The estimation of eigenvalues for linear mechanical 2-DOF system

The problem of eigenvalues estimation and stability for linear mechanical system is considered. The system is under influence of potential, gyroscopic, dissipative, and circulatory forces. Possible variants are described, when the spectrum of the system belongs to the left semi-plane - the conditions of Routh-Hurwitz criterion are fulfilled, but fading of perturbed motions is "too slowly". As an example, the problem about permanent rotations of rigid body under the action of damping torque is considered.