Stochastic dynamics of a nonlinear time-varying spur gear model using an adaptive time-stepping path integration method

Abstract

This study is purposed to enhance conventional spur gear models by moving from a deterministic to a stochastic representation, based on adding a Gaussian white noise in the loading terms while including the effect of both backlash and time-varying mesh stiffness. Meanwhile, an efficient numerical path integration (PI) technique is used to obtain the probability distribution functions (PDFs) of the response. A novel adaptive time-stepping scheme is proposed for the PI method that momentarily increases its accuracy in order to detect the edge of transition when dealing with the non-smoothness due to backlash. The results demonstrate good agreement with Monte Carlo simulation (MCS). It is observed that if the initial distribution covers a region containing the basins of attraction of coexisting solutions, the stationary PDF will be formed around both of their deterministic solutions. The resulting response PDF contains subtle information about the system, which is essential in the design of its parts and its reliability assessment. The generality of the approach is such that the new path integration method can be implemented for the stochastic analysis of other non-smooth random dynamical systems. © 2019 Elsevier Ltd