## 3 Numerical Validation
### Numerical Analysis  
We first analyzed the dynamic response of the intact bridge. Figure 5 illustrates the displacement time histories captured at five distinct observation points (OP-1 through OP-5). Notably, the low-frequency vibration components dominate the bridge's displacement, while the high-frequency components, which manifest as local peaks along the time history curve (encircled in Figure 5), exhibit significantly lower amplitudes compared to their low-frequency counterparts. This observation aligns with previous theoretical studies [29]. Furthermore, earlier research [29] suggests that the first two low-frequency vibration modes are the principal contributors to the bridge's displacement. Consequently, there is an expected phase difference in the time points corresponding to maximum displacement across different observation points [26]. This phase difference (indicated by the dots in Figure 5) is also evident in the numerical simulation results. If the displacement was governed by a single mode component, the time points of maximum displacement would be identical for all observation points. The presence of multiple modes, therefore, is confirmed by the observed phase differences. Figure 4: Finite element model of the simple VBI system. (OP: observation points marked with dots. The stiffness reduction region is marked with a rectangle.) Figure 5: Time histories of displacement at five observation points for the intact bridge. (The dots denote the maximum displacement at each time history curve. The rectangle marks the high–frequency components with relatively lower amplitudes). While leveraging the transfer matrix derived in Section 2.1, a response reconstruction approach for this simple VBI system was proposed using a multiple regression algorithm [38]. The displacements at the two outer observation points (OP-1 and OP-5) were measured, and these values were subsequently employed to reconstruct the responses at the remaining three cross-sections (OP-2, 3, and 4). To estimate the displacement at each of the three middle observation points (OP-2, 3, and 4), as indicated in Equation (7), two regression coefficients were determined. These coefficients correspond to the responses measured at OP-1 and OP-5, respectively. In previous research [26], it has been validated that the low-frequency bridge displacement for the simple VBI system is mainly contributed by the first two low-frequency modes. Therefore, the multiple linear regression for two mode-related coefficients should be appropriate for the response reconstruction. To ascertain the efficacy of our proposed methodology, Figure 6a-c depict comparative analyses of bridge displacement at the three central observation points. These comparisons are made between the reconstructed responses and the simulation outcomes. The simulation results were directly extracted from finite element models, offering a benchmark. Conversely, the reconstructed results were derived from the product of the regression coefficients and the displacement measurements at the two side observation points, OP-1 and OP-5. It is crucial to highlight that while the coefficients were determined based on the intact bridge's characteristics, the displacement time histories at the side points were recorded under conditions simulating a damaged beam. Figure 6: Cont. As delineated in the theoretical derivations of Section 2.2, if the bridge is intact, the reconstructed bridge displacements should closely correspond with the direct simulation outcomes throughout the observation period, with the time history curves of simulation and reconstruction ideally aligning perfectly. However, in the presence of local damage, a divergence between the simulation and reconstruction results is anticipated. Upon scrutinizing Figure 6a-c, discernible discrepancies between the reconstructed and simulated bridge displacements are evident. Notably, the most pronounced deviation is observed at the OP-2 observation point, which is positioned directly above the damaged area.