## 5 Model Validation  
Through the analysis in Section 4, Model 4 should be the most reasonable choice as the wheel rail contact model under earthquake excitations. In order to verify the correctness of the model, the calculation results of wheel-rail contact force based on the model established in this paper were compared with those given in the study of [PERSON] et al. [33]. [PERSON] et al. simplified the vehicle as a system consisting of four rigid bodies, including a body, a frame, and two wheelsets. They considered the lateral, sinking, and rolling movements of each rigid body and assumed that the two wheelsets had the same motion state, establishing a 9-degree of freedom vehicle model. Furthermore, they also considered the lateral and vertical degrees of freedom of the rail, resulting in a comprehensive vehicle dynamic simulation model with a total of 13 degrees of freedom (as shown in Figure 19). The correctness of this model was verified through shaking table tests [39]. Taking five-circle sine waves with frequencies of 0.5 Hz, 0.8 Hz, and 1.5 Hz as excitation, the wheel rail force calculation results are shown in Figures 20-22. It can be seen that the time histories of the vertical and lateral wheel-rail forces calculated based on this article  
Figure 19: The vehicle dynamic analysis model established by [PERSON] et al. [33]: (**a**) vehicle model; (**b**) track model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Running Safety Indices** & **Model 1** & **Model 2** & **Model 3** & **Model 4** \\ \hline Q/P & 0.83 & 0.87 & 0.78 & 0.78 \\ \(\Delta\)P/P & 0.95 & 1.0 & 0.80 & 0.79 \\ \(\sum\)Q (kN) & 228.79 & 250.99 & 49.33 & 46.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Maximum values of the running safety indices based on different models. the selection of wheel-rail contact algorithms under earthquakes. The main conclusions are as follows:  
1. Using different wheel-rail contact algorithms will significantly affect the calculation accuracy of wheel-rail force in the case of flange-root contact under earthquakes. The most significant influence is due to the normal compression algorithm. Using an algorithm based on vertical penetration can lead to a maximum relative error of 339.50% for the case considered in this study. The consideration of the number of wheel-rail contact points also has a notable impact, with a maximum relative error of 35.00% caused by only considering single point contact. The influence of the normal contact stiffness algorithm is the least significant, with a maximum relative calculation error of 23.55% caused by using the empirical formula. 2. Using different wheel-rail contact algorithms will have a significant impact on the indices of running safety assessment under earthquakes. Using wheel-rail normal compression algorithm based on vertical penetration will significantly underestimating the train running safety margin, while only considering the wheel-rail single point contact will overestimate the train running safety margin, and using the wheel-rail normal contact empirical formula has little impact. 3. To ensure the accuracy of running safety assessment of trains under earthquakes, it is recommended to use the normal compression algorithm based on normal penetration and consider multi-point contact in wheel-rail contact modelling. Conceptualization, Z.Z. ; methodology, G.C. and Z.Z. ; formal analysis, G.C. ; writing-original draft preparation, G.C. ; writing-review and editing, W.G. and G.Z. ; supervision, L.J. and B.Y. ; funding acquisition, Z.Z. All authors have read and agreed to the published version of the manuscript. This study was funded by the National Natural Science Foundation of China (Grant No. 52078498), Natural Science Foundation of Hunan Province of China (Grant number: 2022J[30745), Frontier cross research project of Central South University (Grant number: 2023 QYJC006), Science and Technology Research and Development Program Project of China railway group limited (Major Special Project, No.