On the utility of free-running models in researches into manoeuvrability. Part 1 and 2

D.S.n.A. Tr, I: IS I n t.i on No. 21 59 On t h e u t i l i t y o f f r o o - r u n n i n g m o d e l s i n r o a e a r c h e s i n t o m a n o e u v r a b i l i t y ( I ) ( A l e c t u r e d e l i v e r e d a t t h e s p r i n g s e s s i o n o f t h e J . S o c . Nav. A r c h . J a p a n , May 1961) R e c e i v e d on 20th December, 1960 By K e n s a k u NOMOTO. o f t}ie E n g i n e e r i n g D e p a r t m e n t , O s a k a U n i v e r s i t y ; Moniber o f t h e A s s o c i a t i o n . H i s a y o s h i TATANO, o f t h e E n g i n e e r i n g D e p a r t m e n t , O s a k a U n i v e r s i t y ; Member o f th© A s s o c i a t i o n . A k i h i s a MURASE, G r a d u a t e S t u d e n t , O s a k a U n i v e r s i t y ; Member o f t h e A s s o c i a t i o n . I n t r o d u c t i o n R e s e a r c h i n t o t h e p r o b l e m s o f t u r n i n g m a n o e u v r a b i l i t y b y t h e u s e o f f r e e - r u n n i n g m o d e l s was u n d e r t a k e n b e f o r e t h e w a r , b e g i n n i n g w i t h t h e w e l l - k n o w n t e s t s o f D r . A k a z a k i 1 J , and i n r e c e n t y e a r s , b e c a u s e ol' a d e e p e n i n g o f i n t e r e s t i n t h i s p r o b l e m , t h i s t y p e o f t e s t h a s b e e n w i d e l y c a r r i e d o u t t h r o u g h o u t t h e v o r l d 2 ] [ 3 j . ::owever, i n t h i s k i n d o f e x p e r i m e n t , i t i s n o t y e t .. . t e r m i n e d what k i n d o f s t e e r i n g s h o u l d be f i t t e d t o i.e 1 r o e - r u n n i n g model f o r t h e p u r p o s e o f m a k i n g o b s e r v a t i o n s , ar.d what f a c t o r s s h o u l d be t a k e n i n t o a c c o u n t when

::..rrying o u t t h e a n a l y s i s . I t h a s l o n g b e e n o b s o r v e d i.::at t u r n i n g and b a s i c s t e e r i n g t e s t s a r e i m p o r t a n t . However i t i s n o t n e c e s s a r i l y r e a s o n a b l e t o d e c i d e f, .ii s t e a d y t u r n i n g t e s t s o n l y t h e p o r f o r m a n c e o f a s h i p .a t h e u n s t a b l e m o t i o n w h i c h o c c u r s u n d e r a c t u a l r u n n i n g c o n d i t i o n s . Hence v,e a l s o c o n s i d e r a t t h e same t i m e s t e e r i n g t e s t s < u n d e r more u n s t e a d y c o n d i t i o n s t h a n t u r n i n g t e s t s , e . g . t h e

s o - c a l l e d Z t e s t s e t c . T i i e r e a r c a l s o many o t h e r q u e s t i o n s w h i c h a r i s e i n t h e methods o f t e s t a n a l y s i s -e . g . how to o b t a i n a g -e n -e r a l -e x p r -e s s i o n o f m a n o -e u v r a b i l i t y f r o m t h e m o t i o n o f a f r e e - r u n n i n g m o d e l , f o r p a r t i c u l a r t y p e s o f s t e e r i n g , w h e t h e r f o r u n s t e a d y s t e e r i n g o r t u r n i n g t e s t s . 1 I n t h e S h i p b u i l d i n g R e s e a r c h U n i t o f O s a k a U n i v e r s i t y r e s e a r c h h a s b e e n c a r r i e d on c o n t i n u o u s l y o v e r a number o f y e a r s on t u r n i n g m a n o e u v r a b i l i t y u s i n g f r e e -r u n n i n g models,, and -r e s u l t s h a v e b e e n p u b l i s h e d f o -r two o r t h r o e t y p e s o f m e r c h a n t vesBel[k][5] and much t e s t e x p e r i e n c e c o n c e r n i n g methods o f a n a l y s i s and m e t h o d s o f r e s e a r c h i n t o m a n o e u v r a b i l i t y u s i n g f r e e - r u n n i n g m o d e l s h a s b e e n a c c u m u l a t e d . ^ ' I n t h i s k i n d o f r e s e a r c h , i t i s i m p o r t a n t n o t m e r e l y t o i n d i c a t e d i r e c t l y t h e m o t i o n o f t h e s h i p f o r a s p e c i a l k i n d o f s t e e r i n g u s e d i n t h e t e s t s , b u t t o o b t a i n f r o m t h o s e r e s u l t s t h e g e n e r a l m a n o e u v r a b i l i t y c h a r a c t e r i s t i c s o f t h a t s h i p t y p e . By s o , d o i n g i t I s p o s s i b l e t o d e t e r m i n e t h e r e s p o n s e o f t h e s h i p to t h e v a r i o u s m a n o e u v r e s w h i c h a r e c a r r i e d o u t i n p r a c t i c e , and to a n t i c i p a t e t h e a p p l i c a t i o n t o d e s i g n . A g a i n , c o n s i d e r a t i o n o f t h e h y d r o d y n a m i c t r e a t m e n t i n b l a d e - t h e o r y c a l c u l a t i o n s and t h r e e - p a r t p o w e r t e s t s on f r e e - r u n n i n g m o d e l s i n t h e t u r n i n g t a n k may s u p p l e m e n t t h e k n o w l e d g e g a i n e d f r o m t h i s r e s e a r c h . One e f f e c t i v e method f o r t h e m o t i o n o f a f r e e -r u n n i n g model u s i n g e q u a t i o n s o f m o t i o n , i s t h e a n a l y s i s o f f r e q u e n c y r e s p o n s e ( r e s p o n s e a m p l i t u d e o p e r a t o r i n s t a t i s t i c a l c s c l l l a t i o n t h e o r y o r t r a n s m i s s i o n f u n c t i o n ) . On t h i s b a s i s , i f we d e s c r i b e t h e method o f a n a l y s i n g s t e e r i n g t e s t s on f r e e - r u n n i n g m o d e l s , we o b t a i n *

. 3 ¬ 1 ) t h e r e s u l t s w h i c l i f o l l o w . I . e . t h e e q u a t i o n o f motion" a s s u m i n g l i n e a r i t y , i s (1) w h o r e ^ , / ^ , » : t u r n i n g a n g u l a r v e l o c i t y , d r i f t a n g l e a n d : ' r i i d d e r a n j ' l e r u d d e r a n f j l e V, L : s h i p ' s s p e e d a n d l e n g t h i : a p p a r e n t mass o f s h i p l o n g i t u d i n a l l y ^ and l a t e r a l l y , and 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 o f a p p a r e n t moment o f i n e r t i a a b o u t t h e v e r t i c a l a x i s /- 0 ^,C^f3,Crqr^. 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 f o r c e • '^-^ and moments a c t i n g on t h e s h i p : r e s i s t a n c e d e r i v a t i v e C v ^ C A J S • 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 f o r c e and ' * ^ moment e x e r t e d b y s t e e r i n g on t h e r u d d e r I f wo e l i m i n a t e t h e d r i f t a n g l e / ^ a n d a d j u s t

• /

A-P u r t h e r ^ i f we maLke t h e s e n o n - d i m e n s i o n a l , w e o b t a i n K , T 1.2.3 a r e e s t a b l i s h e d f r o m t h e 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 n l y o f t h e o r i g i n a l e q u a t i o n ( l ) Now, i f we l e t ^ ( t ) be t h e s t e e r i n g g i v e n i n t h e f r e e - i n i n n i n g m o d e l , and ^ ( t ) t h e r e s u l t i n g m e a s u r e d t u r n i n g a n g u l a r v e l o c i t y , t h e n t h e t r a n s m i s s i o n f u n c t i o n i s g i v e n by ^uj The r e a l number p a r t ^ ^ o f p o c c u r s b e c a u s e t h e a b o v e two i n t e g r a t i o n s d i v e r g e b u t f o r a s h i p t y p e w h i c h h a s d i r e c t i o n a l s t a b i l i t y , 0 ^ . , -and t h i s i n t e g r a t i o n c o n v e r g e s . A c c o r d i n g l y , llmYMi^+i»)-'-'-^ - ^ é i ) . .._ - — . •-B e c a u s e most s h i p s h a v e d i r e c t i o n a l s t a b i l i t y , we make t h i s e q u a t i o n t h e b a s i s o f o u r a n a l y s i s , and h e n c e f o r t h w r i t e ^ i m Yg ±ii) a s Yg (itlk). I n t h e c a s e o f d i r e c t i o n a l i n s t a b i l i t y , f u n d a m e n t a l l y a s i m i l a r s o l u t i o n i s p o s s i b l e , t a k i n g ^ > / 7 ^ , b u t i n p r a c t i c e many d i f f i c u l t i e s a r i s e due to n o n - l i n e a r e f f e c t s . I f we t a k e t h e <? ( t ) o f t h e u s u a l f o r m by n u m e r i c a l I n t e g r a t i o n o f e q u a t i o n ( 3 ) Y {±6i) i s d e t e r m i n e d s f o r a n y «0 , f r o m m e a s u r e d S ( t ) and B i t ) . F o r s p e c i a l v a l u e f l ^ o f i . e . when = o i n t u r n i n g t e s t s , and f o r t h e s I-o f s i n u s I-o i d a l s t e e r i n g p e r i I-o d s , nI-o i n t e g r a t i I-o n i s p e r f I-o r m e d [ and t h e v a l u e s o f e q u a t i o n ( 3 ) a r e o b t a i n e d d i r e c t l y f r o m ^ \

5 -and _{B . I n a n y c a s e , o v e r w i d e l i m i t s o f <s), i f we o b t a i n} t h e v a l u e o f K ( l + ±vt Tn) - Y„ ( i W ) (1 + i w T j U + i t r t ' o j b e c a u s e h e r e t h e unknown q u a n t i t i e s a r e t h e f o u r "^l» '^2' ^ 3 ' ^* ^® p o s s i b l e t o d e t e r m i n e t h e s e v a l u e s by s i m u l t a n e o u s e q u a t i o n o r t h e method o f l e a s t s q u a r e s . A c c o r d i n g l y , e q u a t i o n ( 2 ) o f t h e m a n o e u v r e m o t i o n f o r t h a t s h i p t y p e i s o b t a i n e d , and i t i s p o s s i b l e t o c a l c u l a t e t h e m o t i o n o f t h e s h i p f o r a n y s t e e r i n g . Again, t h e s e f o u r c o n s t a n t s r e p r e s e n t d i r e c t l y a l l t h e c h a r a c t e r i s t i c s o f t h e s h i p i n t h e v a r i o u s s t e e r i n g m o t i o n s , K r e f e r s t o th© s t r e n g t h o f t h e t u r n i n g power w h i c h c a n be s t e a d i l y d i s p l a y e d , , T^, r©plr©s©nt th© s p e e d o f r e s p o n s e o f t h e s h i p t o s t e e r i n g , a c c o r d i n g t o p h a s e s , and T^^, r e p r e s e n t t h e c o u r s e s t a b i l i t y [ 4 ] . Hence, d e n o t i n g t h e s e f o u r c o n s t a n t s a s t h e i n d i c e s o f m a n o e u v r a b i l i t y , i t i s póssible t o c o n s i d e r them a s a g e n e r a l e x p r e s s i o n o f t h e m a n o e u v r a b i l i t y o f t h a t s h i p . F u r t h e r m o r e , t h e method o f u s i n g | ( i W ) / i n t h e d e t e r m i n a t i o n o f K , T^^ e t c . from Y^ (i«) f o r a w i d e r a n g e o f U i s p o s s i b l e f r o m c o n s i d e r a t i o n o f

A r g Yg (i©) and shows a good c o r r e s p o n d e n c e w i t h t h e t e s t e x a m p l e s , a n d [ ó ] b e c a u s e | Y J i s e a s i e r t o c a l c u l a t e and r e l i a b i l i t y i s good, we u s e t h i s . I f we f o l l o w t h e a b o v e p r i n c i p l e s , i t s h o u l d be p o s s i b l e t o u n d e r s t a n d c o m p l e t e l y th© m a n o e u v r a b i l i t y from o b s e r v a t i o n o f t h e m o t i o n o f t h e s h i p f o r one p a r t i c u l a r t y p e o f s t e e r i n g , b u t i n p r a c t i c e t h e r e a r e c o m p l i c a t i o n s . The r e s u l t s o f t h e c a l c u l a t i o n s o f e q u a t i o n ( 3 ) show t h a t f o r a p a r t i c u l a r m o t i o n a l a r g e v a l u e o f Yg ( i t o ) o c c u r s f o r c e r t a i n l i m i t s o f <d, and a s m a l l v a l u e f o r o t h e r s .

-6-Hence, i t o f t e n happens t h a t l a r g e d e v i a t i o n s i n t h e v a l u e o f Y„ (i<^) o c c u r , even when t h e r e i s s l i g h t l o c a l d i s t u r b a n c e r e s u l t i n g from e x t e r n a l d i s t u r b a n c e due to w i n d , e t c . , and t h e r e s u l t o f t h e i n i t i a l m o t i o n o f t h e s h i p . I n o r d e r to e v a l u a t e t h e f o u r i n d i c e s K, e t c . , w i t h a f a i r d e g r e e o f a c c u r a c y , i t i s n e c e s s a r y t o d e t e r m i n e lY„(ifc>) j f a i r l y a c c u r a t e l y f o r f a i r l y w i d e l i m i t s o f &», b u t i t i s d i f f i c u l t , f o r t h e r e a s o n s g i v e n above, to d e t e r m i n e t h a t from o n l y one p a r t i c u l a r m o t i o n . By r e p e a t i n g m o t i o n s , w i t h v a r i o u s k i n d s o f s t e e r i n g , and c o m b i n i n g t h e r e s u l t s |Yg | may be found w i t h

r e a s o n a b l e a c c u r a c y w i t h i n w i d e l i m i t s o f W. A g a i n , s i n c e t h e v a l u e s o f t h e r e s i s t a n c e d e r i v a t i v e s Cjy^, CyCi change, a c c o r d i n g t o t h e d e v e l o p m e n t o f t h e t u r n i n g m o t i o n , and t h e s p e e d d r o p s i n p r o p o r t i o n to t h e s t e e r i n g , t h e i n d i c e s K, e t c . a l s o change l i k e w i s e . Hence, i t i s n e c e s s a r y to d e t e r m i n e K, T^^ e t c . f o r g i v e n s t r e n g t h s o f t u x m i n g m o t i o n and c o n s e q u e n t l y many r u n s a r e r e q u i r e d . From t h e s e c i r c v i m s t a n c e s , a l l t u r n i n g t e s t s and s i n u s o i d a l s t e e r i n g t e s t s a r e combined t o g e t h e r , and t h e d e t a i l s o f t h e s e a r e g i v e n i n v a r i o u s s e c t i o n s . . I f we c o n s i d e r t h e r e l a t i o n s w i t h t h e hydrodyn£unic t r e a t m e n t o f t h e t u r n i n g s t e e r i n g t o u c h e d upon e a r l i e r , we may hope to p r o g r e s s w i t h t h e method o f a n a l y s i s to f i n d t h e r e s i s t a n c e d e r i v a t i v e and n o t m e r e l y t h e m a n o e u v r a b i l i t y i n d i c e s K, e t c . U s i n g t h e a n a l y s i s f o r t u r n i n g m o t i o n r e f e r r e d t o above f o r d r i f t m o t i o n i f we d e d u c e t h e a p p a r e n t mass o f t h e s h i p by c a l c u l a t i o n and o t h e r methods, i t i s p o s s i b l e to deteznnine t h e r e s i s t a n c e d e r i v a t i v e from t e s t s w i t h f r e e - r u n n i n g m o d e l s . I t i s

u n f o r t u n a t e t h a t i t i s more d i f f i c u l t to f i n d t h e d e r i v a t i v e a c c u r a t e l y by t h i s i n d i r e c t method t h a n by u s i n g t h e t u r n i n g t a n k , b u t i f we d e t e r m i n e t h e d e r i v a t i v e

-7-c l o s e to t h e advan-7-ce p o s i t i o n i n t h i s way, i t has t h e v i r t u e t h a t we o b t a i n a l s o the d e r i v a t i v e f o r a r e a l s h i p and so by combining the t u r n i n g tank t e s t s and hydrodyneunic t r e a t m e n t i t i s p o s s i b l e to e x p e c t developments i n t h i s f i e l d . T h i s type o f a n a l y s i s i s c o n s i d e r e d e s p e c i a l l y u s e f u l i n f i n d i n g out t h e r e l a t i o n s between the s h i p type and m a n o e u v r a b i l i t y . Furthermore,

i n c o n n e c t i o n w i t h t h i s q u e s t i o n , we must a l s o c o n s i d e r i n free-zninning model t e s t s the t o t a l - l e s s - r u d d e r moment proposed by Davidson and S c h i f f , i . e . the method o f f i n d i n g out the hydrodynamic moment a c t i n g on the s h i p i n s t e a d y t u r n i n g , by measuring the d i r e c t p r e s s u r e on the rudder.

Next t h e r e i s the s o - c a l l e d Z t e s t , but t h i s s t e e r i n g method h a s not been s p e c i a l l y a p p l i e d . t o t h e frequency r e s p o n s e a n a l y s i s u s i n g e q u a t i o n ( 3 ) . However t h i s s t e e r i n g c e r t a i n l y r e p r e s e n t s w e l l a l l phases o f s t e e r i n g a r e a l s h i p , and t h e r e i s t h e a t t r a c t i o n t h a t t h e r e a l r e a d y e x i s t s much m a t e r i a l on r e a l s h i p s , s i n c e i t i s p o s s i b l e to c a r r y i t out s i m p l y on them. Hence, u s i n g an e q u a t i o n o f motion made a s s i m p l e a s p o s s i b l e by changing the s t a n d p o i n t a l i t t l e from t h e above a n a l y s i s , we may o a s i l y get a rough i d e a o f the m a n o e u v r a b i l i t y o f t h a t s h i p from one t e s t o n l y . T h i s i s perhaps an a p p r o p r i a t e method o f a n a l y s i s f o r t h i s t e s t . As a s i m p l i f i e d e q u a t i o n , the f o l l o w i n g form was c o n s i d e r e d [k'j from the s t r u c t u r e of t h e t r a n s m i s s i o n f u n c t i o n and e q u a t i o n ( z ) .

T-^^-+i,KS Where T = Tj^ + T^ - T^

T h i s e q u a t i o n was a p p l i e d to t h e r e s u l t s o f Z t e s t s i n many r e a l s h i p s , and i t was confirmed t h a t t h i s i f o r m u l a h a s s u f f i c i e n t a c c u r a c y f o r p r a c t i c a l p u r p o s e s . Hence, t a k i n g the r u d d e r a n g l e S measured by Z t e s t s , and

-8-tho t u r n i n g a n g l e ^ , we t a k e v a l u e s o f K and T which s a t i s f y the above e q u a t i o n s i m u l t a n e o u s l y . I n f r e e -r u n n i n g model t e s t s a l s o , Z t e s t s a -r e v a l u a b l e i n making i t p o s s i b l e to f i n d out e a s i l y the c h a r a c t e r i s t i c s o f a p a r t i c u l a r s h i p type, and i f we measure the speed

c o n t i n u o u s l y d u r i n g t e s t s , changing the rudder angle i n a number o f ways, i t i s p o s s i b l e to c a r r y out even more advanced a n a l y s e s . However, what i s o b t a i n e d by t h i s t e s t i s a rough o u t l i n e o f m a n o e u v r a b i l i t y , and must be used i n c o n j u n c t i o n w i t h the frequency response a n a l y s i s p r e v i o u s l y r e f e r r e d t o , i n r e s e a r c h of any d e t a i l .

L a s t l y we would l i k e to touch on manoeuvre t e s t s w i t h r e a l s h i p s . The f r e e - r u n n i n g model t e s t s r e f e r r e d to i n t h i s paper can b a s i c a l l y be a p p l i e d , as they s t a n d , to r e a l s h i p s ; and, by means of s i m p l e c a l c u l a t i o n s , a s u i t a b l e p a r t o f them can a c t u a l l y be a p p l i e d i n about one hour. I f t h i s k i n d of t e s t i s c a r r i e d out d u r i n g s e a t r i a l s , c l e a r e r and more d e t a i l e d i n f o r m a t i o n about the m a n o e u y r ^ b l l l t y o f a p a r t i c u l a r s h i p can o b t a i n e d then from o r d i n a r y t u r n i n g t e s t s ; i t i s a l s o expected t h a t

important r e s u l t s w i l l be o b t a i n e d c o n c e r n i n g the r e l i a b i l i t y o f f r e e - r u n n i n g s h i p models and the e f f e c t o f s h r i n k a g e

-9¬

1. The d e t e r m i n a t i o n o f I n d i c e s o f m a n o e u v r a b i l i t y by t h e f r e q u e n c y response method.

1.1 T u r n i n g t e s t s

T h i s t e s t a f f o r d s a means o f f i n d i n g K and i t can a l s o be used to determine Yg (iii) W e 0 w i t h a f a i r degree o f a c c u r a c y . , the s t e a d y t u r n i n g r a d i u s , i s

measured f o r v a r i o u s v a l u e s o f rudder a n g l e *"rom a s s m a l l an angle a s p o s s i b l e to a maximum. I f we p l o t the non-dimensional a n g u l a r v e l o c i t y ( L / R g ) a g a i n s t , we o b t a i n F i g . l .

Comparing t h e r u d d e r a n g l e when t h e s h i p i s a d v a n c i n g , we a d j u s t the c u r v e f o r t h e approximate z e r o . We o b t a i n the c i i r v e 5 o *'or ( L / R g ) u s i n g the method o f l e a s t s q u a r e s , w i t h an e q u a t i o n o f the type

5 o ^ a ( L / R g ) • b ( L / R s ) 5

S i n c e t h e r e i s some v a r i a t i o n f o r l e f t and r i g h t t u r n s , we c a r r y out c a l c u l a t i o n s f o r t h e s e s e p a r a t e l y and c a r r y out zero r u d d e r a n g l e c o r r e c t i o n to m i n i m i s e the d i f f e r e n c e i n the c o e f f i c i e n t a . The ( L / R g ) - 5 o c u r v e o f t h e c u b i c e q u a t i o n o b t a i n e d i n t h i s way i s s u f f i c i e n t l y a c c u r a t e a s shown i n t h e f i g u r e . Depending on c i r c u m s t a n c e s , we may

a l s o c o n s i d e r t h e terms o f ( L / R g ) ^ but a t p r e s e n t t h i s does j not appear to be n e c e s s a r y . I t i s i n t e r e s t i n g to note

t h a t t h e t u r n i n g r a d i u s and rudder a n g l e a s a c t u a l l y measured ha'» t h i s k i n d o f s i m p l e shape. T h i s c u r v e i s l i n e a r i f K ' i s

c o n s t a n t and I t s s l o p e i s equal to K'' but s i n c e , w i t h t h e development o f t h e t u r n i n g motion, K ' d e c r e a s e s , we o b t a i n t h i s k i n d o f c u r v e . The s l o p e o f the c u r v e d ( L / R g ) / d i s equal to K i n t h e c o r r e s p o n d i n g (L/R: ) . However i s measured i n r a d i a n s . Hence i f we d i f f e r e n t i a t e t h e c u b i c e q u a t i o n i n ( L / R ) - &O p r e v i o u s l y o b t a i n e d , we o b t a i n = • 3b ( L / R g ) 2 } • I

•10-MNé6<l FULL i •• »4M ; • .«»' •Jir -if If n • >^ ^ ly if if t : "f ^ « 1/ Iff -Jfr Jf J. '^.l-i • yst> 'A' j • .cn '^Hüï •• « •Jf -tf -if ' / 'f ir jy i.

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-11-K'' i s g i v e n by the unbroken l i n e i n F i g . 2. When t h e r e i s a d i s c r e p a n c y between l e f t and r i g h t , an average must be

taken f o r Ts! and i f n e c e s s a r y they can be t r e a t e d s e p a r a t e l y . Next the s l o p e o f the s t r a i g h t l i n e c o n n e c t i n g the

o r i g i n and the p o i n t on the c u r v e ( L / R ^ ) , t u r n i n g by s t e e r i n g from the rudder c e n t r e to the rudder angle

c o r r e s p o n d i n g to t h a t p o i n t , g i v e s a mean v a l u e o f which v a r i e s a c c o r d i n g to the motion. The s t e a d y t u r n i n g a n g u l a r v e l o c i t y ( L / R ^ ) which i s found i s the product o f t h i s

mean K ' and • Mean v a l u e s o f a r e shown by the d o t t e d l i n e i n F i g . 2 .

The most d i r e c t way o f c a l c u l a t i n g R^ i s w i t h compasses s e t a t two p o i n t s on shore and i t i s d e s i r a b l e to c o r r e c t f o r the e f f e c t o f wind and c u r r e n t by r e p e a t i n g t h r e e t i m e s . I f a c c u r a t e measurement o f R^ i s o f prime importance, we can a c c u r a t e l y measure o n l y the b e a r i n g o f the i n t e r c e p t on the t u r n i n g c i r c l e . Because the motion o f the s h i p seen from the compass s t o p s t e m p o r a r i l y i n the v i c i n i t y o f the i n t e r c e p t , i f we use a t e l e s c o p e w i t h

c r o s s - w i r e , m a g n i f i e d s e v e r a l t i m e s , w i t h a f i e l d o f v i s i o n 6 - 10**, i t i s p o s s i b l e to measure the b e a r i n g to an

a c c u r a c y o f s e v e r a l c e n t i m e t r e s , a t the p o s i t i o n o f the s h i p . However, w i t h t h i s method, t h e continuous r e c o r d o f t r a c k i s not t a k e n and a c c o r d i n g l y the speed in advance and t u r n i n g i s not observed. I n o r d e r to take t h i s , i f we

f o l l o w the s h i p w i t h the compass, the a c c u r a c y o f the b e a r i n g d i m i n i s h e s s l i g h t l y , and c a u s e s t r o u b l e i n the a n a l y s i s . The r e s u l t s o f i n v e s t i g a t i n g v a r i o u s methods show t h a t the e a s i e s t to use i s t h a t o f measuring the speed by a t t a c h i n g a s m a l l v e r t i c a l a x i s d i s c speed gauge o f the type shown i n F i g . 3 a t the s h i p ' s c e n t r e o f g r a v i t y , d e t e r m i n i n g R^ b y the b e a r i n g o f the i n t e r c e p t . R^ and the speed i n turning! a r e dc-tormined by t h i s and the e f f o r t o f a n a l y s i s i a s m a l l . Again, i f i t i s n e c e s s a r y to determine the advance t h i s i s

-12-found w i t h f a i r a c c u r a c y from the speed and the t u r n i n g angle measured by gyro.

V i t h a s m a l l r u d d e r a n g l e , the t u r n i n g c i r c l e does not l i e w i t h i n the t e s t w a t e r s u r f a c e but i n t h i s c a s e we

determine ( L / R ) B ^ / ( J - ) f r o n the speed g i v e n by the speed g^uge and B the t u r n i n g a n g u l a r v e l o c i t y measured by gyro and t a k i n g i n t o account the f a c t t h a t ( L / R ) remains c o n s t a n t .

I n o r d e r to be a b l e to e n t e r the s t e a d y t u r n a s

q u i c k l y as p o s s i b l e , a r a t h e r l a r g e rudder angle i s taken a t f i r ; which must soon be changed to the t e s t a n g l e . I n the c a s e o f a s h i p w i t h a low d i r e c t i o n a l s t a b i l i t y , the graph ( L / R ^ ) - So tends to have a l a r g e c u r v a t u r e i n the v i c i n i t y o f the o r i g i n and w i t h t h i s type o f s h i p , by c a r r y i n g out t h i s t e s t w i t h a s m a l l rudder a n g l e o f about 2-5° i t i s p o s s i b l e to determine K ' e x a c t l y c l o s e to the advance p o s i t i o n .

By the above p r o c e d u r e , t h e t u r n i n g index i s o b t a i n e d from o b s e r v a t i o n o f the s t e a d y t u r n i n g motion, and t h i s v a l u e o f K' i s more i m m e d i a t e l y r e l i a b l e than the K ' produced by the s t e e r i n g method. Next, i f we c a r r y out the f r e q u e n c y

a n a l y s i s based on e q u a t i o n (3) i n the e x c e s s p a r t o f the t u r n i n g motion, we o b t a i n Y ^ ( i t j ) a c c u r a t e l y f o r c o m p a r a t i v e l y low v a l u e s o f ». However, to speak c o n c l u s i v e l y , s i n c e a l l the i n d i c e s of m a n o e u v r a b i l i t y a r e determined i n g e n e r a l w i t h good a c c u r a c y from K from s t e a d y t u r n i n g and the a n a l y s i s o f the s i n u s o i d a l steerang r e f e r r e d to below, t h i s f r e q u e n c y a n a l y s i s o f the i n i t i a l

t u r n i n g motion i s not always n e c e s s a r y and i s g e n e r a l l y used as a check on the v a l u e o f Yg i(»), when i s s m a l l . I n

c a r r y i n g out t h i s c a l c u l a t i o n , because t h e r e i s a marked decreas» i n speed, i n the t u r n i n g t e s t s , t h e r e i s g r e a t change i n K, Tj^ e t c . i n e q u a t i o n (2) and i t i s not s u i t a b l e f o r timerbased l i n e a i a n a l y s i s . Hence i n s t e a d o f time t , we use the v a r i a b l e

-13-f o l l o w i n g e q u a t i o n s i m i l a r to e q u a t i o n (3) lim ƒ "/l(f)r»'(eo»»'i-liine'j)4f „ u«r^+i»ioi=^^^^ . ^ o f g l (4) where O ( s ) --VOH:?) ^•<'"'^°(F)>^t^)°nTëj^^^;S^ The I n t e g r a t i o n o f ü ( i c * ) f o r J^, a f t e r e n t e r i n g s t e a d y t u r n i n g i s by Simpson's r u l e but t h e r e a f t e r because

^ ( S ) a (L/Rg) we can I n t e g r a t e the v a r i o u s e q u a t i o n s . The r e s u l t s g i v e ^ (±1t^) = +-^^=^^[(«%ctt»%-sina'a)-Kcat/»+«'it(iD0'i»)]-(£/«f)ii((oi«'it-<»iDa'<») • _ , . ... , i s the v a l u e o f s a t . t h e p o i n t where i t c r o s s e s the s a x i s by e x t e n d i n g the s t r a i g h t l i n e ^ ( s ) a f t e r e n t e r i n g s t e a d y t u r n i n g . T h i s f o r m u l a r a p p l i e s to the c a s e when

the t u r n i n g a n g l e d i s measured and a s i m i l a r formula which a p p l i e s to the c a s e where we measure S can a l s o be made from e q u a t i o n (k).

The i n t e g r a t i o n f o r 5 , I f we make Si the v a l u e o f s r e q u i r e d I n the s t e e r i n g , g i v e s

Again, i n the c a s e o f the u s u a l s t e e r i n g v e l o c i t y , when ö'< 0.3. we o b t a i n w i t h s u f f i c i e n t a c c u r a c y

S (1« ) = - i S o E s p e c i a l l y when O'-^ 0,

-Ik-A c c o r d l n g l y

""/•WJo-i^^o, mean K

From t h i s i t i s c l e a r t h a t the Y

by t h i s method i s the Y^ e s t a b l i s h e d from the mean v a l u e s (i«») o b t a i n e d

o f K, which g r a d u a l l y change t o g e t h e r w i t h motion. T h i s e x p r e s s i o n f o r Y ' i s r e l i a b l e

s m a l l , but i t s a c c u r a c y d e t e r i o r a t e s as W i n c r e a s e s . T h i s i s due to the f a c t t h a t the v a l u e when a oj

i s c o n t r o l l e d o n l y by the s t a b i l i s e d observed the t u r n i n g when W ^ i s i s vez>y r e l i a b l e v a l u e s o f / s t e a d y t u r n i n g and the f a c t t h a t (i«>) and j (i«> ) d e c r e a s e a l o n g w i t h an i n c r e a s e ot 0^ .

F o r time c o r r e c t i o n o f r e c o r d e d Stnd S on s b a s e , we use the symbols o f the speed gauge r e f e r r e d to e a r l i e r , which procediire i s g i v e n i n the a n a l y s i s o f 1;he Z t e s t s . Again t h e r e a r e many c a s e s where some i n i t i a l . motion e x i s t s , b e f o r e e n t e r i n j j the t u r n i n g t e s t , and the i n f l u e n c e o f t h a t must be excluded from the c a l c u l a t i o n . That procedure i s

d i s c u s s e d i n d e t a i l i n the a n a l y s i s o f the s i n u s o i d a l s t e e r i n g . F i g . 5 and F i g . 6 show examples o f the r e s u l i s o f the a n a l y s i s c a r r i e d out f o r a high-speed l i n e r i n the foiiD o f l o g Y,

a g a i n s t l o g <§>^ .

1.2. S i n u s o i d a l s t e e r i n g teèts.

I n t h e s e t e s t s , the r u d d e r i s moved l e f t and r i g h t , i n s i n u s o i d a l form, w i t h a c o n s t a n t a m p l i t u d e and p e r i o d .

(So, h a l f the a m p l i t u d e o f the rudder angle, :..e. th>e

maximum r u d d e r a n g l e , i s stopped a t 1 0 ° 15T, and the p e r i o d extends from about 2 s e e s to 20 ~ ko s e e s , changing i n

r e g u l a r sequence i n each r u n . T h i s s t e e r i n g i s a t y p i c a l method i n f r e q u e n c y r e s p o n s e , and v a r i e s froiji a l a r g e to a f a i r l y s m a l l v a l u e o f and g i v e s Y^ (i«) a c c u r a t e l y , and

-15-•

i n c o n j u n c t i o n w i t h the v a l u e s of K (Yg i n <«) = o) from t u r n i n g t e s t s , makes c l e a r the whole p a t t e r n of the m a n o e u v r a b i l i t y of the p a r t i c u l a r s h i p t y p e . I f we a p p l y e q u a t i o n (3) to the s i n u s o i d a l s t e e r i n g motion f o r a s h i p h a v i n g d i r e c t i o n a l s t a b i l i t y , i t may be e x p r e s s e d i n p o l a r form when

s 2 71/1^ i . e . the 60 ot numerator and denominator s t e e r i n g p e r i o d , the e x c e s s terms d i s a p p e a r and we o b t a i n

I n t h i s e q u a t i o n , 6 i s the h a l f amplitude of ^ a f t e r the s i n u s o i d a l s t e e r i n g motion has become s t e a d y , / i s the phase d i f f e r e n c e of the 5 f o r ^ . T h i s r e s u l t i s determined a l s o by c a l c u l a t i n g th© r e g u l a r s o l u t i o n , s u b s t i t u t i n g the edlnusoidal s t e e r i n g S o S/, s i n 6> t i n the motion e q i i a t l o n ( 2 ) . T h i s *^ r s t e a d y s o l u t i o n g i v e s and hence and a l s o

A c c o r d i n g l y i f we measure ^ the s t e a d y a m p l i t u d e o f the s i n u s o i d a l s t e e r i n g t e s t and take th© r a t i o w i t h So then . Y^Cit;^) j f o r the s t e e r i n g p e r i o d I s i m m e d i a t e l y determined. The d e t e r m i n a t i o n of Yg f o r l a r g e («> may a l s o be e f f e c t e d by t h i s method, f a i r l y a c c u r a t e l y . When the motion o f a f r e e - ^ r u n n i n g model i s measured by a girro speed gauge, ^ i s o b t a i n e d d i r e c t l y and the t u r n i n g a n g l e i s o f t e n measured

-16-••

by means o f a gyro compass. At t h i s time, B i s o b t a i n e d by m u l t i p l y i n g 0^ by 6 h a l f tho amplitude o f ^ . Because i t i s i n the form Y g ' , we put i t i n the non-dimensional

form ) " ^^r/(z ) " ^ • Ev®" "^en t a k i n g the speed d u r i n g advfince, t h e r e i s not too much e r r o r i n V, and i f we use the speed gauge mentioned above, i t i s r e l i a b l e oven whei^

i s l a r g e . The r e c o r d o f yawing, even when T^ i s

c o m p a r a t i v e l y l a r g e , i s markedly s i n u s o i d a l , but c o n f u s i o n o c c u r s when T^ i s s m a l l . I n t h a t c a s e , i f we take an average

v a l u e w i t h a s s h o r t a time as p o s s i b l e , good r e s u l t s a r e o b t a i n e d . Again, i f the mean c o u r s e o f the s i n u s o i d a l motion i s not i n a s t r a i g h t l i n e , t h e r e o c c i i r s a g r a d u a l s l i p .

T h i s should be a s s m a l l a s p o s s i b l e , and the remainder i s c o r r e c t e d a s f o l l o w s

'•'•-{-«vèr)"}

where t a n g u l a r v e l o c i t y o f s l i p o f averoffe c o u r s e B t h a l f - a m p l i t u d e determined from a number o f

peaks and troughs f o l l o w i n g ^

I f t h i s i s i n a d e q u a t e , we may p l o t the & v a l u e s o b t a i n e d and take more r e l i a b l e r e a d i n g s a t i n t e r v a l s a l o n g ^ . MM ma. 0 nu 1 \inaM F i e , k. R e s u l t s o f s i n u s o i d a l and t u r n i n g s i n u s o i d a l 1 s t e e r i n g t e s t s (z)

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1

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1

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- 1 8 ¬

»

F i g s , k, 5, and 6 show examples o f t h e s i n u s o i d a l s t e e r i n g a n a l y s i s .

From t h e s i n u s o i d a l s t e e r i n g t e s t s , i t i s

p o s s i b l e t o determine not o n l y YgCiw) f o r the s t e e r i n g p e r i o d l a l s o Yg f o r an oven s m a l l e r 4^, by u s i n g tho c a l c u l a t i o n s o f e q u a t i o n ( 3 ) . These c a l c u l a t i o n s l i n k t o g e t h e r tho r e s u l t s o f t h e s i n u s o i d a l s t e e r i n g and t h e r e s u l t s o f t h e t v i r n i n g t e s t s , and a r e e f f e c t i v e I n c l a r i f y i n g t h e m a n o e u v r a b i l i t y o f t h a t s h i p t y p e . A f t e r c a u s i n g t h e model to advance, i f we a p p l y e q u a t i o n ( 3 ) to t h e c a s e when s i n u s o i d a l s t e e r i n g b e g i n s we o b t a i n Zl ' ^ * " * ^ * r * l-(»/«,>.r~«*»-'»'n«^) ( c f . F l g . 7 )

Here t^^ i s taken a t t h e peak o f ^ a f t e r tho motion h a s become s t e a d y , and up to t h i s we u s e t h e t r a p e z o i d a l Simpson's r u l e s . A f t e r t ^ , t h e motion I s B l n u s o l d a l and we c a n i n t e g r a t e .

^ i s the average c o u r s e a n g l e a f t e r s t e a d y motion o c c u r s . T h i s e q u a t i o n i s s u i t a b l e when ^ i s known but t h e

f o l l o w i n g e q u a t i o n i s used when we measure ^ .

I n t h i s c a s e t ^ i s taken a s t h e v a l u e a t tho peak o f ^ . i r

i n i t i a l peak o f $ :

1 9

-I f wo o l l m i n a t o w i t h S ( l ( # ) | wo o b t a i n Y s ( l W ) ] , and I f we e l i m i n a t e w i t h V/L we o b t a i n YJ.{±V) . I t I s d e s i r a b l e to tee the speed gauge a l s o f o r V. I n c a r r y i n g out t h i s c a l c u l a t i o n , I t I s u s e f u l to c o n s t r u c t a t a b l e of t r i g o n o m e t r i c v a l u e s s u c h t h a t I n each second Wt has the v a l u e s 1 ° , 2**, 4 ° , 8 ° , 1 5 ° , 3 0 ° . T h i s I s enough f o r most s h i p s .

Esamples o f the a n a l y t i c a l r e s u l t s a r e shown i n P i g s . 5 and 6. I f the v a l u e 6^ ( i o /w^ ) f o r O o f the Yg o b t a i n e d by t h i s method i s e q u a l to an average YL tor via r the aniplitude o f non-dimensional a n g u l a r v e l o c i t y i n those t e s t s , i t i s f a i r l y a c c u r a t e f o r & s 0 » ^ . Such agreement, i n a s h i p w i t h c o m p a r a t i v e l y good s t a b i l i t y , i s s a t i s f a c t o r y but i n a s h i p where s t a b i l i t y i s poor i t i s not n e c e s s a r i l y so. I n t h a t c a s e , a s s e e n from the above e q u a t i o n , when W o < ^ because a g r e e s w i t h the r e l i a b l e v a l u e o b t a i n e d d i r e c t l y from the s i n u s o i d a l s t e e r i n g , t h i s Y g i s s u i t a b l e i n the v i c i n i t y o f < ^ and g e n e r a l l y seems to bo a c c u r a t e , when W/**^ B 0 . 5 — 1 . 0 .

I f we c a r r y out tho t o s t , i t i a d i f f i c u l t to a v o i d produolng some l e f t and r i g h t t u r n i n g motion, however

s h o r t the time b e f o r e the s t e e r i n g i s begun and, i n a d d i t i o n , t h e r e a r e many c a s e s o f mean c o i i r s e s l i p a f t e r the motion has become s t e a d y . These must bo a s few a s p o s s i b l e but b e c a u s e they €U«e not c o m p l e t e l y a b s e n t , we must c o r r e c t f o r t h e s e two i n f l u e n c e s on 6 . I f we c a l l $ ^ t h e r e s i d u a l & o f the i n i t i a l motion, and ^ r the ^ due to the s l i p o f tho mean c o u r s e , then the motion to be e l i m i n a t e d i s

v h i c h i s r e p r e s e n t a t i v e , e z c l i i d i n g the s l i g h t d i f f e r e n c e o f the e x c e s s p a r t . I n t h i s e q u a t i o n , • • 6^ and ^0 » i n i t i a l a n g u l a r v e l o c i t y and a c c e l e r a t i o n , T o Tj^ + -I a n g u l a r v e l o c i t y o f s l i p o f moan c o u r s e $a and oan bo r e a d o f f a l s o from tho r e a d i n g s o f but i f 6^ I s not found from the values tfOit i s too

d i f f i c u l t a t f i r s t . However i f c a r e i s t a k e n when tho t o s t i s c a r r i e d o u t , i t i s p o s s i b l e to do i t i n s u c h a way t h a t the c o r r e c t i o n o f ^ 0 i s not a problem. T^^, l^i i n t h i s c a l c u l a t i o n a r e r e l i a b l e i f we u s e the v a l u e o f K from t h e t u r n i n g t e s t s and tho d i r e c t l y determined from the s i n u s o i d a l s t e e r i n g by t h e method o f s l m p l i f i o a t l o n d e s c r i b e d i n 1 . 5 < One method i s a l s o to make (T^^ ••¬

about 1 . 2 times T from the Z t e s t s , i f i s s m a l l . Ve deduct, from B , ^^•'•^ i n t h e same way a s when

m e a s u r i n g h . I n o r d e r to employ f r e q u e n c y a n a l y s i s w i t h the p r e a r r a n g e d s i n u s o i d a l s t e e r i n g I t i s d e s i r a b l e t o take groups from l e f t and r i g h t s t e e r i n g e a c h a t the same f r e q u e n c y . F o r any one group i s e q u a l , and i n many c a s e s v a l u e s o f ^0 agree w e l l . Because t h e motion f o r s i n u s o i d a l s t e e r i n g I s i n i n v e r s e phase f o r l e f t and r i g h t , i f we tEüce t h e d i f f e r e n c e o f t h e B r e a d i n g s and r u d d e r a n g l e S o f one group, a f a i r l y s m a l l i d e a l measxirement o f ^a»and ^ i s o b t a i n e d . \rs T u r n i n g s i n u s o i d a l s t e e r i n g t e s t s . As a m o d i f i c a t i o n o f the s i n u s o i d a l s t e e r i n g , we. f i r s t t a k e a c o n s t a n t r u d d e r a n g l e and a f t e r t h e t u r n i n g has commenced we c a r r y out s i n u s o i d a l s t e e r i n g a t the

p e r i o d o f t h a t r u d d e r a n g l e . T h i s t e s t i s uaed to determine

\

the v a l u e o f Y ( i t t ) f o r a c o m p a r a t i v e l y s t r o n g t u r n i n g oatlot s

2 1

-when (d i s l a r g e . Combining t h e s e r e s u l t s and the from the t u r n i n g t e s t s , i t i s p o s s i b l e to determine t h e index o f m a n o e u v r a b i l i t y i n tiurning.

The method o f a n a l y s i s c o r r e s p o n d s to s i n u s o i d a l s t e e r i n g , but s i n c e the t u r n i n g s i n u s o i d a l s t e e r i n g can bo c a r r i e d out up to a c o m p a r a t i v e l y l a r g e p e r i o d ,

c a l c u l a t i o n o f frequency a n a l y s i s i s a s a r u l e not n e c e s s a r y . . R e a d i n g o f f ^ from the p a r a l l e l s t r a i g h t l i n e s drawn on

the c u r v e ^ ( t ) , wo determine /^^'(iö,') / . When th© p e r i o d i s s m a l l and yawing i s c o n s i d e r a b l e , we u s e ^ ^ t h © average t u r n i n g a n g u l a r v e l o c i t y read from tho r e c o r d , and put i t i n t h e form ^ - ^ t , and i f we make a magnified p l o t aftc d e d u c t i n g the s t e a d y t u r n i n g i t i s p o s s i b l e to r e a d o f f ^ w i t h good a c c u r a c y . The magnitude o f th© a n g u l a r v a l o c i t y i n t h i s t o s t i s g i v e n by ^oo. ( V / L ) , t a k i n g t h e s t e a d y t u r n i n g a s a b a s i s .

F i g s , k and 5 show a c t u a l ©xafflpl©8 f o r morchant s h i p t y p a s , and from thos© r e s u l t s , when the t x i m i n g i s

/ / /

s t r o n g , K , Tj^ ^ Tg seem to show an a p p r o p r i a t e d e c r e a s e . I f t h i s h a s tho tendency t h a t tho i n c r e a s e o f the t u r n i n g power owing to t h e i n c r e a s e o f r u d d e r a n g l e i s g r a d u a l l y n i a i i f i e d , th© d i r e c t i o n a l s t a b i l i t y on t u r n i n g i s good and the f o l l o w - u p i s r a p i d i n p r o p o r t i o n to tho s t r e n g t h o f t u r n i n g , and c o r r e s p o n d s w i t h t e s t s c a r r i o d out h i t h o r t o .

Henc© i t i s p o s s i b l e to ©stimat© th©s© t©nd©ncies q u a n t i t a t i v e l y by t h i s k i n d o f t e s t . The c l a r i f i c a t i o n o f t h e s e m a t t e r s i n t h e s e c o n d i t i o n s by^ r e p e a t i n g t h i s t e s t and s e l e c t i n g many v a l u e s o f s t r e n g t h o f t u r n i n g a s a b a s i s i s c o n s i d e r e d to b© n©c©88ary to fundamantal r a s s a r c h i n t o th© probloms o f m a n o o u v r a b l l i t y . Thar© a r e p r a c t i c a l d i f f i c u l t i e s i n

a c c u m u l a t i n g a g r e a t number o f r u n s , i n s y s t e m a t i c t e s t s to f i n d o u t t h e c o i m e c t l o n o f t u r n i n g m a n o e u v r a b i l i t y w i t h * rudder a r e a and s h i p type, and t h e d e t e r r o i n a i i o n o f i n d i c e s bf m a n o e u v r a b i l i t y , at. each l e v e l o f t u r n i n g a n g u l a r

-22-v e l o c l t y cannot be consldereid to be a p r a c t i c a l n o u e s s l t y a t the p r e s e n t s t a g e . Hence as a s t a n d a r d f o r t h i s type o f t e s t , t h e s t e a d y r u d d e r a n g l e I s r e s t r i c t e d to about 1 5 ° and, c£u*rylng out a s e r i e s o f t u r n i n g s i n u s o i d a l

s t e e r i n g t e s t s , ve determine the I n d i c e s f o r f a i r l y s t r o n g t u r n i n g motion which i n c o n j u n c t i o n w i t h the i n d i c e s c l o s e to the advance, o b t a i n e d from the t u r n i n g t e s t s and

s i n u s o i d a l s t e e r i n g r e f e r r e d to e a r l i e r , may be t a k e n a s i n d i c a t i v e o f the m a n o e u v r a b i l i t y o f t h e type o f s h i p .

1P u l s e s t e e r i n g t e s t s .

T h i s i s s t e e r i n g i n which, a f t e r the s h i p has been caused to advance, the r u d d e r i s s e t a t an angle o f 10**-^ 1 5 ° f o r a w h i l e , and then the s t e e r i n g i s r e t u r n e d to the c e n t r e . As a r e s u l t o f t h i s the s h i p t u r n s once but a s the a n g u l a r v e l o c i t y soon d e c r e a s e s i t then t a k e s up a new c o u r s e . We determine Y g ( i w ) f o r v • 0 - l / T j ^ u s i n g the

meastired v a l u e s o f ^ and 6 and <f i n e q u a t i o n ( 3 ) . Two o r t h r e e a c c o u n t s have a l r e a d y been p u b l i s h e d S. k\ [ 3 J b u t , s i n c e then, some d e f e c t s i n the method have been observed and s i n c e a way of f i n d i n g out the whole p a t t e r n o f m a n o e u v r a b i l i t y by d e t e r m i n i n g Ts (l4^) e x t e n d i n g

from %f a O to flO w i t h o u t r e c o u r s e to i t has been developed, i t i s probable t h a t t h i s t e s t w i l l not now be used.

As d e f e c t s we n o t e :

(1) I n the p r o c e s s o f r e s t o r i n g to the advance p o s i t i o n , which o c c u p i e s a g r e a t p a r t o f the p r o c e s s , the motion i s complovely p a s s i v e and moreover w i t h t h a t s h i p t y p e , tho r e s u l t s a r e confused, even w i t h s l i g h t e x t e r n a l d i s t u r b a n c e , s i n c e d i r e c t i o n a l s t a b i l i t y i s a t i t s w o r s t when c l o s e to the advance p o s i t i o n . A c c o r d i n g l y , when B o n l y i s measured r a t h e r than B , .the u l t i m a t e c o u r s e i s d i f f i c u l ^ t

to d i s c e r n and the v a l u e o f K i s a l s o u n c e r t a i n . Again t h i s i s o n l y one example t e s t e d , but i n the c a s e o f a f u l l

s h i p typo, thoro i s an example of non-damped yawing b e i n g c a u s e d , thought to bedue to the phenomenon, p e c u l i a r to the s h i p , o f break away f l u t t e r a t the s t e r n , r e s t o r i n g the advance p o s i t i o n a f t e r the c o m p l e t i o n of the s t e e r i n g , i n which ckne t h i s t e s t i s m e a n i n g l e s s . (2) I n the f r e q u e n c y a n a l y s i s of the t u r n i n g t e s t , th© r e l i a b l e K determined from s t e a d y t u r n i n g a u t o m a t i c a l l y r e s t r i c t s Yg f o r and i n the a n a l y s i s of th© s i n u s o i d a l s t e e r i n g Yg from s t e a d y amplitude r e s t i ^ i c t s Yg f o r Wp, but i n p u l s e s t e e r i n g , no r e l i a b l e i n d i c a t i o n of Yg f o r t h e s e e x i s t s f o r any 0 ) .

(3) I n s h i p s which have poor d i r e c t i o n a l s t a b i l i t y , s i n c e K and markedly i n c r e a s e when a p p r o a c h i n g the advance p o s i t i o n , t h e r e i s consequent n o n - l i n e a r e f f e c t i n Yg due to t h i s method, and i t i s d i f f i c u l t t o ^ s p e c ^ f y the l e v e l of a n g u l a r v e l o c i t y s i n c e the motion i s not p e r i o d i c even though i t i s s t e a d y .

1 . 5 D e t e r m i n a t i o n of m a n o e u v r a b i l i t y I n d i c e s K ^ T ^ ^ T^' ,

The i n d i c e s of m a n o e u v r a b i l i t y a r e determined from I Yg (ifd ) | o b t a i n e d by combining the r e s u l t s o f a\\ the above t e s t s . F i r s t we take a s a s t a n d a r d the s i n u s o i d a l s t e e r i n g t e s t s f o r which frequency a n a l y s i s i s c a r r i e d out to determine the i n d e x c l o s e to the advance p o s i t i o n , and we determine from th© r e s u l t s o f t u r n i n g t e s t s the average f o r ( L / R ) e q u a l to ^ tf^, the a m p l i t u d e of the

non-d i m e n s i o n a l a n g u l a r v e l o c i t y . P l o t t i n g l o g W a g a i n s t l o g 1 Y, we d e s c r i b e Che a v e r a g e curve o f (*•** ) L p a s s i n g through each p o i n t o f the s i n u s o i d a l s t e e r i n g f r e q u e n c y a n a l y s i s , and each p o i n t o f the s i n u s o i d a l s t e e r i n g o f a s t i l l l a r g e r . Then f o r s u f f i c i e n t l y s m a l l W , we approximate K

/

determined p r e v i o u s l y , and w i t h a l a r g e W , we c o n s i d e r i t a s a s t r a i g h t l i n e o f s l o p e - 1 . Because ( L / R ) f o r

-2^1-/ .

s i n u s o i d a l s t e e r i n g f o r l a r g e id i s smaBar than the s t a n d a r d ( L / R ) ^ , i t may be c o r r e c t e d by a p p l y i n g

s i n u s o i d a l s t e e r i n g w i t h a l a r g e r rudder a n g l e than the s t a n d a r d , f o r l a r g e «, but the e f f e c t seems to bo s l i g h t . A c c o r d i n g l y , we must c o n s i d e r t h e IVgI c u r v e , now drawn, a s b e i n g f o r a s t a n d a r d ( L / R ) ^ . Vo r e a d o f f

c o r r e s p o n d i n g to t h r e e v a l u e s o f |*' j on the s t r a i g h t l i n o o f s l o p e - 1 , a s u i t a b l e d i s t a n c e below the

h o r i z o n t a l l i n o o f l o g K , and i n between those two.

/ / /

Wo o b t a i n Tj^ , , T. by s o l v i n g s i m u l t a n e o u s l y the

/

e q u a t i o n c o n s t r u c t e d f o r each w t '

T a k i n g more v a l u e s o f w , the l e a s t square method may be used but t h e c a l c u l a t i o n becomes somewhat c o m p l i c a t e d . F o r c o n f i r m a t i o n , when , then i t c o r r e s p o n d s to tho h o r i z o n t a l l i n o o f l o g K\ and we draw a l i n e o f s l o p e - 1 ftpom t h a t t o le • l / T ^ , h o r i z o n t a l up to l/T^'', and a f t e r t h a t a g a i n o f s l o p e - 1 . ThB p r e v i o u s l o g /T^'jcurvo must approximate to t h i s when tt i s ' f a i r l y s m a l l and when i t i s l a r g e . T h i s Bode approximate a n a l y t i c a l l i n e i s e a s i l y deduced from the form of Yg (i«). The p r o c e d u r e f o r d e t e r m i n i n g T^^, Tg^ , T^ ^ from K f o r ( L / R ) under t u r n i n g s i n u s o i d a l s t e e r i n g c o n d i t i o n s i s e x a c t l y tho same. Here ( L / R ) i s on the b a s i s o f s t e a d y t u r n i n g and

' I

K i s the l o c a l K determined from the s l o p e o f the curve fL/rO - from the t u r n i n g t e s t s .

I n t h e i n i t i a l s t a g e s o f tho a n a l y s i s , we

determined from the t u r n i n g t e s t s the s t e a d y a m p l i t u d e o f the s i n u s o i d a l s t e e r i n g , and t r i e d to f i n d the rough v a ^ o o f Tj^ , Tg , T^ w i t h o n l y Yg f o r a c o m p a r a t i v e l y l a r g e U and K . I n t h i s c a s e , tho f o l l o w i n g s i m p l e method I s

u s e f u l . We p l o t tho l o g a r i t h m s o f tho s i n u s o i d a l s t e e r i n g r e s u l t s , a g a i n s t \Y^'\ - w, and p r o d u c i n g tho s t r a i g h t l i n o o f the s l o p e - 1 shown by a l a r g e a» , we^take tho p o i n t s o f i n t e r s e c t i o n w i t h t h e h o r i z o n t a l l i n e o f K , and make »J the c o r r e s p o n d i n g <#)^ ( F i g . 4 ) . From tho Bode a n a l y t i c a l l i n e a p p r o x i m a t i o n , because

° ^3 / ^ l ' "^2^ • *e e l i m i n a t e T^' by u s i n g t h i s , the p r e v i o u s e q u a t i o n becomes .

/ / I • \ I Hence we determine Tj^ , , from j Yg i n the two «l', the s m a l l e r o b t a i n e d from the

s i n u s o i d a l s t e e r i n g , and a much l a r g e r v a l u e . I n t h i s method, tho t e s t tank i s wide and tho Tj^ o f the model i s s m a l l , and when we a r e a b l e to c a r r y out the

s i n u s o i d a l s t o o r i n g f o r t> a p p r o a c h i n g Q W ^ / , the v a l u e s appear to be a c c u r a t e , not m e r e l y approximate. However i t i s n e c e s s a r y to measure tho speed a c c u r a t e l y by u s i n g a speed gauge.

iié' P l a n f o r s t a n d a r d i s a t i o n o f m f t n o e u v r a b i l i t v t e s t s w i t h f r e e - r u n n i n g models bv the

f r e q u e n c y r e s p o n s e method

Summarising the above remarks, we c o n s i d e r s t a n d a r d i s i n g m a n o e u v r a b i l i t y t e s t s w i t h f r e e - r u n n i n g models h> the f r e q u e n c y r e s p o n s e method i n o r d e r to o b v i a t e e f f o r t and i n e f f i c i e n c y r e s u l t i n g from r e p e a t i n g t e s t s . Tho i n d i c e s o f m a n o o v r a b l l i t y c l o s e to the advance p o s i t i o n o b t a i n e d from t h e s e t e s t s show tho c h a r a c t e r i s t i c o f tho s h i p under a u t o m a t i c s t e e r i n g and when k e e p i n g <

- 2 6 .

c o u r s e on voyage, and c a n be used I n t h e a n a l y s i s o f t h e s e problems. They a r e most important f a c t o r s i n t h e m a n o e u v r a b i l i t y o f a merchant s h i p . Again, t h e i n d i c e s under t u r n i n g motion show the c h a r a c t e r i s t i c s o f a s h i p under f a i r l y s t r o n g s t e e r i n g such a s c o l l i s i o n a v o i d a n c e , and change o f d i r e c t i o n and t o g e t h e r w i t h t h e r e s u l t s o f Z t e s t s and t u r n i n g t e s t s g i v e a c l e a r i n d i c a t i o n o f the n o n - l i n e a r e f f e c t i n manoeuvring. 1. C a r r y i n g out l e f t and r i g h t t u r n i n g t e s t s of r u d d e r a n g l e 5 ° , 1 0 ° , 1 5 ° , 2 0 ° , 3 0 ° and 3 5 ° we c a l c u l a t e t h e p r o p e l l e r r.p.m., R by o b s e r v a t i o n o f 8 the b e a r i n g o f tho i n t e r c e p t on t h e t u r n i n g c i r c l e V by speed gauge, and ^ ( t ) a n d S ( t ) . Ve a r r a n g e t h e r e s u l t s i n the form (L/Rg) - S jj and d e t e r m i n e K ' a c c o r d i n g to 1 . 1 . I f advance i s necessairy, we c a l c u l a t e from B

and V . F u r t h e r , i n t h i s t e s t , i t i s d e s i r a b l e to

photograph t h e motion o f t h e s h i p i n t h e r e g i o n o f t h e p o i n t s o f i n t e r s e c t i o n o f t h e two i n t e r c e p t l i n e s , from a p o i n t on the shore, w i t h a c i n e - c a m e r a . The d r i f t

a n g l e i n t u r n i n g i s o b t a i n e d from t h e s e r e s u l t s and c a n be used i n t h e c a l c u l a t i o n o f the r e s i s t a n c e d e r i v a t i v e . 2 . Ve c a r r y out t h e s i n u s o i d a l s t e e r i n g t e s t s s e t t i n g t h e s t a n d a r d r u d d e r a n g l e a t 1 5 ° . Tho p e r i o d i s t a k e n a s s t a n d a r d up to t h e maximum p e r i o d which i t i s p o s s i b l e t o have, from where tho c u r v e l o g

-YJ - log»'' c o i n c i d e s w i t h t h e s t r a i g h t l i n e o f s l o p e -1 and t h e x n t e r v a l i s t a k e n a t about e v e r y e. l ^ *6 . 2 ,

o f l o g V i t h a U'^6H model, i t i s u s u a l f o r T^ = f r o m- 2 ~ 5 seconds to about 20/^^30 s e c o n d s . From t h i s < s e r i e s o f t e s t s , we measure t h e a m p l i t u d e o f t h e s t e a d y yawing and d e t e r m i n e | / f o r tho 6> o f tho s t e e r i n g p e r i o d } and u s i n g K from t h e t u r n i n g t e s t s tho roi:,':h

2 7

-/ -/ I

v a l u o s o f Tj^, T^, BJCO f i r s t determined by t h e s i m p l e method o f 1 . 3 •

3 . S e l e c t i n g two p e r i o d s . o f t h e o r d e r o f a few seconds and t e n s o f seconds, we, d e t e r m i n e the s i n u s o i d a l s t o o r i n g b e g i n n i n g from l e f t and r i g h t , w i t h a rudder a n g l e o f 10 1 5 ° . Wo c a r r y out c a l c u l a t i o n s o f f r e q u e n c y a n a l y s i s f o r t h i s s e r i e s o f t e s t s , and d e t e r m i n e (Yg ( f o r a s m a l l Here t h e i n d i c e s o f m a n o e u v r a b i l i t y

K / , and c l o s e to the advance p o s i t i o n , a r e d e t e r m i n by c a l c u l a t i n g a c c o r d i n g to 1 . 3 .

k. T a k i n g a s s t a n d a r d a s t e a d y rudder a n g l e o f

1 5 ° , and a s i n u s o i d a l s t e e r i n g a n g l e o f 1 0 ~ 1 5 ° , we c a r r y out t u r n i n g s i n u s o i d a l t e s t s . T^ c o r r e s p o n d s to s i n u s o i d a l s t e e r i n g . [Yb''! i s o b t a i n e d f o r tho l a r g o a n g u l a r v e l o c i t y l e v e l from t h e ampllttide o f t h e s t e a d y s i n u s o i d a l motion, and combining t h e r e s u l t s o f the t u r n i n g t e s t s , K, T^, T^. T^ under i h e t u r n i n g motion a r e determined.

C o n c l u s i o n and Acknowledgmentè.

The above completes t h e r e p o r t on f r e e - r u n n i n g model m a n o e u v r a b i l i t y t e s t s by the frequency r e p o n s e method. As r e g a r d s t h e a n a l y s i s o f t h e r e s i s t a n c e d e r i v a t i v e r e f e r r e d to i n t h e i n t r o d u c t i o n , t h e Z t e s t s , and the e x p e r i m e n t a l methods, i t i s hoped to g i v e some account l a t e r i f space p e r m i t s .

Much i n v a l u a b l e and s y m p a t h e t i c h e l p h a s been r e c e i v e d from t h e Kawasaki Heavy E n g i n e e r i n g B a s i c D e s i g n Department. We would l i k e t o take t h i s o p p o r t u n i t y o f e x p r e s s i n g o u r a p p r e c i a t i o n f o r t h i s , a a - a l s o to a niimber of s t u d e n t s o f t h e U n i v e r s i t y o f Osaka, Messrs.Oka, Ota, ^ Furuhlma and Nagayakawa f o r t h e i r e f f o r t s on oxir b e h a l f .

p

2 8

-REFERENCES

S h i g e r u AKAZAKIi E x p e r i m e n t a l B t u d l e s on t h e t u r n i n g o f a Bhip ( B u l l e t i n o f J.S.N.A. Japan, él, 1 9 3 ? " [ 2 . ] HiramitBU SHIBA (SHINAMi), Tokio MIZUNO, T e t a u j i n

TODA ( T O M I D A ) , Haruzo ETAi S t u d i e s o f t h e optimum rudder a r e a , u a i n g s h i p models

(Review o f J.S.N.A. Japan, 103, 1959)

[ 3 . ] R e l e v a n t m a t e r i a l p r e s e n t e d t o the N i n t h N a t i o n a l V a t e r - t a n k Conference, 196O.

[ 4 . ] NOMOTO, TANAKA, HONDA, JilRANO:

The m a n o e u v r a b i l i t y o f s h i p s ( l ) , ( 2 ) Review o f J.S.N. Arch.Japan 9 9 , 101, 1936 and 1957. [3.1 NOMOTO, MOKUNAKA: Model t e s t s on tho r n a n o e u v r a b i l i t v o f l a r g e o i l t a n k e r s Review o f J.S.N.Arch.Japan, [ 6 . ] AklhioaMin^ASE. The d e t e r m i n a t i o n o f i n d i c e s o f m n n o e u v r a b i l l t v by phase measurement

Graduate Review o f Osaka U n i v e r s i t y S h i p b u i l d i n g Department, 1939.

[ 7 . ] D a v i d s o n & S c h i f f , T u r n i n g and Course-keepln>r Q u a l i t i e s at fih<p TVS.N.ATM.E. 1 9 4 é

B.S.R.A. T r a n s l a t i o n ^fo. 2 1 5 9 ( b ) On the u t i l i t y o f f r e e - r u n n i n g models i n r e s e a r c h e s i n t o m a n o e u v r a b i l i t y ( 2 )

( L e c t u r e d e l i v e r e d a t the autumn s e s s i o n o f the J.S.R.A.November I961)

By

Kensaku Norooto, Member o f the A s s o c i a t i o n and o f the F a c u l t y o f E n g i n e e r i n g i n t h e U n i v e r s i t y of Osaka,

H i s a y o s h i Tatano, Member o f t h e A s s o c i a t i o n and o f the F a c u l t y o f E n g i n e e r i n g i n t h e U n i v e r s i t y of Osaka,

and

A k i h i s a Murase, Member o f t h e A s s o c i a t i o n and o f the F a c u l t y o f E n g i n e e r i n g i n the U n i v e r s i t y of Osaka. ( R e c e i v e d 20th June, 1960) 2. The e s t i m a t i o n o f a l l hydrodynamic f o r c e s a c t i n g on a s h i p , from f r e e - r u n n i n g model t e s t s . 2.1 The d e t e r m i n a t i o n o f t h e r e s i s t a n c e d e r i v a t i v e s 'by f r e e - r u n n i n g model t e s t s . Tbe r e s i s t a n c e d e r i v a t i v e s a r e the c o e f f i c i e n t s

^y^> ^nt^^* e q u a t i o n ( l ) used i n d e s c r i b i n g tho f o r c e s a c t i n g on a s h i p , i n terms o f tho l i n e a r f u n c t i o n o f tho d r i f t

a n g l o ^ , t h e t u r n i n g angle v e l o c i t y ^ ( u s u a l l y made non-d i m e n s i o n a l , ( L / R ) o r H ) annon-d t h e runon-dnon-der angle ^ . I n

d e t e r m i n i n g these from t e s t s , the u s e o f the t u r n i n g tank i s t r a d i t i o n a l , and a l s o tho v i b r a t i o n p u l l method i n tho normal tank i s used w i t h i t . The u s e o f the d e r i v a t i v e s i s v e r y r e l i a b l e i n c a l c u l a t i n g the f o r c e s a c t i n g on any s h i p . I n t h e f r e e - r u n n i n g model t e s t s , the e s s e n t i a l aim i s the s t u d y o f the motion o f the s h i p , and s i n c e t h i s i s caused by t h e f o r c e s a c t i n g on the s h i p the o b s e r v a t i o n o f motion must n e c e s s a r i l y e n t a i l much i n f o r a a t i o n about thol f o r c e s , w h i c h giv« r i s e t o a s u b s i d i a r y purpose.

-2-I n theory i t i s p o s s i b l e to determine the

d e r i v a t i v e s from f r e e - r u n n i n g model t e s t s , and i f t h i s can be done, the way i s c l e a r to obtainirg the d e r i v a t i v e s c l o s e to the advance p o s i t i o t v which are not o b t a i n e d w i t h the t u r n i n g tank, and the d e r i v a t i v e s f o r a r e a l s h i p . Again, t h e r e i s tho p r a c t i c a l a t t r a c t i o n t h a t knowledge can be obtained o f t h i s f i e l d w i t h o u t the t u r n i n g tank and the s p e c i a l equipment a s s o c i a t e d w i t h i t . Hence i t may be expected i n t h i s a n a l y s i s t h a t we can combine the two streams o f r e s e a r c h on m a n o e u v r a b i l i t y which have t a k e n p l a c e h i t h e r t o , v i z i by i n v e s t i g a t i n g the f o r c e s a c t i n g on the s h i p , and by o b t a i n i n g more d i r e c t l y a c t u a l d a t a by o b s e r v a t i o n o f the motion.of f r e e - r u n n i n g models and r e a l s h i p s .

However, the f o u r i n d i c e s o f m a n o e u v r a b i l i t y , K , Tj^ , , T j , obtained from s t e a d y t u r n i n g and

s i n u s o i d a l s t e e r i n g , d e s c r i b e the t u r n i n g motion, but do not g i v e any i n f o r m a t i o n about tho d r i f t motion of the s h i p . Hence i n a n a l y s i n g the f o r c e s which cause i t i t i s probably n e c e s s a r y to c o n s i d e r not o n l y the n a t u r e o f the s h i p ' s motion, but a t the same time a l s o the d r i f t motion which always accompanies tho t u r n i n g motion. I n f a c t , i f we s e p a r a t e tho f o r c e o f i n e r t i a , t h e r e a r e s i x c o e f f i c i e n t s which d e s c r i b e the f o r c e s a c t i n g on a s h i p and t h e s e f o r c e s cannot be determined from f o u r c o e f f i c i e n t s . Hence we attempt to d e s c r i b e the d r i f t motion by tho same procedure as t h a t by which we obtained e q u a t i o n ( 2 ) , which d e s c r i b e s the t u r n i n g motion, from the o r i g i n a l e q u a t i o n ( l ) ,

mentioned i n the i n t r o d u c t i o n . I . e . i f we c o n s t r u c t an e q u a t i o n i n ^ by e l i m i n a t i n g 0 from ( l ) , we o b t a i n an e q u a t i o n of t h a same type, w i t h two new i n d i c e s o f m a n o e u v r a b i l i t y a s c o e f f i c i e n t s .