## 2 Bridge Dynamics in a VBI System and the Data Reconstruction Algorithm
### Characteristics of Bridge Dynamics in the VBI System
As illustrated in Equation (13), \(T_{b,1}\) is matrix related only to the global vibration modes. Similarly, one can obtain the transfer matrix corresponding to the low-frequency driving-force-related vibrations [20] from Equation (8) as follows:  
\[\boldsymbol{T_{b,2}}= \begin{bmatrix}sin\frac{\pi x_{2}}{L}&sin\frac{2\pi x_{2 }}{L}\\ sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ \end{bmatrix}\begin{bmatrix}sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ \end{bmatrix} ^{-1} \tag{14}\]  
Figure 2: Three kinds of components in monitoring data. It should be noted that, similarly to the high-frequency components, only the first low-frequency vibration modes were considered in this study. The following apparently applies:  
\[\mathbf{T_{b,1}}=\mathbf{T_{b,2}} \tag{15}\]  
In fact, Equation (15) should also be applicable to other bridges with multiple spans and varying mode shape functions, given that the dynamics of typical beam bridges can always be formulated as a product of modal coordinates and time functions, as depicted in Equation (16):  
\[u_{b}(x,t)=\phi(x)\cdot Y(t) \tag{16}\]  
where \(\phi(x)\) denotes the mode shape function and \(Y(t)\) represents the time-domain function. Analogous to Equations (8) and (9), \(Y(t)\) can be segmented into a high-frequency component attributed to the bridge's free vibration and a low-frequency component related to the driving force, as detailed in Equation (17):  
\[u_{b}(x,t)=\phi(x)\cdot\left[Y_{high}(t)+Y_{low}(t)\right] \tag{17}\]  
where the subscripts \"high\" and \"low\" represent the high-frequency and low-frequency components, respectively. Equation (17) illustrates that for components of identical order across both high-frequency and low-frequency domains, the associated mode shape function \(\phi(x)\) is invariant. As a result, the transfer matrix \(\mathbf{T_{b,1}}\) should perpetually match \(\mathbf{T_{b,2}}\), holding true not solely for the simply supported beam but also for various other general linear bridge configurations. Furthermore, Equation (17) suggests that for any additional monitoring parameters--such as velocity, acceleration, strain, or inclination--while the mode shape function may diverge from the displacement mode shape \(\phi(x)\), the principle that the transfer matrices (which may also diverge from \(\mathbf{T_{b,1}}\) and \(\mathbf{T_{b,2}}\)) for high-frequency and low-frequency responses are identical remains universally applicable. Moreover, as evidenced by prior research [19; 20], Equation (14) underscores a significant attribute. The transfer matrix \(\mathbf{T_{b,1}}\) is dimensionless; consequently, fluctuations in traffic conditions, which serve as the external driving force for the VBI system, and alterations in bridge material stiffness due to temperature changes should not affect the transfer matrix \(\mathbf{T_{b,1}}\). Additionally, the equivalence between \(\mathbf{T_{b,1}}\) and \(\mathbf{T_{b,2}}\) is perpetually maintained, even amidst varying traffic conditions and air temperatures.