# Multi-Criteria Optimization of an Innovative Suspension System for Race Cars
## 4 Synthesis of the Proposed Solution
Equation (21) can be rewritten as:  
\[\rm X_{M}+r_{1}\cdot Y_{M}+r_{2}+Z_{M}+r_{3}=0 \tag{22}\]  
where \(\rm r_{1}\), \(\rm r_{2}\), and \(\rm r_{3}\) are given from the following equations:  
\[r_{1}=\frac{\cos\beta}{\cos\alpha},r_{2}=\frac{\cos\eta}{\cos\alpha},\ r_{3}=-(X _{N}+r_{1}\cdot Y_{N}+r_{2}\cdot Z_{N}) \tag{23}\]For an \"m\" number of finite positions of the point M, by subtracting the first equation (corresponding to the initial position) from the others, the Equation (22) becomes:  
\[\left[\left(X_{M}\right)_{k+1}-\left(X_{M}\right)_{1}\right]+r_{1}*\left[\left(Y _{M}\right)_{k+1}-\left(Y_{M}\right)_{1}\right]+r_{2}*\left[\left(Z_{M}\right)_ {k+1}-\left(Z_{M}\right)_{1}\right]=0,\ k=1,\ldots m-1 \tag{24}\]  
Equation (6) was used to determine the location of M\({}_{0}\) according to the algorithm for determining the center of a spherical surface. The orientation of the revolute joint axis (M\({}_{0}\)-M\({}_{0}\)-) was obtained by (24), which had a similar solution as (8). The presented algorithm was transposed in a computer program by using MATLAB. The actual numerical implementation was reflected in the global coordinates of the points M\({}_{0}\) in which the guiding rocker was connected to the chassis, along with an additional point M\({}_{0^{\prime}}\) by which the revolute joint axis of the rocker was defined, as follows:  
* The front wheel suspension mechanism (Figure 23): X\({}_{\text{M}0}=-323.9\), Y\({}_{\text{M}0}=245.9\), Z\({}_{\text{M}0}=42.9\); X\({}_{\text{M}0^{\prime}}=-318.2\), Y\({}_{\text{M}0^{\prime}}=250.4\), Z\({}_{\text{M}0^{\prime}}=-56.8\) (in mm);
* The rear wheel suspension mechanism (Figure 24): X\({}_{\text{M}0}=-282.1\), Y\({}_{\text{M}0}=152.4\), Z\({}_{\text{M}0}=-1698.3\); X\({}_{\text{M}0^{\prime}}=-267.5\), Y\({}_{\text{M}0^{\prime}}=154.4\), Z\({}_{\text{M}0^{\prime}}=-1599.4\) (in mm). Figure 22: Guidance of the interest point (M) on a semicircle. By the kinematic analysis of the proposed front and rear suspension mechanisms, approximately zero variations of the camber and caster angles were obtained. In addition, the variations of the wheelbase, wheel track, and bump steer angle for the suspension mechanisms with the guiding rocker shown in Figures 23 and 24 were very close to those corresponding to the five-bar mechanism-based suspension systems (the optimal curves in Figures 11-13, and, respectively, Figures 16-18). These very good correlations between results demonstrate the viability of replacing the fictive kinematic restrictions (by which the variations of the camber and caster angles were initially canceled) by the physical geometrical constraints associated with the guiding rockers. As mentioned, most Formula Student race cars use the double wishbone (four-bar) suspension mechanism (schematically represented in Figure 2b). This solution was also implemented on the initial race car developed at Transilvania University of Brasov (Brasov, Romania, BlueStreamline), on both the front and the rear axles, with which the single seater obtained good results in the competitions where it participated (on the circuits from Silverstone, Catalunya, and Varano). Prior to implementation, that double wishbone suspension was kinematically optimized by an algorithm based on DOE techniques and linear regression functions, as presented in previous authors' papers [31,32]. The proposed suspension system solution is an improvement of the classical double wishbone suspension system, by being able to decouple the mentioned contradictory kinematic variations (the issues in which the double wishbone suspension mechanism was deficient), thus improving the dynamic behavior of the suspension (car, in general).