## 4 Vehicle Dynamics Model
### Half-Car Dynamics Model
#### 4.1.3 4-DOF Roll Dynamics Model  
As shown in Figure 5, 4-DOF considered in this model are vertical translation of the sprung mass denoted by \(z_{b}\), roll motion of sprung mass denoted by \(\varphi\) and vertical translation of left and right unsprung mass denoted by \(z_{w1}\) and \(z_{w2}\), respectively. Vertical force balance on sprung mass and two unsprung masses can be expressed as Equations (24)-(26). \[m_{s}\ddot{z}_{b}=F_{s1}+F_{s2} \tag{24}\]  
\[m_{u1}\ddot{z}_{w1}=-F_{s1}-k_{t1}(z_{w1}-z_{r1}) \tag{25}\]  
\[m_{u2}\ddot{z}_{w2}=-F_{s2}-k_{t2}(z_{w2}-z_{r2}) \tag{26}\]  
where \(F_{s1}\) and \(F_{s2}\) are forces on both sides (left and right) of the suspension. Roll dynamics of sprung mass are developed as follows. \[(I_{x}+m_{s}h_{r})\ddot{\phi}=m_{s}(a_{y}\cos\phi+g\sin\phi)h_{r}+\frac{B}{2} (F_{s1}-F_{s2}) \tag{27}\]  
where \(I_{x}\) is moment of inertia along \(X\) axis; \(a_{y}\) is lateral acceleration; \(g\) is gravitational acceleration; Calculation is shown in Section 4.1.2. Dynamics suspension forces are given in Equations (28) and (29). \[F_{s1}=-k_{s1}\bigg{(}z_{b}-z_{w1}-\frac{B\sin\phi}{2}\bigg{)}-c_{p1}\bigg{(} \dot{z}_{b}-\dot{z}_{w1}-\frac{B\dot{\phi}\cos\phi}{2}\bigg{)} \tag{28}\]  
\[F_{s2}=-k_{s2}\bigg{(}z_{b}-z_{w2}+\frac{B\sin\phi}{2}\bigg{)}-c_{p2}\bigg{(} \dot{z}_{b}-\dot{z}_{w2}+\frac{B\dot{\phi}\cos\phi}{2}\bigg{)} \tag{29}\]  
Dynamics tire forces are given by Equations (30) and (31). \[F_{tl}=-k_{t1}(z_{w1}-z_{r1}) \tag{30}\]  
\[F_{tr}=-k_{t2}(z_{w2}-z_{r2}) \tag{31}\]  
where \(F_{tl}\) and \(F_{tr}\) are forces on left and right sides of the tire, respectively. Assuming \(c_{p1}=c_{p2}=c_{p}\) and \(k_{s1}=k_{s2}=k_{s}\), and substituting Equations (28) and (29) into Equation (27), overall equations of motion are obtained as follows. \[ \begin{array}{rl}m_{s}\ddot{z}_{s}=&-k_{s2}\Big{(}z_{b}-z_{w2}+\frac{B\sin \phi}{2}\Big{)}-c_{p2}\bigg{(}\dot{z}_{b}-\dot{z}_{w2}+\frac{B\dot{\phi}\cos \phi}{2}\bigg{)}\\ &-k_{s1}\Big{(}z_{b}-z_{w2}-\frac{B\sin\phi}{2}\Big{)}-c_{p1}\bigg{(}\dot{z}_ {b}-\dot{z}_{w1}-\frac{B\dot{\phi}\cos\phi}{2}\bigg{)}\end{array}  \tag{32}\]  
\[ \begin{array}{rl}m_{w1}\ddot{z}_{w1}=&k_{s1}\Big{(}z_{b}-z_{w2}-\frac{B\sin \phi}{2}\Big{)}+c_{p1}\bigg{(}\dot{z}_{b}-\dot{z}_{w1}-\frac{B\dot{\phi}\cos \phi}{2}\bigg{)}\\ &-k_{t1}(z_{w1}-z_{r1})\end{array}  \tag{33}\]  
\[ \begin{array}{rl}m_{w2}\ddot{z}_{w2}=&-k_{s2}\Big{(}z_{b}-z_{w2}+\frac{B\sin \phi}{2}\Big{)}-c_{p2}\bigg{(}\dot{z}_{b}-\dot{z}_{w2}+\frac{B\dot{\phi}\cos \phi}{2}\bigg{)}\\ &-k_{t2}(z_{w2}-z_{r2})\end{array}  \tag{34}\]\[ \begin{array}{ll}(I_{x}+m_{s}h_{R})\ddot{\phi}=&m_{s}(a_{y}\cos\phi+g\sin\phi)h_{ R}-\frac{k_{s}B^{2}}{2}\sin\phi\\ &-\frac{c_{p}B^{2}}{2}\dot{\phi}\cos\phi\ +\frac{k_{s}B}{2}(z_{w}-z_{r})+\frac{c_{p}B}{2} \big{(}\dot{z}_{w}-\dot{z}_{r}\big{)}\end{array}  \tag{35}\]  
where \(\ddot{z}_{u1}\) and \(\ddot{z}_{u2}\) are acceleration on left and right sides of the suspension, respectively; \(\dot{z}_{w}-\dot{z}_{r}\) is rattle space velocity between velocities of unsprung mass and road profile height. Total tire forces on the ground include both static forces due to weight and dynamic force. Including both static and dynamic forces, total tire forces on the ground are given by  
\[F_{zl}=\frac{m_{s}}{2}g+m_{u1}g-k_{t1}(z_{w1}-z_{r1}) \tag{36}\]  
\[F_{zr}=\frac{m_{s}}{2}g+m_{u2}g-k_{t2}(z_{w2}-z_{r2}) \tag{37}\]  
Since vertical forces \(F_{zl}\) and \(F_{zr}\) are equal to vertical forces on the tire, i.e., \(F_{tl}\) and \(F_{tr}\), Equation (16) can also be expressed as follows. \[ \begin{array}{ll}\mathrm{RI}_{half}=\frac{m_{u2}\ddot{z}_{w2}+F_{s2}+m_{u2}g- m_{u1}\ddot{z}_{w1}-F_{s1}-m_{u1}g}{m_{u2}\ddot{z}_{w2}+F_{s2}+m_{u2}g+m_{u1} \ddot{z}_{w1}+F_{s1}+m_{u1}g}\end{array}  \tag{38}\]  
Considering \(m_{u1}=m_{u2}=m_{u}\), Equation (34) can be further simplified. By substituting Equations (31)-(34) into Equation (38), simplified \(\mathrm{RI}_{half}\) equation can be obtained as follows [16]. \[