# Bridge Damping Extraction Method from Vehicle-Bridge Interaction System Using Double-Beam Model
## 5 Application
### Bridge Damping Ratio Evaluation  
Figure 13 shows the mid-span acceleration data from 12:00 to 12:59 on a certain day, with a sampling frequency of 50 Hz. The data were band-pass filtered to identify the first order frequency and damping ratio. In the filtered data, the free decay response was selected as the sample for damping identification, as shown in Figure 14. The self-oscillation frequency and damping ratio were calculated using the HHT method. The identification results are shown in Table 3. From the table, it can be seen that when the number of vehicles on the bridge increased, the identification self-oscillation frequency decreased and the damping ratio increased. The identified damping ratio ranged from 0.83% to 1.19%, showing a large dispersion.  
\begin{table}
\begin{tabular}{c c c} \hline
**Order** & **Calculated Value (Hz)** & **Measured Value (Hz)** \\ \hline
1 & 0.59 & 0.55 \\
2 & 0.78 & 0.86 \\
3 & 1.23 & 1.31 \\ \hline \end{tabular}
\end{table}
Table 2: First three order frequency of the bridge.  
Figure 13: Mid\(-\)span acceleration data from 12:00 to 12:59.  
We considered the results of the table obtained from the monitoring data (In Figure 14) as the frequency and damping ratio of the VBI system, and we analyzed these data using the bridge damping ratio evaluation method proposed in Section 4. The equivalent traffic flow stiffness was first estimated using the PSO algorithm. The result was 8.15 \(\times\) 10\({}^{3}\), and the convergence of each particle is shown in Figure 15.  
The identification curve of the bridge damping coefficient was obtained as shown in Figure 16. The damping coefficient identification curves of the bridge did not intersect completely at one point, and the range of the intersection point was between 1260 and 1315, corresponding to the first-order damping ratio of the bridge of 0.75%-0.78%. It can be seen that after extracting the bridge damping using the proposed method, the obtained bridge damping dispersion was significantly reduced. The reason for this phenomenon was that  
Figure 14: (a) Acceleration after band\(-\)pass filteringl (b) extracted decay response.  
Figure 15: The objective function value of each particle.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & \multirow{2}{*}{**Frequency (Hz)**} & \multirow{2}{*}{**Damping Ratio (\%)**} & \multirow{2}{*}{**Number of Vehicles on the Bridge**} & \multirow{2}{*}{**Total Mass of Vehicle on Bridge (kg)**} \\  & & & & \\ \hline
1 & 0.5402 & 0.85 & 5 & 6153 \\
2 & 0.5413 & 0.83 & 1 & 2520 \\
3 & 0.5343 & 1.19 & 20 & 26,515 \\
4 & 0.5421 & 0.9 & 15 & 17,288 \\
5 & 0.5426 & 0.87 & 6 & 8278 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Modal parameter identification results.  
the traffic load was random traffic flow under operating conditions, which was not strictly consistent with the assumptions of the simplified VBI model proposed in this paper.  
Sample 2 was calculated with only one vehicle on the bridge, and the damping ratio was 0.83%. We approximated that 0.83% was the real damping ratio of the bridge. It can be seen that the bridge damping identification values of 0.75-0.78% were close to the real values. The above analysis proves that the method proposed in this paper is still valid in engineering applications and can remove the influence of the vehicle-bridge coupling effect on bridge damping evaluation.  
To illustrate the stability of the algorithm proposed in this paper, the acceleration data from 13:00 to 13:59 on the same day were selected. The traffic load at this time had some similarity with that from above, so the equivalent vehicle stiffness obtained earlier was directly used here for identification. The identification results are shown in Figure 17. The intersection range was 1368-1380, corresponding to a bridge damping ratio of 0.81-0.82%, and the results were basically consistent with the previous results.