Bounds on the Extension of Antennas for Stable Spinning Satellites

Abstract

L us consider a force-free satellite spinning about the axis of maximum moment of inertia. If the satellite is rigid, then from rigid-body dynamics we conclude that the rotational motion is stable. Next, let us assume that a pair of thin flexible rods with given tip masses are being deployed slowly, so that they are extended at equal rates but in opposite directions along the spin axis. The question is being asked as to how far can the rods be extended without causing the satellite motion to become unstable. It is assumed that the rate of extension of the rods is sufficiently small that any Coriolis effects that might arise because of the rate of change in length can be ignored. It is also assumed that the mode of deformation of the flexible rods is antisymmetric, with the implication that the satellite mass center does not shift relative to the rigid part. The satellite is shown in Fig. 1. In an early attempt to investigate the flexibility effects on the stability of a force-free spinning satellite, Meirovitch and Nelson considered two mathematical models related to that used here. In fact, the two mathematical models used in Ref. 2 can be obtained as limiting cases of that used here by assuming in one case that the rods act like massless springs and in the other case that there are no tip masses. Stability was investigated in Ref. 2 by means of an infinitesimal analysis. More recently, Meirovitch and Calico have formulated the general problem of stability of motion of a force-free spinning satellite with flexible appendages, with stability being investigated via Liapunov's second method. For stability, the system Hamiltonian must be positive definite. Moreover, because of internal energy dissipation, however slight, the equilibrium is asymptotically stable if the Hamiltonian is positive definite in the neighborhood of the equilibrium point in question. Of particular interest in Ref. 3 is