Chaotic attitude tumbling of an asymmetric gyrostat in a gravitational field

The chaotic attitude tumbling of an asymmetric gyrostat is investigated in detail. The gyrostat has three symmetrical wheels along the principal axes rotating about a fixed 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 Poincaré-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-order Runge-Kutta simulation in terms of the Euler parameters (quaternions). The relationships of the following physical parameters are established: moments of inertia of carriers 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.