 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
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| In English (Mech. Solids): | | 4760 |
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| O. V. Kholostova, "Stability of plane oscillations of a satellite in a circular orbit," Mech. Solids. 43 (2), 185-198 (2008) |
| Year |
2008 |
Volume |
43 |
Number |
2 |
Pages |
185-198 |
| DOI |
10.3103/S0025654408020039 |
| Title |
Stability of plane oscillations of a satellite in a circular orbit |
| Author(s) |
O. V. Kholostova (Moscow Aviation Institute (State University of Aerospace Technologies), Volokolamskoe sh. 4, GSP-3, A-80, Moscow, 125993, Russia, kholostova_o@mail.ru) |
| Abstract |
We study motions of a rigid body (a satellite) about the center of mass in a central Newtonian gravitational field in a circular orbit. There is a known particular motion of the satellite in which one of its principal central axes of inertia is perpendicular to the orbit plane and the satellite itself exhibits plane pendulum-like oscillations about this axis. Under the assumption that the satellite principal central moments of inertia A, B, and C satisfy the relation B = A + C corresponding to the case of a thin plate, we perform rigorous nonlinear analysis of the orbital stability of this motion.
In the plane of the problem parameters, namely, the oscillation amplitude ε and the inertial parameter, there exist countably many domains of orbital stability of the satellite oscillations in the linear approximation. Nonlinear orbital stability analysis was carried out in thirteen of these domains. Isoenergetic reduction of the system of equations of the perturbed motion is performed at the energy level corresponding to the unperturbed periodic motion. Further, using the algorithm developed in [1], we construct the symplectic mapping generated by the equations of the reduced system, normalize it, and analyze the stability. We consider resonance and nonresonance cases. For small values of the oscillation amplitude, we perform analytic investigations; for arbitrary values of ε, numerical analysis is used.
Earlier, numerical analysis of stability of plane pendulum-like motions of a satellite in a circular orbit was performed in several special cases in [1-4]. |
| References |
| 1. | A. P. Markeev,
"Stability of Planar Rotations of a Satellite in a Circular Orbit,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 4, 63-85 (2006)
[Mech. Solids (Engl. Transl.)
41 (4), 46-63 (2006)]. |
| 2. | A. P. Markeev, "Stability of Plane Oscillations and Rotations of a
Satellite in a Circular Orbit," Kosmich. Issled. 13 (3),
322-336 (1975) [Cosmic Res. (Engl. Transl.) 13, 285-298
(1975)]. |
| 3. | A. P. Markeev and A. G. Sokol'skii, "Study of Stability of
Satellite Plane Periodic Motions in a Circular Orbit," Izv. Akad.
Nauk SSSR. Mekh. Tverd. Tela, No. 4, 46-57 (1977) [Mech. Solids]. |
| 4. | A. P. Markeev and B. S. Bardin, "On Stability of Planar
Oscillations and Rotations of a Satellite in an Circular Orbit,"
Celestial Mechanics and Dynamical Astronomy 85 (1), 51-66
(2003). |
| 5. | V. V. Beletskii,
Satellite Motion about the Center of Mass in Gravitational Field
(Izd-vo MGU, Moscow, 1975)
[in Russian]. |
| 6. | A. P. Markeev,
Theoretical Mechanics
(SPC. "Regular & Chaotic Dynamics," Izhevsk, 2001)
[in Russian]. |
| 7. | A. M. Lyapunov,
"On Stability of Motion in a Special Case of the Three-Body Problem,"
in Collected Works
(Izd-vo AN SSSR, Moscow-Leningrad, 1954)
Vol. 1, pp. 327-401
[in Russian]. |
| 8. | V. A. Yakubovich and V. M. Starzhinskii,
Parametric Resonance in Linear Systems
(Nauka, Moscow, 1987)
[in Russian]. |
| 9. | A. P. Markeev,
"On a Multiple Resonance in Linear Hamiltonian Systems,"
Dokl. Ross. Akad. Nauk
402 (3), 339-343 (2005)
[Dokl. Phys. (Engl. Transl.)
50 (5), 278-282 (2005)]. |
| 10. | A. P. Markeev,
Libration Points in Celestial Mechanics and Space Dynamics
(Nauka, Moscow, 1987)
[in Russian]. |
| 11. | V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt,
Mathematical Aspects of Classical and Celestial Mechanics, 2nd ed.
(Springer, Berlin, 1997; Editorial URSS, Moscow, 2002). |
| 12. | J. Glimm,
"Formal Stability of Hamiltonian Systems,"
Comm. Pure Appl. Math.
17 (4), 509-526 (1964). |
| 13. | J. Mozer,
"New Aspects in the Theory of Stability of Hamiltonian Systems,"
Comm. Pure Appl. Math.
11 (1), 81-114 (1958). |
|
| Received |
14 June 2005 |
| Link to Fulltext |
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