e l -e a d i n g t o chaos ( s t r a n g e a t t r a c t o r s ) .
k.2 S t a b i l i t y Domains
We have v i s u a l i z e d r e s u l t s o f our s t a b i l i t y a n a l y s i s i n F i g . 3 by t a k i n g t e n plane s e c t i o n s t h r o u g h t h e parameter space i n such a way t h a t each p o s s i b l e p a i r o f t h e f i v e chosen parameters ^C'yA'l'AU''^»'^ occurs e x a c t l y once. The curves shown i n each graph r e p r e s e n t s t a b i l i t y bound-a r i e s , s e p bound-a r bound-a t i n g t h e s e t o f pbound-arbound-ameter p o i n t s p o s s e s s i n g a t l e a s t one s t a b l e e q u i l i b r i u m s t a t e
( s t a b l e domain) from t h e r e s t ( u n s t a b l e domain). Before d i s c u s s i n g d e t a i l s o f i n d i v i d u a l graphs, t h r e e g e n e r a l remarks are i n o r d e r .
F i r s t , note t h a t o n l y two d i s t i n c t curves are v i s i b l e i n each graph a l t h o u g h f o u r d i f f e r e n t analyses were c a r r i e d out as d e s c r i b e d i n Sec-t i o n h.1. The reason i s t h a t i n c l u d i n g memorya s s o c i memorya t e d f o r c e s f memorya i l e d t o produce memoryany n o t i c e a b l e d i f f e r e n c e ! T h i s i s i n c o n t r a s t t o our f o r -mer f i n d i n g s r e p o r t e d i n J i a n g e t a l . (1987, c f . F i g . h). As s t a t e d i n t h e I n t r o d u c t i o n , an un-f o r t u n a t e programming e r r o r was r e s p o n s i b l e un-f o r g e n e r a t i n g a s p u r i o u s and r a t h e r i n c r e d i b l e memo r y e f f e c t i n memour e a r l i e r c a l c u l a t i memo n s . The r e -v i s e d r e s u l t s a r e , i n f a c t , much l e s s s u r p r i s i n g , because autonomous motions o f t h e moored t a n k e r occur a t e x t r e m e l y low f r e q u e n c i e s , w e l l below t h e appearance o f memory e f f e c t s i n t h e c a l c u -l a t e d requency response, c f . F i g . 2 i n t h e pa-per j u s t c i t e d .
Second, t h e d i f f e r e n c e between s t a b i l i t y b o u n d a r i e s f o r model c o n d i t i o n and s h i p c o n d i t i o n i s not as s i m p l e and s t r a i g h t f o r w a r d as r e -p o r t e d , e.g. by L a t o r r e ( 1 9 8 7 ) . The reason i s t h a t our s c a l e e f f e c t s comprised p a r t i a l l y count e r a c count i n g componencounts. On count h e one hand, l o w e r r e -s i -s t a n c e c o e f f i c i e n t i n -s h i p c o n d i t i o n e n t a i l e d l o w e r mean l i n e t e n s i o n , which i s b a s i c a l l y de-s t a b i l i z i n g . On t h e o t h e r hand, l o w e r v i de-s c o u de-s wake f r a c t i o n i n s h i p c o n d i t i o n I m p l i e d h i g h e r r u d d e r f o r c e s as w e l l as h i g h e r p r o p e l l e r p u l l (and s i d e f o r c e ) a t n e g a t i v e p r o p e l l e r r a t e s , a l l o f which are m o s t l y s t a b i l i z i n g . The o v e r -a l l e f f e c t i s , t h e r e f o r e , -a m b i v -a l e n t -and p -a r t l y c o u n t e r i n t u i t i v e . We t a k e t h i s o p p o r t u n i t y t o d e c l a r e t h a t r e s u l t s r e p o r t e d i n our two p r e v i -ous p a p e r s , a l r e a d y c i t e d , concern o n l y t h e model c o n d i t i o n .
T h i r d , i t might seem s u r p r i s i n g t h a t our s t a b i l i t y domains are not p e r f e c t l y symmetric about yA=0 and 6=0, d e s p i t e t h e u s u a l p o r t s t a r -board symmetry o f t h e t a n k e r h u l l and r u d d e r . The reason i s t h a t t h e i n h e r e n t hydrodynamic asymmetry o f our r i g h t h a n d e d s i n g l e screw was p r o p e r l y i n c l u d e d i n t h e f o u r - q u a d r a n t f o r c e f o r m u l a t i o n s . For i n s t a n c e , w h i l e f i x e d or f r e e -w h e e l i n g , t h e p r o p e l l e r g e n e r a t e d enough s i d e f o r c e t o r e q u i r e a n e u t r a l r u d d e r angle 6-3° a t Vc=2 m/s. T u r n i n g now t o t h e p r i m a r y e f f e c t s o f i n -d i v i -d u a l parameters w i t h i n t h e i r p r a c t i c a l ranges, we see from graphs a,b,c,d t h a t i n c r e a s -i n g c u r r e n t speed VQ -i s d e s t a b -i l -i z -i n g , from graphs a , e , g , i t h a t i n c r e a s i n g r e v e r s e p r o p e l -l e r r a t e -n i s s t a b i -l i z i n g , from graphs b , e , f , j t h a t i n c r e a s i n g r u d d e r d e f l e c t i o n |6| i s s t a -b i l i z i n g , from graphs c , h , i , j t h a t i n c r e a s i n g f a i r l e a d asymmetry |yj^| i s s t a b i l i z i n g , and f r o m graphs d,f,g,h t h a t i n c r e a s i n g mooring l i n e l e n g t h Lj^u i s d e s t a b i l i z i n g . A l l t h e s e e f f e c t s can be u n d e r s t o o d q u a l i t a t i v e l y i n terms o f an i n t e r p l a y o f the s t a b i l i z i n g a c t i o n o f mean l i n e t e n s i o n and t h e d e s t a b i l i z i n g a c t i o n o f hydrodynamic Munk moment i n c o n j u n c t i o n w i t h t h e dominant r o l e o f e q u i l i b r i u m asymmetry.
V/hen any two parameters s i m u l t a n e o u s l y dep a r t from t h e i r r e f e r e n c e v a l u e s , t h e i r i n d i -v i d u a l e f f e c t s are g e n e r a l l y c u m u l a t i -v e , as seen i n graphs a , b , c , d , f , g , h , i . However, p a i r s n,6 and yA,5 d i s p l a y a more complex i n t e r a c t i o n . F i r s t , the s t r a n g e shape o f s t a b i l i t y domains i n graph e r e s u l t s f r o m a s t r o n g hydrodynamic e f -f e c t o -f p r o p e l l e r l o a d i n g on r u d d e r o p e r a t i n g i n t h e s l i p s t r e a m . Rudder e f f e c t i v e n e s s s t e a d i l y i n c r e a s e s w i t h i n c r e a s i n g p o s i t i v e p r o p e l l e r r a t e ( t h e r e b y n a r r o w i n g t h e regime o f i n s t a b i l -i t y ) and d r a m a t -i c a l l y decreases w -i t h -i n c r e a s -i n g n e g a t i v e p r o p e l l e r r a t e i i n t i l t h e r u d d e r i s v i l -t i m a -t e l y s -t u c k i n deadwa-ter beyond n:-5 RPM ( t h e r e b y b l o w i n g up t h e regime o f i n s t a b i l i t y ) . However, f o r s u f f i c i e n t l y l a r g e n e g a t i v e p r o p e l l e r r a t e s , s t a b i l i t y i s r e g a i n e d due t o p r o p e l l e r p u l l a l o n e , i r r e s p e c t i v e o f r u d d e r s e t -t i n g . Nex-t, s -t a b i l i -t y domains i n graph j owe t h e i r l e n t i c u l a r shape t o t h e s i m p l e f a c t t h a t e q u i l i b r i u m asymmetries provoked by f a i r l e a d l o c a t i o n and rudder anlge are a d d i t i v e i n quad-r a n t s one and t h quad-r e e but s u b t quad-r a c t i v e i n quadquad-rants two and f o u r .
F i n a l l y , two minor anomalies deserve c l a r i f i c a t i o n . F i r s t , t h e r e are noseshaped p r o t u b e r -ances o f u n s t a b l e regimes i n graphs a , e , g , i t h a t r e s u l t from t h e sudden breakdown o f r u d d e r e f -f e c t i v e n e s s near i t s s t a l l a n g l e , see Oltmann and Sharma (1984, F i g . 1 0 ) . Second, t h e r e are p e c u l i a r i n d e n t a t i o n s i n t o t h e u n s t a b l e regimes i n t h e f i r s t quadrant o f graph i t h a t a r i s e from non-monotonic v a r i a t i o n o f e q u i l i b r i u m asymmetry under j o i n t a c t i o n o f f a i r l e a d l o c a t i o n and p r o -p e l l e r s i d e f o r c e , t h e l a t t e r r e v e r s i n g i t s s i g n a t t h e zero t h r u s t p o i n t ( h e r e n=8 RPM f o r model c o n d i t i o n and n = l l RPM f o r s h i p c o n d i t i o n ) . h.3 B i f u r c a t i o n L o c i I n S e c t i o n 3.3 we s t a t e d t h a t s t a b i l i t y boundaries i n parameter space are a s s o c i a t e d w i t h q u a l i t a t i v e changes ( s o - c a l l e d b i f u r c a t i o n s ) i n system b e h a v i o r . I t t u r n e d out t h a t a l l bound-a r i e s seen i n F i g . 3 were Hopf b i f u r c bound-a t i o n s . Howe v Howe r , whHowen t h Howe rangHowe o f i n v Howe s t i g a t i o n o f p a r a -meters was s u f f i c i e n t l y extended, saddle-node b i f u r c a t i o n s l e a d i n g t o m u l t i p l e e q u i l i b r i a were a l s o encountered; b u t no p i t c h f o r k s , s i n c e our s h i p i s h y d r o d y n a m i c a l l y asymmetric by v i r t u e o f i t s s i n g l e screw. We demonstrate t h e s e phe-nomena w i t h two sample diagrams: F i g . 1* f o r model c o n d i t i o n and F i g . 5 f o r s h i p c o n d i t i o n .
F i r s t , c o n s i d e r F i g s , l+b and Sb. B a s i c a l l y , t h e y are e n l a r g e d , extended, and separated v e r -s i o n -s o f t h e two -s t a b i l i t y domain-s' a l r e a d y -shown i n F i g . 3 J . But note t h a t f a i r l e a d l o c a t i o n i s now r e p r e s e n t e d on t h e o r d i n a t e by (a l i n e a r s c a l e o f ) l o n g i t u d i n a l c o o r d i n a t e x;^ r a t h e r t h a n t r a n s v e r s e c o o r d i n a t e y^ i n o r d e r t o m a g n i f y t h e r e g i o n s o f i n t e r e s t : 2 0 m <.|y^|<. 23.75 m, compare a l s o F i g . 2 . Moreover, f o r ease o f i n t e r p r e t a t i o n , l o c i o f Hopf b i f u r c a t i o n s ( l o n g -dashed l i n e s ) a r e shown a g a i n s t a backdrop o f t h e a s s o c i a t e d e q u i l i b r i u m m a n i f o l d , i . e . t h e set o f e q u i l i b r i u m s t a t e s XQEyoE'^'E c o r r e s p o n d -i n g t o each p o -i n t -i n parameter p l a n e x^ó. T h -i s i s achieved by p r o j e c t i n g i s o l i n e s o f y^g i n F i g . l)b and, f o r sake o f change, i s o l i n e s o f ipg i n F i g . 5b. I n each graph, observe two c u s p o i d shapes b i t i n g i n t o quadrants two and f o u r ; t h e y enclose r e g i o n s o f t r i p l e e q u i l i b r i a demarcated by saddle-node b i f u r c a t i o n s ( s h o r t - d a s h e d l i n e s ) . A c c o r d i n g l y , t h r e e competing i s o l i n e s o f yQj; ( o r ijfg) e x i s t a t each p o i n t w i t h i n t h e s e i n d e n -t a -t i o n s ; whereas a -t each p o i n -t o u -t s i d e , -t h e e q u i l i b r i u m s t a t e i s unique.
Now, c o n s i d e r F i g s . ka. and 5a. They a r e i n t e n d e d t o r e n d e r t h e s a i d phenomena more p e r -spicuous by r e p l o t t i n g t h e same r e s u l t s i n mixed phase-parameter planes YQE^ ( F i g . ha.) and ii-g^S
( F i g . 5 a ) . P u r s u i n g , from l e f t t o r i g h t , i s o -l i n e s o f t h e r e m a i n i n g parameter xp^, we can c l e a r l y i d e n t i f y t r a n s i t i o n s between s i n g l e and t r i p l e e q u i l i b r i a a t s o - c a l l e d r e v e r s a l p o i n t s . A g a i n , t h e longdashed l i n e s i n d i c a t e Hopf b i -f u r c a t i o n s , here on i s o l i n e s o -f x;^. Moreover, because t h e s u r f a c e r e p r e s e n t i n g t h e e q u i l i b r i u m set i s f o l d e d , we i d e n t i f i e d t h e domain o f Hopf i n s t a b i l i t y by d o t t i n g t h e i s o l i n e s concerned. Observe t h a t each branch o f an i s o l i n e between two r e v e r s a l p o i n t s r e p r e s e n t s a domain o f ( a t l e a s t ) s t a t i c i n s t a b i l i t y , and o f a d d i t i o n a l dy-namic i n s t a b i l i t y when d o t t e d ! A s a l i e n t f e a t u r e h i g h l i g h t e d by F i g s . 1» and 5 i s t h e c l o s e c o r r e l a t i o n between s t a b i l i t y and e q u i l i b r i u m asymmetry, s p e c i a l l y as r e f l e c t e d by h e a d i n g a n g l e : s t a b i l i t y boundaries seem t o hug t h e i s o l i n e s \J)j; = ± 2 ° . F i n a l l y , a p r a c t i c a l l e s s o n t o be l e a r n t from t h e s e graphs i s t h a t , i f s t a b i l i t y i s t o be ensured by a c o m b i n a t i o n o f f a i r l e a d asyrometry and r u d d e r d e f l e c t i o n , t h e most e f f e c t i v e mix i s p o r t f a i r l e a d w i t h s t a r -board r u d d e r ( f o r a r i g h t h a n d e d s i n g l e s c r e w ) . h.h T r a j e c t o r y S i m u l a t i o n s Fourteen l o n g - t e r m t r a j e c t o r y s i m u l a t i o n s , chosen f o r m i s c e l l a n e o u s r e a s o n s , are p r e s e n t e d i n F i g s . 6 t o l i t . F i r s t , c o n s i d e r t h e q u a r t e t o f F i g s . 6 t o 9. Each f i g u r e shows, f o r a p a i r o f cases s e l e c t e d t o i l l u s t r a t e a p a r t i c u l a r p o i n t , t h e f o l l o w i n g i n f o r m a t i o n : l ) t i m e h i s t o r i e s o f c o o r d i n a t e s
^o'yo'^ °^ l i n e t e n s i o n s F^^ over a span o f
1 2 0 m i n , 2 ) t r a j e c t o r i e s o f m i d s h i p p o i n t 0 i n h o r i z o n t a l p l a n e X^YQ over t h e same t i m e span, w i t h schematic deck c o n t o u r s superimposed a t 3 0 min i n t e r v a l s , and 3 ) t a b l e s o f parameter v a l u e s , e q u i l i b r i u m s t a t e s , and e i g e n v a l u e s . F i g s . 6 and 7 ( b o t h f o r model c o n d i t i o n ) a r e r e v i s e d v e r s i o n s o f F i g s . 5 and 7, r e s p e c t i v e l y , o f our
p r e v i o u s paper, c f . J i a n g e t a l . ( 1 9 8 7 ) ; t h e y are i n c l u d e d here t o i l l u s t r a t e t h e two c o r r e c -t i o n s men-tioned a -t -t h e o u -t s e -t . F i g . 6 demon-s t r a t e demon-s t h e i r r e l e v a n c e o f memory e f f e c t demon-s i n n o n l i n e a r motion s i m u l a t i o n s , c o r r o b o r a t i n g and supplementing r e s u l t s o f l i n e a r s t a b i l i t y ana-l y s i s , c f . S e c t i o n h.2. F i g . 7 shows t h a t a s p u r i o u s h i g h f r e q u e n c y o s c i l l a t i o n , p r e v i o u s l y p e r s i s t i n g i n our s i m u l a t i o n s ( i n c o n t r a d i c t i o n t o t h e n e g a t i v e r e a l p a r t o f t h e c o r r e s p o n d i n g e i g e n v a l u e ) , has now d i s a p p e a r e d . I n p a s s i n g , l e t us r e c a l l t h a t t h e o r i g i n a l purpose o f F i g . 7 was t o e x e m p l i f y t h e s t r o n g l y s t a b i l i z i n g e f f e c t o f rudder a p p l i c a t i o n ! F i g s . 8 and 9 r e -p r e s e n t two b o r d e r l i n e cases, i n s -p i r e d by F i g . 3b, f o r purpose o f v a l i d a t i n g t h e a m b i v a l e n t e f f e c t o f s c a l e on s t a b i l i t y ; t r a j e c t o r y simu-l a t i o n s c o n f i r m t h a t , a t S^^, t h e modesimu-l i s s t a b simu-l e and t h e s h i p u n s t a b l e w h i l e , a t Sg, t h e o p p o s i t e h o l d s , as p r e d i c t e d by s t a b i l i t y a n a l y s i s . The p h y s i c a l reason i s t h a t , a t S i , t h e s t a b i l i z i n g e f f e c t o f h i g h e r model r e s i s t a n c e c o e f f i c i e n t p r e v a i l s w h i l e , a t S2, t h e s t a b i l i z i n g e f f e c t of h i g h e r s h i p r u d d e r e f f e c t i v e n e s s p r e v a i l s . N e x t , c o n s i d e r t h e s e t o f F i g s . 10 t o 1 3 . These cases, i n v o l v i n g wind and waves, were s e l e c t e d by t r i a l and e r r o r i n search o f i n t e r -e s t i n g t r a j -e c t o r i -e s . Each s i m u l a t i o n , spanning a p e r i o d o f 15OO m i n , shows: l ) t i m e h i s t o r i e s o f c o o r d i n a t e s XQ,yQ,iii,i)/j^ and o f l i n e t e n s i o n F;^, 2) h o r i z o n t a l t r a c k s o f m i d s h i p p o i n t 0 and o f f a i r l e a d A, and 3) a l t e r n a t i v e t r a j e c t o r i e s i n angle-plane IJ^J^A- U n d e r l y i n g system parameters are documented i n t h e c a p t i o n s . F i g . 10 shows t h e s h i p swaying a p e r i o d i c a l l y between two e q u i -l i b r i a , accompanied by h i g h t e n s i o n peaks i n t h e mooring l i n e , w h i l e F i g . 1 1 shows t h e model ex-e c u t i n g an a p p a r ex-e n t l y p ex-e r i o d i c l i m i t c y c l ex-e undex-er o t h e r w i s e i d e n t i c a l e x t e r n a l c o n d i t i o n s . F i g . 12, a case w i t h s h o r t e n e d mooring l i n e , e x h i b i t s a more e r r a t i c b i s t a b l e w a n d e r i n g , a l s o e n t a i l -i n g h -i g h l -i n e - t e n s -i o n peaks; u n f o r t u n a t e l y , -i t i n v o l v e s o v e r r u n n i n g o f t h e buoy by t h e s h i p . F i g . 1 3 shows t h e s h i p p e r f o r m i n g o b v i o u s l y i r -r e g u l a -r motions w i t h even h i g h e -r peak t e n s i o n s i n " r e g u l a r " waves. We b e l i e v e t h a t , f o r a l l p r a c t i c a l purposes, t h e cases shown i n F i g s . 12 and 13 r e p r e s e n t autonomous and f o r c e d chaos, r e s p e c t i v e l y , a l t h o u g h we have n o t y e t r i g o r -o u s l y pr-oven t h i s u s i n g any -o f t h e c-omputensive t e c h n i q u e s recommended i n t h e l i t e r a t u r e , c f . e.g. Kunick and Steeb ( I 9 8 7 ) .
F i n a l l y , i n F i g . ih a r e j u x t a p o s e d perspec-t i v e views o f f o u r s i m u l a perspec-t e d perspec-t r a j e c perspec-t o r i e s i n t h r e e - d i m e n s i o n a l s t a t e space; t h e y m a n i f e s t s t r i k i n g l y d i f f e r e n t ways an SPM s h i p can behave under r e a l i s t i c c o n d i t i o n s . Graphs a and b ex-e m p l i f y c l a s s i c a l m o t i o n s o f an autonomous sys-tem i n p r o x i m i t y o f a s t a b l e and an u n s t a b l e e q u i l i b r i u m , r e s p e c t i v e l y ; t h e y compare w i t h F i g . 7, except t h a t s h i p c o n d i t i o n r a t h e r t h a n model c o n d i t i o n was s i m u l a t e d . Graphs c and d demonstrate d e t e r m i n i s t i c chaos even more con-v i n c i n g l y than c o r r e s p o n d i n g F i g s . 12 and 13. Paraphrasing a famous quote f r o m t h e B i b l e ( P r o -verbs 30: 1 8 - 1 9 ) , f o n d l y i n v o k e d by s c h o l a r s of seakeeping, we propose t h a t t h e way o f a s h i p can be t o o w o n d e r f u l even i n t h e absence o f a random sea!
6 ACKMOWIEDGMENTS
T h i s work was p a r t i a l l y supported by t h e M i n i s t r y o f Research and Technology (BMFT) o f t h e F e d e r a l Republic o f Germany. We are g r a t e f u l t o Ms. Ingeborg Jurschek f o r her d e d i c a t e d a s s i s t -ance i n p r e p a r i n g t h e m a n u s c r i p t .
7 REFERENCES
Clauss, G., Sükan, M., and S c h e l l i n , T.E. (1982): " D r i f t Forces on Compact O f f s h o r e S t r u c -t u r e s i n Regular and I r r e g u l a r Waves," J o u r n a l o f A p p l i e d Ocean Research, V o l . It, No. k,
pp. 208-218.
Cox, J.V. (1982): "Statmoor - a S i n g l e P o i n t Mooring S t a t i c A n a l y s i s Program," Naval C i v i l E n g i n e e r i n g L a b o r a t o r y , San D i e g o , C a l i -f o r n i a , ' Report No. AD - A 119 979.
Guckenheimer, J . and Holmes, P. { 1 9 8 6 ) : N o n l i n e a r O s c i l l a t i o n s , Dynamical Systems, and B i f u r c a t i o n s o f Vector F i e l d s , A p p l i e d Mathema-t i c a l Sciences, V o l . h2, S p r i n g e r - V e r l a g , New York, B e r l i n , H e i d e l b e r g , Tokyo.
Holden, A.V., ed. (1986): Chaos, P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , New J e r s e y .
Holden, A.V. and Muhamad, M.A. ( 1 9 8 6 ) : "A G r a p h i c a l Zoo o f Strange and P e c u l i a r A t t r a c t o r s , " Chaos, P r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n , New J e r s e y , pp.
15-35-J i a n g , T. and S c h e l l i n , T.E. ( 1 9 8 8 ) : "Mo-t i o n P r e d i c "Mo-t i o n o f a S i n g l e P o i n "Mo-t Moored Tanker S u b j e c t e d t o C u r r e n t , Wind and Waves," Proceedi n g s o f t h e 7 t h I n t . O f f s h o r e MechanProceedics and A r c -t i c E n g i n e e r i n g Symp., American S o c i e -t y o f Mech-a n i c Mech-a l Eng., New York, V o l . 2, pp. 317-326.
J i a n g , T., S c h e l l i n , T.E., and Sharma, S.D. ( 1 9 8 7 ) : "Maneuvering S i m u l a t i o n o f a Tanker Moored i n a Steady Current I n c l u d i n g Hydrody-namic Memory E f f e c t s and S t a b i l i t y A n a l y s i s , " Proceedings o f t h e I n t . Conf. on S h i p Manoeuvr-a b i l i t y , RoyManoeuvr-al I n s t i t u t i o n o f NManoeuvr-avManoeuvr-al A r c h i t e c t s , London, V o l . 1 , Paper No. 25.
Kunick, A. and Steeb, W.H. ( 1 9 8 7 ) : Chaos i n Dynamic Systems, W i s s e n s c h a f t s v e r l a g , Mannheim, Wien, Zürich ( i n German).
L a t o r r e , R. (1987): "Scale E f f e c t i n Towed Barge Course S t a b i l i t y T e s t s , " Proceedings o f t h e I n t . Conf. on Ship M a n o e u v r a b i l i t y , Royal I n s t i t u t i o n o f Naval A r c h i t e c t s , London, V o l . 1 , Paper No. 23.
M a r t h i n s e n , T. ( I 9 8 3 ) : " C a l c u l a t i o n o f S l o w l y V a r y i n g D r i f t Forces," J o u r n a l o f A p p l i e d Ocean Research, V o l . 5, No, 3, pp. ikl-lkk.
OCIMF (1977): P r e d i c t i o n o f Wind and C u r r e n t Loads on VLCCs, O i l Companies I n t . Marine Forum, London.
östergaard, C , S c h e l l i n , T.E., and Sükan, M. ( 1 9 7 9 ) : "On S a f e t y o f O f f s h o r e S t r u c t u r e s : Hydrodynamic C a l c u l a t i o n f o r Compact S t r u c t u r e s , " S c h i f f und Hafen, Hamburg, V o l . 3 1 , No. 1 , pp. 71-76 ( i n German).
Oltmann, P. and Sharma, S.D. (198I)): " S i -m u l a t i o n o f Co-mbined Engine and Rudder Maneuvers Using an Improved Model o f H u l l - P r o p e l l e r - R u d d e r I n t e r a c t i o n s , " Proceedings o f t h e 1 5 t h Symp. on Naval Hydrodynamics, N a t i o n a l Academy P r e s s , Washington, D.C, pp. 83-108.
P a p o u l i a s , F.A. (1987): "Dynamic A n a l y s i s
o f Mooring Systems," Ph.D. D i s s e r t a t i o n , Dept. of Naval A r c h i t e c t u r e and Marine Eng., U n i v e r -s i t y o f M i c h i g a n , Ann A r b o r , M i c h i g a n .
Sharma, S.D. ( I 9 8 2 ) : " D r i f t Angle and Yaw Rate Towing Tests i n Four Quadrants - P a r t 2," S c h i f f und Hafen, V o l . 3^+, pp. 219-222 ( i n German)
Wichers, J.E.W: ( 1 9 8 8 ) : "A S i m u l a t i o n Model f o r a S i n g l e P o i n t Moored Tanker," P u b l i c a t i o n No. 797, M a r i t i m e Research I n s t i t u t e Nether-l a n d s , Wageningen. 8 TABLES Table I P r i n c i p a l p a r t i c u l a r s o f s u b j e c t t a n k e r Length between p e r p e n d i c u l a r s 290.000 m Length c f w a t e r l i n e 296.1tl;6 m Beam 1)7.500 m D r a f t f o r w a r d 16.196 m D r a f t a f t 15.961) m Block c o e f f i c i e n t 0.805 LCB fwd o f m i d s h i p s e c t i o n 7.2lt3 m Radius o f g y r a t i o n ( z - a x i s ) 66.360 m L o n g i t u d i n a l wind a t t a c k area 2 770.0 m2 Transverse wind a t t a c k a r e a 1161.0 m2 P r o p e l l e r diameter 7.910 m P i t c h r a t i o 0.7lt5 Number o f blades 5 Screw sense r i g h t h a n d e d Rudder a r e a 73.500 m^ Rated t u r b i n e power 20 608.0 kW Rated t u r b i n e speed 95.0 RPM Table I I Sample g r a d i e n t s f o r s t a b i l i t y a n a l y s i s Parameters: V(i=2m/s L;;u=75m y;^,6,Q=0
3XQ+A CkN] 3YQ+A CkN] 3NQ+A CkNm]
[ms-2] -.13l)E+5 .OOOE+0 .OOOE+0 8v Cms-2] .OOOE+0 -.139E+6 -.252E+7 Sr [ s - 2 ] .OOOE+0 -.I85E+7 -.59IE+9 3u [ms-1] -.175E+3 .132E+2 .270E+lt 3v [ms-1] -.9l)0E+l -.13l)E+l) -.ll)7E+6 3r Cs-1] -.l62E+lt .l)09E+6 -.116E+8 H o Cm] -.7l)6E+2 .llOE+2 .159E+1) 3no Cm] .976E+I -.i409E+l -.593E+3 3X C l ] .157E+1) .l8i)E+4 .172E+6
Table I I I V a r i a t i o n o f system parameters f o r s t a b i l i t y and b i f u r c a t i o n a n a l y s i s Parameter Reference Examined
v a l u e range
\
Cms-1] 2.0 0 t o 1).0 ^A Cm] 1^5 90 P o r t t o 90 Stbd Cm] 0 -23.75 t o +23.75 ^AU Cm] 75 0 t o 1600 n CRPM] (Q=0) -20 t o +20 6 C°] 0 -30 t o +30-30 i 1 1 1 J _ 1 0 l I 1 I 1 0 1 2 [m/s] 1» 0 1 2 [m/s] 1* ( a ) S e c t i o n a t 5 = 0, = 0, L^y = 75 m ( b ) S e c t i o n a t Q = 0, y^ = 0, L^^j = 75 m
23.75
( c ) S e c t i o n a t 6 = 0, Q = 0, L^y = 75 m ( d ) S e c t i o n a t S = 0, Q = 0, y ^ = 0
F i g . 3 S t a b i l i t y domains i n parameter space f o r model c o n d i t i o n ( ) and s h i p c o n d i t i o n ( )
-20 -10 O n [RPM] 20 (e) S e c t i o n a t VQ = 2m/s, yj^=0, L^y = 75 m
[m] 100 50 O
j
1 J 1 1 1 UNSTABLE 11\ «\
V -
V . • s STABLE 1 1 t 1 r 1 -20 -10 O n [RPM] 20 (g) S e c t i o n a t V^ = 2m/s, 6-0, yj^ = O 20 ^A [m] 10 -10 -20 1 1( j
\ /
\ 1 \ l ' \ y ' / / /\y
UNSTABLE STABLE 1 1 1 1 1 1 120 ^A [m] J l l * 0 11*5 lltO 130 120 -20 -10 O n [RPM] 20 ( i ) S e c t i o n a t V(, = 2m/s, 6 = 0 , Ly^y = 75 m 20 6 [°] _ 1 1 i 1 -L^U [m] l 6 0 0 O 1*00 800 ( f ) S e c t i o n a t VQ = 2 n i / s , Q = 0, y^ = O ^AU [m] 100 50 O ( ' 1 r • 1 I j 1 1 l UNSTABLE | \ 1 \ 1 _ \ / l \'
\ \
/ \ \ / \ \ / / \ \ / / y / ,y 1 STABLE 1 1 I I I I -10 O 10 y^ [ m ] 25 (h) S e c t i o n a t Vp = 2m/s, 6 = 0 , Q = 0 -20 -10 O 6 [O] 20 ( j ) S e c t i o n a t V(. = 2m/s, Q = 0 , L^y = 75 m100 I 1 1 1 1 1 \ 1 1 1 1 r
(a) I s o l i n e s o f parameter v i s u a l i z e d i n phase-parameter p l a n e y^gö
(b) I s o l i n e s o f s t a t e v a r i a b l e y^g v i s u a l i z e d i n parameter p l a n e Xp&
F i g . k Two sample s e c t i o n s t h r o u g h e q u i l i b r i u m m a n i f o l d i n phase-parameter space, showing l o c i o f Hopf ( ) and saddle node ( ) b i f u r c a t i o n s
( F i x e d parameters: Vc = 2 m/s, Q = 0, L^u = 75 m; model c o n d i t i o n )
-20 -10 O 10 20 6 [ O ] 30 (a) I s o l i n e s o f parameter v i s u a l i z e d i n phase-parameter p l a n e fji^S
23.75
P o r t
-30 -20 -10 O 10 20 6 [O] 30 (b) I s o l i n e s o f s t a t e v a r i a b l e ijij, v i s u a l i z e d i n parameter p l a n e x^6
F i g . 5 Two sample s e c t i o n s t h r o u g h e q u i l i b r i u m m a n i f o l d i n phase-parameter space, showing l o c i o f Hopf ( ) and s a d d l e node ( ) b i f u r c a t i o n s
( F i x e d parameters: = 2 m/s, Q = 0, L^U = 75 m; s h i p c o n d i t i o n )
V 800 [ k N ] lao 16» 140 120-100 DO 00-40 20 0 O [m] O -50 -100¬ -150¬ -200 -250¬ -300-100 120 t [ m i n ] O [m] COORDINATES OF FAIRLEAD; XA-145.0 M . YA-0.0 M UNSTRETCHED L I N E LENGTH: 75.0 M CURRENT VELOCITY: 1.5 M/S RUDDER ANGLE: 0.0 DEG PROPELLER RATE AT 0-0: 1.1 RPM EQUILIBRIUM POSITION:
X O E — 2 3 0 . 1 M.YOE-7.7 M,PSIE-0.9 DEG MEAN L I N E TENSION: 146 .1 KN
EIGENVALUES
REAL PART IMAG. PART WITHOUT MEMORY 1 -0.1009D-01 O.OOOOD+00 2 -0.9576D-03 O.OOOOD+00 3 -0.4097D-03 0.17210-01 4 -0.4097D-03 -0.17210-01 5 0.4807D-04 0.46510-02 6 0.4B07D-04 -O.4651D-02 WITH MEMORY 1 -0 .40080+00 0 .96950+00 2 -0 ,4008D+00 -0 .96950+00 3 - 0 4253D+00 0 . 8272D+00 4 -0 .42530+00 -0 .82720+00 5 -0 .39800+00 0 .82840+00 6 -0 39800+00 -0 .82840+00 7 -0. 17370+00 0 .48100+00 8 -0 17370+00 -0 4810D+O0 9 -0, 14470+00 0 29850+00 10 -0. 1447D+00 - 0 . 2985D+00 11 -0. 20450+00 0. 36900+00 12 - 0 . Z045D+00 - 0 . 36900+00 13 - 0 . 10080-01 0. OOOOD+00 14 - 0 . 95890-03 0. OOOOD+00 15 - 0 . 40930-03 0. 1721D-01 16 -0. 40930-03 - 0 . 17210-01 17 0. 49530-04 0. 4650D-02 18 0. 4953D-04 - 0 . 46500-02
F i g . Motion s i m u l a t i o n s w i t h •) and w i t h o u t ( ) memory; model c o n d i t i o n
[ k N ] o [m]
[°]
100 t 120 [ m i n ] *• 2' 0¬ - 2 -4 -6^ -8 •'O [m] [m] -200-150-100 - 5 0 O 50 100 150 200 O [m] .'40 : ;'80 \ .'BO ', .100 \ J20 - ./
\
/'
/ \ :
0 ' \ 'ejo 1 I'ojb '; 120 COORDINATES OF FAIRLEAD: XA-145.0 M , YA-0.0 M UNSTRETCHED L I N E LENGTH: 75.0 M CURRENT VELOCITY: 2.0 M/S RUDDER ANGLE: 0.0 DEGPROPELLER RATE AT 0-0: 1.4 RPM EQUILIBRIUM POSITION:
X O E — 2 3 1 . 1 M,Y0E-9.5 M.PSIE-0.9 DEG »CAN L I N E TENSION: 226.4 KN
EIGENVALUES, WITHOUT kCMORY
COORDINATES Of FAIRLEAD: XA-145.0 M , YA-0.0 M
UNSTRETCHED L I N E LENGTH: 75.0 M CURRENT VELOCITY: 2.0 M/S RUDDER ANGLE: -35.0 DEG PROPELLER RATE AT 0=0: 1.4 RPM EQUILIBRIUM POSITION:
X O E — 2 0 9 . 1 M,Y0£-44.2 M.PSIE-5.2 DEG MEAN L I N E TENSION: 356.9 KN
EIGENVALUES. WITHOUT MEMORY
F i g , 7 M o t i o n s i m u l a t i o n s w i t h (•
REAL PART IMAG. PART REAL PART IMAG. PART
1 -0.1374D-01 0.00000+00 1 -0.1194D-01 0.00000+00
2 -0.1258D-O2 O.OOOOO+OO 2 -0.32300-02 O.OOOOD+00
3 -0.5134D-03 0.2037D-01 3 -0.28340-02 0.34460-01 4 -0.5134D-03 -0.20370-01 4 -0.28340-02 -0.34460-01 6 0,2070D-O3 0.56980-02 5 -0.40250-03 0.5174D-02 6 0.20700-03 -0.56980-02 6 -0.4025D-03 -0.5174D-02 - ) and w i t h o u t ( ) r u d d e r a p p l i c a t i o n ; model c o n d i t i o n 5 5 6
