Approximately analytical technique for random response of LuGre friction system

This manuscript concentrates on the random response evaluation of dynamic friction system, in which the dynamic friction is described by LuGre friction law while the random excitation by Gaussian white noise. The definitive solution can be obtained by solving two coupled equations: a second-order stochastic differential equation with respect to the system displacement and an auxiliary first-order nonsmooth differential equation with respect to the internal friction state. The auxiliary differential equation is first integrated to derive the approximate relation between the friction force and current system states under the assumption of slowly varying amplitude, and then the original problem with definitive solution is approximated by a modified second-order stochastic differential equation with respect to the system displacement and amplitude. With the application of the equivalent non-linearization technique, an equivalent nonlinear system is constructed and solved to obtain the analytical expression of the stationary probability density of system displacement and velocity, which is the approximately analytical solution for the modified system and the original frictional system. Numerical results on the Duffing system with LuGre friction validate the effectiveness and accuracy of the proposed approximately analytical technique.