## 2 Methodology  
### Dynamics Under Moving Vehicles  
Figure 1a shows a fleet of three 1-DOF vehicles (\(M_{1}\), \(M_{2}\), and \(M_{3}\)) that are supported by springs (\(k_{1}\), \(k_{2}\), and \(k_{3}\)) and dashpots (\(c_{1}\), \(c_{2}\), and \(c_{3}\)), and they move on a simply supported beam bridge at a constant velocity (\(V_{e}\)) with an equal space (\(d\)) between them. In Figure 1b, a beam with mass \(m\), flexural rigidity \(EI\), and damping \(c\) is connected to the regular rigid road at the ends without gaps. The 2-DOF vehicle with wheelbase \(L_{c}\) is supported by springs (\(k_{F}\) and \(k_{R}\)) and dashpots (\(c_{F}\) and \(c_{R}\)) from the beam and the regular road with rough profile \(u_{r}\). Without the consideration of the damping properties of these three vehicles and the beam, the beam motion equation could be written as follows [34;35]:  
\[m\ddot{u}(x,t)+EIu^{\prime\prime\prime\prime}(x,t)=\sum_{i=1}^{3}\left[k_{i}(u_{i }-u_{r}V_{e}(t-t_{i1})-u(V_{e}(t-t_{i1}),t))-m_{i}g\right]\delta(x-V_{e}(t-t_{i 1}))\Gamma_{i}(t) \tag{1}\]  
where \(\Gamma_{i}(t)\) is defined below by the Heaviside step function \(H_{e}(t)\) to indicate the status of each vehicle on this bridge. The over-dot and the prime denote the derivative with time and space, respectively. \(\delta\) is the delta function. \[\Gamma_{i}(t)=H_{e}(t-t_{i1})-H_{e}(t-t_{i2}),t_{(i+1),1}=t_{i,1}+\Delta t, \Delta t=\frac{d}{V_{e}} \tag{2}\]  
where \(t_{i1}\) and \(t_{i2}\) are time instants when the vehicle enters and leaves the bridge. The response of this beam could be expressed by the mode superposition method. \[u(x,t)=\sum_{j=1}^{\infty}V_{j}(t)\phi_{j}(x) \tag{3}\]  
The mode shape \(\phi_{j}\), the undamped natural frequency \(\omega_{j}\) of the simply supported beam mode \(j\), and the vehicle's natural frequency \(\omega_{vj}\) are given below. \[\phi_{j}(x)=\sin(\frac{j\pi x}{L}),\omega_{j}=(j\pi)^{2}\sqrt{\frac{EI}{mL^{4} }}\omega_{vj}=\sqrt{\frac{k_{i}}{m_{i}}} \tag{4}\]  
Figure 1: (**a**) Diagram of a simply supported Euler–Bernoulli beam under three moving 1-DOF vehicles (\(M_{1}\), \(M_{2}\), and \(M_{3}\)) supported by springs (\(k_{1}\), \(k_{2}\), and \(k_{3}\)) and dampers (\(c_{1}\), \(c_{2}\), and \(c_{3}\)) from the beam surface (\(u_{r}\)) at the constant velocity \(V_{e}\) toward the right; (**b**) diagram of a 2-DOF vehicle with mass of \(M_{c}\) and a moment of inertia of \(I_{c}\)owing three 1-DOF vehicles on the bridge. Substituting Equation (2) into Equation (1), multiplying by \(\phi_{n}\), and then integrating from 0 to \(L\) yields the following:  
\[ \begin{split}&\ddot{v}_{n}(t)+\omega_{n}^{2}v_{n}(t)-\sum\limits_{i=1 }^{3}\frac{2\omega_{v_{i}}^{2}m_{i}}{mL}[u_{i}-u(V_{e}(t-t_{i1}),t)]\phi_{n}(V_{ e}(t-t_{i1}))\Gamma_{i}(t)\\ &=-\sum\limits_{i=1}^{3}\big{[}\frac{2\omega_{v_{i}}^{2}m_{i}}{mL }u_{r}(V_{e}(t-t_{i1})+\frac{2m_{i}}{mL}g]\phi_{n}(V_{e}(t-t_{i1}))\Delta_{i}(t )\end{split}  \tag{5}\]  
A previous work [36] provided an analytical solution form for this interaction between the vehicle and the beam, and an approximate solution form is given here to demonstrate the contact point's response to each vehicle. With the consideration of the small mass ratio (\(m_{i}/mL\)) between the vehicle and the beam, the third term in the left side of Equation (5) could be discarded [13]. Similarly, compared to the gravity term, the roughness terms \(\omega_{v_{i}}^{2}u_{r}(V_{e}(t-t_{i1}))\) are very small and could also be discarded. Then Equation (5) could be simplified into an approximate form. \[ \begin{split}\ddot{v}_{n}(t)+\omega_{n}^{2}v_{n}(t)=-\sum \limits_{i=1}^{3}\frac{2m_{i}}{mL}g\phi_{n}(V_{e}(t-t_{i1}))\Delta_{i}(t)=- \sum\limits_{i=1}^{3}\frac{2m_{i}}{mL}g\sin(\frac{n\pi V_{e}(t-t_{i1})}{L}) \Gamma_{i}(t)\end{split}  \tag{6}\]  
With zero initial conditions, its solution could be determined, as shown below [13]. \[