# Bridge Damping Extraction Method from Vehicle-Bridge Interaction System Using Double-Beam Model
## 2 Formulation of the Analytical Theory
### Extended Dynamic Stiffness Method  
[PERSON] [20; 21] proposed a method for solving damped double-beam systems using the extended dynamic stiffness method (EDSM). According to the uniformity of the dynamic stiffness matrix between the damped and undamped system, the dynamic stiffness matrix of the damped system is denoted as \(\mathbf{K}(\mathbf{\kappa})\) and the dynamic stiffness matrix of the undamped system is denoted as \(\mathbf{K}(\mathbf{\kappa}_{0})\); for the same order mode there is:  
\[\kappa_{0}=\mathbf{\kappa},|\mathbf{K}(\mathbf{\kappa})|=|\mathbf{K}(\mathbf{\kappa}_{0})|=\mathbf{0} \tag{11}\]  
Substituting \(c=c_{1}=c_{2}=0\) into Equation (5) results in:  
\[\left\{\begin{array}{l}\overline{A}\left(E_{1}I_{1}\kappa_{0}^{4}+k+m_{1} \lambda_{0}^{2}\right)-\overline{B}k=0\\ \overline{B}\left(E_{2}I_{2}\kappa_{0}^{4}+k+m_{2}\lambda_{0}^{2}\right)- \overline{A}k=0\end{array}\right. \tag{12}\]  
According to Equation (11), Equation (13) can be obtained:  
\[\lambda^{2}+\frac{2c\left(\overline{A}-\overline{B}\right)+\left(\overline{A} c_{1}-\overline{B}c_{2}\right)}{\left(\overline{A}m_{1}-\overline{B}m_{2} \right)\lambda_{0}}\lambda\lambda_{0}+\lambda_{0}^{2}=0 \tag{13}\]  
where \(\lambda_{0}=\omega_{0}i\), \(\omega_{0}\) is the modal frequency corresponding to the undamped double beam system. The damping ratio of the double beam system is:  
\[\xi=\frac{2c\left(\overline{A}-\overline{B}\right)+\left(\overline{A}c_{1}- \overline{B}c_{2}\right)}{2\left(\overline{A}m_{1}-\overline{B}m_{2}\right) \omega_{0}} \tag{14}\]  
The solution of Equation (13) is:  
\[\lambda=\xi\lambda_{0}i\pm\sqrt{1\ -\xi^{2}}\lambda_{0}=-\xi\omega_{0}\pm\sqrt{1 \ -\xi^{2}}\omega_{0}i \tag{15}\]  
where the imaginary part of the solution is the self-oscillation frequency of the damped double beam system.  
\[\omega=\sqrt{1\ -\xi^{2}}\omega_{0} \tag{16}\]  
The Wittrick-Williams (W-W) algorithm [22] is a well-known method for calculating frequency using the dynamic stiffness method. Instead of directly solving the frequency equation, the W-W algorithm first determines the upper and lower bounds of any order frequency and then finds the exact frequency via the dichotomy method or [PERSON]'s method. [PERSON] [23; 24] proposed an improved W-W algorithm to conveniently solve the frequency of an undamped double-beam system. The authors of this study used the improved W-W algorithm. The mode shape is calculated using the guided Newton method, and the detailed solution steps of the method can be found in [25; 26].  
By substituting the damping coefficient \(c_{1}=0\) of the upper beam of the double-beam system into Equation (14), the damping ratio of the VBI system is obtained as:  
\[\xi=\frac{2c(\overline{A}-\overline{B})-\overline{B}c_{2}}{2(\overline{A}m_{1} -\overline{B}m_{2})\omega_{0}} \tag{17}\]  
As seen in Equation (17), the damping ratio of the vehicle-bridge coupled system is related to vehicle and bridge amplitudes, vehicle damping coefficient, bridge damping coefficient, and vehicle and bridge masses, where \(\omega_{0}\) is the frequency corresponding to the undamped VBI system, not the frequency of the bridge.