Spinup Dynamics of Gyrostats

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 firstorder 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.