Study of orbital stability for planar oscillations of a magnetized symmetric satellite in a circular orbit

Theoretical mechanics


Аuthors

Bardin B. S.*, Savin A. A.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: bardin@yandex.ru
**e-mail: sashka_savin@mail.ru

Abstract

It is considered the motion of a satellite about its center of mass in a central Newtonian gravitational field. The satellite is a magnetized dynamically symmetrical rigid body, whose typical linear dimensions are small compared to the radius-vector of the mass center. The above assumption allows us to consider the problem in a restricted formulation, that is we assume that the motion of the satellite about the center of gravity does not affect the motion of the center of mass [1]. It is also assumed that the orbit of the center of mass is circular. The orientation of the satellite relative to the orbital coordinate system is described by Euler angles ψ, θ, φ.

In this paper it is assumed that magnetic moment arises from the magnetization of the material of satellite shell by earth’s magnetic field [1, 4, 5]. Earth’s geomagnetic field is described by dipole model [6]. It is assumed that the Earth’s magnetic dipole axis coincides with the axis of rotation (direct dipole), and the orbit of the satellite lies in the equator plane.

Under these assumptions, the equations of motion of the magnetized satellite about the center of mass can be written in Hamiltonian form. One of the generalized coordinates (proper rotation angle φ) is a cyclic coordinate, so the corresponding momentum pφ is the first integral of the motion. Planar periodic motions are possible only if pφ=0. It is assumed that the perturbations satisfy the above equality.

The problem has three parameters: the parameter β, which characterizes the geometry of the masses; the parameter ξ, characterizes the effect of the magnetic field of the Earth; doubled amplitude of planar oscillations Ψ.

Linear analysis of the orbital stability for periodic motions is performed. The above motions are planar pendulum-like oscillations such that the polar principle axis of inertia is located in the plane of the orbit of the mass center. To this end local coordinates are introduced in the neighborhood of the unperturbed periodic motion and linearized equations are written.

Separately, it was considered the special case when amplitudes of the oscillations are small. In this case small parameter (doubled oscillation amplitude) has been introduced. It allowed to study the orbital stability analytically. In particular, using the normal forms theory the explicit formulas describing the boundaries of parametric resonance domains were obtained.

For oscillations with arbitrary amplitudes linear stability analysis based on the numerical integration of the linearized system has been performed. In typical cross-sections of the three-dimensional parameter space the domains of orbital instability and stability have been obtained. The results of numerical analysis of orbital stability are in good agreement with the results of analytical study performed in the above mentioned special case.

Keywords:

Hamiltonian system, periodic orbits, normal form, resonance, action-angel variables, orbital stability

References

  1. Beletskii V.V. Dvizhenie iskusstvennogo sputnika otnositel’no tsentra mass (Motion of an Artificial Satellite about its Center of Mass), Moscow, Nauka, 1965, 416 p.

  2. Markeev A.P. Kosmicheskie issledovaniya, 1975, Vol. 13, no. 3. pp. 322–336.

  3. Landau L.D., Lifshits E.M. Elektrodinamika sploshnykh sred (Electrodynamics of Continuous Media), Moscow, Nauka, 1982, 621 p.

  4. Beletskii V.V., Shlyakhtin A.N. Rezonansnye vrashcheniya sputnika pri vzaimodeistvii magnitnogo i gravitatsionnogo polei (Resonant rotations of a satellite in coupled magnetic and gravitational fields), Moscow, Institut Prikladnoi matematiki AN SSSR, Preprint № 46, 1980, 30 p.

  5. Beletskii V.V., Khentov A.A. Vrashchatel’noe dvizhenie namagnichennogo sputnik (Rotational Motion of Magnetized Satellite), Moscow, Nauka, 1985, 288 p.

  6. Landau L.D., Lifshits E.M. Teoriya polya (The Classical Theory of Fields), Moscow, Nauka, 1988, 509 p.

  7. Bardin B. S. On orbital stability of planar motions of symmetric satellites in cases of first and second order resonances. In Proceedings of the 6th Conference on Celestial Mechanics, Monogr. Real Acad. Ci. Exact. F s.-Qu m. Nat. Zaragoza, 25, pages 59–70, 2004.

  8. Markeev A.P., Bardin B.S. On the stability of planar oscillations and rotations of a satellite in a circular orbit. Celestial Mech. Dynam. Astronom, 2003, no. 85(1), pp. 51–66.

  9. Markeev A.P. Prikladnaya matematika i mekhanika, 2001, no.65 (1), pp. 51–58.

  10. Markeev A.P. Lineinye gamil’tonovy sistemy i nekotorye zadachi ob ustoichivosti dvizheniya sputnika otnositel’no tsentra mass (Linear Hamiltonian Systems and Several Problems of Stability of Satellite Motion about the Center of Mass), Moscow, NITs Institut komp’yuternykh issledovanii, 2009, 396 p.

  11. Markeev A.P. Tochki libratsii v nebesnoi mekhanike i kosmodinamike (Libration Points in Celestial Mechanics and Aflmdynamics). Moscow, Nauka, 1978, 313 p.

  12. Birkhoff Dzh. D. Dinamicheskie sistemy (Dynamical Systems), Izhevsk, Izdatel’skii dom «Udmurtskii universitet», 1999, 408 p.

  13. Giacaglia G.E.O. Metody vozmushchenii dlya nelineinykh sistem (Perturbation methoda in non-linear systems), Moscow, Nauka, 1979, 319 p.

  14. Malkin I.G. Teoriya ustoichivosti dvizheniya (Motion stability theory), Moscow, Nauka, 1966, 532 p.


Download

mai.ru — informational site MAI

Copyright © 2000-2021 by MAI

Вход