## 3 Utilized Design Optimization Algorithms
### Grey Wolf Optimizer (GWO)
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Design Variable** & **LB** & **UB** \\ \hline Stiffness along x and y axes (lb.in) & 100 & 5000 \\ Stiffness on the z-axis (lb.in) & 500 & 10,000 \\ Location of mounts 1 \& 2 (x-axis) & 8 & 12 \\ Location of mounts 1 \& 2 (y-axis) & \(-9\) & \(-5\) \\ Location of mounts 1 \& 2 (z-axis) & \(-7\) & \(-3\) \\ Location of mounts 3\& 4 (x-axis) & \(-17\) & \(-11\) \\ Location of mounts 3 \& 4 (y-axis) & \(-10\) & \(-6\) \\ Location of mounts 3 \& 4 (z-axis) & \(-7\) & \(-3\) \\ Orientation of mount 1 (deg.) & 0 & 50 \\ Orientation of mount 2 (deg.) & 0 & 50 \\ Orientation of mount 3 (deg.) & 0 & 50 \\ Orientation of mount 4 (deg.) & 0 & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Upper and lower bounds of the design variables. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Opt. Algorithm** & \multicolumn{5}{c}{**Damping Coefficients**} \\ \hline GWO & 0.0166 & 0.0198 & 0.0257 & 0.0274 & 0.0327 & 0.0399 \\  & 0.0682 & 0.0692 & 0.1134 & 0.1427 & 0.1475 & 0.3057 \\ \hline GA & 0.0059 & 0.0263 & 0.0286 & 0.0399 & 0.0483 & 0.0519 \\  & 0.0690 & 0.0813 & 0.0944 & 0.1246 & 0.1474 & 0.2925 \\ \hline SQP & 0.021 & 0.0146 & 0.0209 & 0.0257 & 0.0397 & 0.0402 \\  & 0.0448 & 0.1013 & 0.1474 & 0.1495 & 0.2157 & 0.3233 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Optimization results of damping coefficients (12-DOF model—shaking loads only). The decoupled vibration modes for the two models utilizing different optimization strategies are shown in Figures 3-11. It is accomplished by minimizing the global stiffness matrix's off-diagonal terms. We guarantee that the diagonal stiffness matrix terms are the only dominant ones by doing so. This will ensure that the powertrain's rolling motion is kept to a minimum. At lower frequency values, vibration modes are not decoupled as intended. Meanwhile, the vibration modes are cleaner, i.e., when the vibration frequency is higher, and decoupling occurs as shown in Figures 3-8 for model 1 (shaking loads only) and Figures 9-14 for model 2 (combined loading). The modes were not totally separated for low-vibration frequencies, as shown in Figures 3 and 4 specifically when the frequency was in the range of 0-100 Hz except for few shapes where even at low frequency decoupling occurred Figure 3 (mode shape at frequency 10.3316 Hz) and Figure 5 (mode shape at frequency 25.1378 Hz). GA and SQP were also successful in decoupling the modes of vibration but at high computational cost and not forgetting the values of the transmitted force that were not optimal compared to those obtained by GWO. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Design Variables** & **Opt. Alg.** & **Mount 1** & **Mount 2** & **Mount 3** & **Mount 4** \\ \hline \multirow{3}{*}{Optimized Orientation (deg)} & GWO & (0.7, 0.0012, 0) & (\(-0.7\), \(-0.0012\), 0) & (0.0002, 0.017, 0) & (\(-0.0002, -0.017, 0) \\ \cline{2-6}  & GA & (50, 106, 0) & (\(-50\), \(-10.6\), 0) & (50, 108, 0) & (\(-50\), \(-108, 0) \\ \cline{2-6}  & SQP & (0.0005, 3.2, 0) & (\(-0.0005\), \(-3.2\), 0) & (0.00002, 0.0006, 0) & (\(-0.00002, -0.00006, 0) \\ \hline \multirow{3}{*}{Optimized Position (in)} & GWO & (\(8\), \(-5\), \(-3\)) & (\(8\), \(-5\), \(3\)) & (\(-12.3\), \(-7.3\), \(-5\), \(2\)) & (\(-12.3\), \(-7.3\), \(5\), \(2\)) \\ \cline{2-6}  & GA & (11.265, \(-5\), \(-3.0024\)) & (11.265, \(-5\), \(3.0024\)) & (\(-11.036\), \(-6\), \(-3\)) & (\(-11.036\), \(-6\), \(3\)) \\ \cline{2-6}  & SQP & (\(8\), \(-5\), \(-7\)) & (\(8\), \(-5\), \(7\)) & (\(-17\), \(-6\), \(-3\)) & (\(-17\), \(-6\), \(3\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Optimized mounts’ position and orientation (12-DOF model—shaking loads only). The best design variables and objective function for the 12-DOF model considering solely the shaking loads are shown in Tables 3-5. Table 3 reports the optimized mount stiffness along each axis. Tables 4 and 5 report the damped and un-damped natural frequencies and damping coefficients, respectively.