# 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  
The bridge had the following parameters: bridge length \(L=32\) m, flexure stiffness \(E_{2}I_{2}=11.04\ \times 10^{11}\) Nm\({}^{2}\), and bridge mass per unit length \(m_{2}=28,600\) kg/m. The first order frequency of the bridge was 9.5306 Hz. The damping coefficient of the bridge was set to \(c_{2}=0\)Ns/m to visualize the vehicle damping effect on the VBI system. The \(1/4\) vehicle model was used, and the vehicle parameters were as follows: vehicle body mass \(m_{v}=20\) t and suspension system stiffness \(k_{v}=10.14\ \times 10^{6}\) N/m.The undamped self-oscillation frequency of the vehicle was 3.82 Hz. The suspension system damping factor was \(c_{v}=7.64\ \times 10^{4}\ \sim 7.64\ \times 10^{5}\)Ns/m. The uniform traffic flow consisted of four vehicles spaced 8 m apart with the same parameters and traveling at the  
Figure 4: The VBI finite element model. Figure 3: The double\(-\)beam finite element model. same speed. The case study then considered three conditions, as follows: Case 1: uniform traffic flow with a speed of 20 km/h; Case 2: uniform traffic flow with a speed of 40 km/h; and Case 3: random traffic flow with an average speed of 40 km/h. The vehicle parameters and speed of the random traffic in Case 3 were generated using a Gaussian distribution, where the mean value of each distribution was the same as that of the uniform traffic, with a standard deviation of 0.2 times the mean value. The bridge and vehicle parameters described above were simplified to obtain the parameters of the double-beam finite element model using the method outlined in Section 2.1. The impact load was applied to the double-beam finite element model, and the acceleration of the structure was calculated using the Newmark method with a time step of 0.001 s.  
The acceleration of the bridge could be obtained by calculating the response of the VBI system as traffic passes over a bridge. The differential equation of a VBI system can be expressed as:  
\[\left\{ \begin{array}{l}\mathbf{M}_{v}\ddot{\mathbf{Z}}_{v}+\mathbf{C}_{v} \ddot{\mathbf{Z}}_{v}+\mathbf{K}_{v}\mathbf{Z}_{v}=\mathbf{F}_{v}\\ \mathbf{M}_{b}\ddot{\mathbf{Z}}_{b}+\mathbf{C}_{b}\ddot{\mathbf{Z}}_{b}+ \mathbf{K}_{b}\mathbf{Z}_{b}=\mathbf{F}_{b}\end{array} \right. \tag{18}\]  
where the mass, damping, stiffness, and load matrices of the vehicle are \(\mathbf{M}_{v}\), \(\mathbf{C}_{v}\), \(\mathbf{K}_{v}\), and \(\mathbf{F}_{v}\), respectively. The mass, damping, stiffness, and load matrices of the bridge are \(\mathbf{M}_{b}\), \(\mathbf{C}_{b}\), \(\mathbf{K}_{b}\), and \(\mathbf{F}_{b}\), respectively. The axle interaction relationship is constructed through the load vectors, \(\mathbf{F}_{v}\) and \(\mathbf{F}_{b}\), of the two systems. \[\mathbf{F}_{v}=\mathbf{K}_{v}\mathbf{S}+\mathbf{C}_{v}\dot{\mathbf{S}} \tag{19}\]  
\[\mathbf{F}_{b}=\mathbf{K}_{v}(\mathbf{Z}_{v}-\mathbf{S})+\mathbf{C}_{v}\Big{(} \dot{\mathbf{Z}}_{v}-\dot{\mathbf{S}}\Big{)}+\mathbf{G}_{v} \tag{20}\]  
where \(\mathbf{S}\) is the vertical displacement at the location of the contact point between the vehicles and the bridge and \(\mathbf{G}_{v}\), is the gravitational force on the vehicle. While there have been many approaches to calculating VBI system response, this study used the inter-system iterative method (ISI) proposed by [PERSON] [27]. This method enables easy calculated, and its convergence is convenient to control, which is suitable for the VBI calculation problem, especially in cases involving multiple vehicles. Figure 5 shows the decay response of the double-beam finite element model subjected to impact loading in the bridge span. The acceleration response of a bridge obtained by VBI calculations usually had multiple frequency components. To calculate the damping ratio, the acceleration response needed to be band-pass filtered so that only one frequency component was included in the acceleration response.