Comments on "Energy Considerations for Attitude Stability of Dual-Spin Spacecraft"

N a recent Note,1 Prof. Fang obtains criteria for stability of an equilibrium motion of a dual-spin spacecraft by determining whether the equilibrium motion corresponds to the minimum energy state of permissible motions. The equations he employs to determine the minimum energy state treat the dual-spin spacecraft as consisting of a rigid symmetric rotor, body S, which rotates relative to a rigid asymmetric body, body A, in a force-free field. The axis of relative rotation between the two bodies corresponds to the axis of symmetry of the rotor and to a principal axis of body A. It is assumed that no net torque is exerted about this axis between the two bodies. Although admittedly not modeled in his equations, he states that the analysis is made with the assumption that "the bodies A and S possess some degree of flexibility and some internal dissipative mechanism," and leaves the reader with the incorrect impression that his results are generally valid for this case (possibly excepting "highly flexible spacecraft"). In fact, it is possible to make conclusive statements about the stability of the equilibrium solution only for the particular equations being examined regardless of the method of analysis. Extending results for this case to the same equilibrium solution for other, more complex, equations (multibody or flexible body equations) is at best an intuitive and heuristic procedure and can be only approximate. Even though this type of analysis is not rigorous, the stability criteria gained thereby are still useful as a "rule of thumb" to guide more exact analyses or spacecraft design choices. Similar forms of this type of analysis have been performed by lorillo 2 and Likins3 for dual-spin spacecraft. Fang references the Likins' paper and employs his spacecraft model and, to a large extent, notation. His results, however, differ from those of Likins (and lorillo, whom Likins has followed). Fang's results are those that would be obtained for energy dissipation on body A only, and agree with those of Likins for this special case only. The stability of the equilibrium solution coi = 0, co2 = 0, cos5 = const, co3A = const (1) is examined. The stability criterion is given by Likins as (PA/XA) + (Pa/X fl) < 0