Stochastic bifurcation analysis of a friction-damped system with impact and fractional derivative damping

Impact together with friction can be widely found in the mechanical engineering. Although some scholars have investigated the stochastic systems with impact and friction, they only involve the integer-order systems and do not consider the fractional-order cases. In fact, for the system with viscoelastic material, the damping term depends not only on the current time and position but also on the previous states. The memory property of viscoelastic material is characterized by a power-law kernel function which is associated with the fractional derivative. Based on this viewpoint, in this article, we focus on the friction-damped system with fractional derivative damping under Gaussian white noise excitation. We propose an approximate approach to investigate the stochastic response and bifurcation of a fractional-order friction-damped system with the help of variable transformations and stochastic averaging method. One example is employed to verify the effectiveness of the proposed approach. We also explore the stochastic bifurcation phenomenon induced by the fractional order, fractional coefficient and other system parameters through the critical conditions. At last, the difference of bifurcation regions for the fractional-order model and the integer-order model are presented.