Spinup dynamics of gyrostats

Abstract

Attention is given to the spinup dynamics of gyrostats containing a single axisymmetric rotor. Spinup of the rotor is due to a small constant torque applied by a motor on the platform. The dynamics are described by four first- order differential equations, which are put into noncanonical Hamiltonian form. Using conservation of angular momentum and the method of averaging, the equations of motion are reduced to a single scalar first-order equation for the slow evolution of the Hamiltonian. This reduction is formally valid for small spinup torques and in regions of phase space where the unperturbed motion is periodic. The unperturbed separatrices are therefore regions where averaging fails to describe the motion adequately and are also indicative of dramatic changes in the attitude dynamics. Exact solutions to the averaged equation are used to justify further the projection of solutions of the four-dimensional system onto the plane of the slow states. The reduced two-dimensional slow state space is used to construct a single planar diagram that is useful for portraying spinup dynamics. OST artificial satellites contain one or more spinning rotors to provide gyroscopic stability of a desired orientation or attitude of the vehicle. Dual-spin spacecraft use the spin of a large rotor to maintain pointing accuracy of a comparatively small Earth- pointing antenna platform. Bias momentum satellites, on the other hand, use small but rapidly spinning momentum wheels to control the attitude of a large platform. In any case, such a vehicle is put into orbit in an all-spun condition, in which the rotors are locked and the vehicle spins as a single body. After the mission orbit is acquired, an attitude acquisition maneuver is performed in which a combination of external torques from thrusters and internal torques from spinup motors is used to achieve the mission orientation. Some satellites may also regularly perform attitude maneuvers using external and internal torques. In this paper we consider rotational motion during a spinup maneuver using internal torques only. Our model consists of a rigid-body satellite containing a single rigid axisymmetric rotor constrained to relative rotation about its symmetry axis. This model is usually called a gyrostat. The gyro- stat is free of external torques, and the rotor is spun up by a small constant internal torque applied by the platform. We use a noncanon- ical Hamiltonian formulation of the rotational equations of motion to analyze this spinup problem. We also indicate how to apply our approach to multiple-rotor gyrostats. The equations of motion for gyrostats have two well-known in- tegrable cases: the constant-speed rotor case and the zero-axial- torque case. In apparently the only study of a natural gyrostat, Volterra1 introduced a gyrostat model in order to explain preces- sion of Earth's equinoxes and obtained a solution for the angu- lar velocities in terms of Weierstrass's elliptic function. More re- cently, and motivated by artificial satellites, Masaitis2 obtained a solution in terms of Jacobi's elliptic functions for the angular ve- locities of an axial gyrostat (rotor parallel to principal axis) with no axial torque. Interestingly, he suggested using the solution as a basis for a perturbation solution for more complicated prob- lems. At about the same time, Leipholz,3 motivated by flight of aerospace vehicles containing rotating parts, also found the so- lution for this case. Leimanis4 gave an extensive treatment ap- parently based on Masaitis 2 and Leipholz.3 Wittenburg5 treated more general cases where the rotor is not parallel to any prin- cipal axis of the gyrostat. For the axial gyrostat, he solved for the angular velocities in terms of Jacobi's elliptic functions. For the more general cases, he reduced the equations of motion to a