Abstract

The Thomson-Tait-Chetayev theorem states that “if a system with an unstable potential energy is stabilized with gyroscopic forces, then this stability is lost after the addition of an arbitrarily small dissipation”. The importance of this property in growing number of physical examples and engineering applications in the practice is not well good unified and understood, i.e. the destabilizing effect of dissipation needs to be compensated in various gyroscopic devices by applying accelerating forces.

In the present paper an analytical study of the stability behaviour of a specific class of nonlinear autonomous dynamic systems (i.e. RHS of the equations is a square polynomial) with two degrees of freedom is developed. Considering the general case, we find that the system is multi-stationary and has several possible equilibria. The system is investigated with analytical tools coming from Lyapunov-Andronov theory, and our analytical calculations predict that soft (reversible) loss of stability takes place.