Stability of the rotation of Steklov gyro

In both uniform gravitational field and Newtonian central gravitational field, the conditions of stability and instability of the permanent rotation of the Steklov gyro with linear damping or nonlinear damping torques are studied by the Liapunov direct method and the first approximation method. Additionally, some new theorems are used in finding the stability and asymptotic stability of the gyro motion. In the Newtonian central gravitational field for the sake of high accuracy, the second order term of I/R in the expression of potential energy has been considered, where R is the distance between the mass center of the gyro and the center of the earth. Numerical results are presented to show the domains of attraction in two cases by using a cell-to-cell mapping, the effects of types of damping torque and the system parameters on the gyro motion are also discussed in these cases.