## 3 Utilized Design Optimization Algorithms
### Grey Wolf Optimizer (GWO)
The problem has boundary constraints on the engine mount stiffness, constraints on the mount placement based on available space, symmetry requirements on the mount orientation, and lastly a constraint on the powertrain's center of gravity deformation owing to static weight. The objective function \(f_{w}\) is defined by Equation (22). Both \(f_{w}\) and, \(g_{j}\) are functions of the design parameters, \(k_{i},\ r_{i},\ \theta_{i}\). ## 5 Results and Discussion

In this section, the optimization findings are reported and discussed. The results of the optimization algorithms are exhibited and compared to each other to demonstrate that swarm intelligence optimization algorithms can effectively and reliably tackle highly non-linear optimization problems and complicated engineering challenges. Most notably, the two 12-DOF models' optimal design variables were successfully determined, and the transmitted force from the body/frame to the engine, and vice versa, was minimized. Mechanical vibrations were thus isolated and/or reduced. Each method undergoes a total of five computational optimization experiments to guarantee that appropriate and dependable results are obtained. In addition to the tolerance given in the method, the maximum number of iterations is set to 1000 iterations as a stopping condition for these algorithms. Table 1 lists the parameters for the GWO, GA, and SQP algorithms. Table 2 shows the engine mount's optimum design variables, as well as their upper and lower bounds. The achieved design optimization results for the first model, which solely addresses shaking loads, are shown in Tables 3 and 4. The achieved design optimization results for the second model that includes the combined loads are shown in Tables 5 and 6 (shaking and road loads). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Parameters** & **Value/Setting** \\ \hline \multirow{8}{*}{GWO} & Number of wolves & 30 \\ \cline{2-3}  & Max. number of iterations & 1000 \\ \cline{2-3}  & \(\overset{\leftarrow}{a}\) & Decreases linearly from 2 to 0 \\ \cline{2-3}  & \(\text{r}_{1}\) and \(\text{r}_{2}\) & Random numbers in [0, 1] \\ \cline{2-3}  & \(\overset{\leftarrow}{A}\) & A = 2. \(\overset{\leftarrow}{a}\). rand (\(\text{r}_{1}\))\(\overset{\rightarrow}{a}\) \\ \cline{2-3}  & \(\overset{\leftarrow}{C}\) & \(\overset{\leftarrow}{C}\) = 2. rand (\(\text{r}_{2}\)) \\ \hline \multirow{4}{*}{GA} & Population size & 200 \\ \cline{2-3}  & Crossover’s probability & 0.8 \\ \cline{2-3}  & Mutation probability & 0.1 \\ \hline \multirow{4}{*}{SQP} & Starting point & (475, 7500, 12, \(-9\), 0, \(-19\), \(-5\), 0.1, 50, 0.5, 25) \\ \cline{2-3}  & Max. number of iterations & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The set parameters of the employed algorithms. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Opt. Algorithm** & \multicolumn{5}{c}{**Natural Frequencies (Hz)**} \\ \hline GWO & 1.5047 & 10.2841 & 10.5974 & 11.4362 & 19.2999 & 27.9737 \\ (Un-Damped) & 36.2066 & 83.1606 & 103.0303 & 104.9282 & 200.5981 & 1271.338 \\ \hline GWO & 1.4327 & 10.2178 & 10.5918 & 11.4340 & 19.2926 & 27.9086 \\ (Damped) & 36.1198 & 82.3092 & 102.9964 & 104.9138 & 198.4054 & 1270.326 \\ \hline GA & 1.6681 & 9.9124 & 10.7731 & 16.7775 & 24.2217 & 25.1362 \\ (Un-Damped) & 42.3448 & 63.0191 & 103.6622 & 104.1913 & 200.5471 & 1271.340 \\ \hline GA & 1.5951 & 9.8888 & 10.6892 & 16.7717 & 24.1934 & 25.1023 \\ (Damped) & 42.1556 & 62.8106 & 103.6198 & 104.1895 & 198.3563 & 1270.328 \\ \hline SQP & 1.4887 & 8.86195 & 10.9074 & 11.04533 & 20.2400 & 23.4522 \\ (Un-Damped) & 44.8094 & 103.0313 & 104.7901 & 110.6138 & 200.6037 & 1271.304 \\ \hline SQP & 1.4093 & 8.7623 & 10.8985 & 11.0445 & 20.2197 & 23.4471 \\ (Damped) & 44.5790 & 102.9973 & 104.7789 & 108.0104 & 198.4111 & 1270.297 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimization results of natural frequencies (12-DOF model—shaking loads only).