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Dynamics of Satellite With a Tether System

  1. 1. Vladimir S. Aslanov aslanov_vs@mail.ru Dynamics of satellite with a Tether System Theoretical Mechanics Department www.termech.ru Samara State Aerospace University, Russia www.ssau.ru 2012
  2. 2. Statement of the problem The motion about a centre of mass of a spacecraft (satellite) with a elastic heavy tethered system at a orbit is studied. Tethered satellite systems (TTS) includes: - rigid satellite (spacecraft), - elastic heavy tether, - end load. The dynamics of a rotating body studied famous mathematicians of all time as Euler, Poinsot, Lagrange and Kovalevskaya. The research of the dynamics of rotating bodies is very important for numerous applications such as the dynamics of satellites. In this area we note the papers of scientists as Yaroshevsky, Belezky, Rumyantsev, J.Nicolaides, G.Gross et al. Study the behavior of the space tethered systems devoted to the papers: Beletsky and Levin, Williams, Kruijff, Misra, Sidorov, Pirozhenko and others. 2
  3. 3. Example of the Tethered Satellite Systems Scheme of the dynamic deployment of TSS «Foton-М3" №3 – YES2" (2008) Initial Foton-M3 parameters are assumed as follows: Mass 6530 kg Ballistic coefficient 0.0123 m2/kg. Inclination 63 degrees Minimum orbital altitude 262 km Maximum orbital altitude 304 km Tether parameters are assumed as follows: Diameter 0.5 mm Length 30000 m Mass density 0.00018 kg/m Initial Speed of tether deployment 2.58 m/c Mass End Load 12 kg 3
  4. 4. Aims of the research 1. To obtain mathematical models of the plane motion of the satellite of about of mass center under the influence of elastic the tether system. 2. To deduce approximate analytical solutions describing the oscillations of the satellite caused by the change magnitude and direction of the tether force. 3. To build models chaotic behavior of the satellite and to study of the satellite motion under the influence the elastic tether of the chaotic dynamics methods. 4. To find the approximate estimates of the accelerations in the satellite arising from the deployment of the tether. 4
  5. 5. The Lagrange equations Kinetic energy of the TSS 1 1 2 1 T  m(r  r  )   mi i2  C0 (   )2  C1 (   )2   2 2 2        (1) 2 2 i0 2 where ρi  ri  r, i  0,1,2; q j   , , , l , r - generalized coordinates Potential energy 2 mi 3  m1l 2 c W      3  A  B  cos   2 cos 2   (l  l0 )2 (2) i 0 ri 2r0 8r13 2 Lagrange equations of the second kind d L L   Qj   D0 P, l  PD2 dt q j q j  where L  T W - Lagrange function, Qj - nonpotential forces 5
  6. 6. The motion equations The approximate motion equations of the TTS We assume  / l  1, l / r  1 C0  C0  ml cos(   )  ml cos(   )  m sin(   )  f1 (l, , , )  Q    l  (3) ml cos(   )   I  ml cos(   )   f 2 (l, , , , )  Q     (4)    l    Q  sin(   )       f3 (l, , , , )  l m (5)  mr 2  C0  I   C0  I  f3 (l, , r , , )  Q      (6)  3 I 9 4  r    r 2  2  1  3cos 2    4  A  B  cos 2   Qr (7) r 2mr 2mr where m  m0 m2 / m, I  ml 2 6
  7. 7. The motion equations on a elliptic orbit Since the orbital time on a elliptic orbit is relatively short, it may be assumed that the centre of mass remains in an unperturbed Keplerian elliptic orbit. In such a case, the generalized coordinates and are known through p p r     nk 2 n   p 3 1  e cos k d Substitution variable from t to the true anomaly angle θ: dt  n 1  e cos  2 The motion equations Q C0  k   2e  sin    ml cos(   )  k   2e  sin    m sin(   )kl   f1* ( , , , ', l ',)  (8) n2 k 3 Q ml cos(   )  k   2e  sin    I k   f 2* ( , , , ', ', l ',)  (9) n2 k 3 Ql  sin(   )k   kl   f3* ( , , , ', ', l ',)  (10) mn2 k 3 7
  8. 8. The equations of elastic vibrations the tether We assume that the line of action of the tether tension is the center of mass of the spacecraft, then 0 Q 0  The equations of elastic vibrations the tether l 3 e    2 1      sin  cos   2 1     sin  (11) l k k l  l0   1  3cos    l 1   c l e 2 4  l      2 l  sin  2 2 (12) mn k k k 8
  9. 9. The elastic vibrations of tether near the local vertical We assume, that:   O   Motion equations of the elastic tether A B    3 sin  cos   J 1  L  sin   2 L cos    2e 1     sin    (13) kC c L   L  1  3   sin    1     cos   2e  cos  L sin   2 (14) n2 k 4 m    l ml02 where   , L , J  , C  C0  m0  2 l l0 C 9
  10. 10. The approximate analytical solutions The motion equation of the spacecraft under the action of the tension force and the gravitational moment C  T  sin(   )  3n 2 ( B  A) sin  cos   (15) where  -angle between the longitudinal axis of the spacecraft and the local vertical    ( ) - angle between the rope and the local vertical T  T ( ) - tension force A, B, C - inertia moments of the spacecraft 3n 2 ( B  A) sin  cos  - gravitational moment   t - the slow time  - small parameter   CA 10
  11. 11. The approximate analytical solutions The motion equation of the spacecraft under the action of the tension force only   ( )sin   ( ) cos    sin 2  (16) where  ( )   2 ( ) cos  ( ),  ( )   2 ( ) sin  ( ), 3   n 2  B  A / C , 2  2 ( )  T ( ) / C Exact solution in terms of elliptic functions for   0     2arcsin  sn(t  K (k ), k ) (17) 11
  12. 12. The approximate analytical solutions The tension force and its direction change slowly over time T  T ( ),    ( ) The adiabatic invariant J (, k )    E (k )  (1  k 2 ) K  k   h  const   (18) The approximate analytical solutions 2 3 h 1  h  1  h   min,max  t    (t )  2 arcsin   2   (t )  4   (t )   ... (19)  (t ) 2     If   - is small value, then 0  min,max  t    (t )  A0 (20)  (t ) where A0 is the arbitrary constant Micro-acceleration at the point the remote at a distance d from the mass center x0 d  4 Wmax (t )  T0 T (t ) 3/4 (21) C 12
  13. 13. The approximate analytical solutions The simulations for the YES-2 The deployment trajectory of the TTS The deflection angle of the tether from the local vertical and the tension force Oscillations of the spacecraft about Accelerations on mass center the spacecraft to point removed at d = 1m 13
  14. 14. The approximate analytical solutions The linearized equation of the spacecraft motion under the influence of the gravitational torque and the tension force    a( )  c   b( )  0  (22)   B A where a( )  T ( ) cos  ( ), b( )  T ( ) sin  ( ), c  3n 2 0 C C C The approximate solution for the oscillation amplitude of the spacecraft const C T (t ) sin  (t )  max (t )   (23) T (t ) cos  (t )  3n ( B  A) 2 T (t ) cos  (t )  3n2 ( B  A) 14
  15. 15. Chaotic oscillations of the spacecraft with a vertical tether The motion equations of the spacecraft with the elastic vertical tether for a circular orbit A B    3 sin  cos   J 1  L  sin   2 L cos    (24) kC c L  2 4  L  1  3   sin    1     cos  2 (25) n k m   Approximate law of change rope length (δ = 0) L    c / m 1/2 / n, L1   3  2  2  L  L1  0 sin     3  The tether will always be stretched (L> 1) if L0   The equation of the perturbed motion of the spacecraft about its mass center    a sin   c sin  cos     sin  sin   2cos cos   (26) ml0 B A  ml0 L0 where a  , c3 ,  - the small parameter C  m1 2 C  m1 2 C  m1 2 15
  16. 16. Chaotic oscillations of the spacecraft with a vertical tether The equation of the unperturbed motion of the spacecraft    a sin   c sin  cos (27)  2 The energy integral:  W ( )  E 2 Equilibrium position is defined as the roots of the equation c B A 1 3 2  1   cos   sin   0,       (28) a   m2l0 ES  for  *    ,0   0,    *   arccos   1  for the remaining provisions of  *   , 0,  16
  17. 17. Chaotic oscillations of the spacecraft with a vertical tether The types of spacecraft The bifurcation diagram 1  cos  sin   0
  18. 18. Chaotic oscillations of the spacecraft with a vertical tether The homo-heteroclinic trajectories (separatrix solutions) k  c/a Separatrix solutions 1  d  2 d sinh t   1   (t )  2arctg   ,   (t )  ( )     ,   a  c , d   a  c  cosh t  (cosh t )  d 2 2 a 2   1, 2 d cosh t   (t )  2arctg  d sinh t  ,   (t )  ( )    ,   a  c, d  a 1  d 2 sinh 2 t ac  1 2 cosh t 3  0     (t )  2arctg sinh at ,   (t )  ( )    1  sinh 2 t , a  S t   sin  S   (t )  2arctg  tg th  ,   (t )  ( )     ,  2 2 cosh t  cos  S 4  1  1 c2  a2 a  S   arccos    ,   ,d    c c  S t   sin  S   (t )    2arctg  ctg th  ,   (t )  ( )    .  2 2 cosh t  cos  S 5  1  1 c2  a2 a  S   arccos    ,   , d    c c 18
  19. 19. Chaotic oscillations of the spacecraft with a vertical tether Melnikov method The equation of perturbed motion of the spacecraft - a generalized Duffing equation   a sin   c sin  cos   sin sin t   .   (29) Two first-order equations     f1  g1 ,  (30)   f2  g2 ,  (31) where f1   , g1  0, f 2  a sin   c sin  cos  , g 2   sin  sin t    Melnikov function M  (t0 )   ( f1g2  f 2 g1 )dt  M   M  ,   M  ( k )     k ) sin  k ) sin (t  t0 )dt   I k ) sin(t0 ) ( ( (   M  ( k )    ( k ) )2dt   J k ) , k  1,2...5 ( (  The condition of absence of the chaos: M  M 19
  20. 20. Chaotic oscillations of the spacecraft with a vertical tether Improper integrals appearing in Melnikov function for the different motion types sinh 2 sinh 2     I (1)   2d  2 sin 1 d , J   4d 2   (1) 2 d  (cosh   d )  cosh   d   2 2 2 2 sinh 2 cosh 2     I (2)   d  2 sin 2 d , J   4d 2   (2) 2  d  (d sinh   1)  1  d sinh    2 2 2 2 sinh 2   cosh  2   I (3)   sin  2 d , J   4   (3) d   (sinh 2   1) 2  1  sinh 2     2 sinh    sin  S  (4)  (1  d )  2 sin 4 d , J      (4) d  cosh   cos   I   (cosh   d ) 2  S  2 sinh    sin  S  (5)  (1  d )  2 sin 5 d , J      (5) d  cosh   cos   I   (cosh   d ) 2  S  where i   / ,   t 20
  21. 21. Chaotic oscillations of the spacecraft with a vertical tether The Poincare sections Load mass 100kg Load mass 100kg Load mass 20kg  0  0   5  104 21
  22. 22. Chaotic oscillations of the spacecraft with a vertical tether Numerical simulation The TTS parameters: the mass of spacecraft - 6000kg, load weight - 100 kg, p =6621 km, Δ = 2m, E = 5000N, load weight of 100 km, 30 km length of the tether, inertia moments: A = 2500kgm2, B = C = 10000kgm2, the initial velocity load-1m / s (the case k = 2). The Melnikov functions 22
  23. 23. The main results were published in the following papers 1. Aslanov V. S. and Ledkov A. S. Chaotic Oscillations of Spacecraft with an Elastic Radially Oriented Tether, ISSN 00109525, Cosmic Research, 2012, Vol. 50, No. 2, pp. 188–198. 2. Aslanov V.S. Orbital oscillations of an elastic vertically-tethered satellite, Mechanics of Solids, Vol. 46, Number 5, 2011, pp. 657-668, DOI: 10.3103/S0025654411050013. 3. Aslanov V.S. The effect of the elasticity of an orbital tether system on the oscillations of a satellite - Journal of Applied Mathematics and Mechanics 74 (2010) 416–424. 4. Aslanov V. Oscillations of a Spacecraft with a Vertical Elastic Tether, AIP Conference Proceedings 1220, CURRENT THEMES IN ENGINEERING SCIENCE 2009: Selected Presentations at the World Congress on Engineering-2009, Published February 2010; ISBN 978-0-7354-0766-4, One Volume, pp.1-16. 5. Aslanov V. Oscillations of a Spacecraft with a Vertical Tether. Proceedings of the World Congress on Engineering 2009 v. 2, pp. 1827-1831. 6. Aslanov V. The Oscillations of a Spacecraft under the Action of the Tether Tension. Moment and the Gravitational Moment AIP (American Institute of Physics) Conf. Proc. September 1. 2008. v. 1048. 56-59 p. (ISBN: 978-0-7354-0576-9 ) 7. Aslanov V. S. Chaotic behavior of the biharmonic dynamics system. International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 319179, 18 pages doi:10.1155/2009/319179. 2009. 8. Aslanov V. S. The oscillations of a body with an orbital tethered system - Journal of Applied Mathematics and Mechanics 71 (2007) 926–932. 23

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