differentiation of constraints

8/3/2019 Differentiation of Constraints

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Jacket launching simulat ion by different iat ion ofconstraints

L . l t A M B R O

D e t n o r s k e Ve r it a s, P O B o x 3 0 0 , 1 3 2 2 H O v i k , N o r w a y

A m e t h o d i s p r e s e n t e d f o r t r e a t i n g m e c h a n i c a l s y s t e m s w i t h a r b i t r a r y h o l o n o m i e c o n s t r a i n t s b yd i f f e r e n t ia t i n g t h e c o n s t r a i n t s t w i c e . T h i s i s u s ed t o c o m p u t e t h e m o t i o n s w h e n a j a c k c t is l a u n c h e df r o m a b a rg e , a n d i s i n g o o d a g r c c m c n t w i t h m o d c l e x p e r i m e n t s . C l o s c d f o r m u l a e f o r t h e M o r i s o na d d c d m a s s a n d d r a g f o r c c s o n c y l i n d e r s m o v i n g t h r o u g h w a t e r a t r e st a r c g iv e n .

I N T R O D U C T I O N

W h e n a j a c k c t i s l a u n c h e d f r o m a b a rg e , t h r c cd y n a m i c a l l y d i s ti n c t s t a g es m u s t b e d i s t i n g u is h e d . T h ef i rs t s tage i s when the jacke t s l ides on t i l e deck of theb a rg e . T h e s e c o n d s t a g e i s w h e n t h e j a c k e t r o t a t e s a b o u tt h e r o c k c r a r m p i v o t a t t h e b a rg e s t e r n a s i t s l id c s d o w nt h e r o c k e r a r m i n t o t h e s e a . T h e t h i r d s t a g e i s w h e n t i l ejacke t has le f t the barge and i s f ree in the sea .

D u r i n g t h e fi rs t t w o s ta g e s h o l o n o m i c c o n s t r a i n t s a c to n t h e j a c k e t - b a rg e s y s t e m , i n t h e f i r st s t a ge , t il ep e r p e n d i c u l a r d i s t a n c e f r o m t h e b a rg e d e c k t o a f i x e dp o i n t i n t h e j a c k e t , s a y t h e c e n t r e o f m a s s , m u s t b ec o n s t a n t . I n t h e s e c o n d s t a g e , th e p r o j e c t i o n o f t h e v e c t o rf r o m t h e r o c k e r a r m p i v o t t o t h e j a c k e t c e n t r e o f m a s s , s a y,o n t o t h e n o r m a l f r o m t h e j a c k e t c e n t r e o f m a s s t o t h ej a c k e t ' s u r f a c e ' in c o n t a c t w i t h t h e b a rg e m u s t b e c o n s t a n t .C o n s t r a i n t f o r c e s o f e q u a l m a g n i t u d e a n d o p p o s i t e s i g n sa c t o n t h e j a c k e t a n d b a rg e , r e s p e c ti v e l y, d u r i n g t h el a u n ch i n g. T h e c o n s t r a i n t f o rc e s a r e u n k n o w n . T h e m e a n st o f i nd t h e m , a l o n g w i t h t h e m o t i o n o f t h e s y s t e m , isp r o v i d e d b y t h e c o n s t r a i n t e q u a t i o n s .

I n t h e c a u s e o f re d u c i n g t h e j a c k e t - b a rg e e q u a t i o n s o fm o t i o n , w h i c h a r c d i f f e r e n t f o r t h e t h r e e s t a g es , a n d t h ec o n s t r a i n t e q u a t i o n s t o a f o r m w h i c h c o u l d b ep r o g r a m m e d o n a c o m p u t e r , a m e t h o d w a s d e v is e d w h i chc a n b e u s e f u l i n o t h e r p r o b l e m s i n a p p l i e d m e c h a n i c sw h c r c t w o ( o r m o r e ) s y s t e m s w i t h m a n y d e g r c c s o ff r e c d o m i n t e r a c t b y c o n s t r a i n t f o r c e s a n d t h e c o n s t r a i n t s

a r e h o l o n o m i c , i.e . g i v e n b y e q u a t i o n s b e t w e e n t h e d c g r c c so f f re e d o m o f t h e t w o s y s t e m s .T h e i d e a i s t o d i f f e r e n t i a t e t h e c o n s t r a i n t e q u a t i o n s

t w i c c w i t h r c s p e c t t o t i m e , g i v in g a d d i t i o n a l ' e q u a t i o n - o f -m o t i o n - l i k e ' e q u a t i o n s . S o m e o f t h e o r i g in a l e q u a t i o n s o fm o t i o n a r e u s e d t o e x p r e s s th e c o n s t r a i n t f o r c e s i n t e r m so f k n o w n f o r c e s o n t h e s y s te m s a n d t h e ( v e lo c i ti e s a n d )a c c e l e r a t i o n s o f t h e s y s te m s . T h e s e e x p r e s s i o n s a r ei n s e r te d i n t o t h e r e m a i n i n g e q u a t i o n s o f m o t i o n . To g e t h e rw i t h t h e t w i c e . d i f f e re n t i a t e d c o n s t r a i n t e q u a t i o n s o n et h e n h a s a s y s t e m o f s e c o n d - o r d e r d i f f e r e nt i a l e q u a t i o n sw h i c h i s f r e e f r o m u n k n o w n c o n s t r a i n t f o r c e s , a n d t h en u m b e r o f e q u a t i o n s i s e q u a l t o t h e n u m b e r o f d e g r e e s o ff r e e d o m o f th e t w o s y s t e m s .

T h e m e t h o d is c x p la i n c d i n t h e n e x t s e c ti o n . T h e r c st o f

t h e p a p e r a p p l i e s t h is i d e a t o d e r iv e e q u a t i o n s o f m o t i o n

f o r j a c k e t l a u n c h i n g i n a f o r m s u i ta b l e f o r p r o g r a m m i n go n i t c o m p u t e r. T h e p r o g r a m i s r e s t r ic t e d t o m o t i o n i n t w odimens ions , and to qu ie t seas (no waves) . In a se r ies ofA p p e n d i c e s f o r m u l a e a r e g iv e n fo r t h e in t c g r a t c d M o r i s o nh y d r o d y n a m i c a d d e d m a s s a n d d r a g f o r c c s o n ac y l i n d r ic a l j a c k e t m e m b e r, t h e b u o y a n c y o f a b a rg e a n dt h e m o m e n t o f i n e r t ia o f a c y l i n d e r a b o u t a n a r b i t r a r i l yor ien ted ax is .

T h e m c t h o d i s d e s c r i b e d h e r e in s o m c w h a t g r c a t c rd e t a i l t h a n n c c d c d f o r t h e l a u n c h a n a l y s i s , w h e r e t h e r e iso n l y o n e s c a l a r c o n s t r a i n t e q u a t i o n , i n t h e t w o c a s e st r e a t e d . B u t t h e m e t h o d h a s a l r e a d y b e e n s u c c e s s f u l l ya p p l i e d t o a n o t h e r p r o b l e m i n o c e a n e n g i n e e r i n g , w h e r et h e c o n s t r a in t s a r e m o r e c o m p l i c a t c d . T h e p r o b l e m i s t o

s i m u l a t e t h e m o t i o n s o f a v e s se l c o n n e c t e d t o a s i n g le o rm u l t i a r t i c u l a t c d m o o r i n g t o w e r b y .'l r i gi d a r m . T h e r e a r et h e n t w o c o n s t r a i n t s : t h e l e n g t h o f t h e a r m m u s t b ec o n s t a n t a n d t h e t w o c o n n e c t i n g p o i n t s o f t h e ri g id a r m ,v e ss e l b o w a n d t o w e r- l o p , a n d t h e v e s se l c c n t r e o f m a s sshould l i e on a s t ra igh t l ine as seen f rom above . Them e t h o d d e s c r i b e d i n t h e n e x t s e c t i o n i s t h e n r e q u i r e d .

M E T l i O D

C o n s i d e r t w o m e c h a n i c a l s y s t e m s w i t h n ~ a n d n 2 d e g r e e so f f r e e d o m X I = ( x I .. . . x , ,) a n d X 2 = ( x..... , ... x ......) . T h e. \ ,' s wil l usua l ly be p os i t ion coo rd in : t tcs o r ang les , bu t cani n p r i n c ip l e b e a n y k i n d o f g e n e r a l i z e d c o o r d i n a t e . T h ee q u a t i o n s o f m o t i o n o f t h e t w o s y s t e m s , w h e n t h e y a r eu n c o u p l e d a r c , i n m a t r i x f o r m :

A j ) ( = F i A 2 ~ ' 2 = 1 " 2 (1 )

wh ere t i l e squ are mat r ice s A~ and A 2 wi ll conta in add edmass tc rms in addi t ion to mass and iner t ia t cnsors . 1"1 a n dl: 2 a r c k n o w n ( g e n e r a li z e d ) f o r c e s a n d m o m c n t s . I n o c e a nengine er ing appl ica t io ns bo th the F ' s and A 's wi ll usua l lyd e p e n d o n t h e i n s t a n t a n e o u s v a l u e s o f t h e d e g r e e s o ff reed om and the cor resp ond ing v e loc i t ies .~ '~ and )~' ,.

I n t r o d u c e n o w n c g e n e r a l i z e d s c a l a r c o n s t r a i n t f o r c e sFr . . . . , /~ ,, ) which ac t on bo th sys tems . Th e f~; 's cant y p i c a l ly b e t h e m a g n i t u d e s o f c o n s t r a i n t f o r c e v e c to r s ,

0 1 4 1 - l 1 8 7 / 8 2 / 0 3 0 1 5 1 - 0 9 S 2 . 0 0

9 1982 CM L Publications Applied Ocean Re.se~o 'ch, 1982, |~f l . 4 , No. 3151


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J a c k e t h m n c h i n g s i m u l a t i o n b y d i J fe r e n ti a t io n o f c o n s t r a in t s : L . l l a m b r o

w h i c h a c t i n o p p o s i t e d i r e c t i o n s o n t h e t w o s y s t c n l s. T h ec o n s t r a i n t f o r c e s c o u p l e t h e t w o s e t s o f e q u a t i o n s o fmot ion (1) :

A I J ( 'I = F t + G t F c A 2 f ( 2 = F 2 + G 2 F r (2 )

w h e r e G 1 (G2)is a 121X n c( 1 1 2 X I Irr e c t a n g u l a r m a t r i xw h o s e e l e m e n t s a r e f u n c t i o n s o f X 1 ( X 2 ). C o r r e s p o n d i n gt o t h e n c c o n s t r a i n t f o r c e s t h e r e m u s t b e nr s c a l a rc o n s t r a i n t e q u a t i o n s i n v o l v i n g t h e d e g r e e s o f f r e e d o m o fb o t h s y s t e m s :

r~(X1,X2) = 0 i = 1 .... nr (3)

F o r d e f i n i t e n e s s a s s u m e n o w t h a t n t > n o A 1 c a n t h e n b eblo cke d into the sub -blo cks ,,l~t ) an d ,'1(i ) of dim ens ion snr n t and (n t - n , ) n l , respec t ive ly, and Gt can be sp l itin to the n~ x n~ squa re ma t r ix G(~ ) and the 0 ) i -n , ) x n~block G~2):

A , =I - A: I 'I c , V c ; " 1LAf , =L sf ) j (4 )

Pro vide d th a t G~t ) i s non-s ingular, the n , f i r st equa t ion s (2)c a n b e s o l v e d f o r t h e c o n s t r a i n t f o r c e s :

Fr = (G ] t)) - t (Atl t)2, -- Ft l )) (s)

wh ere F] 1) den otes the f i rs t n~ e lem ents of the co l um nv e c t o r F t . T h e s e e x p r e s s i o n s f o r t h e f ~ / s a r e i n s e r te d i n t ot h e r e m a i n i n g e q u a t i o n s ( 2) . T h e n c e q u a t i o n s w h i c h h a v eb e e n u s e d i n o b t a i n i n g e q u a t i o n ( 5) a r e t h e n m a d e u p f o rb y d i f f e r e n t ia t i n g e q u a t i o n ( 3) t w i c e w i t h r e s p e c t t o t i m e .T h e r e s u l t in g s y s t e m o f n t + n 2 s c c o n d - o r d e r d i f f e re n t i a le q u a t i o n s , f r e e f r o m u n k n o w n c o n s t r a i n t f o r c e s , is :

(l i I - - I l c

2

I I I~ . I 1 2

{ AIE )-G] 2)(G (tt))- t A(I ) 0

O R O Rn~ OX t OX2

n2 { -G2(G ~ t ) ) -tA(t) / 1 2 ( ' 2k

I

f r e e d o m in t e rm s o f t h e r e m a i n i n g n t + n , - n r b y s o l vi n gt h e c o n s t r a i n t e q u a t i o n s ( 3) . I f t h e c o n s t r a i n t s a r cc o m p l i c a t e d t h i s i s n o t f e a s i b l e . T h e m a s s m a t r i c c s a n df o r c e s d e p e n d s o n t h e d e g r e e s o f f r e e d o m a n d t h e i r f ir std e r i v a t iv e s , a n d t h e c o u p l i n g m a t r i c e s G t a n dG 2 d e p e n don the degre es of f reedom , so even i f i t is poss ib le toe x p r e s s n , d e g r e e s o f f r e e d o m a s f u n c t i o n s o f t h e r e m a i n i n go n e s , t h e i n s e r t i o n o f t h e s e e x p r e s s i o n s i n t o t h e v a r i o u sm a t r i x e l e m e n t s m i g h t b e l a b o r i o u s a n d c o m p l i c a t e d .Wi t h s u c h a p r o c e d u r e t h e r e w o u l d , h o w e v e r, b e nr fe .w e rd i f f e re n t i a l e q u a t i o n s t o s o l v e th a n i n t h e s c h e m ep r o p o s e d h e r e .

T h e m e t h o d d e s c r i b e d h a s t w o a d w m t a g e s f r o m ap r o g r a m m i n g p o i n t o f v i ew. I t c a n b e a p p l i e d t o s y s t e m sw i t h m a n y d e g r e e s o f f r e e d o m a n d c o m p l i c a t e dcons t ra in t s . I t is a lway s poss ib le to d i ffe ren t ia te thec o n s t r a i n t e q u a t i o n s ( 3 ) t w i c e w h e n t h e c o n s t r a i n t s a r eh o l o n o m i c , e v e n i f i t le a d s t o f o r m i d a b l e e x p r e s s i o n s .O n c e t h e s y s t e m o f e q u a t i o n s ( 6) h a s b e e n p r o g r a m m e d , i tc a n b e s o lv e d b y s t a n d a r d n u m e r i c a l t e c h n i q u e s( R u n g e - K u t t a , H a n m f i n g e t c .) , s t a r t i n g w i t h g i v e n in i ti a lv a l u e s f o r X t , X 2 , A 'I a n d X ' 2 . T h e s e c o n d a d w m t a g e o f

t h e s c h e m e i s t h a t t h e n u m e r i c a l a c c u r a c y o f t h e s o l u t i o no b t a i n e d b y t i m e - s t e p i n t e g r a t i o n c a n b e c h e c k e d c l o s e l y,i n e a c h t i m e - s t e p , b y i n s e r t i n g t h e s o l u t i o n i n t o t h e

c o n s t r a i n t e q u a t i o n s ( 3) . T h i s t e s t i s l o s t i f n t + n 2 - n ~e q u a t i o n s o f m o t i o n a r e d e r i v e d b y e l i m i n a t in g t h ec o n s t r a i n t e q u a t i o n s . T h e c o n s t r a i n t f o r c e s c a n a l s o b ef o u n d i n e a c h t i m e - s t e p f r o m e q u a t i o n ( 5 ) .

D E S C R I P T I O N O F J A C K E T A N D B A R G E

B e f o r e th e i d e a s o f t h e p r e v i o u s s e c t i o n c a n b e a p p l i e d t oj a c k e t l a u n c h i n g , a c o o r d i n a t e s y s t e m f o r th e m o t i o n sm u s t b e i n t r o d u c e d .

F ]2) - G]2) (G] ) ) - )F] ' )

n l " ~ n 2 ~ 2 n

_ I t % .

i ~ t O x , O x ~ x i x J

F2 _ G 2(G(II))- t FU)

(6 )

w h e r e R = ( r t. . .. . r , ) a n d F~2) d e n o t e s t h e n t - n c l as te l e m e n t s o f t h e c o l u m n v e c t o r F t . T h e m a t r i x b l o c k sO R / O X ,a n d O R / O Xh a v e e l e m e n t s?q/Ox~.

T h i s a p p r o a c h i s o p p o s i t e t o t h e s p i r it o f c la s s ic a lm e c h a n i c s t . T h e g o a l t h e r e i s t o i n t r o d u c e n t + n 2 - n ,g e n e r a l i z e d c o o r d i n a t e s s u c h t h a t t h e r e s u l t i n g L a g r a n g eo r H a m i l t o n e q u a t i o n s c a n b e s o l v e d a n a l y t i c a l l y f o ra 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 , a n d s u c h t h a t a l l c o n s t r a i n tf o r c e s d i s a p p e a r. T h u s , f o r th e s i m p l e p e n d u l u m , t h ec o n s t r a i n t f o rc e a lo n g t h e p e n d u l u m a p p e a r s w h e nN e w t o n ' s l a w is d e c o m p o s e d h o r i z o n t a l l y a n d v e r t i c a l ly,b u t d i s a p p e a r s i f t h e d e f l e c t i o n a n g le i s c h o s e n a sg e n e r a l i z e d c o o r d i n a t e . B u t f o r s y s t e m s w i t h m a n yd e g r e e s o f f r e e d o m a n d c o m p l i c a t e d c o n s t r a i n t s i t m i g h tn o t a h v a y s b e p o s s i b l e t o f i n d g e n e r a l i z e d c o o r d i n a t e sw h i c h m a k e t h e c o n s t r a i n t f o r c e s d i s a p p e a r f r o m t h e f i n a le q u a t i o n s o f m o t io n .

A n a l t e r n a t i v e p r o c e d u r e i s t o e x p r e s s n r d e g r e e s o f

O n l y t w o - d i m e n s i o n a l m o t i o n i s c o n s i d e r e d i n t h isp a p e r, t h e c e n t r e s o f m a s s o f a c k e t a n d b a rg e a r e a s s u m e dto m ove in a ver t ica l xz .-p lane . This i s t rue i f bo th bodiesa r e s y m m e t r i c a b o u t t h is p l a ne . S o m e ja c k e t s t r u c t u r e sl a c k t h i s s y m m e t r y.

T h e j a c k e t i s c o n s i d e r e d a s c o n s i s t i n g s o l e ly o fc y l i n d e rs . A c y l i n d e r, o r m e m b e r, i s s pe c i fi e d b y e n d - p o i n tn o d e s , d i a m e t e r a n d t h e w a l l t h i c k n e s s .

A j a c k e t c o o r d i n a t e s y s t e m w i t h o r i g i n a t t h e c e n t r e o fmass i s in t roduc ed . T he z -ax is i s a lon g the ver t ic a l o f thej a c k e t , a n d t h e x z - p l a n e d iv i d e s t h e j a c k e t i n t o t w o e q u a lh a l ve s . I f a m e m b e r g o e s f r o m n o d e p o i n t( X t , Y 1 , Z t )ton o d e p o i n t ( X z , Y 2 , Z 2 ) ,t h e r e s h o u l d b e a n e q u a l m e m b e r( s a m e d i a m e t e r , sa m e t h ic k n e ss ) f r o m ( X t , - Y I , Z t ) t o( X 2 , - Y 2 , Z 2 ) .

T h e m o t i o n i s d e s c r i b e d i n a g l o b a l c o o r d i n a t e s y s t e mw h e r e t h e x y - p l a n e i s o n t h e ( c a lm ) s e a s u r f a c e a n d t h e z -a x i s is v e r t i c a ll y u p w a r d s . T h e d e g r e e s o f f re e d o m a r e t h e

15 2 A p p l i ed O cean Re s ea r c h , 1 982 , Vo l. 4 , No . 3


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J a c k e t h n m c h i n g s i m u l a t i o n b y d i f f e re n t i a ti o n o f c o n s t r a i n ts : L . l l a m l n ' oz[

- - C r

PIVO|

F i g u r e 1 . G l o b a l c o o r d i n a t e s y s t e m a m l b a rg e c e n t r e o fI ll t3SS co ord ina te sys tem

p os i t i o n vec to r s ~ ff 'and ~ fo r t he j a ck e t a nd b a rge cen t r e so f m a s s in t he g l oba l c oo r d i na t e sy s t e m, and t he twop i t ch es 0~ and 0b g iv in g t h e o r i en t a t i o ns o f j a cke t an db a rg e . The re a r e s i x deg ree s o f f r e e dom s i nc e ) )=Yb =0 ,always (Fig. 1) .

A un i fo rm ba rge w id th i s a s sum ed . Th e ba rg e gco m c t ryis spec i f ied by the x .- and z .coo rd in a tcs o f a nu m ber of' p ro f il e po in t s ' i n a co o r d i n a t e sy s t em (X a , Za ) whe re t heor ig in i s a t the b arge cent re o f mass a nd the X~-axis i spa r a l l e l t o t he d eck . Th e ba rge s i l houe t t e shou ld bes t ra ight l ines be tween these prof i le poin ts (F ig . 1) .

I f a j a c ke t me m b er e x t en ds b e twe cn two n ode po in t sw i t h coo rd ina t e vec to r s R~ an d R~ in t he j a c ke t c e n t r e o fmas s co o r d ina t e sy s t em (X j ,Ys, ZA ,t h e n a n a r b i t r a r y p o i n ton t h i s member h a s p os i t i o n vec to r

q u ) = , ~ + ~ l ( 0 / ) - ( R ~ + u ( h ' : - R , ~ ) ) w i t h 0~


4/9

J a c k e t l a u n c h i n g s i m u l a t i o n b y d i f f e r e n t ia t i o n o f c o n s t r a in t s : L . H a m b r o

F ad b J . . J - - J "" , t dA ~ . x ~ + A = z j + A = o O I + B : + F ~ . = + F b . (15)

F~a~- s "" ~ : ~ "" + B ~ + F ~ , (16). :, - A :: xb + A~,=,b + A~oOb

F'~a ~- ~ .. a .. A ~ "". ~ . X b + A . . a ~ + .o O b+ B~ . + F a + F ~ . (17)

w h e r e t h e s y m m e t r i c a d d e d m a s s m a t r i c e s f o r j a c k e t a n db a rg e a r e :

J i B bA 1 = Z C a a r a , a _, . . A, . . - C o ~ a , . . ( m , n =x , z a n d 0 ) ( I 8 )j b

T h e s u m o v e r j ( b ) e x t e n d s o v e r a l l j a c k e t m e m b e r s ( b a rg esec t ions) which a re p ar t ia l ly or f l tl ly subm erge d in the sea .

a ~, , a n d a b , a r c g i v e n b y e q u a t i o n ( A2 ). T h e a d d e d m a s sc o e ffi c i en t C~ m a y v a r y f r o m m e m b e r t o m e m b e r, b u t C a~iSt h e s a m e f o r a l l b a rg e s e c t i o n s . Wi t h s i m i l a r n o t a t i o n :

B , . - ~ C , , b . , B , , , - C . ~ b , ., (m = x , zor 0) (19)j

F a _ ~ - ' t - . j r a a _ aJoin - / ~ ~ " d J j . m F B.. , - C a ~ f/,~,. ( 20 )j b

w h e r e b ~ a n d b bare g iven by eq ua t i on (A3 ) an d fj.a ,,, an dft~a.,, by (B4 ). F in al ly ,

b b b bF~== ~ f j . = F ,= = ~ f ~ = ( 2 1 )

j b

w h e r e t h e b u o y a n c i e s o f i n d i v id u a l j a c k e t m e m b e r s a n dbarge sec t ions a rc g iv.cn by equ a t ion s (CI) a nd (C2) .

T h e e q u a t i o n f o r 0 b i s o b t a i n e d b y t a k i n g m o m e n t sa b o u t t h e j a c k e t - b a rg e c o m m o n c e n t r e o f m a ss . I n t h es l id i n g p h a s e j a c k e t a n d b a rg e r o t a t e a s o n e b o d y - - abod y wi th a vary in g mom ent of iner t ia 1 ,o .:

( 22 )( 1 , o ~0 = I , o ,O b+ i , o ,b o= . / ~d t

w h e r e J i s t h e m o m e n t o f a ll t h e e x te r n a l fo r c e s o n b o t hj a c k e t a n d b a rg e a b o u t t h e j a c k e t - b a rg e c o m m o n c e n t r eo f m a s s . G r a v i t y d o e s n o t c o n t r i b u t e t o J . T h e t o t a lm o m e n t o f i n e r ti a a b o u t a n a x i s p a r a l le l I o t h e y - a x ist h r o u g h t h e b a rg e - j a c k e t c o m m o n c e n t r e o f m a s s i s:

F~ .o + Fn.o + ( ro~X + F0o x je~,~), (24)

I ~ ' a d b and ~ ' a d bw h e r e t h e m o m e n t s - J .0 - ~.0 a r e g i v e n b y e x p a n s i o n ss i m i l a r t o e q u a t i o n s ( 1 5 ) , ( 1 7 ) a n d ( 18 ) a n d b y t h e f o r m u l a e(A4), (A5), (B6), (CI) and (C2).

F 'o j a n d F o~ a r e t h e v e c t o r s f r o m t h e c o m m o n c e n t r e o fm a s s t o t h e j a c k e t a n d b a rg e i n d i v i d u a l c c n t r e s o f m a s s :

_ t n j- - ' % ( F ~ - r ~ ) % . = , ,- ~F 0~ ( 25 )r o J I n j + I l l b

In the s l id ing phase the cons t ra in t con di t ion s ( I 0 ) g ives :

(x - xb)sin0~+ ( z ~ - Z b ) C O s O b = c o n s t a n t = n ~ ,~ + Z , ~

(26)

T h e h o r i z o n t a l c o m p o n e n t o f t h e e q u a t i o n o f m o t i o n ( I 3 )f o r t h e j a c k e t c e n t r e o f m a s s i s u s e d t o e l im i n a t e t h ec o n s t r a i n t f o r c e m a g n i t u d e f . T h e s y s te m o f c o n s t r a i n t -f re e e q u a t i o n s o f m o t i o n , c o r r e s p o n d i n g t o e q u a t i o n ( 6 ) is :

M a.~,~ - F a ( 2 7 )

w h e r e M ~t is a 5 x 5 ma trix , ~ 'a =(~,~j,.~b,~b,0b) an d

F s l ~ - .

bB - + F ~ . +f ~ : - l l t i n 0 o -p ( B ~ + F a , +I V c o s 0 ~ ) -m j g

B~ . F a + B~ + aT ~ . ~ F a , x

B ~ - '- F a " F ~ . + lVsinOb + p(BJ~ + Fa~ ~+ IVcos0b)-- ,nd

B g + F ~ o + b i ~ b . j9 F ~. , + Bo + Fs.o + F.to + k z(B x + Fi. , )

. k 2 ( B ~ + F ~ .,) - k 3 ( a : J + 1 5 . + bj : )

+ ka (B ~ + FaB .: + F~ . , ) -2 mdnb / )b ( (x . i- - xb ) (k j - Xb)

+ (Z~ -- zb)(~.j -- -d )/(m b + m~)

02(( .x-j -- X b)Sin O b + ( : j - - :b)COSOb)

- 20b((.i'~ -.'~b)cOSOb - ( ~ j - -b)sin Oh)

w h e r e g is t h e a c c e l e r a t i o n o f g r a v i t y a n d

I l l f l l l b! tot = ij + ] b + m j + m b( ( ' \ ' b - - X j ) 2 + ( Z b _7.j)2) ( 23 )

T h e f o r m u l a f o r t h e m o m e n t o f i n e r ti a o f a n i n d i v id u a lj a c k e t m e m b e r is g i v e n in A p p e n d i x D ; t h is f o r m u l a ( D I )s u m m e d o v e r t h e j a c k e t g i v en ! r T h e b a rg e m o m e n t o finer t ia lb mus t be spec if ied .

p = (1 + p t a n O b ) / ( t a n O b - I t )

k i = m b(z j- Z b ) /( m b + m f l

k 3 = m b ( x i - % ) / 0 % + m f l

k 2 = n l j k l / n Z b

ka. = mjk 3 /m b

T h e t r a n s p o s e o f th e m a t r i x M a i s:

( 29 )

M r =

-. ,1~:- p( % - ,1~,) nb - , 1 ~ , p ( m ~ -A~,) - A~ e- k ,A~, + k3~: sin0~,

mj- , . l~ r+pA~: -A~: -pA~: -At :o -k tA~ ,+k3A~: : cos0b

0 m ~ - A n, - A ~ : -A,e+k2A,S. s _ L 4 . . I ~ : - s i n 0 b

0 - A L % - A : ~ = -A~.o+k2, tL-~, t"== -r

J J- A : o + p A , o B J- - A ~ o - A ~ o - A . " . o - p A ~ oI

,t B . J 9 8 I/ t o t - A O a - - A o o - k l A ~ o +k2A~o + k 3A~o + k.*A~oI

I x , - . % ) c o s 0 b - ( : i -:blsinOb

( 30 )

15 4 A p p l i e d O c e a n R e s e a r c h , 1 9 8 2 , Vol. 4 . N o . 3


5/9

J a c k e t l a u n c h i n g s i m t d a t io n b y d i f fe r e n t i a ti o n o f c o n s t r a i n t s : L . l l a m b r o

W h e n t h e e q u a t i o n s o f m o t i o n a r e s o l v e d , t h e c o n s t r a i n tf o r c e m a g n i t u d e i s f o u n d f r o m

f= (( m i_ A ~ ,) .~ j_ .~ .. .I "" s F e= : z i - A = o O ~ - B = - ~ .= --

H~:osOb)/(sinO~ -- pcosOb)

T h e f o r m u l a t i o n o f a d y n a m i c c r i t e r i o n o f t h e s t a r t o f th ero ta t ing s tage i s made d i ff icu l t by the fac t tha t the rocker

arm p ivot i s no t a t res t in the g loba l sys tem , nor the cen t reof mass of any o f th e tw o bodies ; i t i s no t e ven a t res t in thej a c k e t . N e c e s s a r y, b u t n o t s u f fi c ie n t c o n d i t i o n s f o r t h es t a r t o f t h e r o t a t i o n s t a g e a r e : (I ) th e j a c k e t c e n t r e o f m a s smust be to the r igh t o f the rock er a rm p ivot , and (2) thes u m o f m o m e n t s o n t h e j a c k e t a b o u t t h e r o c k e r a r m p i v otmust be pos i t ive . I f bo th of these cond i t ions a re sa t i s f ied ,t h e e q u a t i o n s o f m o t i o n f o r th e r o t a t i n g s t a g e a r ei n t e g r a t e d o n e t i m e - s t e p w i t h t h e l as t o b t a i n e d v a l u e s o fpos i t ions , p i tch and ve loc i t ies as in i t ia l va lues . The

w h e r e f i l ( O ) = d f l ( O ) / d O . D e f in e O = O j - t r t - c ~ a n d

s ~ = s i n O - t t cosO s 2 =c osO + psinO

s 3 = s2 (x t , - x~ + X , , ~cosO~ + Z, oa sin 0t,) +

S~(Zb-- Z - - z ,oas inOb +Z,oacos0~)

s 4 ~ " s 2 ( X r o c k C O S O b "] -Z,, ,asin0b) + st( -X , , a s i n O t , +Z,o~tcos0~)

( 3 4 )t h e n

q l = s 2 / s l q 2=S3/SI q3= s.ffs~ (35)

Th e coeff ic ien t o f f r ic t ion in these form ulae i s no tnecessar i ly the sam e as the f r ic t ion coeff ic ien t in thep r e v i o u s s e c t i o n , w h i c h w a s t h e c o e f fi c i en t b e t w e e n b a rg ed e c k a n d j a c k e t , w h e r e a s t h e c o e f f i c ie n t h e r e i s b e t w c e nr o c k e r a r m a n d j a c k e t .

M,o=

-A~:-qx(mi-A~A m s - , I ~ + q t . , l J , : -A~:o+qlA~o

- A ~ o - q ~ ( m j - , l { , ) - A ~ s + q ~ A ~: I ~ - A~oo+ q2A~o 0 0 0

, , ~ - , ~ ;~ - . t ' . , - , r , , , o - A L - A ~ . - ~ o

,n( , ,~ - , -t .' .) - , n a . ' : - , n , t ' . o - M : , , , , - , t L - A ~ o

qsA{:q . d m j - A ~ ) qsA~0 - A ~ - , t ~ l . - Ao~ ,

i~ to l ~ to l- ( 1 , , , d , - - ( n j , c t ) :

r a n s i t i o n t o t h e r o t a t i n g s t a g e i s a s s u m e d t o h a v e t a k e n~lace i f the d i ffc rence be twe en jack e t and barge p i tch

angles has increased . A decrease in the p i tch d i ffe rence ,w h i c h m a y h a p p e n e v e n i f c o n d i t i o n s ( 1) a n d ( 2 ) a r e

sa t i s fied , impl ies tha t the s l id ing equ a t io ns sh ould s t il l bcu s e d. A d e c r e a s e i n p i t c h d i f f e rc n c e m e a n s t h a t t h e ja c k e tr o t a t e s t h r o u g h t h e b a rg e d e c k .

( ~ - r ~ + , 0 ( o ~ ) . P, . . O . ( R ( o ~ ) . E , c O

w h e r e

E Q U AT I O N S O F M O T I O N : R O T AT I N G A N DF R E E

In the ro ta t ing s tage 0 ./ and0 b a r e i n d e p e n d e n t a n dm o m e n t s m u s t b e t a k e n a b o u t t h e t w o c c n t r e s o f m a s s .C o n s t r a i n t a n d f r i c ti o n a l f o rc e s w il l c o n t r i b u t e t o t h e s e

~ t o l(n , ,d : ' ( R i o , ) . ~ , , . , ) . ( ,f ~ ( o , ) . ,L , o

=jack

(36)

m o m e n t s . T h e c o n s t r a i n t - f r e e e q u a t i o n s o f m o t i o n ,c o r r e s p o n d i n g t o e q u a t i o n ( 6 ) , a r e i n t h i s c a s e :

M , o } ( , o = F , o (32)

w h e r e X , o- (Xi ,-i , Oj,xb,-b,Ob)a n d

B [ + F ~ . : + F ~ . : - q , ( B ~ +F ~ . , ) -, n i l

Fro

B ~ + , s J + F ~ J. i .o-- q2(B ,, .

B d J FaB x + F s . , + B x + .n ,

It, "d b l dB: + I"e .: + F 8= + q t (Bx + F~.)- ruby

B ~ + F " t , J .IL0 + F t L o - - q s ( B , + F y ~ )

O ] , t : . , - 2 O A ; . ~- ; i + 0 ~ R ( o ~ ) 9 E , , ~ , ) - ( R ( o i ) . , z ,, c ,)

_ + 0 ~ ( ,s 9~ , ~ k ) ' ( R ( O i ) " , ~ o ~ )

(33)

T h e r c s u h s f r o m t h e r o t a t i n g s t a t e s h o u l d b e c h e c k e d t os e e th a t t h e d if f e r e n c e b e t w e e n ja c k e t a n d b a rg e p i t c ha n g l e s i s g r e a t e r t h a n t h e c o n s t a n t d i f f e re n c e o f t h e f i rs ts t a g e , t h r o u g h o u t t h e r o t a t i o n s t a ge . T h e c o n s t r a i n t f o r c efc i s aga in g iven by equa t ion (31) .

W h e n t h e t r a i li n g n o d e ( s ) o f t h e j a c k e t p a s s e s t h e r o c k e ra r m p i v o t , t h e t w o a r e n o l o n g e r in c o n t a c t a n d t h e s e c o n ds tage of the launc hing i s over.

T h e r o t a t i o n e q u a t i o n s ( 3 2 ) w e r c d e r i v c d b y t a k i n gm o m e n t s a r o u n d t h e i n d i v i d u a l b a rg e a n d j a c k e t c e n t r e sof mass , bu t t i l e s l id ing equa t io ns (27) were fou nd by us ingt h e c o m m o n j a c k e t - b a rg e c e n t r e o f m a s s i n o r d e r t o a v o idm o m e n t s o f t h e s t a ti c a ll y u n d e t e r m i n e d f o rc e s a l o n gt h e d e c k . H e n c e c q u a t i o n ( 2 7 ) c a n n o t b e o b t a i n e d

f r om e q u a t i o n (32 ) b y i m p o si n g t h e c o n s t r a i n t0 = 0 b + c o n s t a n t , a l o n g w i t h s u i ta b l e c o n s t r a i n tm o m e n t s .

In the th i rd phase (bo th bodies f ree in the sea) thee q u a t i o n s o f m o t i o n f o r j a c k e t a n d b a rg e d e c o u p le , a n dt h e r e i s n o l o n g e r a n y c o n s t r a i n t . F o r t h e j a c k e t , t h ee q u a t i o n s o f m o t i o n a r c:

-A ' ,= ,, ,i -A ' : - . -A ' :o [ I i l=- A ' , o - A : 0 I ,- A 0 '0 1 L 0 d /

(37)

i n m a t r i x f o r m . T h e e q u a t i o n s o f m o t i o n f o r th e b a rg e a r ej u s t t h e s a m e , w i t h i n d i c e s b a n d B i n s t e a d o f j a n d d .

A p p l i e d O c e a n R e s e a r c h , 1 9 8 2 , Vol . 4 , N o . 3155


6/9

Jac ket latmchiny simulation by differentiation o f constraints: L . Hambro

I t w a s n e c e s s a r y t o i n t r o d u c e p o t e n t i a l d a m p i n g o n t h eb a rg e w h e n i t m o v e s o n i ts o w n , w i t h o u t t h e j a c k e t , t op r e v e n t u n p h y s i c a l o s c i l la t i o n s , i.e . s u rg e a n d h e a v e f o r c e sp r o p o r t i o n a l t o - x t , a n d - z b , r e s p e c t iv e l y a n d a p i tc hm o m e n t p r o p o r t i o n a l t o - 0 b .

S O I , U T I O N A N D E X A M P L E

A c t u a l l a u n c h i n g o p e r a t i o n s m o s t o f t e n s t a r t b y l e t t i n gw a t e r i n t o t h e b a rg e b a l l a s t t a n k s i n o r d e r t o t r i m t h eb a rg e t o a n a n g l e w h o s e t a n g e n t i s a b o u t e q u a l t o t h ec o e ff ic i e n t o f s l id i n g f r i c ti o n b e t w c e n j a c k e t a n d b a rg ed c c k . T h e b a l l a s t t a n k s m u s t b e f i ll e d s l o w l y t o a v o i d p i t c ho s c i l la t i o n s . S in c e s t a t ic f r i c t i o n is g r e a t e r t h a n d y n a m i cf r ic t ion , the jack e t wi l l s t il l be a t rcs t on th e ba rge dcc ka f t e r t h e t r i m m i n g . T h e w i n c h i s t h e n u s e d t o h a u l t h ej a c k e t a l o n g t h e b a rg e d e c k . T h i s i s p r i m a r i l y t o s e t t h ej a c k e t i n m o t i o n , i .e . t o o v e r c o m e t h e s t a t i c f r ic t i o n , b u tt h e b a rg e t r i m w i l l a l s o i n c r e a s e w h e n t h e j a c k e t i s p u s h e da l o n g t h e d e c k . O n c e t h e j a c k e t is in m o t i o n , t h e c o m p l e t el a u n c h c a n p r o c e e d w i t h o u t w i n c h a s s i s ta n c e . T h e j a c k e ts l ides a long the deck , whose p i tch wi l l increase , un t i l i ts t a r t s t o t i l t a r o u n d t h e s t e r n . T h e r o c k e r a r m s a r em o u n t e d o n p i v o t s a t t h e s t e r n e d g e o f t h e d e c k . T h e i rf u n c t i o n i s t o d i s t r i b u t e t h e l o a d o n t h e j a c k e t d u r i n g t h er o t a t i o n s t a g e o f th e l a u n c h i n g . E v e n t u a l l y t h e ja c k e t w i lll o s e c o n t a c t w i t h t h e r o c k e r a r m s a n d d i v e t h r o u g h t h ewater. Af te r a per iod o f osc i l l a t ions , the jack e t wi l l f ina l lycome to res t f loa t ing in the sea .

I n th e c o m p u t e r l a u n c h s i m u l a t i o n s t h e l o c a t i o n o f t h ec e n t r e o f m a s s, m a s s a n d m o m e n t o f in e r ti a a b o u t t h ec e n t r e o f m a s s o f th e b a rg e a f t e r b a l la s t i n g m u s t b espec i f ied , a lon g wi th the dcs i rcd in i t i a l barge t r im . Thep r o g r a m c a l c u l a t e s t h e b a rg e d r a u g h t a n d t h e p o s i t i o n o ft h e j a c k e t o n t h e d e c k a t t h e s ta r t . Tw o c o n d i t i o n s a r eu s e d : b u o y a n c i e s a n d w e i g h ts s h o u l d b a l a n c e a n d t h e s u mo f m o m e n t s o n b o t h b o d i e s a b o u t t h e b a rg e c e n t re o f m a s ss h o u l d b e z e r o . A d o u b l e i t e r a t iv e p r o c e d u r e i s u s e d, b o t hb a rg e t r i m a n d d r a u g h t a r e a d j u s t e d t o s a t i s fy th e t w oc o n d i t i o n s . T h i s d e t e r m i n e s t h e i n i t i a l c o n d i t i o n s ,cons is ten t w i th equa t ion (10). Al l ve loc it ies a re zero a t thes ta r t .

I f t h e t r i m is t o o s m a l l t o o v e r c o m e t h e s t a t i c f r i c ti o n ,t h e d y n a m i c s t a g e s o f t h e c o m p u t e r s i m u l a t i o n i s s t a r t e db y s w i t c h i n g o n t h e w i n c h f o r c e. T h e s l i d in g e q u a t i o n s o fm o t i o n ( 2 7 ) a r e a t f i r s t i n t e g r a t e d w i t h t h e s t a n d a r dR u n g e - K u t t a m e t h o d 2.

A f t e r t h r e e t i m e - s t e p s w e s w i t c h to H a m m i n g ' s f o u r t ho r d e r p r e d i c t o r - c o r r c c t o r m e t h o d 3 w h i c h r e q u i rc dk n o w l e d g e o f t h e s o l u t i o n s f o r f o u r t im e s . T h e i n it i a lv a l ue s t o g e t h e r w i t h th e R u n g e - K u t t a s o l u ti o n s f o r th r e et i l n e - s t e p s p r o v i d e s t h e s e . H a m m i n g ' s m e t h o d r e q u i r e so n l y t w o e v a l u a t i o n s o f t h e f or c e s a n d m a s s m a t r i x p e rt i m e s t e p , w h e r e a s R u n g e - K u t t a r e q u i r e s f o ur, f o r t h es a m e n u m e r i c a l a c c u r a c y.

I f t h e w i n c h i s s w i t c h e d o f f o r o n d u r i n g t h e f i rs t s ta g e ,t h e i n t e g r a t i o n i s c o n t i n u e d w i t h t h r e e r o u n d s o f R u n g e -K u t t a b e f o r e s w i t c h i n g b a c k t o H a m m i n g . L i k e w i s e ,s t a g e s t w o , t h r e e a n d f o u r ( r o t a t i o n , j a c k e t f r e e i n se a a n db a rg e f r e e i n se a ) a r c s t a r t e d w i t h R u n g e - K u t t a . I n t h e .t r a n s i t i o n f r o m s t a g e o n e t o t w o , t h e r o t a t i o n e q u a t i o n sa r e i n t e g r a t e d o n e t i m e - s t e p b y R u n g e - K u t t a .

T h e c o m p u t e r p r o g r a m h a s b e e n u se d t o s i m u l a te t h el a u n c h o f t h e N o r t h C o r m o r a n t j a c k e t f r o m a b a rg e a n dc o m p a r e d t o r e s u l t s o f m o d e l e x p e r i m e n t s 4 . T h e b a rg e h a s

m a s s 2 9 , 3 5 0 to n s , l e n g t h 1 8 3 m , w i d t h 4 7 m a n d m o u l d e dd e p t h 11 . 5 m . T h e a d d e d m a s s a n d d r a g c o e f f i c i en t s C ffand Cff for the ba rge in equ a t ion s (19) and (20) were sc t to1 .0 a n d 0 .7 , r e s p e c ti v e l y. T h e j a c k e t w a s m o d e l l e d b y 6 5 7members and 430 nodes . I t s overa l l he igh t i s 171 m, thebase i s 74 m by 76 m and the top i s 30 m by 76 m. Al lm e m b e r s w e r e g iv e n th e s a m e a d d e d m a s s a n d d r a gcoeff ic ien ts , C~ and C~ in equ a t io ns (19) and (20), 0 .6 and0 .9 , r e s p e c t iv e l y. T h e c a l c u l a t e d i n a s s a n d m o m e n t o fi n e r t i a o f t h e j a c k e t a r e a b o u t 1 0 % s m a l l e r t h a n t h e a c t u a lv a l u e s, p r o b a b l y d u e t o e x t r a m a t e r i a l a t t h e n o d e s w h i c hi s n o t i n c l u d e d i n t h e p r o g r a m m o d e l . T h e a c t u a l v a l u e swere used in the s imula t ion ; the mass i s 22 ,450 tons . Thec o e ff ic i e n ts o f s t a t ic f r i c ti o n o n d e c k , o f m o v i n g f r i c ti o n o nd e c k a n d o f m o v i n g f r i c ti o n o n t h e r o c k e r a r m s w e r e s e t to0 .1, 0 .5 a nd 0 .05, respec t ive ly. Al l these coeff ic ien ts werec h o s e n t o c o n f o r m w i t h c o n d i t i o n s i n th e m o d e le x p e r i m e n t 4 . T h e p r e c is e v a l u e s o f t h e h y d r o d y n a m i cc o e ff ic i e n ts a r e n o t i m p o r t a n t b e c a u s e t h e s i m u l a t i o nr e s u l ts a r e n o t v e r y s e n s i ti v e to t h e m . T h e h y d r o d y n a m i cf o r c e s w e r e o f t h e o r d e r o f 1 0 ~ o f t h e b u o y a n c i e s , o r l es s.

T h e b a rg e t r i m w a s s e t t o 0 . 0 5 ( r a d i a n s ) , e q u a l t o t h e

c o e ff ic i e n t o f d y n a m i c f r i c t io n . T h e l a u n c h i n g t h e np r o c e e d s s m o o t h l y, w i t h o u t a n y t e n d e n c y t o o s c i l l a t e .C o m p u t e d a n d m o d e l r e s u lt s fo r n in e l a u n c h p a r a m e t e r sa r e g i v e n in Ta b l e 1. T h e j a c k c t m u s t b e h a u l e d t h e' D i s t a n c e t o s e l f -l a un c h .' a l o n g i h e b a rg e d e c k t o i n c r e a s et h e ( e q u i l i b r i u m ) t r i m f r o m 0 . 0 4 t o 0 . 0 5 . T h e r o c k e rr e a c t i o n i s t h e c o n s t r a i n t f o r c e o n t h e j a c k e t d u r i n g t h er o t a t i n g s t a g e , it is d i s t r ib u t c d a l o n g p a r t s o f t h e j a c k e te d g e s b y t h e r o c k e r a r m s . S t a r t i n g f r o m t h e 0 . 0 4 t r i mc o n f i g u r a t i o n , t h e j a c k e t m o v e s t h e ' D i s t a n c e t o r o t a t i o n 'a l o n g t h e b a rg e d e c k b e f o r e r o t a t i o n s t a r t s , a n d t h e ' To t a ld i s t a n c e ' b e f o r e i t l e a ve s t h e b a rg e . I t t a k e s t h e ' Ti m e t or o t a t i o n ' f r o m s e l f - l a u n c h t o t h e s t a r t o f th e r o t a t i o n s t a g e .T h e m a x i m u m j a c k e t S u b m e rg e n c e is t h e l o w e st d e p t h

r e a c h e d b y a n y p o i n t o f t he j a c k e t . T h e m a x i m u m b a rg es u b m e rg e n c e , b e n e a t h t h e w a t e r l i ne , o c c u r s ju s t w h e n t h ej a c k e t l e a v e s th e b a rg e . T h e j a c k e t r e a c h e s i t s l arg e s t p i t c ha n g l e d u r i n g t h e r o t a t i o n s t a g e .

T h e d i f f e r e n c e b e t w e e n c a l c u l a t e d a n d m o d e l v a l u e s f o rt h e d i s t a n c e t o s e l f - l a u n c h i s p r o b a b l y d u e t o i n a c c u r a c yi n t h e m o d e l f r i c ti o n c o e f fi c ie n t . T h e h t rge d i s c r e p a n c y i nt h e t i m e t o r o t a t i o n i s d i s t u r b i n g . T h e v a h l e f r o m t h em o d e l t es t s s e e m s r a t h e r s m a l l a n d m a y b e c a u s e d b ys c a l i n g e r r o r s . B u t t h e d i s c r e p a n c y c o u l d a l s o i n d i c a t et h a t t h e c r i t e r i a u s e d t o d e c i d e w h e n t h e t r a n s i t i o n f r o mt h e s l i d i n g t o t h e r o t a t i o n s t a g e t a k e s p l a c e a r einsuff ic ien t , o r tha t the s l id ing cqua t ions (27) a re

Table 1. Laun ch parameters

Computed Model test

Dis tanc e to Self-launch m) 8.2 11.9Rock er reaction at start of rotation (t) 11925 12990Dista nce to rota tion (m) 103.8 103.0Time to rotation( s ) . 46 26Rocker reaction when acket leavesbarge(t) 5960 5300To tal dista nce (m) 151.5 148Max. jacket subme rgence m) 89.9 80Max. barge subme rgence m) 17.12 18.5Max. jacke ! piteh.(degr.) 13 13

156 Applied Ocean Research, 1982, Vol. 4, No. 3


7/9

J a c k e t h m n c h i n y . s im u l a t i o n b ) " , l iJ f i ,r c n l i a ti o n o f c o n . st r a i n ts : L . l l a m l , r o

i n c o m p a t i b l c w i t h t h e r o t a t i o n e q u a t i o n s ( 3 21 c l o s e t o t h et r a n s i t i o n . F o r t h e r e m a i n i n g s c v c n l a u n c h p a r a m e t e r s ,t h e a g r e e m e n t b e t w e e n c o m p u t e r a n d m o d e l r c s u l t s i swith in 10',7o. Th is is quite sat is fac tory.

T h e t r a j c c t o r i c s f o r t h e s l i d i n g a n d r o t a t i n g s t a g e ssa t i s fied the c ons t r a in (10) to w i th in 0 .3Y,, in evc ry t ime-s tep .

w h e r e

umj~

f 2 = C L f ( ( F ( u ) - % ) v ; ~ ( u ) ) , d uUmin

f ~ = C . (a o c: ~ o + a o : 5 o + a o f ) + b o )

A C K N O W L E D G E M E N TST h i s w o r k w a s p a r t i a l l y f u n d e d b y t h e R o y a l N o r w e g i a nC o u n c i l f o r S c i e n ti f ic a n d I n d u s t r i a l R e s e a r c h . M r M .L i n d a u p r e p a r c d t h e c o m p u t e r m o d e l f o r t h e N o r t hC o r m o r a n t J a c k e t a n d h e l p e d a n a l y s e t h e r e s u l t s . S h e l lhas k ind ly pcrm i t tcd the use of the i r m odel t es t resu l ts .

aox = a~o ao: = a: , aoo = ( 'af t + cd '2 4 c6f3(A4)

b o = O 2 ( c d ' ~ + o a f 2 ) (A5)

C . t = X ~ + Z f - c ~ L 2 c 5= 2 c3 1 ? c6 = 1.2(1 - c.s)

t.7 = ~Cl(.5 C 8 = t ' l t' 6 L 2

A P P E N D I X A

A d d e d m a s s fi Jr c eA c c o r d i n g to M o r i s o net a l . 5 ,t h e a d d e d m a s s f o r c e o n a

j a c k e t m e m b e r o r b a rg e p r o f i le sc c t i o n is :

f = c ~ f r , ( , ,I l d r( . )1v, c [

( A I )

where F( , , ) i s g iven by equa t ions (7) o r (9) and , for anyvec tor V:

~ '~ = f f - ~ ' ( O " ~ ' ) / L 2,~ = r - ' ( l ) - F ( O ) = h t( O ) '( ff 2- - / ~ , )

a n d L = [R ,2 - R ~ [ . T h e i n t e g r a t i o nis o v e r t h e w e t t e d p a r to f th e j a c k e t m e m b e r o r b a rg e p r o f il e s e c t i on . A v c c t o rc a l c u l a t i o n s h o w s t h a t ,

f 2 = C o ( a . ,, ,; .~o + a ~ : . 5 o + a . , ,o (~ + h .O

J~" = C ,,(a:., ,2 o +a : :5 o + a :oO + b : )

w h e r e (Xo ,zo ,O o)d e n o t e s ( x r - . r O . ) o r (Xb,-b,Ob) a n da , ; , = ( I - e ~ / L : ' ) ] ~ ,a ~ = = a : ~ = - e ~ e = f t / L2 , a : = = ( I - e Z / L 2 ) f I .

a ,o = ( g : -e~c t )J~ + e-J) a:o = - (.qx +e :c t ) f~ - e , f , .(A2)

b . = 02 ( ( _ q : , +e . c 2 ) f t- e x c 3 f 2 )

b. = 02(( - 9 : + e :c2) f l -e : r

c t = ( Z I X z - X I Z 2 ) / L ' - c 2 = ( X 2 t X I + Z 2 t X I ) / L 2

( h 3 )

c ~ = ( ) ~ - ) ; ) / c 2 F = g t ( o ) - ,~ ,/

f . = - - L ~ ( u L , - , 7 ~ ,. )n I I

u . .. . and u ,~ , a re 0 and I for a fu l ly subm erge d jack e tm em ber o r barg c scc t ion , and 0 and u~,,~or m,.~ and 1 for ap a r t l y s u b m e rg e d m c m b c r ; u , ,. ~ = - ( z o +g: ) / e : .

f ~ i s n o t z e r o fo r an i n d i v i d u a l j a c k c t m e m b e r, b u t w h e ns u m m e d o v e r t h e j a c k e t i t g i ve s z e r o b e c a u s e o f th ea s s u m e d j a c k c t s y m m e t r y. T h e p i tc h m o m e n t o f I f " i ne q n . (A I ) ] a b o u t t h e j a c k c t o r b a rg e c e n t r e o f m a s s i s:

T h e c o n s t a n t s C. . . . C d o n o t d e p e n d o n t h e p o s i ti o n s a n dv e l o c it i e s s o t h e y d o n o t h a v e t o b e c o m p u t c d i n e a c ht i m e . s t e p w h e n t h e d i f fe r e n t i a l e q u a t i o n s o f m o t i o n a r es o l v e d n u m e r i c a l l y. B u t f t , f2 a n d f 3 d e p e n d o n t h ei n s t a n t a n e o u s c o o r d i n a te s .

N o t i c e t h a t t h e 3 a d d e d m a ss ( a n d m o m c n t o finer t ia ) mat r ix a , f i j = x , - a n d 0 ) i s s y m m e t r i c . T h i se m e rg e s f r o m t h e a l g c b r a , i t i s n o t o b v i o u s apr io r i .

A I ' P E N D I X B

D r a g f i~ r c eT h e M o r i s o n d r a g f o r m u l a i s :

w h e r e

f~ a= C af ~ ( u ) l ~ , ( u ) lIde ' (u) lw e t

( B I )

,~ ( . ) = , . - ( , , ) -< ( b , . k ( . ) ) l L "

; - ( , ,) = ; .- o + ( ~ 7 , 9 ( o ) ) . ( f i , + ,, ( fi ~ - f i ,) )

S i m p l e c a l c u l a t i o n s h o w s t h a t :

( B 2 )

for any vec to ~ whe n A-I i s g iven by equa t i on (8) and~ ' = M ' ( R 2- R t ) a s in A p p e n d i x A . T h i s r c d u c e s t h e d r a gforce in tegra l to :

umjx

f a = Ca l , f ( ff ._ , ff t , ) (a 2 +2(,u + b 2 u Z ) " 2 d u (B3)

Um m

whc rc the vec tors f i" and f la re :

9 d 9

- , - o +

d ~

A p p l i e d O c e a n R e s e a r c h , 1 9 8 2 , Vo l. 4 , N o . 315 7


8/9

J a c k e t l a t m c h i n 9 s i m z d a t i o n b y d i f f e r e n t i a t i o n o f c o n s t r a i n t s : L . l l a m b r o

b y v i r tu e o f e q u a t i o n (B 2) . T h e c o m p o n e n t s o f 5 a n d f l ar e :

a , , = 5 : o + O ( g : - e , , c t ) - e . ~ c , , a t =- ( Y 2 - Y t ) ( c , . + q O )

a~ = Zo - O(g,~ + e: Q ) - e:c v c+. = (e~,Sc + e: ~o )/L2

b,, = Oe= b~ = 0 b~ = - Oe.~

The in tegra t ion in equa t ion (B3) i s s tandard , the resu l t i s :

p r i sms , a l l wi th the wid thw of the barge . O ne ver tex i s a tt h e b a rg e c e n t r e o f m a s s a n d t h e t w o o t h e r a t b a rg e p r o f i lepoin ts . There a re then e igh t d i ffe ren t cases to d i s t inguish ,d e p e n d i n g o n w h e t h e r o r n o t t h e t h r e e v e r t ic e s a r e a b o v eo r b e l o w t h e s e a s u r f a c e , i. e. w h e t h e r t h e i r z - c o o r d i n a t e i nt h e g l o b a l c o o r d i n a t e s y s t e m i s p o si t i ve o r n e g a t i v e .

T h e b u o y a n c y f o r m u l a e f o r t h e b a rg e a r e :

- "

- -C b X c b Av. r

~ , . o - C ~ = 9 ' ; (C2)

f ~ = C d L ( a f l z n+bf lz 2) f a . . = C a L ( a : h n+b :h 2) (B4)

w h e r e

h, = ~,b - z[(bZu + (~'b))(a ~ + 2(5"~ u + b:,,2)t;2 +

~ (ab 2 - ( a ~ ) 2)x

Th e coeff ic ien t C~ i s the sam e for a ll the t r iangu lar p r i smstha t ma ke the barge , bu t th e coeff ic ien ts C/, in equa t io n( C I ) w i ll b e d i f f e re n t f o r j a c k e t m e m b e r s w i t h u n e q u a ld i a m e t e r s .

I n t r o d u c e t h e b a rg e p r i s m q u a n t i t i e s

x t = g x x 2 = g x + e x z t = g z z 2 = g z + e :

ln(b(a2 + 2 ( i~ff)u+ b2u 2) '12 +bZ zt +( aff))]~minm,~,t( B 5 )

h : , = [ 8 9 ~ ( a 2 + 2 ( / / ' ~ u + b 2 , , 2 ) 3 , 2 2: : ? - ( ~ l , , / b2

F o r a l l j a c k e t m e m b e r s w h i c h a r e p a r a l l e l t o t h e( x z ) p l a n eof mo t ion , and for a l l s t ra igh t - l ine barge prof i le sec t ions ,I,'2 = Y~ so tha t a~. =0 . ~,, 5a nd f la re then vec tors in thex z -p l a n e s o t h a t

a2b 2 _ (g. ff )2 = aZb~ + a~b 2 _ 2a: ,a :b=b== 02(~'-b-}2 = 0

s ince fi" ~ ,4 - b - '(~ . ,4 ) /L whe re ,4 = ~o + ~I ' /~ , . So for theb a rg e a n d f o r m a n y j a c k e t m e m b e r s t h e i n t e g r a l h~ isg iven by the f i r s t t e rm in equa t ion (B5) on ly. Thel o g a r i t h m m u s t t h e n b e l e f t o u t s i n c e i t c a n b eind ete rm ina te, 0 In0, wh en Y2 = Yt-

T h e y - c o m p o n c n t o f th e m o m e n t o f t he d r a g f o r c ea b o u t t h e j a c k e t o r b a rg e c e n t r e o f m a s s i s:

f 0 = C a I ( ( t - (u ) -7 o ) X : ~ ( , , ) ) , . l ~ ( , , ) I l d r - ( , O Iv. c [

U m a x

= C a L f ( ( g + u e ) x (~T + uN) , (a 2 + 2(h '-b' )u +b2 u2) t '2d u

U m i n

= 9 . f~ _ 9 , fd + C dL(e : (a f l z 2 +bflz3 _ eAa._h 2 +b:h3))

(B6)

w h e r e

Z O I : Z b " ~ Z I Z o 2 : Z b . . ~ - Z 2 Z I 2 = - - C z Z 2 1 : e z

ut = 1+ Z b / Z t U2 = I +Zb/Z2 Vl =Zo t /Z12

v2 = Z o o ~ Z 2 , a , = " l x , z 2 -- X2Z,[

A , i s the a rea of the t r iangle wi th ver t ices a t F a nd the tw oprof i le po in ts w i th coord ina t e vec tors R,t and R2 in theb a rg e c e n t r e o f m a s s c o o r d i n a t e s y s t e m .

Th e f orm ula e for , ,I ,,+, an d X,b in the e ight ca ses a re:

S ign of

" b " 0 1 2 0 2 / | w e t X{ b

+ + + 0.2. + _

+ + - - - . o 2 , . 1 J ( . 2 . . 1 2 ) ( (2 Z b + Z 1 - - z b z l / z z ) x z - - Z o z X l ) /(3: ,2 .)

+ - + Sameas abov e, but w ith 1 and 2 interchanged.+ ( I - :~/:,:2.)A, ( ( : , 2.+ :~/z~),c, +

(:,z2+:~lz2)~2)l(3(zlz2- z ~ ) )- + + : ~ : 1 , / ( : , : 2 ) - : d x x / z , + x z / z z ) /3- + - ( I - u t t ' 0 A , ( ( I - 3 u n t " t + u ~ t ' t + u x v ~ ) x x +

( 1 - l i t r 0 x 2 ) , ' ( 3 ( I - l t l t ' l ) )

- - + Sam e as above, but with I and 2 interchanged.- - A , (x t + x l ) /3

A P P E N D I X D

A l o m e n t o f in e r t ia o f a o ' l i n d e r a b o u ta n y a x i sAn inf in i te ly th in ro d f rom r~ to r 2 is descr ibed by:

t -(u) = ~ +u (T - , :~) 0~< u~


9/9

J a c l , c t l a u n c h i m j . ~ im u la ti m ~ h y d i f f c r c n t ia t i o n o f c o n s t r a i n t s : L . H a m h r o

2n

= 89 I R, lO{(x~ + v~(OI)2 + (x , + v~(O))2 + (zt + V,(O))" +

0

(x '~ + vJO)) (x2 +t ,~(0)) + (z 2 + v:(0)) ~ +

)) , 2 - 2 - 2 ;,~..+/,~..)(-\'1 + v..(0))(-x2 + e:(0 ))}= i , ~ + - 2 M R ( e l ~ + e 2 , +

I t e r e L - - I r ~ - r ~ l . L e t /~ b e a n a r b i t r a r y u n i t v e c t o r n o tp a r a l l e l t o r ~ - F ~ . T h e n , q u i t e g e n e r a l l y :

~ , ( 6 - ~ ) , c~ , ~ , = ~ ' ,l i~ ,I ; ~ = ( 6 - r, ) ,c , ; ,, ; , , = ~ / I~ , I

A s h o r t c a l c u l a t i o n s h o w s t h a i :

b ~ , + " ~ - 2 -e2~. + e l : + e~Z.-= I + [.!, ' 2 -- Yl )2/L2

i n d e p e n d e n t o f /5. F o r a c y l i n d e r w i t h w a l l s o f f in i t et h i c k n e s s , i n n e r a n d o u t c r r a d i i R~ a n d R E , m a s s M a n d

a x i s f r o m r t o f' 2 , t h e m o m c n t o f i n e r t i a a b o u t t h e y - a x i sis :

t iC , = ~ M I . ,~ + . ,~ + : f + : ~ + x ,.~ -~ + : , : , +

, ( " ' s ) }d R , + R ~ ) I + ( D I )

R E F E R E N C E S

I Goldstein,t l . Classical Mechamc~,Addison-Wesley. 19722 l la, ,dhuuLoJM,*thematiealFu,wtiuus.lEds. M. Ab ram o~ itza nd I. A

Segun). Do~er, N ew York, 196x. p. I~973 l lamming. R. W.Numerical Methods [or Scientists a,,d En,ji,wcrs.

M cG ta~ .Hill, Kogakusha. 1973, p. 3934 C.J.B - Earl and Wright Limited.Phase I Model Test Report. Part I

- Launchm 9,London, Scplcmbcr 19785 MorJson. J. R., O'Brien. M . P.. Johnson. J. W. and Shaaf. S. A. "lhc

forces exerted by surface x~a~eson piles,Tra,~s. Am . P etrol. In st.1950.189. 28-16

A p p l i e d O c e , m R e s e a r c h , 1 9 8 2 , I 'o l. 4 , N o . 3159

differentiation of constraints

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