## 3 Numerical Validation
### Numerical Analysis
In contrast, as the observation points are further removed from the site of damage, the discrepancies between the two datasets gradually decrease, aligning with the theoretical assumption outlined in Section 2.2. For a more detailed analysis, Table 4 presents a comparison of the maximum displacement values obtained from the reconstructed and simulated results. It is evident that the trend of decreasing discrepancy is consistent across various observation points. This pattern underscores the effectiveness of the proposed method, as it demonstrates its capability to accurately capture the impact of damage on bridge responses. Furthermore, at OP-2, depicted in Figure 6a, a marked discrepancy between the reconstructed and simulated results is evident at the time point of maximum displacement, as indicated by the dashed lines. For an undamaged bridge, such a divergence is not expected. In contrast, at OP-4, depicted in Figure 6c, where the divergence between simulation and reconstruction results is the least among the three observation points, this discrepancy becomes negligible. It is hypothesized that these variations may originate from differences in the mode shape functions, which in turn affect the transfer matrix. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Location** & **OP-2 (Near Damage)** & **OP-3** & **OP-4** \\ \hline Simulation (E-3 m) & 0.41 & 1.14 & 1.65 \\ \hline Reconstruction (E-3 m) & 2.09 & 2.43 & 2.16 \\ \hline Difference (\%) & 409.12 & 112.71 & 31.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between reconstructed and simulated displacement for the damaged bridge. Figure 6: Comparison of displacement time history for the damaged bridge between reconstructed (legend as “Reconstruction”) and simulated (legend as “Simulation”) results (the dots denote the largest displacement on each time history curve). (**a**) Comparison at OP-2 (the divergence (0.7 s) in time points corresponding to the maximum displacement between simulation and reconstruction results is marked with a dashed line and double arrow. Maximum displacement is marked with dots). (**b**) Comparison at OP-3 (the divergence in time points is about 0.1 s). (**c**) Comparison at OP-4 (the divergence in time points is not obvious). However, the primary focus of this study is to develop an online monitoring system that can operate effectively amidst the complexities introduced by multiple moving vehicles. In such scenarios, accurately determining the exact time point of maximum amplitude and identifying structural damage presents inherent challenges. As a result, this particular issue is not explored in further depth within the scope of this research. After assessing the sensitivity of the proposed method with respect to the distance between the damage location and the observation point, we proceeded to evaluate its sensitivity to the extent of damage. A series of new finite element models were constructed to simulate the bridge dynamics under varying degrees of damage. The reductions in stiffness were set at 10%, 20%, 30%, 40%, and 50%. The damage index was defined as the difference in maximum displacement between the reconstructed and simulated results at observation point 2 (OP-2). The detailed values for all these damage scenarios are presented in Table 5.  
It is observable that, initially, as the extent of damage increases, the difference in maximum displacement also increases, which corroborates the theoretical derivation in Section 2.2 that the disparity becomes more pronounced with more severe damage. Furthermore, comparing Table 4 with Table 5, the difference at a fixed observation point (i.e., OP-2) due to changes in damage extent is less significant than that caused by changes in the distance between the damaged region and the observation point. This suggests that, for a given observation point, the initial few vibration modes are not highly sensitive to variations in damage extent.