## 3 Evaluation of Mechanical Properties of Two Sub-Mounts in MIMM
In the case of high-stiffness conditions, Case I-(c) and -(d), the original stiffness \(k_{m}\) is much higher than the stiffness \(k_{p}\) such that the two masses, \(M_{M}\) and \(M_{B}\), behave as a rigid body compared to the target electric power plant \(M_{p}\). In addition, \(M_{B}\) was more than ten times larger than \(M_{p}\) and \(M_{M}\). Therefore, the optimal cases, Cases I-(c) and-(d), can be assumed to be two 2-d.o.f. system models with a small mass \(M_{p}\) and a large mass \(M_{M}\)+\(M_{B}\) with a loose connection (small stiffness and damping coefficient) between them. This result differed from that reported in an earlier study [24]. The complete performance indices are summarized in Table 5, and two performance indices (first and second) are shown in Figure 4. The proposed method can significantly decrease the performance \begin{table}
\begin{tabular}{c c c c c} \hline  & **Case I** & **Case II** & **Case III** & **Case IV** \\ \hline Case (a) & 0.11 & 1059.0 & 0.73 & 1059.0 \\ Case (b) & 1.31 & 13,124.0 & 8.67 & 13,132.0 \\ Case (c) & 0.07 & 702.0 & 0.39 & 702.0 \\ Case (d) & 0.07 & 699.0 & 0.39 & 699.0 \\ \hline \end{tabular}
\end{table} 
Table 4: Simulation results for sixteen cases ignoring \(I_{r2}(\omega_{1})\). \begin{table}
\begin{tabular}{c c c c c} \hline  & **Case I** & **Case II** & **Case III** & **Case IV** \\ \hline Case (a) & 0.24 & 1071.0 & 3.97 & 1073.0 \\ Case (b) & 1.44 & 13,136.0 & 12.09 & 13,145.0 \\ Case (c) & 0.20 & 715.0 & 3.63 & 715.0 \\ Case (d) & 0.20 & 711.0 & 3.63 & 715.0 \\ \hline \end{tabular}
\end{table} 
Table 3: Simulation results for sixteen cases. indices by simultaneously assigning two sub-mounts instead of the restricted lower sub-mount scenario. Based on the optimization results derived from the suggested method, all index values were below one, underlining the potential for efficient vibration control using the MIMM across the entire frequency range of interest. Under the same evaluation criterion with two performance indices, increasing the stiffness coefficient, and decreasing the damping coefficient (equivalent to Case II), is best for the lower sub-mount with fixed mechanical properties in the upper sub-mount. The best condition of the lower sub-mount was changed to Case I, considering the variations in the mechanical properties of the upper sub-mount. The calculation of the total performance index decreased considerably when the mechanical properties of the upper sub-mount were adjusted, as listed in Table 2. Therefore, the partial optimum of the MIMM cannot guarantee the global optimum of the MIMM, such that the total tuning of mechanical properties in sub-mounts can achieve the reliable control of vibrations from two different excitation conditions. Hence, the design guidelines for the MIMM can be summarized as follows: First, the stiffness and damping coefficients of the upper sub-mount should be kept low to hinder the transfer of excitation between the supporting system and mass block. Moreover, the static stiffness of the upper sub-mount should be sufficiently robust to uphold the heavy-weight target system at an appropriate height. Second, the stiffness coefficient of the lower sub-mount should be increased without any limitations on the damper characteristics. Generally, elastomers or rubber materials are employed to isolate  
Figure 4: Comparison of the performance indices: (**a**) first, (**b**) second. \begin{table}
\begin{tabular}{c c} \hline \hline
**Optimization Target** & \(\mathbf{I^{T}}\) \\ \hline Lower sub-mount only (Previous study) & 8.2 \\ Upper + Lower sub-mounts (Proposed) & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of the total performance indices for two optimization cases. the mass block from the structure at the floor level with a high stiffness coefficient, which is crucial for the selection of an appropriate lower sub-mount. Third, a simulation was conducted for the supporting system, ensuring that there was no resonance frequency within the frequency range of interest (1-120 Hz).