Probability Density Evolution and Reliability Analysis of Gear Transmission Systems Based on the Path Integration Method

Aimed at dealing with the problems of high reliability solution cost and low solution accuracy under random excitation, especially Gaussian white noise excitation, this paper proposes a probability density evolution and reliability analysis method for nonlinear gear transmission systems under Gaussian white noise excitation based on the path integration method. This method constructs an efficient probability density evolution framework by combining the path integration method, the Chapman–Kolmogorov equation, and the Laplace asymptotic expansion method. Based on Rice’s theory and combined with the adaptive Gauss–Legendre integration method, the transient and cumulative reliability of the system are path integration method calculated. The research results show that in the periodic response state, Gaussian white noise leads to the diffusion of probability density and peak attenuation, and the system reliability presents a two-stage attenuation characteristic. In the chaotic response state, the intrinsic dynamic instability of the system dominates the evolution of the probability density, and the reliability decreases more sharply. Verified by Monte Carlo simulation, the method proposed in this paper significantly outperforms the traditional methods in both computational efficiency and accuracy. The research reveals the coupling effect of Gaussian white noise random excitation and nonlinear dynamics, clarifies the differences in failure mechanisms of gear systems in periodic and chaotic states, and provides a theoretical basis for the dynamic reliability design and life prediction of nonlinear gear transmission systems.