Stochastic Averaging for Quasi-Integrable Hamiltonian Systems With Variable Mass

Abstract

Variable-mass systems become more and more important with the explosive development of micro-and nanotechnologles, and it is crucial to evaluate the influence of mass disturbances on system random responses. This manuscript generalizes the stochastic averaging technique from quasi-integrable Hamiitonian systems to stochastic variable-mass systems. The Hamiitonian equations for variable-mass systems are firstly derived in classical mechanics formulation and are approximately replaced by the associated conservative Hamiitonian equations with disturbances in each equation. The averaged Itô equations with respect to the integrals of motion as slowly variable processes are derived through the stochastic averaging technique. Solving the associated Eokker-Plank-Kolmogorov equation yields the joint probability densities of the integrals of motion. A representative variable-mass oscillator is worked out to demonstrate the application and effectiveness of the generalized stochastic averaging technique; also, the sensitivity of random responses to pivotal system parameters is illustrated.

Stationary Response of a Kind of Nonlinear Stochastic Systems with Variable Mass and Fractional Derivative Damping

“…According to the approximately equivalent Equation ( 18), the corresponding Stratonovich stochastic differential equation can be obtained [18],…”

“…In Equation (24), m H and σ H 1 , σ H 2 are respectively drift and diffusion coefficients, which are introduced in [18] and described by…”

“…In order to better simulate the randomness of the variable mass, many scholars used Gaussian white noise to model the mass disturbance. Wang et al [18] utilized Gaussian white noise to describe the random mass disturbance and proposed the stochastic averaging technique of the variable mass system based on Hamilton theory, which provided a method to analyze the random responses of stochastic systems with Gaussian-white-noise-based variable mass. Then, Qiao et al [19,20] presented a group of stochastic stability conditions of the variable mass system under the Gaussian white noise excitation by using the stochastic averaging method and the maximum Lyapunov exponent method.…”