## 2 Theoretical Analysis  
### Bridge Dynamics of a Simple VBI System  
A typical VBI system consists of a simply supported beam and a sprung-mass vehicle model (shown in Figure 2). The vehicle moves at a constant speed \(v\). The dynamic equation of the vehicle is [42]  
\[m_{v}\ddot{q}_{v}+k_{v}q_{v}=k_{v}\,u\big{|}_{x=vt} \tag{1}\]  
where \(m_{v}\) and \(k_{v}\) denote the sprung mass and the spring stiffness, respectively. \(q_{v}\) is the vertical displacement of the vehicle, and \(\ddot{q}_{v}\) is the corresponding vehicle acceleration. \(u\big{|}_{x=vt}\) is the vertical displacement of the contact point at \(x\) on the time point \(t\). The dynamic equation of the beam is [42]  
\[\overline{m}\ddot{u}+EIu^{mr}=p(x,t) \tag{2a}\] \[p(x,t)=[-m_{v}g+k_{v}\,(q_{v}-u\big{|}_{x=vt})]\delta(x-vt) \tag{2b}\]  
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 \(u^{mr}\) 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 \(x=vt\)[42]. In the referenced study [43], it was determined that the amplitude of bridge vibrations caused by road roughness is significantly smaller compared to those caused by moving vehicles. Consequently, the impact of road roughness, being minimal, was considered negligible and thus excluded from our subsequent analysis. By assuming that the vehicle mass is much  
Figure 1: The overall structure of the whole study. Figure 2: A simple VBI system [41]. smaller than the bridge mass, theoretical modal formulation for bridge dynamics can be derived as [42]  
\[u(x,t)=\phi_{dis}(x)Y_{dis}(t)=\sum_{n}\frac{\Delta_{stn}}{1-S_{n}^{2}}\bigg{\{} sin\frac{n\pi x}{L}\bigg{[}\text{sin}\frac{n\pi\pi vt}{L}-S_{n}\text{sin}\, \omega_{bn}t\bigg{]}\bigg{\}} \tag{3}\]  
where \(\phi_{dis}(x)\) is the mode shape function, \(Y_{dis}(t)\) is the time function, and \(\omega_{bn}\) is the \(n\)-th bridge natural vibration frequency, and [42]  
\[\Delta_{st,n}=\frac{-2m_{v}gL^{3}}{n^{4}\pi^{4}EI} \tag{4a}\] \[S_{n}=\frac{n\pi v}{L\omega_{b,n}}\] (4b) \[\omega_{b,n}=\frac{n^{2}\pi^{2}}{L^{2}}\sqrt{\frac{EI}{\overline{\overline{m}}}} \tag{4c}\]  
Mode Shape Function and Inter-Section Transfer Matrix of Bridge Displacement, Inclination, and Acceleration  
The \(u(x,t)\) in Equation (3) can be divided into a low-frequency component \(u_{low}(x,t)\) (shown in Equation (5a)) and a high-frequency component \(u_{high}(x,t)\) (shown in Equation (5b)) [42]. \[u_{low}(x,t)=\sum_{n}\frac{\Delta_{stn}}{1-S_{n}^{2}}\bigg{[}sin\frac{n\pi x} {L}\text{sin}\,\frac{n\pi vt}{L}\bigg{]} \tag{5a}\] \[\overline{m}\ddot{u}+EIu^{mr}=p(x,t) \tag{2a}\]0  
The division is based on the difference in a frequency range of these two components. The vibration frequencies of the low-frequency responses are \(\frac{n\pi v}{L}\), which are the driving-force frequencies determined by the moving speed of the vehicle, while the frequencies of \(u_{high}(x,t)\) are bridge natural vibration frequencies \(\omega_{b,n}\). According to [PERSON] [42] and [PERSON] [44], the frequencies of the first several driving-force modes are much lower than those of the first natural vibration modes. Moreover, displacement, inclination, and strain are mainly contributed by the first several low-frequency components, while acceleration is mainly contributed by the first several high-frequency components. Specifically, according to Equations (4a) and (5a), for the low-frequency responses, the ratio of contribution of each mode to the whole responses would be \(1/n^{4}\) (i.e., \(1/1\), \(1/2^{4}\), \(1/3^{4}\)...) [41]. Therefore, to ensure the accuracy, the first two modes were considered in this study.