## 3 Simulation Examples
### Residual STFT-Based Mode Shape Extraction
#### 3.2.4 Effect from the Beam Damping Property  
In the field of moving load problems, especially when identifying bridge modal properties, most previous works considered the damping property of the vehicles rather than the bridge itself. However, beam damping could also impact its dynamical characteristic in this limited duration [22; 44]. As shown in Equation (25) with the Rayleigh damping assumption [45],to investigate the beam damping effect on mode extraction, there are six damping cases, including three cases with only the stiffness damping coefficient (\(r_{i}=\beta EI\)) and three cases with both the mass damping coefficient \(r_{e}=\alpha m\) and the stiffness damping coefficient \(r_{i}=\beta EI\) in this section, as shown in Table 3 as well as the corresponding damping ratios for the first three modes. Figure 8: The first three beam mode shapes using residual STFT-based extraction under roughness classes A (very good), B (good), and C (average), compared to the analytical first mode shape (**a**), the second mode shape (**b**), and the third mode shape (**c**), as well as the MAC values. With the velocity of 5 m/s and the road profile of Class A, the STFT of the acceleration residual between the first two trailers (\(u_{1}\) and \(u_{2}\)) is plotted in Figure 9 for the first three cases (Casel to Case 3). Compared to Figure 5c for the undamped case, at the second and especially the third natural frequencies, the peak ridges are less prominent from all the three cases. This indicates that the existence of beam damping could suppress the dynamical response, especially on the peaks near the natural frequencies. Moreover, to consider the effect of traffic flow, two groups of 10 vehicles with an equal space of 5 m are added to the three cases. The vehicles in both groups share the parameters of the one-DOF vehicle in Table 2 and their velocity. However, one vehicle group has the same moving direction as the tractor (toward the right), and the other vehicle group moves in the opposite direction (toward the left). So, when the tractor with three trailers enters \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Case 1** & **Case 2** & **Case 3** & **Case 4** & **Case 5** & **Case 6** \\ \hline \(r_{t}\) (kg-m\({}^{3}\)/s) & 0 & 0 & 0 & 400 & 2000 & 4000 \\ \(r_{t}\) (kg-m\({}^{3}\)/s) & 1.0240 \(\times\) 10\({}^{6}\) & 3.0720 \(\times\) 10\({}^{6}\) & 8.1920 \(\times\) 10\({}^{6}\) & 1.0240 \(\times\) 10\({}^{6}\) & 1.0240 \(\times\) 10\({}^{6}\) \\ The first three & 0.0013 & 0.0038 & 0.0101 & 0.0032 & 0.0112 & 0.0211 \\ damping ratios & 0.0051 & 0.0152 & 0.0404 & 0.0055 & 0.0075 & 0.0100 \\ (\(\xi_{1}\), \(\xi_{2}\), and \(\xi_{3}\)) & 0.0114 & 0.0341 & 0.0910 & 0.0116 & 0.0125 & 0.0136 \\ \hline \end{tabular}
\end{table} 
Table 3: There are only three cases of the stiffness damping coefficient (\(r_{t}=\beta E\)) and three cases of \(r_{t}=\alpha m\) and \(r_{t}=\beta E\)I existing simultaneously. Figure 9: STFT of the acceleration residual for \(\Delta u_{12}\) for the contact points (\(u_{C1}\) and \(u_{C2}\)) for Case 1 (**a**), Case 2 (**b**), and Case 3 (**c**). and leaves the beam, there are well-spaced vehicles moving on the bridge all the time. The extracted mode shapes are plotted for all cases without traffic flow in Figure 10 and their corresponding cases with traffic flow in Figure 11.  
At the first mode, all cases of different damping coefficients, without or with traffic flow, have the accurate extracted mode shapes. The second mode shapes extracted from these cases without or with traffic flow also approximate the analytical ones with a MAC value of larger than 0.95 except for Case 6, which has the largest second modal damping coefficient (\(\xi_{2}\) = 0.0404). The evident deviations at the third mode are caused by beam damping. With the smallest third modal damping ratio (\(\xi_{3}\) = 0.0114), Case 1 (without traffic flow) and Case 4 (with traffic flow) could provide well-fitted mode shapes like the undamped case in Figure 7, with MAC values of 0.946 and 0.943, respectively.