Non-stationary semi-analytical solution of fractional vibro-impact system with multiplicative random excitations

This paper uses the complex fractional moment (CFM) method to construct an efficient semi-analytical approximation framework for studying the probabilistic response of a fractional-order non-smooth vibro-impact system under combined Gaussian colored and white noise excitations. Initially, the Zhuravlev non-smooth coordinate transformation is employed to eliminate the velocity jumps of the system. Then, the stochastic averaging method is applied to derive the Fokker-Plank-Kolmogorov (FPK) equation governing the amplitude. On this basis, the CFM method is introduced and combined with the inverse Mellin transform. A semi-analytical approximation solution to the FPK equation is further derived. To verify the effectiveness of the proposed method, numerical simulations are carried out to analyze the influence of different parameters on the complete probability density evolution of the system. Meanwhile, error analysis is conducted using the absolute error E _abs, the L ₂ norm, and the Kullback-Leibler (KL) divergence. The reconstruction performance of the method under different conditions is evaluated quantitatively. Finally, the evolution law of the steady-state response with parameter variations is further revealed. The applicability of the method for the full process analysis from transient state to steady state is demonstrated.