# Wavelet Time-Frequency Analysis on Bridge Resonance in Train-Track-Bridge Interactive System
## 1 Introduction
### Establishment of Motion Equation for the Track-Bridge Subsystem  
The track-bridge subsystem is presented in Figure 3 with three layers containing the rail, sleepers and bridges from top to bottom, between two of which there is an inserted connective layer of spring-dampers simulating fasteners and ballasts, respectively. For the specific meaning of the parameters, refer to Table 2; they are selected according to the mainstream parameters in the world for the track-bridge subsystem. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Train Subsystem Parameter** & **Value** & **Unit** \\ \hline Car body mass \(m_{\rm c}\) & 40,000 & kg \\ Bogie mass \(m_{\rm b}\) & 3000 & kg \\ Wheel set mass \(m_{\rm w}\) & 2000 & kg \\ Car body inertia \(l_{\rm c}\) & \(2\times 10^{6}\) & kg\(\cdot\)m\({}^{2}\) \\ Bogie inertia \(l_{\rm B}\) & 3000 & kg\(\cdot\)m\({}^{2}\) \\ Primary suspension stiffness \(k_{1}\) & \(1\times 10^{6}\) & N/m \\ Primary suspension damping \(c_{1}\) & 20,000 & N\(\cdot\)s/m \\ Secondary suspension stiffness \(k_{2}\) & \(2\times 10^{5}\) & N/m \\ Secondary suspension damping \(c_{2}\) & 10,000 & N\(\cdot\)s/m \\ Semi–distance of wheel sets \(d_{1}\) & 1.25 & m \\ Semi–distance of bogies \(d_{2}\) & 8.75 & m \\ Distance of vehicle \(l_{\rm v}\) & 25 & m \\ Number of vehicle elements \(n_{\rm v}\) & 8 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values of the train subsystem [21]. In detail, the rail and the bridges with simple support are modeled by the finite element method (FEM) with multiple Euler beam elements, each of which includes four DOFs (left node: \(z_{j}\), \(\varphi_{j}\); right node: \(z_{j+1}\), \(\varphi_{j+1}\)) and considers the Rayleigh damping [22]. Furthermore, the sleepers are modeled as rigid bodies, each of which vibrates with only one vertical DOF. According to the FEM, the shape function will be adopted to interpolate the layers of the fasteners (rail to sleepers) and the ballasts (sleepers to bridges), respectively, in the way of non-nodal connection so as to simplify the model by reducing the DOFS of rail and bridges. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Track–Bridge Subsystem** & **Value** & **Unit** \\
**Parameter** & & \\ \hline Rail elastic modulus \(E_{\text{r}}\) & \(2.1\times 10^{11}\) & Pa \\ Rail inertia \(I_{\text{r}}\) & \(3.2\times 10^{-5}\) & m\({}^{4}\) \\ Rail section area \(S_{\text{r}}\) & \(7.7\times 10^{-3}\) & m\({}^{2}\) \\ Rail material density \(\rho_{\text{r}}\) & \(7850\) & kg/m\({}^{2}\) \\ Beam element length of rail \(I_{\text{re}}\) & \(3.2\) & m \\ Rail length \(I_{\text{r}}\) & \(96\) & m \\ Rail damping ratio \(S_{\text{r}}\) & \(0.01\) & \\ Fastener stiffness \(k_{\text{f}}\) & \(1.2\times 10^{8}\) & N/m \\ Fastener damping \(c_{\text{f}}\) & \(75,000\) & N\(\cdot\)s/m \\ Fastener space \(l_{\text{f}}\) & \(0.64\) & m \\ Sleeper mass \(m_{\text{s}}\) & \(120\) & kg \\ Sleeper space \(l_{\text{s}}\) & \(0.64\) & m \\ Ballast stiffness \(k_{\text{s}}\) & \(1.8\times 10^{8}\) & N/m \\ Ballast damping \(c_{\text{s}}\) & \(5.8\times 10^{4}\) & N\(\cdot\)s/m \\ Bridge elastic modulus \(E_{\text{b}}\) & \(3.5\times 10^{10}\) & Pa \\ Bridge inertia \(l_{\text{b}}\) & \(22.5\) & m\({}^{4}\) \\ Bridge section area \(S_{\text{b}}\) & \(17.5\) & m\({}^{2}\) \\ Bridge material density \(\rho_{\text{b}}\) & \(2500\) & kg/m\({}^{2}\) \\ Beam element length of bridge \(l_{\text{be}}\) & \(3.2\) & m \\ Bridge length \(l_{\text{b}}\) & \(32\) & m \\ Bridge damping ratio \(S_{\text{b}}\) & \(0.03\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter values of track–bridge subsystem [21]. Figure 3: 2D model of track–bridge subsystem: (a) Global view of 2D model of track–bridge subsystem; (b) Beam element \(j\); (c) Local view of track–bridge subsystem. In accordance with the D'Alembert principle, the motion equation of the train can be established in Equations (17) and (18). \[