The stability of mechanical systems subjected to impulsive actions

Abstract

Mechanical systems subjected to an impulsive load at set times are considered. The impulsive forces depend on generalized coordinates and cause variation of the generalized velocities of the system. Equations of this type describe the vibrations of structures due to seismic shocks, the dynamics of systems of rigid bodies on a moving train or a landing aircraft, etc. The instants of the impulsive action may have limit points, as in the case of shocks which attenuate in a geometric progression. Various problems related to the stability of motion will be discussed. First, general properties of solutions with infinitely many impulse times will be established, namely, their existence, uniqueness and the nature of their dependence on the parameters and initial conditions. The results obtained enable in particular, the linearization method to be used to investigate the stability. Particular attention is paid to non-linear Hamiltonian systems with generalized (impulsive) potential. It is shown that such systems possess a cononical phase flow, and KAM-theory may be used to investigate the stability. An important part of such investigations is the problem of constructing the stability domain in the first approximation, the solution of which frequently involves an analysis of Hill's equation. A series of sufficient conditions are obtained for the stability of the trivial solution of Hill's equation with periodic shocks, generalizing well-known criteria which are applicable to smooth systems. The example of a pendulum whose suspension point is given periodic equal vertical impulses is considered in detail.