# Bridge Damping Extraction Method from Vehicle-Bridge Interaction System Using Double-Beam Model
## 3 Verification of the Simplified VBI Model
### Description of the Bridge Model and the Vehicle Model
Figure 6 shows the acceleration response of a bridge system in the bridge mid-span after band-pass filtering when the traffic passed by, the traffic presented a uniform flow, and the speed was 20 km/h. As seen in Figure 6, when there was traffic on the bridge, even if the bridge damping factor was zero, the response of the bridge still decayed similar to that shown Figure 5. When the traffic left the bridge, however, the bridge response became a free vibration without decay. For this study, the damping ratios of the two models were calculated. Then, the formula in Section 2 was used to calculate the damping ratio of the simplified model. Table 1 shows the damping ratio results obtained for each model. The damping ratio calculation results of the double-beam theory and the double-beam FEM showed that the double-beam theory could accurately calculate the damping ratio of the system. The results Case 1 and Case 2, as well as the double-beam theory, showed that the damping ratios were close to each other, which verified the accuracy of the simplified VBI system using the double-beam system. The results of Case 1 and Case 2 also showed that the vehicle speed in the VBI model had little influence on the damping ratio of the system. The random traffic conditions in Case 3 were set up to illustrate the robustness of the simplified method proposed in this paper. The variation pattern of the damping ratio in Case 3 was essentially the same as that in the uniform traffic condition, which indicates \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Vehicle Damping** & **Double-Beam** & **Double-Beam** & **VBI System FEM** & **VBI System FEM** & **VBI System FEM** \\
**Ratio** & **Theory** & **FEM** & **(Case 1)** & **(Case 2)** & **(Case 3)** \\ \hline
0.085 & 0.0046 & 0.0037 & 0.0040 & 0.0043 & 0.0036 \\
0.170 & 0.0089 & 0.0073 & 0.0076 & 0.0073 & 0.0066 \\
0.339 & 0.0155 & 0.0139 & 0.0136 & 0.0134 & 0.0137 \\
0.509 & 0.0196 & 0.0190 & 0.0185 & 0.0180 & 0.0191 \\
0.678 & 0.0216 & 0.0225 & 0.0212 & 0.0209 & 0.0223 \\
0.848 & 0.0220 & 0.0245 & 0.0228 & 0.0213 & 0.0201 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Damping ratio calculation results for each model. Figure 5: Mid\(-\)span acceleration of the double\(-\)beam system. Figure 6: Bridge mid\(-\)span acceleration for VBI model (results after bandpass filtering; vehicle speed was 20 km/h). that the simplified model could still well simulate the damping ratio of the VBI system under some stochastic conditions. Table 1 shows that the damping ratio of the vehicle increased as the damping ratio of the VBI system increased, but there was not a linear relationship between these two variables. When the vehicle damping ratio increased from 0.678 to 0.848, the damping ratio of the VBI system changed very little. To understand the relationship between VBI system damping ratio and the vehicle damping ratio, the simplified model of the VBI system was used to calculate the damping of the VBI system under different vehicle damping ratios. The results are shown in Figure 7.  
The damping ratios of the VBI system are shown in Figure 7 for three traffic conditions: light traffic (\(n=1,\ 2\) ), normal traffic (\(n=3\) ), and heavy traffic (\(n=4\) ). As seen in Figure 7, the damping ratio of the VBI system increased as the traffic volume increased. When the vehicle damping ratio was relatively small, the damping ratio of the VBI system was approximately linear with the vehicle damping ratio. After the vehicle damping ratio was greater than 0.8, however, the damping ratio of the VBI system slightly decreased. The reason is analyzed as follows: Equation (17) shows that the damping ratio of a VBI system is related to the bridge and vehicle amplitudes. When the vehicle damping ratio is large, the relative amplitude of both the vehicle and the bridge decreases. Following additional damping energy dissipation, the vehicle damping ratio is also reduced, resulting in a reduction of the damping ratio of the VBI system.