Response Evaluation and Optimal Control for Stochastically Excited Vibro-Impact System with Hertzdamp Contact

Abstract

PurposeThis manuscript devotes to procedures to evaluate random responses and explore stochastic optimal control for a vibro-impact system, in which the inelastic impact is modeled as Hertzdamp contact model.MethodsThe dimension of governing equation is halved with respect to the total energy of uncontrolled/controlled system by conducting the stochastic averaging of energy envelope. The stationary probability density and then the response statistics of the uncontrolled system are much easier to solve due to a low-dimensional Fokker-Planck-Kolmogorov equation. The original optimal control problem with respect to states of system is converted to that with respect to system total energy. The latter is quite simpler than the former although it only can be regarded as the suboptimal control of the original system. The optimal control with damping property is determined by adopting the stochastic maximum principle and solving simultaneously a set of forward-backward stochastic differential equations.ResultsThe accuracy of the proposed procedure for evaluation of the random response is verified by comparing with Monte-Carlo simulations and the efficiency of the optimal control strategy is also illustrated.ConclusionsThe consistence between the analytical result from energy dissipative balance technique and that from stochastic averaging of energy envelope is discussed, which means the rigorous restriction on stochastic averaging, i.e., light damping and weak excitation, can be removed. The optimal control strategy from stochastic maximum principle and that from stochastic dynamic programming principle holds close connection based on our related discussion.