R e p o r t No, 336
LABORATORIUM VOOR
SCHEEPSBOUWKUNDE
TECHNISCHE H O G E S C H O O L DELFT
r
SOME ASPECTS OF PREDICTION AND SIMULATION OF MANOEUVRES
by
G. van Leeuwen
L
Page: Summary. 1 I n t r o d u c t i o n . 1 The m a t h e m a t i c a l model. 2 D e t e r m i n a t i o n o f c o e f f i c i e n t s . h Examples o f f u l l - s c a l e d a t a a n a l y s i s . 7 C o n c l u s i o n s and recommendations. 9 R e f e r e n c e s . L i s t o f symbols.
Summary.
P r e d i c t i o n and s i m u l a t i o n a r e b o t h dependent f r o m t h e knowledge o f m a t h e m a t i c a l models.
The c h o i c e and t h e f o r m o f t h e model i s d e t e r m i n e d hy t h e s p e c i f i c a p p l i c a t i o n and t h e a c c u r a c y w a n t e d . M o d e l t e c h n i q u e s , as h o r i z o n t a l o s c i l l a t i o n t e s t s , p r o v i d e t h e way t o f i n d t h e c o e f f i c i e n t s o f e x t e n s i v e n o n - l i n e a r m o d e l s , t h o u g h f o r o t h e r p u r p o s e s , as s i m u l a t i o n , t h e r e i s a l s o need o f s i m p l e r , e m p e r i c a l m o d e l s . To d e t e r m i n e t h e c o e f f i c i e n t s o f such m o d e l s , f r e e r u n n i n g f u l l s c a l e - o r model t e s t s a r e n e c e s s a r y .
Some examples o f t h e d e t e r m i n a t i o n o f t h e c o e f f i c i e n t s o f m a t h e m a t i c a l models a r e d i s c u s s e d , u s i n g f o r c e d m o d e l t e s t s as w e l l as f u l l sca.le t e s t s .
I n t r o d u c t i o n •
P r e d i c t i o n and s i m u l a t i o n a r e t w o c o n c e p t s c l o s e l y a l l i e d , t h e p r i n c i p a l f a c t o r i n common i s t h e m a t h e m a t i c a l model.
As f a r as p r e d i c t i o n i s c o n c e r n e d i t i s v e r y o f t e n i n t h e d e s i g n s t a g e o f a s h i p t h a t we want t o know t h e m a n o e u v r i n g p r o p e r t i e s and s t e e r i n g q u a l i t i e s . U s i n g m o d e l t e s t s i t i s p o s s i b l e i n d e e d t o g e t an i n s i g h t i n t h e m a n o e u v r a b i l i t y o f a s h i p . I f t h e t e s t s e x i s t o f f r e e - r u n n i n g m o d e l t e s t s , t h e S p i r a l Test e.g. i r m n e d i a t e l y shows, w h e t h e r t h e s h i p i s s t a b l e o r n o t . A d i f f e r e n t way i s t o p e r f o r m f o r c e d o s c i l l a t i o n t e s t s . I f we t h e n want a l l t h e i n f o r m a t i o n , p r o v i d e d by a S p i r a l T e s t , f i r s t t h e v a r i o u s c o e f f i c i e n t s have t o be d e t e r m i n e d and t h e n t h e s h i p i s s i m u l a t e d on an a n a l o g o r d i g i t a l computer. On t h e o t h e r hand t h e r e a r e many p r o b l e m s i n t h e f i e l d o f m a n o e u v r i n g , i n w h i c h man p l a y s an i m p o r t a n t r o l e .
To examine t h e man-ship s y s t e m , i t i s n e c e s s a r y t o simula.te t h e s h i p on r e a l - t i m e s c a l e , because human b e h a v i o u r , s t e e r i n g a s h i p - m o d e l , i s s e r i o u s l y a f f l i c t e d w i t h s c a l e e f f e c t s .
For t h i s purpose a " S t e e r i n g S i m u l a t o r " i s u s e d , t h e main p r o p e r t y o f w h i c h i s t h e v i s u a l d i s p l a y , . T h i s s i m u l a t o r i s a p p l i e d f o r t r a i n i n g helmsman and t o examine, w h e t h e r a c e r t a i n manoeuvre can he p e r f o r m e d i n a g i v e n vraterway.
The m a t h e m a t i c a l model.
To make p r e d i c t i o n s , based on f o r c e d m o d e l t e s t s , and t o p u t t h e s t e e r i n g s i m u l a t o r i n a c t i o n as w e l l , we need a m a t h e m a t i c a l model.
The most i m p o r t a n t q u e s t i o n i s t h e n :
What do we e x a c t l y want t o p r e d i c t ? The answer t o t h i s q u e s t i o n m a i n l y d e t e r m i n e s t h e f o r m o f t h e model t o be a p p l i e d .
Assuraing we a r e i n t e r e s t e d i n s t i l l w a t e r manoeuvres i n t h e f i r s t p l a c e , t h e n t h e most c o m p l e t e model e x i s t s o f t h r e e n o n - l i n e a r , c o u p l e d f i r s t - o r d e r d i f f e r e n t i a l e q u a t i o n s , d e s c r i b i n g t h e a n g u l a r v e l o c i t y , t h e d r i f t - v e l o c i t y and t h e f o r w a r d v e l o c i t y r e s p e c t i v e l y . See [ l ] , [ 2 ] e.g.
The terms o f t h i s " e l e m e n t a r y model" r e p r e s e n t components o f t h e f o r c e s -and m o m e n t e q u i l i b r i u m -and t h e c o r r e s p o n d i n g c o e f f i c i e n t s can be d e t e r m i n e d by f o r c e d model t e s t s , m e a s u r i n g t h e f o r c e s on t h e model by means o f
dynamometers.
Such a model i s used f o r a l l p r e d i c t i n g and s i m u l a t i n g p u r p o s e s . The l a r g e number o f t e r m s , hO t o 80 e.g., makes i t o f t e n unmanageable, however, and t h a t ' s
why we have been l o o k i n g f o r s i m p l e r , e m p e r i c a l models. The o n l y p u r p o s e
o f t h e s e models i s t o d e s c r i b e t h e m o t i o n s : y a w i n g , s u r g e i n g and, i f n e c e s s a r y , s w a y i n g , as a c c u r a t e as i s d e s i r e d f o r t h e p u r p o s e c o n c e r n e d .
The s i m p l e s t example o f such a m o d e l , i s t h e l i n e a r m o d e l , p r o p o s e d by Nomoto [ 3 ] : T ^ i + ^ ^ K6 ( 1 ) d t 2 B e s i d e s a l a r g e e d u c a t i o n a l v a l u e , t h i s model has a l s o u s e f u l n e s s f o r a r o u g h d e s c r i p t i o n o f t h e m a n o e u v r i n g p r o p e r t i e s o f a s h i p ; i t d e s c r i b e s t h e a n g u l a r v e l o c i t y as a f u n c t i o n o f the, r u d d e r a c t i o n o n l y , however.
I f h i g h e r demands a r e made upon t h e m o d e l , a l s o t h e n o n - l i n e a r e f f e c t s and speed r e d u c t i o n s h o u l d be d e s c r i b e d .
3
-A g r e a t d e a l o f t h e n o n - l i n e a r b e h a v i o u r o f a. c o u r s e - s t a b l e s h i p and,
a t t h e sarae t i m e t h e change o f f o r w a r d speed d u r i n g a manoeuvre, a r e
d e s c r i b e d by t h e model:
" ds-2 ^
T " . ^ + u - = K ^ r - 2 (2^^) u ds u
where u * i s t h e r e l a t i v e s p e e d r e d u c t i o n and s**" t h e d i s t a n c e , covered by
t h e s h i p i n s h i p l e n g t h s [h].
T h i s model i s based upon t h e f a c t t h a t , n e a r l y f o r a l l s h i p s , t h e r e l a t i o n
-s h i p between ^ and t h e r u d d e r a c t i o n i -s c o n -s i d e r a b l y -s i m p l e r - more l i n e a r
as t h a t between t h e a n g u l a r v e l o c i t y ~ and t h e r u d d e r a n g l e , w h i l e t h e
f o r w a r d speed e q u a t i o n i s based upon t h e f a c t , t h a t speed r e d u c t i o n i s m a i n l y
due t o t h e l o n g i t u d i n a l component o f t h e c e n t r i f u g a l f o r c e , wh:
a p p r o x i m a t i o n , can be c o n s i d e r e d p r o p o r t i o n a l t o t h e square o f
due t o t h e l o n g i t u d i n a l component o f t h e c e n t r i f u g a l f o r c e , w h i c h , as a good É i
ds *
Most o f t h e e m p e r i c a l models g i v e no d e s c r i p t i o n o f t h e d r i f t , because d u r i n ,
a non-steady m o t i o n , l i k e a z i g - z a g t e s t , t h i s v a r i a b l e i s n e a r l y i n phase
w i t h t h e a n g u l a r v e l o c i t y , so t h a t as a good a p p r o x i m a t i o n , terms w h i c h
depend on b o t h a n g u l a r - and d r i f t v e l o c i t y can be r e p l a c e d by o n l y a n g u l a r
v e l o c i t y dependent t e r m s . A n o t h e r r e a s o n why a d r i f t e q u a t i o n i s w a n t i n g , i s
t h a t t h e d r i f t i s o f secondary i m p o r t a n c e w i t h r e s p e c t t o t h e c o o r d i n a t e s '
o f t h e s h i p ' s p a t h .
The b e h a v i o u r o f c o u r s e - u n s t a b l e s h i p s cannot p o s s i b l y be d e s c r i b e d w i t h o u t
n o n - l i n e a r t e r m s . The w e l l - k n o w n s~shaped c u r v e , w h i c h i s t h e r e s u l t o f a
S p i r a l Test w i t h a such l i k e s h i p , can s i m p l y be a p p r o x i m a t e d by a c u b i c
p o l y n o m i a l , so t h a t t h e y a w - e q u a t i o n c o u l d be:
T'r' + r ' + 0.3' r ' ^ = K'ö ( 3 )
T h i s e q u a t i o n m i g h t be c o n s i d e r e d an e x t e n s i o n o f Nomoto's l i n e a r model
and was a p p l i e d by N o r r b i n [ 5 ] . For u n s t a b l e s h i p s t h e second e q u a t i o n
m e n t i o n e d ( 2 ^ ) becomes:
m which r = L.-r- . ds
R e l a t e d t o t h e s e n o n - l i n e a r e q u a t i o n s i s t h e second o r d e r e q u a t i o n :
T^jT^ r + (Ti+Tg)!- + H ( r ) = KS + KT^Ö ( 5 )
used hy Bech [ 6 ] , w h i c h i s o r i g i n a t e d f r o m t h e l i n e a r i s e d e l e m e n t a r y s i d e -f o r c e and moment e q u a t i o n s . The second o r d e r o -f i t can he c o n s i d e r e d t h e
consequence o f t h e phase s h i f t between d r i f t and a n g u l a r v e l o c i t y . I n p a r t i c u l a r t h i s e q u a t i o n i s a p p l i e d t o a u t o - p i l o t s . The yaw r a t e dependent f u n c t i o n H, i s e v a l u a t e d f r o m t h e s p i r a l t e s t r e s u l t s .
The o u t s t a n d i n g d i f f e r e n c e between models, used f o r a u t o p i l o t s and t h o s e , w h i c h serve t o d e s c r i b e t h e m a n o e u v r a b i l i t y as a w h o l e , i s t h a t f r o m t h e f i r s t k i n d i s wanted t o d e s c r i b e t h e change o f course d u r i n g a r e l a t i v e l y s m a l l space o f t i m e as a c c u r a t e as p o s s i b l e , w h i l e t h e u s e f u l n e s s o f t h e o t h e r k i n d r a t h e r i s j u d g e d f r o m t h e a b i l i t y t o d e s c r i b e w i t h s u f f i c i e n t a c c u r a c y a l a r g e number o f manoeuvres, w h i c h , one by one, l a s t a r e l a t i v e l y l o n g e r t i m e i n t e r v a l .
A n o t h e r i m p o r t a n t d i f f e r e n c e i s due t o t h e f a c t , t h a t a u t o p i l o t s a r e m a i n l y d e s i g n e d t o keep a s h i p , as good as p o s s i b l e , on a c e r t a i n c o u r s e . D u r i n g t h i s p r o c e s s , o n l y s m a l l yav? r a t e s v r i l l o c c u r , so t h a t f o r s t a b l e s h i p s
no n o n - l i n e a r terms have t o be r e t a i n e d ; i n g e n e r a l no f o r v r a r d speed e q u a t i o n i s n e c e s s a r y , because o f t h e s m a l l c r o s s - c o u p l i n g betV'/een t h e yav? r a t e and
t h i s speed i n t h i s r e g i o n .
D e t e r m i n a t i o n o f c o e f f i c i e n t s .
To d e t e r m i n e t h e c o e f f i c i e n t s o f an e l e m e n t a r y m o d e l , f o r c e s and moment on a shipmodel are measured d u r i n g f o r c e d o s c i l l a t o r y m o t i o n s , e i t h e r pure y a w i n g , p u r e swaying o r a c o m b i n a t i o n o f t h e s e m o t i o n s . The l i n e a r and n o n - l i n e a r hydrodynamic d e r i v a t i v e s a r e d e t e r m i n e d by s p l i t t i n g up t h e s e f o r c e s and moments i n t o tvfo components, t h e one i n phase v r i t h t h e v e l o c i t y and t h e o t h e r i n phase v f i t h t h e a c c e l e r a t i o n o f t h e m o t i o n c o n c e r n e d , v f h i l e t h e a m p l i t u d e s o f t h e s e components a r e c o n s i d e r e d a f u n c t i o n o f t h e
c o r r e s p o n d i n g motion-components. U s i n g a a n a l o g o r d i g i t a l computer, t h e v a r i o u s manoeuvres can be computed.
5 -Below a scheme o f t h i s p r o c e d u r e i s g i v e n . Forced o s c i l l a t i o n t e s t s l o n g i t u d i n a l - , s i d e f o r c e and moment c o e f f i c i e n t s o f math. model t i m e h i s t o r i e s c o m p u t e r o f V , r , Ö, u , X, y ( s o l u t i o n o f o f a l l m a n o e u v r e s d i f f . e q s . ) d e s i r e d d e s i r e d
I n f i g . 1 an example i s g i v e n o f a t u r n i n g c i r c l e , w h i c h i s one o f about f i f t e e n manoeuvres, computed a c c o r d i n g t o t h e above p r o c e d u r e .
The d e t e r m i n a t i o n o f t h e c o e f f i c i e n t s o f e l e m e n t a r y m a t h e m a t i c a l models, u s i n g f u l l s c a l e t e s t s , o r f r e e - r u n n i n g m o d e l t e s t s , i s h a r d l y p o s s i b l e i n g e n e r a l . Assuming e l e m e n t a r y models t o be a b l e t o d e s c r i b e a l l nuances o f m a n o e u v r a b i l i t y , we can s t a t e , t h a t t h e number o f terms i s t o o l a r g e f o r t h i s purpose. The r e s t r i c t e d a c c u r a c y o f measured f u l l s c a l e d a t a i s one o f t h e causes f o r t h i s . I n p a r t i c u l a r t h i s a p p l i e s t o t h e d r i f t - and f o r w a r d v e l o c i t y , as most s h i p s have no a p p r o p r i a t e i n s t r u m e n t s t o measure t h e s e v a r i a b l e s
s u f f i c i e n t l y a c c u r a t e .
B u t , even i f a l l v a r i a b l e s c o u l d be r e c o r d e d a c c u r a t e l y enough, i t w o u l d be a b i g p r o b l e m t o f i n d an a p p r o p r i a t e a l g o r i t h m t o s o l v e t h e l a r g e number o f unknown p a r a m e t e r s .
For t h e s e reasons t h e r e s u l t s o f f u l l - s c a l e manoeuvres a r e r a t h e r t o be used f o r t h e d e t e r m i n a t i o n o f t h e c o e f f i c i e n t s o f e m p e r i c a l models.
The p r o c e d u r e p u r s u e d i n t h i s case i s s c h e m a t i c a l l y shown below:
t i m e h i s t o r i e s o f f u l l s c a l e v a r i a b l e s t i m e h i s t o r i e s o f computed v a r i a b l e s c r i t e r i o n t o improve V i n i t i a l ) s e t o f c o e f f i c i e n t s new s e t o f c o e f -f i c i e n t s s o l u t i o n o f d i f f e -r e n t i a l eqs To s o l v e t h i s p r o b l e m we need an o p t i m i z a t i o n t e c h n i q u e by w h i c h t h e d i f f e r e n c e between t h e measured f u l l - s c a l e v a l u e s and t h e computed v a l u e s o f t h e v a r i a b l e s c o n c e r n e d , i s used t o improve an i n i t i a l l y e s t i m a t e d , s e t o f c o e f f i c i e n t s .
Two q u e s t i o n s o f v i t a l i m p o r t a n c e have t o be answered:
1° What c r i t e r i o n do we use t o j u d g e , w h e t h e r a s e r i e s o f computed manoeuvres i s b e t t e r t h a n a n o t h e r .
2° I n what way a r e t h e c o e f f i c i e n t s t o be changed, so as t o e x p e c t t h a t t h e new s e r i e s o f computed manoeuvres w i l l be b e t t e r t h a n t h e
p r e c e d i n g one.
C o n c e r n i n g t h e f i r s t q u e s t i o n , we n o t i c e t h a t i n g e n e r a l s e v e r a l manoeuvres o f a d i f f e r e n t k i n d a r e i n v o l v e d , w h i l e more o r l e s s on h i s t o r i c a l g r o u n i i s , each k i n d has i t s own j u d g i n g c r i t e r i a .
The most f r e q u e n t l y used t e s t s a r e t h e z i g - z a g t e s t s , t h e s p i r a l t e s t s and t h e t u r n i n g c i r c l e t e s t s .
The z i g - z a g t e s t , o r a m o d i f i e d v e r s i o n o f i t , i s j u d g e d f r o m t h e
change o f t h e c o u r s e w i t h t i m e and f r o m t h e mean change o f t h e f o r w a r d speed; i n g e n e r a l t h e c o n t i n u o u s speed measurements a r e r a t h e r u n r e l i a b l e and s u g g e s t e x t r e m e l y l a r g e l o n g i t u d i n a l a c c e l e r a t i o n s and d e c e l e r a t i o n s .
The r e s u l t s o f a s p i r a l t e s t a r e j u d g e d f r o m t h e v a l u e s o f t h e yaw r a t e and t h e f o r w a r d speed a f t e r each 5^0 degrees t u r n , t h o u g h c o n t i n u o u s r e c o r d i n g s o f t h e s e q u a n t i t i e s c o n t a i n a l o t o f i n f o r m a t i o n .
From a p r a c t i c a l p o i n t o f v i e w t h e i m p o r t a n t v a r i a b l e s o f a t u r n i n g c i r c l e a r e t h e t a c t i c a l d i a m e t e r , t h e advance and t h e t r a n s f e r , w i t h t h e t i m e s
b e l o n g i n g t o them. D u r i n g f u l l - s c a l e t e s t s , i n g e n e r a l one i s c o n f i n e d t o t h e r e c o r d i n g o f y a w - r a t e and speed however. C o n s e q u e n t l y o n l y t h e s e v a r i a b l e s can be c o n s i d e r e d f o r t h e judgement o f t u r n i n g c i r c l e s .
C o n c e r n i n g t h e second q u e s t i o n we have t o use e i t h e r a p r i m i t i v e o r a p a r t l y o f f u l l y a u t o m a t i c t r i a l - a n d - e r r o r method. A c o n s i d e r a b l e s i m p l i f i c a t i o n o f t h e p r o b l e m , w h i c h d o e s n o t alvfays l e a d t o a good r e s u l t however, i s o b t a i n e d i f we s e p a r a t e t h e s h i p ' s a s y m p t o t i c b e h a v i o u r f r o m t h e t r a n s i e n t phase o f manoeuvres. U s u a l l y , t h e s o l u t i o n o f t h e c o e f f i c i e n t s o f t h e s t a t i o n a r y p a r t o f t h e m a t h e m a t i c a l m o d e l , does n o t p r e s e n t many d i f f i c u l t i e s . The d e t e r m i n a t i o n o f t h e v a r i o u s t i m e - c o n s t a n t s i s t h e n t h e r e m a i n i n g p r o b l e m . A c t u a l l y t h i s method i s d i s p u t a b l e , because i t c a n n o t p o s s i b l y l e a d t o an o p t i m a l r e s u l t ( u n l e s s one d e f i n e s such a s o l u t i o n as o p t i m a l . . . ) .
A n o t h e r o h j e c t i o n t o t h i s method i s caused by t h e f a c t t h a t , p a r t i c u l a r l y when t h e model i s more c o m p l i c a t e d , i t o f t e n appears p o s s i b l e t o d e s c r i b e t h e s t a t i o n a r y c h a r a c t e r i s t i c s i n more t h a n one a c c e p t a b l e way. The problems i n v o l v e d , are i l l u s t r a t e d by t h e f o l l o v / i n g s i m p l e example o f two models, w i t h t h e same s t a t i o n a r y c h a r a c t e r i s t i c s :
6 (6)
T* + + r * 3 + ^2 + a^Sr""^ = 6 (?)
y ^ ^ ^ ^ where = 5, aj^ = 3, = 9 - 5 and T = K = 1 .
•X -B o t h " s p i r a l - c u r v e s " s a t i s f y t h e e q u a t i o n r ^ = K , w h i l e t h e t r a n s f e r o f b o t h systems i s q u i t e d i f f e r e n t , as f o l l o w s f r o m f i g u r e 2. C o n s e q u e n t l y f o r an o p t i m a l d e s c r i p t i o n o f a number o f manoeuvres, i t i s n e c e s s a r y t o t r e a t a l l unknown p a r a m e t e r s o f t h e m a t h e m a t i c a l model i n t h e same way i n a c e r t a i n c a l c u l a t i n g p r o c e d u r e .
An example o f such a p r o c e d u r e i s c a l l e d t h e "Simplex method", i n w h i c h , as a f i r s t s t e p , a number o f s e t s o f c o e f f i c i e n t s are e s t i m a t e d . For each s e t o f c o e f f i c i e n t s t h e manoeuvres wanted are computed, w h i l e t h e l e a s t s a t i s f y i n g s e t i s r e p l a c e d by a new one, t h e elements o f w h i c h are l i n e a r c o m b i n a t i o n s o f t h e c o r r e s p o n d i n g ones o f t h e r e m a i n e d s e t s . I n r e f e r e n c e [T ] t h i s method i s d e s c r i b e d i n more d e t a i l .
I n t h e n e x t c h a p t e r t h r e e examples o f t h e a n a l y s i s o f f u l l - s c a l e t r i a l r e s u l t s are d e s c r i b e d s h o r t l y .
Examples o f f u l l - s c a l e d a t a a n a l y s i s .
The f i r s t s h i p was t h e "Compass I s l a n d " , t h e f u l l - s c a l e t r i a l r e s u l t s o f w h i c h are g i v e n i n [ 8 ] .
Because t h i s a n a l y s i s was t h e f i r s t o f t h i s k i n d , t h e s t a r b o a r d t u r n i n g c i r c l e d a t a were used o n l y ; i n p a r t i c u l a r t h e y a w - r a t e and t h e f o r w a r d speed as a f u n c t i o n o f t i m e . The most s i m p l e d e s c r i p t i o n o f t h e s t a t i o n a r y r e l a t i o n between r u d d e r a n g l e and t u r n i n g d i a m e t e r appeared t o be
*
where r ^ ^ r ' ^ ' - r * , 6 = 6 - ÖQ and r**' = . The syimnetry p o i n t o f t h e " s p i r a l c u r v e " i s d e n o t e d by ( r ^ , 6 Q ) .
L i k e w i s e , u s i n g t h e l e a s t - s q u a r e s c r i t e r i o n , t h e s t a t i o n a r y r e l a t i o n
u = r ( 9 )
was f o u n d . F i n a l l y t h e t u r n i n g c i r c l e s were computed u s i n g some e s t i m a t e d
v a l u e s o f b o t h t h e t i m e - c o n s t a n t s . P l o t t i n g t h e mean e r r o r as a f u n c t i o n
o f b o t h t h e s e c o n s t a n t s , t h e b e s t c o m b i n a t i o n was f o u n d . A d d i t i o n a l l y , some
p r a c t i c a l v a r i a b l e s were computed, based on t h e e s t i m a t i o n o f a t i m e
-i n d e p e n d e n t d r -i f t - y a w r a t e r e l a t -i o n V . = -Y^ r ( 1 0 ) I n f i g u r e 3, b o r r o w e d f r o m t h e c o n c e r n i n g r e p o r t [ h], t h e computed and f u l l s c a l e v a l u e s o f t h e v a r i a b l e s a r e compared. The s e c o n d ' s h i p i s a 50.000 t o n s t a n k e r , s i m u l a t e d w i t h a 70 c o e f f i c i e n t s e l e m e n t a r y m o d e l , t h e unknown p a r a m e t e r s o f w h i c h were d e t e r m i n e d by h o r i z o n t a l o s c i l l a t i o n t e s t s . The p r e d i c t i o n s made w i t h t h i s m o d e l , f i t t e d c l o s e l y t o t h e f u l l s c a l e d a t a [ 9 ] , so t h a t , f o r s i m p l i c i t y r e a s o n s , t h e computer o u t p u t
o f seven z i g - z a g t r i a l s and f o u r t e e n t u r n i n g c i r c l e t e s t s was c o n s i d e r e d
t o be n o i s e - l e s s f u l l s c a l e measurements.
The s i m p l i f i e d m o d e l s , used t o a p p r o x i m a t e t h i s " s h i p " , d e s c r i b e d ^ a g a i n ,
-1 J as
i n s t e a d o f - r r and f u r t h e r m o r e t h e l o n g i t u d i n a l speed.
For c o m p a r i s o n p u r p o s e s a l i n e a r and a n o n - l i n e a r model were a p p l i e d .
An a t t e m p t was made t o d e t e r m i n e t h e f o u r c o e f f i c i e n t s o f t h e l i n e a r m o d e l ,
two t i m e c o n s t a n t s and two p r o p o r t i o n a l i t y f a c t o r s , u s i n g t h e S i m p l e x method
j u s t m e n t i o n e d , t h o u g h w i t h o u t r e s u l t s . A f t e r w a r d s t h e p r o p o r t o n a l i t y f a c t o r s
K and were d e r i v e d f r o m t h e c o r r e s p o n d i n g s t a t i o n a r y c h a r a c t e r i s t i c s ,
w h i l e t h e S i m p l e x method was u s e d , t o f i n d b o t h t h e t i m e c o n s t a n t s .
The r e s u l t s a r e shown i n f i g u r e s h, 5 and 6.
The p r o b l e m s i n v o l v e d i n t h e n o n - l i n e a r d e s c r i p t i o n m a i n l y c o n c e r n e d t h e
c h o i c e o f terms t o be a d o p t e d . A f t e r t e s t i n g some n o n - l i n e a r e x p r e s s i o n s
f o r t h e d e s c r i p t i o n o f t h e s p i r a l c u r v e , w i t h o u t good r e s u l t , t h e z i g - z a g t r i a l s
7 9 7
-Assuming t h a t o n l y t h i r d degree t e r m s a r e c o n s i d e r e d , t h e cha.nge o f
s l o p e a t r * = 0 can be caused b y a t e r m rö'^ o n l y . The c o e f f i c i e n t , o f t h i s t e r m c o u l d be e s t i m a t e d f r o m t h e s e p l o t s . F u r t h e r i t was f o u n d t h a t u s i n g t h i s t e r m t h e s p i r a l c u r v e c o u l d be d e s c r i b e d v e r y a c c u r a t e l y , p r o v i d e d t h a t i n a d d i t i o n a t e r m 6 r was a d o p t e d , t o n e u t r a l i z e p a r t l y t h e r e l a t i v e l y 2 l a r g e e f f e c t o f t h e r 6 t e r m . The f i n a l s p i r a l c u r v e e q u a t i o n t h e n became: 3 2 ^2 » 3 T* + a^"* + a* T"" & + 0 , 5 * 6 ? " = K^*" 6 + K * 6 ^ ( I I )
M o r e o v e r , a model had t o be'formed t o d e s c r i b e t h e f o r w a r d speed r e d u c t i o n . A l s o i n t h i s case s e v e r a l p o s s i b i l i t i e s appeared t o e x i s t f o r t h e d e s c r i p t i o n o f t h e s t a t i o n a r y c o n d i t i o n , e.g.:
a^ u»^ + b^ u^2 ^ ^^^>2 ^^^K 6 + e^Ö^ ( 1 2 ^ )
ag u^+ bg u*2 ^ ^ ^ r * 2 ^ ^^^^ g (^2^^) b ^ u ^ ^ = C 3 r * 2 + d 3 r * Ó + e.^&^ ( 1 2 = )
But a g a i n s t most o f them, some p r a c t i c a l o b j e c t i o n s a p p e a r e d t o e x i s t . The b e s t r e s u l t s were o b t a i n e d w i t h an e q u a t i o n o f t h e f o r m : a u * + b u * 2 + c u * 3 ^ 2 (^3) I n f i g u r e 8 t h e a p p r o x i m a t i o n o f b o t h t h e s t a t i o n a r y c h a r a c t e r i s t i c s i s shown. F i n a l l y b o t h t h e t i m e c o n s t a n t s o f t h e system were d e t e r m i n e d e m p e r i c a l l y . I n f i g u r e 9 and 10 an i m p r e s s i o n i s g i v e n o f t h e r e s u l t s . C o n c l u s i o n s and r e c o m m e n d a t i o n s .
From t h e above t h r e e examples, i t seems p o s s i b l e t o f o r m a m a t h e m a t i c a l model based on f u l l s c a l e manoeuvres, and t o d e t e r m i n e i t s c o e f f i c i e n t s . On t h e o t h e r hand t h e r e ' s no q u e s t i o n o f a u n i f o r m , r e a d y ~ f o r - u s e method however.
The d e s i r e d a c c u r a c y o f t h e p r e d i c t i o n s and t h e s i m u l a t i o n s , appeared t o be e s s e n t i a l t o t h e f o r m and c o m p l e x i t y o f t h e m a t h e m a t i c a l model.
The j u d g i n g c r i t e r i a , a p p l i e d f o r t h e above examples, a r e r a t h e r a r b i t r a r y : f o r t h e d e s c r i p t i o n o f t h e s t a t i o n a r y c h a r a c t e r i s t i c s t h e l e a s t - s q u a r e s c r i t e r i o n was u s e d , w h i l e f o r t h e o p t i m i z a t i o n o f t h e " t i m e - c o n s t a n t s " t h e mean sum was used o f t h e r e l a t i v e d e v i a t i o n s i n t h e c o u r s e , t h e t i m e and t h e d i s t a n c e c o v e r e d , i n a number o f c r i t i c a l p o i n t s .
F u r t h e r m o r e , a t t e n t i o n s h o u l d be p a i d t o t h e o b j e c t i o n s t o t h e s e p a r a t i o n o f t h e a s y m p t o t i c b e h a v i o u r f r o m t h e g e n e r a l s o l u t i o n o f t h e s y s t e m . I n t h e near f u t u r e o t h e r o p t i m i z a t i o n t e c h n i q u e s w i l l be t e s t e d , t o t r e a t a l l unknown p a r a m e t e r s o f t h e system i n t h e same way.
I t a l s o seems d e s i r a b l e t o l o o k f o r a manoeuvre, w h i c h i s more s u i t a b l e f o r t h e p u r p o s e o f e v a l u a t i n g m a t h e m a t i c a l m o d e l s , as t h e t r a d i t i o n a l manoeuvres need much t i m e f o r t h e e x e c u t i o n as w e l l as f o r t h e a n a l y s i s .
R e f e r e n c e s .
[ l ] A t k o w i t z , M.A. :
" L e c t u r e s on s h i p h y d r o d y n a m i c s - s t e e r i n g and m a n o e u v r a h i l i t y " HyA r e p o r t no. Hy 5, May 1961^
[ 2 ] N o r r b i n , N.H.:
"Theory and o b s e r v a t i o n s on t h e use o f a m a t h e m a t i c a l model f o r s h i p manoeuvring i n deep and c o n f i n e d w a t e r s " .
SSPA Report no. 68, 1970
[ 3 ] Nomoto, K.:
" A n a l y s i s o f t h e s t a n d a r d m.anoeuvre t e s t o f Kempf and p r o n o s e d s t e e r i n g q u a l i t y i n d i c e s "
Symposium S h i p M a n o e u v r a b i l i t y , Washington lO^O.
[h] Leeuwen, G. v a n :
"A s i m p l i f i e d n o n - l i n e a r model o f a manoeuvring s h i p " U n i v e r s i t y o f Technology D e l f t , r e p o r t no. 262, March I97O
[ 5 ] N o r r b i n , N.H.:
"On t h e d e s i g n and a n a l y s i s o f t h e z i g - z a g t e s t on base o f q u a s i l i n e a r f r e q u e n c y r e s p o n s e "
SSPA r e p o r t B.10i|.3. F o r m a l c o n t r i b u t i o n 10'^h ITTC 1963
[ 6 ] Bech, M.I. and S m i t t L.W.:
"Analogue s i m u l a t i o n o f s h i p manoeuvres" HyA r e p o r t no. Hy l i | , September I969
[7] Koyama, T.:
"An a n a l o g program o f t h e S i m p l e x method"
U n i v e r s i t y o f T e c h n o l o g y D e l f t , r e p o r t no. 307, A p r i l 1971
[ 8 ] Morse, R.V.:
"Manoeuvring c h a r a c t e r i s t i c s o f t h e M a r i n e r t y p e s h i p (uss "Compass I s l a n d ) i n calm seas"
DTMB p u b l . no. G.J. 2233/1019, December I 9 6 I
[ 9 ] Leeuwen, G. v a n a n i J o u r n e e , J . I I . J . :
" P r e d i c t i o n o f s h i p m a n o e u v r a b i l i t y , making use o f model t e s t s ' U n i v e r s i t y o f T e c h n o l o g y D e l f t , r e p o r t no. 288, A p r i l 19T0
i n s t a n t a n e o u s f o r w a r d speed n o n - d i m e n s i o n a l t i m e c o n s t a n t s (T^ = ^ l ' ^ ^ ' ^u ~ ^ "^u*^ n o n - d i m e n s i o n a l p r o p o r t i o n a l i t y c o n s t a n t s r u d d e r a n g l e and r u d d e r r a t e o f t u r n r e s p e c t i v e l y u n i t o f p a t h l e n g t h ( d t = d s ) n o n - d i m e n s i o n a l u n i t o f p a t h l e n g t h ( d t = ds*) course a n g l e yaw r a t e o f s h i p n o n - d i m e n s i o n a l yaw r a t e ( r = y r^) L n o n - d i m e n s i o n a l vaw a c c e l e r a t i o n ( r = ( r * + r * " ^ ) ) i n i t i a l speed o f a manoeuvre speed r e d u c t i o n ( U = UQ + u) t i m e d e r i v a t i v e o f u r e l a t i v e speed r e d u c t i o n ( U - U Q ( 1 + U * ) ) UQ n o n - d i m e n s i o n a l yaw r a t e ( r = — • r ' ) ' . L n o n - d i m e n s i o n a l t i m e c o n s t a n t (T = ^ • T') U n o n - d i m e n s i o n a l p r o p o r t i o n a l i t y c o n s t a n t (K = • K') Ll t i m e c o n s t a n t s o f second o r d e r system n o n - d i m e n s i o n a l c o e f f i c i e n t s o f n o n - l i n e a r yaw e q u a t i o n U2 "3 - u 2 "'3 ' "'4 - ^-h ' «5 " U ™5 = r * - r * I (1^0' ^o^ symmetry p o i n t o f t h e 6Q J ~ "^c ^ p i ^ ^ l c u r v e , n o n - d i m e n s i o n a l sway v e l o c i t y ( v = Uv*)
. (YDS) - 2000 - 1000 (YDS) - 2000 - 1000
MEASURED AND COMPUTED MEASURED AND COMPUTED VALUES OF r'(t) VALUES OF TACTICAL DIAMETER
• T I M E (MINUTES)
MEASURED AND COMPUTED VALUES OF U(t)
-60-30° - 2 0 ° ^ -10°
MEASURED AND COMPUTED VALUES OF TRANSFER F I G . 1
T r +r = K 6
2 TWO D I F F E R E N T MODELS WITH T H E S A M E S T A T I O N A R Y C H A R A C T E R I S T I C S .
F I G . 3 TIME HISTORIES AND X-Y PLOT OF TURNING CIRCLE D2 AT 19° TO STARBOARD AND 100 NOMINAL R.RM.
20V20'
O O
2 5 / 2 0
F I G . 5 APPROXIMATION OF Z I G - Z A G T R I A L S WITH T H E L I N E A R MODEL.
r
„ = Model
- . 5 0 .5
FIG. 8 N O N - L I N E A R APPROXIMATION OF S T A T I O N A R Y C H A R A C T E R I S T I C S .
