## 6 Comparative Performance Evaluation with a Modal Flexibility-Based Damage Detection Approach
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Mode 1**} & \multicolumn{2}{c}{**Mode 2**} & \multicolumn{2}{c}{**Mode 3**} \\ \cline{2-7}  & \(f\) (Hz) & \(\zeta\) (\%) & \(f\) (Hz) & \(\zeta\) (\%) & \(f\) (Hz) & \(\zeta\) (\%) \\ \hline Baseline & 79.40 & 1.44 & 171.17 & 1.59 & 207.76 & 3.99 \\ \hline DS 1 (up to 30 kN) & 52.60 & 2.45 & 149.65 & 4.44 & 161.14 & 4.09 \\ \hline DS 2 (up to 60 kN) & 37.43 & 10.32 & 50.39 & 8.65 & 97.75 & 3.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Modal frequencies identified by ERA. Figure 12: Stabilization diagram for the identified natural frequencies from impact excitation at A\({}_{1}\): (**a**) Baseline, (**b**) DS 1, (**c**) DS 2. Each color represents the set of frequency values obtained for a certain system order. 1 as the potential damage location, leading to a mislocalization in the first damage case (DS1). However, with an increased load and progression of damage in DS2, the method improves and correctly locates the damage around sensor 2. The comparison between the flexibility- and transmissibility-based approaches highlights the advantages of the transmissibility-based method, particularly in detecting and localizing early-stage and distributed damage. In the arch bridge test, the flexibility-based method misidentified the damage location in the first damage case (DS1), where cracks initiated not only at the primary hinge formation zone but also with separations along the spandrel walls on both sides of the crown. These distributed damage patterns reduced the sensitivity of the flexibility-based method, which relies on global stiffness changes. In contrast, the transmissibility-based method, by capturing localized dynamic deviations between consecutive sensors, was able to more reliably indicate damage progression, even when cracks appeared asymmetrically. This effectiveness is further supported by its broader frequency range (0-400 Hz), which captures higher-order dynamics beyond the -200 Hz limit of the flexibility-based approach, allowing for richer and more damage-sensitive features. In DS2, although both methods identified the damage zone between sensors 2 and 3, the flexibility-based method showed a sharper peak due to normalization (index values of 1.00, 0.17, and 0.01 for sensors 2-4), whereas the transmissibility-based approach exhibited a more gradual distribution (indices of 0.72 and 0.21 for sensor pairs 2-3 and 3-4). While this could suggest finer localization from the flexibility index, such sharpness results from the way each metric is scaled and should be interpreted accordingly. The performance of the transmissibility-based method further improved as the load increased and damage became more clearly localized in DS2. Similar trends were observed in the numerical study, where--despite ideal conditions without modal truncation, mode switching, or pairing issues--the flexibility-based method still failed to localize certain damage cases as effectively as the transmissibility-based strategy. In addition to its accuracy, the transmissibility-based method requires significantly less data preprocessing and eliminates the need for complex modal identification procedures, including mode selection and issues like truncated, missing, or closely spaced modes. By removing user interaction and subjective decisions, it becomes far more suitable for automated, robust implementations, especially in noisy, operational, or variable environments where flexibility-based approaches often struggle.