## 2 Methodology  
### Dynamics of a VBI System  
The theoretical dynamics of a simple VBI system were investigated. As depicted in Figure 1, the vehicle was simplified as a moving sprung mass, and the bridge is a simple supported beam. The vehicle moves with a constant speed \(v\). The dynamic equation [35] of the vehicle is illustrated as Equation (1):  
\[m_{v}\bar{q}_{v}+k_{v}q_{v}=k_{v}\left.u\right|_{x=vt} \tag{1}\]  
where \(m_{v}\) denotes the mass of the vehicle, \(k_{v}\) represents the stiffness of the spring, \(q_{v}\) is the vertical displacement of the vehicle, and \(\bar{q}_{v}\) is the corresponding vehicle acceleration. \(u(x,t)\) denotes the vertical displacement of the bridge at \(x\) on the time point \(t\). The initial configuration is that the vehicle is situated at the far end of the beam. The initial beam deflection under gravity was not considered for the purposes of simplification [35]. The displacement of the bridge and the vehicle is in a downward direction. The dynamic equations of the beam are the following [35]:  
\[\overline{m}\ddot{u}+EIu^{\prime\prime\prime}=p(x,t) \tag{2a}\] \[p(x,t)=[-m_{v}g+k_{v}\left(q_{v}-u\left.\right|_{x=vt}\right)]\delta(x-vt) \tag{2b}\]  
where \(\overline{m}\) denotes the mass of the bridge per unit length, \(E\) is the elastic modulus, and \(I\) represents the moment of inertia of the beam cross-section. \(L\) is the span of the beam, and \(p(x,t)\) is the interaction force applied on the bridge through the contact between the tire and bridge. In this simplified model [35], the contact region between the tire and the bridge surface was simplified as a contact point. \(\ddot{u}\) denotes the second derivative of displacement with respect to time, and \(u^{\prime\prime\prime}\) 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, \(x=vt\). By assuming that the vehicle mass was much smaller than the bridge mass, [PERSON] and [PERSON] [35] derived the theoretical dynamic responses of the bridge (illustrated as Equation (3)):  
\[u(x,t)=\sum_{n}\frac{\Delta_{stn}}{1-S_{n}^{2}}\left\{sin\frac{n\pi x}{L}[ \sin\frac{n\pi vt}{L}-S_{n}\sin\omega_{bn}t]\right\} \tag{3}\]  
Figure 1: A simple VBI system. where \(n\) (\(n=1,2,3,\ldots\)) denotes the mode order and \(\omega_{bn}\) is the \(n\)-th bridge natural vibrational frequency. In detail,  
\[\Delta_{st,n} =\frac{-2m_{v}gL^{3}}{n^{4}\pi^{4}EI} \tag{4a}\] \[S_{n} =\frac{n\pi v}{L\omega_{b,n}}\] (4b) \[\omega_{b,n} =\frac{n^{2}\pi^{2}}{L^{2}}\sqrt{\frac{EI}{\overline{m}}} \tag{4c}\]  
in which \(g\) is the gravitational acceleration. The effects of damping on the vehicle and the beam were omitted from the analysis. Equation (3) [35], derived from the simplified model that treats a vehicle as a moving force \(-m_{v}g\), inherently neglects the damping effect of the vehicle. Moreover, as the vehicle traverses the beam, the interaction force persistently stimulates the beam's oscillations. The damping effect of the bridge, in the context of bridge dynamics, is not conspicuously evident. As a result, the damping influence of the bridge was similarly disregarded in this study. The \(u(x,t)\) in Equation (3) was divided as a low-frequency component \(u_{low}(x,t)\) and a high-frequency component \(u_{high}(x,t)\) (shown in Equation (5a) and (5b)). \[u_{low}(x,t)=\sum_{n}\frac{\Delta_{stn}}{1-S_{n}^{2}}\bigg{[}sin\frac{n\pi x}{L }\text{sin}\frac{n\pi vt}{L}\bigg{]} \tag{5a}\] \[u_{high}(x,t)=\sum_{n}\frac{\Delta_{stn}}{1-S_{n}^{2}}\bigg{[}-sin\frac{n\pi x}{L }S_{n}\text{sin}\,\omega_{b,n}t\bigg{]} \tag{5b}\]  
This division is predicated on the variance in the frequency spectrum that distinguishes these two components. For \(u_{low}(x,t)\), the vibrational frequency is \(\frac{n\pi v}{L}\) and is determined by the moving speed of the vehicle. Therefore, this frequency is named the \"Driving-force frequency\" [35].