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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Chaotic oscillations of the spacecraft
with a vertical tether
The types of spacecraft The bifurcation diagram
1  cos  sin   0

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. 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. 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. 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. 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. 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