# Multi-Criteria Optimization of an Innovative Suspension System for Race Cars
## 3 The Kinematic Optimization of the Suspension Mechanism
In this regard, the trajectories of several points on the upper arm were monitored, finally choosing the point that generated more convenient trajectory, which had the following global coordinates (in its initial position) (in [mm]): \(X_{\text{M}}=-432.8\), \(Y_{\text{M}}=274.8\), \(Z_{\text{M}}=38\). The trajectory (\(\Gamma\)) of this point during the kinematic analysis of the front wheel suspension mechanism, which was controlled by the kinematic restriction \(Y_{\text{K}}=25\)-sin(time), can be shown in Figure 14.  
For the rear wheel suspension system, the MBS model was also designed in ADAMS/View (Figure 15). The optimization purpose was, as in the case of the front wheel suspension, to minimize the wheel track, wheelbase, and bump steer variations, while the camber and caster angles variations were canceled in this stage with the help of kinematic restrictions. As for the front suspension system, points A and B were considered as the centers of two spherical joints that link the lower control arm (1) to the chassis, point C was the link between the rocker (5) and the chassis, and point D represented the center of the spherical joint between the toe arm (6) and the chassis. The initial values of the design variables were (in [mm]): DV_1 = \(-341.42\), DV_2 = \(-50.0\), DV_3 = \(-1310.74\), DV_4 = \(-319.62\), DV_5 = \(-50.0\), DV_6 = \(-1617.95\), DV_7 = \(-344.97\), DV_8 = \(211.15\), DV_9 = \(-1743.65\), DV_10 = \(-213.76\), DV_11 = \(54.01\), DV_12 = \(-1720.29\) (with correlations in notations similar to those in Table 1). As mentioned, in the rear wheel suspension optimization process, the same investigation strategy as for the front wheel suspension was used, namely DOE screening (level 2)--interactions--D-optimal. The optimization algorithm used was OptDes-GRG,  
Figure 14: The interest point trajectory during the kinematic analysis of the front suspension. Figure 15: Quarter-car model of the rear wheel suspension system. with the MinTo operator. By running the optimization process, the following optimal values of the design variables were obtained (in [mm]): DV_1 = \(-\)381.42, DV_2 = \(-\)90.0, DV_3 = \(-\)1270.74, DV_4 = \(-\)279.62, DV_5 = \(-\)10.0, DV_6 = \(-\)1657.95, DV_7 = \(-\)384.97, DV_8 = 171.15, DV_9 = \(-\)1783.65, DV_10 = \(-\)173.76, DV_11 = 14.01, DV_12 = \(-\)1760.29. Through the kinematic analysis performed in ADAMS/View, the variation diagrams shown in Figures 16-18 were obtained, for the initial (before optimization) and optimal (after optimization) suspension mechanisms, observing (as in the case of the front suspension) significant reductions in the monitored parameters variations. Further, in a similar way to those presented for the front wheel suspension, the most convenient trajectory (\(\Gamma\)) generated by a point (M) located on the upper arm (3) of the rear wheel suspension mechanism was selected, as shown in Figure 19, where the interest point had the following global coordinates (in the initial position) (in [mm]): XM = \(-\)525.9, YM = 225.9, ZM = \(-\)1663.8. For the chosen trajectory, the global coordinates of the center point M\({}_{0}\) on the chassis were determined according to the synthesis method presented in the next section of the paper. Figure 16: Wheelbase variation (initial—red, and optimal—blue). Figure 17: Wheel track variation (initial—red, and optimal—blue). Figure 10:0 Bump steer variation (initial—red, and optimal—blue). Concluding, it was observed that with the canceling of the camber and caster angle variations (which in this stage were controlled by kinematic restrictions), the wheelbase, wheel track, and bump steer angle variations were significantly reduced, which was not possible in the case of the classic four-bar suspension mechanism. As a result of the implemented optimization procedure, the variations of all kinematic parameters were within the accepted limits for the Formula Student single seater, both for the front and rear wheels, with a positive effect on the ride and handling performance of the car.