Stable tumbling motions of a dual-spin satellite subject to gravitational torques.
The motion of a dual-spin or gyrostat satellite subject to gravitational torques is governed by a set of nonlinear equations. In general, the equilibrium solutions (solutions keeping one side of the satellite facing the Earth at all times) for a satellite in a circular orbit are isolated in the state space. However, for a critical value of the spin rate two continua (or one-parameter families) of equilibria suddenly appear. This paper presents a stability analysis of these families of equilibrium solutions. A satellite in any one of these orientations cannot have any resistence to perturbations sending it along the continuum, since each orientation it passes through is an equilibrium, and cannot supply any restoring torques. Thus, tumbling motions result. The topology of the dynamic potential surface is investigated in the parameter space in order to determine stability properties. It is shown that if the spin axis is aligned with the axis of least moment of inertia, the families of equilibria are stable and tumbling motions will closely follow the continuum. It is also shown that the critical spin rate forms a point of bifurcation in the stability behavior of many of the isolated equilibria.