## 3 Numerical Analysis
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Track Components** & \begin{tabular}{c} **Finite Element** \\ **Type** \\ \end{tabular} & **Related Parameters** & **Value** \\ \hline \multirow{4}{*}{Rails} & \multirow{4}{*}{Beam188} & Modulus of elasticity/(Pa) & \(2.1\times 10^{11}\) \\  & & \begin{tabular}{c} Sectional moment of inertia/(m\({}^{4}\)) \\ Poisson ratio \\ Line Density(kg\(\cdot\)m\({}^{-1}\)) \\ \end{tabular} & \(3.215\times 10^{-5}\) \\  & & \
\begin{tabular}{c} Poisson ratio \\ Line Density(kg\(\cdot\)m\({}^{-1}\)) \\ \end{tabular} & \(0.3\) \\ \hline \multirow{4}{*}{Fasteners} & \multirow{4}{*}{Combin14} & Fastener stiffness/(N\(\cdot\)m\({}^{-1}\)) & \(4\times 10^{7}\) \\  & & Fastener damping/(N\(\cdot\)s\(\cdot\)m\({}^{-1}\)) & \(2.2656\times 10^{4}\) \\  & & Vertical spacing/(m) & \(0.625\) \\ \hline \multirow{4}{*}{Track} & \multirow{4}{*}{Solid45} & Length/(m) & \(4.93\) \\  & & Width/(m) & \(2.4\) \\  & & Thickness/(m) & \(0.2\) \\  & & Modulus of elasticity/(Pa) & \(3.6\times 10^{10}\) \\  & & Poisson’s ratio & \(0.25\) \\  & & Density/(kg\(\cdot\)m\({}^{-3}\)) & \(2500\) \\ \hline \multirow{4}{*}{CA mortar layer} & \multirow{4}{*}{Combin14} & Stiffness/(N\(\cdot\)m\({}^{-1}\)) & \(9.375\times 10^{9}\) \\  & &
\begin{tabular}{c} Damping/(N\(\cdot\)s\(\cdot\)m\({}^{-1}\)) \\ \end{tabular} & \(7.5\times 10^{5}\) \\ \hline \multirow{4}{*}{Base} & \multirow{4}{*}{Solid45} & Modulus of elasticity/(Pa) & \(3.3\times 10^{10}\) \\  & & Poisson’s ratio & \(0.2\) \\  & & Density/(kg\(\cdot\)m\({}^{-3}\)) & \(2500\) \\ \hline \multirow{4}{*}{Bridge} & \multirow{4}{*}{Shell63} & Length/(m) & \(32.5\) \\  & & Thickness of top plate/(m) & \(0.315\) \\ \cline{1-1}  & & Thickness of web/(m) & \(0.480\) \\ \cline{1-1}  & & Thickness of base plate/(m) & \(0.300\) \\ \cline{1-1}  & & Modulus of elasticity/(Pa) & \(3.8\times 10^{10}\) \\ \cline{1-1}  & & [PERSON]’s ratio & \(0.25\) \\ \cline{1-1}  & & Density/(kg\(\cdot\)m\({}^{-3}\)) & \(2500\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: CRTS-I type plate ballastless track–BGB dynamics parameters. Figure 3: High-speed train–slab ballastless track–BGB theoretical calculation model. burden have been verified, to calculate the wheel-rail forces. The wheel-rail normal force is calculated using the nonlinear Hertzian elastic contact theory. The tangential wheel-rail creep forces are first derived using [PERSON]'s linear creep theory, and then modified using the Shen-Hedrick-Elkins nonlinear modification theory. Because this study focuses on a bridge as a tangential railway segment, a single-point wheel-rail contact model is adopted. The entire system can be solved using the multi-step explicit-implicit time integration method [42, 43] (i.e., the train-track system is solved by [PERSON]'s explicit method [44]) with a time step of \(10^{-4}\) s, and the bridge system can be solved using the implicit Newmark-\(\beta\) method with a time step of \(10^{-3}\) s. To calculate the vibrations of the BGB, the irregularity of the high-speed tracks is used to represent the excitation; the foregoing is also used as the acoustic boundary condition in the second step of the analysis. For these calculations, we assume that a CRH380B high-speed train under tare conditions is moving at a speed of 200 km/h. Taking the vibration acceleration response of the BGB as the acoustic boundary condition, the acoustic boundary element method is adopted to determine the acoustic radiation characteristics of the BGB. The theory of the acoustic boundary element method can be seen in [45]. The sound pressure levels calculated and measured at a position 5 m directly below the bridge are shown in Figure 4. The results are in good agreement, and within the studied frequency band. Their trends of changing over time are consistent, and their peak values appear at similar frequencies. Therefore, the acoustic prediction model herein developed for BGBs is reliable.