Relative equilibria and stabilities of spring-connected bodies in a central gravitational field

This paper discusses relative equilibria (or steady motions) and their stabilities for the motions of two spring-connected bodies in a central gravitational field. This two-body system can be regarded as a simplified model for the tethered satellite system (TSS). In the studies of TSS, typical assumptions include: (1) the center of mass and the center of gravity are both located at the massive one of the two end masses; (2) the center of mass moves on a great-circle orbit. In this paper, these assumptions are lifted to derive more exact models for analyses. In particular, for the simple system treated in this paper, it is proved that the nongreat-circle relative equilibria do exist, and hence the above assumption (2) is not always valid. Some fundamental concepts of the dynamics of an arbitrary assembly moving in a central gravitational held are discussed. The notion of radial relative equilibria, which is the familiar station-keeping mode for TSS, is introduced. Their stabilities are analyzed by adopting the reduced energy-momentum method. It is shown that with physically practical configuration, the system at radial relative equilibria is stable if certain conditions are satisfied