Stability of permanent rotations around the principal axis of the Lagrange’s gyroscope with dampers-balancers

The problem of passive stabilization is considered for permanent rotations of Lagrange’s gyroscope. The two balancers are attached to the rigid body by cylindrical viscous-elastic hinges. The necessary and sufficient conditions of asymptotic stability for the linearized motion equations are obtained and analyzed. The estimation of eigenvalues is given. The question of optimal choice of the damper characteristics, which guarantees the fastest damping rate of the perturbed motions is discussed. The damper presented is much more effåctive then the 1-DOF one that was used in [1].