Dynamics of two spring-connected masses in orbit

Abstract

This paper discusses relative equilibria (or steady motions) and their stability for the dynamics of the system of two spring-connected masses in a central gravitational field. The system can be regarded as a simplified model for the Tethered Satellite System (TSS), where the tether is modeled by a (linear or nonlinear) spring. In the previous studies of the TSS problem, it was typically assumed that the center of mass is located at the massive one of the two end-masses, and moves on a great-circle orbit. However, for the simple system treated in this paper, it is proved that nongreat-circle relative equilibria do exist. Some fundamental concepts of the dynamics of an arbitrary assembly moving in a central gravitational field are discussed. The notion of steady motions used in engineering literature is linked with the notion of relative equilibria in geometric mechanics. Numerical computations show some interesting nongreat-circle relative equilibria for the spring-connected system. Radial relative equilibria, which correspond to the station-keeping mode for TSS, are then introduced. Within the framework of symmetry and reduction, their stability properties are investigated by adopting the reduced energy-momentum method, which takes the advantage of the intrinsic symmetry structure. It is shown that for practical configurations, the system at radial relative equilibria is stable if some conditions are satisfied.