## 4 Design Case
### Evaluation of Driving Performance  
The second stage of the design procedure presents the evaluation of driving performance of the MLTD powertrain with the specifications of M and transmission adapted from those of the Nissan Leaf, a commercial electric car [20]. The performance map of M is illustrated in Figure 17 with the rated maximum torque and rated power of 280 N-m and 80 kW, respectively. The values of some major design parameters of the MLTD powertrain are deliberately selected with practical consideration, as listed in Table 2. It should be noted that the values of \(k_{r}\) and \(k_{l}^{\prime}\) are to be so selected that the kinematic relation of Equation (11) must be satisfied. Now, the driving performance of the designed vehicle shown in Figure 18 can be calculated based on Equations (5), (12), and (24) with an additional transmission efficiency, \(\eta_{t}\), taken into account. The designed vehicle is capable of maintaining the cruise speed of 45 km/h with climbing hill at Grade \(=\) 24%, and reaching the maximum speed of 160 km/h. A conceptual 3D model of the designed MLTD based on the parameter setting in Table 2 Is created using computer graphics to help visualize the kinematic feature as shown in Figure 19. The image in Figure 19a with a view from right side reveals that a reduction ratio of GM (\(r_{m}=7.94\)) is given by the gear mesh between the pinion of the output shaft of M and the outer rim of the ring gear (\(r\)) of PR, and the driving torque to WR is transmitted \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline \(Z_{rr}\) & 120 \\ \hline \(Z_{sr}\) & 72 \\ \hline \(Z_{rl}\) & 132 \\ \hline \(Z_{sl}\) & 36 \\ \hline \(k_{r}\) & 1.67 \\ \hline \(k_{l}^{\prime}\) & \(-\)3.67 \\ \hline \(r_{m}\) & 7.94 \\ \hline \(\eta_{t}\) & 0.97 \\ \hline \end{tabular}
\end{table} 
Table 2: Values of major design parameters of the MLTD powertrain. Figure 17: Performance map of M.  
through the planetary carrier (\(c\)) of PR. Another view from right side in Figure 19b shows that the arrangement of the gear mesh between M and PL and transmission of driving torque between PL and WL are analogous to the descriptions for Figure 19a, except that PL is a type of double-pinion, not single-pinion like PR. Furthermore, DM acting as a torque vectoring device is mounted on a balance shaft interconnected with the sun gear (s) of PL and that of PR. The above driving performance is evaluated in a straight-driving condition, and cannot be used for sizing DM since DM is operated most of the time when cornering. Hence, the optimal cornering performances assessed in Section 4.1 are utilized as the operating conditions to find the corresponding operating points of M and DM. In particular, the specifications of DM in terms of rated maximum torque, rated power, and maximum rotational speed are to be determined with the performance map scaled from a practical motor of the Toyota Prius HEV [21]. First, the values of \(a_{y\_max}^{*}\) and \(\zeta_{lr}^{*}\) with respect to \(v\) in the range of 20 to 140 km/h are acquired based on the algorithm in Section 4.1. Since \(R=v^{2}/a_{y}\), the turning radius for the optimal cornering performance, i.e., \(R^{*}\), is obtained, and subsequently substituted into Equations (36)-(38) to calculate the values of \(\alpha_{r}^{*}\), \(\alpha_{r}^{*}\), and \(\delta^{*}\). Then, the values of \(R_{fc}^{*}\) and \(R_{rc}^{*}\) are computed via Equations (44) and (45). Finally, substituting the above results into Equations (41)-(44) yields the solutions of \(R_{rl}^{*}\), \(R_{rr}^{*}\), \(R_{rl}^{*}\), and \(R_{rr}^{*}\), as shown in Figure 20. It  
Figure 19: Conceptual model of MLTD using computer graphics viewed from (**a**) right side and (**b**) left side. Figure 18: Driving performance of the designed vehicle. is found that the differences in turning radius between two left wheels, i.e., front-left and rear-left wheels, are large at low vehicle speed, as seen in Figure 20a, and become smaller when vehicle speed gets higher, as indicated in Figure 20b.