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The motion of a self-excited rigid body Lee, Richard Way Mah 1964

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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.  In  such _a c a s e , e i t h e r a> assumes _a st able p e r i o d i c motion with p e r i o d or the body assumes ja st able pe rmanent rot at ion equal to T = hr 27T o 2 ~ l with , ^ , r _as the angular v e l o c i t y components i n the m  m  Q  0  0  d i r e c t i o n of the  x,  y,  z axes  respectively.  122  BIBLIOGRAPHY 1.  Andronov, A . , C h a i k i n , S o , Theory of o s c i l l a t i o n s , Moscow, 1937, E n g l i s h t r a n s l a t i o n by L e f s c h e t z , S., P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , 1949.  2.  Bodewadt, U . T „ , Der syrametrische K r e i s e l bei z e i t f e s t e r D r e h k r a f t , Math. Z . , 55 (1952), pp. 310-320.  3.  Bogoliubov, N . N . , M i t r o p o l s k y , Y. A . , Asymptotic methods in the theory of n o n - l i n e a r o s c i l l a t i o n s , translated from R u s s i a n , Hindustan P u b l i s h i n g C o r p o r a t i o n , D e l h i ,  I n d i a , 196.1.  4.  Byrd, P. F . , Friedman, M. 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