# Dynamic Modeling and Analysis of a Driving Passenger Vehicle
## 2 Dynamic Vehicle Model
However, experimentally obtaining the mass and mass moment of inertia values of various parts to calculate the inertia parameters of the model shown in Figure 2 is a time-consuming and complicated procedure. Therefore, in this study, the inertial values of the sprung and unsprung masses for the dynamic vehicle model were extracted from the 3-D solid model of the vehicle using commercial CAD software. These are the values: \(m_{s}=1568\) kg, \(m_{f}=47\) kg, \(m_{r}=31\) kg, \(I_{x}=719\) kg\(\cdot\)m\({}^{2}\), and \(I_{y}=2495\) kg\(\cdot\)m\({}^{2}\). The products of inertia were neglected in this study because they were relatively small. The suspension, tire, and anti-roll bar that connect the rigid body and ground defined above were modeled with linear equivalent springs and dampers. The front suspensions vertically connected between the sprung mass and front unsprung masses were modeled with an equivalent stiffness \(k_{f}\) and an equivalent damping coefficient \(c_{f}\). Additionally, the rear suspension vertically connected between the sprung mass and rear unsprung masses was modeled with an equivalent stiffness \(k_{r}\) and equivalent damping coefficient \(c_{r}\). As the tires between the ground and the unsprung masses have very high stiffness, they are modeled with an equivalent stiffness \(k_{t}\) without damping. On the other hand, anti-roll bars, which help reduce vehicle body roll in fast cornering or on irregular roads, are modeled as torsion springs. The bars are long U-shaped metals with hollow cross sections that join the suspensions on the right and left sides of an axle. They are usually attached to the chassis at two points using rubber bushings. The torsional stiffnesses of the front and rear anti-roll bars were assumed to be \(\kappa_{f}\) and \(\kappa_{r}\), respectively [30]. The twist angles of the anti-roll bars were determined by the angular displacements of the axel shafts relative to the sprung mass. The axle shaft has an angular displacement owing  
Figure 2: Dynamic model of a vehicle with seven degrees of freedom. to the difference between the right and left wheels. To help understand this, a dynamic vehicle model viewed from the front is shown in Figure 3 as an example. As the front left and right unsprung masses have vertical displacements \(z_{1}\) and \(z_{2}\), respectively, the angular displacement of the front axle shaft is computed as \(\tan^{-1}[(z_{2}-z_{1})/\)\((w_{1}+w_{2})]\). Therefore, the twist angle of the front anti-roll bar \(\phi_{f}\), is given by  
Figure 1.0  
Similarly, the twist angle of the rear anti-roll bar can be expressed as  
Figure 1.1  
The equivalent stiffness and damping coefficients presented above were calculated using an experimental method. The equivalent stiffness values of the suspensions and tires were obtained through static vertical load-deflection tests [31; 32]. Similarly, the equivalent torsional stiffness values of the anti-roll bars were obtained through torsional load-torsion angular deformation tests. The values for the various equivalent stiffness, equivalent torsional stiffness, and damping coefficients obtained by the tests were as follows: \(k_{f}=44,400\) N/m, \(k_{r}=36,000\) N/m, \(k_{t}=255,487\) N/m, \(\kappa_{f}=62,085\) N\(\cdot\)m/rad, \(\kappa_{r}=58,098\) N\(\cdot\)m/rad, and \(c_{f}=c_{r}=5000\) N\(\cdot\)s/m. The equations of motion for the driving vehicle were derived using the following Lagrange equation:  
\[ \begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial T}{ \partial k_{t}}\right)+\frac{\partial F_{d}}{\partial k_{t}}-\frac{\partial T }{\partial k_{t}}+\frac{\partial V}{\partial k_{t}}=Q_{k}\\ \text{for }k=Z,\;\theta,\;\varphi,\;z_{1},\;z_{2},\;z_{3},\;z_{4}\end{array}  \tag{10}\]  
where the superposed dot represents differentiation with respect to time; \(q_{k}\) stands for the seven generalized coordinates; \(T\), \(V\), \(F_{d}\) and \(Q_{k}\) are the kinetic energy, potential energy, [PERSON]'s dissipation function, and generalized force, respectively.