# Wavelet Time-Frequency Analysis on Bridge Resonance in Train-Track-Bridge Interactive System
## 1 Introduction
### Establishment of Motion Equation for the Track-Bridge Subsystem
\begin{bmatrix}M_{\text{r}}&0&0\\ 0&M_{\text{s}}&0\\ 0&0&M_{\text{b}}\end{bmatrix}\begin{pmatrix}\ddot{u}_{\text{r}}\\ \ddot{u}_{\text{s}}\\ \ddot{u}_{\text{b}}\end{pmatrix} +\begin{bmatrix}C_{\text{r}}&C_{\text{rs}}&0 \\ C_{\text{sr}}&C_{\text{s}}&C_{\text{sb}}\\ 0&C_{\text{bs}}&C_{\text{b}}\end{pmatrix} \begin{pmatrix}\dot{u}_{\text{r}}\\ \dot{u}_{\text{s}}\\ \dot{u}_{\text{b}}\end{pmatrix}\begin{pmatrix}\dot{u}_{\text{s}}\\ 0\\ \dot{u}_{\text{b}}\end{pmatrix} +\begin{bmatrix}K_{\text{r}}&K_{\text{rs}}&0\\ \dot{u}_{\text{s}}\\ 0&K_{\text{bs}}&K_{\text{b}}\end{pmatrix} \begin{pmatrix}\dot{u}_{\text{r}}\\ \dot{u}_{\text{s}}\\ \dot{u}_{\text{b}}\end{pmatrix} = \begin{pmatrix}F_{\text{r}}\\ 0\\ 0\end{pmatrix}  \tag{17}\]  
\[F_{\text{r}}=T_{\text{r}}\Bigl{(}-m_{\text{w}}\ddot{z}_{\text{w}}+c_{1}\Bigl{(} T_{\text{r}}^{\text{T}}\dot{u}_{\text{r}}-\dot{z}_{\text{w}}\Bigr{)}+k_{1}\Bigl{(}T_{ \text{r}}^{\text{T}}u_{\text{r}}-z_{\text{w}}\Bigr{)}+(m_{\text{w}}+\frac{m_ {\text{b}}}{2}+\frac{m_{\text{c}}}{4})g\Bigr{)} \tag{18}\]  
The mass matrix \(M_{\text{b}}\) and stiffness matrix \(K_{\text{b}}\) of the bridge can be formed by withdrawing the nonsupport elements from the matrix \(K_{\text{be10}}\) and \(M_{\text{be10}}\) assembled in the way of Equation (18) with \(M_{\text{be}}\) and \(K_{\text{be}}\) for beam elements. Based on Raleigh damping, the damping matrix \(C_{\text{b}}\) of the bridge can be expressed in Equation (17) by combination of \(M_{\text{b}}\) and \(K_{\text{b}}\) with \(\tilde{c}_{\text{b}}\), damping ratio of the bridge, and \(\omega_{\text{b1}}\) and \(\omega_{\text{b2}}\), the first and second order angular frequencies of the bridge, respectively. The formulation of \(M_{\text{r}},K_{\text{r}}\) and \(C_{\text{r}}\) for the rail can be conducted by similar means to the bridge. \[K_{\text{be10}}=\sum_{j=1}^{10} \begin{bmatrix}0_{4\times(2j-2)}&\mathbf{I}_{4 \times 4}&0_{4\times(20-2j)}\end{bmatrix} ^{\text{T}}K_{\text{be}}\Bigl{[}0_{4 \times(2j-2)}&\mathbf{I}_{4\times 4}&0_{4\times(20-2j)}\Bigr{]} \tag{19}\]  
\[K_{\text{be}}=\frac{E_{\text{b}}I_{\text{b}}}{I_{\text{be}}^{3}} \begin{bmatrix} 12&6l_{\text{be}}&-12&6l_{\text{be}}\\ 6l_{\text{be}}&4l_{\text{be}}^{2}&-6l_{\text{be}}&2l_{\text{be}}^{2}\\ -12&-6l_{\text{be}}&12&-6l_{\text{be}}\\ 6l_{\text{be}}&2l_{\text{be}}^{2}&-6l_{\text{be}}&4l_{\text{be}}^{2}\end{bmatrix}  \tag{20}\]  
\[M_{\text{be}}=\frac{\rho_{\text{b}}S_{\text{b}}l_{\text{be}}}{420} \begin{bmatrix} 156&22l_{\text{be}}&54&-13l_{\text{be}}\\ 22l_{\text{be}}&4l_{\text{be}}^{2}&13l_{\text{be}}&-3l_{\text{be}}^{2}\\ 54&13l_{\text{be}}&156&-22l_{\text{be}}\\ -13l_{\text{be}}&-3l_{\text{be}}^{2}&-22l_{\text{be}}&4l_{\text{be}}^{2}\end{bmatrix}  \tag{21}\]  
\[C_{\text{b}}=\frac{2\tilde{c}_{\text{b}}\omega_{\text{b1}}\omega_{\text{b2}}}{ \omega_{\text{b1}}+\omega_{\text{b2}}}M_{\text{b}}+\frac{2\tilde{c}_{\text{b}} }{\omega_{\text{b1}}+\omega_{\text{b2}}}K_{\text{b}} \tag{22}\]  
Moreover, \(K_{\text{rs}}\) and \(C_{\text{rs}}\), the transpose of which are \(K_{\text{sb}}\) and \(C_{\text{sb}}\), reflect in Equation (17) the interaction of layers and are formulated through the interpolation of the shape function which can be expressed in Equation (23) with the local distance \(x\) on a certain bridge or rail element. \[\mathbf{N}(x)= \begin{bmatrix}1-\frac{3\lambda^{2}}{l_{\text{be}}^{2}}+\frac{2 \lambda^{3}}{l_{\text{be}}^{3}}&x-\frac{2\lambda^{2}}{l_{\text{be}}^{2}}+\frac {\lambda^{3}}{l_{\text{be}}^{2}}&\frac{3\lambda^{2}}{l_{\text{be}}^{2}}-\frac{2 \lambda^{3}}{l_{\text{be}}^{3}}&-\frac{\lambda^{2}}{l_{\text{be}}^{2}}+\frac{ \lambda^{3}}{l_{\text{be}}^{2}}\end{bmatrix}