## 3 Introduction to the Generalized Inverse Cascade Method
### Optimal Intervals for Design Variables Based on Generalized Inverse Calculation Solving  
Based on the performance orientation of this case vehicle, the weighted force transmission rate should be less than 1400 N-s\({}^{2}\)/m, and the weighted decoupling rate should be greater than 80% (the design requirements for these two objectives can vary for different vehicle performance preferences). In order to ensure that any combination of the design variables in the value interval can make the objective function meet the design requirements, a confidence level of 100% and a Pareto coefficient of 0.02 are set. Tables 3 and 4 show the Pareto optimal solution set #1 and the Pareto optimal solution set #2 for each design variable, which are solved by applying the generalized inverse cascade method, respectively. Theoretically, any combination of stiffness interval values can make the objective functions meet the design requirements of vibration isolation rate and the decoupling rate. However, it is impossible to exhaust all combinations for verification. A representative combination of stiffness values (as shown in Table 5) is selected and substituted into the ADAMS simulation model to calculate the force transmission rates and the decoupling rates. The results are shown in Tables 6 and 7. As can be seen from Table 6, the force transmission rate in all directions and the weighted force transmission rates of the mounting system have been reduced after the optimized design. The weighted force transmission rates for different stiffness values meet the target requirement of 100% less than 1400 N-s\({}^{2}\)/m, which improves the vibration isolation performance of the powertrain mounting system. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Lower Limit of** & **Upper Limit of** \\
**Design Variable** & **the Optimization Interval** & **the Optimization Interval** \\  & **(N/mm)** & **(N/mm)** \\ \hline S\({}_{\text{FL,X}}\) & 28 & 36 \\ S\({}_{\text{FL,Y}}\) & 39 & 51 \\ S\({}_{\text{FL,Z}}\) & 256 & 311 \\ S\({}_{\text{FR,X}}\) & 28 & 41 \\ S\({}_{\text{FR,Y}}\) & 40 & 50 \\ S\({}_{\text{FR,Z}}\) & 265 & 304 \\ S\({}_{\text{RR,X}}\) & 57 & 66 \\ S\({}_{\text{RR,Y}}\) & 203 & 213 \\ S\({}_{\text{RR,Z}}\) & 199 & 325 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Design variable optimal intervals—Pareto optimal solution set #2. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Lower Limit of** & **Upper Limit of** \\
**Design Variable** & **the Optimization Interval** & **the Optimization Interval** \\  & **(N/mm)** & **(N/mm)** \\ \hline S\({}_{\text{FL,X}}\) & 28 & 38 \\ S\({}_{\text{FL,Y}}\) & 36 & 49 \\ S\({}_{\text{FL,Z}}\) & 233 & 271 \\ S\({}_{\text{FR,X}}\) & 28 & 35 \\ S\({}_{\text{FR,Y}}\) & 35 & 48 \\ S\({}_{\text{FR,Z}}\) & 229 & 282 \\ S\({}_{\text{RR,X}}\) & 51 & 62 \\ S\({}_{\text{RR,Y}}\) & 172 & 207 \\ S\({}_{\text{RR,Z}}\) & 178 & 327 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Design variable optimal intervals—Pareto optimal solution set #1. As shown in Table 7, the decoupling degree of the main vibration modes for the first six orders of rigid body modes is enhanced compared to the original state. The weighted decoupling rates for all four combinations of stiffness values are above 80%, which meets the target requirement of a 100% confidence level and reduces the coupled vibration of the mounting system to a greater extent than the original weighted decoupling rate of 72.26%. The comparison from Tables 6 and 7 shows that typical combinations of parameters within the mount stiffness intervals, which are based on the generalized inverse cascade