# Multi-Criteria Optimization of an Innovative Suspension System for Race Cars
## 3 The Kinematic Optimization of the Suspension Mechanism
With the purpose of identifying the DOE investigation strategy that provided the optimum results from the point of view of the regression functions accuracy, several investigation strategies were tested, namely: DOE screening (level 2)--linear--Plackett Burman; DOE screening (level 2)--linear--fractional factorial; DOE response surface--Linear--Latin hypercube; DOE response surface--interactions--D-optimal; DOE screening (level 2)--interactions--D-optimal [41]. The results thus obtained highlighted that the first 4 techniques mentioned above could be used under the condition of refining the regression models, but the DOE screening (level 2)--interactions--D-optimal strategy could be used without the need for regression refining. Therefore, the optimization study of the suspension mechanism was based on this strategy. The regression model generated by the selected strategy allowed us to consider the interaction effects between factors, which were captured through terms in the model that consisted of products of factors, as presented in Equation (3). The D-Optimal design created a model that minimized the uncertainty of the coefficients, this being characterized by its flexibility, allowing the total number of runs in an experiment to be specified, supplementing with runs from other experiments, and indicating different levels for each factor. Based on the fact that all the evaluation parameters fell within acceptable limits, the regression functions generated by this model were valid (accurate), without any changes. As mentioned above, the kinematic optimization purpose was to minimize the three responses, corresponding to the wheel track, wheelbase, and bump steer variations. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **X (mm)** & **Y (mm)** & **Z (mm)** \\ \hline A & DV\_1 & DV\_2 & DV\_3 \\ B & DV\_4 & DV\_5 & DV\_6 \\ C & DV\_7 & DV\_8 & DV\_9 \\ D & DV\_10 & DV\_11 & DV\_12 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Front axle suspension system design variables. Figure 10: Quarter-car model of the front wheel suspension system. The algorithm used in the kinematic optimization of the suspension system was OptDes-generalized reduced gradient (GRG), provided with ADAMS/Insight, which required that factors have range limits, since it worked in scaled space [42]. This algorithm used the MinTo function, which constrained the response (goal) to be as close to the target value as possible (\"0\"). The optimum values of the design variables were obtained by running the tuning to the \"0\" variations of the targets. These values were as follows (in [mm]): DV_1 = \(-\)230.37, DV_2 = \(-\)1.5, DV_3 = 68.37, DV_4 = \(-\)233.5, DV_5 = \(-\)1.5, DV_6 = \(-\)319.43, DV_7 = \(-\)257.64, DV_8 = 254.89, DV_9 = 78.21, DV_10 = \(-\)248.5, DV_11 = 61.5, DV_12 = 175.01. With these values, through the kinematic analysis performed in ADAMS/View, the variations of the interest parameters for the optimal (after optimization) suspension mechanism were obtained (Figures 11-13). To compare, the results corresponding to the initial mechanism (before optimization) are also presented, noting a significant improvement of all the parameters, which demonstrated the viability/usefulness of the implemented optimization algorithm. Figure 11: Wheelbase variation (initial—red, and optimal—blue). Figure 12: Wheel track variation (initial—red, and optimal—blue). Figure 13: Bump steer variation (initial—red, and optimal—blue). According to the optimization flowchart shown in Figure 8, the next step after the kinematic optimization of the suspension mechanism consisted of determining the location of a point (M) on the upper control arm whose trajectory during the simulation (depending on the imposed wheel vertical travel) was close to a convenient circular or spherical one, so that later it would be easy to determine the center (M0) of this trajectory on the chassis (for notations, see the schematic planar model in Figure 7), as presented in \"Section 4\" of the work.