## 3 Numerical Validation of the Damping Estimation Methodology
### Validation of the Application of the Prony Method
#### 3.3.3 Application of the Prony Method  
The results presented in this section refer to the scenarios described above and correspond to the responses obtained at the bridge deck midspan, at the track centre. A summary table for all the scenarios is presented later at the end of this section. The goal is to achieve modal parameters for the train passage similar to those imposed to the numerical model using the Prony method. Figure 9 presents the record for the Alfa Pendular train at a speed of 135 km/h, and the irregularity of September 2019, in which the free vibration period for which the train loads start to lose influence in the bridge response, as described in Section 3.2, is highlighted in grey and isolated in Figure 9, along a comparison with the LD method. The perfect overlapping between the original record and that obtained with the Prony method shows the accuracy of the method in representing the free vibration response. It can be seen in Figure 9 that the LD method is inefficient to fit an exponential decay to the original record. In this case, the estimates of the damping coefficient are not accurate in comparison to the Prony method, with errors of 2.4% in relation to the first fundamental frequency, and 13.2% in relation to the second fundament frequency. Moreover, it is interesting to observe a \"breathing effect\" induced by the presence of the two fundamental frequencies of the bridge, which can clearly be detected in the Fast Fourier Transform (FFT) spectrum depicted in Figure 9. Finally, Figure 9 presents the decomposition of the Prony record obtained before, illustrated by the two main sinusoid records that compose the original Prony record. For these sinusoids, it can be seen that the frequencies that were obtained by the Prony method are identical to the main frequencies of the numerical model, as well as the damping coefficients that are very similar to those imposed in the model, \(\xi=2.12\%\) (see Figure 7). A second example is presented in Figure 10 for the same speed of 135 km/h, but this time for a more recent irregularity measured on October 2020. The free vibration period considered for the application of the Prony method is also highlighted. As well as the previous case, the Prony method performed with high accuracy, overlapping the original record, as can be seen in Figure 10, along the comparison with the LD method. In this case, the free vibration record is controlled by only one main frequency, as illustrated in the FFT spectrum in Figure 10, and the LD method has a better performance in comparison to the previous case; however, the damping coefficient estimates present damping coefficient values with an error of 6.4%, in comparison to the Prony method and to that imposed in the numerical model. Despite having only one main frequency that is controlling the record, the free vibration record does not present a clearly damped sinusoid because  
Figure 8: Vertical irregularities used in the dynamic analyses: (**a**) left rail, (**b**) right rail. of minor interferences of other frequencies in the record, which are also represented in the FFT spectrum. The main frequency that composes the Prony record is illustrated in Figure 10d, and, once again, it can be seen that the method estimates the damping coefficient corresponding to the second fundamental frequency with great accuracy (see imposed damping of \(\xi=2.12\%\) in Figure 7d). A summary table with all the results corresponding to all the scenarios used to validate the method is presented in Table 3. For all of them, and independently of the number of controlling modes in each scenario (one or two), the damping coefficients and main frequencies obtained with the Prony method are in a very good agreement with those imposed in the numerical model, which confirms the accuracy and validation of the method.