# Bridge Damping Extraction Method from Vehicle-Bridge Interaction System Using Double-Beam Model
## 2 Formulation of the Analytical Theory
### Governing Differential Equations of Double-Beam System  
The elasticity modulus of the two beams of the double-beam system are \(E_{1}\) and \(E_{2}\), the cross-sectional moments of inertia are \(I_{1}\) and \(I_{2}\), the masses per unit length are \(m_{1}\) and \(m_{2}\), and the damping coefficients are \(c_{1}\) and \(c_{2}\). The stiffness and damping coefficients of the connection layer are \(k\) and \(c\), respectively. The governing differential equations (GDE) of this double-beam system can be expressed as:  
\[\left\{ \begin{array}{l}E_{1}I_{1}\frac{\partial^{4}u_{1}}{\partial x^{4}}+m _{1}\frac{\partial^{2}u_{1}}{\partial x^{2}}+c_{1}\frac{u_{1}}{\partial t}+c \frac{\partial(u_{1}-u_{2})}{\partial t}+k(u_{1}-u_{2})=0\\ E_{2}I_{2}\frac{\partial^{4}u_{2}}{\partial x^{4}}+m_{2}\frac{\partial^{2}u_{2 }}{\partial t^{2}}+c_{2}\frac{u_{2}}{\partial t}+c\frac{\partial(u_{2}-u_{1}) }{\partial t}+k(u_{2}-u_{1})=0\end{array} \right. \tag{2}\]  
where \(u_{1}\) and \(u_{2}\) are the displacement values of the two beams. The variable separation can be performed in the Laplace domain as follows:  
\[u_{1}(x,t)=\varphi_{1}(x)e^{\lambda t},\ u_{2}(x,t)=\varphi_{2}(x)e^{\lambda t} \tag{3}\]  
where \(\varphi_{1}(x)\) and \(\varphi_{2}(x)\) represent the mode shape functions of the two beams and can be analytically expressed as:  
\[\varphi_{1}(x)=\overline{A}e^{xx},\ \varphi_{2}(x)=\overline{B}e^{xx} \tag{4}\]  
where \(\overline{A}\) and \(\overline{B}\) are undetermined coefficients related to the boundary condition. By substituting Equations (3) and (4) into the double-beam system GDE Equation (2), one can obtain the following:  
\[\left\{ \begin{array}{l}\overline{A}\left(E_{1}I_{1}x^{4}+k+\lambda(c_{1}+c )+m_{1}\lambda^{2}\right)-\overline{B}(k+c\lambda)=0\\ \overline{B}\left(E_{2}I_{2}x^{4}+k+\lambda(c_{2}+c)+m_{2}\lambda^{2}\right)- \overline{A}(k+c\lambda)=0\end{array} \right. \tag{5}\]  
The matrix form of Equation (5) is:  
\[\left( \begin{array}{cc}E_{1}I_{1}x^{4}+k+\lambda(c_{1}+c)+m_{1}\lambda^{2}& -k-c\lambda\\ -k-c\lambda&E_{2}I_{2}x^{4}+k+\lambda(c_{2}+c)+m_{2}\lambda^{2}\end{array}  \right)\left( \begin{array}{c}\overline{A}\\ \overline{B}\end{array} \right)=\left( \begin{array}{c}0\\ 0\end{array} \right) \tag{6}\]  
For this equation to have a non-trivial solution, the value of the coefficient matrix determinant of \(\overline{A}\) and \(\overline{B}\) is zero. As a result, a characteristic equation for an 8 th order polynomial with respect to \(\kappa\) can be obtained as follows:  
\[\eta_{3}\kappa^{8}+\eta_{2}\kappa^{4}+\eta_{1}=0 \tag{7}\]  
where  
\[\eta_{3}=E_{1}I_{1}E_{2}I_{2}\]  
Figure 2: Simplified schematic of a VBI system. \[\eta_{2} =E_{1}I_{1}\left(k+\lambda(c_{1}+c)+m_{2}\lambda^{2}\right)+E_{2}I_{2} \left(k+\lambda(c_{2}+c)+m_{1}\lambda^{2}\right)\] \[\eta_{1} =\left(k+\lambda(c_{1}+c)+m_{1}\lambda^{2}\right)\left(k+\lambda(c _{2}+c)+m_{2}\lambda^{2}\right)-\left(k+c\lambda\right)^{2}\] The solution of the characteristic equation is \[\pm\kappa_{1},\ \pm\kappa_{2},\ \pm\kappa_{3},\ \pm\kappa_{4},\] and the general solution of Equation (3) can be expressed as: \[ \begin{split}\varphi_{1}(x)=\overline{A}_{1}e^{\kappa_{1}x}+ \overline{A}_{2}e^{-\kappa_{1}x}+\overline{A}_{3}e^{\kappa_{2}x}+\overline{A }_{4}e^{-\kappa_{2}x}+\overline{A}_{5}e^{\kappa_{3}x}+\overline{A}_{6}e^{- \kappa_{3}x}+\overline{A}_{7}e^{\kappa_{4}x}+\overline{A}_{8}e^{-\kappa_{4}x} \\ =\sum\limits_{j=1}^{8}\left(\overline{A}_{2j-1}e^{\kappa_{j}x}+ \overline{A}_{2}je^{-\kappa_{j}x}\right)\end{split}  \tag{8}\] \[ \begin{split}\varphi_{2}(x)=\overline{B}_{1}e^{\kappa_{1}x}+ \overline{B}_{2}e^{-\kappa_{1}x}+\overline{B}_{3}e^{\kappa_{2}x}+\overline{B }_{4}e^{-\kappa_{2}x}+\overline{B}_{5}e^{\kappa_{3}x}+\overline{B}_{6}e^{- \kappa_{3}x}+\overline{B}_{7}e^{\kappa_{4}x}+\overline{B}_{8}e^{-\kappa_{4}x} \\ =\sum\limits_{j=1}^{8}\left(\overline{B}_{2j-1}e^{\kappa_{j}x}+ \overline{B}_{2j}e^{-\kappa_{j}x}\right)\end{split}