## 2 Theoretical Analysis  
### Transfer Matrix of Bridge Responses in a VBI System  
When a vehicle traverses a bridge, it engages in an interaction with the bridge structure, collectively known as the VBI system [28]. For the purposes of simplification, the vehicle can be modeled as a moving sprung mass. The typical VBI system, comprising the vehicle and a simply supported beam, is depicted in Figure 1[27]. In this system, the bridge dynamics can be calculated with the following equations (Equations (1) and (2)) [29]:  
\[\overline{m}\ddot{u}+EIu^{\prime\prime\prime\prime}=p(x,t) \tag{1}\]  
\[p(x,t)=[-m_{v}g+k_{v}\left(q_{v}-u\left|{}_{x=vt}\right.\right)]\delta(x-vt) \tag{2}\]  
where \(\overline{m}\) denotes the unit mass of the bridge per meter. \(E\) and \(I\) are the stiffness and moment of inertia of the beam cross-section, respectively. \(L\) is the span of the beam. \(p(x,t)\) is the interaction force applied on the bridge through the contact between the tire and bridge. \(\ddot{u}\) is the second derivative of displacement with respect to time, and \(u^{\prime\prime\prime\prime}\) denotes the fourth derivative of displacement with respect to \(x\). \(g\) is the gravitational acceleration, and \(\delta(x-vt)\) is the Dirac delta function evaluated at the contact point \(x=vt\)[29]. By assuming that the vehicle mass is much smaller than the bridge mass, the bridge dynamics can be solved as follows [29]:  
\[u(x,t)=\sum_{n}\frac{\Delta_{stn}}{1-S_{n}^{2}}\{sin\frac{n\pi x}{L}[\sin \frac{n\pi vt}{L}-S_{n}\sin\omega_{bn}t]\} \tag{3}\]  
where \(\omega_{bn}\) is the \(n\)-th bridge natural vibration frequency, and [29]  
\[\Delta_{st,n}=\frac{-2m_{v}gL^{3}}{n^{4}\pi^{4}EI} \tag{4a}\]  
\[S_{n}=\frac{n\pi v}{L\omega_{b,n}} \tag{4b}\]  
\[\omega_{b,n}=\frac{n^{2}\pi^{2}}{L^{2}}\sqrt{\frac{EI}{\overline{m}}} \tag{4c}\]  
In this study, we focus on the low-frequency responses, i.e., the responses corresponding to the driving force, which contribute most to the bridge displacement and inclination [29]. According to previous research, the low-frequency bridge responses should be [29]  
\[u_{low}(x,t)=\sum_{n}\frac{\Delta_{stn}}{1-S_{n}^{2}}[sin\frac{n\pi x}{L}\sin \frac{n\pi vt}{L}] \tag{4}\]  
and the transfer function, i.e., the mapping relationship for responses between three cross-sections \(x_{1}\), \(x_{2}\), and \(x_{3}\) can be derived from Equation (5) as [26]  
\[T_{low}= \begin{bmatrix}sin\frac{\pi x_{2}}{L}&sin\frac{2\pi x_{2}}{L}\\ sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ \end{bmatrix}\begin{bmatrix}sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{2}}{L}\\ \end{bmatrix} ^{-1} \tag{5}\]  
It is important to note that, given that the first few vibration modes account for the majority of the total bridge responses, only the first two modes were incorporated into the transfer matrix. This approach has been validated in existing research and proven effective in practical engineering applications [26]. Figure 1: A simple VBI system [27]. Equation (6) can be further transformed with modal parameter expression  
\[T_{low}= \begin{bmatrix}\phi_{1}(x_{2})&\phi_{2}(x_{2})\\ \phi_{1}(x_{3})&\phi_{2}(x_{3})\end{bmatrix}\begin{bmatrix}\phi_{1}(x_{1})&\phi_ {2}(x_{1})\\ \phi_{1}(x_{2})&\phi_{2}(x_{2})\end{bmatrix} ^{-1} \tag{6}\]  
where \(\phi_{n}(x)\) is the corresponding modal amplitude of the \(n\)-th bridge mode shape at the location \(x\). It should be noted that the transfer matrix corresponding to bridge inclination can also be derived as  
\[T_{low,incl}= \begin{bmatrix}\phi_{1,incl}(x_{2})&\phi_{2,incl}(x_{2})\\ \phi_{1,incl}(x_{3})&\phi_{2,incl}(x_{3})\end{bmatrix}\begin{bmatrix}\phi_{1,incl }(x_{1})&\phi_{2,incl}(x_{1})\\ \phi_{1,incl}(x_{2})&\phi_{2,incl}(x_{2})\end{bmatrix} ^{-1} \tag{7}\]  
where the subscript _incl_ represents the mode shape functions and transfer matrix corresponding to bridge inclination. It should be noted that \(T_{low}\) and \(T_{low,incl}\) are determined by bridge mode shapes and independent of varying external vehicular loading.