## 3 Sensitivity Analysis and Genetic Algorithms for Design Optimization of the MRE-Based ATDVA
### Application of Sensitivity Analysis and Genetic Algorithms for Optimization of the MRE-Based ATDVA
The sensitivity analysis revealed that the sensitivity of design parameters was lowest for \(k\) (which was removed for that reason), highest for \(R_{S}\) and similar for \(a\), \(b\), \(c\), \(d\) and \(h\); accordingly, the selected design parameters included \(a\), \(b\), \(c\), \(d\), \(h\) and \(R_{S}\). Before determination of the design parameter \(k\), the mass of the dynamic vibration absorber was determined based on the lowest frequency of 110 Hz in the target range for vibration reduction and the MRE's stiffness at the time of 0 A application. The reason was that the MRE-based ATDVA would work to reduce the vibration at 110 Hz under the excitation of 0 A application, and the frequency for vibration reduction would increase with increasing MRE stiffness with the application of a current. Therefore, the MRE stiffness at the time of 0 A application and the lowest frequency of 110 Hz in the target range for vibration reduction were substituted into Equation (20) to determine the mass of the dynamic vibration absorber. \[k={(2\mu f)}^{2}m \tag{20}\]  
The mass calculated from Equation (20) was 1.55 kg, and design parameters \(a\), \(b\), \(c\), \(d\), \(h\) and \(R_{S}\) were calculated using the genetic algorithm. The value of \(k\) was determined by substituting the values of the design parameters into Equation (19), which represents a mass boundary condition of the MRE-based ATDVA. For the application of seven design parameters in Table 4 to the genetic algorithm, the parameters are binarized into the boundary conditions for Equation (19) and then expressed as a matrix. This is what we call the propagation process. The subsequent selection process calculates the magnetic field strength using Equation (14) and the probability of selection from the magnetic field strength. Genes with high selection probability are selected and left behind. This selection procedure is called roulette wheel selection. The crossover process determines crossover points randomly using a simple crossover to accelerate the rate of convergence and allows different genes to cross over at the random points. Finally, the mutation process mutates genes with a 1% probability for a global search effect. Through the processes described above, the genetic algorithm finds optimal design parameters for the MRE-based ATDVA to have a maximum magnetic field strength. For size constraints, \(R_{S}\) was set to be no larger than the MRE (\(R_{S}\leq 15\) mm) for a combination with the coil, and d was larger than the radius (\(>\)52.5 mm) of the coil. Because parameters \(b\) and \(c\) determine the cross-sectional areas of both magnetic paths, the larger their size, the greater the magnetic resistance. Therefore, the lower values are more favorable. The value of \(b\) was set to the smallest whole number larger than \(R_{S}\) because it influences the cross-sectional area of the lower magnetic path as well. The value of h was set to be greater than the coil's radius (\(h>52.5\) mm). Table 4 lists the parameters necessary for optimization and their optimized values. The optimization results showed that the magnetic field strength improved from 20.45 mT to 30.35 mT with \(a=120\) mm, \(b=37\) mm, \(c=29\) mm, \(d=80.5\) mm, \(h=95\) mm, \(k=300\) layers and \(R_{s}=13\) mm. The final configuration is shown in Figure 17. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Design Parameters** & **Initial Value** & **Optimized Value** \\ \hline \(a\) & 120 mm & 120 mm \\ \(b\) & 37 mm & 37 mm \\ \(c\) & 29 mm & 29 mm \\ \(d\) & 52.5 mm & 80.5 mm \\ \(h\) & 52.5 mm & 95 mm \\ \(k\) & 280 layer & 300 layer \\ \(R_{s}\) & 10 mm & 13 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Initial and optimized value of design parameters.