Dynamics of spinning flexible satellites

Abstract

This investigation is concerned with the dynamic characteristics and stability of a spinning rigid body with a number of flexible parts. The system is hybrid in the sense that it is described by coordinates depending on time alone and coordinates depending on spatial position and time. The space-dependent coordinates are discretized by the assumed-modes method based on the Rayleigh-Ritz approach. The matrix form of the linearized equations of motion, which is of gyroscopic type, can be reduced to a general matrix multiplied by the state vector. The constrained system Hamiltonian is employed as a Liapunov function for stability analysis. It is shown that for stable nontrivial equilibrium the mass matrix and the stiffness matrix must be positive definite. In this case, the general matrix multiplied the state vector becomes skew-symmetric and its eigenvalues are complex conjugate pure imaginary.

The method has been applied to the simplified model of the European Space Agency's GEOS spacecraft to obtain explicit forms of the equations of motion and stability criteria in terms of system parameters. It is found that the motion about the equilibrium is stable but the fundamental frequency is lower than the spin rate. As a by-product, it is shown that neglecting the motion of the mass center is immaterial as far as the stability is concerned, except in the case in which the points of attachment of the flexible parts are off-set along the spin axis relative to the mass center of the spacecraft. In the latter case, the effect of the motion of the mass center must be examined carefully.