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IssuesArchive of Issues2006-4pp.46-63

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A. P. Markeev, "Stability of planar rotations of a satellite in a circular orbit," Mech. Solids. 41 (4), 46-63 (2006)
Year 2006 Volume 41 Number 4 Pages 46-63
Title Stability of planar rotations of a satellite in a circular orbit
Author(s) A. P. Markeev (Moscow)
Abstract We study the stability of planar motions of a satellite relative to the center of mass in a central Newtonian gravitational field. The satellite is a rigid body whose geometry of mass corresponds to a plate (C=A+B, where A, B, and C are the principal central moments of inertia). The orbit of the center of mass is circular. In the unperturbed motion, one of the principal central axes of inertia of the satellite is perpendicular to the orbit plane, and the satellite itself rotates around this axis at an arbitrary angular velocity. The problem on the stability of this rotation is solved in the nonlinear setting. To this end, we propose a constructive algorithm for studying the orbital stability of periodic motions of autonomous Hamiltonian systems. This algorithm is based on a special construction method and the subsequent analysis of the symplectic mapping generated by the equations of perturbed motion on the energy level corresponding to the unperturbed periodic motion. We perform an analytic study in the case of a nearly dynamically symmetric satellite. Numerical analysis is used for arbitrary parameter values (the inertial characteristic and the angular rotation velocity of the satellite).
References
1.  V. V. Beletskii, Artificial Satellite Motion Relative to the Center of Mass [in Russian], Nauka, Moscow, 1965.
2.  T. R. Kane and D. J. Shippy, "Attitude stability of a spinning unsymmetrical satellite in a circular orbit," J. Astronaut. Sci., Vol. 10, No. 4, pp. 114-119, 1963.
3.  T. R. Kane, "Attitude stability of Earth-pointing satellites," AIAA Journal, Vol. 3, No. 4, pp. 726-731, 1965.
4.  L. Meirovitch and F. Wallace, "Attitude instability regions of a spinning unsymmetrical satellite in a circular orbit," J. Astronaut. Sci., Vol. 14, No. 14, pp. 123-133, 1967.
5.  V. V. Sidorenko and A. I. Neishtadt, "Study of stability of long-periodic planar motions of a satellite in a circular orbit," Kosmicheskie Issledovaniya, Vol. 38, No. 3, pp. 307-321, 2000.
6.  A. P. Markeev, "Stability of planar oscillations and rotations of a satellite in a circular orbit," Kosmicheskie Issledovaniya, Vol. 13, No. 3, pp. 307-321, 1975.
7.  A. P. Markeev and A. G. Sokolskii, "Studies of stability of planar periodic motions of a satellite in a circular orbit," Izv. AN SSSR. MTT [Mechanics of Solids], No. 4, pp. 46-57, 1977.
8.  A. P. Markeev and B. S. Bardin, "On the stability of planar oscillations and rotations of a satellite in a circular orbit," Celest. Mech. and Dynam. Astronomy, Vol. 85, No. 1, pp. 51-66, 2003.
9.  A. P. Markeev, "An algorithm for normalizing the Hamiltonian system in the problem of orbital stability of periodic motions," PMM [Applied Mathematics and Mechanics], Vol. 66, No. 6, pp. 929-938.
10.  A. M. Lyapunov, "On stability of motion in a special case of the three-body problem," in Collected Works [in Russian], Vol. 1, pp. 327-401, Izd-vo AN SSSR, Moscow-Leningrad, 1954.
11.  I. G. Malkin, Theory of Stability of Motion [in Russian], Nauka, Moscow, 1966.
12.  A. P. Markeev, Libration Points in Celestial Mechanics and Space Dynamics [in Russian], Nauka, Moscow, 1978.
13.  V. A. Yakubovich and V. M. Starzhinskii, Parametric Resonance in Linear Systems [in Russian], Nauka, Moscow, 1987.
14.  G. E. O. Giacaglia, Perturbation Methods in Nonlinear Systems, Springer, Berlin, 1972.
15.  V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Editorial URSS, Moscow, 2002.
16.  A. P. Markeev, "To the problem of stability of Lagrangian solutions of the restricted three-body problem," PMM [Applied Mathematics and Mechanics], Vol. 37, No. 4, pp. 753-757, 1973.
17.  J. Moser, "New aspects of stability of Hamiltonian systems," Comm. Pure Appl. Math., Vol. 11, No. 1, pp. 81-114, 1958.
18.  A. P. Markeev, Theoretical Mechanics [in Russian], NITs Regular and Chaotic Dynamics, Izhevsk, 2001.
19.  J. Glimm, "Formal stability of Hamiltonian systems," Comm. Pure Appl. Math., Vol. 17, No. 4, pp. 509-526, 1964.
20.  A. M. Zhuravskii, Reference Book in Elliptic Functions [in Russian], Isz-vo AN SSSR, Moscow-Leningrad, 1941.
Received 01 October 2004
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