# Wavelet Time-Frequency Analysis on Bridge Resonance in Train-Track-Bridge Interactive System
## 1 Introduction
### Establishment of Motion Equation for the Train Subsystem  
The train subsystem is modeled by several independent vehicle elements shown in Figure 2, each of which includes a car body, two bogies, four wheel sets, and a two-layer spring-damper suspension system by referring to Table 1. The assumptions are made to simplify the analysis with enough accuracy. Namely, the interaction is neglected among vehicle elements. The rigid car body, bogies, and wheel sets in each vehicle element are adopted. The vehicle element is a linear system; namely, the mass, damping, and stiffness matrices of the vehicle element are constant. The train runs on the track at a constant speed. The wheels and rails fit snugly. In this way, each vehicle element includes six independent degrees of freedom (DOFs) for the car body (\(z_{cj}\), \(\varphi_{ci}\)) and bogies (\(z_{\mathrm{b1l}}\), \(\varphi_{\mathrm{b1l}}\), \(z_{\mathrm{b2l}}\), \(\varphi_{\mathrm{b2l}}\)), and with four dependent DOFs (\(z_{\mathrm{w1i}}\), \(z_{\mathrm{w2i}}\), \(z_{\mathrm{w3i}}\), \(z_{\mathrm{w4i}}\)) for the wheel sets reflecting the vertical motion states of four contact points, where \(z\) is the vertical displacement; \(\varphi\) is the pitch torsional displacement; the numbers of the bogies and the wheel sets represent the front and rear orders; and subscript \(i\) denotes the order of certain vehicle elements.  
Figure 2: 2D model of the vehicle element.  
In accordance with the D'Alembert principle, the motion equation of the train can be established in Equation (11).  
\[\mathbf{M}_{\rm t}\ddot{\mathbf{u}}_{\rm t}+\mathbf{C}_{\rm t}\dot{\mathbf{u}}_{\rm t}+\mathbf{K}_{ \rm t}\mathbf{u}_{\rm t}=\mathbf{T}_{\rm t}(k_{1}\mathbf{z}_{\rm w}+c_{1}\dot{\mathbf{z}}_{\rm w}) \tag{11}\]  
where \(\mathbf{M}_{\rm t}\), \(\mathbf{C}_{\rm t}\), \(\mathbf{K}_{\rm t}\) and \(\mathbf{T}_{\rm t}\) signify, respectively, the mass, the damping, the stiffness and projector matrices of the train and can be formulated by diagonally assembling the \(\mathbf{M}_{\rm ve}\), \(\mathbf{C}_{\rm ve}\), \(\mathbf{K}_{\rm ve}\) and \(\mathbf{T}_{\rm ve}\) of vehicle elements in the same way as Equation (13). Similar to \(\mathbf{K}_{\rm ve}\) in Equation (14), \(\mathbf{C}_{\rm ve}\) can be formulated by replacing the spring factor \(k\) with damping factor \(c\).  
\[\mathbf{M}_{\rm ve}=\text{diag}\big{[}\mathbf{m}_{\rm c}\quad l_{\rm c}\quad\mathbf{m}_{ \rm b}\quad\mathbf{I}_{\rm B}\quad\mathbf{m}_{\rm b}\quad\mathbf{I}_{\rm B}\big{]} \tag{12}\]  
\[\mathbf{M}_{\rm t}=\text{diag}\big{[}\mathbf{M}_{\rm ve}\quad\mathbf{M}_{\rm ve}\quad \cdots\quad\mathbf{M}_{\rm ve}\big{]} \tag{13}\]  
\[\mathbf{K}_{\rm ve}=\begin{bmatrix}2k_{2}&0&-k_{2}&0&-k_{2}&0\\ 0&2k_{2}d_{2}^{2}&-k_{2}d_{2}&0&k_{2}d_{2}&0\\ -k_{2}&-k_{2}d_{2}&2k_{1}+k_{2}&0&0&0\\ 0&0&0&2k_{1}d_{1}^{2}&0&0\\ -k_{2}&k_{2}d_{2}&0&0&2k_{1}+k_{2}&0\\ 0&0&0&0&0&2k_{1}d_{1}^{2}\end{bmatrix} \tag{14}\]  
\[\mathbf{T}_{\rm ve}=\begin{bmatrix}0&0&1&d_{1}&0&0\\ 0&0&1&-d_{1}&0&0\\ 0&0&0&0&1&d_{1}\\ 0&0&0&0&1&-d_{1}\end{bmatrix}^{\rm T} \tag{15}\]  
With the application of CWT in Equation (2) and its computational properties of linearity and derivation in Equations (7) and (9), the motion equation of the train can be redefined as Equation (16) in the CWT domain.  
\[\mathbf{M}_{\rm t}\frac{\partial^{2}WT_{\mathbf{u}_{\rm t}}(a,b)}{\partial b^{2}}+ \mathbf{C}_{\rm t}\frac{\partial WT_{\mathbf{u}_{\rm t}}(a,b)}{\partial b}+\mathbf{K}_{ \rm t}WT_{\mathbf{u}_{\rm t}}(a,b)=\mathbf{T}_{\rm t}\bigg{(}k_{1}WT_{z_{\rm w}}(a,b)+ c_{1}\frac{\partial WT_{z_{\rm w}}(a,b)}{\partial b}\bigg{)} \tag{16}\]