## 4 Influence Factors on Dynamic Responses of Vehicle-Track-Subgrade System in Transition Section
### Change Rule of Dynamic Response along the Longitudinal Direction of the Line  
Based on the vehicle-track-subgrade coupling model of the transition section, the dynamic performance of vehicles under the above different track irregularity conditions is calculated, and the  
Figure 8: Comparison of measured and numerical results of dynamic stress amplitude on the surface of subgrade bed. (**a**) Dynamic displacement; (**b**) dynamic stress. Figure 7: Schematic diagram of dynamic load application. results are shown in Tables 1 and 2. The change rules between differential settlement, bending angle change, stiffness ratio and length of transition section and the surface dynamic response of transition section subgrade bed along the line direction are shown in Figures 9-12. It can be seen from Figure 9 that the maximum amplitude of dynamic stress and displacement on the bed surface does not vary significantly with the stiffness. However, the amplitude varies significantly in the longitudinal direction of the railway line, especially in the transition section. The higher the stiffness of subgrade is, the smaller the dynamic displacement on the bed surface is, and the higher the dynamic stress and the dynamic stress on the bed surface are. Table 1 shows that under the power spectral density (PSD) irregular conditions, the absolute rigid foundation is not suitable for high-speed vehicles. This is because the calculated dynamic response in the case of a rigid subgrade exceeds the relevant regulations, such as those specified in 95J01-L and 95J01-M. In comparison, for the flexible subgrade that meets the requirements of the high-speed railway ballastless track specification, the dynamic response obtained by the simulation calculation does not exceed the various prescribed values mentioned above. This shows that it is safe for high-speed vehicles to run on flexible subgrades under PSD irregular conditions. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{3}{*}{**Working Condition**} & \multicolumn{3}{c}{**Acceleration of Train**} & \multirow{3}{*}{\begin{tabular}{c} **Rate of Wheel** \\ **Body (m/s2)** \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} **V/Y\({}_{\text{Dexilment}}\)** \\ **Q/P** \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} **Rail Wheel Force** \\ **(kN)** \\ \end{tabular} } \\ \cline{4-7}  & & & **Lateral** & & & **Q/P** & & **Vertical** & **Lateral** \\ \hline \multirow{4}{*}{Rigid} & \multirow{4}{*}{\begin{tabular}{c} Transition \\ length \\ \end{tabular} } & 10 & – & 0.893 & 0.120 & 0.010 & 115.3 & – \\  & 15 & – & 0.582 & 0.110 & 0.010 & 115.8 & \\  & 20 & – & 0.424 & 0.105 & 0.010 & 116.4 & – \\  & 25 & – & 0.405 & 0.063 & 0.010 & 118.2 & – \\  & 30 & 0.018 & 0.406 & 0.054 & 0.011 & 121.3 & – \\ \hline \multirow{4}{*}{\begin{tabular}{c} Rigid \\ \end{tabular} } & 1/1000 & 0.033 & 1.035 & 0.124 & 0.015 & 123.8 & – \\  & 3/1000 & -0.145 & 4.413 & 0.330 & 0.050 & 142.7 & – \\  & 4/1000 & 0.152 & 4.327 & 0.522 & 0.069 & 150.2 & – \\ \hline \multirow{4}{*}{\
\begin{tabular}{c} Flexible \\ \end{tabular} } & PSD & 0.349 & 0.500 & 1.469 & 2.041 & 140.6 & 193.7 \\  &
\begin{tabular}{c} PSD \\ Bending angle of \\ 0.75/1000 \\ \end{tabular} & 0.341 & 0.520 & 0.448 & 0.229 & 135.7 & 41.9 \\ \hline \multicolumn{7}{l}{Note: Y\({}_{\text{Dexilment}}\) is the maximum allowable value of the lateral displacement between wheel and rail, which is \(-0.045\) m according to the requirements of the German Railway code. Y/Y\({}_{\text{Dexilment}}\) is the ratio of the lateral displacement between wheel and rail to the maximum allowable lateral displacement between wheel and rail. Derailment coefficient = Q/P, where Q is the lateral force, and P is the vertical force. * Bending angle indicates the ratio of differential settlement to transition length. * PSD (power spectral density) indicates the power spectral density of track irregularity.} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of dynamic performance of four kinds of vehicles with irregularities.