## 3 Analysis and Calculation Parameters
### Earthquake Excitation Model  
For high-speed railways, earthquakes not only directly affect the operation of high-speed trains and pose a threat of derailment to passengers, but also affect the construction and maintenance of high-speed railway lines. Once an earthquake occurs, it causes serious damage to infrastructures such as tracks and viaducts, greatly increasing maintenance costs and operational losses. Based on the above situation, it is of great significance to consider the seismic influence in vehicle-bridge vibration analysis. Due to the limited number of existing seismic records, it is not enough to solely study the dynamic response of the system under earthquakes using existing records. Therefore, this article uses artificial seismic waves as seismic excitations. The mathematical model of seismic waves is simulated by using trigonometric series synthesis [35], and the artificial seismic wave is synthesized by using the simulated standard response spectrum. The acceleration equation of seismic ground motion is expressed in Equation (10):  
\[\ddot{q}(t)=I(t)\cdot\sum_{k=1}^{n}A(\omega_{k})\sin(\omega_{k}+\varphi_{k}) \tag{10}\]  
\[I(t)=\left\{ \begin{array}{ll}\left(\frac{t}{t_{1}}\right)^{2}&0\leq t\leq t _{1}\\ 1&t_{1}\leq t\leq t_{2}\\ e^{-c(t-t_{2})}&t_{2}\leq t\end{array} \right. \tag{11}\]  
\[A(\omega_{k})=\sqrt{4S_{g}(\omega)\Delta\omega} \tag{12}\]  
\[ \begin{array}{l}\omega_{k}=\omega_{1}+\left(k-\frac{1}{2}\right)\Delta\omega \\ \Delta\omega=\frac{\omega_{k}-\omega_{l}}{n},k=1,2,\cdots n\end{array}  \tag{13}\]  
where \(I(t)\) denotes the intensity envelope function, which describes the process of earthquakes from appearance, enhancement, stationarity, and weakening, as shown in Figure 5. The values of each parameter of the intensity envelope function are referred to in Ref. [35]. \(c\), \(A(\omega_{k})\), \(S_{g}(\omega)\), \(\omega_{k}\), \(\varphi_{k}\), \(\omega_{u}\), \(\omega_{l}\) denote the attenuation coefficient of the weakening section, the amplitude of corresponding \(\omega_{k}\), the spectral density function of ground motion acceler \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Working Condition Type** & \(\mathbf{H_{1}}\) & \(\mathbf{H_{2}}\) & \(\mathbf{H_{3}}\) & \(\mathbf{H_{4}}\) \\ \hline Equal-height pier (m) & 8 & 8 & 8 & 8 \\ Gradually increasing (m) & 8 & 10 & 12 & 14 \\ Sharply increasing (m) & 8 & 8 & 14 & 14 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Pier height parameters under different working conditions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Definitions** & **Values** & **Units** \\ \hline \(E_{b}\) & Elastic modulus & \(3.45\times 10^{10}\) & N/m\({}^{2}\) \\ \(I\) & Mass moment of inertia of cross section & 12.744 & m\({}^{4}\) \\ \(\mu\) & Poisson’s ratio & 0.2 & / \\ \(\overline{m}_{b}\) & Mass per unit length & \(2.972\times 10^{4}\) & kg/m \\ \(L_{x}\) & Length of element & 3.2 & \(m\) \\ \(\zeta\) & Damping ratio & 0.03 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Some parameters of the bridge. ation power, the k-th frequency, the random phase angle evenly distributed and mutually independent within \([0\,,\,2\pi]\), and the upper and lower limits of frequency, respectively. In this paper, the ground vibration power spectral density function model adopts the Clough-Penzien model that considers the low-frequency components.