# Dynamic Modeling and Analysis of a Driving Passenger Vehicle
## 2 Dynamic Vehicle Model
The centrifugal force applied to a turning vehicle can be obtained by substituting Equation (4) with Equation (1). \[F=\frac{m_{t}V^{2}\delta}{L} \tag{5}\]  
The average steering angle of the front wheels is determined by the steering ratio and steering wheel angle, which can be expressed by the following equation [28]:  
\[\delta_{s}=i_{s}\delta \tag{6}\]  
where \(\delta_{s}\) is the angle of the steering wheel and \(i_{s}\) is the steering ratio of the steering wheel angle to the steering angle of the front wheels. The steering angle ratio \(i_{s}\) is determined by the gear ratio of the steering linkage connecting the steering wheel and the front wheel, which is 15 for the target vehicle of this study. Finally, the centrifugal force applied to a turning vehicle is expressed as follows: vehicle mass \(m_{t}\), vehicle speed \(V\), wheelbase distance \(L\), steering wheel angle \(\delta_{s}\), and steering ratio \(i_{s}\). \[F=\frac{m_{t}V^{2}\delta_{s}}{i_{s}L} \tag{7}\]  
Next, a dynamic vehicle model with seven degrees of freedom was established to analyze the behavior of a driving vehicle. As shown in Figure 2, the vehicle was modeled with one sprung mass and four unsprung masses, for a total of five rigid bodies, and the suspension, anti-roll bar, and tires were modeled as springs and dampers. In Figure 2, the vehicle's forward direction is the negative \(x\)-axis direction. In addition, the \(y\)- and \(z\)-axes are parallel to the left and right directions and the up and down directions of the vehicle, respectively. The sprung mass is the total mass of a vehicle supported by the suspension, which includes the body, frame, engine, transmission, and internal components. The unsprung mass includes wheel axles, wheel bearings, wheel hubs, tires, drive shafts, springs, shock absorbers, and suspension links. The motion of the sprung mass is described by the bounce displacement of the mass center \(Z\), the pitch angle \(\varphi\), and the roll angle \(\theta\), while the motions of the unsprung masses are described by the vertical displacements of \(z_{1}\), \(z_{2}\), \(z_{3}\) and \(z_{4}\). In this study, the x-direction motion of the sprung mass is defined as a constant vehicle speed \(V\). Since the vibration and yaw motion in the \(x\) and \(y\) directions are ignored, the motion of the sprung mass is defined by the bounce displacement, pitch angle, and roll angle. As shown in Figure 2, the sprung mass has a roll motion that rotates about a roll axis parallel to the \(x\)-axis at a distance \(h=476\) mm vertically from the center of mass [29]. This considers the phenomenon that the lateral centrifugal force acting on the center of mass of the vehicle body located at a distance \(h\) from the roll axis generates a roll moment for the vehicle body when the vehicle turns to the right. The position of the roll axis changes with a very complex mechanism depending on the kinematics of the suspension links. However, in this study, the relative position of the roll axis to the body is considered fixed to establish a simple and efficient model. Therefore, the roll moment \(M_{r}\) shown in Figure 2 is expressed as the product of the centrifugal force \(F\) applied to the vehicle and the distance \(h\) from the roll axis to the mass center, that is, \(M_{r}=Fh\). The sprung mass has mass \(m_{s}\), mass moment of inertia about the center of mass \(I_{x}\), and mass moment of inertia about the \(y\)-axis \(I_{y}\). The mass center of the sprung mass \(G\) is located at a distance \(l_{1}=1.07\) m from the front of the vehicle, \(l_{2}=1.579\) m from the rear, \(w_{1}=0.734\) m from the left end of the vehicle, and \(w_{2}=0.824\) m from the right end of the vehicle. The front unsprung masses with mass \(m_{f}\) have vertical displacements \(z_{1}\) and \(z_{2}\), while the rear unsprung masses with mass \(m_{r}\) have displacements \(z_{3}\) and \(z_{4}\).