# 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} C_{vij2311}\dot{R}_{vij2311}\dot{R}_{vij2311} =C_{vij2311}\left(\dot{Z}_{vcij}+b_{ij}\dot{\varphi}_{vcij}-l_{ij1}\dot{ \theta}_{vcij}-\dot{Z}_{vvnij11}\right)\delta\big{(}Z_{vcij}+b_{ij}\varphi_{vc ij}-l_{ij1}\theta_{vcij}-Z_{vwij11}\big{)}\\ =(C_{vij2311}\dot{Z}_{vcij}\delta Z_{vcij}+C_{vij2311}b_{ij}\dot{Z }_{vcij}\delta\varphi_{vcij}-c_{vij2311}l_{ij1}\dot{Z}_{vcij}\delta\varphi_{ pcij}-C_{vij2311}\dot{Z}_{vcij}\delta Z_{vwij11}+\\ C_{vij2311}b_{ij}\dot{\varphi}_{vcij}\delta Z_{vcij}+C_{vij2311}b_{ ij}\dot{\varphi}_{vcij}\delta\varphi_{vcij}-C_{vij2311}b_{ij}\dot{\varphi}_{vcij} \delta\theta_{vcij}-C_{vij2311}b_{ij}\dot{\varphi}_{vcij}\delta Z_{vwij11}-\\ C_{vij2311}l_{ij1}\dot{\theta}_{vcij}\delta Z_{vcij}-C_{vij2311}l_{ ij1}b_{vij}\delta\theta_{vcij}+C_{vij2311}l_{ij1}\dot{\theta}_{vcij}+C_{vij2311}l_{ ij1}\delta\varphi_{vcij11}-\\ C_{vij2311}\dot{Z}_{vwij11}\delta Z_{vcij}-C_{vij2311}b_{ij}\dot{Z}_{ vwij11}\delta\varphi_{vcij}+C_{vij2311}l_{ij1}\dot{Z}_{vwij11}\delta\theta_{vcij}+C_{vij2311} \dot{Z}_{vwij11}\delta Z_{vwij11}\end{split} \] (18) where, the polynomial of the right side consists of monomials similar to \(ci\dot{u}\delta u\), \(c\) is the constant coefficient of the monomial, \(i\) is the speed corresponding to the independent DOF of the vehicle, and \(u\) is the displacement corresponding to the independent degree of freedom of the vehicle. According to the \"right seating\" rule [19], the coefficient \(c\) is placed in the matrix corresponding to the \"row\" of DOF related to \(\delta u\) and the \"column\" of DOF related to \(i\), resulting in the following submatrix form: \[\begin{array}{ccccc}\dot{Z}_{vcij}&\dot{\varphi}_{vcij}&\dot{\theta}_{vcij} &\dot{Z}_{vwij11}\\ \delta Z_{vcij}&\left[ \begin{array}{ccccc}C_{vij2311}&C_{vij2231}b_{ij}&-C_{ vij2311}l_{ij1}&-C_{vij2311}\\ C_{vij2311}b_{ij}&C_{vij2311}b_{ij}&-C_{vij2311}l_{ij1}&-C_{vij2311}b_{ij}\\ -C_{vij2311}l_{ij1}&-C_{vij2311}b_{ij1}&C_{vij2311}l_{ij1}&C_{vij2311}l_{ij1}\\ -C_{vij2311}&-C_{vij2311}b_{ij}&C_{vij2311}l_{ij1}&C_{vij2311}\end{array} \right]\] (19) Similarly, by substituting Equations (14)-(16) into Equation (17), respectively, a submatrix form similar to Equation (19) is achieved. The damping matrix of a single two-axle vehicle is obtained via superimposing and combining all submatrices. By collecting the stiffness matrices of all vehicles, the damping matrix \(C_{v}\) of the vehicle fleet is obtained. * Formation of quality matrix The mass matrix of a vehicle is related to the virtual work achieved via inertial forces, which is determined by the acceleration at each mass point. Each two-axle vehicle has a body and four wheels for five rigid body mass points. The mass matrix of a single two-axle vehicle is obtained, as shown in Equation (20). \[M_{v}= \begin{bmatrix}M_{vcij}&&&&\\ &M_{vcij}&&&&\\ &&J_{vcaij}&&&&\\ &&&J_{vcaij}&&&&\\ &&&&M_{vunij11}&&&&\\ &&&&M_{vunij12}&&&&\\ &&&&M_{vunij12}&&&&\\ &&&&M_{vunij21}&&&&\\ &&&&M_{vunij21}&&&&\\ &&&&M_{vunij22}&&\\ &&&&M_{vunij22}&&\\ &&&&M_{vunij22}&&\\ &&&&M_{vunij22}&&\\ &&&&M_{vunij22}&&\\ \end{bmatrix}  \tag{20}\]  
Combining all of the car quality matrices, the vehicle quality matrix \(M_{v}\) can be obtained.