# Attitude dynamics of slowly spinning axi-symmetric satellites under the influence of gravity gradient Torques

- Paper ID
IAF-69-35

- author
- company
The University of British Columbia

- country
Canada

- year
1969

- abstract
Modem satellite systems, capable of performing sophisticated on-board experiments, usually demand a corresponding degree of sophistication in attitude control to overcome a variety of disturbing influences. Ideally, the attitude control of a satellite should be accomplished with the minimum expenditure of energy since space and weight are at a premium aboard instrument packed space vehicles. In many applications where attitude control requirements are not too severe, passive techniques involving no expenditure of stored energy have proven to be adequate. Among the methods belonging to this category, those utilizing gravity gradient and/or gyroscopic effects are quite common. Depending on the satellite configuration, the gravity gradient torque may reinforce or oppose spin stabilization. The gravitational moment is always present, except for the case of a spherical satellite, and its effect is particularly significant where the rate of spin is small. Thus a study of the attitude dynamics of a slowly spinning satellite in a gravitational field should lead to information of considerable practical significance. In the field of slowly spinning satellites, early work has been restricted to the study of systems undergoing circular orbital motion. Thomson1 presented a stability criterion in terms of spin and inertia parameters using linearized analysis. Subsequently, Kane et al.2 applied the criterion to obtain a stability chart in terms of these parameters. The stability of a spinning unsymmetric satellite was investigated by Kane and Shippy3 applying Floquet theory to a linearized model of the system. The large amplitude librational motion of a spinning satellite in a circular orbit was studied by Pringle4 using a Liapounov type of analysis. Positions of equilibrium as well as bounds of motion, i.e. séparatrices, about these positions were obtained. The attempt by Rane and Barba5 to analyze motion in an elliptic orbit should be mentioned here. By applying Floquet theory to a linearized system, they developed a procedure for testing the stability, in the small, of a spinning satellite undergoing orbital motion of arbitrary eccentricity. Recently Wallace and Meirovitch6 investigated the same problem by performing asymptotic analysis on linear and low order non-linear systems. Non-linear stiffening effects and resonance regions were observed; however, there is an element of doubt about the validity of these results since even conclusions based on the linearized analysis are not in general agreement the with those of Kane and Barba,5 particularly in the negative spin regime. The roll dynamics of a spinning axi-symmetiic satellite was studied by the authors using the approximate WKBJ7 as well as the numerical8 approaches. The analysis emphasized the usefulness of limiting invariant surfaces and periodic solutions of the system. This paper investigates a more general situation involving a system with three degrees of freedom in attitude. Stability charts are presented in greater detail than those found in current literature. Through the use of the Hamiltonian function, a variation of the invariant surface concept9”12 is used to investigate motion in the large for the case of e = 0. It provides all possible combinations of disturbances to which a satellite may be subjected at any point in its orbit without causing it to be unstable. The concept of principal cross- sections is introduced enabling the maximum response of the system to be ascertained given an arbitrary disturbance.