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Dynamics of ships and floating platforms Ele, Abraham Y. 1989

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D Y N A M I C S O F S H I P S A N D F L O A T I N G P L A T F O R M S BY ABRAHAM Y. ELE B.Sc, American University in Cairo, 1986 A THESI8 SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1989 © Abraham Y. Ele, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ^ firrd I c A (_ £ W C /rj &&Zl A/ G The University of British Columbia Vancouver, Canada Date Zo/oV/g"!  DE-6 (2/88) ABSTRACT Three d i m e n s i o n a l angu la r mot ions of s h i p s and f l o a t i n g p l a t f o r m s are i n v e s t i g a t e d u s i n g E u l e r ' s dynamica l e q u a t i o n s . The s o l u t i o n of E u l e r ' s e q u a t i o n s p r e d i c t two new o s c i l l a t o r y mot ions , namely r o l l and p i t c h t o g e t h e r wi th a c o n s t a n t yawing or t u r n i n g - r a t e . U n l i k e c l a s s i c a l r o l l and p i t c h the new motions have a common p e r i o d of o s c i l l a t i o n which depends on the c h a r a c t e r i s t i c s of the s h i p and the magnitude of the t u r n i n g - r a t e . The p o s s i b i l i t y t h a t g y r o s c o p i c c o u p l i n g between r o l l i n g and p i t c h i n g induce yawing i s c o n s i d e r e d f o r unpowered p l a t f o r m s s i t t i n g i n o b l i q u e r e g u l a r waves. The r o t a t i o n a l k i n e t i c energy shows t h a t f o r some sea s t a t e s , t h e r e c o u l d be a non t r i v i a l t u r n i n g - r a t e which min imizes the peak energy so t h a t the p l a t f o r m r o l l s and p i t c h e s w h i l e t u r n i n g a t a c o n s t a n t - r a t e . Resonance takes p l a c e when the f requency of the wave c o i n c i d e s w i t h the f requency of the coup led mot ions i n s t i l l water . For s h i p s maneuvering i n r e g u l a r waves, t h e r e are i n f i n i t e comb ina t ions of the t u r n i n g - r a t e and the f requency of the wave which c o u l d cause r e s o n a n c e . Damping, however s m a l l i t may be , c o n s i d e r a b l y reduces the ampl i tude a t r e s o n a n c e . R e g a r d l e s s of i t s o r i e n t a t i o n w i th r e s p e c t to the d i r e c t i o n of wave p r o p a g a t i o n , s imp le r o l l i n g or p i t c h i n g of an unbalanced - i i -body i s not p o s s i b l e , i n s t e a d , the re i s a combined r o l l - p i t c h mot ion hav ing a common f r equency , which does not appear to have been i n v e s t i g a t e d p r e v i o u s l y . - i i i -TABLE OF CONTENTS Abstract i i Table of Contents i v L i s t of Figures v Nomenclature v i Acknowledgements v i i i 1. INTRODUCTION 1 E. SHIP THEORY 8 2-1 Elementary Hydrostatics 8 2-1-1 Equilibruim and s t a b i l i t y 8 2-1-2 Couples Acting on a T i l t e d Floating Body 9 2-2 Theory of Small O s c i l l a t i o n s 12 2-2-1 Unresisted motion in s t i l l water 14 2-2-2 Unresisted motion among waves 16 2-2-3 Motion in a Resisting medium 19 3. EQUATIONS OF MOTION 2 3 4. COUPLED MOTIONS 28 4-1 Undamped Motion of a Balanced Body 28 4-1-1 In S t i l l Water 28 4-1-2 In Regular Waves 34 4-1-3 Energy Consideration for Platforms in Waves 40 4-2 Damped Motion of a Balanced Body 44 4-3 Motion of Unbalanced Body 54 5. CONCLUSION 6 0 6. NOTES 6 4 7. BIBLIOGRAPHY 65 - l v -LIST OF FIGURES F i g u r e 1 Body axes and the degrees-of-freedom of a s h i p . 66 F i g u r e 2 A t i l t e d f l o a t i n g body. 66 F i g u r e 3 Couples a c t i n g on t i l t e d f l o a t i n g o b j e c t . 66 F i g u r e 4 Coordinates to d e s c r i b e s h i p motion i n waves; e a r t h - f i x e d axes Xt, , p r i n c i p a l i n e r t i a l axes X~ and p r i n c i p a l water plane axes X.. 67 F i g u r e 5 E u l e r i a n a n g l e s . 67 F i g u r e 6 Angles s u i t a b l e f o r d e s c r i b i n g s h i p motions with l a r g e yaw;rotating t r i a d g , g z and g g i n i t i a l l y c o i n c i d e n t with f i x e d t r i a d G , G and G . 68 1 2 3 F i g u r e 7 T r a j e c t o r i e s T=0. c o 2 / c o 2 versus c o 2 / c o 2 and K / c o . 69 2 1 2 1 F i g u r e 8 N a t u r a l frequency of g y r o s c o p i c coupled motion versus i n i t i a l c o n d i t i o n s . R= (e°/e°) 2( 1 - K ) / ( l + K ) . 70 1 2 2 1 F i g u r e 9 Wave frequency and p l a t f o r m o r i e n t a t i o n versus the t u r n i n g - r a t e t h a t minimizes the peak energy. 71 F i g u r e 10 Wave slope and p l a t f o r m o r i e n t a t i o n versus the t u r n i n g - r a t e t h a t minimizes the peak energy. 72 F i g u r e 11 R o l l amplitude to wave amplitude r a t i o versus the damping r a t i o and K C O 2 / T q . 73 F i g u r e 12 P i t c h amplitude to wave amplitude r a t i o versus the damping r a t i o and K c o 2 / T o . 74 F i g u r e 13 Roll-yaw Phase s h i f t versus damping r a t i o and K w z / T . 75 o F i g u r e 14 Roll-heave phase s h i f t v ersus damping r a t i o and K w 2 / T . 76 o F i g u r e 15 Amplitude r a t i o when K and T o have the same s i g n . 77 F i g u r e 16 The e f f e c t of imbalance on r o l l response. T * = ( w Z - c o Z ) ( c o Z - w Z ) + ( c o Z / K - c o Z ) ( c o Z K - c o 2 )tan 2/? 78 1 2 1 3 2 3 -v-NOMENCLATURE B , B Q Centre of. buoyancy. C Rate or speed of t u r n i n g about the v e r t i c a l . C Rate of t u r n i n g t h a t minimizes the peak k i n e t i c energy. C c e n t r e of f l o a t a t i o n . F D D i f f e r e n t i a l o p e r a t o r . E K i n e t i c energy. G-'/9-/9. Base v e c t o r s . t X, V Gil (1=1,2) t r a n s v e r s e and l o n g i t u d i n a l m e t a c e n t e r i c h e i g h t s r e s p e c t i v e l y . GZ R i g h t i n g Lever. i Dummy index r e p r e s e n t i n g t h r e e - d i m e n t i o n a l r o t a t i o n ; e.g.,1=1 r o l , i = 2 p i t c h and i=3 yaw. I p r i n c i p a l moments of i n e r t i a about X.. i x. (1=1,2) P r i n c i p a l area moments of i n e r t i a of the water pla n e . K Dlmensionless i n e r t i a s ; K =(I -I )/I and K =(l -I )/I i ' 1 2 3 1 2 3 1 2 V W K = ( l + K i ) ( K z - l ) C 2 . L. Damping moment c o e f f i c i e n t . 1 D i r e c t i o n cosine m a t r i x . Tnn M. E x t e r n a l moment, X. ,r M^  Moment of r e s i s t a n c e or damping moment M. Wave induced hydrodynamic moment, X, .c M Wave Induced constant t i l t i n g moment t time. T = (w -co ) (o> -co ) . O. 1 2 - v i -T =T + Kco Z . o T ( w Z / K -co 2) (coZK -coZ) . O 1 3 Z 3 <x Ins tantaneous ang le of s u r f a c e waves. « Maximum angle of s u r f a c e wave. ft Angle of imba lance . A Weight or bouyancy of a f l o a t i n g body;sometimes i t i s c a l l e d d i s p l a c e m e n t . s Wave to s t r u c t u r a l f r equency r a t i o , v O r i e n t a t i o n w i th r e s p e c t to the d i r e c t i o n of p ropaga t i on of waves. V,4>,& E u l e r i a n a n g l e s . r.x Phase a n g l e . r M a g n i f i c a t i o n f a c t o r or s t r a c t u r a l ampl i tude to wave ampl i tude r a t i o . & i Angu la r d i sp l acement used to d e s c r i b e r o t a t i o n f o r a s h i p w i th l a rge yaw. Oc I n c l i n a t i o n at t = 0 . - v i i -ACKNOWLEDGEMENTS I would l i k e to thank my s u p e r v i s o r , p r o f e s s o r Henry Vaughan,for h i s guidance i n the a n a l y s i s and f i n a l p r e p a r a t i o n of t h i s t h e s i s . I g r e a t l y a p p r e c i a t e a l l h i s support. Thanks i s a l s o to a l l my teachers f o r t r a i n i n g me s i n c e my c h i l d h o o d . S p e c i a l thanks i s to World U n i v e r s i t y S e r v i c e of Canada (WUSC) f o r sponsoring the f i r s t year of my s t u d i e s a t U.B.C.. - v i i i -INTRODUCTION 1 INTRODUCTION The c l a s s i c a l work on the mot ion of f l o a t i n g o b j e c t s In waves was w r i t t e n i n 1861 by W. Froude and i s e n t i t l e d "On the r o l l i n g of s h i p s " . 1 In t h i s paper Froude showed t h a t f o r a g i ven I n i t i a l i n c l i n a t i o n and angu la r v e l o c i t y , the subsequent p o s i t i o n of a s h i p In s t i l l water may be expressed In terms of the n a t u r a l p e r i o d . Fu r the rmore , by c o n s i d e r i n g the l i n e a r d imens ions of the s h i p to be s m a l l i n compar ison to the l eng th of the wave, Froude was ab l e to e x p l a i n the behav iour of a r o l l i n g s h i p i n a seaway; i n p a r t i c u l a r , the f i r s t mathemat ica l e x p l a n a t i o n of the phenomenon of resonance i s due to h im. A few decades l a t e r K r i l o f f g e n e r a l i z e d F r o u d e ' s t heo r y by t r e a t i n g a s h i p as a r i g i d body i n space hav ing s i x deg rees-o f- f r eedom, of which fou r (namely: heave, r o l l , p i t c h and yaw) are o s c i l l a t o r y . He e s t a b l i s h e d s i x coup led d i f f e r e n t i a l equa t ions f o r the s i x motions ( three t r a n s l a t i o n s and th ree r o t a t i o n s ) , which he was subsequen t l y ab l e to uncouple by making some s i m p l i f y i n g a s s u m p t i o n s . 2 For those who are f a m i l i a r w i th gy roscopes , i t Is a w e l l known f a c t t ha t s imu l taneous r o t a t i o n i n two p e r p e n d i c u l a r p lanes are f e l t i n a p lane a t r i g h t a n g l e s . That e f f e c t can be produced wi thout the presence of an a c t u a l g y roscope , as i n the case when the s t a b i l i t y of a f l y i n g machine i s c o n s i d e r e d . A f l o a t i n g 1 INTRODUCTION 2 o b j e c t has three r o t a t i o n a l degrees-of-£reedom (namely: r o l l , p i t c h and yaw) and the simultaneous execution of two of these motions must be f e l t i n the t h i r d plane; t h i s phenomenon, even though i t may have been observed by many has never been g i v e n due a t t e n t i o n . In 1920, while s t u d y i n g the r o l l i n g of model s h i p s i n waves, Suyehiro observed the tendency o£ models to yaw i n one d i r e c t i o n or a n o t h e r . 3 The tendency t o yaw, he thought, was generated by asymmetric d i s t r i b u t i o n o£ wave pressure around the h u l l . In order to get r i d of such source of yawing, he repeated the experiment with a h e m i s p h e r i c a l model and found out th a t the behaviour of the l a t t e r was s i m i l a r to th a t of the s h i p l i k e model. Regarding t h i s problem of yawing, Suyehiro found a mathematical s o l u t i o n i n which he e x p l a i n e d how, depending on the range of the frequency of the wave, a s h i p advancing i n ob l i q u e r e g u l a r waves c o u l d t u r n bow i n t o the sea i n some i n s t a n c e s or t u r n broad s i d e to the waves and expose h e r s e l f to heavy r o l l i n g i n some other i n s t a n c e s . However, a few t h i n g s need to be s a i d about h i s method of s o l u t i o n . F i r s t l y , he used a c o o r d i n a t e system f i x e d to the ground. Since the r e l a t i v e geometry of the body with r e s p e c t to the c o o r d i n a t e system was changing, the mass moments and the products of i n e r t i a were not co n s t a n t . Secondly, the gyroscopic terms are ne g l e c t e d in the f i r s t tvo o£ the three equations thereby g i v i n g two uncoupled e q u a t i o n s . 4 As we w i l l see l a t e r the equations are i n t e r l a c e d through g y r o s c o p i c 2 INTRODUCTION 3 c o u p l i n g and g y r o s c o p i c e f f e c t s must be r e t a i n e d i n the s o l u t i o n . Goodman extended S u y e h i r o ' s a n a l y s i s to the case of a s h i p i n random s e a s . He conc luded h i s a n a l y s i s by showing t h a t fo r some sea s t a t e the s h i p tends to head sea and fo r some other sea s t a t e i t tends to beam s e a . A l o t of r e s e a r c h has been conducted on sea keep ing t h e o r i e s f o r the l a s t t h i r t y y e a r s . S t r i p theo ry has been used e f f e c t i v e l y to p r e d i c t mot ions and loads of s h i p s and p l a t f o r m s . The s t r i p t heo r y t echn ique of e v a l u a t i n g s h i p mot ion i n v o l v e s s u b d i v i d i n g the s h i p i n t o a f i n i t e number of s t r i p s . Each s t r i p i s i n v e s t i g a t e d by u s i n g a v e l o c i t y p o t e n t i a l which must s a t i s f y the l o c a l boundary c o n d i t i o n s on the body and other c o n d i t i o n s f o r out go ing waves. The s o l u t i o n i s n u m e r i c a l l y i n t e g r a t e d over the h u l l s u r f a c e to o b t a i n hydrodynamic f o r c e s and moments a c t i n g on the complete h u l l . A thorough rev iew of s t r i p theo ry can be found i n the paper by Kim, Chou and Tien.** In t h i s paper coup led heave-p i t ch and sway-rol l-yaw equa t ions were s o l v e d to c a l c u l a t e the response of s h i p s to o b l i q u e r e g u l a r waves. Us ing E u l e r ' s dynamica l e q u a t i o n s , Vaughan examined g y r o s c o p i c c o u p l i n g between r o l l , p i t c h and yaw. 7 N e g l e c t i n g v i s c o u s damping and under the assumpt ion of s m a l l ampl i tude mot ion , the th ree d imens iona l equa t i ons were s o l v e d c o m p l e t e l y . I f the r a t e of t u r n i n g or yaw i s c o n s t a n t , a c o n d i t i o n which i s s a t i s f i e d by the assumpt ion of s m a l l ampl i tude mot i on , the 3 INTRODUCTION 4 s o l u t i o n p r e d i c t s two new motions a t a s i n g l e f r equency and a qua r t e r of a c y c l e a p a r t . In a d d i t i o n to the w e l l known r o l l - h e a v e and p i t ch-heave coup led mot ions , he e s t a b l i s h e d tha t f o r a s h i p making a c o n t r o l l e d t u r n i n waves both r o l l and p i t c h a t e GOUpled t-0 yaw, l £ i t were no t fo r r e sonance , however, the c o u p l i n g e f f e c t of t u r n i n g on r o l l i n g and p i t c h i n g c o u l d g e n e r a l l y be c o n s i d e r e d s m a l l . I t has been observed on a number of o c c a s i o n s , t h a t f l o a t i n g o b j e c t s r i d i n g on o b l i q u e waves t u r n i n one d i r e c t i o n or another depending on the c h a r a c t e r i s t i c s of the f l o a t i n g body and the f requency of the wave. 8 A l s o , i t has been shown t h a t the yawing moment due to asymmetr ic d i s t r i b u t i o n of wave p r e s su re around the h u l l may not be the cause of the t u r n i n g . P Fu r the rmore , the magnitude of the yawing moment i s q u i t e sma l l f o r c o n v e n t i o n a l h u l l forms and i s zero fo r most o f f s h o r e p l a t f o r m s because of symmet r i ca l deck s t r u c t u r e . Under the i n f l u e n c e of a s e t of cop lana r f o r c e s the a x i s of s p i n of a gyroscope w i l l t r y to move i n a d i r e c t i o n a t r i g h t - a n g l e s to the d i r e c t i o n of the f o r c e s ; i f i t i s o b s t r u c t e d from a t t a i n i n g i t s want, i t w i l l f i g h t back f u r i o u s l y . T h i s seeming ly odd behav iour of gy roscopes i s r e f e r r e d to as g y r o s c o p i c c o u p l i n g . As mentioned e a r l i e r , the phenomenon of g y r o s c o p i c c o u p l i n g i s not r e s t r i c t e d to gyroscopes or other o b j e c t s of r e v o l u t i o n ; the e f f e c t i s f e l t whenever a body excutes 4 INTRODUCTION 5 mot ion i n two mu tua l l y p e r p e n d i c u l a r p lanes s i m u l t a n e o u s l y . Thus a s h i p r o l l i n g about the l o n g i t u d i n a l a x i s which i s a t the same t ime t u r n i n g about the v e r t i c a l , as when she i s be ing s t e e r e d onto a c o u r s e , must exper i ence p i t c h i n g . On the o ther hand, s imu l taneous r o l l i n g and p i t c h i n g , as i n the case of an unpowered p l a t f o r m s i t t i n g i n ob l i que waves, shou ld induce yawing . The o b j e c t i v e of t h i s s tudy i s to I n ve s t i g a t e the g y r o s c o p i c c o u p l i n g be tween r o l l , p i t c h and yaw w i th emphasis on the response a t resonance and f i n d i n g ways of r e l a t i n g the r a t e of yawing to the c h a r a c t e r i s t i c s of the f l o a t i n g body and the parameters of the wave. Chapter two g i v e s some t h e o r e t i c a l background; f i r s t , h y d r o s t a t i c s and s t a b i l i t y of f l o a t i n g ob j e c t s i s r ev i ewed , then the t heo r y of sma l l o s c i l l a t i o n , w i th emphasis on r o l l and p i t c h i s h i g h l i g h t e d . In chapte r 3 E u l e r ' s dynamica l equa t ions are w r i t t e n down but i n s t ead of u s i n g E u l e r i a n a n g l e s , which are more s u i t a b l e fo r d e s c r i b i n g the mot ion of gyroscopes r a t h e r than the angu la r mot ion of s h i p s , a d i f f e r e n t se t of ang les i s i n t roduced fo r use i n the th ree d l m e n t i o n a l e q u a t i o n s . Coupled mot ions of a ba lanced f l o a t i n g body (where the p r i n c i p a l axes of i n e r t i a a t G are p a r a l l e l to the p r i n c i p a l water p lane axes a t the cen te r of f l o a t a t i o n ) i s c o n s i d e r e d i n the f i r s t p a r t of chapte r 4. under the assumpt ion of sma l l 5 INTRODUCTION 6 ampl i tude motion E u l e r ' s equa t ions are reduced to a s e t o£ o r d i n a r y d i f f e r e n t i a l equa t ions w i th cons tan t c o e f f i c i e n t s , the s o l u t i o n of which p r e d i c t s two new o s c i l l a t o r y mot ions toge the r w i th a cons t an t r a t e of t u r n i n g . The o s c i l l a t o r y mot ions are about the l o n g i t u d i n a l and t r a n s v e r s e axes ; n e v e r t h e l e s s , u n l i k e c l a s s i c a l r o l l and p i t c h both mot ions have a common n a t u r a l p e r i o d of o s c i l l a t i o n . The p e r i o d of o s c i l l a t i o n i s determined by the c h a r a c t e r i s t i c s of the f l o a t i n g ob j e c t and the magnitude of the r a t e of t u r n i n g . Us ing F r o u d e ' s p o s t u l a t e f o r wave moments, wave induced motions are i n v e s t i g a t e d . Except a t resonance the ampl i tude of the coup led mot ions are g e n e r a l l y s m a l l . For a s h i p maneuvering i n waves, the re i s a t u r n i n g r a t e f o r which resonance may take p l a ce depending on the o r i e n t a t i o n of the s h i p w i th r e s p e c t to the d i r e c t i o n of p ropaga t i on of the waves. Resonance can a l s o occur at ze ro t u r n i n g r a t e , i f the f requency of the wave matches the f r equency of r o l l or p i t c h . From energy c o n s i d e r a t i o n fo r an unpowered p l a t f o r m s i t t i n g i n o b l i q u e r e g u l a r waves, i t i s shown tha t the re may be a r a t e of t u r n i n g , d i f f e r e n t from z e r o , which min imizes the peak r o t a t i o n a l k i n e t i c energy . The r e s u l t a l s o shows tha t damping, however s m a l l i t may be , s u b s t a n c i a l l y reduces the ampl i tude a t r e sonance . F i n a l l y , the e f f e c t of Imbalance on the ampl i tude of the coup led mot ion i s i n v e s t i g a t e d ; a d y n a m i c a l l y unbalanced s h i p i s 6 INTRODUCTION 7 one In which the p r i n c i p a l axes o£ i n e r t i a a t G are not p a r a l l e l to the p r i n c i p a l water p lane axes a t C . s imple r o l l i n g or p i t c h i n g of an unbalanced s h i p are shown to be i m p o s s i b l e . Though the response ampl i tude tends to i n c r ease w i th the imba lance , the i n c r ease i s i n s i g n i f i c a n t up to an imbalance of 7 THEORY 8 2 SHIP THEORY 1 0 2-1 E lementary H y d r o s t a t i c s : The f a m i l i a r law of A rch imede ' s s t a t e s tha t when a s o l i d i s immersed i n a l i q u i d , i t expe r i ences an u p t h r u s t equa l to the weight of the d i s p l a c e d l i q u i d . T h i s up th rus t i s c a l l e d the buoyancy of the o b j e c t ; and t h u s , f o r a body t o be f l o a t i n g f r e e l y i n a f l u i d i t s weight must be equa l to the buoyancy. The l i n e of a c t i o n of the buoyancy f o r c e always passes through the c e n t r o i d of the submerged volume or the cen te r of buoyancy. 2-1-1 E q u i l i b r u i m and S t a b i l i t y : A r i g i d body i s i n a s t a t e of e q u i l i b r u i m i f a l l the f o r c e s a c t i n g on i t a re ze ro and the r e s u l t a n t moment of the f o r c e s i s a l s o z e r o . A f r e e l y f l o a t i n g ob j e c t i s under the i n f l u e n c e of two equa l and oppos i t e f o r c e s , i t s own weight and the buoyancy f o r c e ; hence , e q u i l i b r u i m e x i s t s on l y i f these f o r c e s ac t a l ong the same l i n e . For a f l o a t i n g body t h i s l i n e must be v e r t i c a l . A body i s s a i d to posses p o s i t i v e s t a b i l i t y i f s u b j e c t to a sma l l d i s t u r b a n c e from the s t a t e of e q u i l i b r u i m , i t tends to r e t u r n to t h a t s t a t e ; i f the body posses n e u t r a l s t a b l i t y , i t w i l l remain i n i t s new p o s i t i o n a f t e r the d i s t u r b a n c e . I f the d e v i a t i o n from e q u i l i b r u i m tends to i n c r ease a f t e r the d i s t u r b a n c e , then the body i s s a i d to be i n a s t a t e of uns tab l e 8 THEORY 9 e q u i l i b r u i m or to posses nega t i ve s t a b i l i t y . A c c o r d i n g to the axes d e f i n e d i n f i g u r e 1 (the p o s s i t i v e d i r e c t i o n s a re i n d i c a t e d by the a r row ) , the gene r a l mot ion of a s h i p can be r e s o l v e d i n to th ree t r a n s l a t i o n a l and th ree r o t a t i o n a l components. T r a n s l a t i o n a l d i s t u r b a n c e s a l o n g x- and y-ax i s l ead to no r e s u l t a n t f o r c e so tha t the body i s i n n e u t r a l e q u i l i b r u i m f o r these types of d i s t u r b a n c e s . T r a n s l a t i o n a long the z - a x i s , be i t In the p o s i t i v e or nega t i ve s e n s e , i s countered by buoyancy f o r c e which w i l l tend to move the body i n oppos i t e d i r e c t i o n , t end ing to r e s t o r e e q u i l i b r u i m ; thus e q u i l i b r u i m i s s t a b l e f o r t h i s k ind of d i s t u r b a n c e . While r o t a t i o n about the z - a x l s , yawing, produces no r e s u l t a n t f o r c e or moment so t ha t n e u t r a l e q u i l i b r u i m e x i s t s fo r t h i s type of d i s t u r b a n c e , no g e n e r a l i z a t i o n can be made r e g a r d i n g the r o t a t i o n about the x- or the y - a x i s . The s h i p may d i s p l a y s t a b l e , n e u t r a l or uns t ab l e e q u i l i b r u i m . 2-1-2 coup le A c t i n g on a T i l t e d F l o a t i n g Body: i f a f l o a t i n g body of an a r b i t r a r y shape (see f i g u r e 2) i s t i l t e d wi thout change of d i sp l a cemen t ( I . e . , the weight of d i s p l a c e d l i q u i d remains the same be fo re and a f t e r r o t a t i o n ) , the volume of emerged and immersed wedges remain the same. T h i s c o n d i t i o n can be ma thema t i c a l l y expressed as f o l l o w s : fy (x e) dx = fy (x e) dx (1) 9 THEORY 10 I .e . . fy x dx = fy x dx , which i s the c o n d i t i o n f o r the cen te r of ' J J F F JU A A a rea of the water p l a n e ; t h u s , the ob j e c t must r o t a t e about the l i n e p a s s i n g through the c e n t r o i d of water p lane a rea or cen te r of f l o a t a t i o n (c ). F" Cons ide r a sma l l r o t a t i o n e of the i r r e g u l a r shaped body shown i n f i g u r e 3 and l e t the r o t a t i o n be about one of the p r i n c i p a l water p lane a x i s th rough c ( i n t h i s case the i n e r t i a ma t r i x w i l l reduce to i t s p r i n c i p a l va lues and thus to produce r o t a t i o n about one of the a x i s we need to a p p l y moment about the same a x i s o n l y ) . The cen te r of buoyancy, B Q , which was l y i n g on the same v e r t i c a l l i n e w i th the cen te r of g r a v i t y , G, w i l l move to a new p o s s i t i o n , B. S ince the r o t a t i o n i s under the c o n d i t i o n of cons tan t d i s p l a c e m e n t , the magnitude of the buoyancy f o r c e remains the same; f u r the rmore , the weight and buoyancy f o r c e s , which were a c t i n g v e r t i c a l l y under the s t a t e of e q u i l i b r u i m , con t inue to do so a f t e r r o t a t i o n . In g e n e r a l , the two f o r c e s are sepa ra ted so t ha t the body i s sub j e c t ed to a moment AGZ, where GZ i s the p e r p e n d i c u l a r d i s t a n c e between the f o r ce s and A i s the weight of the body. As shown i n the d iagram, t h i s moment tends to r e s t o r e the body to the e q u i l i b r u i m p o s i t i o n and i t must, of c o u r s e , be equa l and oppos i t e to the e x t e r n a l l y a p p l i e d moment. The coup le i s termed the r i g h t i n g moment and GZ the r i g h t i n g l e v e r . Another important p o i n t to observe i s t ha t f o r the r o t a t i o n 10 THEORY 11 about one of the p r i n c i p a l water p lane axes , the cen te r of buoyancy always l i e s under M (the p o i n t of i n t e r s e c t i o n of the l i n e of a c t i o n of buoyancy f o r c e w i th the z - a x l s , see f i g u r e 3) . As 9 i s d i m i n i s h e d M approaches a l i m i t i n g p o s i t i o n c a l l e d metacen te r . For sma l l v a lues of © i t f o l l o w s tha t GZ=GM s i n © =* GM© (2) The e x t e r n a l moment r e q u i r e d to ho ld the body i n t h i s t i l t e d p o s i t i o n i s g i v en by M=AGZ=AGM© and the oppos ing r i g h t i n g or h y d r o s t a t i c moment by MH=-AGM© (3) The work done to g i ve the body i n f i g u r e 3 the sma l l t i l t of e degrees i s 9 e W=J* M 69 = J AGM© d© (4) o o dW The f i r s t d e r i v a t i v e ^ van i shes a t ©=0 and the second d e r i v a t i v e of W wi th r e s p e c t to 0 i s from which i t f o l l o w s tha t fo r r o t a t i o n about e i t h e r of the p r i n c i p a l a x i s the e q u i l i b r u i m of the body i s s t a b l e i f GM>0, uns tab l e i f GM<0 and n e u t r a l i f GM=0. T h i s important parameter gove rn ing the e q u i l i b r u i m of f l o a t i n g ob j e c t s i s c a l l e d me t a cen t e r i c h e i g h t . I t i s easy to show t h a t , i f the body i s r o t a t e d w i thout change of d i s p l a c e m e n t , then BM = BG+GM =J/V. Where J=moment of i n e r t i a of the water p lane and V=submerged volume. A s h i p has two BMs, the t r a n s v e r s e BM f o r r o t a t i o n about l o n g i t u d i n a l a x i s 11 THEORY 12 p a s s i n g through the cen te r of f l o a t a t i o n of the water p lane and the l o n g i t u d i n a l BM f o r r o t a t i o n about the t r a n s v e r s e a x i s . I f we d e s i g i n a t e the r o t a t i o n about the l o n g i t u d i n a l a x i s by •© and the one about the t r a n s v e r s e a x i s by Q^, then equa t i on 3 g i v e s MH= -AGM & 1 i i MH= -AGM B (5a) 2 2 2 MH= 0 3 I f the p r i n c i p a l water p lane axes c o i n c i d e wi th the p r i n c i p a l axes of i n e r t i a f o r the body then the p r i n c i p l e of angu la r momentum shows t h a t : i f the body i s r o t a t e d about the l o n g i t u d i n a l a x i s and then r e l e a s e d i t w i l l o s c i l l a t e i n r o l l w i th f requency u>± g i v en by w =V(AGM /I ) (5b) S i m i l a r l y f o r r o t a t i o n about the t r a n s v e r s e a x i s the n a t u r a l f r equency i n p i t c h i s g i v en by co =V(AGM /I ) (5c) 2 2 2 The h y d r o s t a t i c moments can then be r e w r i t t e n as MH= - w z i e , M h= - w z i e and MH= 0 (5d) 1 1 1 1 ' 2 2 2 2 3 2-2 Theory of Smal l O s c i l l a t i o n s As p o i n t e d out e a r l i e r , a s h i p may be regarded as a r i g i d body ac ted upon by a complex se t of wave f o r c e s which va r y w i th t ime . I t s mot ion may be r e s o l v e d i n t o th ree t r a n s l a t o r y components a l o n g o r thogona l axes f i x e d i n the s h i p and th ree r o t a r y components about these axes , see f i g u r e 1. The t r a n s l a t o r y mot ion a r e : (1) f o r - a n d - a f t d r i f t i n g p a r a l l e l to the 12 THEORY 13 l o n g i t u d i n a l a x i s o£ the s h i p or s u r g e ; (2) d r i f t i n g l a t e r a l l y p a r a l l e l to the t r a n s v e r s e a x i s or sway; and (3) heav ing p a r a l l e l to the v e r t i c a l a x i s . The r o t a r y motions about these th ree axes are r o l l i n g , p i t c h i n g and yawing r e s p e c t i v e l y . For s m a l l d i s t u r b a n c e , no r e s t o r i n g f o r c e or moment i s encountered i n s u r g e , sway and yaw modes; f o r the rema in ing t h r e e modes of mot ion , movement i s opposed by a f o r c e or moment p r o v i d e d the s h i p i s s t a b l e i n t h a t mode. Analogous to a s imple s p r i n g sys tem, the magnitude of the oppos ing f o r c e and moment v a r i e s l i n e a r l y w i th the d i s p l a c e m e n t from e q u i l i b r u i m p o s i t i o n , so long as the d i s t u r b a n c e remains s m a l l . In t h i s s e c t i o n , the d i s c u s s i o n w i l l focus on r o l l , p i t c h and yaw modes of m o t i o n . I f each mode i s c o n s i d e r e d s e p a r a t e l y , the e q u a t i o n s g o v e r n i n g r o l l and p i t c h i n s t i l l water a re s i m i l a r to the e q u a t i o n g o v e r n i n g the mot ion of a mass on a s p r i n g ; hence, f o r s m a l l u n r e s i s t e d d i s t u r b a n c e i n e i t h e r r o l l or p i t c h modes the s h i p excu tes s imp le harmonic mot ion . I t i s worth n o t i n g , however, t h a t such a s o l u t i o n i s v a l i d o n l y i f the mot ion i s i n a s i n g l e p l a n e , I . e . , I f the a x i s about which the s h i p i s r o t a t i n g i s s t a t i o n a r y . S imultaneous mot ions i n m u t u a l l y p e r p e n d i c u l a r p l anes (such as s imu l taneous r o l l i n g and p i t c h i n g ) couse g y r o s c o p i c c o u p l i n g and thus a d i f f e r e n t type of response Is to be e x p e c t e d ; i n f a c t , the o b j e c t i v e of t h i s t h e s i s , as mentioned e a r l i e r , i s t o i n v e s t i g a t e the g y r o s c o p i c c o u p l i n g between the 13 THEORY 14 three angular motions. 2-2-1 u n r e s i s t e d Motion In s t i l l Water: I£ a s h i p i s i n c l i n e d i n s t i l l water and then r e l e a s e d , a r i g h t i n g moment equal i n magnitude to the i n c l i n i n g moment opposes i t . I f damping i s negl e c t e d the p o t e n t i a l energy of the s h i p a f t e r i n c l i n a t i o n i s equal to the work done by e x t e r n a l moment. Upon removal of the e x t e r n a l moment, the r i g h t i n g moment r o t a t e s the s h i p to e q u i l i b r u i m or the u p r i g h t p o s i t i o n and a t t h i s p o i n t a l l the p o t e n t i a l energy i s converted to k i n e t i c energy. The r o t a t i o n continues to the other s i d e of the v e r t i c a l with c o n v e r s i o n of k i n e t i c energy to p o t e n t i a l energy. T h i s process i s repeated over and over a g a i n with the s h i p o s c i l l a t i n g from s i d e to s i d e i n simple harmonic motion with constant amplitude. R o l l i n g : l e t the Instantaneous i n c l i n a t i o n of the s h i p to the v e r t i c a l be 9 . I f damping i s n e g l e c t e d , Newton's law of motion g i v e s d z © I 1 = M 1 d t 2 The o n l y moment a c t i n g on the s h i p i n s t i l l water i s the r i g h t i n g moment given by equation 5, hence d 2 s 1 + wz6> = 0 (6) d t 2 T h i s i s a d i f f e r e n t i a l equation denoting simple harmonic motion with frequency to=V(AGM/I ). For s m a l l 9 , w i s independent of i i i i i 9 and hence such r o l l i n g i s isochronous. Since the p e r i o d of 14 THEORY 15 r o l l i s the same r e g a r d l e s s o£ the amp l i t ude , the average angu la r v e l o c i t y v a r i e s d i r e c t l y as the a m p l i t u d e . I f the i n i t i a l c o n d i t i o n s a t t=0 are £ ±(0) = G° and 9^(0) = 0°, the s o l u t i o n of equa t i on (6) becomes e° $ = & ° Cos w t + - S i n » t (7) 1 1 1 CO 1 1 I f the s h i p i s a s s igned i n i t i a l c o n d i t i o n s such t h a t a t t=0, © (0)= 0 and & (0)=&°, we o b t a i n 1 1 1 e° e = - S i n w t ( 8 ) 1 CO 1 I S ince the mot ion i s harmonic , the angu la r v e l o c i t y i s a maximum when the i n c l i n a t i o n i s ze ro and i s j t imes the mean angu la r v e l o c i t y , wh i ch , f o r an extreme i n c l i n a t i o n of 0 , i s 20 u /n. m' m i Hence 9 = 9 S i n co t i m i I f , when t=0, the i n c l i n a t i o n i s equa l to 6 and 6 (0)=0, m i ' e = e cos co t i m i P i t c h i n g : the equa t i on of mot ion which d e s c r i b e s p i t c h i n g i s d 2 e —J + "2v° <9> a t T h i s Is s i m i l a r to equa t i on (6) which governs the r o l l i n g mot ion . However, the f requency of o s c i l l a t i o n i n p i t c h i n g i s co 2 =V(AGM 2 / I 2 ) . S ince the equa t i on gove rn ing p i t c h i n g and r o l l i n g i n r e g u l a r waves are a l s o s i m i l a r , o n l y r o l l i n g i n beam seas w i l l be c o n s i d e r e d i n the f o l l o w i n g s e c t i o n . The d i s c u s s i o n , muta t i s mutand is , can be extended to p i t c h i n g of s h i p s i n head s e a s . 15 THEORY 16 2-2-2 U n r e s i s t e d Mot ion Among Waves: The e f f e c t of wave induced f o r c e s and moments on f l o a t i n g s t r u c t u r e s i s d i f f i c u l t to d e s c r i b e m a t h e m a t i c a l l y ; a number of assumpt ions are u s u a l l y made i n o rder to approximate the r e a l mot ion . Forexample , more o f t e n than n o t , the wave d i s t r i b u t i o n encountered i s random i n nature and the p r o f i l e of an i n d i v i d u a l wave i s t r o c h o l d a l ; however, i n o r d e r to o b t a i n a c l o s e d form s o l u t i o n to the gove rn ing d i f f e r e n t i a l e q u a t i o n , we take a r e g u l a r wave of s i n u s o i d a l p r o f i l e as our f o r c i n g f u n c t i o n . The s o l u t i o n thus ob ta ined Is not e x a c t ; n e v e r t h e l e s s , s i n c e the response of a s h i p to random seas i s the sum of the response to i n d i v i d u a l f r equency components, the behav iour i n a r e g u l a r wave can g i ve us an idea of how the s h i p would per form i n an open s e a . In the f o l l o w i n g d i s c u s s i o n Froude*s hypo thes i s w i l l be u sed ; i t w i l l be assumed t h a t ; (1) the p re s su re d i s t r i b u t i o n w i t h i n the wave system i s u n a f f e c t e d by the presence of the s h i p ; (2) the e f f e c t i v e wave s lope passes through the cen te r of bouyancy of the s h i p ; (3) the r e s u l t a n t of the water p ressu re a c t s normal to the wave s u r f a c e . The s o l u t i o n i s r easonab le i f the wave l e n g t h i s much b igge r than the b read th of the s h i p . R o l l i n g : Based on the a fo rement ioned assumpt ions , the equa t i on of mot ion of a s h i p r o l l i n g i n waves may be w r i t t e n as d 2 * 1 + - « ) = o ( 1 0 ) d t 2 1 1 16 THEORY 17 The instantaneous wave slope a i s given by « = « Q S i n <»t (11) Where « i s the maximum angle between the h o r i z o n t a l and the s u r f a c e wave and « = frequency of the s u r f a c e wave. I f the i n i t i a l v e l o c i t y and the I n i t i a l i n c l i n a t i o n are give n by 9 (0) = 9° and 9 (0) =9° r e s p e c t i v e l y , the g e n e r a l 1 1 1 1 s o l u t i o n of equation (10) becomes a 9 = - { S i n cot - - S i n co t } + 1 1- « V c o t ) 2 9° { -± S i n co t + 9° cos cot } (12) CO 1 1 1 1 The second term on the r i g h t - h a n d - s i d e of equation (12) i s s i m i l a r to equ a t i o n (7) and g i v e s i n c l i n a t i o n due to u n r e s i s t e d r o l l i n g i n s t i l l water; t h i s term vanishes i f the s h i p i s e r r e c t (©°= 0) and s t i l l (9°= 0) when the wave a c t i o n s t a r t e d . The f i r s t term on the r i g h t - h a n d - s i d e given by ex e = 2 { S i n cot - - S i n cot } (13) 1 . . . . . 2 CO 1 1- (co/co ) 1 1 d e s c r i b e s the i n c l i n a t i o n due to the a c t i o n of the wave. Regarding equation (13) there are s e v e r a l cases to c o n s i d e r ; f i r s t we examine what happens when the p e r i o d of the wave matches the n a t u r a l p e r i o d of the s h i p . Equation (13) reduces t o 0/0; usi n g l ' H o p i t a l ' s r u l e to eva l u a t e the l i m i t we o b t a i n , a t « = co^  ot 9 = —{ Sinco t — co t Cosco t } I 2 i i i Even though the i n c l i n a t i o n i s constant when the s h i p i s 17 THEORY 18 l o c a t e d on the p o s i t i o n of maximum wave s l o p e , each s u c c e s s i v e i n c l i n a t i o n on the ho l low or c r e s t i s tret /2 g r ea t e r than the i n c l i n a t i o n on the p rev i ous ho l low or c r e s t . For eve ry wave c y c l e , i s i n c r e a s e d by TTQ^ and r e g a r d l e s s ' of the i n i t i a l c o n d i t i o n a t t=0 the mot ion , i n u n r e s i s t i n g medium, u n i f o r m l y Inc reases to i n f i n i t y . Suppose t h a t the s h i p i s r o l l i n g i n s t i l l water such tha t #°= 0 and &°= a w / ' d - w 2 / w Z ) when the wave a c t i o n s t a r t e d . 1 1 O 1 S u b s t i t u t i n g the above va lues of 9° and 9° i n t o (12 ) , we have f o r © , the ampl i tude of f o r c e d r o l l i n g a 9 = S in cot (14) 1 F 1- (co / c o ^ 2 COCK and 9 = - Cos cot (15) 1- (co /co^) At t=0, the angu la r v e l o c i t y of f o r c e d r o l l i n g i s g i v en by c o a 9° = 1- (co/co ) When « i < w, 9°^ i s p o s i t i v e and the s h i p i s r o l l i n g away from the wave c r e s t . On the other hand, i f the p e r i o d of the wave i s such tha t co <co , 9° i s nega t i ve and the s h i p r o l l s i n t o the wave. 1 I F E The maximum i n c l i n a t i o n to the wave normal i s the same i n each c a s e , but when & ° i s nega t i ve the i n c l i n a t i o n to the t rue ' I F v e r t i c a l i s v e r y much l e s s than when 9°^ i s p o s i t i v e . Equa t i on (14) shows tha t resonance occurs when w = w and when w i s v e r y sma l l compared to « .9 =- ±« . •* r 1 I F O 18 THEORY 19 2-2-3 Mot ion In a R e s i s t i n g Medium: Regard less o£ the c h a r a c t e r i s t i c s of the s h i p , the f o l l o w i n g sources of r e s i s t a n c e to r o l l i n g and p i t c h i n g may be i d e n t i f i e d : (1) the f r i c t i o n a l r e s i s t a n c e of the water on the wetted s u r f a c e . (2) r e s i s t a n c e due to a c c e l e r a t i o n of the water s e t i n mot ion by the immersed pa r t of the s h i p . (3) the g e n e r a t i o n of water waves by the s h i p ' s r o t a t i o n and (4) r e s i s t a n c e due to the a c t i o n of the a i r on the above water p a r t s of the s k i n . The r e s i s t a n c e o f f e r e d by the a i r i s ex t reme ly s m a l l to be c o n s i d e r e d h e r e . The f r i c t i o n a l r e s i s t a n c e and the r e s i s t a n c e due to the a c c e l e r a t i o n of the water va r y as the square of the angu la r v e l o c i t y . T h i s may be c o n t r a s t e d wi th the r e s i s t a n c e due to the g e n e r a t i o n of water waves, which v a r i e s as the angu la r v e l o c i t y . S ince a l l the d i f f e r e n t forms of r e s i s t a n c e are f u n c t i o n s of ^ and (g^ ) z , the e x p r e s s i o n fo r moment of r e s i s t a n c e i s R o l l i n g : Damping opposes the mo t i on , hence the e x p r e s s i o n fo r the damping moment, equa t i on (16 ) , may be i n c o r p o r a t e d i n t o the equa t i on of mo t i on , equa t i on (6 ) , as f o l l o w s d 2 © d© I 1 + L + co2© = 0 (17) * d t 2 1 d t Note t ha t on the b a s i s of sma l l ness of © the second term on the i 19 THEORY 20 r i g h t - h a n d - s i d e of equa t i on (16) has been d ropped . Equa t i on (17) can be expressed i n s tandard form d 2 © d© I 1 + 2K co + co 2© = 0 (17) ' d t 2 1 1 dt Where the damping r a t i o , C ± - 1^/2(0 I . For i n i t i a l c o n d i t i o n © ± ( 0 ) = © ° and © ^ ( 0 ) = © ° , the s o l u t i o n of equa t i on (17) becomes © V co © ° ©= E«(F ( -C « t ) { ©°Cos co t + — — 1 1 1 S i n co t} l i i d co d d Where co = c o V d - t ; ) i s the f r equency of damped f r e e o s c i l l a t i o n . d 1 1 For a s h i p among waves, a f o r c i n g f u n c t i o n i s i n t r o d u c e d on the r i g h t - h a n d - s i d e of equa t i on ( 1 7 ) ; F o l l o w i n g F r o u d e ' s hypo thes i s f o r wave moments, we have d 2 © d© 1 + 2C ^ - + co 2© = co2ot S i n cot (18) d t 2 1 1 d t The s o l u t i o n of equa t i on (18) has two components, r e s i s t e d r o l l i n g i n s t i l l water and the motion due to wave a c t i o n . The former d i m i n i s h e s i n g e o m e t r i c a l p r o g r e s s i o n and i s e x t i n g u i s h e d i n no t ime ; hence , the motion i s ma in l y a t the wave f r equency , © t = r « o s i n ( w t - r ) (19) 2 Where tan r = 2% ^e/ ); f r equency r a t i o , s = co/co^; and m a g n i f i c a t i o n f a c t o r , r=l/-/{ (l-<sz) + ( 4^ 2 C Z ) >. One of the e f f e c t s of damping i s to reduce ampl i tude at r esonance . As we saw e a r l i e r , when the f requency of the wave matches the n a t u r a l f r equency of a s h i p o s c i l a t i n g i n u n r e s i s t i n g medium, the ampl i tude i n c r e a s e s by equa l Increments to i n f i n i t y , in a r e s i s t i n g medium, however, the maximum ang le of r o l l , wh i le 20 THEORY 21 great, i s f i n i t e . At resonance s=l and the phase s h i f t , y=n/2; equation (19) shows t h a t the r o l l amplitude i s quarter of a c y c l e out of phase with the wave moment. © i s maximum when t=nny<*>, where n i s an i n t e g e r . Hence a t resonance, the maximum i n c l i n a t i o n s occur at wave c r e s t s and hollows. Since the superposed f r e e o s c i l l a t i o n i s damped out p r e t t y q u i c k l y , there must be some time the amplitude a t t a i n s a constant value of Tc< o. P r i o r to such time the maximum angles are gre a t but f i n i t e . The i n c l i n a t i o n s of s h i p s of ve r y s h o r t p e r i o d or high n a t u r a l frequency, i s of the same order of magnitude as the i n c l i n a t i o n of the wave; i n t h i s case yta 0, Trsl and hence equation (19) shows that the maximum i n c l i n a t i o n s occur where the wave slope i s g r e a t e s t , i . e . , a t the mid height of the wave and are equal to « o . The s h i p i s e r r e c t i n the c r e s t s and troughs. For s h i p s of long p e r i o d , © i s always s m a l l and has the maximum 1 value at mid h e i g h t s as w e l l . Yawing: Rotary motion of a s h i p about the v e r t i c a l a x i s approximately through i t s center of g r a v i t y , i . e . , d e v i a t i o n from a s t r a i g h t course i n s t e e r i n g , i s c a l l e d yawing. There are three types of f o r c e s and yawing moments: (1) the s t a t i c pressure of the water, which o f t e n i s not a t the same l e v e l on the two s i d e s of the s h i p . 21 THEORY 22 (2) dynamic p ressu re f o r c e s caused by the r o t a t i o n a l motion of the water i n waves. (3) the g y r o s t a t l c coup le due to i m p o s i t i o n of r o l l i n g motion on a p i t c h i n g s h i p or v i s e v e r s a . The f i r s t case a r i s e s when the s h i p i s advanc ing o b l i q u e l y to the d i r e c t i o n of wave p r o p a g a t i o n . The wave p r o f i l e s on each s i d e of the s h i p are then d i f f e r e n t so t ha t the l o n g i t u d i n a l p o s i t i o n s of the c e n t e r s of p r e s su re of the wetted s i d e s are a l s o d i f f e r e n t . Hence, a coup le p roduc ing r o t a t i o n about the v e r t i c a l a x i s i s i n t r o d u c e d . The s i g n of t h i s coup le changes as the waves move pas t the s h i p , so t ha t the r o t a r y mot ion becomes o s c l l l a t l o n a l hav ing the same p e r i o d as the wave. Dynamic p re s su re f o r c e s caused by r o t a t i o n a l mot ion of water p a r t i c l e s i n t r o c h o i d a l waves a l s o produce yawing i n t he apparent-wave p e r i o d . I f a s h i p i s r o l l i n g and p i t c h i n g s i m u l t a n e o u s l y , l i k e when advanc ing o b l i q u e l y to the wave d i r e c t i o n , the a x i s of r o l l i s not a f i x e d h o r i z o n t a l l i n e i n space but an a x i s which i t s e l f o s c i l l a t e s an amount equa l to the angu la r ampl i tude of p i t c h i n g . T h i s o s c i l l a t i o n of the a x i s of r o l l s e t s up a g y r o s t a t i c coup le which causes yawing. 22 EQUATIONS OF MOTION 23 3 EQUATIONS OF MOTION K i n e m a t i c s ; The motion of a r i g i d body i s known i f a t any g i ven t i m e , the po s i t i o n of eve ry p o i n t of the body can be l o c a t e d i n space . C h a s l e ' s theorem s t a t e s t ha t the most g e n e r a l d i sp l acement of a r i g i d body may be reduced to t ha t of a t r a n s l a t i o n , f o l l owed by a r o t a t i o n . Thus , to determine the M a b s o l u t e p o s i t i o n , X , ( i = l , 3) of any p o i n t M of the r i g i d body i n f i g u r e 4, we o n l y need to know the r e l a t i v e p o s i t i o n of G to 0 toge the r w i th the o r i e n t a t i o n of the body f i x e d a x i s , X^, w i th r e s p e c t to the e a r t h f i x e d a x i s , X ,^ . S ince t r a n s l a t i o n can be c o n s i d e r e d s e p a r a t e l y , we move the two a x i s to a common o r i g i n i n order to s tudy r o t a t i o n s . Nine d i r e c t i o n c o s i n e s i n the form of 1.,., where ^ J 1., =Cos(X., OX.), are r e q u i r e d i n order to d e s c r i b e the o r i e n t a t i o n of the body f i x e d axes X.. However, the n ine d i r e c t i o n c o s i n e s are not a l l i ndependent ; a w e l l known a l g e b r i a c r e l a t i o n s h i p i n the form, V i V * 6 i ' k ' ( 2 1 ) where 6^k, i s the Kronecker d e l t a , e x i s t s between them. Because of i t s i n t e rdependen t e lements the d i r e c t i o n cos i ne ma t r i x i s not a s u i t a b l e cho i c e f o r use as c o o r d i n a t e . The a l t e r n a t i v e method of d e s c r i b i n g t h r ee-d imens i ona l r o t a t i o n or body o r i e n t a t i o n i s by u s i n g E u l e r i a n a n g l e s . E u l e r i a n ang les d e s c r i b e the o r i e n t a t i o n of one c a r t e s i a n c o o r d i n a t e system w i th r e s p e c t to 23 EQUATIONS OF MOTION 24 a n o t h e r . A s e t of E u l e r i a n ang les may be d e f i n e d as f o l l o w s (the o r d e r i n g of the r o t a t i o n s i s impo r t an t , s i n c e l a r g e r o t a t i o n s are not commutat i ve ) . (1) F i r s t , a p o s i t i v e r o t a t i o n <p of the body about X g , a x i s i s taken to y i e l d an i n t e rmed i a t e p o s i t i o n where body- f i xed c o o r d i n a t e s , which p r e v i o u s l y c o i n c i d e wi th f i x e d d i r e c t i o n m.,, are now the c o o r d i n a t e a x e s , r e s p e c t i v e l y . The X^ a x i s i s c a l l e d the l i n e of nodes (see f i g u r e 5 ) . (2) Then a p o s i t i v e r o t a t i o n © about the new X^ a x i s , the l i n e of nodes , produces a new i n t e rmed i a t e p o s i t i o n where the body- f i xed axes X' becomes the X" c o o r d i n a t e s e t . (3) A f i n a l p o s i t i v e r o t a t i o n y about the new X" a x i s a produces the f i n a l o r i e n t a t i o n of the body where the body f i x e d axes are now the X^  c o o r d i n a t e sys tem. S ince X , c a r t e s i a n c o o r d i n a t e system has a f i x e d d i r e c t i o n i n space , the angu la r v e l o c i t y of the body i s Q = y + '<p + 0 (22) or i n terms of r e c t a n g u l a r components = © Cosy + 4> S in© S i n y O = -© S i n y + 4> S in© Cosy CI = y + 4> COS© 3 Eventhough E u l e r s ang les are v e r y s u i t a b l e f o r the s tudy of the mot ion of g y r o s c o p i c ins t ruments and s p i n n i n g t o p s , where the a x i s of s p i n , n u t a t i o n and p r e c e s s i o n can be choosen to c o i n c i d e wi th the d i r e c t i o n of y , ©, and <p r e s p e c t i v e l y (see f i g u r e 5 ) , 24 EQUATIONS OF MOTION 2 5 they are not the s i m p l e s t s e t to d e s c r i b e the r o l l i n g and p i t c h i n g mot ions of s h i p s . System of Ang les fo r Sh ips w i th Large Yaw; T h i s s e c t i o n d e a l s w i th an a l t e r n a t i v e method of d e s c r i b i n g t h r e e - d i m e n s i o n a l r o t a t i o n or body o r i e n t a t i o n The se t of o rdered r o t a t i o n a r e : (1) a r o t a t i o n 6 about X , a x i s 3 9' (2) a r o t a i o n © 2 about X'z a x i s (3) a r o t a t i o n 9 about X" a x i s i i Note t ha t X' and X" are i n t e rmed i a t e p o s i t i o n s taken by body f i x e d r o t a t i n g axes , X , i n i t i a l l y p a r a l l e l to e a r t h f i x e d axes X t , . T h i s s e t i s v e r y s u i t a b l e f o r a n a l y s i n g s h i p mot ion i n which r o l l and p i t c h are s m a l l but yaw may be l a r g e . ©^and © z are r o t a t i o n s about l o n g i t u d i n a l and t r a n s v e r s e a x i s r e s p e c t i v e l y ; hence , f o r sma l l ampl i tude mot ion they c l o s e l y r ep r e sen t r o l l and p i t c h . Yaw, 6 , i s measured w i th r e s p e c t to f i x e d d i r e c t i o n , the d i r e c t i o n of p ropaga t i on of waves. I f we denote the base v e c t o r s f o r X . , , X' and X" by G , 1 1 X. X. g t and g. r e s p e c t i v e l y , the t r a n s f o r m a t i o n mat r ix between g , g 2 , g , and G . G „ , G^ , (see f i g u r e (6)) i s g i ven by 1 3 l z 3 mn Cos© Cos© Cos© S in© S in© -Cos© S in© Cos© S in© Cos© -S in© S in© 3 2 3 2 1 1 3 3 2 1 3 1 Cos© Sin© Cos© Cos© +Sin© S in© S in© S in© S in© S in© -Cos© S in© 2 8 3 1 3 2 1 3 2 1 3 1 - S in© Cos© S in© Cos© Cos© 2 2 1 2 l which f o r s m a l l ang les s i m p l i f i e s to 25 EQUATIONS OF MOTION 26 1 = mn cose - S i n e 3 Sin© co s e 3 3 -© © 2 1 e cose -e s i n e 2 3 1 3 e s i n e -e cose 2 3 1 3 (23) The angu la r v e l o c i t y i s g i v en by a = ^ + e2gz + e±q± (24) F i g u r e (6) shows tha t g = g Cos© -g S in© . hence , f o r s m a l l © 2 2 1 3 1 1 V V l (25) Now, we may use equa t i on (23) to r e l a t e G f c to g f c . G =Cose g -S in© g + (e Cose -e S in© )g 1 9 3 1 3^2 2 3 1 3 ^3 G =Sin© g +Cos© g + (© S in© -© Cos© )q 2 3 1 3 2 2 3 1 3 3 3 (26) G =-© g + © g + g 3 2 3 1 1^2 ^3 The f i n a l e x p r e s s i o n f o r the angu la r v e l o c i t y i s ob t a i ned by s u b s t i t u t i n g equa t i on (26) i n t o equa t i on (24 ) . For s m a l l ©^and © , o becomes 2' 1 3 2->^l ^ 3 1 2-*^ 2 3^3 6 = (0 - e e - £> Q \q + (& +e e + e & }q + e q (27) 1 3 2 3 2-> ^1 ^2 3 1 3 1 ^ 2 3^3 \ *• ' I Note t h a t , g ± , g z , g g are body- f i xed r o t a t i n g t r i a d i n i t i a l l y c o i n c i d e n t w i th e a r t h - f i x e d t r i a d G±f G 2 and G g r e s p e c t i v e l y . Equa t ions of mo t i on : The p r i n c i p l e of angu la r momentum shows t h a t , f o r body f i x e d p r i n c i p a l axes r e f e r e n c e d to a c en te r of mass p o i n t , the moment equa t i on i n th ree d imens i ona l r o t a t i o n of a r i g i d body i s g i v en by M = I o + f l -I a 1 1 1 3 2-* 2 3 M = i Ci + f i -i I n n (28) 2 2 2 1 3-* 1 3 M = I 6 + f i - i )o n 3 3 3 ^ 2 1 ^ 1 2 26 EQUATIONS OF MOTION 27 These equa t i ons are c a l l e d E u l e r ' s equa t ions fo r body- f i xed p r i n c i p a l a x e s . M ,^ M g / M g and 1 ,^ I2, Ig are e x t e r n a l moments and p r i n c i p a l i n e r t i a s , r e s p e c t i v e l y , a long the p r i n c i p a l d i r e c t i o n 1, 2 and 3. S u b s t i t u t i n g f o r O and O from equa t i on (27) i n t o e q u a t i o n (28) and d ropp ing terms c o n t a i n i n g the p roduc ts of & and & 2 on the b a s i s of t h e i r s m a l l n e s s , we o b t a i n ft = & - e e - f l+ K ) © 9 1 1 3 2 l J 3 2 $1 = 9 + 9 9 + f l - K ")© 9 (29) 2 2 3 1 *- 2 ^ 3 1 M = I 9 3 3 3 Where M.= M./I.and K , K a re nond imens iona l q u a n t i t i e s g i v en by K = (I - I )/I and K = (I - I )/I (30) 1 Z 3 1 Z 3 1 Z 27 COUPLED MOTIONS 28 4 COUPLED MOTIONS I £ a f l o a t i n g ob j e c t i s t i l t e d and then r e l e a s e d , i t o s c i l l a t e s about a l i n e p a s s i n g through the cen te r of f l o a t a t i o n , C^, (see e q u a t i o n (1)) p rov i ded t h a t e q u i l i b r u i m i s s t a b l e i n t ha t mode; t h i s l i n e shou ld be the p r i n c i p a l water p lane a x i s , i f the h y d r o s t a t i c r e s t o r i n g coup les are to be g i ven by equa t i on (5 ) . On the o ther hand, E u l e r ' s equa t i ons (see (28)) are w r i t t e n f o r p r i n c i p a l axes of i n e r t i a f i x e d a t G. S ince we want to s tudy r o t a t i o n , the l o c a t i o n of the o r i g i n of the two axes s e t i s not of g rea t impor tance ; hence we assume tha t the cen te r of f l o a t a t i o n and the c en t e r of g r a v i t y are c o i n c i d e n t . The o r i e n t a t i o n of the l n e r t i a l axes w i th r e s p e c t to the p r i n c i p a l water p lane axes shou ld be c o n s i d e r e d , however. In t h i s chapter coup led motions of (1) ba l anced body, where the two axes se t are p a r a l l e l and (2) unbalanced body, where the two s e t s are not p a r a l l e l w i l l be I n v e s t i g a t e d . 4-1 Undamped Mot ion of a Ba lanced Body 4-1-1 In S t i l l Water: For d i s t u r b a n c e i n s t i l l water , the on l y e x t e r n a l moments a c t i n g on the body are the h y d r o s t a t i c r e s t o r i n g or the r i g h t i n g moments, M , M and M . T h e r e f o r e , M , M and M i n equa t i on (29) take the h y d r o s t a t i c va lues g i v en by equa t i on (5 ) . 28 COUPLED MOTIONS 29 e - e e - fi+ K }e e + o>"e = o 1 3 2 1 ^ 3 2 1 1 & + # 9 + f l - K 1© © + w 2 © = 0 a a i < - 2-> a l 2 2 I © = 0 3 3 The t h i r d of the above equa t ions i s s a t i s f i e d i f the r a t e of t u r n i n g about the v e r t i c a l i s a c o n s t a n t , i . e . , © g= 0 or © g = C . S u b s t i t u t i n g t h i s va lue i n t o the f i r s t two e q u a t i o n s , we o b t a i n l i n e a r o r d i n a r y d i f f e r e n t i a l e q u a t i o n coup led i n ©±and e as f o l l o w s 9 - f l+ K )C9 + c o Z © = 0 1 *- 2 1 1 9 + f l - K ~]C9 + c o 2 © = 0 2 V 2J 1 2 2 Equa t i on (31) can be w r i t t e n i n a mat r ix format ( D z + c o 2 ) -(1+ K j C D (1- K j C D (31) ( D 2 + c o 2 ) 2 '0" o. or P (D)© = 0. Where the d i f f e r e n t i a l o p e r a t o r , D = d /dt assume a s o l u t i o n of the form _ _ ml 9 = Re We (33) S ince D n ( e m t )= m n e T n t , i t f o l l o w s tha t the r e s u l t of a p p l y i n g any po l ynomia l ope ra to r P(D) to e m i i s s i m p l y to m u l t i p l y e m t by P(m). Hence, P(m)Re n i t= 0 o r , d i v i d i n g out the s c a l a r f a c t o r emt, P(m)R = 0, which has a non t r i v i a l s o l u t i o n i f |P(m)|= 0. The expans ion of t h i s de te rminant y i e l d s a po l ynomia l i n m (34) the so c a l l e d c h a r a c t e r i s t i c equa t i on of the d i f f e r e n t i a l equa t i on m*+ m 2 ( c o 2 + c o Z - ( l + K ) ( K -1 )C Z )+ c o Z c o 2 = 0 2 1 1 2 1 2 (31) . For s t a b l e s o l u t i o n the r o o t s m. ( j = l , 4) of equa t i on (34) shou ld be i n complex con jugate p a i r s . Moreover , s i n c e the motion i s u n r e s i s t e d , we expect the r o o t s to be p u r e l y imag ina r y . Le t 29 COUPLED MOTIONS 30 the r o o t s be denoted by m = iQ. and m =-iQ ; f o r eve ry r oo t m and l x. x. x, i. m ,^ equa t i on (33) d e f i n e s a c o r r e s p o n d i n g complex con juga te v e c t o r . Rand R. Now, c o n s i d e r the f i r s t r oo t m = i f i , which when s u b s t i t u t e d i n t o the p o l y n o m i a l , P(m), y i e l d s ( w z - ft2) i i i (1- K )Cft -v f l+ K "ICO ( w 2 - ft2) 2 1 M - f v 12^ (35) r A i and r^a r e a r b i t r a r y c o n s t a n t s , hence , we take r = 1 and get r = - i ( c o 2 - n z ) / Q i ( l + K 4 ) C ; s i m i l a r l y , f o r m , s e t t i n g r = 1 g i v e s r = i ( « z - a2)/a (1+ K )C, 3 12 1 1 1 1 The elements of the v e c t o r co respond ing to the other con jugate r o o t , m z and m^, can be ob ta ined by s i m i l a r method; so t h a t 1 2 R = l ft (1+ K )C l l i (<o - ft ) • = -R and R = • 1 1 1 z i (co — O ) ft (1+ K )C 2 2 F i n a l l y , we s u b s t i t u t e fo r R and m i n t o the s o l u t i o n f o r 9, equa t i on (33 ) . There are two s o l u t i o n s c o r r e s p o n d i n g to each p a i r of the r o o t s of the c h a r a c t e r i s t i c e q u a t i o n ; s i n c e they are independent t h e i r s u p e r p o s i t i o n i s a l s o a s o l u t i o n . Now we may expand the g e n e r a l s o l u t i o n u s i n g E u l e r s fo rmula to o b t a i n 9 = A-{Re (R i )CosO i t- Im ( R j s i n ^ t ^ + B-{ Im (R i ) cosO i t+ Re (R±) s inft t + E^Re (R z )Cosft 2 t- im (R 2 ) s in f t 2 t ^ + F-{ Im (R z )Cosft 2t+ Re (R 2 ) s in f t 2 t ^ A, B, E and F are cons t an t s which depend on how the mot ions are i n i t i a t e d . S u b s t i t u t i n g fo r the r e a l and imag inary p a r t s of 30 COUPLED MOTIONS 31 v e c t o r s R and R Into the s o l u t i o n g i v e s 1 z © = ACosO t + BSinft t + ECosO t + FSinft t 1 , ' * \ z < 3 6 > (« - a£) (« - or) O = — - - ( A S i n O t - BCosQt) + — - - (ESinQ t - FCosO t) 2 a (1+ K )C 1 1 Ci (1+ K )C 2 2 1 1 2 1 Now, we l e t |m| = P i n e q u a t i o n (34) and denote the r e s u l t i n g equation by T, so t h a t T = P2+ P(co2+ w2-K)+ « V = 0 (37) 2 1 1 2 where K = (1+K^) ( K 2 ~ l ) C 2. Equation (37) i s q u a d r a t i c i n P and i t s s o l u t i o n depends on two parameters, namely co^w^/c^and K= ( 1 + K i ) ( K z ~ l ) C 2 . The s o l u t i o n may be expressed as 2P = l+w2-K±/{ ( K - Q - w J 2 ) ( K - d + w J 2 ) } (38) By d e f i n i t i o n P i s a r e a l number, hence the r a d i c a l i n equation (38) must be p o s i t i v e . Since P i s the square of the frequency of o s c i l l a t i o n , P must a l s o be p o s i t i v e . F i g u r e (7) i s the p l o t of P versus K=K/w 4for s p e c i f i e d v a l u e s of o . T h i s f i g u r e shows t h a t the nature of the f r e e o s c i l l a t i o n s I 3 depend, among other t h i n g s , on the s i g n of K. When K i s p o s i t i v e the s h i p o s c i l l a t e s a t a frequency between the n a t u r a l frequency of r o l l , co , and p i t c h , co , and the frequency f a l l s o u t s i d e co and 1 2 1 coz when K i s n e g a t i v e . Furthermore, r o l l i n g and p i t c h i n g motions, whose f r e q u e n c i e s are g i v e n by w^and w zas d e f i n e d i n e q u a t i o n (5), are not p o s s i b l e unless the t u r n i n g r a t e , C=0. That i s to say, i f the s h i p i s d i s t u r b e d about the l o n g i t u d i n a l a x i s o n l y , i t w i l l o s c i l l a t e a t the n a t u r a l frequency of r o l l or co^ ; on the other hand, i f the d i s t u r b a n c e i s about the t r a n s v e r s e a x i s , the motion 31 COUPLED MOTIONS 32 would be a t the n a t u r a l frequency i n p i t c h or w^. However, i f there i s a simultaneous d i s t u r b a n c e about the l o n g i t u d i n a l and the t r a n s v e r s e axes, g y r o s c o p i c c o u p l i n g i s I n e v i t a b l e and hence r o l l i n g and p i t c h i n g w i l l have the same p e r i o d of o s c i l l a t i o n . Since C may not be zero equation (38) shows t h a t the frequency of the new motion i s d i f f e r e n t from e i t h e r to or co . 1 z Suppose a t t=0 the s h i p or the p l a t f o r m i s giv e n an i n i t i a l o i n c l i n a t i o n &^(Q)=9^ about the l o n g i t u d i n a l a x i s together with an angular v e l o c i t y ^zi0)=9^ about the t r a n s v e r s e a x i s . Two of the four c o n s t a n t s i n equation (36) namely B and F w i l l v a n i s h and the remaining two w i l l assume the f o l l o w i n g v a l u e s ©°(1+K )C- e°(to 2-ft 2) ©°(<o 2-o 2 ) - ©°(1+K ) A = -5 i 1 1 2 E = 1 1 1 2 l - (39a) ( o 2 - o 2 ) (ft^-ft2) Equation (36) g i v e s 6>°(l+K )C- 0 ° ( « z - a z ) ©°(<o z-a z ) - e°(i+K )C _ 2 1 1 1 2 „ ^ , , 1 1 1 Z 1 „ & = Cos ft t + Cos ft t 1 (o 2-a 2) 1 ( Q 2 - Q 2 ) 2 2 1 2 1 ( w Z - ft2) 9°(1+K )C- © ° ( w Z - f t Z ) © = s i n ft t + 2 ft (1+ K )C (ft 2 - f t 2 ) 1 1 1 2 1 (co 2 - ft2) 6 > ° ( w 2-ft 2) - ©°(1+K )C * 2 1 1 1 2 1 • ^ a. , _ „ , . s i n ft t (39b) ft (1+ K )C (^-^!) Z 2 1 2 1 I t was shown i n s e c t i o n 2-2 t h a t s m a l l amplitude motions of a s h i p are s i n u s o i d a l . An i n i t i a l i n c l i n a t i o n of 9° about the l o n g i t u d i n a l a x i s w i l l cause the s h i p to o s c i l l a t e a t the n a t u r a l p e r i o d of r o l l ; the maximum amplitude f o r undamped motion w i l l remain a t I f the s t a r t i n g c o n d i t i o n i s an impulse, say 9° 32 COUPLED MOTIONS 33 about the t r a n s v e r s e axis, a g a i n the motion would be sinusoidal but a t the n a t u r a l p e r i o d of p i t c h ; the maximum amplitude of the speed i n the l a t t e r case remains a t Eventhough, i t i s hard to set a tanker p i t c h i n g i n s t i l l water, the above o b s e r v a t i o n c l o s e l y d e s c r i b e s the behaviour of models. N e v e r t h l e s s , equation (39), which d e s c r i b e s the coupled motion i n s t i l l water are non p e r i o d i c and one wonders whether or not p e r i o d i c motions are p o s s i b l e . The answer i s yes, provided that the r a t e of t u r n i n g i s give n by C = e°(o>z- C?)/(1+K)e° or C = © G ( c o 2 - fi2)/(l+K )9° (40) 1 1 1 1 2 1 1 2 1 2 I f C takes the value g i v e n by the f i r s t of equation (40), e and © 2 reduce t o the f o l l o w i n g simple e x p r e s s i o n s e = e° cos at e = (©°/o ) s i n at (41) 1 1 1 2 2 1 1 9^ and 6^ are s i n u s o i d a l , having a common frequency Q^ , and they are q u a r t e r of a c y c l e a p a r t . Since the maximum amplitude of undamped o s c i l l a t i o n s remain c o n s t a n t , the r a t i o , &Jez i s i n v a r i a b l e and i s equal to ©"/©g. In other words, equation (41) says t h a t i f the s h i p i s s e t i n motion a t t=0 such t h a t e {0)=&° i i and &^0)=e°f i t w i l l r o l l and p i t c h while t u r n i n g a t a uniform r a t e g i v e n by equation (40). The gen e r a l s o l u t i o n , equation (36), from which equation (39b) i s obtained, i s a s u p e r p o s i t i o n of two motions. The two motions are independent and t h e r e f o r e , are i n d i v i d u a l l y s o l u t i o n s of d i f f e r e n t i a l equation (31). I f we had cons i d e r e d the two motions s e p a r a t e l y , we would have a r r i v e d at the same c o n c l u s i o n d e r i v e d from equations (40) and (41). 33 COUPLED MOTIONS 34 In equation (38) we saw t h a t the frequency of o s c i l l a t i o n depends on co^  and K, i . e . , on the product of (1-K z)(l+K ) and C. Since C i s the f u n c t i o n of the I n i t i a l c o n d i t i o n s , as giv e n by equation (40), we can r e l a t e the frequency to the i n i t i a l c o n d i t i o n s a l s o ; t h i s i s done by s u b s t i t u t i n g f o r C from equation (40) i n t o e q u a t i o n (37), which g i v e s (w z -P )[(l-P)+P (w 2-p)R]=0 (42) Where R - ^ / e ^ d - K )/(l+K ) and P i s the square of the frequency of o s c i l l a t i o n . One of the s o l u t i o n s of equation (42), P=<*\, coresponds to C=0. The second s o l u t i o n i s (l-P)+P(w Z-P)R=0 (43) Equat i o n (43) i s p l o t t e d i n f i g u r e (8); the f i g u r e shows t h a t when R approaches 0, the frequency of o s c i l l a t i o n approaches the n a t u r a l frequency of p i t c h whereas when R becomes l a r g e the frequency of o s c i l l a t i o n becomes c l o s e r to the n a t u r a l frequency of r o l l . In other words, when the i n i t i a l angle of h e e l , i s n e g l i g i b l y s m a l l compared to the i n i t i a l speed i n the t r a n s v e r s e d i r e c t i o n (R-»0), the o s c i l l a t i o n would be e s s e n t i a l y about the t r a n s v e r s e a x i s , p i t c h . When 0° i s l a r g e compared to &°, (R-»<x) and the o s c i l l a t i o n would be about the l o n g i t u d i n a l a x i s or r o l l . In both cases the c l a s s i c a l case i s r e t r i e v e d (see s e c t i o n 2-2) s i n c e the two motions are uncoupled and thus C=0. 4-1-2 In Regular Waves: For s h i p s maneuvering i n waves, the e x t e r n a l moments (M , M zand Mg) of equation (29) c o n s i s t of a time independent component and a time v a r y i n g component i n a d d i t i o n to 34 COUPLED MOTIONS 35 the h y d r o s t a t i c r e s t o r i n g couple g i v e n by equation ( 5 ) . The e x i s t e n c e of a time independent component and the r e s u l t i n g constant t i l t of pla t f o r m s have been e s t a b l i s h e d by Mart i n and The time v a r y i n g component of the wave Induced moment depends on the assumed wave p r o f i l e . In t h i s study, Froude's p o s t u l a t e f o r wave moments i s f o l l o w e d ; the approach i s s u f f i c i e n t l y a c c u r a t e f o r the present study because here, the o b j e c t i v e i s to explo r e the nature of g y r o s c o p i c coupled motions r a t h e r than the e v a l u a t i o n of the exact magnitude of the rasponse. Assuming a s i n u s o i d a l wave p r o f i l e and f o r waves of long p e r i o d , the wave sl o p e may be taken a s : a = a S i n wt (44) o where « i s the maximum angle between the h o r i z o n t a l and the o s u r f a c e wave and co i s the frequency of propagation. The r e s u l t a n t moments a c t i n g on a body are giv e n by M = M°+ MH+ M S i n cot 1 i i i M = M°+ MH+ M S i n w t (45) 2 2 2 2 M = 0 3 M°and M° are the time independent couples due to wave a c t i o n ; M^and M", the h y d r o s t a t i c r e s t o r i n g couples, are g i v e n by equation ( 5 ) . A c c o r d i n g to Froude's h y p o t h e s i s f o r wave moments, which i s reasonable i f the wave l e n g t h i s l a r g e compared to the breadth of the s h i p , the maximum amplitude of the time v a r y i n g component can 35 COUPLED MOTIONS 36 be taken as — z — z M = l a w Smv and M = I « <*> Cosy ( 4 6 ) 1 1 O 1 2 2 O 2 F i n a l l y , we s u b s t i t u t e the e x p r e s s i o n f o r the e x t e r n a l moment given by e q u a t i o n ( 4 5 ) i n t o e q u a t i o n ( 2 9 ) . Since M 3 =0, the l a s t of equation ( 2 9 ) admits uniform r a t e of t u r n i n g as a s o l u t i o n , i . e . , © a = 0 and hence O = C. The f i r s t two equations then g i v e M°/I + M /I s i n w t = & - fl+ K Ice + w z e 1 1 1 1 1 -^ 1 J 2 1 1 M°/I + M/I S i n <*>t = & - (K -l)c& + <o2© 2 2 2 2 2 2 ^ 1 2 2 2 1 ' 2 or ( 4 7 ) .2 . Z , A. . ^ - i O [(D + « 4 ) - ( l + K j C D 1 f M i / I i + M 1 / I 1 s i n w t "1 [ 1 - K j C D (D2+ co 2) J {©J tl£/I2+ M 2 / I z S i n cot j We m u l t i p l y the f i r s t e quation by (D 2+co 2) and the second equation by DC(1+K ), so t h a t summing the r e s u l t i n g equations w i l l e l i m i n a t e 9 . 2 (D 2+<o 2) { (M°/I )+ (M /I ) S i n <ot}+ DC(1+K ) { (M°/I )+ ( M / I ) S i n <ot} 2 1 1 1 1 1 2 Z Z Z = [(D 2+ co 2)(D 2+ «*)-KD a]© d\> d Z © ( 4 8 ) 1 . 1- Z . 2 „ . . 2 2_ + (CO +CO -K ) + CO CO Q = d t * d t 2 * 2 (co Z-co 2) ( M / I ) S i n cot + (C(l+K )coM / I )Cos cot+ M°coZ/I 2 1 1 1 2 2 1 2 1 Equation ( 4 8 ) i s a f o u r t h order non homogeneous o r d i n a r y d i f f e r e n t i a l e q uation with constant c o e f f i c i e n t s ; the homogeneous s o l u t i o n has been found i n s e c t i o n 4 - 1 - 1 and we seek a p a r t i c u l a r s o l u t i o n of the form O = 0° + A Cos cot + B S i n cot ( 4 9 ) i i i i 36 COUPLED MOTIONS 37 S u b s t i t u t i n g e^and i t s d e r i v a t i v e s i n t o equation (48) and equating the c o e f f i c i e n t s of l i k e terms on both s i d e s of the e q u a l i t y s i g n , we o b t a i n t o C(l+K )M c * C t o(l+K )Cosy A = 1 2 = _2 i (50) l I T T z 2 2 — 2 2 2 O ( t o - t o )M ot t o ( t o - t o ) S i n y M B = —=-= = = and 9 = i I T T i _ 2 ± I to i i Where T= ( t o 2 - t o z ) ( < o 2 - t o z ) + K t o z (see equ a t i o n ( 3 7 ) ) . Though M and M are f u n c t i o n s of time, i t i s assumed that t h e i r v a r i a t i o n over one wave c y c l e i s s m a l l so t h a t an average value may be taken. Equation (49) c o n s i s t s of two motions. One i s coupled to the r a t e of yaw, C; s i n c e i t vanishes when c=0, we w i l l c a l l i t r o l l - y a w coupled response. The second component of 9^ which i s g i v e n by reduces to the c l a s s i c a l value when C=0;13 we w i l l r e f e r to the l a t t e r component as r o l l - h e a v e response. 6 w i l l be simply r e f e r e d to as r o l l . R e t u rning to equation (47) b r i e f l y and e l i m i n a t i n g e by s i m i l a r method we used e a r l i e r t o e l i m i n a t e 9 , we o b t a i n the f o l l o w i n g d i f f e r e n t i a l e q u a t i o n : 6*9 d 2 © + -(fc>z+«2-K) + t*u?e = dt dt (»f-» Z) (M /I ) S i n cot + (C(K - l ) c o M / I )Cos cot+ M°co Z/I (51) 2 2 2 2 x x 2 x 2 A p a r t i c u l a r s o l u t i o n 9= 9% A Cos tot + B S i n tot (52) 2 2 2 2 would s a t i s f y (51), i f the co n s t a n t s A , B , and 9° are giv e n by 37 COUPLED MOTIONS 38 c o C(K -DM a C c o c o 2(K - l ) S i n y / A = I i = (53) 2 I T T V ' l 2 2 — 2 2 2 O (to -to )M ot <«> (to -to )Cosy M _ 1 2 0 2 1 , 2 B = —=r-= = and 9 = a I T T a . a a I <*> 2 2 F o l l o w i n g the above d e f i n i t i o n , we may c a l l A 2 pitch-yaw coupled response and B z pitch-heave coupled response; 9^, the g e n e r a l motion, w i l l be c a l l e d p i t c h . The g e n e r a l s o l u t i o n i s the combination of the homogeneous s o l u t i o n , equation (36), and the p a r t i c u l a r i n t e g r a l s , equations (49) and (52). i n t h i s s e c t i o n o n l y f o r c e d motion, the motion a t the frequency of the wave, w i l l be d i s c u s s e d . The f r e e o s c i l l a t i o n vanishes i f proper i n i t i a l c o n d i t i o n s are assumed. i n a d d i t i o n to the constant t i l t , the e f f e c t of which can be c o n s i d e r e d s e p a r a t e l y , each f o r c e d motion c o n s i s t s of two motions which are q u a r t e r of a c y c l e a p a r t . Roll-yaw and pitch-yaw coupled responses, which are g i v e n by A^and A 2, are 90° out of phase with the wave; these two motions a t t a i n t h e i r maximum when cot=0, i . e . , when the wave slope i s z e r o . Hence, we expect the r o t a t i o n a l k i n e t i c energy to be a maximum on wave c r e s t s and hollows. T r i g n o m e t r i c r e l a t i o n s a l l o w us to w r i t e 9^ = R^Cosfwt-X^) and 9 =R Cos(tot-x ); where 2 2 2 ' R = VA2+ B 2 = -£-V(C(l+K )coto2Cosv)2+(w2(«2-w2)Sinv)2 I i i J, i 2 i 2 38 COUPLED MOTIONS 39 R i « W i « = A2+ B 2 = - £ - V ( C ( K - l ) c o c o 2 S i n v ) 2 + ( t o 2 ( c o 2 - c o 2 ) C o s v ) 2 ( 5 4 X = t a n 1 {co 2 ( c o 2 - c o 2 ) t a n y / C ( 1+K^)co 2co} and X = tan {co (co -co ' ) / C ( K - l ) c o c o t a n y } 2 2 1 2 1 Suppose a s h i p Is s i t t i n g p a r a l l e l to the wave c r e s t s , i . e . , v=90°. The l o n g i t u d i n a l a x i s remains h o r i z o n t a l and thus no gy r o s c o p i c c o u p l i n g would be expected, C=0; the motion of the s h i p i s d e s c r i b e d by z a. co 9 = ° 1 S i n <ot , 9 =0 ( 5 5 ) 1 2 2 ' 2 ' CO —CO 1 T h i s i s the c l a s s i c a l r e s u l t g i v e n by equation (13). When *><w , B i s p o s i t i v e and the s h i p i s r o l l i n g away from the wave c r e s t and v i c e v e r s a when co <«. On the other hand, c o n s i d e r a s h i p maneuvering i n waves, c r o s s i n g the p o s i t i o n y=90°at uniform r a t e of t u r n i n g £ g = C . In t h i s case, the motion i s giv e n by o t co Z ( co 2 —<o2 ) o i Ccoco2 ( K — 1 ) €>=-?-± i S i n c o t , 9^=— 1 C o s c o t ( 5 6 ) For s m a l l C, i f co i s n e i t h e r c l o s e to co nor co , 9 and 9 ' 1 2' 1 2 are w e l l behaved. When co i s c l o s e t o e i t h e r co^or co z, T may become s m a l l ; t h i s w i l l r e s u l t i n l a r g e response amplitude. Moreover, T may va n i s h i f co c o i n c i d e s with one of the eige n v a l u e s g i v e n by f i g u r e 7, r e s u l t i n g i n the resonance of the o s c i l l a t i n g system. I t i s important t o note t h a t s i n c e y=90°, e 2 i s not the co n v e n t i o n a l p i t c h i n g but a g y r o s c o p i c coupled motion which i s q u i t e n o t i c a b l e a t resonance. 39 COUPLED MOTIONS 40 F i n a l l y , suppose a p l a t f o r m i s r i d i n g on a wave of ve r y long p e r i o d compared to the n a t u r a l p e r i o d of r o l l , i . e . , to « a^. Since co <co , equ a t i o n (54) shows t h a t the maximum response amplitudes are approximately g i v e n by R 4 = « o S i n v and R z = « o C o s y (57) In t h i s case , r o l l - y a w and pitch-yaw coupled responses, which are give n by A ^ n d A 2 r e s p e c t i v e l y , are n e g l i g i b l y s m a l l ; no t u r n i n g i s expected. We may note while p a s s i n g t h a t e q u a t i o n (57) g i v e s W. Froude's " e f f e c t i v e " wave s l o p e encountered by a s h i p r i d i n g a t an o r i e n t a t i o n of y degrees to the wave d i r e c t i o n . 4-1-3 Energy C o n s i d e r a t i o n f o r Pl a t f o r m s i n Waves The t o t a l k i n e t i c energy, E, of a r i g i d body i s g i v e n by the ex p r e s s i o n E = i{MV 2 + I n z+ i n z+ i a2} (58) 2 om 1 1 2 2 8 8 1 , 1 and I are the p r i n c i p a l i n e r t i a s of the body and V i s the 1 2 3 cm t r a n s l a t i o n a l v e l o c i t y of the ce n t r e of mass. For the sake of st u d y i n g r o t a t i o n , we may f i x the center of mass r e f e r e n c e p o i n t so t h a t V i s z e r o . S u b s t i t u t i n g f o r ft , ft and ft from equation cm 3 1 2 3 (27) i n t o (58), we o b t a i n the e x p r e s s i o n f o r the r o t a t i o n a l k i n e t i c energy: E = J{i (& -e e ) z+ i ( 9 + 9 9 ) z+ i 9zy (59) 2 1 1 8 2 2 2 8 1 8 8 As p o i n t e d out e a r l i e r coupled r o l l - y a w and pitch-yaw are maximum on wave c r e s t s and hollows; s i n c e the most un s t a b l e 40 COUPLED MOTIONS 41 p o s i t i o n f o r a f l o a t i n g o b j e c t to occupy i s on the wave c r e s t s , i t i s reasonable to assume t h a t coupled motions i n i t i a t e on wave c r e s t s , i . e . , when Sincot i s zero (see equation ( 4 4 ) ) . The k i n e t i c energy when the s h i p occupies t h i s p o s i t i o n i s I f M to N Z f(co Z-co Z) 2-2C Z(co Z-co Z) (K -1)+C*(K -I)2} g _ 1 I 1 I V 2 2 2 2 J (ri co -v 4. 2 1 I . I T 2 X 2 f M 2 W Y ((w Z-« 2) Z+2C Z(co Z-<o Z) <K +1)+C 4(K +1) Z) I 2-1 — ) : = +—C ( 6 0 ) i m a 2 a J T Equation (60) g i v e s the peak k i n e t i c energy and we would l i k e to f i n d out wether or not C=0 g i v e s the p l a t f o r m the minimum energy. I f the k i n e t i c energy i s a minimum f o r a value of C d i f f e r e n t from zero, then the p l a t f o r m would adopt t h i s t u r n i n g r a t e . The d e r r i v a t i v e of E with r e s p e c t to C z i s M « - | 3 r -(coZ-<oZ)(K -1)+C Z(K - l ) 2 E = ' 1 2 2 2 T ((<o Z-co 2) 2-2C 2(co Z-co Z) (K -1) +C*(K - 1 ) 2 ) . - (K, + l ) (K„-l)co2 2 — 1 * 1 1 2 m 3 T M co -.2 f (co 2-w 2)(K +1)+C 2(K +1) 2 " . i f — h  :— <61> ( n w •x* r iw —to i v i\. -i ? „ 2 , 2 2 w „ , , 4 „ 4 , „ , t 2> (( <o 2-co 2) 2+2C 2(<o 2-co 2) (K +1) +C*(K +1 )Z)-v I - (K +i)(K - i ) c o — — t i i i l + 4 1 8 Ta 1 2 2 Z , , „ , i „ Z , „ , .2 f M « ^ f (K -1) ( « ; - « ) ( K - 1 ) - C * ( K -1)' E = 1± j - j - j I —-§ + 4(K j+l)(K„-l)co 2 .a A S _a T 2-^2_ y-, 2 2 . 2 „ „Z , 2 2, f(K +1)(K -1 )w J 3 ("(to -to ) -2C (to -co ) (K -1) +C (K - I T ] 1 2 -* 2 2 2 2  J T 4 } f M w l i ! f ( K + l ) 2 , (coZ-co2) (K +1)+C Z(K +1) 2 2 ' I a J I T 2 1 2 T a f(K +1)(K -1 )to 2} Z3 f(co 2-to 2 ) 2+2C 2(co 2-co 2) (K +1) +C 4(K +1) Z) ^ 1 2 J 1 1 1 1 ' (62) T* 41 COUPLED MOTIONS 4 2 I t i s d i f f i c u l t to make g e n e r a l c o n c l u s i o n s from equations (60) and (61). N e v e r t h e l e s s , a value of C d i f f e r e n t from zero, may be found f o r which E =0; the s i g n of the second d e r r i v a t i v e , equation (62), should be checked t o f i n d out i f the t u r n i n g r a t e obtained would minimize the peak energy of the p l a t f o r m . We i l l u s t r a t e t h i s by examining the energy equations, equations (60)-(62), f o r the case i n which I /I =0.9, I /I =0.1; equation(30) g i v e s K =1.0 and K2=-0.8. At C=0 equ a t i o n (61) becomes M « -s2 (w 2-to 2) z(to Z+K to2) E = I <{ -4- V d-K_) — ^ - + r " i r : = ^ l ^ r l ( 1 " K a ) { M to -v 2 yw ~T~ } — t  ^ ( wZ-to Z) 2(w Z-K toZ) I 2 2 + ^ (63) 2 J T o Where T =(to 2 -w 2 ) (to 2 -w 2 ) . Since <*> i s g e n e r a l l y g r e a t e r than to and K z i s negative E i s always p o s i t i v e i f oKw^ or co>co2; any departure from C=0 would i n c r e a s e E (see equation (60)), hence C=0 i s an ab s o l u t e minimum. T h e r e f o r e , t u r n i n g i s not i n i t i a t e d when OKCO^ or w>w z. E i s always negative i f w i<to<w2, however. Hence, a value of C d i f f e r e n t from zero c o u l d be found f o r which E =0. We may denote t h i s value of c, which minimizes the peak energy, by C ; thus, the p l a t f o r m would r o l l and p i t c h while t u r n i n g a t the r a t e of C . —* F i g u r e 9 i s a graph of C a g a i n s t the o r i e n t a t i o n of the p l a t f o r m with r e s p e c t to the d i r e c t i o n of propagation of the waves, y. The numerical v a l u e s were obtained f o r 0^=0.5 and the 42 COUPLED MOTIONS 43 o — maximum wave angle <*0=6 • Consider the curve f o r which 0=0.8, C*decreases as y i n c r e a s e s . Suppose y=40°when the waves reached the p l a t f o r m , i t would r o l l and p i t c h while t u r n i n g a t the r a t e of C*=0.26. The t u r n i n g stops a t y=82°, i . e . , a f t e r the p l a t f o r m has almost turned broad s i d e t o the wave. When wsrw^or <*>=«2, equation (54) p r e d i c t s i n f i n i t e v alues f o r R^and R 2 hence an i n f i n i t e value f o r E, i f C=0. I t goes without s a y i n g t h a t , f o r these two cases, the t o t a l energy i s l e s s i f the p l a t f o r m turns a t any r a t e g r e a t e r than zero. At y>=50°, forexample, C*=0.36 i n both cases; the p l a t f o r m t u r n s to head sea,C >0, and to beam sea, C <0, when <»>=<«> and w=<->2 r e s p e c t i v e l y . I n f a c t C i s negative f o r a l l the f r e q u e n c i e s i n the neighbour hood of a^and p o s i t i v e f o r f r e q u e n c i e s c l o s e t o co^ . T h i s r e s u l t has been demonistrated by experiment. F i g u r e 10 shows the e f f e c t of wave slope on the r a t e of t u r n i n g , C . As would be expected, C i n c r e a s e s with c*o. Since the p e r i o d of the wave i s kept c o n s t a n t , i n c r e a s i n g « would be the same as i n c r e a s i n g the maximum wave he i g h t ; t h e r e f o r e , the l a r g e r the wave amplitude, the more pronounced the coupled motions would be. Before c l o s i n g t h i s s e c t i o n i t i s important to note t h a t as i n a l l o s c i l l a t i n g systems i n which damping i s ne g l e c t e d the value of C obtained from f i g u r e (9) and (10) i s hi g h . In p r a c t i c e 43 COUPLED MOTIONS 4 4 C may be about t en times s m a l l e r than what i s p r e d i c t e d here. N e v e r t h e l e s s , the tr e n d of the motion i s c o r r e c t l y p r e d i c t e d . 4-2 Damped Motion of a Balanced Body The p h y s i c a l nature of r e s i s t a n c e to motion of s h i p s was d i s c u s s e d i n s e c t i o n 2-2-3; the r e s i s t a n c e o f f e r e d by water c o n s i s t s of f r i c t i o n a l , form r e s i s t a n c e and the r e s i s t a n c e due to the g e n e r a t i o n of water waves. In g e n e r a l , the moment of r e s i s t a n c e due to angular motion of s h i p s i s giv e n by equation ( 1 5 ) . However, c o n s i s t e n t with our assumption of s m a l l amplitude motion the term p r o p o r t i o n a l to the square of the v e l o c i t y may be dropped, g i v i n g Mr= -L 9 g - L 9 g ( 6 4 ) 1 131 2 2 J2 Where L , and L 2 a r e the damping c o e f f i c i e n t s i n r o l l and p i t c h r e s p e c t i v e l y . R o t a t i o n i n h o r i z o n t a l plane generates l i t t l e or no waves and s i n c e we are c o n s i d e r i n g r e s i s t a n c e due to ge n e r a t i o n of waves only, damping i n g g d i r e c t i o n i s n e g l e c t e d . Force Free Motion: In s t i l l water, the moment a c t i n g on the p l a t f o r m c o n s i s t s of the h y d r o s t a t i c r e s t o r i n g couple g i v e n by equation ( 5 ) and the damping moment of equation ( 6 4 ) ; hence Mt= M^+M". S u b s t i t u t i n g f o r M^nd M zin equation ( 2 9 ) and r e a r r a n g i n g the r e s u l t i n g equation we o b t a i n 9 - (1+ K ")C© +2C w 9 + t o 2 © = 0 4 4 COUPLED MOTIONS 45 e + f l - K \ce +2C co © + co 2© = 0 2 ^ 2^  1 2 2 2 2 2 Note that s i n c e M gis zero, the t h i r d of equation (29) admits a constant t u r n i n g r a t e , C, as a s o l u t i o n . Equation (65) i s i n a standard form and as i n v i b r a t i o n s t u d i e s £.= L./2l.co. . F o l l o w i n g i i i i s i m i l a r method of s o l u t i o n used i n s e c t i o n 4-1, i t can be shown t h a t the c h a r a c t e r i s t i c equation of the d i f f e r e n t i a l equation (65) i s m*+m32(C co +C co )+mz(co2+coz+4C 17 co co -K)+m2(C <*> <oZ+r co co z)+co zco z=0 (66) 1 1 2 2 2 1 1^  2 1 2 1 1 2 ^ 2 2 1 1 2 N ' having s t a b l e s o l u t i o n s which are a p a i r of complex conjugate r o o t s of the form m =-17 -id m =-C _,, m =-C -id 1 1^ Id' 1 1^ Id' 2 2^ 2d' m =-C +id . ft. =-/i.-t72 ft., i s the damped n a t u r a l p e r i o d and ft. i s 2 2 2 d t d l i ' * * i g i v e n by equation (38). The most ge n e r a l s o l u t i o n s of equation (65) are 9 =E^(P(-c7 Q t ) (ACosO t+BSinCi t)+E^P(- t7 Q t ) (ECosfi t+FSinQ u t ) x 1 1 Id Id 2 2 2d 2d E«(P(-c n t ) r r -, V V 2 i 2 ^ 0 L { w i - 1 ) - e r ! ( 1 - 2 w 4 ) - ^ i ( w ! - ° i d ) (ACosO t+BSinO t ) 2 (K +1)C (C -O 2 ) LL 1 ± D * 1 i i i id J id id 1 1 Id + r 2 C 2 " (co -l)+ft r 2 ( l - 2 c o )+ft ( co 2 -f t 2 )1 (ASinft t - BCosft t ) \ |_ 1 Id 1 Id 1 1 Id 1 id J id Id J E«P(-t7 n t) f r - i + ± - 2 — - \ 2 C / » 2 (co - l ) - C 2 ( l - 2 c o )-C ( c o 2 - 0 2 ) (ECosft t+FSinft t ) (K + l ) C K Z - f t 2 ) LL 2 2 d 1 2 1 Z 2 2d J 2d 2d 1 2 2 d + [ 2 C ^ 2 d ( c o i - l ) + f t 2 / 2 ( l - 2 < o J + f t 2 d ( c o 2 - f t 2 d ) ] ( E S i n f t 2 d t - F C o s f t 2 d t ) } (67) Each s o l u t i o n i s a s u p e r p o s i t i o n of two motions; s i n c e they are independent, the motion a t the f i r s t frequency, O i d, and the motion a t the second frequency, ft ..can i n d i v i d u a l l y s a t i s f y (65) 2d 45 COUPLED MOTIONS 46 a l s o . At t h i s p o i n t , i t may be p o i n t e d out t h a t e q u a t i o n (67) reduces to (36) i f both Ctand C 2are zero. The e f f e c t of damping i s to reduce the amplitude e x p o n e n t i a l y with time. The damped n a t u r a l p e r i o d of o s c i l l a t i o n i s a l s o s m a l l e r than i t s undamped c o u n t e r p a r t ; s i n c e the damping r a t i o i s a s m a l l q u a n t i t y , the change i n the p e r i o d of o s c i l l a t i o n i s i n s i g n i f i c a n t , however. Motion i n Regular waves: The e f f e c t of waves on s h i p s i s Incorporated i n t o the d i f f e r e n t i a l equation of motion by i n t r o d u c i n g the e x p r e s s i o n f o r wave moments on the r i g h t - h a n d - s i d e of the e q u a t i o n of motion, equation (65) © - (1+ K )c© +2C w © + coz© = M°/I +(M / I ) S i n cot 1 ±J 2 i l l 1 1 1 1 1 1 i c o \ l o o ) 9 + f l - K }C© +2< co 9 + co2© = M ° / I +(M / I ) S i n cot 2 V. 2-* 1 2 2 2 2 2 2 2 2 2 The homogeneous s o l u t i o n i s g i v e n by equation (67); we seek a p a r t i c u l a r s o l u t i o n of the form © = ©°+ A Cos cot + B S i n w t (69) 1 1 1 i © = ©° + A Cos cot + B S i n cot 2 2 2 2 For e q u a t i o n (69) to s a t i s f y (68), the requirements a r e : A = I B = l A = 2 T co{ 2tT co M / I + 1 2 2 1 1 (K +1)CM / I } 1 2 2 + T (co 2-co 2)M / I 2 2 1 1 T T ( o Z - c o Z ) M / I -1 2 1 1 T {2C co M /1 2 2 2 1 1 + (K +1)CM / I } 1 2 2 T T co{2C co M / I + 1 1 1 2 2 (K - 1)CM / I } 2 1 1 + T (co Z-co Z)M / I 2 1 2 2 T T (co 2-co 2)M / I -1 1 2 2 T { 2C co M / I 2 1 1 2 2 + (K - 1)CM / I } 2 1 1 T B 2= ^ " ^ * ' ' = (70) 46 COUPLED MOTIONS 47 o o z o o z A AZ 9 =M /I to , S =M /I co and T= T +T where 1 1 1 1 2 2 Z 2 1 2 T = w*-w a (w s +w 2 +4C C co to - K ) +coatoa and T = 2{(( w +( « )w3-(C cow Z +( co co2 )co} 2 4 >1 1 ^2 2 v ^ l 1 2 ^ 2 2 l ' The g e n e r a l s o l u t i o n of (68) i s the s u p e r p o s i t i o n of the f r e e o s c i l l a t i o n , g i v e n by (67), and the f o r c e d response of equation (69). N e v e r t h e l e s s , we can s p e c i f y the i n i t i a l c o n d i t i o n s such t h a t the motion a t ^ d i s zero; moreover, damping, however s m a l l i t may be d i m i n i s h e s the s t i l l water motion i n g e o m e t r i c a l p r o g r e s s i o n . T h e r e f o r e , a f t e r a s h o r t time, the response c o n s i s t s of the motion conforming to the frequency of the wave, to, only. Since the wave induced constant t i l t may be co n s i d e r e d s e p a r a t e l y , we w i l l c o n c e n t r a t e on st u d y i n g the harmonic response. To t h i s end we w i l l r e w r i t e the f o r c e d response as O = R Cos(cot-\ ) i i i © z= R2Cos(<ot-X2) (71) R. =-/A2+ B 2 and tanX.=B./A.. We s u b s t i t u t e f o r M and M from x. x. x. x. x, x. 1 2 equation (46 ) i n t o equation (70) and then (71) and w r i t e R± and R 2 i n a more s u i t a b l e form f o r a n a l y s i s R f{ (co Z-co Z ) c o ZSiny} Z + {2C co co ZSiny+ (1+K )Ceo 2Cosy} Z w Z ~ l 1 / Z 1_ ] 2 1 2 2 1 1 2 [ I ao \{ T Q (1 +Kco2 / T o ) - 4C ± C 2 « ± <V> 2 >2+4{(C1coi+C2co2)co3"(C ±<*±<*l+C ) **> 2 J (72) R f{ ( c o 2 - c o z ) c o 2cosv} Z + {2C co to zCosy+(K -l ) C c o 2Sinv } 2 c o 2 V / z 2_ 1 1 2 1 1 2 2 1 I a o \{ T Q (1+Kco 2/T 0 ) -4< 4C2co4<o2co2} 2 + 4 { ( K 2<*>2 )co 3- (C1»1»*+C2<i»2«a)«)2J I f C1=C2=0, R ±and R 2 i n equation (72) reduce t o t h e i r undamped 47 COUPLED MOTIONS 48 co u n t e r p a r t o£ equation (54). Some g e n e r a l i n t e r p r e t a t i o n of (72) can be made by p l o t t i n g i t . In f i g u r e 11 and 12, the o r d i n a t e i s the r a t i o of maximum angular response to the maximum sl o p e of the wave, what i s c a l l e d the a m p l i f i c a t i o n f a c t o r , whereas the a b c i s s a i s the non dimensional group Kco2/T ; T = ( « z - w 2 ) (1-co 2), co = « /to , <o= to/co and ^ * o' o I ' 1 1 2 ' 2 as d e f i n e d e a r l i e r K=(l+K )(K -1)C 2 where C=C/<o . The reason f o r 1 2 2 2 p l o t t i n g the a b c i s s a as Kto /T , the frequency r a t i o , i s t h a t as i n v i b r a t i o n s t u d i e s , the o s c i l l a t i n g system resonates when t h i s q u a n t i t y i s u n i t y . We are i n t e r e s t e d i n i n v e s t i g a t i n g the e f f e c t of C and the damping on the m a g n i f i c a t i o n f a c t o r , hence i n f i g u r e s 11 and 12, the numerical v a l u e s are obtained f o r y=30°, K =1.0, K 2=-0.8, to=0.4, to=0.5. These f i g u r e s show t h a t f o r a s h i p maneuvering i n ob l i q u e r e g u l a r waves, there i s a combination of C and to which co u l d r e s u l t i n a severe response. For example, i n f i g u r e 11 the curve f o r C =0.05 and C =0.1 may be i n t e r p r e t e d as f o l l o w s : i f a t z s h i p having to =0.5, K =1.0 and K =-0.8 encounters a wave of 3 1 ' 1 2 frequency to=0.4, and i f a t the i n s t a n t the heading angle y=30°it i s making a c o n t r o l l e d t u r n such t h a t K<o2/To=-1, the maximum hee l would be twelve times the maximum wave s l o p e . That i s to say, i f « =5° the maximum hee l would be about 60°. I f the c o u p l i n g e f f e c t of t u r n i n g i s n e g l e c t e d , the r o l l and p i t c h responses are gi v e n by the Y - i n t e r c e p t of f i g u r e 11 and 12 r e s p e c t i v e l y ; thus, 48 COUPLED MOTIONS 49 f o r t h i s p a r t i c u l a r example, r e g a r d l e s s of the magnitude of C the c l a s s i c a l t h e o r y p r e d i c t s the response would be R 4 = 1 . 3 5 6 o » o . There are i n f i n i t e combinations of to and C which can r e s u l t i n resonance. The p r o x i m i t y of to to to and « determines the •* 1 2 magnitude of the t u r n i n g r a t e and the r e s u l t i n g response. When to i s c l o s e t o to^or t o 2 , T q d i m i n i s h e s and the s l i g h t e s t change of course may r e s u l t i n an extremely l a r g e angular response. On the other hand, when to i s f a r from both to and to . T i s b i g and the 1 2' O s h i p has to be s t e e r e d a t a f a s t e r r a t e f o r K t o z / T o to approach u n i t y . In other words, the f a r t h e r to i s removed from to and to , 1 2 ' the l e s s l i k e l y the system would re s o n a t e . We a l s o note t h a t resonance can take p l a c e o n l y i f K and T o are of opp o s i t e s i g n ; f i g u r e 7 shows t h a t f o r the frequency of the wave to match one of the e i g e n v a l u e s , co must be between co and co i f (K + 1 ) ( K - 1 ) i s 3 ' 1 2 1 2 p o s i t i v e whereas to must not f a l l between to and to i f (K + 1 ) ( K - 1 ) 1 2 1 2 i s n e g a t i v e . T h i s i s the reason f o r p l o t t i n g the a b c i s s a i n f i g u r e 1 1 and 12 as - K t o Z / T o . As i n other o s c i l l a t i n g systems the e f f e c t of damping i s to d i m i n i s h the response a t resonance by a c o n s i d e r a b l e amount. Changing t^from 0.02 to 0.05 and i ^ f r o m 0.07 to 0.1 decreases the response a t resonance by a f a c t o r of 2. When the wave frequency c o i n c i d e s with one of the value s g i v e n by f i g u r e 7 (see the curve f o r t o ^ O . 5 ) , K t o 2 / T o = - 1 and the amplitude f o r undamped case goes to i n f i n i t y . In p r a c t i c e the amplitude a t resonance i n c r e a s e s with 49 COUPLED MOTIONS 50 t ime but w i l l be f i n i t e ; when the energy absorbed by damping i s e x a c t l y e q u a l t o t h a t s u p p l i e d by wave per p e r i o d , maximum ampl i tude has been reached and no f u r t h e r i n c r e a s e i s p o s s i b l e . F i g u r e 12 shows the p i t c h r e s p o n s e ; I t Is s i m i l a r t o f i g u r e 11 and the f o r g o l n i n g d i s c u s s i o n a p p l i e s to i t a l s o . The peak p i t c h response i s g e n e r a l l y s m a l l e r than the c o r r e s p o n d i n g r o l l r e s p o n s e , however. T h i s may be a t t r i b u t e d to the f a c t t h a t l a r g e r damping r a t i o i s assumed i n p i t c h . The f i g u r e s a l s o show t h a t the two motions share a common n a t u r a l f requency d i f f e r e n t from both co^and w^as l ong as C*0; and as we have seen e a r l i e r , the peak k i n e t i c energy d i c t a t e s t h a t f o r a s h i p or unpowered p l a t f o r m s i t t i n g i n o b l i q u e r e g u l a r waves, C may not a lways be z e r o . Fu r thermore , a s h i p may be making a c o n t r o l l e d t u r n i n waves and hence t h e r e a re a l o t of t imes i n p r a c t i c e when C^O. So f a r we have e s t a b l i s h e d t h a t both r o l l and p i t c h resonate when K« 2 /T o =- l . T h i s i s i n c o n t r a s t to the c l a s s i c a l t h e o r y which assumes t h a t the two motions are independent ; i t i s assumed t h a t resonance i s observed o n l y i f co=w4 or w=u>2 r e g a r d l e s s of the speed a t which the s h i p i s be ing s t e e r e d , on the o ther hand, l e t us assume a s h i p i s t r a v e l l i n g a l o n g a d e f i n i t e c o u r s e . I f i t encounters a wave of f r equency co=o> , the r o l l amp l i tude would grow w i t h t ime to a h igh l e v e l r e g a r d l e s s of the o r i e n t a t i o n of the s h i p s i n c e C=0. In t h i s case i t would be b e t t e r to s t e e r the s h i p u n t i l i t s bow i s i n t o the wave. I f w matches « , however, the 50 COUPLED MOTIONS 5 1 opposite should be done; I.e., the s h i p should be s t e e r e d u n t i l i t i s p a r a l l e l to the wave c r e s t s . Once a g a i n r e f e r to e q u a t i o n (72); i f ^ a n d C 2are zero R± and R reduce to ( 5 4 ) 2 a j —-« R = -£-/(C(l+K )oxo 2Cosy) 2+(to 2(to 2-o 2)SinyO i R a= - ^ V ( C ( K a - l ) * ^ 2 s i n v ' ) 2 + ( w 2 ( w 2 - w 2 ) C o s v ' ) 2 ( 5 4 ) As i n d i c a t e d before the system resonates when K<->2/To = - l ; suppose to does not f a l l between co and co , i . e . , T i s p o s i t i v e . When Kco 2/T 1 2' ' O * O i s s l i g h t l y l e s s than - 1 , the a m p l i f i c a t i o n f a c t o r i s s u b s t a n c i a l l y g r e a t e r than u n i t y and hence equation ( 5 4 ) shows t h a t R 4and R 2are g r e a t e r than « and have the same s i g n . The r e s u l t i s heavy response i n both modes s i n c e the I n c l i n a t i o n of the s h i p to the t r u e normal i s the sum of the i n c l i n a t i o n of the wave normal to the true normal, « o , and the i n c l i n a t i o n of the s h i p to the wave normal. On the other hand, i f K<o 2/T o i s s l i g h t l y g r e a t e r than - 1 , R^and R zare g r e a t e r than oi^but have opposite s i g n ; moderate response i s expected i n the l a t t e r case. I f <o l i e s between to^and co z, T o i s negative and l a r g e r response amplitude i s 2 expected when K<o / T i s s l i g h t l y g r e a t e r than - 1 . F i g u r e 1 5 g i v e s R^nd R 2 f o r co=0.6. Since the product (K + 1 ) ( K - 1 ) i s negative and co i s between to and co no resonance i s 1 2 1 2 a n t i c i p a t e d . The f i g u r e shows t h a t the p i t c h response decreases as the r a t e of t u r n i n g , C, Increases whereas the r o l l response shows a s l i g h t i n c r e a s e a t the b e g i n i n g and then s t a r t s 5 1 COUPLED MOTIONS 52 d e c r e a s i n g . In other words, i f two i d e n i t c a l s h i p s were maneuvering i n o b l i q u e waves and the sea c o n d i t i o n i s such tha t K « z / T o i s p o s i t i v e , the one t u r n i n g a t a s lower r a t e would exper i ance a h ighe r response i n p i t c h . Two o p t i o n s e x i s t f o r an unpowered p l a t f o r m s i t t i n g i n o b l i q u e r e g u l a r waves: i t e i t h e r o s c i l l a t e s w i thout chang ing o r i e n t a t i o n w i th r e s p e c t to the d i r e c t i o n of the p ropaga t i on of waves or r o l l s and p i t c h e s wh i l e t u r n i n g a t a cons t an t r a t e , we would expect the p l a t f o r m to adopt the o p t i o n which would minimize the peak r o t a t i o n a l k i n e t i c ene rgy ; k i n e t i c energy i s a f u n c t i o n of speed and speed i s p r o p o r t i o n a l t o l i n e a r a m p l i t u d e . F i g u r e 16 shows tha t i f the p l a t f o r m adopts the f i r s t o p t i o n then the combina t ion of l^and Rzmay not g i ve the p l a t f o r m the l e a s t energy eventhough C=0. T h i s i s aga in the r e s u l t ob ta ined i n s e c t i o n 4 - 2 - 1 (see f i g u r e 9 ) . © a n d 9 can each be r e s o l v e d i n t o two separa te mo t i ons . 1 2 (K +l)wCM M / • 9 = 1 — - S i n ( « t - v ) + \— V(co Z -co 2 ) 2 +(2cof c.> ) z S i n ( « t - x ) I T I T 2 1 (K -1)<OCM 2 1 „ . , . . . 2 / , 2 2 v 2 . , „ „ .2 M / 2 / . 2 2.2. + v(co -CO ) +1 G = S m ( w t - r ) V(co co  (2c< co ) S m ( w t - r ) 2 _ -1 /2 1 _ m i / 2 1 2 2 3 I T I T 1 2 where T (l+Kco 2/T )-At; C , » « » Z T . O 0 1 2 1 2 1 i n*> \ tanr = = — (73) 1 2{(t; co +C co )co 3-(t: co coZ+t: co coZ) co} T 1 1 2 2 1 1 2 2 Z 1 2 tanr = {T 2C « co + T ( « Z - « 2 ) }/{T 2C « w-T (co 2 -w 2 )} and 2 1 2 2 2 2 2 2 2 1 2 52 COUPLED MOTIONS 53 tan?' = IT 2 ( w « + T ( w Z - w Z)}/{T 2C w co-T {</-</)} 3 1 1 1 2 1 2 1 1 1 1 The f i r s t e x p r e s s i o n on the r i g h t - h a n d - s i d e of the f i r s t of equation (73) i s the r o l l - y a w coupled response, whereas the second term g i v e s the r o l l - h e a v e coupled response which reduces to c l a s s i c a l r o l l g i v e n by equation (18) i f C=0. F i g u r e 13 and 14 giv e j-^and rz as a f u n c t i o n of C f o r v a r i o u s values of the damping r a t i o s . When C±=C2=0 the r o l l - y a w coupled response i s 90°out of phase with the wave, i . e . , the maximum takes p l a c e on wave c r e s t s and hollows, which i s the most unstable p o s i t i o n f o r f l o a t i n g o b j e c t s t o occupy. As C^and C 2 i n c r e a s e the p o s i t i o n a t which the maximum coupled response occur moves away from wave c r e s t s and hollows. T h e r e f o r e i n a d d i t i o n to r e d u c i n g the response amplitude at resonance, damping i n c r e a s e s d i r e c t i o n a l s t a b l i t y a l s o . The coupled response i s almost i n phase with the wave a t resonance. The r o l l - h e a v e response, the second term i n the e x p r e s s i o n f o r 9±, i s i n phase with the wave i f C4=C2=0; the amplitude of t h i s component of e a t t a i n s i t s maximum on the p o s i t i o n of maximum wave s l o p e , and i s zero on wave c r e s t s and hollows. Damping s h i f t s the p o s i t i o n of maximum r o l l response towards the wave c r e s t s and hollows. For example, when Kw Z/T o=-0.3, f i g u r e 15 shows t h a t y =0 i f K =C =0, and r =20° i f K =0.05 and K =0.1. The 2 t 2 ' 2 1 2 r o l l - h e a v e amplitude i s q u a r t e r of a c y c l e out of phase with the wave a t resonance. 53 COUPLED MOTIONS 54 4-3 Motion of Unbalanced Body F i g u r e 4 shows two s e t s of body f i x e d axes; the p r i n c i p a l i n e r t i a l axes are denoted by X~ and the p r i n c i p a l water plane axes by Xt. So f a r the d i s c u s s i o n has been r e s t r i c t e d t o the case when X~ i s p a r a l l e l t o x t . In g e n e r a l , X~ do not c o i n c i d e with Xt and the s h i p i s s a i d to be d y n a m i c a l l y unbalanced. The angular v e l o c i t y g i v e n by equation (27) and the h y d r o s t a t i c r e s t o r i n g couples of equation (29) are d e s c r i b e d i n terms of g L, the base v e c t o r f o r X . In order to use them i n E u l e r ' s e q u a t i o n s , equation (28), which are w r i t t e n f o r p r i n c i p a l axes of i n e r t i a , we need t o t r a n s f o r m both d and M H from gjto g~, the base v e c t o r f o r x~ . From f i g u r e 4 i t can be seen t h a t g,is r e l a t e d to g~ by Where the t r a n s f o r m a t i o n matrix 1 £• = Cos(X^GX l). Using 1*^ we may r e f e r the angular v e l o c i t y and the h y d r o s t a t i c r e s t o r i n g couples to p r i n c i p a l i n e r t i a l axes a =((& - e e )i ~ + (e + e e ) i ~ + e i - O g -1 2 3 11 2 3 1 21 3 a i J ' l + ({& - 9 9 )1 - + (9 + 9 9 ) 1 ^ + 9 1 ~}q~ 1 2 3 12 2 3 1 22 3 az-*^2 + ((9 - 9 9 ) l - + ( 9 + 9 9 ) l ~ + 9 1 ~^g~ 1 2 3 13 2 3 1 23 3 33J 3 3 MH= -(coZ£ I 1 - + w2g» I 1 -)g--(w 2© I 1 - + « 2£ I 1 - ) g ~ 1 1 1 11 2 2 2 21 J l 1 1 1 12 2 2 2 22 2 - ( t o 2 © I 1 - + co 2© I 1 ~ ) g -1 1 1 13 2 2 2 23 3 3 (75) (76) 54 COUPLED MOTIONS 55 S and Sz are 3 t i l l measured with r e s p e c t to the p r i n c i p a l water plane axes and t h e r e f o r e r e p r e s e n t t r u e r o l l and p i t c h r e s p e c t i v e l y . The s o l u t i o n of E u l e r ' s equations f o r the g e n e r a l case would be q u i t e complicated s i n c e M has components i n a l l the three d i r e c t i o n s . We w i l l i n v e s t i g a t e a s p e c i a l case i n which g~ remains v e r t i c a l . Denoting the angle between X^nd X^ by ft i t f o l l o w s t h a t 1 A = Cos/?,l A = Sin/?,1 - =-Sin/?,l A =Cos(3,1 A = 1 - = 0 and 1 A =1 (77) 11 12 ' 21 ' 2 2 1 ' 13 32 33 We are i n v e s t i g a t i n g motions i n which r o l and p i t c h are s m a l l but yaw may be l a r g e , hence we w i l l drop terms c o n t a i n i n g higher powers of © ±and Q . The t h i r d of equation (28), i s s a t i s f i e d i f the r a t e of t u r n i n g i s a co n s t a n t , ^3=C; with t h i s p r o v i s o the f i r s t two of E u l e r i a n equations give M / C o s ^ - ( « e I -»f* i tan©+L,© -L © tan©)= e I -e j tan©-(I - I J C © , ± 1 x x z z z 1 x z z x i z a . Z 3 Z - ( I -I )C© tantf-I C© -I C© tantf 2 3 1 1 2 1 1 M /Cosft-(<»ze I + * o 2 © I tantf+L © +L & tan©)= © I +© I tan©-(I -I )C© 2 2 2 2 1 1 1 2 2 1 1 2 2 1 2 3 1 1 + (I -I )C© tan^+I C© -I C© tanft (78) 3 1 2 2 1 2 2 M,and M a r e e x t e r n a l hydrodynamic moments and L and L stand f o r X Z X £ the damping moment c o e f f i c i e n t s i n X ±and X z d i r e c t i o n s r e s p e c t i v e l y . Now assume simple r o l l i n g occurs i n s t i l l water, t h a t i s e q u i v a l e n t to s a y i n g © 2 = © g = 0 . Since M^and M zare zero equation(78) shows t h a t 55 COUPLED MOTIONS 56 - (co 20 I +L © )= 6' I and - ( c o 2 © I +L © ) tan/9 = © I tan/9 * * * * * * * 4 4 * 4 4 4 2 T h i s c o n d i t i o n s can be s a t i s f i e d i f e i t h e r (3=0 or I =1 and both 1 2 cases correspond to d y n a m i c a l l y balanced s h i p . Hence, pure r o l l i n g or p i t c h i n g of an unbalanced body Is i m p o s s i b l e . The homogeneous s o l u t i o n of equation (78) can be obtained by s i m i l a r method used i n s e c t i o n 4-1-1. S e t t i n g the e x t e r n a l moments to zero i n equation ( 7 8 ) , g i v e s the c h a r a c t e r i s t i c equation m*(l+tan 2/3) + m3{ 2C1<'>1( 1+ tan 2/VK 3) + 2C2<*>2( 1+K gtan 2/?) } +m 2{co2(l+ tan 2/9/K ) + co 2(l+K tan 2/?)- K(l+tan 2/?) + 4C C co co (l+tan 2/?)} 1 3 2 3 1 2 1 2 +m{2C c o c o 2 ( l + tan 2/?) + 2C co co 2( l+tan 2/?) } + co 2co 2( l+tan 2/?) = 0 (79) 2 2 1 4 1 2 1 2 Kg= I /I . It ft i s s e t to zero e q u a t i o n (79) reduces t o i t s c o u n t e r p a r t f o r balanced body, equation (66). When ft i s a sm a l l angle, tan ft i s sm a l l compared t o u n i t y ; then, the e f f e c t of the imbalance on the eigenvalue seems to be I n s i g n i f i c a n t . To I n v e s t i g a t e the e f f e c t of ft f a r t h e r , we l e t =0 i n equation ( 7 9). O b t a i n i n g m*(l+tan 2/?) +m 2{co2(l+ tan 2/?/K ) + <oz(l+K tan 2/?)- K(l+tan 2/?)} 1 9 2 3 +co 2co 2(l+tan 2/?) = 0 (80) Equation (80) corresponds t o equation (34) f o r balanced body; the l a t t e r e q uation shows t h a t i f c=0, the frequency of o s c i l l a t i o n i s e i t h e r to or w . For i n s t a n c e , i f a balanced body i s t i l t e d about the l o n g i t u d i n a l a x i s and r e l e a s e d , i t would r o l l 56 COUPLED MOTIONS 57 with frequency to^and i f the d i s t u r b a n c e i s about the t r a n s v e r s e a x i s , i t would p i t c h with frequency <*z. I f we l e t C=0 i n equation (80) , however, we o b t a i n two f r e q u e n c i e s d i f f e r e n t from both co^and <A>z. T h e r e f o r e , as we saw e a r l i e r r o l l i n g and p i t c h i n g motions of an unbalanced body are interdependent and simple r o l l i n g or simple p i t c h i n g i s i m p o s s i b l e . The e x t e r n a l moments are giv e n by M JCosft= M°-M°tanf?+(M -M t a n ^ ) S i n <ot 1 1 2 1 2 M2/Costf= M 2+M°tan^+(M 2+M 4tan©)Sin cot (81) M°and M°are the constant t i l t i n g moments and M and M are giv e n by 1 2 3 1 2 ^ •* equation (46). S u b s t i t u t i n g (81) i n t o (78) , we o b t a i n the f o l l o w i n g l i n e a r d i f f e r e n t i a l equations i n © ±and © 2 0 - e tantf + 9 {2C <*> -(K + l)Ctan/?} -9 {2C « K tan/? + (K +1)C} 1 2 1 1 1 1 2 1 2 3 1 + to2© + co2 9 K tan/? = {M°-M°tan/?+ (M -M tan/?) S i n cot}/I 1 1 2 2 3 1 2 1 2 1 0+e tan/? + 0 (2C » -(1-K )Ctan/?} + 9 {2C <*> tan/?/K - (K -1)C} 2 1 2 2 2 2 i l l . 3 2 + <o2© + © coztan^/K = {M°+M°tan©+(M +Mtan©)Sin *»t}/I, (82) 2 2 1 1 3 2 1 2 1 2 A p a r t i c u l a r s o l u t i o n of (82) can be of the form 0 = ©°+ A Cos cot + B S i n wt (83) i i i I 0 = 0° + A Cos cot + B S i n cot 2 2 2 2 For equation (82) to be i d e n t i c a l l y s a t i s f i e d , the con s t a n t s 9°, A , A , B , B and 9° must be g i v e n by l ' 2' Z' 1 2 A = (T'{(CO 2-CO 2)P + (co2K -co 2)P tan/?} 1 * - 1 2 1 2 3 2 -T'co { \2K <o -(1-K )Ctan/?]P + [ 2f » K tan/5+ (1+K ) C ] P } ) / T ' 2 2 2 2 1 2 2 3 1 Z 57 COUPLED MOTIONS 58 B±= (T 2{ (0^-^)91+ (o>2K3-co')Pztan/3} -T^co { [2C 2 c o z-(l - K z)Ctan/3]P i+ [ 2C zto zK 3tan/?+(1+K^ ) C ] P^})/T' A = fa'{ (co Z-co Z)P - ( (co Z /K ) - to Z)P tan/?} 2 ^ - 1 1 2 1 3 1 -T'CO { [2C w-(l+K )Ctan/?]P - [ 2C » tan/3/K - (K -1)C]P } V T ' 2 11 1 2 11 3 2 i. J Bz= ( T 2 { ( " 2 - " 2 ) P 2 - ((«*/K g)-*» z)P 1tan/9) +T'CO { [2C «-(l+ K )Ctan/?]P - [2C w tan^/K - (K -1)C]P } V T 1 1 1 1 2 11 3 2 l - ' ©°= M°/« ZI and <?°=M°/<o 2l (84) 1 1 1 1 2 2 2 2 The other c o n s t a n t s are T'= «*( l+tan 2/?) + co2co2( l+tan 2/?) + co 2{co 2(l+ t a n 2 / ? / K ) 1 1 2 1 3 + co 2(l+K tan 2/?)- K(l+tan 2/?) + 4 ( ( « « (l+tan 2/?)} 2 3 1 2 1 2 T ' = co3{2t; « (1+ t a n 2 / ? / K ) + 2C <o (l+ K tan 2/?)} 2 1 1 3 2 2 3 + co{2C coa>2(l+ tan 2/?) + 2C co to 2(l+tan 2/?) } 2 2 1 1 1 2 T'= T' 2+T' 2, P =(1/1 )(M-Mtan/?), P = (1/1 ) (M +M tan/?) I l l 1 1 2 2 2 2 1 Dropping ^ a n d © z f o r the time being, we can w r i t e the harmonic p a r t of the s o l u t i o n i n a form s u i t a b l e f o r a n a l y s i s 9 = R Cos(cot-0 ) and 9 = R Cos(«t-t£ ) where 1 1 1 2 2 2 O i=/f ( co 2 - co 2 ) (w zSiny-« 2tan/?CosyK ) + (<o2Sinytan/?/K +w 2Cosy) (« 2K -co2)tan/?V I I 2 1 2 3 1 3 2 2 3 O V. + w2f(w2Sinv-<«>2tan/?CosvK ) (2C c o - ( l - K )Ctan/?) v 1 2 3 2 2 2 -v 1 + (co 2sinytan/?/K +co2Cosy) (2C co tan/?/K +(1+K )Ctan/?)") 2 }• 1 3 2 2 2 3 1 -•'j .1/2 58 COUPLED MOTIONS 59 R t 2 / 2 2 2 2 2 2 2 2 —=i ( ( « - < * > ) (co cosv+w tan/3siny/K )-(<*> Siny-<o K Cosytanf?) (co /K -CO )tan/?Y o + co Z ((co Zcosy+« ztan^Siny/K A) (2C1<^1-(l+Ki)Ctan/9) ' 1/2 R. = R.T and tan<A = A. /B. R / « i s plotted in figure 16 (note that R /<* has a similar 1 o z o graph). Resonance takes place when the frequency of the wave, w , coincides with one of the eigenvalues given by equation (80), i. e . , when Kco Z(l+tan z/?)/(T +f tan 2 f t )=-1' , T =(COZ/K -CO Z)(CO ZK - c o z ) . o o o i a z a If ft=0f the abcissa of figure 16 reduces to i t s counterpart for balanced body as given by figure 1 1 . The response increases with the imbalance, ft. If two id e n t i c a l ships are subjected to the same sea conditions, the one with larger imbalance would experience bigger response amplitude. Nevertheless, as figure 16 shows the e f f e c t of ft i s n e g l i g i b l y small. 59 CONCLUSION 60 5. CONCLUSION The e f f o r t , through out t h i s study, has been to i n v e s t i g a t e coupled motions which were h i t h e r t o overlooked. Though the mathematics was kept simple, the p r e d i c t i o n of the trends of the coupled motions were reasonable. The problem was formulated with the a i d of E u l e r ' s dynamical e q u a t i o n s . Instead of the usual E u l e r i a n a n g l e s , however, a d i f f e r e n t s e t of r o t a t i o n s were Introduced and under the assumption of s m a l l amplitude o s c i l l a t i o n s the governing equations reduced to l i n e a r o r d i n a r y d i f f e r e n t i a l equations which were s o l v e d e x a c t l y . A t u r n i n g r a t e , C, was i n t r o d u c e d and r e t a i n e d throughout the i n v e s t i g a t i o n ; the e x i s t e n c e of C c o u l d be a t t r i b u t e d t o , (1) the g y r o s c o p i c c o u p l i n g between r o l l and p i t c h which may induce t u r n i n g , or (2) i t may be s p e c i f i e d as i n the case of a s h i p maneuvering i n waves. In a d d i t i o n to the aforementioned r e s t r i c t i o n t h a t the amplitudes of the motions are s m a l l , Froude's p o s t u l a t e f o r wave moments was used; the l a t t e r assumption e n t a i l s t h a t the dimensions of the s h i p be s m a l l compared to the l e n g t h of the wave. Moreover, i t was assumed t h a t the pressure d i s t r i b u t i o n i n the wave i s not a f f e c t e d by the presence of the s h i p . Notwithstanding these s i m p l i f y i n g assumptions, the f o l l o w i n g c o n c l u s i o n s can be drawn: 60 CONCLUSION 61 (1) When r o l l and p i t c h are coupled to yaw, they have a common p e r i o d of o s c i l l a t i o n i n s t i l l water; the p e r i o d depends on the c h a r a c t e r i s t i c s of the s h i p and the r a t e of t u r n i n g (see equation (38)), i n c o n t r a d i c t i o n t o the b e l i e f i n c l a s s i c a l theory of s h i p motion t h a t r o l l and p i t c h have two d i s t i n c t n a t u r a l f r e q u e n c i e s of o s c i l l a t i o n governed o n l y by the displacement and the m e t a c e n t r i c height of the s t r u c t u r e . (2) I f the s t i l l water motions are taken as p e r i o d i c , the r a t e of t u r n i n g and the frequency of o s c i l l a t i o n can be r e l a t e d to the i n i t i a l c o n d i t i o n s (see equations 40-43). (3) Forced o s c i l l a t i o n s c o n s i s t of two components each. The f i r s t i s coupled to the r a t e of t u r n i n g , C, and may be termed r o l l - y a w , about l o n g i t u d i n a l a x i s , and pitch-yaw, about t r a n s v e r s e a x i s ; these components vanishes i f C i s zero. The second components, which are named r o l l - h e a v e and pitch-heave, reduce to the s o l u t i o n of F r o u d e - K r i l o f f problem i f the r a t e of t u r n i n g i s zero (see equation 49-50, and 73). (4) Undamped r o l l - y a w and pitch-yaw coupled responses are quar t e r of a c y c l e out of phase with the wave, whereas r o l l - h e a v e and pitch-heave are i n phase with the wave. Damping s h i f t s the p o s i t i o n of maximum r o l l - y a w and pitch-yaw coupled responses away from wave c r e s t s and hollows; s i n c e wave c r e s t s are the most unstable p o s i t i o n f o r s h i p s to occupy, i t co u l d be argued t h a t damping i n c r e a s e s d i r e c t i o n a l s t a b i l i t y . A l s o , damping i n t r o d u c e s phase l a g between r o l l - h e a v e , pitch-heave and the wave; these two motions are i n phase with the wave when undamped (see f i g u r e 13 61 CONCLUSION 62 and 14). (5) From energy c o n s i d e r a t i o n s , i t was shown t h a t there may be a r a t e of t u r n i n g , d i f f e r e n t from zero, which minimizes the peak r o t a t i o n a l k i n e t i c energy of an unpowered p l a t f o r m s i t t i n g i n o b l i q u e waves. Numerical v a l u e s f o r C were obtained f o r a p a r t i c u l a r p l a t f o r m with a f a i r l y square deck s t r u c t u r e (see f i g u r e s 9 & 10). For a s p e c i f i e d o r i e n t a t i o n , the magnitude of the r a t e of t u r n i n g i s found to i n c r e a s e as the wave frequency approaches w^and « 2 (where w^and « 2 are the f r e q u e n c i e s of c l a s s i c a l r o l l and p i t c h r e s p e c t i v e l y ) . Moreover, the p l a t f o r m always t u r n s away from the p o s i t i o n of resonance, thus when co i s c l o s e to co^ i t turns to head sea and when co i s i n the neighbourhood of to 2 i t turns to beam sea. (6) In both r o l l and p i t c h modes, resonance takes p l a c e when Kco Z/T Q= -1 or T=0. For a g i v e n s h i p , there are i n f i n i t e v alues of co and C which can make T v a n i s h (see f i g u r e ( 7 ) ) . In c l a s s i c a l t h e o r y resonance i s p r e d i c t e d i f a^co^or co=coz only; and the c h a r a c t e r i s t i c l a r g e response a t resonance i s f e l t i n r o l l and p i t c h independently. While on the other hand, the present study shows t h a t as long as C*0, the n a t u r a l f r e q u e n c i e s of r o l l and p i t c h are the same and when resonance takes p l a c e i t i s f e l t i n both planes (see f i g u r e 12, 13 and 16). The system resonates whenever T=0, the case co=co or w=co i s j u s t a p a r t i c u l a r one. 1 2 (7) Simple r o l l i n g and p i t c h i n g of unbalanced s h i p s or p l a t f o r m s i s found to be Impossible. Moreover, the angular response amplitude i n c r e a s e s with an i n c r e a s e i n imbalance; the 62 CONCLUSION 63 e f f e c t of the imbalance i s i n s i g n i f f l e a n t , however. I t i s b e l i e v e d t h a t g y r o s c o p i c coupled motions of marine v e h i c l e s are n e g l i g i b l y s m a l l to be of any concern. T h i s b e l i e f has not been based on a thorough examination of the problem. Suyehiro was the f i r s t person to address the problem and h i s a n a l y s i s was a l i t t l e f a u l t y . Vaughan has done some work on t h i s problem and t h a t i s where we stand as long as r e s e a r c h i s concerned. Indeed the g y r o s c o p i c e f f e c t i s s m a l l ; n e v e r t h e l e s s , i t should not be overlooked when the frequency of the wave i s c l o s e to e i t h e r w or w . I t does account f o r the p r o g r e s s i v e 1 2 " s t e p p i n g round" of s h i p s i n waves, and a l s o to some new coupled motions. As mentioned e a r l i e r there are two sources of t u r n i n g v i s a v i s , induced t u r n i n g due to g y r o s c o p i c c o u p l i n g between r o l l and p i t c h and c o n t r o l l e d t u r n i n g as i n the case of a s h i p being s t e e r e d onto a new course. Induced t u r n i n g i s of no concern f o r s h i p s c r u i s i n g i n waves because i t i s under c o n t r o l of the A u t o p i l o t . For unpowered p l a t f o r m s s i t t i n g i n o b l i q u e waves, however, the induced t u r n i n g can c r e a t e a d d i t i o n a l loads on the anchor l i n e s which i n t u r n can impart t o r s i o n a l s t r e s s e s In the main deck s t r u c t u r e , which should be co n s i d e r e d d u r i n g the design s t a g e . 63 NOTES 64 NOTES I W. Froude, "On the R o l l i n g of Sh i p s , " Trans. INA IL (1861): 208. Z A. K r i l o f f , "A General Theory of the O s c i l l a t i o n of Ships on waves," Trans.INA XL (1898): 144. 3 K Suyehiro, "Yawing of Ships Caused by O s c i l l a t i o n Amongst Waves," Trans.INA 62 (1920): 93-101. * Suyehiro, 97. Theodore R. Goodman, "Gyroscopic Yawing of Ships i n Random Seas," J o u r n a l of Ship Research 9 (1965): 179-182, tf C.H.Kim,Frank S.Chou and David T i e n , "Motion and Hydrodynamic Loads of a Ship Advancing i n Oblique Waves," Trans. SNAME 88 (1980): 225-256. 7 H.Vaughan, "Three Dimensional Motion of Ships and Pla t f o r m s In Waves," Trans. RINA 127 (1985): 247-256. 8 Vaughan, 254-5, see a l s o Suyehiro, 93-101. P Suyehiro, 94. 1 0 T h i s chapter i s taken from s e v e r a l standard t e x t s on naval a r c h i t e c t u r e I I T h i s system of angles i s developed by Vaughan. 12 J.Martin,C.Kuo, " C a l c u l a t i o n of The Steady T i l t of semi-submersibles In Regular waves," Trans. RINA 121 (1979): 87-101. 1 3 C l a s s i c a l r o l and p i t c h r e f e r s to the s o l u t i o n of F r o u d e - K r y l o f f problem as d i s c u s s e d i n chapter 2. 1 4 Froude, 194-5 . Vaughan, 253-5 BIBLIOGRAPHY 65 BIBLIOGRAPHY Eda /Haruzo. " D i r e c t i o n a l S t a b i l i t y and C o n t r o l of Ships i n Waves." J o u r n a l of Ship Research 16 (1972) :205-219 . Froude,W. "On the R o l l i n g of Sh ips . ".Trans.INA IL 1861:180-229. Goodman,Theodore R. "Gyroscopic Yawing of Ships i n Random Seas." J o u r n a l of Ship Research 9 (1965) :179-182. K r l l o f f , A . "A General Theory of the O s c i l l a t i o n s of A Ship on Waves.", Trans INA XL (1898):135-196. Kim,C.H.,Chou,Frank S. and Tie n , D a v i d . "Motion and Hydrodynamic Loads of a Ship Advancing i n Oblique Waves." Trans.SNAME 88 (1980) :225-256 . M a r t i n , J . and Kuo , c . " c a l c u l a t i o n f o r the Steady T i l t of Semi-submersibles i n r e g u l a r waves." Trans.RINA 121 (1979): 87-101. o g i l v i e , T . F . and Tuck,E.O. A R a t i o n a l S t r i p Theory of Ship Motion. Rept. 0 1 3 , 1 9 6 9 : U n i v e r s i t y of Michigan.Ann Arbor,Michigan. Patel,M.H. "On the Wave Induced Motion of Semlsubmerslbles." Trans.RINA 125 (1983) :221-228 . Rawson,K.J. and Tuper,E.C. B a s i c Ship Theory. 3rd ed. 2 V o l s . , London:Longman,1983. Suyehiro,K. "Yawing of Ships Caused by O s c i l l a t i o n Amongst Waves.", Trans.INA 62 (1920)-.93-101. Torby,B.J. Advanced Dynamics For Eng i n e e r s . 1s t ed.,New York:HRW, 1984 . Vaughan,H. "Three Dimensional Motion of Ships and P l a t f o r m s i n Waves." Trans.RINA 127 (1985) :247-256 .256 . Wylie,C.R. D i f f e r e n t i a l E q u a t i o n s . New York:McGraw-Hill,1979. 65 JS u Surge F i g u r e 1 Body axes and the degrees-o£-£reedo» of a s h i p . F i g u r e 2 A t i l t e d f l o a t i n g body V~±4 e F i g u r e 3 Couples a c t i n g on t i l t e d f l o a t i n g o b j e c t . 66 Figure 4 Coordinates to describe ship motion i n vaves;earth fi x e d axes X l # , p r i n c i p a l i n e r t i a l axes and p r i n c i p a l vater plane axes X . X .X„,and X: are v e r t i c a l . V 9 9 9 Figure 6 Angles suitable for describing ship motions with large yav;rotatlng t r i a d g ^ a n d g 3 I n i t i a l l y coincident with fixed t r i a d G ,G and G 68 Trajectories T=0. r* i i 1 1 1 he— i 1 1 1 1 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 K/co, Figure 7 Trajectories T=0. w 2 / w 2 versus c o 2 / w 2 and K/co ; 2 1 2 l' 2 K=(K 4+1)(K 2-1)C 69 Frequency of Oscillation Versus Initial Condition I I I I M l l | I I I I I I I I I I I I I I I I I I I I l l l l 0.001 0.01 0.1 1 10 100 1000 Log R Figure 8 Natural frequency of gyroscopic coupled motion I n i t i a l conditions.R=(e°/e°) z(1-K )/(l+K ) 1 2 2 1 70 Wave Frequency and Platform Orientation Versus The Rate of Turning that Minimizes The peak energy. in 6 ip Deg. Figure 9 Wave frequency and platform orientation versus turning rate that minimizes the peak energy. 71 Wave Slope and Platform Orientation Versus The Rate of Turning that Minimizes the Peak Energy. 3 \ O Tp Deg. Figure 10 Wave slope and platform orientation versus the turning rate that minimizes the peak energy. 72 Roll Response in Oblique Regular Waves Figure 11 R o l l amplitude to wave amplitude r a t i o versus damping r a t i o and K« z/T ; T = ( w 2 - < * > z ) ( c o 2 - < * > 2 ) . O O 1 2 7 3 Figure 12 Pitch amplitude to wave amplitude r a t i o versus damping r a t i o and K» z/T . 74 Coupled Response Phase Shift Versus Damping Ratio o 1.5 Figure 13 Roll-yaw Phase s h i f t versus damping r a t i o and K c» 2 75 Roll Phase Shift Versus Damping Ratio Figure 14 Roll-heave phase s h i f t versus damping r a t i o and K«' 76 Resultant Roll and Pitch Response When K C J 2 / T 0 is Positive t 1 1 p 1 1-0 0.5 1 1.5 2 2.5 KcoJ/T0 Figure 15 Amplitude r a t i o when K and T have the same s 77 Roll Response of Unbalanced Body in Oblique Regular Waves 0.5 1 1.5 2 K,=l K, =-0.8 uJvt=0 5 u/u, =0 4 f =30' {,=002 y u « « if \% \* 7 2.5 Figure 16 The e f f e c t of imbalance on r o l l response. T = (co -co ) (co -co ) + (co /K -co ) (co K -co )tan ft 1 2 1 8 2 8 78 

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