The non-linear oscillations of a satellite with third-order resonance

Abstract

Plane non-linear oscillations of an artificial satellite—a rigid body—about its centre of mass in an elliptical orbit of small eccentricity are considered. It is assumed that three times the frequency of small oscillations of the satellite in a circular orbit is close to the frequency of revolution of its centre of mass. Methods of classical perturbation theory are used to reduce the problem to that of a model system, described by a Hamiltonian which is characteristic for problems involving the motion of Hamiltonian systems with one degree of freedom in the case of third-order resonance. A detailed analysis of such systems is carried out. The theory of periodic Poincare motions and KAM-theory are used to transfer the results for the model system to the complete system and to apply them to the problem of satellite motion. The question of the existence, number and stability of periodic motions with period equal to three times the period of revolution of the centre of mass of the satellite in orbit is considered, depending on the inertial parameter of the satellite and the eccentricity of the orbit. It is shown that motions of the satellite beginning in a certain neighbourhood of its eccentricity oscillations are bounded, and an estimate is given for the size of that neighbourhood.