Free motion of an asymetric gyrostat studied in the phase plane

The motion of the asymmetric gyrostat consisting of an asymmetrical carrier and an axisymmetric rotor, rotating about a fixed point under the action of the gravitational force. If the rotor of the gyrostat is locked, it does not have any effect on the dynamic behavior of the gyrostat. For purposes of investigating the effect of the rotor on the motion of the gyrostat, Deprit’s canonical are introduced to establish the hamiltonian structure for this problem. Motion equations of the free gyrostat with small rotor asymmetry and small internal moment are obtained in Anoyer-Deprit variables. Control law for the internal moment is proposed that eliminates the possibility of the separatrix chaos. Numerical simulation shows the efficiency of the proposed control. When the coefficient of the gravitational torque is zero, the problem reduces to torque-free motion of the gyrostat. The torque-free motion of the asymmetrical gyrostat may then be described by a one-degree of freedom Hamiltonian system. In this paper we have studied the movement and stability of an asymmetric gyrostat in the phase plane. We have used linear stability analysis to determine the stability of equilibrium of the gyrostat. The number of equilibria changes as angular momentum is varied.