# Analysis of Vehicle-Bridge Coupling Vibration Characteristics of Curved Girder Bridges
## 2 Vehicle-Bridge Coupled Vibration of Curved Girder Bridges
### Vibration Equation of Vehicle Model
\begin{split} K_{vij2211}R_{vij2211}\delta R_{vij2211}=K_{vij2211} \left(Y_{vcij}+H_{ij1}\varphi_{vcij}+l_{ij1}\phi_{vcij}-Y_{vwij11}\right) \delta\left(Y_{vcij}+H_{ij1}\varphi_{vcij}+l_{ij1}\phi_{vcij}-Y_{vwij11}\right) \\ =\left(K_{vij2211}Y_{vcij}\delta Y_{vcij}+K_{vij2211}H_{ij1}Y_{vcij} \delta Y_{vcij}+K_{vij2211}l_{ij1}Y_{vcij}\delta Y_{vcij}-K_{vij2211}Y_{vcij} \delta Y_{vuij11}+\\ K_{vij2211}H_{ij1}\varphi_{vcij}+K_{vij2211}H_{ij1}\varphi_{vcij}\delta Y _{vcij}+K_{vij2211}H_{ij1}l_{ij1}\varphi_{vcij}\delta\phi_{vcij}-K_{vij2211}H_{ ij1}\varphi_{vcij}\delta Y_{vuij11}+\\ K_{vij2211}l_{ij1}\phi_{vcij}\delta Y_{vcij}+K_{vij2211}l_{ij1}H_{ij1} \phi_{vcij}+K_{vij2211}l_{ij1}\phi_{vcij}\delta\varphi_{vcij}-K_{vij2211}l_{ij1 }\phi_{vcij}\delta Y_{vuij11}-\\ K_{vij2211}Y_{vwij11}\delta Y_{vcij}-K_{vij2211}H_{ij1}Y_{vwij11}\delta \varphi_{vcij}-K_{vij2211}l_{ij1}Y_{vwij11}\delta\phi_{vcij}+K_{vij2211}Y_{vwij1 1}\delta Y_{vwij11}\end{split}  \tag{11}\]  
where the polynomial on the right side consists of monomials similar to \(ku\delta u\), \(k\) is the constant coefficient of the monomial, and \(u\) is the displacement corresponding to the independent degree of freedom of the vehicle. According to the \"right seating\" rule [19], the coefficient \(k\) is placed in the matrix corresponding to the \"row\" of DOF related to \(\delta u\) and the \"column\" of DOF related to \(u\), resulting in the following submatrix form:  
\[\begin{split}\delta Y_{vcij}&\varphi_{vcij}\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad  
Similarly, by substituting Equations (7)-(9) into Equation (10), respectively, a submatrix form similar to Equation (12) is achieved. The stiffness matrix of a single two-axle vehicle is obtained by superimposing and combining all submatrices. By collecting the stiffness matrices of all vehicles, the stiffness matrix of the vehicle fleet is obtained.  
* Formation of damping matrix The damping matrix of a vehicle is related to the virtual work made by the damping force, which is determined via the relative velocity between the two ends of the damping element. Each two-axle vehicle has four lateral and vertical damping elements between the vehicle body and the wheels. Taking the vertical damping element as an example, we can obtain: \[\dot{R}_{vij2311} =\dot{Z}_{rcij}+b_{ij}\varphi_{vcij}-l_{ij1}\dot{\theta}_{vcij}- \dot{Z}_{vwij11}\] (13) \[\dot{R}_{vij2311} =\dot{Z}_{vcij}-b_{ij}\dot{\varphi}_{vcij}-l_{ij1}\dot{\theta}_{vc ij}-\dot{Z}_{vwij11}\] (14) \[\dot{R}_{vij2311} =\dot{Z}_{vcij}+b_{ij}\dot{\varphi}_{vcij}+l_{ij1}\dot{\theta}_{ vcij}-\dot{Z}_{vwij11}\] (15) \[\dot{R}_{vij2311} =\dot{Z}_{vcij}-b_{ij}\dot{\varphi}_{vcij}+l_{ij1}\dot{\theta}_{ vcij}-\dot{Z}_{vwij11}\] (16) The virtual work achieved via the damping force of the vertical damping element was: \[ \begin{split}\delta W^{V}_{S3}=\sum_{k=1m=1}^{2}& \left(C_{vij23 km}\dot{R}_{vij23 km}\delta R_{vij23 km}\right)=\sum_{k=1m=1}^{2} \sum_{m=1}^{2}\{C_{vij23 km}\dot{Z}_{vcij}+(-1)^{k+1}l_{ijk}\dot{\theta}_{vcij}- \dot{Z}_{vwijkm}\}\big{[}\delta Y_{vcij}\\ +H_{ij1}\delta\varphi_{vcij}+(-1)^{k+1}l_{ijk}\delta\varphi_{vcij} -\delta Y_{vwijkm}\big{]}\end{split} \] (17) Taking the vertical spring of the left wheel (\(m=1\)) of the front axle (\(k=1\)) as an example, substituting Equation (6) into Equation (10), there is: \[