## 4 Modeling and Simulation
### Motion Equation of the Vehicle  
The transport vehicle is a complex multi-body system, which has not only an interaction force and relative displacement between the components, but also an interaction between the tire and the road. Therefore, it is very difficult to establish an accurate dynamic model of the whole vehicle. According to the structure of the transport vehicle studied, the following assumptions can be made when building a vehicle motion model [16].  
The compressibility of the liquid can be considered as a spring, and the throttle effect of the valve and the leakage of the hydraulic cylinder can be considered as damping. The elasticity of the frame, the load-bearing platform, and the driver's cab is much less than that of the suspension system,and they are regarded as rigid bodies without considering elastic deformation. The vibration influence of the diesel engine, transmission shaft, and gearbox on the whole vehicle is ignored. Because only the suspension system of the vehicle is studied, the influence of the steering system is not considered. The center of gravity of goods is assumed to remain unchanged under full load condition.  
The simplified dynamic model of the vehicle is modeled as shown in Figure 8, and \(M\) is the mass of the vehicle load-bearing platform, \(m_{i}\) is the mass of wheel, \(k_{i}\) is the stiffness of the hydraulic suspension, \(k_{li}\) is the stiffness of the tire, \(\zeta_{i}\) is the damping of the hydraulic suspension, \(z_{i0}\) is the vertical displacement of the connection point, \(z_{i}\) is the vertical displacement of the tire center, \(y_{i}\) is the unevenness of the road, \(Z_{1}\) is the vertical displacement of the mass center, \(\theta_{x}\), and \(\theta_{y}\) is the angle of rotation around the \(X\)-axis and \(Y\)-axis.  
According to [PERSON]'s law and the dynamic model of the vehicle, the force and vertical displacement of the connection point can be obtained as follows [17].  
\[F_{i}=k_{i}[z_{i}-z_{i0}]+ci[\dot{z}_{i}-\dot{z}_{i0}](i=1,2,3,4) \tag{21}\]  
\[\left\{\begin{array}{l}z_{10}=Z_{1}-0.5L_{1}\theta_{x}-0.5L_{2}\theta_{y}\\ z_{20}=Z_{1}+0.5L_{1}\theta_{x}-0.5L_{2}\theta_{y}\\ z_{30}=Z_{1}-0.5L_{1}\theta_{x}+0.5L_{2}\theta_{y}\\ z_{40}=Z_{1}+0.5L_{1}\theta_{x}+0.5L_{2}\theta_{y}\\ \end{array}\right. \tag{22}\]  
According to Equations (21) and (22), the three direction motion equations of the vehicle load-bearing platform can be obtained as Equation (23). The motion equation of the unsprung mass is obtained as Equation (24).  
\[\left\{\begin{array}{l}M\ddot{Z}_{1}=\sum F_{i}\\ I_{X}\ddot{\theta}_{x}=0.5[(F_{2}+F_{4})-(F_{1}+F_{3})]L_{1}\\ I_{Y}\ddot{\theta}_{y}=0.5[(F_{3}+F_{4})-(F_{1}+F_{2})]L_{2}\\ \end{array}\right. \tag{23}\]  
\[m_{i}\ddot{z}_{i}+\zeta_{i}(z_{i}-z_{i0})+k_{i}(z_{i}-z_{i0})+k_{it}(z_{i}-y_{ i})=0(i=1,2,3,4) \tag{24}\]  
where \(I_{X}\) is the rotation inertia of the platform about the \(X\)-axis, \(I_{Y}\) is the rotation inertia of the platform about the \(Y\)-axis.  
The vibration equation of the whole system can be obtained as follows.  
\[[M]\ddot{Z}+[\zeta]\dot{Z}+[K]Z=[K_{t}]Y \tag{25}\]  
where \(M\) is the mass matrix, \(\zeta\) is the damping matrix, \(K\) is the stiffness matrix, \(K_{t}\) is the tire stiffness matrix, \(Q\) is the unevenness matrix of the road.  
Figure 8: The dynamic model of the vehicle.