On Melnikov's method in the study of chaotic motions of gyrostat

Abstract

The homoclinic solutions of the attitude motion of a gyrostat with wheels along its principal axes are formulated using the procedure developed by Wittenburg. The generic Melnikov function for the attitude motion of a gyrostat subject to small non-linear damping torques and small periodic torques is derived. The derivation is based on the chaotic theory for a 'one and half' degrees of freedom system established by Wiggins and Shaw. The conditions for the physical parameters at the onset of the homoclinic solutions are given in detail. The chaotic criteria are investigated using Melnikov's function. In particular, the conditions of the physical parameters (moments of inertia, damping coefficients, amplitudes and frequencies of external excitation, moments of momentum of wheels) that ensure the convergence of Melnikov's integral are also worked out. In order to acquire some knowledge of long-term behaviours of the chaotic attitude motion, the fourth-order Runge-Kutta numerical algorithms are utilized to simulate its dynamics. The numerical experiment shows that the chaotic motions of the gyrostat are bounded, non-periodic and sensitive to initial conditions. One of the practical mechanical models of the gyrostat is the artificial satellite. The attitude motion of the three-axis stabilized gyrostat satellite, whose torque-free motion is the homoclinic orbits, will oscillate chaotically when it is subjected to the appropriate external disturbances.