To the stability of non-conservative systems in one degenerate case

Abstract

We study stability of the mechanical systems subjected to dissipative, gyroscopic, potential and non-conservative positional forces. We start from consideration of non-stationary system under the action of all forces, which are listed above in the case when matrix of the gyroscopic forces is constant (degenerate case). By means of the construction of the Lyapunov function we obtain stability condition. Derived results extend the stability conditions of the stationary system [S.A. Agafonov, 2003]. The stability of non-conservative system in the absence of potential forces is examined and condition of asymptotic stability is obtained. We examine the influence of the linear dissipative forces on the stability of the circulatory system. The estimation of the limit value if these forces for the stable circulatory system to be asymptotically stable is obtained. We also consider circulatory system with two degrees of freedom under the action of non-linear dissipative forces. Original system is transformed to the normal form up to the third order. The internal resonance of the fourth order is considered as well. It is shown that the stable circulatory system becomes asymptotically stable.