THE MOTION OF A SELF-EXCITED RIGID BODY by RICHARD WAY MAH LEE M.A.Sc, University of B r i t i s h Columbia, 1958 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Matheraat i cs We accept t h i s t h e s i s as conforming to required standard THE UNIVERSITY OF BRITISH COLUMBIA April, 1964 the In presenting the this thesis i n partial fulfilment of r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h Columbia, I agree that the L i b r a r y a v a i l a b l e f o r r e f e r e n c e and s t u d y . mission f o r extensive s h a l l make i t f r e e l y I f u r t h e r agree that per- copying of t h i s thesis f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r b y his representatives. I t i s understood that cation o ft h i s thesis f o r f i n a n c i a l gain w i t h o u t my w r i t t e n Department o f permission® Mathematics The U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r 8 , Canada copying or p u b l i - s h a l l n o t be a l l o w e d The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of RICHARD B.Sc, M.A.Sc., LEE The U n i v e r s i t y The U n i v e r s i t y o f Manitoba, 1956 o f B r i t i s h Columbia, 1958 MONDAY, MAY 4, 1964, a t 10:00 A.M. IN ROOM 125, ARTS BUILDING COMMITTEE IN CHARGE Chairman: F.H. Soward A.H. C a y f o r d E. Leimanis E . Macskasy External Institute J.F. D.C. Murdoch G. P a r k i n s o n Scott-Thomas Examiner; J.B. Diaz for Fluid and A p p l i e d University Dynamics Mathematics, o f Maryland THE MOTION OF A SELF-EXCITED RIGID BODY ABSTRACT T h i s t h e s i s d i s c u s s e s the motion of a r i g i d body about a f i x e d p o i n t s u b j e c t t o a torque a r i s i n g from i n t e r n a l r e a c t i o n . Such a body i s c a l l e d s e l f " e x c i t e d , and i t s motion i s governed by E u l e r s dynamical equations o ? F i r s t , i n S e c t i o n 2 we c o n s i d e r the case of a torque v e c t o r which i s f i x e d in. d i r e c t i o n along the l a r g e s t or s m a l l e s t p r i n c i p a l a x i s of the body and has a component in. the chosen a x i s equal, t o a c o n s t a n t p l u s a p e r t u r b a t i o n , term t h a t i s p r o p o r t i o n a l t o ;the square of the modulus of the s p i n v e c t o r G O ( p , q r ) „ I t i s shown t h a t E u l e r ' s equations can be i n t e g r a t e d i n terms of a v a r i a b l e <f> , i n t r o d u c e d by means of a differential relation. F u r t h e r quadrature and i n v e r sion y i e l d p q and r as f u n c t i o n s of the time to Using the method of phase-piane a n a l y s i s , we show t h a t the s p i n v e c t o r can perform a v a r i e t y of -motions w i t h r e s p e c t t o the b o d y - f i x e d trihedral° In p a r t i c u l a r , when the p e r t u r b a t i o n i s zero, we i n f e r from the c o r r e s p o n d i n g phase-plane t r a j e c t o r i e s t h a t the s p i n v e c t o r can perform asymptotic motions of the f i r s t and second k i n d s and p e r i o d i c motions about permanent axes l y i n g i n the p r i n c i p a l p l a n e p e r p e n d i c u l a r t o the torque v e c t o r . Some of the r e s u l t s f o r t h i s case were a l s o obtained by Grammel, u s i n g d i f f e r e n t method. In the g e n e r a l case, when the p e r t u r b a t i o n i s not zero, these motions are p r e s e r v e d . However, a second type of p e r i o d i c motion e x i s t s ; i t occurs about the p r i n c i p a l a x i s c o n t a i n i n g the torque v e c t o r , the p r i n c i p a l a x i s i t s e l f b e i n g a d i r e c t i o n of s t a b l e permanent rotation. s 3 } s In S e c t i o n 3 we c o n s i d e r the same problem w i t h the torque v e c t o r a c t i n g along the m i d d l e p r i n c i p a l a x i s . Using the methods of the p r e v i o u s S e c t i o n , we show t h a t o5 can assume p e r i o d i c motions as w e l l as asymp t o t i c motions of v a r i o u s k i n d s . The p e r i o d i c motions e s t a b l i s h e d i n these two S e c t i o n s are then computed i n S e c t i o n 4 as power s e r i e s i n a small parameter. F i n a l l y 5 i n S e c t i o n ,5 the motion of a symmetric r i g i d body moving i n a v i s c o u s medium s u b j e c t t o a time-dependent t o r q u e i s studied„ I t s motion i s compared w i t h t h a t i n a vacuum. We show f i r s t t h a t p q , and r can be expressed in, terms of c e r t a i n i n t e g r a l s . For t h e s p e c i a l case where t h e self-excitement i s s time-independent and f i x e d i n direction w i t h i n the body, these i n t e g r a l s can be reduced t o t h e g e n e r a l i z e d s i n e and c o s i n e i n t e g r a l s , T h e i r values can be computed from asymptotic and power s e r i e s which a r e developed i n the same Section. The asymptotic b e h a v i o r of t h e s p i n v e c t o r " i s then discussed;, y i e l d i n g q u a l i t a t i v e r e s u l t s which a r e summarized i n t h r e e theorems. GRADUATE STUDIES Field of Study; Mathematics Nonlinear D i f f e r e n t i a l Equations I E . Leimanis Theory of F u n c t i o n s of a R e a l Variable Nonlinear D i f f e r e n t i a l Equations I I Topology Theory D. Derry J . F . Scott-Thomas S. of F u n c t i o n s C e l e s t i a l Mechanics Kobayashi F. E. Brauer Leimanis Related Studies: Unsteady Flow H y d r a u l i c s E. Ruus Elasticity J.A. Jacobs T h e o r e t i c a l Mechanics J.C. Savage ii ABSTRACT This fixed Such thesis point subject a body by E u l e r * s pal the motion to a torque i s called equal i n Section o f t h e body, to a constant 2 we consider and has plus of the modulus is shown Euler's variable the time t. the spin pect to the body-fixed vector i s zero, princi- we infer the spin can p e r f o r m plane using principal itself axis being d i f f e r e n t methods. containing i n terms case were exists; the torque a d i r e c t i o n of stable Fur- about permanent resper- motions permanent torque obtained case, are preserved. vector, of phase-plane also i t occurs a show with to the In the g e n e r a l motions of relation. asymptotic perpendicular Gramme1, of p e r i o d i c motion r ) . It a n a l y s i s , we and p e r i o d i c m o t i o n s these q, a n d r as f u n c t i o n s the corresponding Some o f t h e r e s u l t s f o r t h i s i s not zero, i s proportional o8(p, integrated axis I n p a r t i c u l a r , when t h e vector i n the p r i n c i p a l type that vector p, q, vector. a second term a v a r i e t y of motions frbm and second k i n d s perturbation i n the chosen of phase-plane trihedral. of lying or smallest c a n be can p e r f o r m that axes the largest by means o f a d i f f e r e n t i a l trajectories the f i r s t vector of the spin the method that turbation i s governed of a torque and i n v e r s i o n y i e l d Using a the case a component equations tp$ i n t r o d u c e d quadrature about internal reaction. i t smotion a perturbation the square that from body equations. to ther arising i s f i x e d i n d i r e c t i o n along axis of a r i g i d s e l f - e x c i t e d , and dynamical First, which discusses about when t h e However, the the p r i n c i p a l rotation. by axis i i i In S e c t i o n vector of acting w motions computed i n Section torque terms We principal show t h a t motions i n a viscous I t s motion show f i r s t that of c e r t a i n i n t e g r a l s . computed from asymptotic in t h e same S e c t i o n . is then marized discussed, i n three these and f i x e d in a where t h e i n t e g r a l s c a n be r e d u c e d integrals. The a s y m p t o t i c theorems. that i n direction with- Their values and power s e r i e s which yielding rigid time-dependent F o r t h e s p e c i a l case show t h a t and c o s i n e parameter. p , q , a n d r c a n be e x p r e s s e d i n in sine to a The are then of a symmetric i s compared.with i s time-independent generalized t h e methods kinds. i n a small subject self-excitement t h e b o d y , we the torque Using two S e c t i o n s 5 the motion medium axis. of various 4 as p o w e r s e r i e s i n Section with 8 can assume p e r i o d i c established, i n these i s studied. vacuum. we as a s y m p t o t i c periodic body moving t h e same p r o b l e m the middle Section, as w e l l Finally, consider e along the previous motions 3 behavior qualitative to the c a n be are developed of the spin r e s u l t s which vector a r e sum- vi ACKNOWLEDGEMENTS The for author suggesting throughout tion the is wishes t o thank the topic of this the author*s of this National graduate h i s a d v i s e r , D r . E. t h e s i s and f o r guidance studies the financial and d u r i n g thesis. Also, Research C o u n c i l o f Canada through appreciated. Leimanis, given the prepara- assistance given i t s summer by grant iv TABLE OF CONTENTS Page SECTION 1 INTRODUCTION 1 1.1 E u l e r s Dynamical Equations 1 1.2 The P o s i t i o n of the R i g i d Body i n Space 3 SECTION 2 , TORQUE VECTOR FIXED ALONG THE LARGEST OR SMALLEST PRINCIPAL AXIS 2.1 Equations 2.2 Torque Vector F i x e d Along the Largest of Motion P r i n c i p a l Axis 2.3 Case SECTION 3 B r i e f Summary and Remarks 3.1 Equations 3«2 A Q u a l i t a t i v e D i s c u s s i o n of the Motion of Motion and T h e i r I n t e g r a t i o n of the S p i n Vector 24 43 45 45 52 The Motion of the Spin Vector i n the Unperturbed 3.4 13 TORQUE VECTOR FIXED ALONG THE MIDDLE PRINCIPAL AXIS 3.3 12 The Motion of the S p i n Vector i n the Perturbed Case 2.6 7 The Motion of the Spin Vector i n the Unperturbed 2.5 6 A Q u a l i t a t i v e D i s c u s s i o n of the Motion of the Spin Vector 2.4 6 Case 55 The Motion of the Spin Vector i n the Perturbed Case 60 Page SECTION 3«5 General Remarks and C o n c l u s i o n s 4 PERIODIC 4«1 General 4»2 Periodic Solutions 74 SOLUTIONS 76 Considerations Vector Fixed 76 i n t h e Case o f t h e Torque Along the Largest Principal Axis 4.3 79 Periodic Solutions Vector Fixed i n t h e Case of t h e Torque Along the Middle Principal Axis SECTION 5 82 S E L F - E X C I T E D SYMMETRIC R I G I D BODY IN A V I S C O U S MEDIUM 87 5.1 Equations of Motion 5.2 The A n g u l a r 5«3 Time-Independent Torque Vector F i x e d i n V e l o c i t y o f t h e R i g i d Body Direction 5.4 5.5 BIBLIOGRAPHY W i t h i n t h e Body The I n t e g r a l s Re w > 87 s i ( x , w) 93 a n d c i ( x , w) 0 The A s y m p t o t i c 89 99 Motion of the Spin Vector 107 122 1 SECTION 1 INTRODUCTION 1.1 Euler's This body thesis about reaction. Grammel Dynamical contains a fixed Such [11], equations. point a rigid a study of the motion subject With of this type space rigid-body dynamics. rotation o f a space vehicle introductory Section, and some d e f i n i t i o n s the thesis. we this, due t o t h r u s t from internal by dynamical self-propulsion interest, especially i s the problem of misalignment. In this present the equations of motion which a brief given. Figure by E u l e r ' s are of great An e x a m p l e shall rigid self-excited of internal and n o t a t i o n s Following called i s governed the development of a to a torque a r i s i n g body h a s been and i t s motion systems, problems in Equations 1.1 will be u s e d throughout resume o f o u r work will be 2 In F i g u r e 1.1, coordinate systems l e t OXYZ and Oxyz be r i g h t - h a n d r e c t a n g u l a r fixed respectively i n space at the p o i n t and i n the r i g i d body along the d i r e c t i o n s 0 of the p r i n c i p a l axes. Suppose p, q, and r are the components of the angular v e l o c i t y v e c t o r io along the x, y, and z axes. M z Then i f A, B, C and M . M , A y are the p r i n c i p a l moments of i n e r t i a and the components of the torque v e c t o r 2$ a l o n g the same axes, E u l e r ' s equations may the take form (1.1) A'P - (B - C)qr « M x Bq • (A - C)rp = M y Cf - (A - B)pq - M z Here and throughout our work, dots i n d i c a t e differentiation with respect to the time t . For tinguish the p o s i t i o n two of the torque v e c t o r i n the body we cases, depending whether 3 i s f i x e d i n the body or moves i n i t a c c o r d i n g to some p r e s c r i b e d law. respectively i n the body. These are c a l l e d s e l f - e x c i t e m e n t with f i x e d or moving d i r e c t i o n In a d d i t i o n , nt i s s a i d to be time-dependent time-independent not. dis- with- or a c c o r d i n g as i t s modulus changes with time or For a time-independent self-excitement fixed in direction w i t h i n the body, the components M , M , and M become c o n s t a n t s . x y z Consequently, i n t h i s s p e c i a l case we have one of the s i m p l e s t g e n e r a l i z a t i o n s of the Euler-Po&>»©* r i g i d body. case of motion of a heavy In f a c t , the problem of motion of a s e l f - e x c i t e d r i g i d body s u b j e c t to a time-independent torque v e c t o r f i x e d i n d i r e c t i o n w i t h i n the body and the problem of motion of a heavy 3 rigid in body nature. reduced of i n the Euler-Lagrange case In the l a t t e r , from J a c o b i * s of i t i s known ones c a n be f o u n d . theory of the l a s t be r e d u c e d the s o l u t i o n first integral In the former, multiplier to quadratures, i f a f i r s t that similar c a n be independent i t follows the solution integral can independent t i s obtained. 1.2 The P o s i t i o n , o f t h e R i g i d Body Figure The E u l e r locate are that to quadratures because a f o u r t h t h e t h r e e known also are mathematically Angles 0, uniquely the position known t o be related p (1.2) 0, 1.2 <f and 0 cos 0 + f i n Figure i n space. t o t h e components = t> s i n 0 s i n < f = as d e f i n e d o f t h e body q = (5 B i n O c o s f r i n Space + 0 cos^ - 0 sincf These 1.2 angles o f B8 b y t h e e q u a t i o n s : 4 The solution tion of (l.l) o f t h e body i n space m" do n o t c o n t a i n without as taking functions can always In the at any time the Euler (1.2) i n t o be r e d u c e d vector acts account. with component arbitrary we s h a l l consider along the largest I t i s shown a n d an i n v e r s i o n , discuss also The former the case gave first the case or smallest i n which principal 2 that cp. E u l e r V s e q u a t i o n s c a n be When t h i s i s followed t h e motion o f to i n t h e l a r g e . the subcases: i s t h e case by a t h e n p , q , a n d r c a n be e x p r e s s e d a s Using t h e method o f phase-plane ( i ) \L Q - 0, asymmetric analysis, we a r e In p a r t i c u l a r , we ( i i ) | i = 0, M = 0. c o n s i d e r e d by Gramrael of a force-free [11]; the l a t t e r i s gryoscope, a very eloquent geometric discussion. analytical since i t 0 quadrature to study (l.l) to quadratures[ 2 ] . i n terms o f a v a r i a b l e able we c a n s o l v e (1.2) p r e s e n t s no d i f f i c u l t y integrated of t . I f t h e components o f e q u a l t o M • M-|w| , w h e r e M a n d p. a r e constants. functions the posi- When p , q , a n d r a r e k n o w n i"* i axis t . angles e x p l i c i t l y , o f t> s o l v i n g the following, torque a n d (1.2) c o m p l e t e l y d e t e r m i n e s t o which Poinsot Our t o p o l o g i c a l - method p r o v i d e s a s i m p l e and u n i f i e d approach t o these problems. In S e c t i o n but with We o b t a i n case. the torque vector results that The p e r i o d i c Sections small 3 we c o n s i d e r are then parameter. acting differ motions t h e same p r o b l e m along the middle p r i n c i p a l significantly which as i n S e c t i o n from are established computed i n S e c t i o n 4 as power 2 axis. the previous i n t h e s e two series in a 5 Finally, symmetric i n the last Section rigid time-dependent in fixed Bodewadt problem For torque. The c a s e [2]. He be e x p r e s s e d the problem integrals. Fresnel discussion within showed the corresponding metric subject i n direction could show t h a t the i n a viscous a f r i c t i o n l e s s medium vector by body we medium t h e b o d y was the general t o an rigid first arbitrary body 5 then motions of of the m e d i u m , we a class ends w i t h m. torque integrals. by t h e g e n e r a l i z e d form which a moving considered solution of the Fresnel i n a viscous integrals of the asymptotic subject of a to a time-independent c a n be s o l v e d Section the motion of a symmetric i n terms situation These integrals. that study shall trigono- includes qualitative 6 SECTION 2 o TORQUE VECTOR F I X E D ALONG THE OR 2.1 Equations of In this SMALLEST P R I N C I P A L we rigid body lying along the largest axis That a n d U. • 0 explicit Following this, a qualitative For shall of r o t a t i o n a r e done dynamical we of motions. directions term chosen which i s pro- vector show t h a t component shall able to be of a variable i s given of the the end-point can o f t h e body w i l l be by g i v i n g a phase-plane analysis with perform discussion, also CM. take the and r i n terms discussion 3 shall In the course of t h i s permanent established. of a A l l related system. the present i t i s convenient to take Euler's form p (2.1) a s s u m p t i o n , we We i n the of the spin shall vector axis. component a perturbation formulas, f o r p, q, to which a variety the with self-excited to a torque principal of the modulus Under t h i s of a o f t h e e n d - p o i n t o f 35 i n t h e b o d y - f i x e d t r i h e d r a l , respect these subject a r e c o n s t a n t s , we q be M + p- |et)| motion in point the motion or smallest to the square i s ,i f M obtain «p. a fixed equal to a constant plus portional to consider "A i s t i m e - d e p e n d e n t assume t h a t AXIS Motion Section about LARGEST - aqr = q + b r p = in f y - . c p q = ra equations 7 H e r e we have l e t B-C , a - — > b A-C - — ' ° A-B ~ = and M M M y B ' X x m In addition, that A > = without B > C so V m A~» any that loss the z m of z C~ = generality, inertia we b, shall numbers a, and Largest Principal assume c are positive. 2.2 Torque Vector If x - a x i s denotes (2.1) the with m = y ra Fixed Along =0 (2.2) = M the largest principal Axis axis, system becomes z p where m the ^o [i - -jj— Introducing the - aqr = m q • brp = 0 r = 0 - cpq are + | «i | (JU . . constants. variable tp by means o f the differential relation dtp = we may w r i t e the second and pdt third equations as ^3 • b r = 0 dtp dr d^ " The be general put solution i n the form of this c q = 0 system of equations can immediately 8 q = a-jjb c o s y b c r = tp - c t 2(S / F s i n y b c ep (2.3) The arbitrary values q o r rigid the body body as satisfies q are Ci^ a n d and as on this the sions, (2.4) we <f&c f determined by the obtain the = 0, then q about with ra + |xp following magnitudes i s the of value of p at m an = r = 0. the Accordingly, x-axis, fixed angular velocity p in which 2 expressions and t = (i) For m > 0, \i. > 0 (ii) For m > 0, [i < 0 |i» f o r p, In each of these coth p ) » m < 0, \i > expres- arctanh c o t h (y-rap.t + a r c For depending 0. • (iii) initial equation relative p t = zero, i n space, $> From o.^ a r e r at both cos rotates permanently well the of o 0.^ and the (X constants and If i v ^ s i n y b c cp • 2^ a 0 arctanh Po > l?l 9 (iv) (v) For m < 0 , \i < 0 For u, = 0 p = mt • p o (vi) For m = 0 , \i t 0 p" Having - p" o p * 0 1 1 - [it i f p £ 0 • o ifp = 0 o r d i s p o s e d of t h i s p a r t i c u l a r case, we assume i n what f o l l o w s t h a t at l e a s t one of the constants and <x^ i s d i f f e r e n t from z e r o . A c c o r d i n g to (2.3), t h i s i m p l i e s that at l e a s t the i n i t i a l values q Q and T Q i s non-zero. Expressions one of (2.3) may now be put i n the form q = 67b~ cos (Tbctp + a ) (2.5) V r = o^c" sin(^b*b(p + O ) Q l~2 2 2 where 6 = /a, • a„ and a = arctan — V 1 2 o a be determined are constants which can values q and r • o o to express p as a f u n c t i o n of cp. If of from the i n i t i a l we s u b s t i t u t e e x p r e s s i o n s (2.2) and make use of the f a c t p • (2.6) we o b t a i n the second We proceed now (2.5) i n t o the f i r s t -.2 |oo| that = p 2 + q order equation * y.6 [b c o s ( , / E c f + a ) 2 2 Q * c sin (ybc<p + CT )] 2 Q 2 equation 2 • r and 10 that and by i ssatisfied . Letting making use o f the i d e n t i t i e s 1 + cos 2 cos fl x 2 1-cos 2 ( ^ . 2„ sin cp^ we can w r i t e 2<p ( 2 . 6 ) as dt J" • c d sin 2((p + x ) where 2 ' • J i ^ y - I * * * * <>">] °*i = ^(b-c) 2 arctan ] Further, put ^2 - 2 ^ 1 «1> + t = y2d Then equation Finally (2.7) reduces t o d\y l e t t i n g1 (2.9) li, l m = we o b t a i n the l = 2 1 ( 7\T~ J. U 6 b . / b c j ti,6 c./bc 2 H ( */ r a dC p 2 f b c * ? 2 + ^ 5* 2 d*T f o l l o w i n g system of first o r d e r equations equivalent 11 to (2.8). d l p — (2.10) = m • s i n ^ x 2 • d <P 2 dT Furthermore, " Pi i t i s clear that = 2(ybc"<f + 4> 2 (2.11) • CTo 0 i > m (2.10) we h a v e t h e l i n e a r e q u a t i o n From system i n p^ » 2 d l p (2.12) the 2 - 2 i i ^ = 2m • 2 sincp l P 1 i n t e g r a t i o n o f which 2 (2.13) Here p 2 yields ^ l f 2 2 x 2 (S^xsincpg = C e x 1 i san a r b i t r a r y c o n s t a n t , from the i n i t i a l 2 values • cos<j? ) 2 whose m a g n i t u d e o f p^ and(p # 2 m l — c a n be f o u n d s a y p^Q a n d <p20* W e have then (2.H) The C - e x ^ component p f o l l o w s (2.15) U P^ 2 • --^-2(2n1sin(p20 i*4M-^ (2.11) now f r o m ^ P =^c 0 2 2 If L J ( f ) denotes (2.15) b y r e p l a c i n g c p (2.16) 2 s i n 0 1 the expression with p - obtained the f i r s t tJUW • — * and (2.13) —,(2^1 1 * coscf^) 9 7 < i from equation • cosc&J T < J - ~ ^1 the right of of (2.11), then 12 Thus w i t h Euler*s In that, (2.5) equations order t be q, # to express = i Jxj(f) denoted Sections hedral. the the For we f u n c t i o n s of t we f first note dt, q, and in (2.17). c|> a s a f u n c t i o n of to t. except we of shall Let the are Motion motion system determine having to of complexity latter i n the steps evaluate will special formulas (2.5) we cases note the and of when that Vector motion body-fixed say the the behavior to the invert the t r i of determined The time in motion (2.10). use the of the of integral e x p r e s s i o n U(f), r e q u i r e some k i n d o f type. From the Spin i s completely allows the to the with simply of to of concerned sometimes the result. t» r e l a t i v e show, t h i s us r e p l a c e (^ by desired trajectories Because method, the 2.5, we later r without these ( 2 , 1 6 ) i f we obtain end-point method general and D i s c u s s i o n of 2.3 phase-plane p, we brevity shall phase-plane tic (2.5) (2.18), large of by as . by Qualitative In As integrating t A a». in Cf- ( f ( ) expression the succeeded r i n terras of ^ (2.17) p r o v i d e s in equations 2.3 and them (2.18) Then have ff i n v e r s i o n of this ( 2 . 1 6 ) we for p d q5 = ± since (2.17) The and integral in numerical i s an ellip- 13 2 2 2 2 6 c 6^b that i s , t h e p r o j e c t i o n o f the end-point will always with major this plane. relative tic l i e on an e l l i p s e and minor semi-axes Consequently, _JL? 2 6 b At t h e same t i m e or the equivalent + (2.10), motion according tively 2«4 o f 5) w i l l that follow. a s \i i s z e r o move o f an ellip- p^. Because p i s a behavior o f p^ and thus (2.10). We s h a l l These and perturbed t o (2.6) o f p i s comIn order t o o f p we s h a l l From t h i s distinguish will cases. give knowledge two be c a l l e d cases respec- The f o r m e r will be first. of the Spin When t h e r i g i d acted upon o n l y along the largest Graramel always according o f p^ i s determined. or not. Q The M o t i o n that vary contains a n a l y s i s of system the unperturbed discussed axes i n _JL? _ , 2 6 c which the possible variations the o f m must on t h e s u r f a c e t h e component p must known w h e n e v e r a phase-plane by trihedral at t h e o r i g i n and t h e two p r i n c i p a l m u l t i p l e of p^, the q u a l i t a t i v e pletely see along t h e yz plane whose e q u a t i o n i s (2.19) scalar center the end-point to the body-fixed cylinder with of w onto [11]. relative body i n t h e Unperturbed Case i s f r e e o f p e r t u r b a t i o n , i ti s then by a t o r q u e principal Using Vector vector of constant axis. This problem magnitude was fixed considered methods different from ours, he showed to the body-fixed trihedral i» c a n h a v e three types of m o t i o n . Two other p e r i o d i c . We o f t h e s e he give here called a complete lem by t h e methods o f p h a s e - p l a n e follow as c o n s e q u e n c e s (2.10) f o r a., = 0. possible types of motion o f nl t a k e s on values apparently prisingly simple. the ditions the be inertia We numbers shall also be o f permanent f o r the e x i s t e n c e able The stability discussed. will of system component initial c. Such t u r n s out t o be to i n d i c a t e an sur- i n the unbounded m o t i o n s . axes of p e r i o d i c to the a, b, and ' * o f t h e body and motions o f t h e s e two of w are More- the con- easily types of motions From a l l t h e s e r e s u l t s , we prob- t o s t u d y the w can p e r f o r m , when t h e r e g i o n s o f bounded and locations established. also the trajectories values in proportion the of t h i s His r e s u l t s a method a l l o w s us complicated i n t e r r e l a t i o n s h i p phase-plane over, that different q , r and o' o such discussion analysis. of the phase-plane Also, a s y m p t o t i c , and can later can observe —• 2 the e f f e c t For due to the p e r t u r b a t i o n » 0, system t e r m M-|n>| . 0 (2.10) becomes dp, = m, dT (2.20) d <P 2 dT This pair system. defined ~ - For Clearly, i the t r a j e c t o r i e s i n t h e phase dynamical plane are by m-.+sincp,, - 4 ' - ^ 2 1 ) such p of e q u a t i o n s r e p r e s e n t s a c o n s e r v a t i v e dp. C 2 • sincp2 a dynamical system an e n e r g y integral exists which can 15 be (2.21) obtained from by t h e s e p a r a t i o n of variables. This g i ves 2 l (2.22) m where E i s t h e energy the dynamical system (2.20) nator which ° s C f 5 = E 2 The e q u i l i b r i u m correspond to thesingular occur at points that c constant. of theexpression taneously, * " lT°2 P o f (2.21) on t h e ( p ~ a x i s i s , at points m 1 points where t h e n u m e r a t o r on t h e r i g h t o f system and t h e denomi- vanish simul- where 2 + sincp positions of = 0. 2 Obviously i f |m^| > 1, structure o f t h e p h a s e p l a n e d e p e n d s o n t h e v a l u e s o f m^, we first t h e case 0 < consider 0 (i) < |m | 1 The (2.20) has then They o c c u r i n p a i r s , thecp ~axis. 2 These p o i n t s mum a n d m i n i m u m p o t e n t i a l the system energy, energy 2.2 values o f m^. They they spaced energy energy this while o f minimum at positions V versus ^ These t r a j e c t o r i e s they cross o f 2TT a l o n g Figure 2.1 trajectories o f maximum shows t h e orthogonally. 1. f o r t h e same are curves of constant 2 Since potential f o r 0 < m^ < the q? ~axis, and except axis of maxi- o f t h e dynamical system. are saddle points. about number o f s i n g u l a r a t an i n t e r v a l at positions shows t h e p h a s e - p l a n e are symmetric points 1. an i n f i n i t e arecenters, of thepotential Figure Ast h e correspond to thepositions i s conservative, these points potential graph |m^| < can o c c u r . 1 < system points. no s u c h p o i n t s energy. at the singular \i 1 = 0, 0 < Figure 2.2 < 1 17 The loops i n Figure curves o f (p equations through passes (2.23) the saddle point through 2 S i s given separatrices, with that the spin i t s modulus t w e e n t» a n d t h e x - a x i s reason, originated t» t e n d s t o zero vector approaches the i n the yz Inside regions return + C O S C value of the separatrix formed o f p^ tends t» a p p r o a c h e s approaches (see Figure two b r a n c h e s by two to con- infinity. p i s also to i n f i n i t y , eventually respect the these P2S^ of p^, c l e a r l y and a f i n i t e with unbounded. the positive f o r the angle zero. be- For the f o r any motion 2.2) of each that separatrix. F^ a n d F^, p ^ and C j ^ t e n d value, s a y c P2s to the body-fixed r a s t -• + » . trihedral vector 6yb lying f2 vector F that by c o s C tending on t h e b r a n c h respectively spin I f ( j ^ g denotes to the p o s i t i v e x-axis H o w e v e r , on t h e o t h e r The the condition the equation the magnitude p i s a scalar multiple x-axis from any t r a j e c t o r y G i n t h e r e g i o n s implies same " m i n d i c a t e d by t h e h e a v y points* S, t h e n 2 secutive Since c a n be o b t a i n e d ?\ = l i ( f ~ f 2 5 ^ Along This 2.2, at the saddle 2 that pass of the separatrices, "J + 6^c cos sin H plane. the closed of periodic t o t h e same branches motions, values of the separatrices, since after a we h a v e t h e t h e components p, q and r a l l period 18 where the line trajectory. i n t e g r a l i s taken along Note, however, that the p e r i o d i c i t y of fixed t r i h e d r a l does not n e c e s s a r i l y rigid body closed i s periodic branch Motions rotations the well addition separatrix the periodic to singular body, i s thus angular as i n s p a c e . rotations It i s clear of the r i g i d a constant since are unstable, that a rigid the r i g i d velocity about an a x i s body the are represent Thus, while of the inside motions zero. Moreover, i n t h e body- t h i s motion that points closed stable. permanent p, q, and r a r e c o n s t a n t s , at the saddle at the centers t o t h e one on t h e x - a x i s , rotations mean t h a t i n space. corresponding derivative with as of each the corresponding these subject body fixed points they rotates i n t h e body the permanent are s t a b l e . are the only t o such and a torque In permanent vector may have. Grammel [11] c a l l e d asymptotic motions motions f o r which tending to i n f i n i t y . by w approaches We the x-axis see that these the t r a j e c t o r i e s G i n the regions separatrices called these a finite these [3] h a s i n d i c a t e d Finally, cated motions are represented Physically, with bounded vector are represented by two consecutive the periodic by t h e c l o s e d separatrix. motions f i x e d i n the yz plane. by t h e b r a n c h e s motions that of the second kind will Obviously In f a c t , separatrix. Bogoliubov the p r o b a b i l i t y of t h e i r occurrence motions of w mentioned t r a j e c t o r i e s about the by Grammel centers. He f o r which F, and Fg o f each rarely occur. kind i t s modulus motions a n d by t h e t r a j e c t o r i e s F on e a c h asymptotic a» a p p r o a c h e s of the f i r s t are is zero* indi- 19 Figure -1 < m, < energy. as 2.4 0, and The before, except m |m, | = the For coalesce point 2v have 2.3 f o r the motions into a one center. o f ra, t h e They the ( f ^ - a x i s . phase plane to no closed vector w The of p o t e n t i a l essentially of x-axis. the the same first kind However, approaches f o r m, for are clearly can be with i n the e i t h e r the 1 and spaced m, yz no tending a curve plane. of zero, zero. can For we can say which |ra,| > in Figures can 1, 2.7 that negative however, and 2.8. exist. x-axis of x-axis exist no there The or the a latter practically permanent still approach Since occurrence The 2.6 respectively. the negative. and separatrices to of thus 2.5 As the saddle will Figures = -1 pair intervals a s y m p t o t i c a l l y e i t h e r the plane shown at p e r i o d i c motions p o s i t i v e or unstable. as = slope yz a probability found, equally i n each is neither , approaches i n the which minimum p o i n t s . w h e t h e r ra, i s p o s i t i v e o r axes singular points p o t e n t i a l energy trajectories, vector has are f are about graph motions l H o s p i t a l s r u l e the singular points motion are for considerably. singular point, the constant motion negative change values According spin of trajectories corresponding asymptotic the n e i t h e r maximum n o r show t h e the types approaches these nor along phase-plane = 1 ImJ (ii) Figure possible w h e r e now 1, shows the 35 depending rotations for |m,| = 1 such d i r e c t i o n s 20 Figure 2.4 21 F i g u r e 2.6 22 jm1| > (iii) 1 S i n c e now there are no s i n g u l a r p o i n t s , the phase plane w i l l go to i n f i n i t y . k i n d can be found. shown i n F i g u r e s 2.7 and 2 . 8 . absolute The phase planes are Furthermore because m^ = • ' ^ ' " , 6 aybc 2 m value can be made l a r g e r than 1 by making 6 , which depends s o l e l y together in As a r e s u l t o n l y asymptotic motions of the f i r s t its a l l trajectories on q Q and r , s u f f i c i e n t l y Q small. This fact, with the t o p o l o g i c a l s t r u c t u r e of the phase plane for |m^| > 1 proves that the d i r e c t i o n of permanent r o t a t i o n along the x - a x i s (iv) is stable. = 0 S i n c e now m^ = ra = 0, asymmetric gyroscope. we have the case of a f o r c e - f r e e The motions of c» are d e s c r i b e d by the polhodes on the Poinsot e l l i p s o i d . here that However, we want to these motions can also be i n f e r r e d from the ponding phase-plane t r a j e c t o r i e s shown i n F i g u r e 2 . 9 , show corresthus p r o v i d i n g a u n i f i e d approach to a l l these problems. R e f e r r i n g to F i g u r e 2 . 9 , unstable we have at a saddle p o i n t an r o t a t i o n about the middle p r i n c i p a l axis, and at a center a s t a b l e permanent r o t a t i o n about the smallest axis. For at these p o i n t s from (2.5) principal and (2.11) we have res- pectively p = r = 0, q = 6,/b cos —^~ «= constant p = q = 0, r = bjc = constant and s i n —— C l e a r l y these are s o l u t i o n s of E u l e r * s equation f o r m = 0. F i g u r e 2.9 24 The c l o s e d t r a j e c t o r i e s i n s i d e the s e p a r a t r i x correspond to p e r i o d i c motions about the smallest principal axis as e x h i b i t e d on the Poinsot e l l i p s o i d , and those curves o u t s i d e the separatrix correspond to p e r i o d i c motions about the l a r g e s t p r i n c i p a l axis, since on each one of these p , i s p e r i o d i c with a p e r i o d 27T. the s e p a r a t r i x must have as i t s Thus corresponding part on the Poinsot e l l i p s o i d the s e p a r a t i n g polhodes. T h i s then completes the d i s c u s s i o n on the t o p o l o g i c a l s t r u c t u r e of the phase plane of (2.20). critical and - 1 , values of m, are 1, 0, We n o t i c e that since at these the values the t o p o l o g i c a l s t r u c t u r e s are changed r a d i c a l l y . 2.5 The Motion of the Spin Vector i n the Perturbed Case T u r n i n g to the p e r t u r b e d case, we s h a l l assume without any l o s s of g e n e r a l i t y that |x, = -v < 0. of the l a r g e s t it so that principal axis is negative. is For the p o s i t i v e d i r e c t i o n a r b i t r a r y ; we may always The t o p o l o g i c a l s t r u c t u r e of select the phase plane i s then d e f i n e d by the equation 2 dp, i , *sincp -Vp. -r^ = — idcp p, equivalent 0 (2.24) 2 or i t s d l p — U ' 2 5 d ) = m, + .incp 2 2 - vp, ? 2 dr p l Thus the s i n g u l a r p o i n t s occur along t h e C p ~ a x i s at p o i n t s where 2 m, • s i n c p 2 vanishes. Thus, t h e i r l o c a t i o n s are the same as i n 25 the unperturbed remain symmetric (2.24), in case. points, 0 < m 1 also the phase-plane the cpg *** * f° - the equation they < 1 about In addition, 8 i s invariant. r w e trajectories r e p l a c e p ^ by - p ^ Except at the singular cross the opg-axis o r t h o g o n a l l y . We assume that first. 0 < ra < 1 (i) 2 Then the singular (2.26) ?2 0 denotes If p o i n t s occur at = arcsin(-m.) the principal value 0 T2 • o f ( 2 . 2 6 ) , we have 2kir or <P -0 2 where k i s an To ^2 = for © + • (2k + l ) v integer. determine the nature of the singular 2k7T, we c o n s i d e r t h e e q u a t i o n s (2.25) at these points. Making p o i n t s at of f i r s t approximation the transformation of coordinates (0 we f i n d + 2k7r) that order the equations of f i r s t approximation dn (2.27) 2 or higher = at these i n 5. points Consequently, become 26 The characteristic equation \ be r e a l . According (2.27) i s 0 = 0 - cos 2 g i v e s X = t ,/cos 0. which of -TT — < 9 < 0, Because to Poincare's criterion, the roots these must p o i n t s are saddle p o i n t s . On first the other hand, approximation * t c f = _ e + 2 (2k + 1)T the equations of are = -§co e jft S (2.28) dT The c h a r a c t e r i s t i c equation A. 2 which gives \ Accordingly, However, the In Figures Figure follow increasing sects there 2.10 and l a r g e v a l u e s possibility we be of t h e phase p l a n e , V. small singular 0 < ra, < 1 For and The p d i n t s must becomes now • cos© = 0 as t h e t r a j e c t o r i e s singular tures = ±,/-cosQ. these ' roots points are thus are e i t h e r are symmetric pure foci about are three d e p e n d i n g on t h e r e l a t i v e we 2.10 e s t a b l i s h e d as f o l l o w s : the v e r t i c a l o f ra, i s shown i n Figure T, t h a t leaves the saddle I t must continue remaining 2.15. In Figure 2.12 p o i n t S, f o r to the right and inter- through the center at f , , because i n dp, the i n t e r v a l ( S C _ ) , -r—- > 0 f o r s u f f i c i e n t l y s m a l l v a l u e s o f 1 <c aT p , as i n d i c a t e d i n F i g u r e 2.11. 1 # line values The o t h e r plane o f T» struc- show t h e p h a s e p l a n e s f o r of the phase values the C ^ - a x i s , possible topological of V r e s p e c t i v e l y . the trajectory or centers. centers. a n d 2.14 c a n be imaginary. 27 For small values also trajectory leaves intersects interval (c The 2 # If can be s e e n V 2 l at o P l = s l P f 2 positions (2.13) p at f because * 9 s l C 2 s2 = of T now i n t h e 0 * o f f ^ and f depend 2 f ^ as shown the values on t h e v a l u e o f 2.12. i nFigure o f p^ at these two points c m 2 e 2 (-2vsincp c 1+4V + cos<p ) + — c , - o P C values aatt ff ^ . "2Vf r = through , f g i s below by computing 2 = f o r decreasing 2 < From e q u a t i o n P the vertical dp, S) < 0. relative m V. and T^ t h a t 2yi-mY 2.10 Figure The m. of V < C C s2 " *< 2V e - — 2 2 m (- 2 v ^ n ? c • cos f c ) • l — This 28 Here C ^ and C g trajectories and c p we c are the corresponding g 2 that leave i s the value the saddle of cp & 2 values of points t the center on t h e and C,,. From respectively, equation (2.14), have 2 V < C . = e si — 2 2( - 2 v s i n ( p Psl • cos(p ) - l— m s l sl •4V and 2vcp „r £ C in s2 = ee 8 1+4V w h i c h cp ^ a n d cp ^ s Since n 2 l(- at the saddle 2 2 S J. S rfC difference p . - p s From l s this 1 V > — , three m. relative topological In tinues o f (p 2 V f = = s i n C C O Ps2 ~ = S f s2 c / 2V( Ps2 2 V 2 l r a M C / m to the form l - A we s e e t h a t p ^ - p g g 2 V V ^ - ? (l+4v )v 2 2 < 0 a c c o r d i n g as l m Thus f 1 i s above and c o i n c i d e w i t h f f 2.12 to the right i ff^ lies and crosses i fv < — v -- give 2 1 . three f 2 These different the trajectory the v e r t i c a l T m i fV = — , below plane. above hand, t h e t r a j e c t o r y cut thecp ~axis 2 2 s t r u c t u r e s o f t h e phase 2 a t S^ a n d S . 2 = p o s i t i o n s o f f ^ and f Figure On t h e o t h e r must ^ , " IT ) S V equation iP 2 COS( c a n be s i m p l i f i e d Q 2 I—T>- 0. - 2vyi-ra if + points c o s <PB1 " T°s2 are the values g sin f s l the 2 V s i n 2 through continues o r t h o g o n a l l y a t some p o i n t S T^ 2 at g^. to the left g 2 con- between and S^ 29 F i g u r e 2.12 30 and Cg. Since the trajectories <p2~axis, a r e f l e c t i o n trajectories occur point consecutive the trajectory g , and never (2.13) s h o w s , The vertical on which + ^ = However, (2.13), w h i c h 0 p - 2 v c = C e P p where = arctan ^3—] J "v~~ ' = a x i 8 • Con- from t h e p, = D which large value /™ • cuts the of p, decrease The e x i s t e n c e o f t h e s e p a r a t r i x ° a * n n e n b e e s t a b l i s h e d from F con- equation may a l s o b e w r i t t e n a s P, 10 the line i t c a n a l s o be e s t a b l i s h e d f r o m (2.29) P about t h e <p 2 ~ will E v e n t u a l l y , as e q u a t i o n the trajectory l ^ 2 ^ the corresponding to the right 8 S, at s u f f i c i e n t l y 2 i r give points along - 3 oscillate about t h e T h e same p i c t u r e t h e GOg ** * D to the right. P^C^ tinuity. will T, c o n t i n u e s of p, along through we f o l l o w axis saddle crosses i twill values as this i n the lower-half plane. at other sequently, about are symmetric c o n s '' : a n 0 - ra sin(<f> y • 't *" or t h e * oJ initial C, i s z e r o . + i ~ v a l u e s <^20 In such a case = ~ 3' CT we may write (2.30) and p - ± / -f the expression - • yL-^ under s in(q> the radical l o f Cp^ i f v < — ' / " " ~2 ' </ l + 0 ) sign i spositive for a l l (2.30) p r o v i d e s then t h e m values 2 equations From in Figure separatrix Formula 1 - m o f t h e two s e p a r a t r i c e s F and F , i n F i g u r e t h e above 2.10 d i s c u s s i o n , t h e geometry i s now c l e a r . F will oscillate o f t h e phase The t r a j e c t o r i e s about the line 2.10. p, = lying /— plane above t h e as ^ "* ' 0 0 31 since the c|?2 "* On tories the first the other lying first infinity below term a s (p Because 2ir period phase of the phase i t i - take the may 2.10 are and shown this the now centers closed trajectories a x i s of as c p Intuitively cp The -* OB. The there C. i s true F^. to cylindrical the p ^ - a x i s values and a right the phase coincide in the section this plane in this the phase the cylinder. saddle One point S a continuum branch s t i l l of o f u5. second F^ spirals i s s t a b l e and trajectory spirals limit F^ F^ the encircles : However, spirals as cycles of unstable, toward of closed t o w a r d F^ are c a l l e d point and o f t h e s e p a r a t r i x f r o m S now and F into separatrices F p e r i o d i c motions branches Obviously near the trajectories The is still represent of F every now diverges trajectories points the point encircling C and arc of wrapping saddle -» «>, w h i l e , t h e kind [ l ] . - opposite into the the since 2.13. separatrices F a neighborhood by cylinder. a l l the other 2 along trajec- done f o r s m a l l a c y l i n d e r t o be measured the be the as i s p e r i o d i c with consider will zero to dominates to o r i g i n a t e d from the which t h e two This constructed About toward F second system. curves C. centre appropriate in Figure space the of be around that 2 s along tend (2.25) side of approaches o f p^ will (2.29) of hand coordinate separatrix one -<»• (2.29) of the values right 2 In are -* of t h i s right s e p a r a t r i x F^ c y l i n d e r as ( j ? . Figure S, the i n (p , space space on 2 We the right curvilinear of 2 the hand, the space V . t e r m on the since in i t , while the 32 33 The 2.13* the motions At the body. As of m can singular before, be easily points we find we addition to these periodic c a s e , we represented by the spin the elliptic cylinder rotation. in terms the vector and a continuum now », of limit two cycles of the ones, and F,. returns about to i t s o r i g i n a l Furthermore, the axis q, functions (2.9), (2.11) and on of position components p, Jacobian e l l i p t i c From e x p r e s s i o n s the lies those the as (2.30) motions, surface this after and In unper- I n t h e s e two whose e n d - p o i n t a l w a y s rotates i n the namely, of permanent motions. existed new Figure rotations stable periodic that F from the permanent stable motions establish cylinder, of the inferred i n a neighborhood rotation turbed have now of elliptic one r can complete be expressed follows. we find that (2.31) and (2.32) Here Cp value on p the 2 0 Q i s the initial is positive, right This value of Cp 2 the p o s i t i v e at t = sign 0, and in front i f the of integral = <?2 * " 3 be For making integral i s taken. I can the initial reduced to use of a normal the elliptic identity integral ) of the first kind. sin((p and 2 • a - i 3 the s u b s t i t u t i o n we may write I (2.33) I = d© k sin 0 2 2 where 4V + 2v m Because If we assume t h a t m. v < 2 1. < we l e t F(<f, k) = f we , k d© 0 ^k s i n © 2 2 obtain . 7T (2.34) t = + d ( l^ 2 Inverting (2.35) where and <p a - ^ Jl+k^J V the expression - , am d e n o t e s (2.5), from 2 m 3 - f (2.31) and (2.35) that It follows t h e components p, q, of the Jacobian elliptic func- sn and c n . Moreover, given have amplitude function. r c a n be e x p r e s s e d i n t e r m s tions we • 2 am the Jacobian (2.11), i n (2.34), i n terms the period T of these periodic of the complete elliptic motions integral c a n be of the first 35 kind v K r = do 2 yy/l-k 8in 0' •'O For clearly r 2 Making use integer, = /2d 2 i r d the theory F represented i s a stable by F the motion the end-point in of = 2 n K ± F ^?2' i n t e g r a l s , - we n limit cycle, However, by F^. Under away to i n f i n i t y or the spin the p e r i o d i c motion this any from i s not the case f o r slight this motion; the negative or less vector likely can t e n d disturbance, the spin x-axis a constant the the Similarly, i f the spin v e c t o r 08 i s d i s t u r b e d f r o m a p e r i o d i c motion about i n the yz plane represented by F. F. i n the yz plane, i t either the a x i s of the s t a b l e permanent or tends The vector t o assume by rotation other vector with i t s represented permanent a n have motion unstable tion ^ v asymptotically either the yz plane, periodic l o f <8 s p i r a l s tending k 2 i s stable. represented approaches magnitude F(n7T 1 <f ' of e l l i p t i c 2 Since ?2 I-0 of the i d e n t i t y from " 2 (2.32) from T a 2 t o assume possibility axis assumes rota- the p e r i o d i c motion of w returning to i t s 36 original p o s i t i o n i s not l i k e l y From F i g u r e (2.30), i fp 2.13 > Q again / we to see t h a t , *• • :.- - f ~ occur. , (8 a s s u m e s - zg) 2 i n view of expression asymptotically , i t V W T approaches the negative infinity. In other of the negative axis, tually fixed vicinity Finally, then other sufficiently the x-axis. that i fp of motion i n the v i c i n i t y l a r g e p r o j e c t i o n on in this direction i s unable positive to pull can between t h e above this and evenself- «8 ba.ck t o x - a x i s , as i n t h e u n p e r t u r b e d lies Q lies The c o n s t a n t the x-axis of the positive types i t s modulus t e n d i n g t o i fw i n i t i a l l y with along we n o t e with t o move s p i r a l l y coincides with excitement the words, x-axis i t continues x-axis 1 two case. values, occur. l v > —•• m We the turn now t o t h e c a s e trajectory T^ indicated i n Figure through the center tinues to the right some p o i n t continues S^. A reflection the at f ^ l y i n g lower-half trajectories to the l e f t about plane. of the(p -axis 2 cuts the v e r t i c a l From h e r e 2 the(p -axis 2 gives T^ con- o r t h o g o n a l l y at trajectory their mentioned. i n Figure through counterparts in Thus, t h e g l o b a l p i c t u r e o f the phase-plane we n o t e appeared p e r i o d i c motions now and i n t e r s e c t s the v e r t i c a l i s t h e one g i v e n a n d F^, w h i c h 2.14 The o t h e r 2 As p r e v i o u s l y below f » cross and S » In p a r t i c u l a r F and must between 2.14 v-. i n Figure that 2.14* the absence i n Figure of the second 2.10, kind. of the separatrices implies the Further, non-existence i f we consider 37 2 -vp, appearing some k i n d Figures of disturbances, axis, we c o n c l u d e 2.10 a n d 2.14 w i t h essentially bance, o n t h e r i g h t o f ( 2 . 2 5 ) a s a moment change the spin tends Figure the nature vector, from 2.2 t h a t of motion. instead due t o a comparison o f this moment c a n For a large of tending now t o t h e n e g a t i v e - h a l f , term distur- t o t h e p o s i t i v e x- a complete reversal of thepoints f , and f di rect i o n • m In the limiting case l v = — , 2 in 2/1-m, Figure 2.12 c o i n c i d e . trajectories separatrix T, a n d shown (2.36) p 2 values the 2 (2.36), the planes above Furthermore, vicinity initial oscillates means t h a t = v^"V we o b s e r v e that that large direction; ultimately 16(1 - m2) «5 t e n d s to a large as they perturbation will o f »8 o n oscillate - by t r a j e c t o r i e s G, lying t o the negative i f t» l i e s component p ^ , i t w i l l that between t h e the end-point initially o f e i t h e r t h e p o s i t i v e o r t h e'negative sufficiently of the t h e.separatrix, i n view of represented t h e s e p a r a t r i x , we f i n d axis. the equation with + o^)] cylinder will x = 0 and x f o rmotions 2 p, eventually This of the e l l i p t i c However, below G lying 2 C, a s s o c i a t e d 2.15 c a n be w r i t t e n as v 0 and ^ 1 6(1-m ) • between Thus, = 2 / l - m [ l - sin(<p (2.29) and surface the constant i s now z e r o . i n Figure On t r a j e c t o r i e s formulas Moreover, remain x-axis forever x- i n the with i n the i s , t h e p o s i t i v e moment o r moment d u e cannot reverse i nsituations illustrated the direction of rotation, i n Figures 2.10 a n d 2.14» 38 Figure 2.14 Figure 2.15 3 9 Note also that the motion represented by the s e p a r a t r i x i s not a p e r i o d i c motion of the second k i n d , a saddle p o i n t i s infinite. l i m i t i n g case any s l i g h t tially (ii) different since the time of approach to F i n a l l y } we may mention that in this d i s t u r b a n c e of the system gives an essen- t o p o l o g i c a l p i c t u r e of the phase p l a n e . -1 < m, < 0 As before we f i n d that the s i n g u l a r p o i n t s occur along the (p2~axis at p o i n t s where (2.37) <p = a r c s i n ( - m ) . 1 2 In the same way, i f conclude that 0 denotes the p r i n c i p a l the s i n g u l a r p o i n t s (p = -© + (2k + l)ir 2 at c p are saddle p o i n t s 2 value of 2.16. This f o l l o w s from the f a c t the t r a j e c t o r y T^ i n Figure 2.16 C 2 and centers intersects We see 2 do not d i f f e r e s s e n t i a l l y motions of 35 f o r these two cases are of the At t h i s p o i n t , we may mention that B l - <5£ [m - v& ./bc"(b • 2 shown i n Figure non-positive the v e r t i c a l through that the phase-plane from those of F i g u r e which are given f o r -1 < m, < 0 and |JL, = 0. (2.38) respectively. that since m, i s at a p o i n t below the one f o r T » trajectories and = © + 2k7T The t o p o l o g i c a l p i c t u r e of the phase plane i s ( 2 . 3 7 ) , we 2.4, T h e r e f o r e , the same types. because c)] 2 m, can become negative if 6 m r e p r e s e n t i n g the constant is sufficiently seIf-excitement large, is even though a very large p o s i - 2 tive number. 1 Since 6 depends s o l e l y on the values Q and r , we uo see of immediately how these i n i t i a l values can a f f e c t the motions Wm |m | = 1 (iii) 1 After to see the above d e t a i l e d d i s c u s s i o n s , it i s not d i f f i c u l t the t o p o l o g i c a l s t r u c t u r e s of the phase planes corresponding motions of ni f o r other p o s s i b l e For m^ equal to 1 and - 1 , the phase planes 2.17 and 2.18 r e s p e c t i v e l y . and the values of m^. are shown i n F i g u r e s In each Figure we f i n d that p a i r of saddle p o i n t and center in the p r e v i o u s cases every coalesces i n t o one s i n g u l a r p o i n t C S . We note first kind. i n p a r t i c u l a r the absence of p e r i o d i c motion of For m^ = 1, we s t i l l have the s t a b l e p e r i o d i c motions of the second k i n d . of m^ should be c o n s i d e r e d c r i t i c a l d i s t u r b a n c e of our dynamical system w i l l pictures, (iv) and consequently w i l l and unstable However, these two i n the the sense that values any s l i g h t change the t o p o l o g i c a l a l t e r the types of motion. |m | > 1 1 In t h i s case since there no c l o s e d t r a j e c t o r i e s tend to i n f i n i t y . are no s i n g u l a r p o i n t s , i n the phase p l a n e . exist Every t r a j e c t o r y must In Figure 2.19 we observe that f o r m^ < -1 no p e r i o d i c motion can e x i s t , i r r e s p e c t i v e of the i n i t i a l and u5 tends to the negative Q second kind represented by the s e p a r a t r i c e s F and F^ in F i g u r e 2.20 are found. its x-axis, value P . For m^ > 1, p e r i o d i c motions of the a value such that there c o r r e s p o n d i n g p^-value l i e s If P q is above the s e p a r a t r i x F.. , (5 assumes or tends to assume the p e r i o d i c motion p., For m, = 0, » -v < 0, -1 < ra, < 0 t h e s a d d l e p o i n t s and t h e c e n t e r s o c c u r odd m u l t i p l e s of W r e s p e c t i v e l y Figure 2.16 at even and 42 ^i, = -V < 0, m, F i g u r e 2.20 > 1 43 represented that these the by p e r i o d i c motions Jacobian It the case i s also |m^| larger r 2.6 elliptic f o l l o w s from that and I f b e l o w , 08 t e n d s F. permanent rotation For 1 by We Summary have thus m" a c t i n g a l o n g face of locus the on tially on on basis the which way we That the of the ultimately assume that the M + H- (cf (2.5) (2.8). C and p by p i r i n terms 2.19, f o r the (2.38) whose we of 2.20 perturbed can value qualitative > a and make depends on q Q and of scalar forms, u-^. treating of solution 3, fixed (2.19). Its essen- of (2.10), whose along sur- obtained system p. the (2.8). of to They were problems problems the depending m u l t i p l e of when equal «5 m o v e d o n equation the torque a component trajectories a variety of by d i s c u s s i o n of a self-excited having various o f m^ a number o f the by defined that component 2 to to Figures x-axis end-point phase-plane in brief are the values also covered i s , there duced x-axis could take relative mention have the cylinder surface is satisfied We a complete showed t h a t elliptic the and m a k i n g d, s p i n v e c t o r ro g e n e r a t e d We 0 Note Remarks given vector M- lro| • the expressed x-axis. cn. in expression merely and of + and negative small, motion M the along the be (2.38) expression , sufficiently o' Brief always f u n c t i o n s sn stable. than can to problem of in similar this nature. s o l u t i o n s are For example, i f the x-axis, is rewe equal 2 + r ), i t is easily satisfies Consequently an the seen equation motion of that which w q can in this and be r are reduced case given by i n form is clear from to 44 our previous sidered the discussions. problem In i n which a recent the paper, torque Grammel [13] acted along vector conthe 2 x-axis with stants. the It i s clear preceding lated curves, by based us have p, q, cription the he on (2.5) and r time t. and ( 2 . 1 6 ) we motion of the of body can be reduced express the Euler space, i n terms to angles, of i n v e r s i o n of formula for any t. value Because vector lies solutions of of the along can be the the motion. Thus However, i f m" a c t s are such q, new be and determining that of determine from the preceding quantities that i s presented a method w i l l not give us this the ones in this to body by [2]. With these angles torque the simply intera particular axis, i n the in functions z-axis, solutions. problem the this c h a r a c t e r i z e such principal the i f the that of of f axis, des- the integrals we for requires i n terms Euler's equations, consider complete position then drawn basis. i t is possible the can calcu- formulas i n Bodewadt s paper middle will the in Thus f u n c t i o n s o f cp, as as were (2.25). i n space contain ( 2 . 1 7 ) we problem that explicit expressed r as the We body quadratures, of of Nevertheless i n space here a more g e n e r a l obtained along involved. section• appropriate no on con- numerically conclusions smallest principal obtained changing axis, form of trajectories rigid positive same s i t u a t i o n the is indicated the \i a r e a series have expressions a method and results the p, M the some o f phase-plane However, h a v i n g Such have f u n c t i o n s of ^• problem o f <p . we in putting his the position where B a s e d on obtained the as of that example. succeeded In M - up component i . e . the New case. y- functions following 45 3 SECTION TORQUE 3•1 Equations If with the VECTOR F I X E D ALONG THE MIDDLE P R I N C I P A L A X I S of Motion the torque component and T h e i r I n t e g r a t i o n vector acts along the middle principal t o M + M< l«i| > E u l e r ' s e q u a t i o n s equal 0 then axis have form - aqr = 0 p (3.1) q + b r p = m + u-|m| 2 f - cpq = 0 M ^o ^ w h e r e now m = — , u. = — . Again cp b y m e a n s o f t h e d i f f e r e n t i a l introducing a variable expression (3-2) d<p - q d t we o b t a i n from the f i r s t and t h i r d d? " a = r dr d^ " °P whose g e n e r a l = &p7a p (3-3) r where solutions m a^fc and values o f (3.1) the «s r = 0 , system 0 „ = 0 c a n be p u t i n t h e f o r m c o s h y a c cp + o^*/* s i n h , / a c cp s i n h , / a c c p + d^Jc are arbitrary cosh^/ac cp constants determined by t h e i n i t i a l p , r , a n d CP at t = 0. *o o To When b o t h p equations and and t h e second are zero, equation equations (3*3) o f (3»l) g i v e s give 46 (3.4) q In such a case the middle p r i n c i p a l they both s a t i s f y Henceforth, different values p the Q r i g i d body r o t a t e s axis. permanently about we s h a l l assume that Q since an i d e n t i c a l e q u a t i o n . from z e r o ; e q u i v a l e n t l y and r the ( 3 » 4 ) can be obtained The s o l u t i o n of ( 2 . 4 ) i n S e c t i o n 2 by r e p l a c i n g p with q, from formulas is =m - \iq is non-zero. at l e a s t one of at l e a s t one of the a's the initial The component q must then satisfy equation q = m - (3-5) brp + u. (p identities sinh ( /acf) cosh 2./ac <P - 1 2 c o s h (Jac <p) cosh 2 /ac <P + 1 2 2 A 2 f ( 3 . 5 ) i n the we may w r i t e (3.6) + r ) given by ( 3 « 3 ) i n ( 3 « 5 ) and making S u b s t i t u t i n g the expressions use of the + q q- |iq = m form + k . s i n h 2,/ac cp + k c o s h 2,/ac <p 2 where m (3.7) o = m * 2^ a k, - l i a ^ C a k 2 " a i " 2^ a * c) - = ^(a + c ) ( a S i n c e d<p = q d t , ^ c 2 (2 a • a ) 2 + a 2 } - bo^a^/lc" from ( 3 . 6 ) we have f o r <p the equation 47 2 (3.8) To integrate we first = (3»8) (3.8) values 1. 2,/ac <p 2 ' equation purpose, relative * kjBinh^/accp + k c o s h m D \ and t o f a c i l i t a t e later t h e dependent we c o n s i d e r t h r e e and independent cases, 1 d e p e n d i n g on t h e o f k^ a n d k « 2 assumption, we h a v e 2 /accp+ k c o s h k 2,/accp = e ^ ^ i n h (2,/ac (f) + a ) 1 2 where = M - 4 k 2 cr, = a r c t a n h - — 1 k x £^ Equation = s g n k^ ( 3 « & ) assumes then ^ (3.9) - ^(dT) 2 = % t h e form + £ d sinh(2y*a^f + 1 1 Let A[ 2o\/ac Then ( 3 . 9 ) becomes (3.10) - J --1M^2 dT 27^ 2 Further variables. g k sinh i order I k J > |k | Under t h i s d discussions, t o an e q u i v a l e n t s y s t e m o f f i r s t by c h a n g i n g this Case - reduce equations For 2 — | dt putting - • V i„h ? 2 a) ± 48 (3.11) 2yac ra = o ra, d( ?2 dT = i q we o b t a i n t h e f o l l o w i n g lent d ' l q equiva- = m, > e l S inhcp 2 * 2 ,a { 1 1 2 Obviously we l = q have <P = (3.13) _ q (3.12) d q (3.14) ( ? " a,) 2 r-dj = j _ Q i the first l -5— equation order "1 - 2M- q 1 equation 2 i n q, follows. 1 - 2m, + 2e inh<p l S 2 c a n be i n t e g r a t e d t o o b t a i n i c the constant *?20 a t t = 0 . linear 2 2\i^ (3.15) where equations d dr~ This order <P 1 2 ) From of f i r s t t o (3-10) — ( 3 system e 2C • |(2^ sinhCp 1 c, i s determined In the i n t e g r a t i o n Solution contrary to this limiting case of (3.15). we assumption 2 + cosh(p ) 2 by t h e i n i t i a l assume m ^ values that c a n be c o n s i d e r e d °;,Q# £ as a 49 Case 2. In jk | < |k j 1 this k^inh 2 case, we h a v e 2Jacd{+ kgcosh y before, w i t h cp '~2 2~ k ~k^, 2 2Jac(p = £ d c o s h (2jTccp + 2 2 l = arctanh — a) 2 k a 2 = 2,/acCp + a , and e and t = 2 1T 2 = sgn k « As 2 e q u a t i o n (3.8) 1 V 2d yac 2 takes t h e form d Cp / fo\ 2 Corresponding d d l q ( 3 ' 1 7 d<p ) (3»12) t o equations now t h e f o l l o w i n g — ™ 2 system = m equivalent t o + e coshcp 2 i nthe preceding 2 2 case, we h a v e (3.16): 2 ^ * 2 dT" = q i m H e r e u., = ^ , 2,/ac = 2,yac d d ^ 1 2 2 , M o r e o v e r , we h a v e 2 2 2/ac 2 Equations (3.IS) from where ^ dq- which (3.19) (3.17) '2 lead - 2ii q x 2 to the first » 2m 2 order + 2e coshcp 2 linear equation 2 we o b t a i n 2u.,cp qf= c e c^ i s t h econstant 2e • m ? ^ r ( 2 l i coshCO of integration. + sinhcp ) 2 ~ i n q^ 50 Case 3. Jk j = | kg | * 0 x Finally k l B inh we may write 2/acCp • k g C o s h 2yac(p= ( s g n k,) | k , | e ^ 2 = -(sgn (3«&) Equation then becomes 4;. (3.20) „ . k )|k |e" '/ 2 1 8 *P 0 a c 1 'P i fk , = k i f k , = -kg respectively . o ( s g n k l )| k l |.v«q> and - n(f) (3.21) If we l e t C P 2 - m = 2,/ac@, t = 0 , V (3.21) - o 2 ( k^lkjI.-V"? n ===== , e q u a t i o n s i i ^ T i s g k (3.20) and become and (3.23) Further ^ dT putting respectively u., = d q 2 ) d T / U ^ the following t o (3-22) equivalent (3.24) - - ^ f — 2yac^ 2 l m^ ^ IT = q i = l A - ( B P '"1 o = -J-J;—jm m^ systems d< of f i r s t and (3.23). ?2 + (sgn k,)e + k j e " ^ ?2 , -p— = q, ^ i 2 q i f order 2 we have equations 51 d l q = -£~T (3.25) d T In each 2 = , -?2 (sgn k ) e + - = -" - . q = 2yac d the - 2u.,q In In q both 2 - C l equations 0^ from * m ^ -- ^ (3«25) - ^ c ^ i s an a r b i t r a r y c o n s t a n t . express Let this formulas (3.15), ( 3 . 1 9 ) , (3*27) q as a f u n c t i o n multiple be d e n o t e d o f Cp, s i n c e of q^ a n d c p 2 symbolically or i n each i s a linear function by q = t V(cp) the inversion (3.30) that i s . equation 2 2 • ( s g n k,) from q i s a scalar obtained order yields • ( s g n k,) f e (3-29) Then • (sgn k ) 2 e ^ < x 2 ac the f i r s t ^1 P9 - c e ( 3 . 2 8 ) we may o f cp . = 2m of which Accordingly, case we o b t a i n t h e same w a y we h a v e (3.28) l 3 0 4 2 q, > 11 2 t2 integration (3.27) 2 k. ^ / V «/ * (3.24) system ^1 (3.26) q i q case, From ^ l 1 of the i n t e g r a l t - , from -flfa (3.2) a l l o w s us t o e x p r e s s cp a s a f u n c t i o n of t ; 52 (3.3D <p-<p(t) This e x p r e s s i o n is then the s o l u t i o n of in formulas ( 3 . 3 ) and ( 3 - 2 9 ) # of t, 3.2 -Putting we o b t a i n p , q and r as (3«3l) functions A Q u a l i t a t i v e D i s c u s s i o n of the Motion of the Spin Vector 3 « 2 to 3 « 4 we s h a l l give In S e c t i o n s fixed t r i h e d r a l , a qualitative the end-point of 3> with respect s i o n of motions of that (3.8). u s i n g the phase-plane method. these motions are e s s e n t i a l l y the torque v e c t o r acts along the different largest discus- to the body- It w i l l be shown from the ones when or s m a l l e s t principal axi s. In the first p l a c e we note from equations ( 3 . 3 ) that (3.32) or (3.33) This means that the p r o j e c t i o n of the end-point of t» onto xz plane always l i e s on one of the (3.34) ^ - ^ - a l a and (3.35) c a conjugate h y p e r b o l a s : 2 if |ot 1 > | a | if laj <|a | 1 2 2 a* 2 the 53 These h y p e r b o l a s a r e i l l u s t r a t e d which 08 i s the orthogonal projection X z motion its end-point cylinders f always defined l i l a = lies l ^!' 0 on t h e s u r f a c e (3.34) by must 3«1 i n the xz plane. be s u c h that o f one o f t h e h y p e r b o l i c (3-35). and equation i n Figure o f (?) o n t o o f 35 i n t h e b o d y - f i x e d t r i h e d r a l The I graphically (3.32) degenerates into the equations (3.36) • = 0 and (3-37) The -f- - ~ = 0 conjugate hyperbolas degenerate Figure into the straight l i n e s (see 3.2) (3-38) z = x * - s * and (3.39) In such a case, the vector w will always l i e a l o n g one o f xz these straight a way t h a t lines, while the spin i t s end-point w i l l planes, say I and I I , d e f i n e d Clearly these planes hodes of a force formulas (3«3) free remain vector t8 w i l l forever respectively by move i n such i n one o f t h e two (3.38) and (3«39)« a r e t h e two p l a n e s o f t h e s e p a r a t i n g p o l asymmetric c a n now be w r i t t e n gyroscope. as Furthermore, because 54 Figure 3.2 55 e P =- a y * e (3.40) i ^ c i fa = a. 1 and , (3.41) p V = a - jac cp * e r = -a /cTe~J T, ifa ac 1( the spin of l tt v e c t o r i» w i l l in either = if a a 2' a n n e according to the algebraic the upper o r the l o w e r - h a l f of plane * t h e r the upper sign I i f or the lower-half of plane I I 2 locus of the end-point take a variety consider first of forms, the r i g i d fixed along the middle locus i s a closed also o f t» on t h e s e determined the unperturbed where is 2 - -a . ± The can * d remain =-a x a closed body U- case surfaces or planes by e q u a t i o n = 0, Q that (3«8)» i s ,the We case i s subject to a constant self-excitement principal curve axis. We on t h e c y l i n d e r s , curve on t h e two p l a n e s shall show that and w i t h one the exception of the s e p a r a t i n g p o l - hodes. 3 »3 The In Motion this of the Spin Vector case, expressions m i n the Unperturbed (3»7) for m, Q l k If tion form ^ (3»8) l 0 ^ ! * then can then 2 ( 2 ^ 1 = 2s a 2 become 2 = -bJTc o. a x |k^| > be and k = m o 2 k K k^, Case | k 1, reduced 2 to J 2 a n d s g n k^ (3«12), i s negative. which now assumes Equathe 56 dq 1 — ( 3'42) The behaviors of q and cp w i l l those of q , andcpg expressions a r determined, since e S i n c e equations an energy \ system they are r e l a t e d by (3«42) occurs on the c p g - a x i s a r c s i n h m, = represent (3.42) a c o n s e r v a t i v e dynamical integral q\ Here E i s * ( the - m f?2 c o s h c + r a c 1 p * 2 the p o t e n t i a l energy of the c$ ?2^ = energy c o n s t a n t , v = is as where Cp^ exists. be known as soon (3»13)« s i n g u l a r p o i n t of at the p o i n t system, sinhcp D qualitative The = m, - = 2 " r a l c o s h s i n h c and the expression °f 2 system. + E Since P 2 2 d V U dcp = C ° 8 h C P2 2 the p o t e n t i a l energy curve i s is thus a center with a l l other t r a j e c t o r i e s surrounding i t , as shown in Figure Consequently, of concave upward. the locus of the the c o r r e s p o n d i n g c y l i n d e r i s i n the phase plane 3«3« end-point of 35 on the always words, 35 assumes a p e r i o d i c motion. The s i n g u l a r p o i n t a closed curve. The s i n g u l a r p o i n t surface In o t h e r represents m, < 0 ra, = 0 Figure ra, > 0 3.3 a d i r e c t i o n of permanent r o t a t i o n i n each c y l i n d e r . more, this Figure also indicates along the middle p r i n c i p a l free is the permanent r o t a t i o n unstable as in the force- case. If the axis that Further- |a,| = I 2 I ' A then w moves i n one of the two planes of separating polhodes, and equation dq, -e dT (3.43) d ^ dT ( 3 . 8 ) can be reduced to = q. if a, - a if a, - - a 2 or dq. dr" (3.44) = m. • e d ^ dT The s i n g u l a r p o i n t s of the 2 above systems occur r e s p e c t i v e l y along the Cp--axis at p o i n t s where 58 (3.45) m 3 - e * = 0 m 3 • e or (3.46) = 0 2 These two systems have r e s p e c t i v e l y (3.47) 2 q l ~ 3$2 * m e 2 the energy = integrals E or 1 2 2 l (3.48) " q m „ . 3^2 * ~^2 e E In F i g u r e s 3 » 4 and 3 * 5 are given the phase-plane = 2" for = to the c o r r e s p o n d i n g p o t e n t i a l energy curves or to (3.45) to In and trajectories subcases Naturally, ding to (d) y-axis, and (e) reference equations of each Figure are given the (e), (3»4C) phase- when the systems have no s i n g u l a r p o i n t s . both every t r a j e c t o r y tends | q | and |<f | 1 | q | increases thus tends [u5j -» <*>„ These can be e s t a b l i s h e d with i n such s i t u a t i o n s , subcases quently, a (3.48). plane t r a j e c t o r i e s In _ t end to i n f i n i t y . 2 indefinitely, and ( 3 . 4 I ) , and at the On the other hand, or the p o s i t i v e i n subcases Conse- same time, P and r approach z e r o . toward the negative to i n f i n i t y . accor- The spin v e c t o r y-axis with (d) o5 also tends to the but with a f i n i t e modulus, since now the t r a j e c t o r i e s these F i g u r e s approach a s y m p t o t i c a l l y h o r i z o n t a l l i n e s phase p l a n e . In the i n the l a t t e r two subcases the motions of «$ are also well-known from the Poinsot ellipsoid. From F i g u r e s 3 " 4 and 3 . 5 we see that ^ i n g e n e r a l assumes p e r i o d i c motions i n the planes of the s e p a r a t i n g polhodes, when the (3«43) systems in and ( 3 » 4 4 ) have no s i n g u l a r p o i n t s . except 59 (d) m 0 = 0 force-free Figure (e) 3.4 m a < 0 = a 6 0 3.4 The Motion of the S p i n Vector i n the Perturbed Case Without any l o s s of g e n e r a l i t y , we assume that [i i s tive, i.e. H = -|p.J. additional nega- Furthermore, we s h a l l f i r s t make the assumption that |a,j ^ | | . T h i s i m p l i e s that end-point of the s p i n v e c t o r moves on the surface of one of the the conjugate h y p e r b o l i c c y l i n d e r s . Case 1. |a,| ^ |a | 2 The constants k k , and kg become now 1 = -^i(a 2 - c ) 2 a + (3-49) k • c)(a 2 1 a 2 - ^ ( a a 2 + 2 versus „ • a ) - ^ S £ 2a u 1 2 the graphs of k , and j |JU | f o r a , and CL^ h a v i n g the same and opposite It f o l l o w s (3.50) ) 2 In F i g u r e s 3*6 and 3*7 we give r e s p e c t i v e l y k 2 from the k, - k 2 signs. difference = | [ In-1 (a • c) - by'a^J(a 1 that i n each Figure the two s t r a i g h t lines a,,) 2 intersect at the point l y i n g below the together with (3.51) we see and i f if > Moreover, from these two Figues (3«50) and the sum k, • k that |ti.|-axis. 2 = - ^[|n|(a < [i.^, then then + c) 1 |k,| > |k 1 |k j > |k,| 2 + bjkll(a 2 * a ) 2 2 and sgn k , negative, and sgn kg n e g a t i v e . 61 sgn a , sgn a , = -sgn = sgn F i g u r e 3.6 Let Figure 3.7 us suppose which governs the first that |u,| < u,^. Equation ( 3 . 8 ) , behavior of q and op, can be reduced to (3-12), which now assumes the form dq. dr (3.52) d < P 1 2 dT = m _ ~ where JA, = - V = ~ 1M-1 l " q . s i n h ( P " 2 V l q l j n addition, the constant 2,/ac (3.53) is always l e s s than 1/2. For i f r , and A the ratios definitions of moments of of the inertia denote A and ^ , then a c c o r d i n g to i n e r t i a numbers a, b, b 2(a+c) 2 1 l 2 r (r +l)-r and c we have r 1 respectively 2 2 " the 62 Because of the r 2 assumption A > B > C, i t = r^ + h , h > 0, 2 1 2 (a+c) r 2 l 2 , , -n r +h(r -i; A which can be w r i t t e n where The ^2 arctanh * (-2v) assume the Making the the equations + a ~ 2 at l i n e a r transformation x of f i r s t approximation at the p o i n t form d? (3.55) = "? c o s h <P 2e ^ 9 dT c h a r a c t e r i s t i c equation A. S i n c e the t r a j e c t o r i e s singular point is 2 e = 0 c h a r a c t e r i s t i c roots A. = + ^ -cosh Cp are symmetric about the c p ~ a x i s , 2 2 . e the a center. other t r a j e c t o r i e s surrounding t h i s is + cosh c p 2 from which we o b t a i n the All * J system occurs on the cp ~axis do The (3*15) given i n equation D n = q we f i n d that is * . cosh(f s i n g u l a r p o i n t of the = a r c s i n h m^« 2 as qf = c e = 1 2 1 s o l u t i o n q^ i n terms of Cp (3-54) > 1, that and 1 The follows point i n the phase plane are c l o s e d as shown in F i g u r e 3 " 8 . curves For i n s t a n c e , in F i g u r e 3 . 8 ( a ) i f we f o l l o w the t r a j e c t o r y through the p o i n t f , it will continue to the r i g h t as T i n c r e a s e s and must intersect 63 at f^ the v e r t i c a l through the s i n g u l a r p o i n t . cannot cross the ( p ~ a x i s between the p o i n t s f 2 - dq of the v e r t i c a l , y to the l e f t of q^. From f^ i t Q This t r a j e c t o r y and f^, because x > 0 for s u f f i c i e n t l y small values continues to the r i g h t and cuts the d o r t h o g o n a l l y at g^, ¥ ~axis 2 l q as to the r i g h t of the v e r t i c a l , is always l e s s than z e r o . If T, it we now f o l l o w the same t r a j e c t o r y from f can e i t h e r cut the cp^-axis at g,>, f o r e v e r to the l e f t v < the l a t t e r without 2 side can assume only non-negative large negative S i n c e the t r a j e c t o r i e s about t h i s if is (3«54) v a l u e , while the sufficiently values. are symmetric about the <P -axis, 2 small. we s h a l l see that p e r i o d i c |n| > motions can e x i s t only f o r s m a l l values of q . o | kg | > | k j j , equation ( 3 » & ) can be reduced to d q a that p e r i o d i c motions of UD are p r e s e r v e d On the other hand i f —l left axis gives then the c l o s e d t r a j e c t o r y Thus we see Q However, i f cannot happen, f o r then the r i g h t side of assume an a r b i t r a r i l y through f . as shown, or continue c r o s s i n g the cp ~axis. will reflection for decreasing = m - cosh f 2 2 Since now - v q2 x (3.56) dcp 2 dr~ = q l The ..solution q^ i n terms of (p u.^ r e p l a c e d by - v and £ The 2 i s given by formula = sgn k s i n g u l a r p o i n t s of p o i n t s where 2 2 (3«56) (3«19) with negative. occur along the f ~ a x i s at 2 64 cosh cp g = 0 . > 1 there For the o r i g i n are two such p o i n t s , and e q u i d i s t a n t mined from the equations - ; = i dT points, l " P i 2 ' and £ = <P - c p i ' 2 (3.57) ^ ®' the saddle n = 1 , 2) denotes the values of cp2 (i equation of T h e i r nature can be d e t e r - of f i r s t approximation " where c f ^ from i t . ^ = q^. 2 singular point i s singular From the c h a r a c t e r i s t i c Hence at and at c p ^ < 0 a a center, phase-plane t r a j e c t o r i e s 2 from the are shown i n Figure 3 « 9 » above r e s u l t s On the t r a j e c t o r i e s G^ and G2, q^ -• h y p e r b o l i c sine | r | , and | q | -• 0 0 . v e c t o r i n the xz p l a n e . 0 0 a s <p 2 Because ~*~ a,B functions On the other hand, (3«19)' and equation i n F i g u r e 3 « 9 that o r i g i n a t e d on the t r a j e c t o r i e s since ± and h y p e r b o l i c cosine any of the t r a j e c t o r i e s stable, the t we f i n d that A. = ± y - s i n h c p , ^ . These are e s t a b l i s h e d |p|, a point. The on one on each side of the are unbounded, tend to i n f i n i t y , i n any motion that G^ and F , a> tends to a f i n i t e However, such motions are c l e a r l y un- under s l i g h t d i s t u r b a n c e , eo w i l l assume e i t h e r a p e r i o d i c motion about a permanent axis of r o t a t i o n or an asympt o t i c motion, say of the t h i r d k i n d , and t» i n which | p | , |<j|# tends to c o i n c i d e with one of the two asymptotes | r | •* <», of conjugate h y p e r b o l a s . In F i g u r e s 3 . 1 0 and 3 . 1 1 are given the phase planes = 1 and ra^ < 1 . (See page 6 8 . ) for the 65 m 2 > 1 Figure 3.9 66 In into t h e c a s e ra^ = 1 t h e s a d d l e p o i n t a single motions point only and t h i r d (3«56) f o r system In asymptotic motion k, and sgn kg both Figures (3.58), to 3.12 these Case kinds. If m of the t h i r d = \i, , negative. depending The p o s s i b l e vector singular m c a n now kind. we h a v e |k,| = Since equation w h e t h e r m^ |kg | and (3«8) trajectories motions asymptotic < 1, n o 2 The s p i n the phase-plane a n d 3.13 non-positive. 35 c a n a s s u m e o n l y can o c c u r . t h e l i m i t i n g case be r e d u c e d or at t h e o r i g i n . of t h e second assume sgn point and t h e c e n t e r c o a l e s c e c a n now a r e shown i n i s greater o f at f o l l o w than easily zero from Figures. \<x 2. (0..J » Let us assume \ now t h a t the spin v e c t o r 35 l i e s one o f t h e two p l a n e s o f t h e s e p a r a t i n g P This on means ra~ o = ±1— that — polhodes. initially and a c c o r d i n g l y a, = + ag. Furthermore, as n o t e d o in Section 3«2, Equation we replace defined t» w i l l (3.8) [i, b y - V , always c a n now —d q±. ^2 2 = -2 im + ( s g n k..)e - V-1 q. -<P 2 (3.59)" in be r e d u c e d the trajectories or two i n one o f t h e two p l a n e s . to (3.24) i n t h e phase or (3.25). plane If are then by t h e e q u a t i o n (3-58) We remain shall dq. ^± m -(sgn k.)e - -2 see that planes I and I I w i l l be a 2 2 -Vq.. i the phase-plane equations are e s s e n t i a l l y if a, - i f a, - - a trajectories different. different. Thus 2 defined by t h e s e t h e b e h a v i o r o f 35 67 First plane I . let A c c o r d i n g to (3.58) can be w r i t t e n q (3.60) Further, 2 - Then tS always moves i n us assume that 0.^ = a^. C l e equation (3••7), sgn k^ i s n e g a t i v e . S o l u t i o n s of as -2vf ' - T ^ f e ^ m ^ . (3«58) has a s i n g u l a r p o i n t on the < p 2 ~ i ax 1 p r o v i d e d that m^ > 0. 2e = In s a t m_ 3 L e t t i n g § = cp <Pe _ 2 a n ^ d 2 = q l ' W e o b t a i n at the s i n g u l a r point the equations of f i r s t approximation - do dT ' 2 The corresponding c h a r a c t e r i s t i c equation i s A, gives the roots A, = • y-m^. + m^ = 0, which S i n c e m^ > 0 and the trajectories are symmetric about the cPg-axis as i n d i c a t e d by (3.58), the singular point is a center. The phase-plane trajectories for m^ > 0 and ra^ < 0 are shown in F i g u r e s 3»12 and 3«13 r e s p e c tively. In F i g u r e 3.12 the s e p a r a t r i x F i s that of i n equation initial tion c^ i s (3»60) values zero. a it n d determined by the is possible q 10 s o t h at t to s e l e c t n e constant a suitable of 2 e m 2 which shows that q^ -• ± I — as cp^ "* ~ ' 00 set integra- Thus the equation of the s e p a r a t r i x ^ _2_ *2 l " v " 1+2V q fact becomes 68 Figure m 3 > Figure 3.10 0 3.12 Figure t n> 3 3.11 < Figure 0 3.13 69 In the unperturbed case n = 0, force-free axis w i l l p u l l the s p i n v e c t o r back to the p o s i t i v e - h a l f large, arbitrarily m approaches the negative y - a x i s large p o s i t i v e moment. motions only f o r s u f f i c i e n t l y the s e l f - e x c i t e m e n t y-axis without M is It any o b s t r u c t i o n . Q is fo'r any can assume p e r i o d i c s m a l l values of q . Obviously i f 3) w i l l approach the negative, and In the p e r t u r b e d case As seen from Figure 3 « 1 2 , i f q no longer t r u e . sufficiently coincide moment M along the middle p r i n c i p a l w to perform a p e r i o d i c motion. is to as observed i n F i g u r e 3«4# However, any a r b i t r a r i l y s m a l l p o s i t i v e this spin vector uo of a asymmetric gyroscope has the tendency with the negative y - a x i s . forces the This f a c t is negative exhibited in Figure 3 . 1 3 If a, = ~ 2' a [i =-2yacV, (3-61) If of 2 1 V < b 1 * ^ Y ' ) ^ ~2~' then sgn k , i s a c can then be w r i t t e n q 2 p r o v i d e d that first Since the equation determining k , becomes now Equation of II. k , = jk~c~ a [ 2 v ( a + c) - b] = -kg (3.59) (3.62) then 35 w i l l move i n the plane = C l 2 ___|_ e e As b e f o r e , approximation at the roots X = + yra^ • Thus the The s o l u t i o n as 2 + _ J ( 3 . 5 9 ) has a s i n g u l a r point m^ < 0. negative. on the<pg-axis we o b t a i n f o r the singular point s i n g u l a r point is the at equations characteristic a center. From 70 this fact tories and equation (3»62) i t follows that a l l other in F i g u r e 3*14 are c l o s e d curves surrounding the trajecsingular point o If for we compare t h i s the plane unperturbed case, we see that p e r i o d i c motions of 35 in values it is is reasonable |(x| t h i s of to expect that small. in general w i l l a n d - (3-63) That trajectories in increase indefinitely, but value , us assume now that V > 2(a+c) be p r e s e r v e d . However, because of the negative damping term - v q ^ the magnitude of (?) cannot approaches the positive f o r m^ > 0 and small shown by the phase-plane F i g u r e 3»15 f o r m^ > 0. Equation sufficiently tends to c o i n c i d e with the tendency a c t u a l l y so i s Let is in the unperturbed case the spin v e c t o r of a f o r c e - asymmetric gyroscope y-axis, this corresponding Figure 3.5 II are p r e s e r v e d p r o v i d e d that Since free Figure with the s q C l n e k is l P° iti e. s + large so The s o l u t i o n q^ i s v -2vq> 2 _|_ sufficiently that given by -? m 2 _J ^ + e (3.59) has a s i n g u l a r point on the 9 ~ a x i s at 2 <Pde - = ln<i-) m^ if m^ > 0. The c h a r a c t e r i s t i c roots of the equations approximation at the the s i n g u l a r p o i n t phase plane The of are A. = + ^m^. a saddle p o i n t . are shown in F i g u r e 3«l6. s e p a r a t r i x subdivides motion. totically is above point of first Consequently, The t r a j e c t o r i e s in the (See page 73«) the phase plane i n t o four regions In the t h i r d and the f o u r t h r e g i o n , 3 tends asympto the negative y - a x i s with modulus t e n d i n g to infinity. F i g u r e 3.15 72 In the remaining two r e g i o n s , 35 tends to the p o s i t i v e with a f i n i t e y-axis magnitude. When m^ < 0 , no s i n g u l a r point occurs i n the phase p l a n e . The constant negative non-positive self-excitement damping term completely overcomes any tendency of 35 to c o i n c i d e with the p o s i t i v e y - a x i s , initial negative coupled with the p o s i t i o n is i n plane I I , and 35, wherever e v e n t u a l l y c o i n c i d e s with the Figure 3»17 i l l u s t r a t e s y-axis. its the phase-plane t raj e c t o r i e s . In the rare s i t u a t i o n when V = - / • — r , the k, and kg are z e r o . ,2,„ where ||J-| = b | (3-65) , ^ - , a c ^ = & dt = q dt Equation ra , v 2 " to =m or e q u i v a l e n t l y to the _ m o Jrf (3«8) reduces constants system l„l„2 I N q q T h i s system can be i n t e g r a t e d to o b t a i n the equation (3-66) q 2 - c e" l 'l 2 M 1 < P , c, constant d e f i n i n g the phase-plane t r a j e c t o r i e s . These are shown in F i g u r e 3.18. As seen from F i g u r e 3 « 1 8 ( c ) , when m^ > 0, 35 i n general tends to the p o s i t i v e y - a x i s with a f i n i t e magnitude. The m (a+c) exception occurs when the i n i t i a l value q < o— bs/ac 1 73 ra„ > 1 m„ = 1 Figure 3.16 Figure 3.18 0 < rn, < 1 74 the s p i n v e c t o r then tends to an axis l y i n g i n plane II and p e r p e n d i c u l a r to the y - a x i s . For m < 0, Q mentioned axis without any e x c e p t i o n . ra = 0 (See F i g u r e 3 . 1 8 ( h ) ) , Q the p o s i t i v e y - a x i s i f q mentioned axis if q Q it tends to the above In the c r i t i c a l case we f i n d that w tends to zero along is positive, is negative. and to the above Furthermore, in this critical o case the whole <P-axis represents s t a t e s of unstable equilibrium motion. 3»5 General Remarks and Conclusions The above remarks conclude the d i s c u s s i o n of the motion of a self-excited asymmetric r i g i d body subject to a time- dependent torque vector f i x e d along the middle p r i n c i p a l axis 2 with component equal to M + U. |(3| . Q Section, As i n the p r e c e d i n g we have shown the p o s s i b i l i t y of i n t e g r a t i n g E u l e r * s equations i n terms of an a u x i l i a r y v a r i a b l e Cp. If t h i s is f o l l o w e d by a quadrature and an i n v e r s i o n , we are able to express p , q , and r as f u n c t i o n s of the time t . Using the method of phase-plane a n a l y s i s , we were able to i l l u s t r a t e the different types of motion which m c o u l d assume with respect the b o d y - f i x e d t r i h e d r a l . In p a r t i c u l a r , we showed that to the end-point of 35 always had to move e i t h e r on the s u r f a c e of one of the conjugate h y p e r b o l i c c y l i n d e r s or on one of the two planes of the s e p a r a t i n g polhodes of a f o r c e - f r e e asymmetric gyroscope. F u r t h e r i t was shown that m c o u l d assume p e r i o d i c motions and asymptotic motions of the f i r s t , and t h i r d kinds. stable second, 75 Having o b t a i n e d p, q, and r as f u n c t i o n s pute the E u l e r angles as f u n c t i o n s Finally, slight 3, along the we would l i k e to mention the p o s s i b i l i t y if we assume that largest, equal r e s p e c t i v e l y f (P» 9 r ) * obtain p, 2 ( P » in d e t a i l , fore, )q 2 # o r However, and g, (i as b e f o r e , we s h a l l to f , ( q , f-jCp* a) q and r as f u n c t i o n s and i n v e r s i o n s o functions r the torque + r) v e c t o r 18 i s = 1, + <?,(9# of t by means of fixed axis with r ) p , or quadratures g e n e r a l nature of 2 , 3 ) we are not the q u a l i t a t i v e 2 and 2 , then we can <3J(P, because of the of a in Sections or m i d d l e , or s m a l l e s t p r i n c i p a l components 2 of cp» e x t e n s i o n of the problems d i s c u s s e d Namely, of (p, we can com- the able to discuss behavior of i3. There- leave such problems with these remarks. 76 SECTION 4 PERIODIC SOLUTIONS 4.1 General C o n s i d e r a t i o n s In S e c t i o n s 2 and 3 we e s t a b l i s h e d that under suitable conditions the spin v e c t o r m of an asymmetric gyroscope, subject a time-dependent to the p r i n c i p a l ciently axes, torque vector f i x e d along one of can assume p e r i o d i c motions s m a l l neighborhood of r e p r e s e n t e d by a center this Section a stable in a s u f f i - permanent r o t a t i o n , i n the c o r r e s p o n d i n g phase p l a n e . we are concerned with the* c o n s t r u c t i o n of periodic solutions, u s i n g the method of P o i n c a r e . For t h i s us consider f i r s t purpose l e t the In these differential equation (4.1) to which a l l of our l a t e r equations k > 0, p\ ( i = 2, 3, 4 ) , and equation belongs it is Here are a r b i t r a r y c o n s t a n t s . to a g e n e r a l c l a s s of. d i f f e r e n t i a l known [24] that reduce to the t r i v i a l equations phase plane of in a s u f f i c i e n t l y small neighbor- In the hood of the o r i g i n the equation possesses p e r i o d i c which are a n a l y t i c f u n c t i o n s This (0, ~ ) known as Lyapunov-systems [24]« (4«l), can be reduced. of the solutions i n t i a l value r) of solution 0 = ~ = 0 for on more, the p e r i o d of each of these s o l u t i o n s 0=0. ' is also 0, and Further- an a n a l y t i c 21T f u n c t i o n of r> , r e d u c i n g to — when r~) = 0. the c o n s t r u c t i o n of these p e r i o d i c s o l u t i o n s We s h a l l proceed to for »9 £ 0. 77 Since ( 4 « l ) is autonomous, there is no l o s s of generality d0 in assuming that the i n i t i a l value of — the p e r i o d of the p e r i o d i c s o l u t i o n value rj . initial where h , (i determined = 1, 2, • • • ) h rj 3 are constant • h r ) + h rf x + hy) 2 ( 4 « l ) assumes the (4.4) * (1+h^+h 3 Clearly, f, v (l-h 2 the 2 v l V h 2 0 ...) \2l^2 • ...) 0* (^2 2 ?3 0 3 ^4 ) ^lldS) 0 \+ 4 31* ^2 41 0 as a f u n c t i o n of /d0\ s Thus we s h a l l seek the s o l u t i o n 0 s + f) 0 (s) 2 2 +r) 0^(s) 3 are p e r i o d i c f u n c t i o n s d0.(O) 1 values 0^0) = + ... of p e r i o d 2w which have ~ — - = 0, i = 2, 3, S u b s t i t u t i n g the e x p r e s s i o n of 2 is form 0 = f) cos coefficients + 2 the p e r i o d i c s o l u t i o n ( 4 « 4 ) i n the initial make a change of .y -..) * and of p e r i o d 2 7 T . where 0 . ( s ) the to be form ds the coefficients letting T = (4.5) + --.) 3 ) we s h a l l f i r s t then equation of the later. time s c a l e by analytic Denote by T 0 c or r e sponding to h r * 2 2 F o l l o w i n g Malkin [ 2 4 ] * (4.3) zero. We may w r i t e T = |2T(1 + h,0 + (4.2) is ( 4 « 5 ) in l i k e powers of ••• • ( 4 « 4 ) and equating n , we o b t a i n 78 ^ • • 2 For this cient ^5*2-; + - - ^ 2 - ~ l 2s equation to have a s o l u t i o n of p e r i o d 2ir, the of cos s must be z e r o . For V 2 the f u n c t i o n 0^(s) coeffi- We obtain then we have cos s — 24k whence we f i n d h as 50 2 l O T + * 12k 4 2 - - r * - Icos 3 s \ 24k 1 2 before 2 2 .4 5P„ K 2 K o , , 2 ^1 M1 P\ 2 3 , 2 u 6 (4.7) \576k P 576k- 4 2 36k P + 18k 4 ^2 192k r 4 2 - 192k* 1 2 2 2 ^1 " 9 ^1 ) 2 1 V * c o s i 2 0 x ' 8 1 3 ^1 5 H, 5 + -57—) cos 3 s . 32k* 192k * J ^2 P 1 h 79 The constant coefficient of h^ i s o b t a i n e d by equating to zero s i n the equation f o r 0 . ( s ) . cos the This then k yields We can proceed i n the same way as above to o b t a i n h i g h e r order approximations,; however, and c o m p l i c a t e d . the the computations Thus we s h a l l go no f u r t h e r , required periodic solution approximately by formulas 0 and i t s (4»2), (4»3) long and c o n s i d e r that p e r i o d T are an< 3 (4-5) Let us r e t u r n now to the problem d i s c u s s e d 4.2 soon become to expressed (4.8). in Section 2. P e r i o d i c S o l u t i o n s in the Case of the Torque Vector F i x e d Along the Largest P r i n c i p a l Axis It was shown that i n terms of the by the first differential (L 9) (4.9) the variables equation of components of w can be <p or cp^ s i n c e (2.1l). the two are related F u r t h e r , cp ^ s a t i s f i e s the equation ?2\ "^ a2 * - » ! • s i n cp • * l {-^ d < 2 2 dT ' When jm^| < 1, points of it was e s t a b l i s h e d that ( 4 « 9 ) o c c u r r e d on the cp^-axis, singular points were centers which represent periodic solutions is these p e r i o d i c s o l u t i o n s , to expressed construct described in Section 4»1» the singular and some of surrounded by c l o s e d of (4.9). these trajectories Our purpose here u s i n g the P u t t i n g these s o l u t i o n s theory in the 80 e x p r e s s i o n s f o r p, q , and r , defined in Section 2, we s h a l l o b t a i n the r as p e r i o d i c f u n c t i o n s and i n v e r t i n g the the center 2 (4.10) - —| + / l - m d 2 cp = ^ sin 0 = m - 1 Then (4.9) cos 0 then equation (4.10) (4.11) • ^ ^ write B down, from formulas 0. Hence, the a sufficiently cos c p ~ fl f? l . / ^ f £ (4.2), (4»l), (4*3) - ^ g we can 2 and ( 4 « 5 ) to 0 c o r r e s p o n d i n g to the 2c f (4.8), i n i t i a l value rj r e q u i r e d p e r i o d i c s o l u t i o n Cp , o c c u r r i n g i n = (l-m ) (l = (p . ^ ( f immediately 2 following center, 4 + ")cos + s h r i ) • h ^ 2 2 +") 0 (s) 2 2 + h^ 3 + +^ 0 (s) 3 3 and formulas: 1 Cp equal 3 small neighborhood of the 2 is 41 + p e r i o d T are determined by the S 2c - jj equation with the p e r i o d i c s o l u t i o n of becomes assumes the form , . Comparing t h i s of 0 + ^("jf) Here we have used the f a c t that at the center j 2 to - , / l - r a ^ . If we r e p l a c e s i n 0 and cos 0 by c o s 0 = 1 avoid e v a l u a t i n g a singular point + 0. 2 sin 0 = 0 p , q , and (2.17). in 2 and l e t as components Such procedures value of cp at c type, of t . integral by cp^ the Denote and r e p l a c i n g T by tj2d, . . . ) ~ 1 + . . . T its 81 f 29m, 2 274m, m. 7==^ • t 2 ^ - ra 192(l-m ) v, h 2 ° = ,„„ 5 m l 2^ + 48(l-m ) h„ n 2 5 m , j 5 , n *l 24,/l-m + 16 2 3 TJ]~2 - 144(l-m 3 l m l 32,/l-m 2 = 2 1) 3 / 2 3 s S S it ) ° l ^ J c o s 2s 1 2 l ra l O 2 2 187i-mf 36(l-m ) h 1C 576,/l-mJ m, 9 o T 2 / + ( i u. * 192~ * 216 -"1 2" **1 \ 576(l-mJ) (4.12) , 1—5- • • ^ 2 1 - 192 1 l m _ c o s ~ + m _ ^ + 2 ^1 24 ) 2 n 144(l-m ) 2 ^ ~ 1 n 2 2 „3 /-—jl V-x ~ 9 3/l^f 1 2" ~ 12 48yi-m^ T In " ( J ^ l / 4 1 + h lH t h e above f o r m u l a s periodic solution Further, i n case pal ( axis, + ^ h 3 1 3 + — i f we p u t u.^ = 0, we corresponding the torque + )• o b t a i n the to the unperturbed i s fixed the c o r r e s p o n d i n g p e r i o d i c case. along the s m a l l e s t solutions can be princi- obtained 82 by simply i n t e r c h a n g i n g the mentioned i n S e c t i o n appropriate q u a n t i t i e s , as 2.60 P e r i o d i c S o l u t i o n s i n the Case of the Torque Vector F i x e d Along the Middle P r i n c i p a l A x i s A•3 Here we need to c o n s i d e r a number of the behavior of 35 depends not only on the 35 r e l a t i v e to the two planes differential slightly equations different satisfied i n form, marize the negative, 1. The i n i t i a l (i) [i^ = - V , (see Cp ^ 0 2 (1 • m ) ( l 2 4 = Cf> , • 'jcos 2c 2 are any l o s s of as i n S e c t i o n 3»4 that V > 0. in planes I and II (3.52)) equation + h^ • h H 2 2 + h r) 3 3 s + >") 0 (s) + r j 0 (s) 2 3 2 the we s h a l l simply sum- I s - Although the Moreover, without p o s i t i o n of 55 not | ^ | < \i but are i d e n t i c a l with the ones accordingly. we s h a l l assume i.e. p o s i t i o n of the steps taken to o b t a i n Hence, i n what f o l l o w s results generality, 2, 3)« by the v a r i a b l e cp corresponding p e r i o d i c solutions in S e c t i o n 4 * 2 . initial because of the s e p a r a t i n g polhodes values of [ X , and m^ ( i = 1, also on the subcases, + ...) + ••- = arc s i n h m s is 83 \l2(l+m ) 2jl+*\ f l 2 ? 4 2 3 2 9 m V576(l 2 + + m 5 7 ^ ^ ? • l ;V l & J l ^ (4.13) • 2 - u l 27 " 7 \l92(l*m ) 1 1 \ 192 2 = = ^ V + 32,/l+mJ 2 6 v 2l l m 1 9 2 2 — V2 . ) cos 2s m 2 ra X v 1 * V - -i-, cos s + ) / ^ V36(l+m ) ( 3 2 + 4 1 9 2 C O S 3 8 / h, = 0 2 ^ 48(l+m ) 2 = 24^1^\ + + h ^ 1 3 —TTT? = 144(l m*) J + m 48,/l+m T - M , % i / 4 Putting J,/ * * 2 s = (m <P 2 Cp 2c = cp 2 2 c c ( 1 (See - 1) 1 / 4 + rjcos - arc cosh j — 7 + V 2 23 + V o 371 m* + v y w 12 + h i n h + V = 0 i n the > H- l m 1 7 ^ M 144/1 -ni* . ± _ corresponding p e r i o d i c (ii) ° 1 6 2 3 5 M " 6 2^ + h 3 n 3 + s o l u t i o n s i n the " ] s + ^ 0 (s) 2 2 2 c > 0 the unperturbed c a s e . (3.56)) + h,r) + h ^ cp m above expressions we o b t a i n equation (1 2 + hy) 2 + rj 0 (s) 3 3 3 + -..) + ••• - 1 T, m > 1 2 84 "2 v , /2 . ~ 6 _ l c o s 1 9 0,(s) = - ( — — • -rr= I v+ v 2 2ymJ-l \l2(m^-l) 3 2 s / 29nu 274m ift^ ? 9 2 m 2 m 22 - -£v^ ) cos 2s ' l&7m|-l --v 36(m -l) ^i 2 A L 2 m 2 2 9 + v— ' m v 1 9 2 ( m - l ) " ^fifl 2 h 1 5 m 2 5 m + = 3 5 m 2 144(m -l) (iii) 2^i/4 In the 2. vf + 4 1 9 c o s 3 s 2 ( 1 1_ 6 ° 2 1 , m h i n + h 2^ 2 l i m i t i n g case following 2 v 2 + 2 v 3 37m -1 2 * 6 m + 144(m -l) 3 / 2 can be obtained from the an i d e n t i c a l v 3 + 2 ( m 2 24Vm|-7 48(m -l) 2 2 h 2 + = 0 - h + — -. " 2 u 9 2 + h nh n 3 + 0 3 _ m 2 4^m*-l •••) = u^, p e r i o d i c case 2 s i n c e solutions satisfies equation. The i n i t i a l p o s i t i o n oftt)i n plane I (See _ v_ page 68) 1 2 85 s f = Jm^(l = f 2 9 c • c = In m 2 (4.15) 2 * hfl + h^ ^cos * h^ 2 + f^ ^ ^ r s a + xi - i 3 - | v 2 + T Putting _ ) ^ ( ) 3 s 1 T, m > 3 0 ••• + 3 1 _5 W * 24 * = — ( 1 183 2 • |ZA v • ^ v . cos v + 576 216 y ) > ^ 1 • v 2 2\ 36 18 " 9 ) 2 r 2 ...) 3 0 (.) " (- h + 3 V C O S m / 1 v -^96 M 96 "•' 3232 v v 2 s + + + 2 v \ 24y ° 2 + C 2 v * T • • h r) 2 2 v = 0 we o b t a i n the + h r^> * •-.) corresponding p e r i o d i c solu- t i o n s i n the unperturbed c a s e . 3« The i n t i a l p o s i t i o n (i) II (see page 69) < H - M c s = 9 of «S in plane y-m^U = cp 2 CP_ 2c • h,r) + h r) ^cos + h.^ 2 • ...) T, < 0 s + rj 0 (s) • ry>0 ( ) - i -§») ~ . . -r-ii * i « . 2 8 * ••• - In f- — • (.) a - i - j . s 3 s (4.16) U-io; * y( s) ; - ^ ( - 11-2 • 2 - 2 v 3A / '7 1 + h l " 2 n 3 " " 18 9 V y / 2 2\ " 2 C O S 2 s + 1 9 6 2 4 " 7 2 27T -» 2 4 V + 2 j 2 + 2 4 C O S 6 v 1 2 2 8 3 9 3 ( l +11,0 + h r) + h^rj + . . .) 2 3 2 3 The c o r r e s p o n d i n g p e r i o d i c s o l u t i o n s i n the case are o b t a i n e d by p u t t i n g (ii) + v v \ _j> 1 T = 3 6 • 216 ^ v ^ cos s v 0 1 h ' v - 576 \576 8 When | ^ | > [i, unperturbed V = 0, no p e r i o d i c s o l u t i o n s exist. 87 SECTION 5 SELF-EXCITED SYMMETRIC RIGID BODY IN A VISCOUS MEDIUM 5.1 Equations of Motion We c o n s i d e r here the motion of a symmetric r i g i d body about a f i x e d p o i n t in a v i s c o u s medium subject time-dependent It w i l l ponents self-excitement• be shown that of i« can be o b t a i n e d by q u a d r a t u r e s . when the self-excitement w i t h i n the is constant body, these components g e n e r a l i z e d sine and cosine computed from i n f i n i t e to an a r b i t r a r y the com- In p a r t i c u l a r and f i x e d in d i r e c t i o n are e x p r e s s i b l e integrals, i n terms of whose values s e r i e s developed i n S e c t i o n S e c t i o n 5 ends with a d i s c u s s i o n of the can be 5«4« asymptotic motions of «5 r e l a t i v e motions are f u r t h e r compared with the corresponding ones when friction to the b o d y - f i x e d t r i h e d r a l . is neglected. We s h a l l see tween the two are s i g n i f i c a n t . are summarized i n three During the final in and this Section. include, In a viscous of t h i s case of also The r e s u l t s discussion t h e s i s we d i s - d i s c u s s e d the problem considered here were o b t a i n e d up the equations the r e s u l t s of motion of independently, of [25]. a body moving in generated by the movement of the a force-free be- theorems. medium, we need to c o n s i d e r the nature of . . r e s i s t i n g force the The r e s u l t s i n a d d i t i o n to o t h e r s , setting that the d i f f e r e n c e s stage of w r i t i n g t h i s covered that Merkin [25] These the body. For symmetric r i g i d body r o t a t i n g about 88 a fixed point, K l e i n and Suramerfeld [16] force was e q u i v a l e n t assumed that the drag to a torque vector with two one along the symmetric axis components, and the other along the orthogonal p r o j e c t i o n of m on the e q u a t o r i a l p l a n e . These components were assumed to be p r o p o r t i o n a l to the components of (3 in these di rec'tions. In our study we s h a l l f o l l o w t h i s axis be the symmetric a x i s . components by -V-Q* a n neglected Then we may denote assume p o s i t i v e , + 2 where M^(t) ( i = 1, 2, 3) the s e l f - e x c i t e d are continuous (B - C)qr - M ( t ) - p^p Bq • (A - C)rp - M ( t ) - u^q of t , Then the is equations 2 = M (t) M (t) = —g -, F^(t) 2 functions 1 h form is 3 Ap - + the friction M (t)lc - P r Q ^ B-C A-C . *1 *1 L e t t i n g - j - = —g- ' » X" " B~ " # 2 or i f form Cr F (t) resisting and u.^ are con- q moment a c t i n g on the body. of motion take the T the z- suppose a = M j C O * • M (t)7 (5.2) Let the zero. Further, (5.1) where U . d -u.^(p • q ) , s t a n t s which we w i l l suggestion. v u ^o C~ " _ l f ^ U ) =~J M a n A d M (t) = —* , we may write equations ( t ) ' ( 5 « 2 ) in 89 p - hqr = F ( t ) - Q hrp = P ( t ) - vq = P (t) - [xr 1 (5.3) 2 r 3 When the s e l f - e x c i t e d Vp torque vector i s fixed in d i r e c t i o n i n the body as w e l l , we may put F (t) = ^F(t) = F (t) « g^F(t) = A. F(t) F (t) = ^F(t) = A. F ( t ) 1 (5.4) 2 3 Here a , , a^, vector. In a d d i t i o n , i f F(t) is a constant. the a c t i n g moment i s fixed time-independent, Otherwise the torque v e c t o r w i l l move i n position at any time t i s given by e i t h e r ( 5 « l ) (5.4). Equations 3 (p , Q 2 and Q-^ denote the d i r e c t i o n c o s i n e s of the the body; i t s or ^(t) o q , r ) Q of the body. Q (5«3) together with the i n i t i a l angular velocity at t = 0 determine u n i q u e l y the angular v e l o c i t y We proceed then to the i n t e g r a t i o n of these equations. 5.2 The Angular V e l o c i t y of the R i g i d Body (5«3) can be i n t e g r a t e d d i r e c t l y The t h i r d equation of to give (5.5) where r = r e o _ l i t + R(t) 90 R(t) Multiplying this r = t e-^ the second to the f i r s t o n e , we " ( t s ) F.(s)ds equation (5«3) of by i and adding obtain TT * 5ir = T T (5.6) w h e r e IT = p • i q , TT = F , + i F 1 are complex-valued the general <6 functions • 1 R) O o f t« This linear equation has solution ~~S (5.7) , and P = v + i h ( r e " ^ 0 ir=7re " • e 1 ° 5 t f 1 0 J t | 5(s)ds JQ e S ( s ^Tf(s)ds p H e r e *5 •, J. real and i m a g i n a r y formulas We for p note 7r e ~ "5, e'^.J n d = P o parts and o of + i q • o (5*7) we By shall = e t t e 5 — vt * (p cos^-*g sin5) o ,(B) 5l(S + ie o TT(s)ds |(s)ds -* J| V 0 v ( t - s i j . e~ t V ( t " s ) explicit ) 1 S = f 0 (-p^in^s * q c o s s ) o { F , (s)cos[5(s)-3(t )] F (s)sin[3(s)-S(t)]}ds 2 + t 5(t) — vt {F (s) inC3(s)-5(t)j F (s)cos[^(s)-5(t)]}ds where (5.9) obtain the q. • separating first: Q (5.8) a h [ r e^ 8 * R(s)]ds 2 91 ~2 Letting 6 » J p 6e o and CT = arctan ~ " , we have o ~2 + q P * s i n ("5 V • CT) 0 (5.10) (i) - F (s)sin[3(s)-5(t)]}ds 6e Vt • T * • F (s)cos[5(s)-5(t)]}ds 2 c o s ("5 + cr) - ( - ){F (s)sin[3(s)-3(t)] v e t s 1 2 The Torque Vector F i x e d i n D i r e c t i o n Within the Body In the above a n a l y s i s , the s e l f - e x c i t e d i s not r e s t r i c t e d i n any way. torque v e c t o r If we now assume that m i s f i x e d i n d i r e c t i o n r e l a t i v e to the body so that (5.4) h o l d s , then p o Se - Vt sin(s+cr) 6 J e" t 1 v ( t o " F(s)sin[5(s)-S(t)-CT ]ds s ) 1 (5.U) q = 6e *cos(^+cr) V • 6 j t e" V ( t " F(s)cos[^(s)-5(t)-o ]ds s ) 1 r~2—2~ where 6^ = / \^ * ^"2 / ^i\ a n d °1 ~ a r c t a n 1~ / " From (5.5) the expression f o r r becomes (5.12) r -= r e"^ • \, f ( T ~ >F (s )ds S 92 (ii) The R e s i s t a n c e of the Medium N e g l i g i b l e When a r i g i d body moves i n a medium which we may assume frictionless, then . (5.13) h[r + R(s)]ds (5.10) and (5.5) g i v e . p = 6 s i n ( 3 +a) t • F, (s)cos[s(s)-3(t)]ds - J* t . (5.14) J t n 0 x w i t h i n the body, if the '0 F (s)cos[?(s)-3(t)]dj 2 ds 0 In a d d i t i o n , 2 F ( s ) s i n [ 5 ( s ) - 5 ( t )]ds t + F (s)sin[^(s)-3(t)]ds t q = 6 cos(-5+o) • I For (5*9) t ~S(t) = f Formulas = 0 , and from equation = the torque vector 18 i s fixed in direction above formulas can be f u r t h e r simplified. now we have from (5.11) and (5.12) r p = 6 sin("5+ ) " 6, J CT (5.15) 1 o F(s)sin[3(s)-5(t)-o ]ds 1 J q - 6 cos(s+cr) t • 6, F (S)COB[ s(s)-5(t)- ]ds r * f F ( s ) ds ° 3 J0 n These formulas then p r o v i d e the a r i g i d body under the torque v e c t o r a c t i o n of a i angular v e l o c i t y a time-dependent f i x e d i n d i r e c t i o n w i t h i n the body. components of self-excited 93 In a l l of the cases c o n s i d e r e d above, given i n terms of some i n t e g r a l s . the s o l u t i o n s are The p o s s i b i l i t y of e v a l u a t i n g them by means of elementary or t a b u l a t e d f u n c t i o n s course, if on the f u n c t i o n s F ^ , F ^ , and F ^ , or on the f u n c t i o n F the s e l f - e x c i t e d the f o l l o w i n g , for torque v e c t o r i s f i x e d i n d i r e c t i o n . we s h a l l c o n s i d e r t h i s p o s s i b i l i t y the case of a constant w i t h i n the body. self-excitement Afterward, to t h i s In in detail fixed in direction we s h a l l d i s c u s s asymptotic motions of 3 subject 5.3 depends, of q u a l i t a t i v e l y the type of t o r q u e . Time-Independent Torque Vector F i x e d i n D i r e c t i o n Within the Body Denote by M the modulus of the torque v e c t o r . 2 l ra formulas (5«ll) V t q = 6e~ cos(5+a)+m 1 2 J Vt 12 J to f p «= 6 sin(^+a) - m 1 2 q = 6 cos(^a) 1 2 r = r o * mt + m J e~ e" ^ V In case \i. and v are n e g l i g i b l e , (5.17) m • r n e n f r o m and (5.12) we have p = 6 e " s i n (-s+a) - m (5.16) 2~ * 2° Let t ~ ^ s i n [ 5 ( s ) - 3 ( t )-CT^]ds s t _ s ^cos[5(s)-5(t)-a ]ds 1 these expressions sin["S(s)-^(t)-a ]ds 1 co«[T(s)-3(t)-o ]ds 1 reduce 94 For the two i n t e g r a l s appearing i n ( 5 . 1 7 ) , Bodewadt [2] showed that they g e n e r a l l y could be expressed i n terms of Fresnel integrals. For the c o r r e s p o n d i n g i n t e g r a l s shall the l a t e r show that appearing i n ( 5 . 1 6 ) , we i n the general case they can be expressed in terms of i n t e g r a l s of the t r i g o n o m e t r i c type which i n c l u d e s the F r e s n e l integrals. We proceed now to d i s c u s s values of h, r Case 1. h ^ 0, r = 0, m^ = 0 Q (5.9) and (5.16) we obtain / l S i (5.18) depending on the and m „ . ° 3• From equations fc four cases, _ / m q - ^q 2\ m v t + ~vt ^ 2 m - — j e o l • ~ r = 0 Case I I . h f 0, T Q f 0, m^ = 0 Under these c o n d i t i o n s , we have (5.19) r = r e _ l i t o and -5(t) = h ( l x hr where h.. = —-— . i [i e _ t i t ) Without any l o s s of g e n e r a l i t y , i n the f o l l o w i n g computations that h^ > 0 s i n c e d i r e c t i o n of the z - a x i s may be s e l e c t e d we can assume the so that hr positive > 0. o 95 For the i n t e g r a l appearing i n the f i r s t ( 5 . 1 6 ) , p u t t i n g 0 = h,e (5.20) f e0 t J • v ( t e - 1 . ^ r sin?) vs , -M-S , e cosh e ds t r 1 "cos0 J* sinh,e ^ ds V+ The l a t t e r two i n t e g r a l s f o l l o w i n g two we may write # sin[5(s)-3(t)-a ]ds s ) -vt - e - o, equation of s 0 can be expressed i n terms of the functions: . i• ((x . w)\ - - P J °° s i n u du (5.21) . ,„ c i (Ux,, - |- J , wj -= w) 0 0 X cos u , • — - — du u which we may c o n s i d e r r e s p e c t i v e l y cosine integrals. integrals. Note that as the g e n e r a l i z e d sine and f o r w = 1/2 they become the F r e s n e l We s h a l l allow w to take complex v a l u e s . tence of these i n t e g r a l s The e x i s - and the methods of e v a l u a t i n g them w i l l be c o n s i d e r e d i n S e c t i o n 5«4» In the meantime, we continue to show how they are used i n e x p r e s s i n g the components p and q . If we make a change of v a r i a b l e by means of the equation —s u = h,e of r the f i r s t (5*20) become and second i n t e g r a l s appearing on the respectively J e cosh e" ' ds = h [ c i O ^ a ^ - c i (h^'^a,) J" e s i n h e " " d s = h [ s i ( h ^ a ^ - e i (h^e"*^ ,a,)] Vs M 8 1 2 ] (5.22) Vs lis 1 1 2 right 96 v where a.. = 1 • 77, h_ = __1 . * * \i Thus we may w r i t e 1 (5.23) P = 6 e " s i n ( - s * a ) - m e ~ h [ c i ( h , a ) - c i (h e" ' ,a )]sin0 V t V t M 1 2 * m 12 ~ e V t h 2'- 2 ( s i h 1 * 1 a 1 )~ s i 1 t 1 1 (h e" ' a )]cos0 P t 1 # 1 For the component q we f i n d i n a s i m i l a r way that (5.24) q = 6 e ~ c o s (-s+a)+m e~ h [ci ( h , a ) - c i ( h e ~ ' a ) ] c o s 0 Vt Vt | J 12 +m Case I I I . 2 e" 1 2 V t (5.25) r - (r - ^2) e"^ Suppose f i r s t t = ge" " 1 r* I J r I e 0 c * J e f J e B 3 - o - g t 1 x o vs 4 s 2 g £ 0. + 1 2 1 1 ^ 0 s g i s c o . Putting x v s T # ( 5 « 9 ) and ( 5 « 1 6 ) we now have -lit 1 t 1 h [si(h ,a )-si(he"^, and ~S(t) = g ( l - e ~ ^ ) + g.^ where g = = 1 h < j= 0, m^ ± 0 From formulas T 1 s g e -IAS . -M-s, s i n g^s cos ge V s c o s g,s 1 ds r s i n ge ) d s ia t 1^ = J e 0 V s s i n g^s s i n ge ^ we observe that S d s ^ r -~ ~ —/ ^- , g T. m 1 = T^h. )]sin0. 97 J t J t e " v ( t s ) - ( - ) v e " sin[T(s)--s(t)-(j t 8 C 0 S [ ( )_ (t)T Consequently, s I 0 i ; L ]cls - ]ds e" = e" V t (l +I )sin^+e" 1 V t V t 4 ( l + I )cosf-e" 1 V t 4 V t q = 6e" V t 2 3 (l -I )sin<f 2 f o r the components p , q , we have from p = 6e" (l -I )cosf 3 (5.16) s i n (^+o)-ni e"" [ ( l + I ) s i n f + (l -I )cos<f] Vt 12 1 4 2 3 (5.26) The c o s ('5 o-)+m e" [ (I , + 1 ^ ) 0 0 8 ^ - ( i g - I ^ J s i m f ] + Vt 12 integrals I , to I. can be expressed i n terms of the 1 4 g e n e r a l i z e d t r i g o n o m e t r i c i n t e g r a l s as f o l l o w s : Multiplying I * < • i l t I, r = J ( t T 2 by i 2 v + i g S (5 .27) I 2 can s If we make the M be w r i t t e n i g l t ) 1 t 1 g f a c t o r on the r i g h t as _y _y -lneg) Putting substitution '[ci(eg,w )-ci(ege" ' ,w )] V . l which w. = 1 + 7- + —rr~, and the f i r s t _y we o b t a i n — -US . cosEge ds eg > 0 . i in ) to I , , then • H x i e where 6 = sgn g so that u = ege~^ , and adding t h i s V. tea)* / g l , P + i sinf-T^lneg 98 ci(eg w ) # = U x +i V A ci ( e g e ' ^ w ^ and equating the real h <5- > i 28 In a similar = Ug + i V g and i m a g i n a r y p a r t s - « < i - V - « •= , 3 ( 0 , • , ( u 2 2 x - u ) 2 i • ( v 3 a V V - l o f (5-27), we h a v e v ) 2 w a y i f we l e t si(eg, W l ) = U +i V 3 siCege'^w,) = U 4 3 •iV^ then (5 29) ^ i Thus, provide J* e t " ] - e[ ? 3 formulas c a s e g = 0, of elementary t J *'VV " «3<W . V U e~- where (n -u ) * a (v -v )] 3 4 2 these functions. ) ~ 'cos[T(s)-5(t )-o ]ds / 2 • g ,2~ , Qg = / V p 0. c a n be e v a l u a t e d , - " ^ s i n (o^+CTg) + = 1 6g (5.28) a n d (5.29) in F o r now we h a v e ; S with components 1 (t A p a n d q when g f / \ " sin[T(s)-T(t)-a ]ds s 3 (5.26) t o g e t h e r t h e components In terms 4 = , [ , 1 "Q^COS l arctan — . -Vt — s i n (g^t 2 e~ V t " ^ 3 — ° s (g,t 0 (aj^+CTg) + +0^0 + c + 1 g = c —vt , r _ , <= Oe sxn (g^+aj- 12 ra ^ — It f o l l o w s that -vt . v , e s i n (g,t *o *0 ) ( f + v m + 1 2 (5.30) l ^ l 2 *—2—2— m V q = 6e -vt ag 12 - v t 2 " l l cos(g.t+a)+T—e cos(g.t•CT-.+OO•—~—2— m V 2 m g V+g* m 99 Case I V . h = 0 This is (5«3) equations ( ,. 5.4 the case of dynamic symmetry. we o b t a i n , . i . ( , 3 1 ) D i r e c t l y from The I n t e g r a l s . i ) . - . t o si(x w) and c i ( x , ff w). Re w > 0 v In the problem under c o n s i d e r a t i o n Re w = 1 + — , v C ^1 * where the r a t i o — = — 77— . If LI = u, or i f they do not O differ much, then f o r an elongated v greater than C the Re w i s between 1 and 2. now C i s greater known [18] for that r a t i o 77 i s \r r i g i d body where A i s much l e s s than 1; the For a f l a t - d i s k e d r i g i d body where than A, the Re w i s the accordingly, integrals si(x, greater than 2. w) and c i ( x , w) It is converge Re w > 0. In the f o l l o w i n g computations, let us assume that the V r a t i o 77 i s not an i n t e g e r so that we can write w = K + w* where K i s a positive integer and w = a + i{3, 0 < a < 1. Then repeated i n t e g r a t i o n by p a r t s y i e l d s f o r K odd 1 K+l (-1) c i (x, w) =P (x) s i n x-Q(x)cos (5.32) 2 X+ ( ' - i j,".'. 'ttlSi (x,w w w T )-A(w K^l si(x,w)=P(x)cos x-Q(x)sin x+ ( '~'i'j'.".". w* ^ w Ci ( x , w » )-B(w 100 and f o r K even K c i ( x , w ) = P ( x ) s i n x-Q(x)cos X + j^ <> - [ Ci (x, w • ) - B( w* ) ] t K (5.33) ()^.. t[Si(x,w')-A(w')] s i ( x , w ) = - P ( x ) c o s x-Q(x)sinx+ w In these e q u a t i o n ^ ^ 2 / . \n + l 2n-w P(X) I (:.r..(w-2n) = n=l (5.34) i fK odd W K 2 V (-l) * x ° L (w-l)...(w-2n) n=l = n 1 2 n W i f K e v e n *>i 2 Q ( X ) " - (_ )n + l 2n-l-w I n=l (w-~.j../( -2n+l) (5.35) n i f K o d d W K 2 (, )"+l 2n-l-w (w-l)-..(w-2n l) 1 1 n=l and Si(x,w») J (5.36) = [ 0 K e v e n = f °° 0 u -S-^P du u 0 < Re w» < 1 du 0 < Re w' < 1 x u J J f W : A(w») i + = f p. Ci(x,w») x W du 0 < Re w' < 1 -Sff-i du u 0 < Re w* < 1 W (5.37) B(w») « f 0 J W 101 It is known [18] A(w») that f o r 0 < Re w' < 1 TT - 2r(w )sin(w»|) f 1T B(w») = Here P(w ) f Si(x, w) f denotes the gamma f u n c t i o n of w . The integrals r i g h t of equations f and C i ( x , w ) f appearing on the (5»32) and (5«33) have been s t u d i e d f o r w [18, 19], However, and t h e i r values for w T Si(x, w) T integrals have been p a r t i a l l y are needed f o r the f o l l o w i n g develop and C i ( x , w ) f si(x, r e a l by K r e y s z i g tabulated. complex no such t a b l e s have yet S i n c e these values we s h a l l i n the f compute t h e i r v a l u e s . w). of s e r i e s f o r the and asymptotic w) and c i ( x , solutions appeared [ 6 ] o Case III, integrals s e r i e s f o r the related From these s e r i e s we may In a d d i t i o n , estimates of e r r o r s w i l l be p r o v i d e d when by n e c e s s i t y summations of the given series are c a r r i e d to only N terms. (i) T a y l o r s Expansion f o r S i ( x . t From the infinite series n=l termwise integration yields 00 w) and C i ( x . w), 0 < Re w < 1 102 If w * O • i £ , the s e r i e s real on the r i g h t can be separated i n t o and imaginary p a r t s . For by putting x w = x e a * 1, X where x^ = |3lnx, and n = /(2n-a) n A = arc tan n we may w r i t e . x , T sin u , J — d 0 or (5 . J 38) U + ji 2 1 JL \2n-a " , ,xn+l 2 n » a ^ n ~ l ^ V (-1J x e " I (8.-1)1 ^ „ n= l i A X ^ • - _ In (- l- ^ - l ) m B ° " ( V ' l > o • ( , . sn + 1 2n-a " I -&-iW'-»<v-i> i n = l For the second i n t e g r a l , the same procedures give * „ x 0 ™ U / . xii / , 4 V -' A l l 2n*l-a J a J. L ' n=0 n . * n / , \n 2n+l-a n=0 where now fl* A* The error «,/ (2n*l-a) n 2 + 0 = a r c t a n f x .ft\2n + l - a ; ; above procedures w i l l also y i e l d an estimate when the summation of the s e r i e s of the (5*38) or ( 5 « 3 9 ) is 103 c a r r i e d only to N terras. N sin u we let / .xn+1u 2n-l-w 1-1J i n which R„ i s N the remainder a f t e r then terrawise i n t e g r a t i o n r AiS-Jl J0 ,uw d v X Q I " I U WR where M i s „ x p —w u R„ d u . D U an upper bound on the d e r i v a t i v e of sin u. Since t h i s s i n u, M can be taken as 1. ' V 1 |u Thus with N I = u -ti" that 2N+1-U I N' $ ( 2 N ) l ( 2 N U - a ) E series (5 - 39) we f i n d that 2N-tt , | E | < ( N-i):(2N-a) N 2 value of derivative -w, we may a s s e r t we have absolute u ^ T2N7l 2 (5.41) 2n-l-w N From Lagrange*s remainder f o r m u l a , For the u n we have I<J (5.40) sine gives • - n the Nth term i n the = y -1)° £ (2n-i;X JO ^. ' '' n= l u n Clearly, -w n= l U series, _V" For i f is the 2Nth either + s i n u or 104 Because of the f a c t o r i a l terms i n the denominators of the above two e x p r e s s i o n s , we can expect the s e r i e s (5.38) and (5.39) to converge s u f f i c i e n t l y (ii) si(x, fast Asymptotic S e r i e s f o r small values of for s i ( x . For l a r g e values of x, w), ci(x, series. parts (5.42) w). w) are best e v a l u a t e d by means of w), gives iin w u Re w > 0 the values of the two For the i n t e g r a l s i ( x , J w) and c i ( x , x. integrals asymptotic repeated i n t e g r a t i o n by N-l 2 (-l) w(w+l)««»(w+2n-l) w+2n n du - • y cos x n=l N-l 2 I ( - l ) * w ( w + l ) « " (w+2n-2) sin w+2n-l n 1 x n=1 N+l + (-1) 2 w(w+l)-..(w+N+l) J — ^ d u . u 2 X if 1_ w N-2 2 r ( - l ) w ( w + l ) « " (w+2n-l) _w+2n n=l n + 2 I N odd. (-l) * w(wH)»«(wt2n-2) w+2n-l n cos x 1 sin x n= l • (-1) w(w l ) . - . ( w N - l ) J du 2 + + X u if N even. 105 If we l e t H ( , x w ) = i _ + y (-l)%(w+l)...(w+2n-l) n=l 7 \ ' A X W ; V ( - l ) ' ' w ( w « - l ) " » (w+2n-2) ~ L w*2n-l n=l n > 1 X then CD (5.43) The J ^ ^ ^ ~—^— u x ^ H(w, asymptotic nature of t h i s x • G(x, w)sin x W)COS expansion can be seen from the c o n s i d e r a t i o n of the remainder term. remainder a f t e r the sum of the f i r s t If R„ denotes the N + 1 terms, then from (5.42) |R I N - |w(w + l ) . (w+N)[°° ~ j ^ d u l . X u i f N + 1 odd = |w(w+l).«.(w+N)j ^ " " d u | i f N • 1 even - x u 8 H w In e i t h e r case, we have .00 |R I < |w(w*l)---(wN)|J~ - ^ T l N = |w(w+l)•»•(w*N)| From th© r e l a t i o n (5.44) for Rw • 1) - wP(w) the gamma f u n c t i o n , we may w r i t e - (N+a) X M + a 106 (5.45) that that (H*a) We N t h e v a l u e s o f t h e gamma f u n c t i o n a convenient The N+a N, |x R | - 0 as x -• <» f o r e v e r y N. w have been e x t e n s i v e l y is - r(w) which i m p l i e s note t P ( W * N ^ I ) f o r complex t a b u l a t e d [ 2 7 ] so t h a t may arguments equation (5.45) estimate of the remainders series H ( x , w) and G ( x , w) can be s e p a r a t e d real and i m a g i n a r y p a r t s as f o l l o w s : with relation ( 5 * 4 4 ) , we may A We o b s e r v e into first that write re)* "- *" 2 1 Putting -n_(w) = |r(w) | A(w) x w = a r g p(w) a = x e 1 w h e r e x , = 0 1 n x , we h a v e then OD H(x,w) * n + £ =0 " ^ i ^ "^(w) ( l ) n 2 n ) c o » ^ ( w ^ n ^ A ( w ) - x l 3 31 i J ^ ^^ 1 2 n ),in[ (w.2n)-A(w)-x ] A 1 n«=0 (5.46) G(x,w) = Y ^, n =l ( " l ) n ^Ii"" l~L(w)x v ' l ) co8[A(w^2n-l)-A(w)-x ] 107 For the second i n t e g r a l c i ( x , can be o b t a i n e d i n the same way. It w), is an asymptotic series found that -00 I J (5.47) d C X °w * "du ^ - H ( x , w ) s i n x • G(x,w)cos x u with an estimate of the Finally i f we remainder given again by (5.45). let H(x, w) = + iH 2 G(x, w) = + iG 2 then s i ( x , w ) „ - H c o s x - G . s i n x - i(H_cos x + G s i n x 1 (5.48) 5«5 0 1 c i ( x , w ) ^, 2. . . 2 1 H^sin x - G^cos x + i ( H s i n x - G cos x 2 2 The Asymptotic Motion of the Spin Vector When a r i g i d body r o t a t e s angular v e l o c i t y w i l l statement i s Theo rem: in a viscous medium, i n g e n e r a l be l i m i t e d by f r i c t i o n . substantiated by the f o l l o w i n g The angular v e l o c i t y of _a s e l f - e x c i t e d i f the modulus o f the s e l f - e x c i t e d This boundness symmetric r i g i d body r o t a t i n g about a f i xed po in t i n _a viscous bounded, its medium i s to rque v e c t o r i s bounded. The t r u t h of t h i s expressions theorem follows f o r p, q and r . easily For from equation from the (5.10), we have 108 |p| < |6e- V t S in(r*a)| I fV Jo • l j V + . , - V t < Oe where _ • e - V t P* M* I e M t Vs . ds 1 I n t e g r a t i n g the expressions M* + — (1-e < 6e which i n turn i m p l i e s M» |P! < Similarly, 6 + < 6 +~ From equation r If ( t B } " " F . ( . ) c o « [ < f ( . )-5(t ) ] d s } F 2 _ -vt • e MI M I ( , p 1 ) e 8 i n ^ ( 8 ) - 5 ( t ) ] d s Vs , d» on the r i g h t , we f i n d M' + — (1-e ) that ) that M» * ~{p for a l l t > 0 we o b t a i n M» |q| V " ( s ) | < M« . | F ( B ) | < Ml, | |p| ( t V < M» + ~ f o r a l l t > 0. ( 5 * 5 ) , we have -M-t, ,-lit r e + J 0 F ( t ) < M», then 3 M» < Ir I + -4 rl It is easily f o r a l l t > 0. seen that friction is neglected. rotation is this Moreover, even though the speed of bounded, the s p i n v e c t o r may s t i l l wander i n space without approaching any l i m i t . excitement is theorem need not h o l d i f Nevertheless, if the self- time-independent and f i x e d in d i r e c t i o n r e l a t i v e to the body, we s h a l l show i n the f o l l o w i n g that 3? approaches a s y m p t o t i c a l l y a constant vector i n the b o d y - f i x e d t r i h e d r a l . 109 The r i g i d body i t s e l f then assumes a s y m p t o t i c a l l y a uniform r o t a t i o n about an axis f i x e d i n the body as w e l l as i n We s h a l l c o n s i d e r t h i s space. in four cases as in S e c t i o n 5»3» and in doing so we s h a l l also compare these motions with corresponding ones in which f r i c t i o n i s n e g l e c t e d . essential Case I . h * 0, r = 0, m^ = 0 Q r = 0, the motion of the end-point of 55 takes place wholly in the equatorial plane. p - hqr = m^ - / l (~, 2 —, Vp 2 f m The system Q • hpr = m - vq (5.49) m There are differences. Since with the the = - Lir assumption nu = 0 possesses a unique s i n g u l a r point \ Ol in the pq p l a n e . t r a j e c t o r y i n t h i s plane i s singular point represents rotation is A c c o r d i n g to (5.18) every r e c t i l i n e a r and approaches as i l l u s t r a t e d in Figure 5 . 1 . the This p o i n t a s t a t e of permanent r o t a t i o n of the body, and t h i s stable, as we s h a l l see in the d i s c u s s i o n of Case II. When f r i c t i o n i s n e g l i g i b l e , and the end-point of 35, as b e f o r e , But the phase p i c t u r e i s r remains i d e n t i c a l l y always essentially lies From (5.17) we have = P Q + mt q - q o + mt P 2 2 in the pq p l a n e . different. system has no s i n g u l a r p o i n t on the pq p l a n e , zero, For now the unless m^ = = 0. 110 i m p l y i n g that lines i n the pq plane the t r a j e c t o r i e s are p a r a l l e l t e n d i n g to i n f i n i t y . These are shown i n F i g u r e 5«2. This example v e r i f i e s our previous angular v e l o c i t y need not be bounded i f even though the modulus of the statement that f r i c t i o n is self-excited the neglected, torque v e c t o r is bounded. Case I I . h * 0, X t 0, m^ = 0 q T u r n i n g t o the situation f r i c t i o n l e B S describe. r = r . Q second case, we c o n s i d e r f i r s t s i n c e t h e With L> = V = 0 and I Q Thus, t h e end-point of motion 3), f o l l o w i n g t h i s o b t a i n e d from (5-49) by system, t o (5.19) gives i n i t i a l l y lying plane z = r , w i l l remain f o r e v e r in Q simpler i s then $ 0, equation the plane. p u t t i n g i n t h e The V, |X, a n d m^ z e r o . p - hqr = m, q + hrp = m (5.50) =0 r has in the phase consisting 2 space of the (p, q , " l qr = - m r m Furthermore, since of singular space curve defined as the the two h y p e r b o l i c c y l i n d e r s pr = r) a set 2 T — points intersection of q 112 £ (5.51) this _2 m q m. curve l i e s wholly on the plane d e f i n e d by (5•51)• nature of t h i s The curve i n the plane can be seen i f we r o t a t e - m the pq plane about the r - a x i s by an angle 0 = arc tan In the r o t a t e d coordinate system, 2 « the equation of the curve is given by m q m ' (5.52) 2 q* s i n Q + p* cos 9 = h"rT Here p ' » q * , and r is l cos Q - p ' s i n 0 = 1 equivalent T denote the new c o o r d i n a t e s . Equation (5«52) to q» r r p' and hence t h i s = k' = constant = 0 curve i s a h y p e r b o l a as i l l u s t r a t e d i n Figure 5»3» in which the p * - a x i s i s p e r p e n d i c u l a r to the paper. This curve i n t e r s e c t s / m -m point I - , , o o 2 the plane r = r (^ 0) at the \ 1 r I • I The t r a j e c t o r i e s continuum of c o n c e n t r i c c i r c l e s point. Q For from equations p = 6 sin(hr t+a) Q with the in this center plane form a at the singular (5.17)/ we have + ^ ~ { ] m , s i n ( h r t ) + m [ 1-cos ( h r t ) ]] o, o 2 Q 113 m A, 2 = arctan q m o h~ + o then 2 m P - sin(hr t Q + A ) o In the same way, we have l q + T hr m These l a s t = A . c o s i h r t + A,) 1 o 1 o two equations radius define p a r a m e t r i c a l l y a c i r c l e of Consequently, i n the plane z = r , the end-point t» moves in a c i r c l e . The phase t r a j e c t o r i e s of i n the plane r = T Q are shown i n F i g u r e 5»U* Moreover, a continuous rotation. since r is constant, these c i r c l e s family of p e r i o d i c motions about a s t a b l e The p e r i o d of each of these motions T = which v a r i e s represent is permanent clearly 2 i r |hr | ' o' i n v e r s e l y with r f o r a given mass d i s t r i b u t i o n , o r e p r e s e n t e d by the II, we have the Theorem : constant From the results of Cases I and following For _a s e l f - e x c i t e d about _a f i x e d p o i n t independent h. symmetri c r i g i d body r o t a t i n g i n _a f r i c t i o n l e s s torque v e c t o r i s medium, i f the t ime- f i x e d i n the e q u a t o r i a l p l a n e , mot ion of the end-point of t» t akes p i ace who 11 y i n the plane z = r . In the phase the plane r = r Q Q space the initial (p, q , r ) , the t r a j e c t o r i e s form a continuum o f concent r i c c i r c l e s with in 114 Figure 5«4 115 canter nig -m^ s i n g u l a r point ( — , — , r ) , r e p r e s e n t i n g _a o o at the s t a t e of st able permanent being c o n s i d e r e d at parallel infinity, If r when r o a force-free the center degenerate into £ 0 the end-point z-axis, symmetric r i g i d body, we know of t» also moves i n a plane and i t s locus concentric c i r c l e s with center on the that one e f f e c t geometrically of the center of in t h i s z-axiso plane these c i r c l e s form Thus we observe a time-independent excitement with f i x e d d i r e c t i o n r e l a t i v e of = 0, these c i r c l e s case of p e r p e n d i c u l a r to the shift Q lines. In the that rot at i o n . self- to the body i s to a new l o c a t i o n to in the plane motion. Consider now the The motion of now, since the s i t u a t i o n when f r i c t i o n end-point r = r 9 Q negligible. of m becomes more c o m p l i c a t e d . the motion no longer takes p l a c e plane p e r p e n d i c u l a r to the z - a x i s . approaches i s not Instead, the For in a end-point a s y m p t o t i c a l l y the xy p l a n e . As mentioned i n Case I , the system (5*49) with m^ = 0 has l 2 l y one s i n g u l a r p o i n t ( ~ » ~ , o) • On the b a s i s of the lemma only one s i n g u l a r p o i n t TJ i n page 1 1 7 *nd formulas (5«16), we observe that as t -» a> the m components p, q , m l m m and r approach r e s p e c t i v e l y the values 2 — , —, and 0. approaches the Case I I I . If Thus, every t r a j e c t o r y i n the phase space singular point. h * 0, friction m^ * 0 is neglected the system (5.49) has no s i n g u l a r 116 point. Every t r a j e c t o r y i n the phase infinity. As the two i n t e g r a l s since on the r i g h t of these two kinds of | r | increases functions indefinitely v e c t o r aj approaches z-axis, r) tends to (5.17). (5.17) can be expressed and the the magnitudes of p and q are thus hand, (p, q , The components of m are now given by formulas terms of the F r e s n e l i n t e g r a l s and space in trigonometric-functions, are bounded f o r t > 0, bounded a l s o . when t -» » , On the other and as a r e s u l t a s y m p t o t i c a l l y the p o s i t i v e or the negative a c c o r d i n g as the a l g e b r a i c sign of m^ i s p o s i t i v e or negative. However, i f the body i s due to f r i c t i o n , the system f u r t h e r acted upon by a torque (5«A9) w i l l then possess a unique s i n g u l a r p o i n t which can be o b t a i n e d by s o l v i n g th© system of equations Vp - hqr - m^ = 0 (5-53) <? v + hpr - m = 0 2 - m^ = 0 LIT The s o l u t i o n of (5-53) i s m p = m r The l * 2 l 5 5— V m g , where g, = —~-* 2 - l l V m g V- f o l l o w i n g lemma w i l l tend a s y m p t o t i c a l l y to t h i s show that point. a l l other trajectories 117 Lemma: For "s(t) = g^t - g e (ii) J Proof ; f 9~ - -—^(V.illCTn+giCO.CTi) = f e~ ^ ~*^cos[3(s)-3(t)-a,]ds - g gCvcoaoT-qi^iaai,) v +gj_ t V 9 ( t - *g." 1 yields ^(°>-5( >- i^ 8 i n t jV ^ ( v t •jV The l -vt TT^- •V" x C t I n t e g r a t i o n by p a r t s sincr. J V ~"^»in[^(»)-S(t)- > °» » s t - » v ]dB J .- (i) + g, V > °# _ p , t v ( t v ) 8 a cos[5(.)-5(t)-a ]d 1 ( "" co.[3(.)-5(t)-a 3d. ) 1 t h i r d term on the r i g h t approaches zero i s t •* » . For we h ave |e- J .- *V t t ( v ) 8 co.^(.)-5(t)-a ]d»| 1 •- J .- » - d. Vt < t ( i v)B 0 -Vt S i n c e the r i g h t hand s i d e tends hand s i d e does a l s o . Consequently, we may w r i t e g. (5.54) The J x to zero when t -» <», the -sinai * -^J 2 " —v same procedures y i e l d + e l ' w h e r a e i •* 0 " 1 "* left 118 J coso"i ~ 7^ - • 7~ c £ 2 JJ g e s ^(°)-^ )-a ] t 1 - g1 • o jV^'Kin^s)-^)-*^ V t J t e -v(t- ) 8 8 i n [ ^ ( B ) _^ ( t ) _ C T i - l d s "0 V Because the second and t h i r d terras tend to zero as t -* • , we may wri te g. (5.55) J J"l = - 2 Solving for cosCT, - and "~2^~2 ^ V s — g — • * 2 from (5-54), i n C T l + g i C O S a l^ + ' w h e r e e 2 "* 0 a s 1 "* (5.55), we o b t a i n E 3 * where -» 0 as t -» «o r and Jo 2 = ' o' o V +g 1 2 ( V C O S C T I - g . s i n g . ) + e , where £ , -» 0 as t -* » . 2 4 Thus from formulas (5•16) and the above lemma, we have as t -* < * > vm +g m P = 2 2 v +g^ 1 V m q = 2 1 2 2- l l g m 2 v^+gj S i n c e a l l other t r a j e c t o r i e s the tend to the s i n g u l a r p o i n t , s t a t e of permanent r o t a t i o n represented by t h i s point clearly stable. The'above p i c t u r e of the phase t r a j e c t o r i e s can be compared with the streamlines incompressible is of a t h r e e - d i m e n s i o n a l flow a l l tending to a p o i n t - s i n k . steady 119 In the case when the i n i t i a l p o s i t i o n of the end-point of 3 i n the plane z = ~ , then equation m t» l i e s will remain f o r e v e r this means that m plane r » 77*% will in tRis plane. In the phase space which contains it (p, q , r ) every t r a j e c t o r y o r i g i n a t e d at any p o i n t i n the the s i n g u l a r p o i n t of the system, remain f o r e v e r i n t h i s p l a n e . first (5o25) shows that Furthermore, m u l t i p l y i n g the equation of (5•30) by i and adding t h i s to the second one, we have °2- l l\ ^ . ( g V l m v i y l 2 \ . . -vt < V ° g 1 m = e The If m 1 2 -vt i(» t* *(l ) r oe - 0l e 6 2 term i n the bracket can be w r i t t e n as i < T 6e -Vt ) l m 1 2 (o +a ) / l l' i -j—.e 2 1 m 2 g 2 v r a =I 2—2" \ v^+g* \ / • q I* i(7 \ V r a we l e t 6^ and o*^ denote r e s p e c t i v e l y argument of t h i s / (q \ vector, 2 2 / V**g* / l + g l r a 2 * ol 2 v *q P 2 7 A 1 the modulus and the then / v \ ~ p \ m i ^ i m 2 2 2 \ / 3 = 6 V^+g, / -vt i C ^ i * * ^ ) e e •> m T h i s equation shows that the t r a j e c t o r i e s s p i r a l toward the s i n g u l a r p o i n t , Case I V . In i n the plane r = -~ as i l l u s t r a t e d i n F i g u r e 5»5» h = 0 the case of dynamic symmetry and n e g l i g i b l e we have from ( 5 » 3 ) friction, 2 120 P P a g, > o q Figure P=P Q • mt q = r = r These expressions indefinitely considered. the phase + o x m„t show that at a uniform Again t h i s increase the angular v e l o c i t y increases rate. will For equations space 5.5 be r e s t r i c t e d when f r i c t i o n (5«3l) are r e c t i l i n e a r , show that is a l l trajectories and tend a s y m p t o t i c a l l y to the only s i n g u l a r p o i n t The q u a l i t a t i v e conveniently Theorem: results obtained i n S e c t i o n 5*5 summarized i n t o the For a s e l f - e x c i t e d following symmetrie r i g i d body rot at i n g about _a f i x e d p o i n t i n a v i s c o u s body approaches medium, the mot ion of the asymptot i c a l l y a uniform rot at ion about an axi s f i x e d i n the body as we .11 as i n space. velocity can be components along the x, y, The angular z axes of the body-fixed in 121 V m trihedral are respect j v e l y l l 2 5 5—* + g m V m V^+g~ If increases lies f r i ct ion i s n e g l e c t e d , indefinitely, the g m M M * angular v e l o c i t y except i n the i n the e q u a t o r i a l p l ane with the side t h i s plane 2~ l l 3 o o—' TL ' i n general case when the torque vector i n i t i a l pos it'ion of ts\ out- and the body i s not dynamically symmetric. 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Lecture notes ( 1 9 6 0 - 6 1 ) , U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada. , N o n - l i n e a r d i f f e r e n t i a l e q u a t i o n s . Lecture notes ( 1 9 6 0 - 6 1 ) , U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada. 22. mechanics, , Minorsky, N . , Dynamics and n o n - l i n e a r John Wiley and Sons, I n c . , New York, 1 9 5 8 . 23. M a c M i l l a n , W. D . , Dynamics of r i g i d b o d i e s , P u b l i c a t i o n s , I n c . , New York, 1 9 3 6 . 24. M a l k i n , I . G . , Some problems i n the theory of n o n - l i n e a r o s c i l l a t i o n s , A E C - t r - 3 7 6 6 , Book 1 and 2 , O f f i c e of T e c h n i c a l S e r v i c e s , Washington 2 5 , D. C . 25. M e r k i n , D. R . , On the theory of s e l f - e x c i t i n g g y r o s c o p e s , t r a n s l a t e d from R u s s i a n , O f f i c e of T e c h n i c a l S e r v i c e s , Washington 2 5 , D. C . 26. 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The motion of a self-excited rigid body Lee, Richard Way Mah 1964
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Title | The motion of a self-excited rigid body |
Creator |
Lee, Richard Way Mah |
Publisher | University of British Columbia |
Date Issued | 1964 |
Description | This thesis discusses the motion of a rigid body about a fixed point subject to a torque arising from internal reaction. Such a body is called self-excited, and its motion is governed by Euler’s dynamical equations. First, in Section 2 we consider the case of a torque vector which is fixed in direction along the largest or smallest principal axis of the body, and has a component in the chosen axis equal to a constant plus a perturbation term that is proportional to the square of the modulus of the spin vector [character omitted] (p,q,r). It is shown that Euler's equations can be integrated in terms of a variable φ, introduced by means of a differential relation. Further quadrature and inversion yield p,q, and r as functions of the time t. Using the method of phase-plane analysis, we show that the spin vector can perform a variety of motions with respect to the body-fixed trihedral. In particular, when the perturbation is zero, we infer from the corresponding phase-plane trajectories that the spin vector can perform asymptotic motions of the first and second kinds and periodic motions about permanent axes lying in the principal plane perpendicular to the torque vector. Some of the results for this case were also obtained by Grammel, using different method. In the general case, when the perturbation is not zero, these motions are preserved. However, a second type of periodic motion exists; it occurs about the principal axis containing the torque vector, the principal axis itself being a direction of stable permanent rotation. In Section 3 we consider the same problem with the torque vector acting along the middle principal axis. Using the methods of the previous Section, we show that [character omitted] can assume periodic motions as well as asymptotic motions of various kinds. The periodic motions established in these two Sections are then computed in Section 4 as power series in a small parameter. Finally, in Section 5 the motion of a symmetric rigid body moving in a viscous medium subject to a time-dependent torque is studied. Its motion is compared with that in a vacuum. We show first that p,q, and r can be expressed in terms of certain integrals. For the special case where the self-excitement is time-independent and fixed in direction within the body, these integrals can be reduced to the generalized sine and cosine integrals. Their values can be computed from asymptotic and power series which are developed in the same Section. The asymptotic behavior of the spin vector is then discussed, yielding qualitative results which are summarized in three theorems. |
Subject |
Dynamics, Rigid. Motion. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-10-17 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080543 |
URI | http://hdl.handle.net/2429/38029 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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