Stochastic Dynamic Analysis of Vehicle–Railway–Bridge Systems via the Dimension-Reduced Probability Density Evolution Equation (DR-PDEE) Framework

This study presents an efficient stochastic dynamic analysis framework for the vehicle–railway–bridge coupled system based on the recently developed dimension-reduced probability density evolution equation (DR-PDEE) method. The complex coupled system is modeled through a partitioned computational approach employing a non-iterative heterogeneous coupling scheme. In the modelling, the vehicle, rail, and bridge are treated as three interconnected subsystems linked by two interfaces with shared nodes. This formulation enables accurate representation of subsystem interactions while ensuring high computational efficiency. Uncertainties arising from track irregularity, wheel-rail contact, and system parameters are explicitly considered. The DR-PDEE is then employed to quantify the probability evolution of key degrees of freedom of the vehicle, rail, and bridge, and the obtained displacement and acceleration responses exhibit pronounced nonstationary and non-Gaussian characteristics. Particularly, for cases where the quantity of interest is the acceleration response, a one-dimensional DR-PDEE scheme based on a finite-difference treatment is proposed, further extending the applicability of the method to general engineering problems. Comparisons with Monte Carlo simulation results verify the accuracy and efficiency of the proposed framework. This work represents the first application of the DR-PDEE to the vehicle–railway–bridge system, offering a physics-driven and computationally efficient framework for uncertainty propagation and reliability assessment in complex railway dynamics.