## 2 Research Methodology
\[\left|\left[M_{g}\right]^{-1}\left[K_{g}^{i,j}\right]-\alpha I_{g}\right|=0 \text{ with }i\in[1,n]\text{ and }j\in[1,s]\] (3) where \(\left[M_{g}\right]^{-1}\) represents the inverse of the coupled mass matrix, which is independent of the position of the vehicle; \(\left[K_{g}^{i,j}\right]\) corresponds to the coupled stiffness matrix when a vehicle with a set of properties \(j\) is located at position \(i\) over the bridge; \(\alpha\) is the vector of eigenvalues and \(I_{g}\), the identity matrix. Finally, \(s\) represents the size of the fleet considered in the eigenvalue analysis. By solving Equation (3) for each value of \(i\) and \(j\) and keeping the first element of \(\alpha\), it is possible to obtain \(n\) vectors of \(s\) components that characterize the statistical distribution of the first forced eigenfrequency of the healthy bridge at each vehicle location for the selected fleet configuration. 5. The process described in step (4) is repeated here for a wide range of damage scenarios. In addition to varying the properties of the vehicle based on typical values for the fleet configuration, the stiffness of the bridge is varied according to the parameters of a damage model (i.e., crack location and severity). A specific set of parameters of the damage model leads to a specific set of statistical distributions of the first frequency for each location of the vehicle on the bridge. Experimental data for the development of damage models is limited mostly to beams tested in a lab, and there is a large degree of uncertainty surrounding existing damage or its evolution for a complex structure. For example, the true shape of an existing loss of bending stiffness associated with a particular damage location in a bridge (i.e., a crack) and the assumed mathematical model to characterize a similar damage at the same location may differ considerably. If the width of the shape is narrower in reality than in the mathematical model, then the methodology will be prone to underestimate the depth. Overall, it is expected that small changes in the shape would affect mainly the higher frequencies of the bridge, whereas the first frequency would only change to a small extent. Thus, predictions from the proposed methodology should still be able to point a bridge inspector in the right direction. 6. The acceleration response of the bridge is measured for \(p\) crossings of random trucks within the selected fleet configuration. The latter entails the availability of a WIM system immediately before or after the bridge to identify those trucks that belong to the fleet. Here, the method is tested with theoretical accelerations due to the unavailability of field tests. The interaction between the vehicle and the bridge is implemented using MATLAB R2022a [29]. Therefore, the acceleration response is simulated by integrating the equations of motion of the coupled VBI system defined in Equation (4). \[\big{[}M_{g}\big{]}\big{\{}\ddot{u}_{g}\big{\}}+\big{[}C_{g}\big{]}\big{\{} \dot{u}_{g}\big{\}}+\big{[}K_{g}\big{]}\big{\{}u_{g}\big{\}}=\big{\{}F_{g}\big{\}}\] (4) where \(\big{[}M_{g}\big{]}\), \(\big{[}C_{g}\big{]}\), and \(\big{[}K_{g}\big{]}\) represent the global mass matrix, global damping matrix, and global stiffness matrix, respectively; \(\big{\{}F_{g}\big{\}}\) is the global force vector; and \(\big{\{}u_{g}\big{\}}\) is the unknown displacement vector. The symbols. and.. indicate the first and second derivative with respect to time, respectively. The Newmark-beta integration approach (with \(\gamma=0.5\) and \(\beta=0.25\)) is used to obtain the bridge response, and more specifically, the acceleration of the mid-span section. Details about the integration are referred to [PERSON] [28]. 7. The STFT of the acceleration response is calculated for each of the \(p\) crossings, as per Equation (5). It should be noted that other signal processing tools can potentially be used instead of the STFT here to obtain the time-frequency representation of the acceleration signal, i.e., Hilbert Huang Transform [30], wavelets [31], etc.