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This paper presents a vehicle operation safety evaluation model; to this end, a nonlinear vehicle-track coupled dynamic system stochastic analysis model under random irregularity excitations based on probability density evolution method was developed. The nonlinear coupled vehicle-track dynamic system is used to accurately describe the wheel-rail contact state. The stochastic function-spectral representation is used to simulate the random track irregularity in the time domain for the first time; consequently, the frequency components in the irregularity are preserved and random variables are reduced. In the process of evaluating the safety of train operation, the probability evolution, reliability of evaluation indices for different limit values, and evaluation indices for different probability limits are calculated for more accurate evaluation. The dynamic model and safety evaluation method was verified using the Zhai-model and Monte Carlo method. The results show that, when the probability guarantee is increased, the running safety index of the vehicle increases more rapidly with running speed and the left/right wheel-rail derailment coefficient increases rapidly at running speeds above 400 km/h. The computational model provides a novel direction for vehicle operation safety evaluation.

A three-dimensional rail-bridge coupling element of unequal lengths in which the length of the rail element is shorter than that of the bridge element is presented in this paper to investigate the spatial dynamic responses of a train-track-bridge interaction system. Formulation of stiffness and damping matrices for the fastener, ballast, and bearing, as well as the three-dimensional equations of motion in matrix form for a train-track-bridge interaction system using the proposed element are derived in detail using the energy principle. The accuracy of the proposed three-dimensional rail-bridge coupling element is verified using the existing twodimensional element. Three examples of a seven-span continuous beam bridge are shown: the first investigates the influence of the efficiency and accuracy of the lengths of the rail and bridge elements on the spatial dynamic responses of the train-track-bridge interaction system, and the other two illustrate the influence of two types of track models and two types of wheel-rail interaction models on the dynamic responses of the system. Results show that (1) the proposed rail-bridge coupling element is not only able to help conserve calculation time, but it also gives satisfactory results when investigating the spatial dynamic responses of a train-trackbridge interaction system; (2) the double-layer track model is more accurate in comparison with the single-layer track model, particularly in relation to vibrations of bridge and rail; and (3) the no-jump wheel-rail interaction model is generally reliable and efficient in predicting the dynamic responses of a train-trackbridge interaction system.

Compared with ordinary railways, the curve radius of tram lines tends to be smaller, with minimum values of only 30 m. Therefore, wheel–rail interaction is more intense and complicated in sections of small radius tram line curves. Using a stochastic variable sample set based on a generalized probability density evolution method, the stochastic variable–spectrum representation method was used to generate a time-domain sample set of stochastic track irregularities. By inputting the stochastic set of track irregularities into a tram-track coupled dynamic system model, the stochastic dynamic response of the coupled dynamic system can be obtained. Moreover, by substituting the stochastic dynamic system response into the generalized probability density evolution formula, the process of probability density evolution of each evaluation index can be obtained by the finite difference method. Finally, the dynamic response of the tram-track coupled dynamic system can be evaluated by the probability distribution of each index. By setting a series of specific groove rail wear values, a tram-track coupled dynamic analysis was carried out, and compared with the specification requirements, vehicle safety limits under different wear values were obtained. This research has great engineering value for guiding the routine maintenance of small radius curve sections of trams.

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