Stability analysis of equilibrium positions of nonlinear mechanical systems with nonstationary leading parameter at the potential forces

Certain classes of nonlinear mechanical systems described by the Lagrange differential equations of the second kind with nonstationary evolution of potential forces resulting in their domination are studied. This evolution is defined by a time-varying parameter at the vector of potential forces. It is assumed that the parameter value unlimitedly increases with time. Along with potential forces, gyroscopic and essentially nonlinear dissipative forces act on the examined systems. First, we assume that dissipative forces are determined by the homogeneous Rayleigh function, and after that the case when dissipative forces depend not only on generalized velocities but also on generalized coordinates is investigated. By the use of the Lyapunov direct method and the differential inequalities method, sufficient conditions of the asymptotic stability of the trivial equilibrium position both with respect to all variables and with respect to part of the variables are determined. Furthermore, we study the case when the dissipative forces do not act on the considered system. It is shown that the approaches suggested in this paper allow us to obtain conditions of the asymptotic stability of the equilibrium position with respect to the generalized cooordinates. Compared with known results, these conditions extend types of evolution laws of potential forces for which one can guarantee the asymptotic stability. Two examples are presented to demonstrate the effectiveness of the developed approaches. Bibliogr. 23.