## 3 Numerical Validation of the Damping Estimation Methodology
### Sensitivity Analysis in Relation to the Free Vibration Period
#### 3.2.3. Receptance at the Midspan  
Although the results shown in the previous section clearly demonstrate that the bridge response is still affected by the train loads even when they leave it, the analyses were purely static. To obtain a more comprehensive conclusion which also includes the dynamic effects, receptance tests were numerically simulated to understand to which extent the dynamic response of the bridge can be affected by loads that already left it due to the continuity effects provided by the track. To achieve this, impact excitations have been applied on the rail at different distances from the deck's extremity (see in Figure 4 the load application points P1 to P3 distanced 0.6 m, 1.8 m, and 3.0 m away from the deck's extremity) and the receptance functions were obtained based on the displacements at the midspan. The excitation consisted of a typical hammer impact load of 2000 N, as shown in Figure 4.  
The receptance functions, which consist of the ratio between the displacements obtained at the midspan (see Figure 5) and the impact load, both in the frequency domain, for the different impact locations and for the four analyzed bridges are plotted in Figure 6. The figure also shows the identification of the main peaks, which are consistent with the frequency values shown in Table 2 for first and third global vertical vibration modes (the second mode is not visible because the midspan modal coordinate is null). As can be observed, for all the bridges, the receptance function is practically null when the impact is applied at the point P3 located 3.0 m away from the deck's extremity, which is coherent with the results obtained through the influence line analysis. Thus, it can be admitted that the free vibration period does not starts immediately when the last train axle leaves the bridge, but only when it is already distanced approximately 3.0 m away from the bridge  
\begin{table}
\begin{tabular}{c c c c c} \hline Span length \(L\) (m) & 7.0 & 11.0 & 11.5 & 23.5 \\ Mass of the bridge \(m_{L}\) (t/m) & 16.51 & 17.62 & 16.86 & 17.99 \\ Concrete elasticity modulus \(E_{c}\) (GPa) & 36.10 & 36.10 & 36.10 & 36.10 \\ Moment of inertia \(l\) (m\({}^{4}\)) & 0.196 & 0.294 & 0.280 & 0.499 \\ \hline \end{tabular}
\end{table}
Table 1: Main parameters used in the bridges’ numerical models.  
\begin{table}
\begin{tabular}{c c c c c} \hline Span length \(L\) (m) & 7.0 & 11.0 & 11.5 & 23.5 \\ First frequency \(f_{1}\) (Hz) & 20.03 & 9.49 & 8.61 & 2.19 \\ Second frequency \(f_{1}\) (Hz) & 53.42 & 31.76 & 29.34 & 8.22 \\ Third frequency \(f_{1}\) (Hz) & 84.33 & 56.60 & 53.35 & 17.58 \\ \hline \end{tabular}
\end{table}
Table 2: Numerical modal frequencies of the four analysed bridges.  
extremity. Such an assumption will be used in this work in the next sections for damping estimation purposes based on railway traffic forced vibration responses.  
Figure 4: Acceptance test simulated numerically: (**a**) impact load and (**b**) impact locations.  
Figure 3: Influence line corresponding to the midspan displacement: bridge with (**a**) 7.00 m, (**b**) 11.00 m, (**c**) 11.50 m, and (**d**) 23.50 m span.