## 2 Vehicle-Bridge Interaction System  
The mathematical model for the VBI system is shown in Figure 1. The vehicle is modeled as a one degree-of-freedom oscillator moving over a simply-supported beam. The beam is assumed as a Euler-Bernoulli beam with constant cross-sections. Damping of the entire VBI system, and road surface roughness are also neglected. In the mathematical model, the vehicle moves along the \(x\)-axis of the beam to point x at time \(t\). In Figure 1, \(v\), \(k_{v}\), and \(m_{v}\) is the speed, stiffness, and mass of the vehicle, respectively; \(m_{b}\), \(EI\), and \(L\) is the unit mass, section stiffness, and span of the bridge, respectively; \(y_{v}\), \(y_{b}\), and \(y_{w}\) is the vertical displacement of vehicle, bridge, and wheel, respectively; and \(g\) is the for acceleration due to gravity. The equations of motion at time \(t\) for the vehicle and bridge can be written each as follows,  
\[m_{v}\ddot{y}_{v}+k_{v}(y_{v}-y_{b,x=vt})=0 \tag{1}\]  
\[m_{b}\frac{\partial^{2}y_{b}(x,t)}{\partial t^{2}}+EI\frac{\partial^{4}y_{b} (x,t)}{\partial x^{4}}=f_{N}\delta(x-vt) \tag{2}\]  
where \(f_{N}=m_{v}\big{(}g-\ddot{y}_{v}\big{)}+m_{w}\big{(}g-\ddot{y}_{w}\big{)}\). The assumptions made in this study are: 1. The bridge follows [PERSON]'s law, [PERSON]'s hypothesis, [PERSON]'s principle, and small-deflection theory; 2. the wheels of the vehicle stay on the deck constantly while it passes through the bridge; 3. the bridge is initially at rest before the vehicle starts; and 4. the vehicle speed and its path are constant numbers and the suspension spring is considered to be linearly elastic. By making the above equation dimensionless, the dimensionless frequencies of vehicles and bridges can be expressed as:  
\[\Omega_{v}=\sqrt{\frac{m_{b}}{EI}}L^{2}\omega_{v} \tag{3}\]  
\[\Omega_{bm}=\sqrt{\frac{m_{b}}{EI}}L^{2}\omega_{bn}=\sqrt{\frac{m_{b}}{EI}}L^{ 2}\Big{(}\frac{n\pi}{L}\Big{)}^{2}\sqrt{\frac{EI}{m_{b}}}=(n\pi)^{2}. \tag{4}\]  
Using modal superposition, \(\alpha_{yb}(s,\tau)=\sum\limits_{n=1}^{\infty}H_{n}(\tau)\sin(n\pi s)\), where \(\sin(n\pi s)\) and \(H_{n}(\tau)\) is the \(n\)-th mode shape and mode displacement, respectively. Substituting the aforementioned expressions into Equations (1) and (2), multiplying both sides of Equation (2) by \(\sin(n\pi s)\) and integrating \(s\) from 0 to 1, we obtain:  
\[\ddot{H}_{m}(t)+2\alpha_{mw}\sum\limits_{n=1}^{\infty}\ddot{H}_{n}(\tau)\sin( n\pi s)\sin(m\pi s)+4\alpha_{mw}\alpha_{v}\sum\limits_{n=1}^{\infty}(n\pi) \dot{H}_{n}(\tau)\cos(n\pi s)\sin(m\pi s)\] \[+\Omega_{bm}^{2}H_{m}(\tau)-2\alpha_{mw}\alpha_{v}^{2}\sum \limits_{n=1}^{\infty}(n\pi)^{2}H_{n}(\tau)\sin(n\pi s)\sin(m\pi s) \tag{5}\]  
\[+2\alpha_{mv}\Omega_{v}^{2}\sum\limits_{n=1}^{\infty}H_{n}(\tau)\sin(n\pi s) \sin(m\pi s)-2\alpha_{mw}\Omega_{v}^{2}\alpha_{yv}\sin(m\pi s)=2(\alpha_{mw}+ \alpha_{mw})g^{*}\sin(m\pi s)\]  
\[\frac{\partial^{2}\alpha_{yv}}{\partial\tau^{2}}+\Omega_{v}^{2}\alpha_{yv}- \Omega_{v}^{2}\sum\limits_{n=1}^{\infty}H_{n}(\tau)\sin(n\pi s)=0 \tag{6}\]  
where, \(\alpha_{yb}\), \(\alpha_{yv}\), \(\alpha_{mw}\), \(\alpha_{mw}\), \(\alpha_{v}\), \(s\), \(\tau\), and \(g^{*}\) are the dimensionless parameters of \(y_{b}\), \(y_{v}\), \(m_{v}\), \(m_{w}\), \(v\), \(x\), \(t\), and \(g\), respectively. Their expressions are detailed in Appendix A. The above equations can be solved by using the stepwise integration method. It is generally believed that the mass of the vehicle is far less than the mass of the bridge, which means the impact of the vehicle vibration on bridge vibration is ignored. Thus, we have \(m_{v}\ll m_{b}L\), \(m_{w}\ll m_{b}L\), namely \(\alpha_{mv}\to 0\), \(\alpha_{mw}\to 0\). Equation (5) can be simplified as:  
\[\ddot{H}_{m}(\tau)+\Omega_{bm}^{2}H_{m}(\tau)=2(\alpha_{mv}+\alpha_{mw})g^{*} \sin(m\pi s). \tag{7}\]  
Under zero initial conditions, we obtain:  
\[m_{v}\ddot{y}_{v}+k_{v}(y_{v}-y_{b,x=vt})=0 \tag{1}\]0  
\[m_{v}\ddot{y}_{v}+k_{v}(y_{v}-y_{b,x=vt})=0 \tag{1}\]1 The mathematical model for the vehicle-bridge interaction (VBI) system.