LABORATORIUM VOOR
SCHEEPSBOUWKUNDE
TECHNISCHE HOGESCHOOL DELFT
r n
A SIIVIPLIFIED NON-LINEAR MODEL OF A f/IANOEUVRINGSHIP
by
G. van Leeuwen
L J
Summary.
I n t h i s paper two s i m p l e n o n - l i n e a r s e t s o f e q u a t i o n s o f m o t i o n a r e d i s c u s s e d , d e s c r i b i n g t h e b e h a v i o u r o f t h e manoeuvring s h i p w i t h c o n s t a n t power c o n d i t i o n . Both s e t s a r e c o n s i d e r e d expansions o f t h e f i r s t - o r d e r system concept
based by Nomoto 1
The f i r s t model depends on t h e e l e m e n t a r y e q u a t i o n s o f m o t i o n w h i l e f o r t h e d e s c r i p t i o n o f t h e s t e a d y s t a t e o f t h e m o t i o n an e m p i r i c a l method i s g i v e n .
I n t h e second model t h e d e r i v a t i v e o f t h e a n g u l a r v e l o c i t y i s r e p l a c e d by t h e d e r i v a t i v e o f t h e r a t i o o f a n g u l a r v e l o c i t y and f o r w a r d speed. A nev? symbol i s i n t r o d u c e d t o i n d i c a t e t h a t q u a n t i t i e s a r e n o n - d i m e n s i o n a l i s e d w i t h t h e i n s t a n t a n e o u s f o r w a r d speed.
A t t e n t i o n i s p a i d t o t h e r e l a t i o n and t h e d i f f e r e n c e between t h e s e models and t h e model g i v e n by A b k o w i t z .
Both models can be a p p l i e d f o r s i m u l a t i o n purposes as w e l l as f o r t h e a n a l y s i s o f f u l l - s c a l e manoeuvres.
From t h e l a t t e r an example i s g i v e n , u s i n g t h e t u r n i n g c i r c l e - d a t a o f t h e USS "Compass I s l a n d " ['s) .
The d e s c r i p t i o n o f t h e s e d a t a w i t h t h e second model l e a d s t o a c c u r a t e r e s u l t s . The c a l c u l a t i o n method i s d e s c r i b e d s h o r t l y .
F i n a l l y some f i g u r e s a r e g i v e n w h i c h i l l u s t r a t e c l e a r l y t h e i m p o r t a n c e o f t h e n o n - l i n e a r terms o f t h e A b k o w i t z model d u r i n g a z i g - z a g manoeuvre. These f i g u r e s a l s o i n d i c a t e t h a t a more s i m p l e m a t h e m a t i c a l model m i g h t be used.
I n t r o d u c t i o n .
The e q u a t i o n s o f m o t i o n o f a manoeuvring s h i p as t h e y are g i v e n by Davidson and S c h i f f (^hj can be a p p l i e d t o p r e d i c t manoeuvres w i t h s m a l l helm a n g l e i n a r a t h e r a c c u r a t e way, p r o v i d e d t h e s h i p i s s t a b l e . However, i f manoeuvres a r e p e r f o r m e d w i t h l a r g e r helm a n g l e s t h e n an a p p r o x i m a t i o n w i t h t h e s e
e q u a t i o n s i s n o t p o s s i b l e anymore, n o t even when t h e s h i p i s s t a b l e . T h i s m a t h e m a t i c a l model m i g h t be c o n s i d e r e d t h e base o f Nomoto's l i n e a r second o r d e r model [^5^ , t h e p r i n c i p a a l p r o p e r t i e s o f vfhich a r e a l r e a d y d i s c u s s e d by Davidson and S c h i f f . U n f o r t u n a t e l y Nomoto does n o t c o n s i d e r the f o r w a r d speed e q u a t i o n g i v e n by t h e s e a u t h o r s .
The l i n e a r f i r s t o r d e r model p r o p o s e d by Nomoto j^l'^ appeared a u s e f u l s u p e r s e s s i o n o f t h e second o r d e r model. T h i s m i g h t be d e c l a r e d by t h e f a c t t h a t i n u n s t e a d y m o t i o n s t h e phase l a g between t h e d r i f t and t h e yaw r a t e i s o n l y s m a l l . The p r i n c i p a l m e r i t o f t h e f i r s t o r d e r model i s t h e g r e a t s i m p l i c i t y and t h a t i s why i t has a l s o many i n s t r u c t i v e p r o p e r t i e s . I t i s p o s s i b l e t o s i m u l a t e i n a r a t h e r a c c u r a t e way one t u r n i n g c i r c l e or z i g - z a g manoeuvre, p r o v i d e d a method i s used t o d e r i v e t h e t i m e ¬ and p r o p o r t i o n a l i t y c o n s t a n t s i n an o p t i m a l way.
A d i s a d v a n t a g e however i s t h e change o f t h e s e c o e f f i c i e n t s when o t h e r manoeuvres a r e a n a l y s e d , so t h a t i t i s d i f f i c u l t t o p r e d i c t t h e s h i p ' s b e h a v i o u r i n an a r b i t r a r y manoeuvre.
One o f t h e reasons f o r t h e s e change o f c o e f f i c i e n t s i s t h a t no f o r w a r d speed e q u a t i o n i s c o n s i d e r e d , b u t a l s o n o n - l i n e a r i t i e s i n t h e t u r n i n g r e s i s t a n c e p l a y an i m p o r t a n t r o l e .
Besides t h i s development t h e r e i s a n o t h e r which extends t h e model o f Davidson and S c h i f f w i t h some n o n - l i n e a r terms e.g. Eda and Crane ( 6 ' ] , N o r r b i n j^T^ and o t h e r s . I n g e n e r a l t h e l o n g i t u d i n a l f o r c e e q u a t i o n i s added a g a i n . The most extended s e t o f e q u a t i o n s i s g i v e n by Abkovfitz j^2^ who c o n s i d e r s t h e T a y l o r s e r i e s o f f o r c e s and moments. T h i s s e t o f
e q u a t i o n s seems a b l e t o d e s c r i b e a l l s i g n i f i c a n t phenomena r a t h e r
a c c u r a t e l y b u t t h e g r e a t number o f c o e f f i c i e n t s i s an i m p o r t a n t d i s a d v a n t a g e . Among o t h e r reasons t h i s i s caused by t h e f a c t t h a t t h e e q u a t i o n s a r e
n o n - d i m e n s i o n a l i s e d w i t h t h e advance speed.
I n p r a c t i c e i t i s n e a r l y i m p o s s i b l e t o d e t e r m i n e a l l t h e s e c o e f f i c i e n t s f r o m f u l l - s c a l e t r i a l s . Only w i t h model t e s t s e,g, 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 t h i s appears p o s s i b l e t h o u g h a l s o i n t h i s case t h e r e a r e s e v e r a l p r o b l e m s ,
2.
Now we can p u t t h e q u e s t i o n w h i c h demands s h o u l d he made upon a m a t h e m a t i c a l model o f a m a n o e u v r i n g s h i p , i f we t e n t a t i v e l y c o n s i d e r manoeuvres i n s t i l l w a t e r o n l y , w h i l e f u r t h e r t h e f u e l s u p p l y o f t h e engines i s supposed t o be unchanged d u r i n g a manoeuvre.
These demands a r e s i m m a r i s e d as f o l l o w s :
1. The m a t h e m a t i c a l model s h o u l d be a b l e t o p r e d i c t each manoeuvre i n an a c c u r a t e way.
2. I t s h o u l d be p o s s i b l e t o 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 t h e m a t h e m a t i c a l model f r o m f u l l - s c a l e manoeuvres such as t u r n i n g c i r c l e t e s t s and
a r b i t r a r y z i g - z a g t e s t s .
3. The model has t o be employable f o r s i m u l a t i o n p u r p o s e s .
ad 1 . C o n c e r n i n g t h e f i r s t demand i t has t o be s p e c i f i e d what i s " a c c u r a t e " . I t seems p r a c t i c a l f o r i n s t a n c e t o c l a i m an a c c u r a c y o f t h e p r e d i c t i o n o f t h e 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 and course o f f i v e p e r c e n t f o r a l l p o s s i b l e manoeuvres, compared w i t h t h e f u l l - s c a l e d a t a .
Because t h e o t h e r v a r i a b l e s as yaw r a t e , d r i f t and f o r w a r d speed a r e f u n c t i o n s o f t h e c o o r d i n a t e s and t h e course a l s o t h e i r a c c u r a c y i s f i x e d . ad 2. The demand t h a t i t s h o u l d be p o s s i b l e t o d e t e r m i n e t h e c o e f f i c i e n t s f r o m f u l l - s c a l e d a t a r e q u i r e s t h a t t h e model has t o be as s i m p l e as p o s s i b l e . The l a r g e r t h e model t h e s m a l l e r t h e d e t e r m i n a b i l i t y o f the c o e f f i c i e n t s , due t o t h e r e s t r i c t e d a c c u r a c y o f f u l l - s c a l e
measurements. I n p a r t i c u l a r t h e f o r w a r d speed measurements a r e r a t h e r i n a c c u r a t e i n g e n e r a l so t h a t i t i s no use t o adopt a c o m p l i c a t e d f o r w a r d speed e q u a t i o n .
ad 3. The model's u s e f u l n e s s f o r s i m u l a t i o n p u r p o s e s seldom w i l l be a p r o b l e m because d i g i t a l o r a n a l o g computers a r e a v a i l a b l e i n g e n e r a l . However i f t h e model w o u l d be used on l i n e w i t h an a u t o p i l o t i t s s i m p l i c i t y i s i m p o r t a n t f o r p r a c t i c a l r e a s o n s .
An a d d i t i o n a l demand w h i c h c o u l d have been made on t h e m a t h e m a t i c a l model i s t h a t t h e c o e f f i c i e n t s s h o u l d have a c l e a r p h y s i c a l meaning.
T h i s i s t h e case f o r i n s t a n c e , w i t h t h e c o e f f i c i e n t s o f t h e A h k o w i t z model: t h e y are t h e p a r t i a l d e r i v a t i v e s o f t h e hydrodynamic f o r c e s and moments. C o n c e r n i n g t h e s e c o e f f i c i e n t s one c o u l d e x p e c t t h a t a t a t i m e i t w i l l be p o s s i b l e t h a t t h e i r r e l a t i o n w i t h t h e s h i p ' s f o r m can be d e t e r m i n e d , so t h a t i t i s a l s o p o s s i b l e t o p r e s c r i b e t h e manoeuvring p r o p e r t i e s o f a new s h i p t o be b u i l d , r a t h e r t h a n t h i s m i g h t be e x p e c t e d f r o m t h e c o e f f i c i e n t s o f a more o r l e s s e m p i r i c a l model. T h i s demand was n o t made h e r e however because t h i s i s o u t o f t h e scope o f t h i s paper.
The main t h o u g h t s w h i c h a r e t h e base o f t h e two m a t h e m a t i c a l models d i s c u s s e d h e r e a r e t h e f o l l o w i n g . The l i n e a 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 d r i f t and t h e yaw r a t e as a f u n c t i o n o f t h e r u d d e r a n g l e can be used t o d e s c r i b e manoeuvres w i t h s m a l l r u d d e r a n g l e s b u t t h e u s e f u l n e s s d i m i n i s h e s i f manoeuvres a r e p e r f o r m e d d u r i n g w h i c h l a r g e r y a w - r a t e s o c c u r . I f t h e s e e q u a t i o n s a r e used u n t i l l t h e d i f f e r e n c e s between measured and p r e d i c t e d v a l u e s o f t h e v a r i a b l e s i n v o l v e d become l a r g e r t h a n e.g. f i v e p e r c e n t t h a n i t i s f o u n d t h a t t h e d r i f t e q u a t i o n i n t h i s range i s n o t o f g r e a t i m p o r t a n c e , i n o t h e r words t h e a c c u r a c y o f t h e p r e d i c t e d c o o r d i n a t e s i s i n f l u e n c e d by t h e d r i f t f o r a s m a l l amount o n l y . The c r o s s - c o u p l i n g e f f e c t i n t h e yaw e q u a t i o n can be e l i m i n a t e d
i n t h i s " s e m i - l i n e a r " range i f we adopt a t i m e independant r e l a t i o n between yaw r a t e and d r i f t . I n t h i s way o n l y one e q u a t i o n r e m a i n s , d e s c r i b i n g
the yaw r a t e as a f u n c t i o n o f t i m e and t h e r u d d e r a n g l e .
The c o e f f i c i e n t s o f t h i s e q u a t i o n cannot l o n g e r be c o n s i d e r e d t o be t h e p a r t i a l d e r i v a t i v e s o f t h e hydrodynamic moment, a c t u a l l y t h e y must be c o n s i d e r e d f u n c t i o n s o f them.
Itese c o n s i d e r a t i o n s can be used t o d e c l a r e why t h e l i n e a r f i r s t o r d e r model o f Nomoto i s employable i n a c e r t a i n r a n g e .
To o b t a i n a p r e d i c t i o n o f manoeuvres w i t h t h e same a c c u r a c y i n a range i n w h i c h t h e yaw r a t e reaches l a r g e r v a l u e s , i n g e n e r a l t h e m o t i o n s become more v i o l e n t , we s h o u l d wonder why t h e l i n e a r 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 f a i l s . Then i t appears t h a t t h e d e s c r i p t i o n o f two v a r i a b l e s i s n o t c o r r e c t . I n t h e f i r s t p l a c e , o f c o u r s e , t h e yaw r a t e i t s e l f w h i c h i s n o t p r o p o r t i o n a l t o t h e r u d d e r a n g l e and f u r t h e r i t appears t h a t t h e f o r w a r d speed cannot be c o n s i d e r e d c o n s t a n t n o t even as an a p p r o x i m a t i o n .
F i n a l l y t h e e q u a t i o n c a n n o t be used a t a l l when t h e s h i p i s n o t s t a b l e There a r e no i n d i c a t i o n s however t h a t a d r i f t e q u a t i o n i s n e c e s s a r y . The c o n c l u s i o n s w h i c h f o l l o w f r o m t h i s a r e t h a t i n t h e f i r s t p l a c e a n o n - l i n e a r t e r m s h o u l d be added t o t h e yaw e q u a t i o n and t h a t a s i m p l e f o r w a r d speed e q u a t i o n i s needed.
Summarising t h e above c o n s i d e r a t i o n s we o r i g i n a l l y have t h e Nomoto e q u a t i o n :
Tr + r = K6
and i f more v i o l e n t manoeuvres have t o be d e s c r i b e d t h e system c o u l d be:
I t must be n o t e d however t h a t Nomoto, when n o n - d i m e n s i o n a l i s i n g t h e yaw e q u a t i o n , uses t h e i n s t a n t a n e o u s speed. I f t h i s i s done t h e l i n e a r 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 can be a p p l i e d i n a l a r g e r range o f manoeuvres. I n my o p i n i o n however t o o l i t t l e a t t e n t i o n i s p a i d t o t h i s f a c t , v r t i i l e t h e o r i g i n a l n o t a t i o n u s i n g t h e p r i m e ( ' ) , as a l r e a d y used by Davidson and S c h i f f , when n o n - d i m e n s i o n a l i s i n g w i t h t h e i n s t a n t a n e o u s speed i s c o n f u s i n g because i n a l a t e r s t a g e t h i s s i g n i s o n l y used
i f q u a n t i t i e s a r e n o n - d i m e n s i o n a l i s e d w i t h t h e i n i t i a l speed. I n t h e n e x t c h a p t e r t h i s p r o b l e m w i l l be c o n s i d e r e d i n g r e a t e r d e t a i l . Tr + r + a r = K6 T u + u u 1 )
1. Basic f o r m u l a e .
The e q u a t i o n s o f m o t i o n s w i t h r e s p e c t t o t h e body-axes c o o r d i n a t e s system can be w r i t t e n as f o l l o w s : (see a l s o f i g . l ) .
(m - Y O V + (m - X^)U^r = Y ( l ^ ) ( I - N - ) r = N ( 1 ^ )
zz . r
(m - X^)u - (ra - Y ^ ) v r = X ( l'^)
I n t h i s s e t o f e q u a t i o n s t h e hydrodynamic f o r c e and moment components
w h i c h depend on t h e mass and t h e added mass a r e p u t on t h e l e f t hand s i d e s . Cross c o u p l i n g terms depending on t h e added mass a r e c o n s i d e r e d t o remain i n t h e r i g h t hand s i d e s . I n t h e f o l l o w i n g c h a p t e r s t h e s e e q u a t i o n s w i l l be c a l l e d t h e sway- o r d r i f t e q u a t i o n , t h e yaw e q u a t i o n and t h e f o r w a r d speed e q u a t i o n r e s p e c t i v e l y . T h i s has been done because t h e purpose o f t h i s paper i s n o t an a t t e m p t t o d e r i v e a s e t o f e q u a t i o n s f o r an o p t i m a l
d e s c r i p t i o n o f t h e f o r c e s and moments a c t i n g on t h e manoeuvring s h i p b u t r a t h e r t o g e t a s u i t a b l e s e t o f e q u a t i o n s which a p p r o x i m a t e s t h e s h i p s ' b e h a v i o u r i n a way t h a t i s a c c u r a t e enough f o r p r a c t i c a l p u r p o s e s .
2. The f i r s t n o n - l i n e a r model (Model l ) .
2. 1 . The yaw e q u a t i o n .
F i r s t we v r i l l c o n s i d e r t h e yaw e q u a t i o n . I n g e n e r a l t h e r i g h t hand s i d e o f t h i s e q u a t i o n w i l l be a f u n c t i o n o f t h e v a r i a b l e s v , r , 6 and u.
P r o v i d e d a t i m e - i n d e p e n d e n t r e l a t i o n s h i p between t h e v a r i a b l e s v and r e x i s t s , w h i c h means t h a t t h e phase l a g between v and r d u r i n g n o n - s t e a d y motions i s c o n s i d e r e d o f l i t t l e i m p o r t a n c e f o r t h e b e h a v i o u r o f t h e manoeuvring s h i p t h e r i g h t hand s i d e o f t h e yaw e q u a t i o n i s t h e n o n l y a f u n c t i o n o f r , 6 and u , t h u s
(1^2 - N ; ) . r = N ( r , 6 , u ) ( 2 ) To d e t e r m i n e t h e r i g h t hand s i d e we may a n a l y s e f u l l s c a l e manoeuvres,
i n p a r t i c u l a r t h e s t e a d y s t a t e s o f t u r n i n g c i r c l e t e s t s .
However c o n s i d e r i n g t h e r e l a t i o n between rJU^ and 6^ t h e n o n - l i n e a r c h a r a c t e r o f t e n n e a r l y d i s a p p e a r s . T h i s means t h a t L.rAJ i s a more c o n v e n i e n t
v a r i a b l e t o be c o n s i d e r e d as r i s .
I n f i g u r e s 2 t o T some examples o f r - 6 and L . r AJ - & c u r v e s
° ^ C O c e o
a r e g i v e n w h i c h i l l u s t r a t e t h i s .
I t w i l l be shown t h a t t h i s phenomena i s i n accordance v r i t h t h e assumption t h a t f o r c e s and moments a r e n e a r l y p r o p o r t i o n a l t o t h e square o f t h e
speed, i f t h e v a r i a b l e s , n o n - d i m e n s i o n a l i s e d w i t h t h i s speed, do n o t change. P r o v i d e d t h e m o m e n t - e q u i l i b r i u m d u r i n g a s h o r t t i m e i n t e r v a l i s g i v e n by t h e e q u a t i o n : ( I - N" ) 'r = N,r + N^r"^ + W, ö ( 3 ) zz r 1 3 4 When we c o n s i d e r t h e moment-part = Eyr ih)
t h e n t h e assumption t h a t moments a r e p r o p o r t i o n a l t o t h e square o f t h e speed means t h a t
= N^^ . ^ p l f h ^ . r ^ ( 5 )
where r*^ i s d e f i n e d as Lr/U and i s a c o n s t a n t .
I n o t h e r words t h e moment p a r t M,^ i s p r o p o r t i o n a l t o t h e square o f t h e speed and a l s o p r o p o r t i o n a l t o t h e v a r i a b l e r** . As i t i s a m a t t e r o f f a c t t h a t t h e f o r w a r d speed i n t h e s t e a d y s t a t e o f a t u r n i n g c i r c l e w i t h l a r g e r u d d e r a n g l e i s c o n s i d e r a b l y l e s s t h a n the i n i t i a l speed, i t i s c l e a r t h a t t h e a n g u l a r v e l o c i t y w i l l s u r e l y n o t be p r o p o r t i o n a l t o t h e r u d d e r a n g l e . T h i s means t h a t t h e n o n - l i n e a r c h a r a c t e r o f an r - 6 c u r v e s h o u l d n o t be c o n s i d e r e d as a n o n - l i n e a r i t y c o f r o m a h y d r o d y n a m i c a l p o i n t o f v i e w .
The d i f f e r e n c e between r ' ( o r r ) and r a p p e a r s c l e a r l y f r o m
f i g u r e 8 where t h e t i m e h i s t o r i e s o f t h e s e q u a n t i t i e s a r e p l o t t e d , d e r i v e d f r o m a t u r n i n g c i r c l e t e s t w i t h t h e USS "Compass I s l a n d " a t 3 5 degrees r u d d e r a n g l e .
I n t h e f o l l o w i n g t r e a t m e n t e q u a t i o n ( 3 ) i s used and may be c o n s i d e r e d an e x p a n s i o n o f t h e Nomoto model. I t w i l l be shown t h a t t h i s e q u a t i o n , can a l s o be used t o d e s c r i b e u n s t a b l e s h i p s . 2 3 D i v i d i n g each t e r m o f t h e e q u a t i o n by u L we g e t : r = • - r + — — + — — <5 ( 6 ) o r where W^*" , N^* and a r e c o n s t a n t s .
T h i s e q u a t i o n may a l s o be w r i t t e n i n terms o f T and K, used by Nomoto ( l ) 1
D i v i s i o n by |N j g i v e s
T*(^f r + r * + a^r*^^K*6
A p p a r e n t l y t h e + s i g n s h o u l d be used f o r s t a b l e s h i p s w h i l e t h e - s i g n i n d i c a t e s t h a t t h e s h i p i s u n s t a b l e .
On f i g u r e 9 some examples o f s t a b l e - and u n s t a b l e r * - 6^ c h a r a c t e r i s t i c s a r e g i v e n .
I n most cases e q u a t i o n ( 7 ) w i l l do t o d e s c r i b e t h e s t e a d y s t a t e s o f t u r n i n g c i r c l e t e s t s , t h o u g h sometimes a s i g n i f i c a n t d i f f e r e n c e between p o r t - and s t a r b o a r d o c c u r s . I n t h a t case t h e e q u a t i o n s h o u l d be changed by t h e t r a n s f o r m a t i o n s : 7 ^ = r * - r * a 5 = 6 - 6 a so t h a t t h e e q u a t i o n can be w r i t t e n as;
8 . I n terms o f T and K, e q u a t i o n ( 9 ) i s : . (kf r + + a* r* 2 + „^r^^ ^ ^ j ^ - ^ \U/ - 2 3 o An example o f an a s y m m e t r i c a l r * - curve i s g i v e n on f i g u r e 3 , d e r i v e d f r o m t h e USS "Compass I s l a n d " t u r n i n g c i r c l e d a t a a t 2 0 k n o t s . 2 . 2 . The f o r w a r d s p e e d - e q u a t i o n . The base f o r t h i s e q u a t i o n i s t h e l o n g i t u d i n a l f o r c e - e q u a t i o n ( l c ) ' : i (m - X^)u - (m - Y ^ ) v r = X ( u , v , r , 6 ) ( i c ) C o n s i d e r i n g t h e d a t a o f v , r and U o f t u r n i n g c i r c l e t e s t s i t s h o u l d be p o s s i b l e t o f i n d a s i m p l e e x p r e s s i o n f o r t h e r i g h t hand s i d e o f t h i s e q u a t i o n . The most s i m p l e f o r m m i g h t be however:
(m - X^)u - (m - Y ^ ) v r = X^.u
w h i c h means t h a t t h e f i n a l speed-loss o f t u r n i n g c i r c l e s i s 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 c e n t r i f u g a l a c c e l e r a t i o n .
On t h e one s i d e t h i s seems a b i t queer because one i s used t o p l o t t h i s speed-loss as a f u n c t i o n o f t h e r u d d e r a n g l e and n o t as a f u n c t i o n o f t h e c e n t r i f u g a l a c c e l e r a t i o n . Though t h e r u d d e r a n g l e i t s e l f a l s o causes
some speed l o s s , i t i s p o s s i b l e t o d e s c r i b e t h e speed l o s s i n a r a t h e r a c c u r a t e way o n l y as a f u n c t i o n o f t h e c e n t r i f u g a l f o r c e . T h i s has a l r e a d y been n o t e d b y Davidson and S c h i f f i n 1 9 ^ 6 [h] .
A n e x t s t e p i s t o adopt a t i m e - i n d e p e n d e n t r e l a t i o n s h i p between r and
V j u s t as was done f o r t h e y a w - e q u a t i o n so t h a t speed l o s s i s o n l y a
2 f u n c t i o n o f r .
The g e n e r a l f o r m o f t h e f o r w a r d speed e q u a t i o n t h e n becomes:
(m - X ^ ) i i = X^u + X^r^ ( 1 2 )
C o n s i d e r i n g e q u a t i o n ( 1 2 ) t o be a 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
2 2 i t f o l l o w s t h a t t h e r e s h o u l d be a phase l a g between r and u as r
i s t h e i n p u t o f t h e system. T h i s phase l a g i s a c t u a l l y f o u n d i n t i m e h i s t o r i e s o f t u r n i n g c i r c l e t e s t s .
A p r o b l e m c o n c e r n i n g t h i s f o r w a r d speed e q u a t i o n i s whether t h e v a r i a b l e s i n t h i s case a l s o have t o be r e f e r r e d t o t h e i n s t a n t a n e o u s speed.As t h e p r o p e l l e r c h a r a c t e r i s t i c s and t h e s t r a i g h t l i n e r e s i s t a n c e a r e t h e main f a c t o r s i n t h i s p r o b l e m , w h i l e t h e i r r e l a t i o n t o t h e i n s t a n t a n e o u s speed i s r a t h e r c o m p l i c a t e d i n g e n e r a l , t h e s i m p l e s t s o l u t i o n may be f o u n d when n o n - d i m e n s i o n a l i s i n g w i t h t h e i n i t i a l speed w h i c h has o n l y l i t t l e p h y s i c a l meaning.
For s i m u l a t i o n purposes e q u a t i o n ( 1 2 ) , n o n - d i m e n s i o n a l i s e d w i t h t h e i n i t i a l speed, w i l l g i v e an u s e f u l q u a l i t a t i v e r e s u l t b u t f o r t h e a n a l y s i s o f f u l l s c a l e manoeuvres a l t e r n a t i v e forms m i g h t be more c o n v e n i e n t .
J u s t as has been done w i t h t h e y a w - e q u a t i o n , a l s o t h i s e q u a t i o n can be w r i t t e n i n a f o r m w i t h a t i m e c o n s t a n t . D i v i d i n g e q u a t i o n ( 1 2 ) by X
I
we g e t : 2 . 3 . The d r i f t - e q u a t i o n . From a h y d r o d y n a m i c a l p o i n t o f v i e w t h e sway-motion i s o f e s s e n t i a l i m p o r t a n c e b u t i f o n l y a p r e d i c t i o n o f t h e s h i p ' s p o s i t i o n i s w a n t e d , a v e r y a c c u r a t e d e s c r i p t i o n o f i t i s n o t n e c e s s a r y .F u r t h e r i t i s known f r o m e x p e r i m e n t s t h a t t h e phase l a g between d r i f t and yaw d u r i n g o s c i l l a t i n g m o t i o n s such as z i g z a g and s i n u s r e s p o n s e -t e s -t s i s r a -t h e r s m a l l .
C o n s e q u e n t l y a t i m e independent r e l a t i o n between v and r w i l l be a s u i t a b l e a p p r o x i m a t i o n . Thus: T .Ü + u = K r u u (13) 3 y = - y r - Y r and a l s o : V ^ = - y f r * - ( 1 5 ) where Y I and y ? a r e n o n - d i m e n s i o n a l c o n s t a n t s .
1 0 .
I f f u l l - s c a l e measurements o f t h e d r i f t a r e known s u f f i c i e n t l y a c c u r a t e , t h e s e c o e f f i c i e n t s can he d e t e r m i n e d . When t h e measurements a r e l e s s a c c u r a t e a l i n e a r r e l a t i o n can p o s s i b l y be o b t a i n e d and when no
measurements a r e a v a i l a b l e , an e s t i m a t i o n o f t h e p i v o t - p o i n t p o s i t i o n p r o v i d e s a v a l u e o f y^j '•
. 2< - Y * < . 5 o r 1 . 0
The l a r g e r v a l u e s o f y|* may be f o u n d f o r super t a n k e r s .
I n f i g u r e s 11 and 1 2 two examples a r e g i v e n o f t h e r e l a t i o n between
V * and T* a c c o r d i n g t o e q u a t i o n ( 1 5 )
-I n t h e f i r s t f i g u r e , c o n c e r n i n g t h e t u r n i n g c i r c l e d a t a o f t h e USS "Compass I s l a n d " a t 2 0 k n o t s t h e n o n - l i n e a r t e r m i s zero w h i l e t h e second a r e d e t e r m i n e d f r o m f u l l - s c a l e manoeuvres o f a
t a n k e r .
2.h. The complete s e t o f e q u a t i o n s o f m o t i o n . (Model l ) .
When e q u a t i o n s ( 1 0 ) , (13) and ( 1 5 ) a r e combined we have model I :
Model I . , • K 2 * *3 r + r + r + a^r + a* = o ( 1 0 ) T .Ü + u u -u-v + y ^ r * *3 + Y^r = 0 2 Ku r ( 1 5 ) ( 1 3 )
2 . 5 . Comparison o f Model I w i t h t h e A b k o w i t z model.
I f t h e y a w - e q u a t i o n ( 1 2 ) o f model I as g i v e n i n c h a p t e r 2 . 1 . i s m u l t i -p l i e d by t h e r a t i o o f i n s t a n t a n e o u s s-peed and t h e i n i t i a l s-peed w h i l e t h e d i f f e r e n c e betvreen U and i s n e g l e c t e d , t h e n we have:
( l ' - N . ' ) . r ' = r ' ( I + u') + N*— + N,, ö(l + u ' ) ^ ( 1 6 ) W z r / 1 3 ^ ^ ^, U
The terms i n t h i s e q u a t i o n m i g h t be compared w i t h t h e c o r r e s p o n d i n g terms o f t h e A b k o w i t z model.
I t must be emphasized however t h a t a q u a l i t a t i v e comparison i s n o t c o r r e c t because t h e terms i n t h e r i g h t hand s i d e o f model I r a t h e r must be c o n s i d e r e d t o have t h e same e f f e c t as a l l terms i n t h e A b k o w i t z model t o g e t h e r . T h i s i s f u r t h e r i l l u s t r a t e d on f i g u r e 10
i n w h i c h a Kempf manoeuvre o f a t a n k e r , s i m u l a t e d by an e i g h t y c o e f f i c i e n t s A b k o w i t z model, i s a p p r o x i m a t e d by one o f t h e s i m p l e n o n - l i n e a r models d i s c u s s e d h e r e . The c o n t r i b u t i o n s o f t h e p r i n c i p a l terms a r e p l o t t e d i n b o t h cases w h i l e t h e maximum v a l u e s o f t h e s m a l l e r terms a r e summarized i n t a b l e I .
Comparison o f t h e f o r w a r d speed e q u a t i o n s i s a l s o p o s s i b l e i f t h e above c o n s i d e r a t i o n s a r e k e p t i n mind.
The d r i f t e q u a t i o n s cannot p o s s i b l y be compared because i n model I no t i m e d e r i v a t i v e o f t h e d r i f t i s c o n s i d e r e d .
12.
3. The second n o n - l i n e a r model (Model I I ) .
I t has been shov^n i n s e c t i o n 2 t h a t i t i s more c o n v e n i e n t t o use t h e r a t i o o f t h e yaw r a t e and t h e f o r v m r d speed as a v a r i a b l e i n s t e a d o f t h e yaw r a t e i t s e l f . The n o n - d i m e n s i o n a l v a l u e o f t h i s r a t i o , r'**, can be c o n s i d e r e d t h e d e r i v a t i v e o f t h e course t o t h e d i s t a n c e
c o v e r e d by t h e s h i p , n o n - d i m e n s i o n a l i s e d by t h e s h i p l e n g t h , t h u s :
r*-
fj
,t
( . 7 )Going on i n t h e same way we have: ^2
r = r = * (18) ds*
So i t i s v r o r t h w h i l e t o adopt a l i n e a r r e l a t i o n between r , r and <5.
There a r e i m p o r t a n t i n d i c a t i o n s however t h a t make i t s i g n i f i c a n t t o c o n s i d e r t h e r e l a t i o n between t h e s e t h r e e v a r i a b l e s i n g r e a t e r d e t a i l .
C o n s i d e r i n g t h e t i m e h i s t o r i e s o f r , r and 6 o f one t u r n i n g c i r c l e , o r , w h i c h i s t h e same, r ' , r ' and 6, i t appears t h a t i n g e n e r a l t h e
a n g u l a r v e l o c i t y has an o v e r s h o o t w h i c h , dependent o f t h e r u d d e r a n g l e , may r e a c h v a l u e s o f 50 t o 100^ o f t h e s t a t i o n a r y y a w - r a t e . I n t h e n e x t f i g u r e t h e p r i n c i p a l c h a r a c t e r i s t i c s o f t h e s e v a r i a b l e s a r e s k e t c h e d . T y p i c a l t i m e h i s t o r i e s o f y a w - r a t e and - a c c e l e r a t i o n o f a t u r n i n g c i r c l e manoeuvre
As r ( t ^ ) = r ( t g ) = 0, w h i l e r ' ( t ^ ) >> r'(t,^) a l i n e a r r e l a t i o n o f t h e f o r m :
T r + r = K Ö ( 1 9 )
i s n o t p o s s i b l e . C o n s i d e r i n g r/U and i t s t i m e - d e r i v a t i v e t h e n i t i s n o t e s s e n t i a l i m p o s s i b l e t h a t such a r e l a t i o n e x i s t s as t h i s v a r i a b l e has no s i g n i f i c a n t o v e r s h o o t i n g e n e r a l . The n e x t f i g u r e shows a t y p i c a l p l o t o f t h e v a r i a b l e s concerned.
The shape of t h e r/U c u r v e can e a s i l y be a p p r o x i m a t e d by a s i m p l e e x p o n e n t i a l f u n c t i o n w h i l e t h e a p p r o x i m a t i o n o f r w i l l be much more c o m p l i c a t e d . C o n s e q u e n t l y i t i s more c o n v e n i e n t t o adopt an e q u a t i o n o f t h e f o r m :
t h e s o l u t i o n o f w h i c h f o r c o n s t a n t r u d d e r a n g l e f i t s t h e above t i m e h i s t o r i e s p e r f e c t l y .
These c o n s i d e r a t i o n s a r e t h e base f o r model I I i n w h i c h , as a p a r t i -c u l a r i t y , n o t t h e t i m e i s used as a r e f e r e n -c e , b u t t h e d i s t a n -c e c o v e r e d by t h e s h i p i n s h i p l e n g t h s .
3 . 1 . The y a w - e q u a t i o n .
T h i s e q u a t i o n d e s c r i b e s t h e s t e a d y s t a t e s o f t u r n i n g c i r c l e s i n t h e same way as t h e c o r r e s p o n d i n g e q u a t i o n o f model I , t h u s
+ r * + a J r- * 2 + r^^^ = K'^Ö ( 2 1 )
I n t h e t r a n s i e n t phase o f manoeuvres a t e r m T . r i s added so t h a t t h e y a w - e q u a t i o n i s :
T * r * + r*+ a* r*^ + r ^ ^ + a* = 1 * 6 ( 2 2 )
The d i f f e r e n c e between t h i s e q u a t i o n and t h e c o r r e s p o n d i n g e q u a t i o n o f model I f o l l o w s f r o m t h e e v a l u a t i o n o f r :
Lr
d r * L U /L\ . * LÓ
^ = di-^=ïï • -dF" iïïi ^ - ;^ ^^^^
3 . 2 . The f o r w a r d speed e q u a t i o n .
T h i s e q u a t i o n i s based upon t h e a s s u m p t i o n t h a t t h e r e l a t i v e speed r e d u c t i o n i n t h e s t e a d y s t a t e o f a t u r n i n g c i r c l e i s p r o p o r t i o n a l t o t h e squa.re o f t h e n o n - d i m e n s i o n a l y a w - r a t e r . I n o t h e r w o r d s , t h i s speed r e d u c t i o n i s p r o p o r t i o n a l t o t h e square o f t h e i n v e r s e o f t h e s t e a d y t u r n i n g r a d i u s . Thus: o D e f i n i n g i n t h i s case u = — and u s i n g s as t h e i n d e p e n d e n t v a r i a b l e t h e complete e q u a t i o n becomes: ( 2 5 )
3 . 3 . The d r i f t - e q u a t i o n .
The same e q u a t i o n i s used as i n Model I , t h u s
i t - . * * , - v t - n B
V + Y ^ r + = 0 ( 1 5 )
. The complete s e t o f e q u a t i o n s o f m o t i o n (Model I I ) :
Combining e q u a t i o n s ( 2 2 ) , ( 1 5 ) and ( 2 5 ) we have
Model 1 1 . T * r * + r* + a* r""^ + a* r*^ + a * — 2 3 o ( 2 2 ) V * + Y* i - " ^ + Y 3 1-*^ = 0 ( 1 5 ) T* ü % u * u = r* 2 u ( 2 5 )
3 . 5 . Comparison o f model I I w i t h t h e A b k o w i t z model.
A comparison between t h e s e models i s o n l y p o s s i b l e i f t h e e x p r e s s i o n f o r d r V d s * as g i v e n i n e q u a t i o n ( 2 3 ) i s used whereas t h e t e r m c o n t a i n i n g t h e f o r w a r d speed a c c e l e r a t i o n o f t h i s e q u a t i o n i s e l i m i n a t e d by
s u b s t i t u t i n g t h e e x p r e s s i o n f o r t h i s a c c e l e r a t i o n as may be d e r i v e d f r o m e q u a t i o n ( 2 5 ) . The r e s u l t s o f t h i s p r o c e d u r e i s model I I , n o n - d i m e n s i o n a l i s e d w i t h t h e i n i t i a l speed U^.
The terms i n t h i s m o d i f i c a t i o n o n l y c o n t a i n t h e v a r i a b l e s r ' , u' and ö
whereas t h e c o e f f i c i e n t s o f t h e y a w - e q u a t i o n a r e composed o f t h e c o e f f i c i e n t s o f b o t h t h e yaw- and f o r v j a r d speed e q u a t i o n i n t h e o r i g i n a l f o r m .
A comparison o f 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 o f t h e A b k o w i t z model i s n o t p o s s i b l e f o r t h e same reasons as mentioned i n cha.pter 2 . 5
-1 6 . A n a l y s i s o f f u l l - s c a l e manoeuvres. The c o m p u t a t i o n - p r o c e d u r e . The method o f a n a l y s i n g f u l l - s c a l e d a t a i s i l l u s t r a t e d by 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 model I I , u s i n g t h e t u r n i n g c i r c l e d a t a o f t h e USS "Compass I s l a n d " a t 20 k n o t s , g i v e n i n .
F i g u r e s 2 - 3 3 and 2 - 37 o f t h a t paper were used t o 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 t h e yaw and speed e q u a t i o n w h i l e t h e d a t a o f s t e a d y -t u r n i n g d r i f -t a n g l e s o f -t a b l e 2 - 1 were used -t o f i n d -t h e c o e f f i c i e n -t s o f t h e d r i f t - e q u a t i o n .
The measured v a l u e s o f a n g u l a r v e l o c i t y and f o r w a r d v e l o c i t y were f i t t e d and d i f f e r e n t i a t e d by a computer-program (M09). About 5 0 p o i n t s p e r t u r n i n g c i r c l e were used. The s t a t i o n a r y v a l u e s o f 3, r and U p r o v i d e d t h e c o e f f i c i e n t s and u s i n g a l e a s t squares a n a l y s i s . I n t h e same way t h e c o e f f i c i e n t K* v^as d e t e r m i n e d . U s i n g t h e d e r i v a t i v e s o f r * and u * as w e l l as t h e known c o e f f i c i e n t s , t h e r e m a i n i n g " t i m e
-c o n s t a n t s " TJ*" and T* were f o u n d . (Program no. M03).
F i n a l l y t h e v a l u e s o f 3, r and U were computed by s o l v i n g t h e d i f f e r e n t i a l e q u a t i o n s by means o f t h e Runge-Kutta p r o c e d u r e ( P r o g r . no. M 0 8 ) .
A scheme o f t h e complete p r o c e d u r e i s g i v e n i n f i g u r e 13. C o n c e r n i n g t h e computer-program M03 i t i s m e n t i o n e d t h a t t h e
d e s c r i p t i o n o f an a r b i t r a r y s e t o f measured d a t a i s done by a c h o i c e o f a number o f terms o f t h e f o r m a . v ^ . r"'' . ö"^ . u'^ ,
P
where p 4 7 0 and k + l + m + n ^ i + . A p a r t i c u l a r p r o p e r t y o f t h e program i s t h a t t h e c o e f f i c i e n t s w h i c h g i v e no s i g n i f i c a n t c o n t r i b u t i o n are o m i t t e d w h i l e t h e r e m a i n i n g c o e f f i c i e n t s are computed a g a i n . The r e s u l t s o f t h e "Compass I s l a n d " - d a t a a n a l y s i s . On f i g u r e 2 t h e s t e a d y t u r n i n g r a t e , measured on f u l l - s c a l e , i s p l o t t e d a g a i n s t t h e r u d d e r a n g l e . A p p a r e n t l y t h i s c u r v e shows a v e r y l a r g e asyrmnetry-ef f e c t .
P l o t t i n g however rJ* t h i s e f f e c t n e a r l y d i s a p p e a r s . The l e a s t squares a n s l y s i s gave t h e f o l l o w i n g a p p r o x i m a t i o n (see f i g . 3 ) :
6 = .ook - . T 0 2 r* - .hTO r^- 2 . 1 2 8 r* ^ o r K* = - 1 . 1 | 2 ^ a * o = - . 0 0 6 = +1 ( t h e s h i p i s s t a b l e ) = + . 6 6 9 o,* = + 3 . 0 3 0 U s i n g t h e s e c o e f f i c i e n t s , T * was f o u n d : T* = 2.123. E x p r e s s i n g t h e s e c o e f f i c i e n t s i n t o n o n - d i m e n s i o n a l moment c o e f f i c i e n t s I I _ 5 1 ) and assuming I - N. = 8 7 . 1 0 Zz r we f i n d : N^* = - 5 8 . 3 . 1 0 " ^ N * = + . 2 " o N * = - 1 + 1 . 0 " N * = - 2 7 . 1 4 " = - I 2 U . 2 " On f i g u r e 1^+ t h e f i n a l speed-loss o f t h e f u l l - s c a l e t r i a l s i s p l o t t e d v e r s u s t h e r u d d e r a n g l e . T h i s i s n e a r l y a s t r a i g h t l i n e . On f i g u r e 1 5 •it 2 t h e s p e e d - l o s s i s p l o t t e d v e r s u s r ^ , t h e l a . t t e r v a l u e s r e a d up from, f i g u r e 3 . A l s o t h i s r e l a t i o n i s l i n e a r and i s a p p r o x i m a t e d by: u * = - 2 . 0 6 6 r c t h u s K* = - 2 , 0 6 6 . * 2 u U s i n g t h i s v a l u e and t h e s - d e r i v a t i v e s o f t h e speed we f i n d T * = 2 . 6 7 7 . u The c o r r e s p o n d i n g n o n - d i m e n s i o n a l f o r c e d e r i v a t i v e s , based on m' _ X, = 8 8 0 , " 1 0
' a r e :
u X* = - 3 2 9 . 1 0 " ^ X* = - 6 7 9 . . 1 0 " ^ These v a l u e s a r e g i v e n i n r e f e r e n c e 91 8 .
The I ' e l a t i o n s h i p between v and r was dei'ived f r o m t h e f i n a l v a l u e s o f d r i f t , speed and y a w - r a t e : v * = _ 0 . ^ 3 2 r * so t h a t = .1+32 = 0 On f i g u r e 11 a p l o t o f v v e r s u s r i s g i v e n , c c
The c o e f f i c i e n t s o f model I I , based on t h e f u l l - s c a l e d a t a o f v^ , r ( t ) , 6 ( t ) and U ( t ) c o n c e r n i n g t h e r i g h t t u r n i n g c i r c l e s a t 2 0 k n o t s o f t h e USS "Compass I s l a n d " a r e t h u s : USS "Compass I s l a n d " R i g h t t u r n i n g c i r c l e s T * = 2 , . 1 2 3 a * 0 . 0 0 6 o a 1 a 2 * = + 1 . 0 0 0 a 1 a 2 * = + 0 . 6 6 9 = + 3 . 0 3 0 — 1 .1+21+ - - 0 . ^ 3 2 T = 2 . 6 7 7 K u 2 . 0 6 6
The r e s u l t s o f t h e computed manoeuvres w i t h t h e s e c o e f f i c i e n t s a r e g i v e n on f i g u r e s 1 6 and 1 7 showing t i m e - h i s t o r i e s o f r and U. On f i g u r e s I 8 up t o 2 2 some secundary q u a n t i t i e s a r e g i v e n t o be compared w i t h t h e f u l l - s c a l e v a l u e s . These a r e : I F i n a l d i a m e t e r I I T a c t i c a l d i a m e t e r I I I Advance IV T r a n s f e r V L o c a t i o n o f P i v o t P o i n t vs r u d d e r a n g l e 11 It
I t must be emphasized however t h a t t h e s e q u a n t i t i e s a r e n o t used f o r t h e c o m p u t a t i o n o f t h e c o e f f i c i e n t s : To g e t an o p t i m a l f i t o f t h e p l o t s o f X and Y one s h o u l d a l s o ^ ^ ^ o o use t h e d a t a o f X ( t ) and Y ( t ) . o o
T h i s was n o t t h e purpose f o r t h i s paper.
C o n c l u d i n g r e m a r k s .
The a p p r o x i m a t i o n o f t h e manoeuvring p r o p e r t i e s o f a s h i p by a s i m p l e n o n - l i n e a r s e t o f e q u a t i o n s o f m o t i o n seems t o be v e r y u s e f u l . U s i n g d i g i t a l computer t e c h n i q u e s i t w i l l be p o s s i b l e t o d e r i v e t h e c o e f f i c i e n t s o f such models f r o m f u l l - s c a l e manoeuvres, p r o v i d e d t h e measurements o f y a w - r a t e and f o r w a r d speed a r e
s u f f i c i e n t l y a c c u r a t e . P a r t i c u l a r l y c o n c e r n i n g t h e f o r v r a r d speed i t o f t e n appears t h a t t h e phase as w e l l as t h e magnitude g i v e n by t h e s a l l o g a r e u n r e l i a b l e , t h e l a t t e r b e i n g i n f l u e n c e d by t h e a-symmetric p o s i t i o n o f t h e s a l l o g .
At t h e moment o t h e r f u l l s c a l e d a t a a r e a n a l y s e d t o 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 model I I and i t may be e x p e c t e d t h a t t h i s work w i l l p r o v i d e f u r t h e r i n f o r m a t i o n about t h e m e r i t s o f t h i s model.
R e f e r e n c e s . 1 Nomoto, K. On t h e s t e e r i n g q u a l i t i e s o f s h i p s ISP J u l y 1 9 5 7 2"j A b k o w i t z , M.A. L e c t u r e s on S h i p Hydrodynamics - S t e e r i n g and M a n o e u v r a b i l i t y HyA R e p o r t H y - 5 , December I 9 6 I + 3^ Morse, R.V. and P r i c e , D. 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
hj D a v i d s o n , K.S.M. and S c h i f f , L . J . T u r n i n g and course k e e p i n g q u a l i t i e s SNAME I 9 I+ 6
1^5^ Nomoto, K.
Frequency response r e s e a r c h on s t e e r i n g q u a l i t i e s o f s h i p s Techn. Rep. Osaka U n i v e r s i t y , v o l . 8 , no. 29^+
Eda, H. and Crane, C L .
Research on s h i p c o n t r o l l a b i l i t y ( s t e e r i n g c h a r a c t e r i s t i c s o f a cargo s h i p ) D.L. r e p o r t . Stevens I n s t i t u t e o f T e c h n o l o g y , 1902. 7 N o r r b i n , N.H. P r o v e t s t e k n i k och a n a l y s S t a t e n s S k e p p s p r o v n i n g s a n s t a l t . June I 9 6 5
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Nomoto, K. Some n o t e s on t h e a p p l i c a t i o n o f K-T a n a l y s i s t o m a r g i n a l l y s t a b l e s h i p s .Leeuwen, G. van and G l a n s d o r p , CO.
E x p e r i m e n t a l d e t e r m i n a t i o n o f l i n e a r and n o n - l i n e a r l a t e r a l hydrodynamic d e r i v a t i v e s o f a " M a r i n e r " - t y p e s h i p model R e p o r t no. 1^15, March 1 9 6 6 , S h i p b u i l d i n g Department U n i v e r s i t y o f Technology, D e l f t
m mass o f t h e s h i p
I moment o f i n e r t i a o f t h e s h i p
zz ^ f o r c e i n space bounded c o o r d i n a t e system
moment i n space bounded c o o r d i n a t e system f o r c e i n space bounded c o o r d i n a t e system Y l a t e r a l f o r c e , p o s i t i v e t o s t a r b o a r d N moment, p o s i t i v e t o t h e r i g h t X l o n g i t u d i n a l f o r c e Y. added mass i n l a t e r a l d i r e c t i o n V N. added moment o f i n e r t i a r X. added mass i n l o n g i t u d i n a l d i r e c t i o n u i n i t i a l f o r w a r d speed o f t h e s h i p speed i n l o n g i t u d i n a l d i r e c t i o n U i n s t a n t a n e o u s speed ( v e c t o r ) L l e n g t h o f t h e s h i p between p.p. ds = U d t V sway v e l o c i t y r a n g u l a r v e l o c i t y 6 . r u d d e r a n g l e , p o s i t i v e t o p o r t u speed l o s s i n l o n g i t u d i n a l d i r e c t i o n ( u = U - U ) X O V s w a y - a c c e l e r a t i o n r a n g u l a r a c c e l e r a t i o n Ü a c c e l e r a t i o n i n l o n g i t u d i n a l d i r e c t i o n ds' v' u = U d t / L o = v/U o = r . L/U c ds' d_ ds'
L i s t o f symbols ( c o n t i n u e d ) d s * =Udt/L V * =v/U =Lr/U = 1 ^ . ^ u * =u/U o r u/U • *• a ^ = d ^ ^ p . * d ^e d ^ d^*
' =
7^ 2 ^ . * d u =—i u ds"* ds" 0,1,2,3,14 P r o p o r t i o n a l i t y f a c t o r s i n t h e yaw e q u a t i o n 1>2 P r o p o r t i o n a l i t y f a c t o r s i n t h e f o r w a r d speed e q u a t i o n N ^ =N^/^puV
X* = X /^pUL^ ' - ' I ,1
= N^/^pUL-^ X^^=X^/^pL^ - - / i p L ^ 6 ^2 = N2/ipL = W^U/ipL N j * = Nj^/^pU^L^ 1,3 P r o p o r t i o n a l i t y f a c t o r s i n t h e d r i f t e q u a t i o n m' - Y^' = (m - Y^) / ^pL^ I - w/ = ( I - N. ) / i p L ^ zz,r
zz r !; m' - X^ = (m - X^) / gpL T * = (T ' - N.') / !N*| zz r 1 K* = N^/ |N*|C
= (m- - X. ) / - X f - x^/ - x ; (W&S 7 5 0 5 )T A B L E I '=K'5 a; a; K* 0) •1 . 0 - 1 4 0 0 s
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.1 • 2 0 ^ - 2 - 8 0 0I
(3) .1 . 6 1 2 - 5 - 6 0 0 E 0 . 1 - - 6 8 6 -1 -10-20 . 5 - 6 0 0 yI
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-1 - 6 1 2 . 2 - 8 0 0 yI
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-1 - 4 0 8 . 1 - 4 0 0k
-5 +•1 • 5o(RAD)FIG. 9 EXAMPLES OF STABLE AND UNSTABLE TURNING CIRCLE CHARACTERISTICS
TERM MAX. VALUE TERM MAX. VALUE N'rvv -rV'^ 11-5 N 5 5 'B' 2-7 N'rrr T ' ' 9-6 N'v56u-v'6V 1-5 Nrvvu-r'v'^u' 7-5 N'vvv •v''' Nrrru-r'V 6-8 N'örru'Br'V 1-2 Nv56 -v'B'
i,7
N'raeu-r'B'u' 1-2 N'err -Br''A-S
N'sesu-B^u' •9 Nvrr -Vr'' 4-3 N65U -B^u" •9N'svv -Bv'^ 4 0 Nvvvu'V'^u' •9 N'v -v' 3-3 Nr65 'T'B^ •6 N'vrru-v'r'V 2-9 N'vr5 -v'r'S
Nsse •B"' 2-8
(See also fig. 10)
MAXIMUM CONTRIBUTION OF NON-LINEAR TERMS
M 0 9 Vc — r { t ) — C D 6 ( t ) — U ( t ) — C . D l I M 0 9 I - V c' - I 5 ( s ) - -u'(s-)- l-rV)- -Cf{s')-L S . C S . M03 u; r'(s' [si M03 M03 M03 + C S . v'+fr*+gr hu'+ju'+r* . 3 •f-•b¬ ——c¬ d¬ —e- —a-Ms')¬ hj -M 0 8 RUNGE KUTTA V*= r'= 5*= u*= x'= M 0 8 - V , V , V ' -r.r'.r* - 6 -u,u',u* -v,v',v* -r.r'.r* -ü.ü'.ü* -x,x' - y . y ' '<i> -P -u - t c D L.S. C S .
CURVE FITTING PROCEDURE DIFFERENTIATING PROCEDURE LEAST-SQUARES PROCEDURE COEFFICIENT-SELECTION PROCEDURE
FIG.13
SCHEME OF THE COMPUTING PROCEDURE
• • • „ 6o=-35° O " O O 5„=-10° O A O D „Compass Island" Model n •TIME (MINUTES)
FIG.16 MEASURED AND COMPUTED VALUES OF r ' ( t )
„Compass Island" Model n
- 6 0 - 3 0 ° -20° -10° FIG. 18 MEASURED AND COMPUTED
VALUES OF FINAL DIAMETER
„Compass Island" Model n (YDS) • 2000 - 1000 -5o - 3 0 ° (YDS) 2000 1000 -20" -10" FIG.19 MEASURED AND COMPUTED
