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On the attitude dynamics of slowly spinning axisymmetric satellites under the influence of gravity gradient torques Neilson, John Emery

Abstract

The dynamics of slowly spinning axisymmetric satellites under the influence of gravity gradient torque is investigated using analytical and numerical techniques. Particular emphasis is on motion near the equilibrium position in which the spin axis is normal to the orbital plane. The problem is studied in increasing orders of difficulty. Phase I deals with the response and stability of a simplified model free to librate in roll while the more general problem is treated in Phase II. Phase I serves as a proving ground for techniques to be used in subsequent analysis. A closed form solution is obtained in terms of elliptic functions for the autonomous case. In general, for non-circular orbits, motion in the large is studied using the concept of the invariant solution surface. These surfaces, obtained numerically, reveal the nature of motion in the large in terms of the dominant periodic solutions and allow one to determine the limits of oscillatory motion in terms of the state parameters. Floquet theory is employed in conjunction with numerical solutions of the linearized equations of motion to study stability in the small. This technique is extended to assess the variational stability of the dominant periodic motions in the large. Phase II investigates a more general model with three degrees of freedom in attitude motion. The presence of an ignorable coordinate gives a fourth order, non-autonomous system for an elliptic trajectory. Motion in the small is studied extensively, again using Floquet theory, and stability charts suitable for design purposes are presented. The invariant surface concept is successfully extended to the study of the autonomous case in the large. Methods are developed for determining the maximum response to a given disturbance resulting in a set of charts which are useful in assessing the effects of non-linearities and the validity of the analysis in the small. Procedures are explained for determining periodic solutions of the problem, as well as their stability, for arbitrary eccentricity. The analysis suggests the possibility of attitude instability during spin-up operations. It is shown that stable motion can be established by providing either a positive or negative spin to the satellite with the former preferrable. Given sufficient spin any configuration, even those with an adverse gravity gradient effect, can be stabilized. Eccentricity affects the attitude motion of a satellite adversely as regions of unstable motion increase in size and number with it.

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