Averaging of Random Vibrations in Mechanical Systems in the Sense of Ito, Stratonovich, and Sussmann

Abstract

In this paper, we investigate a stochastic averaging principle for a large class of mechanical systems in the presence of random vibrations. We show that a known deterministic averaging principle for affine connection control systems with large-amplitude high-frequency inputs also holds for non-periodic stochastic inputs. The randomly vibrating mechanical system is described by a stochastic differential equation whose solutions do not depend on its interpretation in the sense of Ito, Stratonovich, or Sussmann. We also show that solutions of this stochastic differential equation can be directly computed from a single ordinary differential equation. We illustrate our theoretical results by the example of stochastic source seeking with a nonholonomic vehicle.