# Bridge Damping Extraction Method from Vehicle-Bridge Interaction System Using Double-Beam Model
## 4 Bridge Damping Ratio Extraction Method
### Equivalent Traffic Flow Damping and Bridge Damping  
According to the relationship of mode shape \(\overline{B}=t\overline{A}\), the damping ratio of a VBI system can be written as:  
\[\xi=\frac{2c(1\ -t)-\overline{t}c_{2}}{2(m_{1}-tm_{2})\omega_{0}} \tag{22}\]  
where \(t\) denotes the ratio of bridge vibration to vehicle vibration amplitude. The value of \(t\) can be calculated using the double-beam theory after obtaining an estimate of \(k_{v}\), combining the undamped frequency \(\omega_{0}\) of a VBI system and the damping ratio \(\xi\). According to Equation (10), four values of \(t_{i},\ i=1\ \ \sim 4\ \ \text{can}\) be calculated; for simplified VBI systems \(E_{1}I_{1}=0\,\)\(t_{i}\) values are the same, so it is abbreviated here as \(t\). The formula for \(t\) is:  
\[t=\frac{k+\lambda c+m_{1}\lambda^{2}}{k+c\lambda} \tag{23}\]Therefore, the bridge damping factor is:  
\[c_{2}=\frac{2c(1-t)-2(m_{1}-tm_{2})\omega_{0}\xi}{t} \tag{24}\]  
It is also necessary to determine the equivalent traffic damping factor \(c\) in order to obtain an estimate of the bridge damping factor \(c_{2}\). The equivalent traffic flow damping factor can be selected as follows. When the number of vehicles on the bridge is \(n\), a series of different test values of \(c_{v}\) are used to calculate the equivalent traffic flow coefficient. Then, a series of different identification values of bridge damping coefficient are obtained using Equation (24). The curve of bridge damping coefficient identification value is drawn with \(c_{v}\). When the vehicle number \(n\) is different, the obtained curves are also different. Because the bridge damping coefficient remains unchanged, these curves will intersect at one point. The horizontal coordinate corresponding to the intersection is the average vehicle damping coefficient \(c_{v}\), and the vertical coordinate is the estimated value of the bridge damping coefficient \(c_{2}\).  
The process of the bridge damping ratio evaluation method proposed in this paper can be summarized as follows:  
1. The frequency and damping ratio of a VBI system is identified using the OMA method based on bridge monitoring data. This process considers the number of vehicles and vehicle masses to be known in the presence of a WIM system.
2. The corresponding undamped frequency of the VBI system is calculated using Formula (16).
3. The best estimation of average vehicle stiffness is obtained by PSO, taking the absolute value of the slope of the line obtained by linear least squares between vehicle bridge system frequency and vehicle number as the objective function. Then, the equivalent vehicle flow stiffness is calculated.
4. The bridge damping coefficient curves under different vehicle numbers are solved using a series of average vehicle damping test values.
5. The horizontal coordinate of the intersection of the curves is the estimated value of the average vehicle damping, and the vertical coordinate is the estimated value of the bridge damping coefficient.  
A flow chart of the proposed bridge damping evaluation method is shown in Figure 8.  
Figure 8: Bridge damping ratio evaluation flow chart.