# Analysis of Vehicle-Bridge Coupling Vibration Characteristics of Curved Girder Bridges
## 2 Vehicle-Bridge Coupled Vibration of Curved Girder Bridges
### Automotive Dynamic Analysis Model  
Vehicle models can be represented using rigid bodies, springs, dampers, and other elements. The main parameters include the number of rigid bodies, the number of springs,the mass of the rigid bodies, and the corresponding mass moments of inertia. By adjusting the combination of these parameters, various shapes of car models can be simulated. Take a two-axis four-wheel car as an example [22]; it can be divided into five rigid bodies: the body and four wheels. These rigid bodies are connected through elastic and damping elements, as shown in Figure 1.  
To simplify the calculation, considering that when the vehicle is traveling with a constant speed in a straight line, the vibration of each rigid body in the vehicle in the direction of the car's travel has a negligible influence on both the bridge's vertical and lateral vibration, the degrees of freedom (DOF) of each rigid body in the direction of the car's motion can be ignored in the calculation. Therefore, the car body only considers five degrees of freedom; each wheel pair has two independent DOFs along the Y-axis displacement (Y) and the Z-axis displacement (Z). That is, four wheels have a total of eight DOFs. A two-axis four-wheel car has a total of 13 DOFs. Its displacement vector can be expressed as:  
\[ \begin{split}& V_{vij}\\ &=\left[Y_{vcij}\;Z_{vcij}\;\phi_{vcij}\;\phi_{vcij}\;\phi_{vcij}\;Y_ {vuij11}\;Z_{vuij11}\;Y_{vuij12}\;Z_{vuij12}\;Y_{vuij21}\;Z_{vuij21}\;Y_{vuij22} \;Z_{vaij22}\right]^{T}\end{split}  \tag{1}\]  
where:\(Y_{vcij},\;Z_{vcij},\;\phi_{vcij},\;\theta_{vcij}\;\)_and_\(\;\phi_{vcij}\) are the displacement of the y vehicle on the \(i\) lane along Y-axis, the displacement along Z-axis, the rotation angle around X-axis, the rotation angle around X-axis, the rotation angle around Y-axis, and the rotation angle around the Z-axis; and \(Y_{vuij11},\;Z_{vuij11}\) are the displacement of the right wheel and the left wheel of the front axle (the first axle) of the \(j\) vehicle on the \(i\) lane along Y-axis and Z-axis; and \(Y_{vuij22},\;Z_{vuij22}\) and \(Y_{vuij21},\;Z_{vuij21}\) are the displacement of the right wheel and the left wheel of the rear axle (the second axle) of \(j\) vehicle on the \(i\) lane along Y-axis and Z-axis. In Table 1, the vehicle model's parameters are shown. Assuming that the car fits perfectly on the bridge's top deck, the contact between the wheel and the top deck is point contact. In this case, the contact point displacement between the wheel and the bridge deck is affected by the displacement and the roughness. Based on the above assumptions, take the vertical spring as an example to derive the relative displacement expression of the above two types of springs. Assuming there are \(L_{\
u}\) lanes and \(n_{\
u}\) vehicles on the \(i\) lane, the relative displacement generated at the two ends of the vertical spring of the first type of spring (the spring connecting the vehicle body and the wheel) on the \(j\) vehicle of the \(i\) lane can be expressed as:  
\[R_{vij23 km}=Z_{vcij}+(-1)^{m+1}b_{ij}\phi_{vcij}-(-1)^{k+1}l_{ijk}\theta_{vcij} -Z_{vuijkm} \tag{2}\]  
where, \(\
u\) is the vehicle; \(c\) is the vehicle body; \(w\) is the wheel; \(i\) is the lane number; \(j\) is the vehicle number; \(2\) is the second type spring (first type spring); \(3\) is the vertical spring; \(k\) is the axle number (\(k=1,\,2,\,1\) is the front axle and \(2\) is the rear axle); \(m\) is the wheel number (\(m=1,\,2,\,1\) in the forward direction, \(1\) is the left wheel and \(2\) is the right wheel); \(R_{vij23 km}\) is  
Figure 1: Vehicle spatial vibration analysis model. the relative displacement of the second type vertical spring on the \(m\) wheel on \(k\) axle of \(j\) car on \(i\) lane; and \(Z_{vwjkm}\) is the vertical displacement on the \(m\) wheel on \(k\) axle of \(j\) car on \(i\) lane.