# Dynamic Response Analysis of Asphalt Pavement under Pavement-Unevenness Excitation
## 2 Solution of Vehicle Load under Pavement-Unevenness Excitation
### The Maneuverable Load of the Whole-Vehicle Model Is Solved  
In this paper, a seven-degree-of-freedom vehicle model is adopted, and the four-wheel pavement unevenness is used as an incentive input to the vehicle to solve the random load exerted by the wheel on the road surface. The vehicle-dynamics model is shown in Figure 1. In the figure, \(m_{b}\) is the body mass, and \(m_{1}\), \(m_{2}\), \(m_{3}\), and \(m_{4}\) are the mass of the four wheels, respectively. \(k_{1}\), \(k_{t2}\), \(k_{t3}\), and \(k_{t4}\) are the elastic stiffness of the four tires. \(k_{k1}\), \(k_{k2}\), \(k_{t3}\), and \(k_{t4}\) are the elastic stiffness of the suspension on the wheel, respectively. \(c_{s1}\), \(c_{s2}\), \(c_{s3}\), and \(c_{s4}\) are the damping of the suspension on the wheel respectively. \(z_{1}\), \(z_{2}\), \(z_{3}\), and \(z_{4}\) are the vertical displacements of the four tires, respectively. \(z_{b1}\), \(z_{b2}\), \(z_{b3}\), and \(z_{b4}\) are the vertical displacement of the suspension on the tire, that is, the displacement at the four corners of the body. \(z_{b}\) is the vertical displacement at the center of mass of the body; \(q_{1}\), \(q_{2}\), \(q_{3}\), and \(q_{4}\) are the excitation of pavement unevenness at four wheels, respectively. \(a\) is the distance between the front car and the center of mass of the vehicle; \(b\) is the distance between the left and right-front wheels; \(B_{r}\) is the distance between the left- and right-rear wheels; \(\theta_{b}\) is the pitch displacement at the center of mass of the car body; \(\varphi\) is the lateral angle displacement at the body center of mass; \(v\) stands for direction of travel. The dynamic equations of the system include the vertical-motion equation of the body's center of mass, the motion equation of the body's pitch and roll, and the vertical-motion equation of four-wheel masses. Vertical-motion equation of body centroid:  
\[ \begin{array}{l}m_{b}\ddot{z}_{b}+c_{1}(\dot{z}_{b1}-\dot{z}_{1})+k_{s1}(z_{b1 }-z_{1})+c_{s2}(\dot{z}_{b2}-\dot{z}_{2})+k_{s2}(z_{b2}-z_{2})+\\ c_{s3}(\dot{z}_{b3}-\dot{z}_{3})+k_{s3}(z_{t3}-z_{3})+c_{s4}(\dot{z}_{b4}-\dot{ z}_{4})+k_{s4}(z_{b4}-z_{4})=0\end{array}  \tag{15}\]  
Body-pitch-motion equation:  
\[ \begin{array}{l}I_{p}\ddot{\theta}_{b}-[c_{s1}(\dot{z}_{b1}-\dot{z}_{1})+k_{ s1}(z_{b1}-z_{1})]\cdot a-[c_{s2}(\dot{z}_{b2}-\dot{z}_{2})+k_{s2}(z_{b2}-z_{2})] \cdot a+\\ [c_{s3}(\dot{z}_{b3}-\dot{z}_{3})+k_{s3}(z_{b3}-z_{3})]\cdot b+[c_{s4}(\dot{z} _{b4}-\dot{z}_{4})+k_{s4}(z_{b4}-z_{4})]\cdot b=0\end{array}  \tag{16}\]  
Body roll motion equation:  
\[ \begin{array}{l}I_{r}\ddot{\varphi}+[c_{s1}(\dot{z}_{b1}-\dot{z}_{1})+k_{s1} (z_{b1}-z_{1})]\cdot\frac{B_{f}}{2}-[c_{s2}(\dot{z}_{b2}-\dot{z}_{2})+k_{s2}( z_{b2}-z_{2})]\cdot\frac{B_{f}}{2}+\\ [c_{s3}(\dot{z}_{b3}-\dot{z}_{3})+k_{s3}(z_{b3}-z_{3})]\cdot\frac{B_{r}}{2}-[c_ {s4}(\dot{z}_{b4}-\dot{z}_{4})+k_{s4}(z_{b4}-z_{4})]\cdot\frac{B_{r}}{2}=0\end{array}  \tag{17}\]  
Equation of vertical motion of four-wheel masses:  
\[ \begin{array}{l}m_{1}\ddot{z}_{1}-c_{s1}(\dot{z}_{b1}-\dot{z}_{1})-k_{s1}(z_ {b1}-z_{1})+k_{t1}(z_{1}-q_{1})=0\\ m_{2}\ddot{z}_{2}-c_{s2}(\dot{z}_{b2}-\dot{z}_{2})-k_{s2}(z_{b2}-z_{2})+k_{t2}( z_{2}-q_{2})=0\\ m_{3}\ddot{z}_{3}-c_{s3}(\dot{z}_{b3}-\dot{z}_{3})-k_{s3}(z_{b3}-z_{3})+k_{t3}( z_{3}-q_{3})=0\\ m_{4}\ddot{z}_{4}-c_{s4}(\dot{z}_{b4}-\dot{z}_{4})-k_{s4}(z_{b4}-z_{4})+k_{s4}( z_{b4}-q_{4})=0\end{array}  \tag{18}\]  
The relationship between suspension displacement and body-mass center displacement is as follows:  
\[ \begin{array}{l}z_{b1}=z_{b}-a\cdot\theta_{b}-\frac{B_{f}}{2}\cdot\varphi\\ z_{b2}=z_{b}-b\cdot\theta_{b}+\frac{B_{f}}{2}\cdot\varphi\\ z_{b3}=z_{b}+b\cdot\theta_{b}+\frac{B_{f}}{2}\cdot\varphi\\ z_{b4}=z_{b}+a\cdot\theta_{b}-\frac{B_{r}}{2}\cdot\varphi\end{array}