About stability and stabilization of mechanical systems

of general type

S.A.Agafonov

N.E. Bauman Moscow State Technical University

5, 2^nd Baumanskaya st., Moscow, 105005, Russia

In this paper two problems of stability are considered. In the first problem the stability of mechanical systems in which dissipative, gyroscopic, potential and positional non-conservative forces act is investigated. The stability is analyzed in case when the potential energy has maximum in origin. The condition for asymptotic stability in terms of system using the Lyapunov function is obtained.

The presence of positional non-conservative forces considerably complicates the stability problem and excludes direct application of classical Kelvin-Chetayev theorems.

In the second problem stabilization of the gyroscopic system by means of non-linear dissipative and positional non-conservative forces is solved.

Stability of gyroscopic system with two degrees of freedom is attained with the help of gyroscopic stabilization (instability degree is equal to two). In system terms the asymptotic stability conditions are obtained. The cases when the gyroscopic stabilization is being destroyed are pointed out as well.