# 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  
In the dynamic analysis model of the above car, taking the equilibrium state of the car under the action of its weight as the initial state, based on the virtual work conducted on the car's inertial force, elastic force, and damping force, the vibration equation of the car can be obtained according to the principle of the following virtual work:  
\[M_{v}\ddot{v}_{v}+C_{v}\ddot{v}_{v}+K_{v}v_{v}=F_{v} \tag{5}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Description** \\ \hline \(M_{ccij},\ J_{yczij}\) & The rigid mass of the vehicle body of the \(j\) car on the \(i\) lane; \\ \(J_{yczij},\ J_{ycyij}\) & The lateral inertia of the vehicle body; \\ \(K_{vj2211},\ K_{vij212},\ K_{vij221},\ K_{vij2221},\ K_{vij2222}\) & The longitudinal rotational inertia of the vehicle body. \\ \(K_{vij2311},\ K_{vij2312},\ K_{vij2321},\ K_{vij2321},\ K_{vij2322}\) & The lateral stiffness coefficient of secondary suspension. \\ \(C_{vj2211},\ C_{vj2212},\ C_{vj2212},\ C_{vj2221},\ C_{vj2222}\) & The lateral damping constant of each wheel. \\ \(C_{vj211},\ C_{vj2232},\ C_{vj2321},\ C_{vj2322}\) & The vertical damping constant of each wheel. \\ \(M_{vij11},\ M_{vij12},\ M_{vvij21},\ M_{vvij22}\) & The vehicle’s unsprung mass. \\ \(K_{vij1211},\ K_{vij1212},\ K_{vij1221},\ K_{vij1221},\ K_{vij1222}\) & The lateral stiffness coefficient of the vehicle’s wheel. \\ \(K_{vij1311},\ K_{vij1312},\ K_{vij1321},\ K_{vij1322}\) & The vertical stiffness coefficient of the vehicle’s wheel. \\ \(C_{vj1211},\ C_{vj1212},\ C_{vj1221},\ C_{vj1221},\ C_{vj1222}\) & The lateral damping constant of the vehicle’s wheel. \\ \(C_{vj1311},\ C_{vj1312},\ C_{vj1321},\ C_{vj1322}\) & The vertical damping constant of the vehicle’s wheel. where, \(M_{v},~{}C_{v},~{}and~{}K_{v}\) are the mass, damping, and stiffness matrix for the vehicle, respectively; \(v_{v}\), \(\tilde{v}_{v}\), and \(\tilde{v}_{v}\) are the absolute displacement, velocity, and acceleration arrays of the vehicle, respectively; and \(F_{v}\) is the array of external loads on the vehicle and is mainly the force between the bridge and the wheel contact point, without considering the vehicle's weight. Here, the vibration equation of the vehicle described above is established in its coordinate system. However, when the vehicle runs on a curved bridge, the coordinate system changes depending on the vehicle's location. Therefore, in each time step, the motion equation of the car needs to be converted to the global coordinate system. * Formation of stiffness matrix  
The stiffness matrix of a vehicle is related to the virtual work via the elastic forces, which depends on the relative deformation of the two ends of the spring. Each two-axle vehicle has four lateral and four vertical springs between the vehicle body and the wheels. Taking the lateral spring as an example, we can obtain. \[R_{vij2211}=Y_{vcij}+H_{ij1}\varphi_{vcij}+l_{ij1}\phi_{vcij}-Y_{vwij11} \tag{6}\]  
\[R_{vij2212}=Y_{vcij}+H_{ij1}\varphi_{vcij}+l_{ij1}\phi_{vcij}-Y_{vwij11} \tag{7}\]  
\[R_{vij2221}=Y_{vcij}+H_{ij2}\varphi_{vcij}+l_{ij2}\varphi_{vcij}-Y_{vwij21} \tag{8}\]  
\[R_{vij2222}=Y_{vcij}+H_{ij2}\varphi_{vcij}+l_{ij2}\varphi_{vcij}-Y_{vwij22} \tag{9}\]  
The virtual work made via the elastic force of the horizontal spring is:  
\[ \begin{split}\delta W_{S2}^{V}=\sum_{k=1m=1}^{2}\sum_{\begin{array} []{c}\left(K_{vij22 km}R_{vij22 km}R_{vij22 km}R_{vij22 km}\right)=\sum_{k=1m=1}^{ 2}\sum_{\{K_{vij22 km}\left[Y_{vcij}+H_{ij1}\varphi_{vcij}\right.\end{array} \right.}\\ +(-1)^{k+1}l_{ijk}\phi_{vcij}-Y_{vwijkm}\left]\left[\delta Y_{vcij}+H_{ij1} \delta\varphi_{vcij}\right.\\ +(-1)^{k+1}l_{ijk}\phi_{vcij}-\delta Y_{vwijkm}\left]\right\}\end{split}  \tag{10}\]  
Taking the lateral spring of the left wheel (\(m=1\)) of the front axle (\(k=1\)) as an example, substituting Equation (6) into Equation (10), there is:  
\[