## 6 Comparative Performance Evaluation with a Modal Flexibility-Based Damage Detection Approach
Similarly, the mode shapes were identified with very high accuracy, achieving a minimum Modal Assurance Criterion (MAC) value of 0.9998 between the exact and the identified mode shapes. Following the formulation given in Equation (13), the flexibility matrix was computed, and the relative changes in its diagonal entries were calculated for all damage cases. These values were then normalized with respect to the largest value to obtain the damage index, denoted as DI\({}_{\Delta F}\). The resulting damage indices for all damage cases are presented in Figure 11.  
Figure 11: Damage localization for the fixed-fixed beam using the flexibility-based approach. \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline \multirow{2}{*}{**State**} & \multicolumn{2}{l}{**Damage btw**} & \multicolumn{6}{c}{**Identified Natural Frequencies (Hz)**} \\  & \multicolumn{2}{l}{**Sensors**} & \multicolumn{1}{c}{**Mode 1**} & \multicolumn{1}{c}{**Mode 2**} & \multicolumn{1}{c}{**Mode 3**} & \multicolumn{1}{c}{**Mode 4**} & \multicolumn{1}{c}{**Mode 5**} & \multicolumn{1}{c}{**Mode 6**} & \multicolumn{1}{c}{**Mode 7**} & \multicolumn{1}{c}{**Mode 8**} \\ \hline Baseline & - & 1.000 & 2.754 & 5.388 & 8.846 & 12.983 & 17.425 & 21.446 & 24  
It is clear from this figure that although the true damage location appears among the potential damage locations, the modal flexibility-based approach fails to isolate it as distinctly as the proposed transmissibility-based method especially for D. Case 1 and D. Case 7. It should be emphasized that in this numerical case, no modal truncation was applied, and issues such as mode selection and pairing for consistency, mode switching, missing or closely-spaced modes were not present. Therefore, a straightforward application of the modal flexibility-based approach was possible, yet its damage localization performance still remained inferior to that of the transmissibility-based strategy. The next comparison was conducted using the experimental dataset from the arch structure, where the preprocessing stage required significant user interaction and engineering judgment, particularly for modal identification. Impulse response functions obtained from vertical acceleration measurements (A\({}_{1}\), A\({}_{2}\), A\({}_{3}\), A\({}_{4}\), A\({}_{6}\)) served as inputs to the ERA, and multiple runs of the algorithm were performed using the impact test data to reliably identify the true system modes. Figure 12 illustrates a representative example of this process in the form of stabilization diagrams, which plot the identified frequencies against varying system orders. Stable, physically meaningful modes were selected based on their consistent appearance across increasing model orders, while unstable or spurious modes were systematically eliminated. In addition, multiple impact tests were used to identify mode shapes separately for each run, and the MAC values between these identified shapes were computed. Mode shapes exhibiting consistently high MAC values across different runs were considered repeatable and reliable, further supporting the validity of the selected modes. Table 6 summarizes the modal parameters for the consistent modes selected across the baseline and the two damaged states, after loading to 30 kN and 60 kN. Using these identified modes, flexibility matrices for both the undamaged and damaged states were synthesized, and the damage indices were computed based on the relative changes in the diagonal entries of the flexibility matrix, following the same procedure as in the numerical case. Figure 13 presents the computed damage indices for these cases. Based on this figure, the flexibility-based approach initially misidentifi