## 7 Conclusions  
Level III damage detection methods can be achieved, in practice, using several dynamic parameters and a few modes of a bridge, yet their implementation is scarce due to the typically high number of sensors involved. The value of the frequency associated with the first mode of vibration of a bridge is easily extracted with only one or a few measurement sensors, but it is insufficient for locating and quantifying damage. Traditionally, a unique value of frequency is retained for a mode while taking a relatively long measurement period into account. However, this paper has demonstrated that the time-history of instantaneous forced frequencies associated with the different positions of a truck on a bridge stores information that makes damage identification possible. Using eigenvalue analysis, a database of bivariate distributions of the first bridge eigenfrequencies associated with a truck fleet when the 1 st axle of the vehicle is located at two different positions on the bridge was derived for different locations and severities of a crack. These distributions allow the visualization of how the frequencies of the bridge vary with crack location and crack severity, and they have accounted for some of the uncertainties related to the dynamic properties of a specific truck configuration. Instantaneous frequencies were calculated at the same two locations by applying a Hann-based STFT to the acceleration response of the bridge traversed by random trucks of the selected fleet. The instantaneous frequencies served to successfully locate and quantify two cracks when plotted over the wide range of bivariate distributions available from eigenvalue analysis. The Mahalanobis distance was proposed to discern scenarios where more than one solution was feasible. Figure 17: Min value rank of Mahalanobis distance for scenarios: (**a**) 15% crack at 10 m, and (**b**) 5% crack at 14 m.  
The theoretical testing was limited to two damage scenarios (15% and 5% cracks located at 1/3 and 7/15 of the total span respectively), two speeds (1 m/s and 10 m/s) and two bridge spans (15 m and 30 m) with a smooth profile. Therefore, the 18 local damage scenarios included in the eigenvalue database covered the two crack scenarios tested in the two bridges. On the one hand, edge effects had a more profound impact in a shorter span, where the requirements for a lower speed are more strict in order to ensure a portion of uncorrupted frequencies sufficiently long. On the other hand, similar damages led to more uncertainty in the predictions for the longer span, where the size of the traffic fleet was proven to be a factor. It is acknowledged that there are multiple degradation processes that affect bridges other than cracks. In practice, there could be damage scenarios for which the eigenvalue database does not have associated distributions. As a result of the ideal testing conditions, only discrepancies between eigenvalue and transient analyses can move the measured frequencies away from the true distribution. It was shown that these discrepancies significantly increase for higher speeds and rougher profiles. Differences between theoretical models and reality, as well as noise in the measurements, also deteriorate the accuracy of the results. As a suggestion for further research, the methodology needs to be assessed with other damage scenarios, higher speeds, rough road profiles, modeling errors, and experimental data. Conceptualization, A.G., K.F. and M.C. ; Formal analysis, A.G., K.F. and M.C. ; Funding acquisition, A.G.; Investigation, A.G., K.F. and M.C. ; Methodology, A.G., K.F. and M.C. ; Project administration, A.G.; Resources, A.G.; Supervision, A.G.; Validation, K.F. ; Visualization, A.G., K.F. and M.C. ; Writing--original draft, K.F. ; Writing--review & editing, A.G. and M.C. All authors have read and agreed to the published version of the manuscript. This research was funded by Science Foundation Ireland, grant number 16/US/I3277. Not applicable.