## 2 Numerical Simulation
### The Basic Theory of the Vehicle-Bridge Interaction System  
The VBI system consists of two sub-systems: a vehicle system and a bridge system. The vehicle system's inputs are the road unevenness and the bridge vibration components, while its output is the vehicle vibration. On the other hand, the bridge system's input is the contact force due to the vehicle vibration, while its output is the bridge vibration. The outputs of one system are the inputs of the opposite system. Therefore, the vehicle and bridge form an interaction system. The conceptual diagram of this system is shown in Figure 2.  
First, the equations of motion of the vehicle are given by  
\[\mathbf{M}_{v}\ddot{z}(t)+\mathbf{C}_{v}\dot{z}(t)+\mathbf{K}_{v}\mathbf{z}(t )=\mathbf{F}_{v}(t) \tag{1}\]  
where  
\[\mathbf{M}_{v}= \begin{bmatrix}\frac{d_{1}m_{s}}{d_{1}+d_{2}}&\frac{d_{1}m_{s}} {d_{1}+d_{2}}&0&0\\ \frac{1}{d_{1}+d_{2}}&\frac{1}{d_{1}+d_{2}}&0&0\\ 0&0&m_{u1}&0\\ 0&0&0&m_{u2}\end{bmatrix} = \begin{bmatrix}m_{s1}&m_{s2}&0&0\\ \frac{1}{d_{1}+d_{2}}&\frac{1}{d_{1}+d_{2}}&0&0\\ 0&0&m_{u1}&0\\ 0&0&0&m_{u2}\end{bmatrix}  \tag{2}\]  
\[\mathbf{C}_{v}= \begin{bmatrix}c_{s1}&c_{s2}&-c_{s1}&-c_{s2}\\ d_{1}c_{s1}&-d_{2}c_{s2}&-d_{1}c_{s1}&d_{2}c_{s2}\\ -c_{s1}&0&c_{s1}&0\\ 0&-c_{s2}&0&c_{s2}\end{bmatrix}  \tag{3}\]  
\[\mathbf{K}_{v}= \begin{bmatrix}k_{s1}&k_{s2}&-k_{s1}&-k_{s2}\\ d_{1}k_{s1}&-d_{2}k_{s2}&-d_{1}k_{s1}&d_{2}k_{s2}\\ -k_{s1}&0&k_{s1}+k_{u1}&0\\ 0&-k_{s2}&0&k_{s2}+k_{u2}\end{bmatrix}  \tag{4}\]  
\[\mathbf{z}(t)= \begin{bmatrix}z_{s1}(t)&z_{s2}(t)&z_{u1}(t)&z_{u2}(t)\end{bmatrix}  ^{T} \tag{5}\]The vertical vibration displacements of sprung-mass and unsprung-mass at the \(i\)-th wheel are \(z_{si}(t)\) and \(z_{ui}(t)\), respectively. The operators \(\left(\ \dot{}\ \right)\) and \(\left(\ \dot{}\ \right)\) represent the first- and second-order time derivatives. The measured data is \(\tilde{z}(t)\). Let \(d_{i}\) be the distance from the gravity point G to the \(i\)-th suspension. The mass of the vehicle body is denoted by \(m_{s}\). The inertia of the vehicle body can be given by \(I=m_{s}d_{1}d_{2}\). The damping \(c_{si}\) and spring stiffness \(k_{si}\) characterize the \(i\)-th suspension. The mass and stiffness of the tires are \(m_{ui}\) and \(k_{ui}\). The forced vertical displacement under the \(i\)-th wheel is called the input profile \(u_{i}(t)\) in this paper. Let \(\mathbf{M}_{v}\), \(\mathbf{C}_{v}\), and \(\mathbf{K}_{v}\) be the vehicle mass, damping, and stiffness matrices, respectively. \(\mathbf{F}_{v}\) represents the external force vector acting on the vehicle system. This vehicle model is generally called a half-car model. The bridge is modeled as an Euler-Bernoulli beam and, therefore, \(\rho A(x)\) and \(EI(x)\) denote the mass per unit length and flexural rigidity of the beam, respectively. Possible aging-caused changes in the bridge system are modeled as the reduction of the bending stiffness \(EI(x)\) in the numerical simulations. The equation of motion of the bridge can be expressed as  
\[\rho A\ddot{y}(x,\ t)+\frac{\partial^{2}}{\partial x^{2}}EI\bigg{(}\frac{ \partial^{2}}{\partial x^{2}}y(x,\ t)\bigg{)}=\sum_{i=1}^{2}\delta(x-x_{i}(t) )P_{i}(t)+\delta(x)R_{\text{A}}(t)+\delta(x-L)R_{\text{B}}(t) \tag{7}\]  
where \(y(x,\ t)\) is the bridge vibration, while \(P_{i}(t)\) denotes the contact force of \(i\)-th tire. \(\delta(x)\) is Dirac's delta function. \(R_{\text{A}}\) and \(R_{\text{B}}\) are the reaction forces acting at the supports. The position of \(i\)-th wheel is given by \(x_{i}(t)\). To simulate the dynamic responses of the bridge, FEM is adopted in this study.