## 6 Discussion  
### Testing of a 15 m Bridge  
This section shows how eigenvalue analysis can be exploited to locate and quantify damage based on the changes in the forced frequencies \(F_{3}\) and \(F_{9}\). For this purpose, Figure 12 compares the forced frequencies of a fleet composed of \(p\) = 1000 3-axle trucks to the associated eigenfrequencies for two damage scenarios. The forced frequency versus vehicle position for a 15% crack at 5 m is illustrated in Figure 12a. The solid and dashed lines represent the eigenvalue and the mean frequency value (\(\mu\)) of the truck fleet, respectively. Additionally, the shaded area surrounding the dashed line indicates three times the standard deviation (\(\sigma\)), i.e., \([\mu-3\sigma,\mu+3\sigma]\), which means that 99.7% of the estimated forced frequencies are located within the highlighted area and frequency range [5.48, 5.56] Hz. Figure 12b makes the same comparison for the 5% crack at 7 m. As expected, the frequency range, [5.58, 5.66] Hz, is different for this second scenario, but the overall pattern remains.  
The points along the two vertical lines signaling the frequencies \(F_{3}\) and \(F_{9}\) in Figure 12 were selected for damage localization and quantification. Figure 13 plots the values of \(F_{3}\) and \(F_{9}\) obtained for the fleet of 3-axle trucks (\(p\) = 1000) against the eigenfrequency database presented in Figure 6. Most of the points lie inside the ellipse from eigenvalue analysis corresponding to the true values of severity and location for both damage scenarios. For the smaller damage scenario, the ellipses associated with 5% crack at 6 m and 5% crack at 7 m are relatively close, but there is no difficulty in distinguishing the true solution. More challenging scenarios will display clusters of points falling within overlapping ellipses.  
The Mahalanobis distance provides a robust method to quantify the discordancy between frequencies in known-damage scenarios from eigenvalue analysis and unknown scenarios from transient analysis or measurements (steps (9) and (10) of the methodology). Figure 14 ranks the first 8 minimum values of the mean Mahalanobis distance for the two tested cracks when using \(p\) = 1000 trucks. As expected, the first-ranked scenario matches the true damage severity and location in both scenarios. The size of the traffic fleet was varied, \(p\) = 1000, 500, 250, 100, 50, 25 and 10 trucks, with no impact on accuracy. The insensitivity of the results to the size of the fleet is attributed to the separation between the distributions illustrated in Figure 13, and to the proximity of the measured frequencies to the ellipse holding the true solution.  
Figure 11: Instantaneous frequencies by STFT vs. location of the 1 st axle for scenarios: (**a**) 15% crack at 5 m, and (**b**) 5% crack at 7 m.  
Figure 12: Comparison between transient forced-frequencies due to the crossing of 1000 trucks and eigenfrequencies for scenarios: (**a**) 15% crack at 5 m, and (**b**) 5% crack at 7 m.  
Figure 13: _Cont._  
Figure 14: Min value rank of Mahalanobis distance for scenarios: (**a**) 15% crack at 5 m, and (**b**) 5% crack at 7 m.  
Figure 13: Comparison between transient forced-frequencies (\(F_{3}\), \(F_{9}\)) due to the crossing of 1000 trucks and eigenvalue database for scenarios: (**a**) 15% crack at 5 m, and (**b**) 5% crack at 7 m.