## 6 Comparative Performance Evaluation with a Modal Flexibility-Based Damage Detection Approach  
To evaluate the performance of the proposed transmissibility-based machine learning strategy, a comparative analysis was performed using a well-established modal flexibility-based damage detection method using the same datasets presented in the previous sections. This section presents a brief overview of the modal flexibility-based approach and the results of both the numerical simulations and the experimental studies, focusing on the accuracy, sensitivity, and damage localization capability. Modal flexibility has been widely adopted in structural health monitoring due to its higher sensitivity to stiffness reductions, particularly in the lower modes [61, 62, 63, 64, 65, 66, 67, 68]. As highlighted in prior studies, natural frequency-based methods often lack sensitivity to minor or localized damage, as the frequency shifts caused by such damage typically fall within the range of estimation uncertainty. Mode shape-based methods, while offering somewhat improved sensitivity, face limitations due to the challenges of accurate mode pairing and matching, especially in noisy or complex systems. Furthermore, localized damage does not always induce distinct or spatially localized changes in mode shapes, limiting their effectiveness for reliable damage localization. The flexibility matrix is computed using:  
\[F=\Phi\Lambda^{-1}\Phi^{T} \tag{15}\]  
where \(\Phi\) is the mass-normalized mode shape matrix and \(\Lambda=\text{diag}\big{(}\omega_{1}^{2},\omega_{2}^{2}\ldots,\omega_{N}^{2} \big{)}\) contains the squared natural frequencies. In practical applications, especially in experimental or operational scenarios where input excitation is unknown, the extracted mode shapes are not mass-normalized but instead arbitrarily scaled. That is,  
\[\Phi_{i}^{mass}=\alpha_{i}\Phi_{i}^{true} \tag{16}\]  
Despite this, for damage localization purposes, relative changes in the diagonal entries of the flexibility matrix remain meaningful as long as the same set of modes and consistent scaling are applied across undamaged and damaged states. It is important to note that modal flexibility-based methods require several critical preprocessing and system identification steps before damage detection can be performed. In this study, the modal parameters were identified using the Eigenvalue Realization Algorithm (ERA), a time-domain system identification technique that constructs the system's state-space model from impulse response functions or free-decay responses. ERA provides estimates of natural frequencies and corresponding mode shapes, which can be subse  
Figure 10: Damage localization index along **(a)** the longitudinal axis (spanwise) and **(b)** the lateral axis (widthwise). quently used to form the flexibility matrix. In this approach, impulse response functions serve as inputs to the ERA, and multiple runs must be conducted to ensure mode stability and mitigate the effects of noise. Additionally, selecting the appropriate model order for constructing the system matrix and discriminating between physical system modes and spurious modes require careful evaluation and engineering judgment. To ensure comparability between structural states, consistent mode selection across undamaged and damaged conditions is essential. Moreover, the effects of modal truncation on the synthesized flexibility matrices must be carefully considered since it limits the capability for detecting localized and subtle damage. These additional steps introduce further user-dependent decisions, making it challenging to automate the entire process and potentially affecting the robustness of the damage identification outcome. The first comparison was performed on the numerical case study of the fixed-fixed beam. The modal frequencies identified using the ERA are listed in Table 5. A comparison with the exact values provided in Table 2 shows excellent agreement, with frequency identification errors remaining below 1%.