Chaotic Attitude Tumbling of an Asymmetric Gyrostat in a Gravitational Field

Abstract

The chaotic attitude tumbling of an asymmetric gyrostat is investigated in detail. The gyrostat has three sym- metrical wheels along the principal axes rotating about a e xed point under the action of either the gravity torques or the gravity-gradient torques. With the use of the Deprit canonical variables, the Euler attitude equations are transformed into Hamiltonian form. This makes the Poincare-Arnold-Melnikov (PAM) function developed by Holmes and Marsden applicable. The physical parameters triggering the chaotic attitude are established. The analytical results are checked by using the fourth-orderRunge -Kutta simulation in terms of the Euler parameters (quaternions ). The relationships of the following physical parameters are established: moments of inertia of carri- ers and wheels, positions of the mass center, kinetic energy and moment of momentum of the torque-free gyrostat, and initial attitude leading to chaotic motion. The results show that the PAM function is a powerful analytical tool for the treatment of the dynamics of nonlinear gyrostat orientations.