## 2 Methodology
### Dynamics Under Moving Vehicles
\begin{split} v_{n}(t)=-\frac{1}{1-q_{n}^{2}}\sum\limits_{i=1}^{3 }\frac{2L^{3}m_{i}g}{Eln^{4}\pi^{4}}[\sin(\frac{n\pi V_{e}(t-t_{i1})}{L})-q_{ n}\sin(\omega_{n}(t-t_{i1}))]\Gamma_{i}(t),q_{n}=\frac{n\pi V_{e}}{L\omega_{n}} \end{split}  \tag{7}\]  
Meanwhile, the beam surface roughness \(u_{r}\) should be added to the response of the contact point with the time-varied beam location \(x_{k}=V_{e}(t-t_{k1})\) of each vehicle (\(k=1,2,3\)). \[ \begin{split}& u_{c,k}(t)=\sum\limits_{j=1}^{\infty}-\frac{1}{1-q_{ j}^{2}}\sum\limits_{i=1}^{3}\frac{L^{3}m_{i}g}{Eln^{4}\pi^{4}}[\cos(\frac{j\pi V _{e}(t_{k1}-t_{i1})}{L})-\cos(\frac{2j\pi V_{e}t}{L}+\theta_{k,i})\\ &-q_{j}[\cos((\omega_{j}-\frac{j\pi V_{e}}{L})t+\theta_{k,i,-}^{ \omega_{j}})-\cos((\omega_{j}t+\frac{j\pi V_{e}}{L})t+\theta_{k,i,+}^{\omega_ {j}})]\Gamma_{i}(t)+u_{r}(V_{e}(t-t_{k1}))=R_{k}(t)+u_{r}(V_{e}(t-t_{k1}))\\ &\theta_{k,i}^{V_{e}}=-\frac{j\pi V_{e}(t_{i1}+t_{k1})}{L}, \theta_{k,i,-}^{\omega_{j}}=\frac{j\pi V_{e}t_{k1}}{L}-\omega_{j}t_{i1}, \theta_{k,i,+}^{\omega_{j}}=-(\frac{j\pi V_{e}t_{k1}}{L}+\omega_{j}t_{i1}) \end{split}  \tag{8}\]  
There are three main frequency response components for one beam mode (\(j\)): the driving frequency (\(2j\pi V_{e}/L\)) and the two frequencies (\(\omega_{j}-j\pi V_{e}/L\) and \(\omega_{j}+j\pi V_{e}/L\)) centered at the beam natural frequency (\(\omega_{j}\)) and shifted by the half driving frequencies (\(j\pi V_{e}/L\)). Due to the absence of vehicle frequency components, the response of the contact point could be a better measurement than the responses of the vehicle body [10]. Normally, the moving velocity (\(V_{e}\)) can be kept low to guarantee that the half driving frequencies (\(j\pi V_{e}/L\)) are much smaller than the beam frequency (\(\omega_{j}\)). Then the shift in the half driving frequencies (\(j\pi V_{e}/L\)) could be reduced efficiently. On the other hand, for ratio \(q_{n}\) in Equation (7), this ratio decreases as the velocity decreases. Then the contact point's response will be dominated by the component of the driving frequencies that are close to the static load case. The roughness term (\(u_{r}\)) on the right side of Equation (8) could be removed by subtracting two consecutive contact points such as \(u_{c,1}\) and \(u_{c,2}\) at their durations on the beam. \[ \begin{split}&\Delta u_{1,2}(t)=u_{c,1}(t)-u_{c,2}(t+\Delta t)=R_{1}(t )+u_{r}(V_{e}(t-t_{11}))-R_{2}(t+\Delta t)+u_{r}(V_{e}(t+\Delta t-t_{11}- \Delta t))\\ &=R_{1}(t)-R_{2}(t+\Delta t)\end{split}  \tag{9}\]This residual response form could reveal that the beam vibrates directly at these beam natural frequencies. The terms about the two frequencies (\(\omega_{i}-j\pi V_{e}/L\) and \(\omega_{i}+j\pi V_{e}/L\)) are denoted below. \[B_{j}^{k,i}(t)=[\cos(\omega)t-\frac{j\pi V_{e}t}{L}+\frac{j\pi V_{e}t_{k1}}{L}- \omega)t_{11})-\cos(\omega)t+\frac{j\pi V_{e}t}{L}-\frac{j\pi V_{e}t_{k1}}{L}- \omega jt_{11})] \tag{10}\]  
From the residual expression of Equation (9), the subtraction \(\Delta u_{12}(t)\) contains the term \(B_{j}^{1,i}(t)-B_{j}^{2,i}(t+\Delta t)\), which could be rewritten after some operations. \[B_{j}^{1,i}(t)-B_{j}^{2,i}(t+\Delta t)=2(\sin(\omega)t-\omega_{i}t_{11})+\sin( \omega)t-\omega_{j}t_{i1}+\omega_{j}\Delta t))\sin(\frac{j\pi V_{e}t}{L}-\frac {j\pi V_{e}t_{11}}{L}) \tag{11}\]  
Then the ratio between the subtraction terms of \(\Delta u_{12}(t)\) and \(\Delta u_{23}(t)\) becomes as follows. \[\frac{B_{j}^{1,i}(t)-B_{j}^{2,j}(t+\Delta t)}{B_{j}^{2,i}(t)-B_{j}^{3,i}(t+ \Delta t)}=\frac{\sin(\frac{j\pi V_{e}(t-t_{11})}{L})}{\sin(\frac{i\pi V_{e}( t-t_{11})}{L})}=\frac{\sin(\frac{j\pi V_{e}(t-t_{11})}{L})}{\sin(\frac{i\pi V_{e}( t-t_{11})}{L})} \tag{12}\]  
Considering Equation (4), both the numerator and the denominator in Equation (12) present the \(j\)th beam mode shape, but with a lag of \(V_{e}\Delta t\) as the distance between the contact points.