# Dynamic Modeling and Analysis of a Driving Passenger Vehicle
## 2 Dynamic Vehicle Model
Among the generalized coordinates, the three coordinates (\(Z\), \(\theta\), and \(\varphi\)) are for the motion of the sprung mass, whereas the four coordinates (\(z_{1}\), \(z_{2}\), \(z_{3}\), \(z_{4}\)) are for the motions of the unsprung masses. First, the kinetic energy of the dynamic vehicle model can be expressed as the following equation:  
\[T=\frac{1}{2}m_{s}\dot{Z}^{2}+\frac{1}{2}l_{x}\dot{\theta}^{2}+\frac{1}{2}l_{ y}\dot{\varphi}^{2}+\frac{1}{2}m_{f}\biggl{(}\dot{z}_{1}^{2}+\dot{z}_{2}^{2} \biggr{)}+\frac{1}{2}m_{f}\biggl{(}\dot{z}_{3}^{2}+\dot{z}_{4}^{2}\biggr{)} \tag{11}\]  
Figure 3: Front view of the dynamic model of the vehicle. where the first term is for the translational motion of the sprung mass, the second and third terms are for the rotational motion of the sprung mass, and the fourth and fifth terms are for the translational motions of the unsprung masses. Next, the potential energy owing to the deformations of the springs and torsional springs in the dynamic vehicle model may be given by  
\[V=\frac{1}{2}k_{t}\sum_{i=1}^{4}z_{i}^{2}+\frac{1}{2}k_{f}\left(d_{1}^{2}+d_{2} ^{2}\right)+\frac{1}{2}k_{r}\left(d_{3}^{2}+d_{4}^{2}\right)+\frac{1}{2}\kappa_ {f}\phi_{f}^{2}+\frac{1}{2}\kappa_{f}\phi_{r}^{2} \tag{12}\]  
where the first term is the potential energy of the tire, the second and third terms are the potential energies of the suspensions, and the fourth and fifth terms are the potential energies of the anti-roll bars. In the second and third terms, \(d_{1}\), \(d_{2}\), \(d_{3}\), and \(d_{4}\) are the suspension deformations at the left front wheel, right front wheel, left rear wheel, and right rear wheel, respectively. The suspension deformation at each wheel, \(d_{i}\) for \(i=1\), \(2\), \(3\), \(4\), is determined by subtracting the displacement of the unsprung mass from the \(z\)-directional displacement of the sprung mass at the point corresponding to each suspension. The displacement of the sprung mass at the left-front suspension, \(\mathbf{x}_{1}\), displacement at the right-front suspension, \(\mathbf{x}_{2}\), displacement at the left- rear suspension, \(\mathbf{x}_{3}\), and displacement at the right- rear suspension, \(\mathbf{x}_{4}\), can be determined by  
\[\mathbf{x}_{i}=\mathbf{x}_{G}+\mathbf{T}_{\varphi}\mathbf{T}_{\theta}\mathbf{ P}_{i}\text{ for }i=1,\ 2,\ 3,\ 4 \tag{13}\]  
where \(\mathbf{x}_{G}\) is the displacement of the mass center of the sprung mass, \(\mathbf{P}_{i}\) is the position vector of the point on the spring mass corresponding to each suspension, and \(\mathbf{T}_{\theta}\) and \(\mathbf{T}_{\varphi}\) are the rotation matrices about the \(x\) and \(y\) axes, respectively. The vectors and matrices in Equation (13) are given by  
\[\mathbf{x}_{G}=\left\{ \begin{array}{c}0\\ 0\\ Z\end{array} \right\}\text{, }\mathbf{P}_{1}=\left\{ \begin{array}{c}-l_{1}\\ -w_{1}\\ 0\end{array} \right\}\text{, }\mathbf{P}_{2}=\left\{ \begin{array}{c}-l_{1}\\ w_{2}\\ 0\end{array} \right\}\text{, }\mathbf{P}_{3}=\left\{ \begin{array}{c}l_{2}\\ -w_{1}\\ 0\end{array} \right\}\text{, }\mathbf{P}_{4}=\left\{ \begin{array}{c}l_{2}\\ w_{2}\\ 0\end{array} \right\} \tag{14}\]  
\[\mathbf{T}_{\theta}=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos\theta&\sin\theta\\ 0&-\sin\theta&\cos\theta\end{array} \right]\text{, }\mathbf{T}_{\varphi}=\left[ \begin{array}{ccc}\cos\varphi&0&-\sin\varphi\\ 0&1&0\\ \sin\varphi&0&\cos\varphi\end{array} \right] \tag{15}\]  
In other words, the suspension deformations were obtained by subtracting the displacements of the corresponding unsprung mass \(z_{i}\) from the \(z\)-directional components of \(\mathbf{x}_{i}\). Consequently, the suspension deformations can be expressed as  
\[ \begin{array}{l}d_{1}=Z-z_{1}-w_{1}\sin\theta\cos\varphi+l_{1}\sin\varphi \text{, }d_{2}=Z-z_{2}+w_{2}\sin\theta\cos\varphi+l_{1}\sin\varphi\text{, }\\ d_{3}=Z-z_{3}-w_{1}\sin\theta\cos\varphi-l_{2}\sin\varphi\text{, }d_{4}=Z-z_{4}+w_{2}\sin\theta\cos\varphi-l_{2}\sin\varphi\end{array}