## 1 Introduction
### Simulation Method Based on Random Vehicle-Bridge Coupling Vibration  
The classical half-vehicle model is adopted to describe the vibration characteristics of the vehicle. Based on the modal synthesis method and the mode superposition method, the modal synthesis equation of the bridge structure with generalized coordinates and vehicle degrees of freedom is established as shown in Equation (1). \[ \begin{array}{c}\left(\begin{array}{cc}[l]&[0]\\ [0]&[M_{v}]\end{array}\right)\left(\begin{array}{c}\left\{\tilde{q}_{r}\right\} \\ \tilde{Z}_{v}\end{array}\right)+\left(\begin{array}{cc}[2\omega\xi]+[\Phi]^{T} [N]^{T}c_{ti}[N][\Phi]&-[\Phi]^{T}[N]c_{ti}\\ -[\Phi][N]c_{ti}&[C_{v}]\end{array}\right)\left(\begin{array}{c}\left\{\dot{ q}_{r}\right\}\\ \dot{Z}_{v}\end{array}\right)\\ +\left(\begin{array}{cc}[\omega^{2}]+[\Phi]^{T}[N]^{T}k_{ti}[N][\Phi]&-[ \Phi]^{T}[N]k_{ti}-V[\Phi]^{T}[N_{x}]c_{ti}\\ -[\Phi][N]k_{ti}&[K_{v}]\end{array}\right)\left(\begin{array}{c}\left\{q_{r }\right\}\\ \dot{Z}_{v}\end{array}\right)=\left(\begin{array}{c}[\Phi]^{T}\{F_{bv}^{VIB }\}\\ F_{v}^{G}+F_{bv}^{VIB}\end{array}\right)\end{array}  \tag{1}\]  
in which \([I]\) denotes the n-order unit matrix; \([M_{v}]\), \([C_{v}]\) and \([K_{v}]\) are the mass of the vehicle and the damping and stiffness matrix of the vehicle suspension, respectively; \(\omega\) and \(\xi\) are a certain order frequency and damping ratio of the bridge structure, respectively; \([\Phi]\) is the modal coordinate corresponding to the frequency of the bridge structure; \([N]\) is the interpolation function for the vehicle tire movement to the bridge structure; \(c_{ti}\) and \(k_{ti}\) represent the vehicle tire damping and stiffness, respectively; \(F_{bv}^{VIB}\) is the interaction force matrix between the vehicle and the bridge; \(F_{v}^{G}\) is the force matrix assigned to the wheel of the vehicle; \(\{\dot{q}_{r}\}\), \(\{\dot{q}_{r}\}\) and \(\{q_{r}\}\) are the acceleration, velocity and displacement vectors under the r-order modal coordinates of the bridge structure, respectively; \(\ddot{Z}_{v}\), \(\dot{Z}_{v}\) and \(Z_{v}\) are response matrices of the vehicle acceleration, velocity and displacement, respectively. In addition, the pavement roughness is considered as well, and the function adopted is shown as Equation (2). \[r(x)=\sum_{k=1}^{N}\sqrt{2\Delta n\varphi(n_{0})\left(\frac{n_{k}}{n_{0}} \right)^{-W}}\cos(2\pi n_{k}x+\theta_{k}) \tag{2}\]  
where \(x\) is the horizontal displacement of pavement, \(\varphi(n_{0})\) is the pavement roughness coefficient, \(n_{0}\) is the reference frequency, \(W\) is the frequency index and \(\theta_{k}\) is a random number. Cellular Automata (CA) analysis theory is introduced to simulate the random traffic flow, and the analysis model is established with MATLAB [16]. The traffic lanes of the bridge are discretized into continuous cellular spaces, and the function for describing pavement  
Figure 2: Distribution Regularity of Concrete Aggregate. (**a**) Before Disturbance; (**b**) After Disturbance. roughness is integrated into the cell space [17]. The process of random vehicle-bridge coupling vibration simulation is shown in Figure 3. The modal matrix and frequency of bridge structure can be obtained by finite element analysis. The modal vector is interpolated and stored into the cell space together with the structural frequency. The known vehicle characteristics and traffic characteristics parameters are used to update the vehicle position Based on the statistics of natural frequency and model, vibration response analysis of the bridge is carried out based on the theory of random vehicle-bridge coupling vibration. In this model, the surface roughness level is considered as Level B, and only one fast lane and one slow lane are opened on the bridge.