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- 1. CHAOTIC MOTIONS OF TETHERED SATELLITES WITH LOW THRUST Vladimir S. Aslanov1, Arun K. Misra2, Vadim V. Yudintsev1 1Samara National Research University, Samara, Russia 2McGill University, Montreal, QC, Canada 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 1
- 2. OUTLINE 1. System description and assumptions 2. Motion equations 3. Stationary solutions 4. Chaotic motion due to eccentricity of the orbit 5. Chaotic motion due to out-of-plane oscillations 6. Conclusion 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 2
- 3. SYSTEM DESCRIPTION • Chaotic behaviour of the system (Space Tug + Tether + Debris) is considered • The following disturbances are assumed to affect the in-plane motion of the tether • Orbital eccentricity • Out-of-plane motion of the tether 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 3
- 4. ASSUMPTIONS • The tug and debris are point masses • Tug’s thrust is low • 𝑀𝑐 𝑷 is comparable to 𝑀𝑔𝑟𝑎𝑣 • The orbit is not changed • |R|= const • e = const • Tether length is constant • 𝐶1 𝐶2 = 𝑙 = const 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 4
- 5. MOTION EQUATIONS • In-plane motion cos2 𝛾 𝛼 + 𝜗 − 2 𝛾 𝛼 + 𝜗 tan 𝛾 + 3 2 𝜇 𝑅3 sin 2𝛼 = 𝑄 𝛼 𝑙2 𝑚12 • Out-of-plane motion 𝛾 + 𝛼 + 𝜗 2 + 3 𝜇 𝑅3 cos2 𝛼 sin 𝛾 cos 𝛾 = 𝑄 𝛾 𝑙2 𝑚12 • Generalized forces 𝑄 𝛼 = 𝑙 𝑚2 cos 𝛾 cos 𝛼 𝑚12 𝑃, 𝑄 𝛾 = − 𝑙𝑚2 sin 𝛾 sin 𝛼 𝑚12 𝑃 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 5
- 6. DIMENSIONLESS EQUATIONS • In-plane motion cos2 𝛾 𝛼′′ − 2𝛾′ tan 𝛾 + 𝐾 𝛼′ + 1 + 3 2 𝐺 sin 2𝛼 = 𝑄 𝛼 • Out of plane motion 𝛾′′ − 𝐾𝛾′ + 𝛼′ + 1 2 + 3𝐺cos2 𝛼 sin 𝛾 cos 𝛾 = 𝑄 𝛾 where 𝛼′′ = 𝑑2 𝛼 𝜕𝜗2 , 𝛾′′ = 𝑑2 𝛾 𝜕𝜗2 , 𝐺 = 1 1 + 𝑒 cos 𝜗 , 𝐾 = 2 𝑒 sin ϑ 1 + 𝑒 cos 𝜗 • Generalized forces 𝑄 𝛼 = 𝑄 𝛼 𝑙2 𝑚12 ⋅ 𝐺4 𝑝3 𝜇 , 𝑄 𝛾 = 𝑄 𝛾 𝑙2 𝑚12 ⋅ 𝐺4 𝑝3 𝜇 , 𝑚12 = 𝑚1 𝑚2 𝑚1 + 𝑚2 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 6
- 7. STATIONARY MOTIONS In-plane motion in a circular orbit 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 7
- 8. STATIONARY MOTION • For in-plane motion in a circular orbit (e=0, 𝛾 =0) 𝛼′′ = 𝑃 𝑚1 𝑙0 𝜔2 cos 𝛼 − 3 2 sin 2𝛼 = 𝑚 𝛼 • First integral 𝛼′ 2 2 + 𝑊 𝛼 = 𝐸 Torque 𝑚 𝛼 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 8
- 9. POTENTIAL ENERGY • First Integral 𝛼′ 2 2 + 𝑊 𝛼 = 𝐸 • Potential energy 𝑊 𝛼 = −𝑎 sin 𝛼 − 𝑏 cos2 𝛼 where 𝑎 = 𝑃 𝑚1 𝑙0 𝜔2 , 𝑏 = 3 2 , 𝜔 = 𝜇 𝑅3 Potential energy 𝑊 𝛼 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 9 a=2
- 10. POTENTIAL ENERGY • First Integral 𝛼′ 2 2 + 𝑊 𝛼 = 𝐸 • Potential energy 𝑊 𝛼 = −𝑎 sin 𝛼 − 𝑏 cos2 𝛼 where 𝑎 = 𝑃 𝑚1 𝑙0 𝜔2 , 𝑏 = 3 2 , 𝜔 = 𝜇 𝑅3 Potential energy 𝑊 𝛼 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 10 a=4
- 11. STABLE POINT • If 𝑃 𝑚1 𝑙0 𝜔2 > 3 • Stable position only 𝑎 𝑠 = 𝜋 2 Bifurcation diagram 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 11 𝑎 𝑠
- 12. UNSTABLE POINTS • If 𝑃 𝑚1 𝑙0 𝜔2 < 3 • Stable positions 𝑎 𝑠1 = asin 𝑎 2𝑏 , 𝑎 𝑠2 = 𝜋 − asin 𝑎 2𝑏 • Unstable position 𝑎 𝑢 = asin 𝑎 2𝑏 Bifurcation diagram 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 12 𝑎 𝑠1 𝑎 𝑠2 𝑎 𝑢
- 13. ORBIT ECCENTRICITY 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 13
- 14. CHAOTIC MOTION • Parameters 𝑎 = 7371 km, 𝑒 = 0.05, 𝑙0 = 100 m 𝑚1 = 500 kg, 𝑚2 = 3000 kg • Tug’s thrust 𝑃 = 0.1 N → 𝑎 = 𝑃 𝑚1 𝑙0 𝜔 𝑚𝑖𝑛 2 < 3 • Orbital rate 𝜔 𝑚𝑖𝑛 = 𝜇𝑝−3 1 − 𝑒 ≈ 0.0009 𝑠−1 𝜔 𝑚𝑎𝑥 = 𝜇𝑝−3 1 + 𝑒 ≈ 0.0011 𝑠−1 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 14
- 15. REGULAR MOTION • Parameters 𝑎 = 7371 km, 𝑒 = 0.05, 𝑙0 = 100 m 𝑚1 = 500 kg, 𝑚2 = 3000 kg • Tug’s thrust 𝑃 = 0.2 N → 𝑎 = 𝑃 𝑚1 𝑙0 𝜔 𝑚𝑖𝑛 2 > 3 • Orbital rate 𝜔 𝑚𝑖𝑛 = 𝜇𝑝−3 1 − 𝑒 ≈ 0.0009 𝑠−1 𝜔 𝑚𝑎𝑥 = 𝜇𝑝−3 1 + 𝑒 ≈ 0.0011 𝑠−1 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 15 Poincare section
- 16. LARGEST LYAPUNOV EXPONENT • Lyapunov exponents for trajectory • 𝛼0 = 𝜋/2, 𝛼0 = 0 • 𝛾0 = 0.00, 𝛾0 = 0 • The largest Lyapunov exponent tends to zero if 𝑃 = 0.2 N. • For 𝑃 = 0.1 N it tends to a positive value, about 0.08 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 16
- 17. OUT-OF-PLANE OSCILLATIONS 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 17
- 18. CHAOTIC MOTION • Parameters 𝑎 = 7371 km, 𝑒 = 0, 𝑙0 = 100 m 𝑚1 = 500 kg, 𝑚2 = 3000 kg, 𝛾0 = 0.1 • Tug’s thrust 𝑃 = 0.1 N → 𝑎 = 𝑃 𝑚1 𝑙0 𝜔2 < 3 • Orbital rate 𝜔 = 𝜇𝑝−3 ≈ 0.0009 𝑠−1 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 18
- 19. REGULAR MOTION • Parameters 𝑎 = 7371 km, 𝑒 = 0.05, 𝑙0 = 100 m 𝑚1 = 500 kg, 𝑚2 = 3000 kg, 𝛾0 = 0.1 • Tug’s thrust 𝑃 = 0.2 N → 𝑎 = 𝑃 𝑚1 𝑙0 𝜔 𝑚𝑖𝑛 2 > 3 • Orbital rate 𝜔 = 𝜇𝑝−3 ≈ 0.0009 𝑠−1 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 19
- 20. LARGEST LYAPUNOV EXPONENT • Lyapunov exponents for trajectory • 𝛼0 = 𝜋/2, 𝛼0 = 0 • 𝛾0 = 0.10, 𝛾0 = 0 • The largest Lyapunov exponent tends to zero if 𝑃 = 0.2 N. • For 𝑃 = 0.1 N it tends to a positive value, about 0.1 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 20
- 21. CONCLUSION • The pitch motion of the system is perturbed by the • out-of-plane roll motion of the tether and • change in the gravity gradient due to motion of the center of mass of the system in an elliptic orbit • The orbital motion of the center of mass of the system and out-of-plane oscillation of the tether can cause chaos if there is an unstable equilibrium. • The choice of the thrust and mass of the space tug, as well as the tether length, should be such as to satisfy the condition for in-plane motion in an elliptic orbit 𝑃 𝑚1 𝜇 𝑙0 1 − 𝑒 4 𝑝3 > 3 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 21
- 22. Thank you! Vladimir S. Aslanov (aslanov_vs@mail.ru) Arun K. Misra (arun.misra@mcgill.ca) Vadim V. Yudintsev (yudintsev@classmech.ru) 67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 22

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