Energy Balance for Random Vibrations of Piecewise-Conservative Systems

Vibrations of systems with instantaneous or stepwise energy losses, e.g., due to impacts with imperfect rebounds, dry friction forces(s) (in which case the losses may be treated as instantaneous ones by appropriate introduction of the response energy) and/or active feedback “bang-bang” control of the systems' response are considered. Response of such (non-linear) systems to a white-noise random excitation is considered for the case where there are no other response energy losses. Thus, a simple linear energy growth with time between “jumps” is observed. Explicit expressions for the expected response energy are derived by direct application of the stochastic differential equations calculus, which contains the expected time interval between two consecutive jumps. The latter may be predicted as a solution to the relevant first-passage problem. Perturbational analysis of the relevant PDE for this problem for a certain vibroimpact system demonstrated the possibility for using the solution to the corresponding free vibration problem as a zero order approximation. The method is applied to an s.d.o.f. system with a feedback inertia control, designed according to a certain previously introduced “generalized reversed swings law”. Extensive Monte-Carlo simulation results are presented for this system as well as for several previously analyzed ones: system with impacts; system with dry friction; system with stiffness control; pendulum with controlled length. The results are compared with those due to the asymptotic stochastic averaging approach. Both methods are shown to provide adequate accuracy far beyond the expected applicability range of the asymptotic approach (which requires both excitation intensity and losses to be small), with direct energy balance being generally superior.