# Analysis and Optimization of Driveline Bushing for Lateral Ride Vibration under Shock Excitation
## 4 Relationship between Seat Rail Vibration and Bushing Parameters
### Response Function  
Using the regression method, the relationships between the responses (VDVs at the driver seat rail in different directions) and the design parameters (\(k_{x\_FL},k_{y\_FL},k_{x\_FR},k_{y\_FR},\)\(k_{x\_RR}\) and \(k_{y\_RR}\)) were fitted by quadratic polynomials. The fitted response functions for the VDVs at the driver seat rail in the three translational directions are given as follows:  
\[\begin{array}{ll}\text{VDV}_{x\_FL}&=0.7870-0.0149k_{x\_FL}-0.0222k_{y\_FL}+0. 0586k_{x\_FR}+0.0048k_{y\_FR}+0.0012k_{x\_RR}+0.0425k_{y\_RR}\\ -0.0009k_{x\_FL}^{2}-0.0045k_{y\_FL}^{2}+0.0148k_{y\_FR}^{2}-0.0127k_{y\_FR}^{2} -0.0023k_{x\_RR}^{2}-0.0166k_{y\_RR}^{2}\\ +0.0145k_{x\_FL}k_{y\_PL}-0.0217k_{x\_FL}k_{y\_FR}+0.0158k_{x\_FL}k_{y\_FR}+0.0025 k_{x\_FL}k_{x\_RR}+0.0040k_{x\_FL}k_{y\_RR}\\ -0.0071k_{x\_FL}k_{x\_FR}+0.006k_{y\_FL}k_{y\_FR}+0.0057k_{y\_FL}k_{x\_RR}-0.0016 k_{y\_FL}k_{y\_RR}\\ -0.0117k_{x\_FR}k_{y\_FR}-0.0083k_{x\_FR}k_{x\_RR}+0.0042k_{x\_FR}k_{y\_RR}\\ +0.0066k_{x\_FR}k_{x\_FL}-0.0112k_{y\_FR}k_{y\_RR}-0.0008k_{x\_RR}k_{y\_RR}\\ \end{array} \tag{3}\]  
\[\begin{array}{ll}\text{VDV}_{y\_FL}&=0.623-0.210k_{x\_FL}-0.164k_{y\_FL}+0. 136k_{x\_FR}-0.158k_{y\_FR}-0.161k_{x\_RR}-0.181k_{y\_RR}\\ -0.0552k_{x\_FL}^{2}+0.0039k_{y\_FL}^{2}+0.0345k_{x\_FR}^{2}-0.0004k_{y\_FR}^{2 }+0.0316k_{x\_RR}^{2}+0.0803k_{y\_RR}^{2}\\ +0.0624k_{x\_FL}k_{y\_FL}+0.0301k_{x\_FL}k_{x\_FR}+0.0682k_{x\_FL}k_{y\_RR}+0.0 2039k_{x\_FL}k_{x\_RR}-0.0039k_{x\_FL}k_{y\_RR}\\ -0.0476k_{y\_FL}k_{x\_FR}+0.0340k_{y\_FL}k_{x\_FR}+0.0200k_{y\_FL}k_{x\_RR}+0. 0111k_{y\_FL}k_{y\_RR}\\ -0.0533k_{x\_FR}k_{y\_FR}-0.0089k_{x\_FR}k_{x\_RR}+0.0070k_{x\_FR}k_{y\_RR}\\ +0.0190k_{y\_FR}k_{x\_PL}+0.0106k_{y\_FR}k_{y\_RR}+0.0020k_{x\_RR}k_{y\_RR} \\ \end{array} \tag{4}\]  
\[\begin{array}{ll}\text{VDV}_{z\_FL}&=0.59-0.150k_{x\_FL}-0.130k_{y\_FL}-0.008 k_{x\_FR}-0.124k_{y\_FR}-0.083k_{x\_RR}-0.124k_{y\_RR}\\ +0.0229k_{x\_FL}^{2}+0.0224k_{y\_FL}^{2}+0.0105k_{x\_FR}^{2}+0.0246k_{y\_FR}^{2 }+0.0164k_{x\_RR}^{2}+0.0622k_{y\_RR}^{2}\\ +0.0316k_{x\_FL}k_{y\_FR}+0.0042k_{x\_FL}k_{x\_RR}+0.0284k_{x\_FL}k_{y\_RR}+0. 0167k_{x\_FL}k_{y\_RR}-0.0040k_{x\_FL}k_{y\_RR}\\ -0.0041k_{y\_FL}k_{x\_FR}+0.0260k_{y\_FL}k_{y\_FR}+0.0124k_{y\_FL}k_{x\_RR}+0. 0011k_{y\_FL}k_{y\_RR}\\ -0.0133k_{x\_FR}k_{y\_FR}-0.0035k_{x\_FR}k_{x\_RR}-0.0028k_{x\_FR}k_{y\_RR}\\ +0.0135k_{y\_FR}k_{x\_FL}+0.0012k_{y\_FR}k_{y\_RR}-0.0014k_{x\_RR}k_{y\_RR} \\ \end{array} \tag{5}\]  
A set of five random combinations of the six design parameters were generated to test the applicability of the fitted response functions, as listed in Table 2. The test samples were never used in the fitting procedure. The VDVs at the driver seat rail were obtained applying the test parameters into the MBD model, and the corresponding VDVs were also computed using the fitted response functions (Equations (3)--(5)), as given in Table 2.  
The relative errors between the DR VDVs obtained from the response functions and the MBD simulations were all less than 6%. The results show that the direct relationships obtained through the RSM gave good predictions of the VDV at DR, which means the response functions were suitable for the following optimization.