## 3 Steady-State Models of Vehicle Dynamics for Lateral Torque Vectoring
Here, the roll effect of springs in the suspension system on the left-right weight transfer is considered. The geometry of roll axis of the suspension system is illustrated in Figure 9, where \(h_{s}\) is the distance in vertical direction between G and the roll axis, \(h_{r}\) is the height of rear roll center, and \(h_{f}\) is the height of the front roll axis. Furthermore, the free-body diagrams of the sprung mass, front axle, and rear axle are shown in Figures 10 and 11a,b, respectively, where \(g\), \(F\), \(M\), \(S\), and \(\phi\) indicate gravitational acceleration, force, moment, distance between left and right springs, and roll angle with the subscript \(s\), \(f\), and \(r\) denoting the springs, front axle, and rear axle, respectively, and, moreover, \(fl\), \(fr\), \(rl\), and \(rr\) representing front-left, front-rear, rear-left, and rear-rear, respectively. Specifically, \(F_{rcf}\) and \(F_{rcr}\) are the forces acting on the front and rear roll centers. Figure 7: Effect of left–right torque vectoring for two-axle model. Now, summing the moments about the center of gravity G with respect to the free-body diagram of the sprung mass in Figure 8 gives an equation of motion written as  
\[-ma_{y}h_{s}\text{cos}\phi=mgh_{s}\text{sin}\phi+\left(F_{sf}-F_{sf}\right)\frac {S_{f}}{2}\text{cos}\phi+\left(F_{srl}-F_{srr}\right)\frac{S_{r}}{2}\text{cos} \phi-\left(M_{sf}+M_{sr}\right) \tag{27}\]  
\[F_{yf}l_{f}-F_{yr}l_{r}+M_{g}=0 \tag{22}\]0 Some definitions of forces used in four-wheel vehicle model. \[F_{yf}l_{f}-F_{yr}l_{r}+M_{g}=0 \tag{22}\]1 Free-body diagram of the sprung mass of vehicle. \[F_{yf}l_{f}-F_{yr}l_{r}+M_{g}=0 \tag{22}\]2 Geometry of roll axis of the suspension system. By assuming that \(\sin\!\phi\approx\phi\) and \(\cos\!\phi\approx 1\), and using a model for describing the characteristics of the suspension springs, Equation (27) can be arranged to an equation indicating the relation of roll angle \(\phi\) and lateral acceleration \(a_{y}\) shown below:  
\[F_{yf}l_{f}-F_{yr}l_{r}+M_{g}=0 \tag{22}\]3  
where \(K_{\phi f}\) and \(K_{\phi r}\) are the roll stiffness of front and rear axles defined as  
\[F_{yf}l_{f}-F_{yr}l_{r}+M_{g}=0 \tag{22}\]4  
\[F_{yf}l_{f}-F_{yr}l_{r}+M_{g}=0 \tag{22}\]5  
where \(K_{stabf}\) and \(K_{stabr}\) are the roll stiffness of front and rear stabilizers, respectively; and \(K_{sf}\) and \(K_{sr}\) the spring stiffness of front and rear springs, respectively. Now, by summing the moments about roll center of front axle with respect to the free-body diagram of front axle in Figure 9, the equation of motion is written as  
\[F_{yf}l_{f}-F_{yr}l_{r}+M_{g}=0 \tag{22}\]6  
Connecting with Equations (22) and (23) and defining the weight transfer of the front axle \(\Delta F_{zf}\) and that of the rear axle \(\Delta F_{zr}\),we obtain  
\[F_{yf}l_{f}-F_{yr}l_{r}+M_{g}=0 \tag{22}\]7  
\[F_{yf}l_{f}-F_{yr}l_{r}+M_{g}=0 \tag{22}\]8  
Furthermore, by summing the pitch moment about the center of gravity G and rearranging, the relations between left-right weight transfer and lateral acceleration due to roll moment are established and expressed as  
\[F_{yf}l_{f}-F_{yr}l_{r}+M_{g}=0 \tag{22}\]9  
\[F_{yf}+F_{yr}=ma_{y} \tag{23}\]0  
\[F_{yf}+F_{yr}=ma_{y} \tag{23}\]1 Free-body diagram of the (**a**) rear axle and (**b**) front axle of vehicle. As the maximum lateral acceleration is obtained with respect to various driving conditions and lateral torque vectoring ratio, it is imperative to estimate the rotational speed of each wheel in order that the operating states for M and DM can be properly predicted. Thereby, sizing for M and DM can be proceeded at the preliminary stage of this conceptual design. First, the two-axle model shown in Figure 12 is employed to calculate the slip angle of front and rear wheels [19]. \(\alpha_{f}\) and \(\alpha_{r}\) are as expressed below:  
\[F_{yf}+F_{yr}=ma_{y} \tag{23}\]2  
\[F_{yf}+F_{yr}=ma_{y} \tag{23}\]3  
where \(C_{af}\) and \(C_{ar}\) are the rolling stiffness of front and rear wheels, respectively, in N/rad.