The damping momentum’s influence on the stability of the non-symmetrical gyro’s permanent rotations

The problem of the damping momentum influence on the stability of permanent rotations of the rigid body with fixed point is considered. The necessary and sufficient conditions of the stability are found, which give restrictions for mass distribution of the body, the angular velocity of the rotation and the damping coefficient. It is shown that the motion for downside position of the gyro becomes asymptotically stable. The influence of the damping for the upside rotations may be two-fold: gyroscopically stabilized motion may lose its stability, or may become asymptotically stable. The noteworthy is the fact that the last stabilization effect is realizable for the nonsymmetric gyro only.