## 3 Sensitivity Analysis and Genetic Algorithms for Design Optimization of the MRE-Based ATDVA  
### Evaluation of Variations in Shear Modulus for MRE Design  
The MRE's shear stiffness was derived by measuring its shear modulus. The derived shear stiffness was used as the stiffness part of the MRE-based dynamic vibration absorber. The shear modulus was measured using the system presented by [PERSON] et al. [21]. The rate of variation of the MRE's shear stiffness was measured to design the stiffness part of the MRE-based ATDVA. The specimen of MRE was spherical in shape with a radius of 15 mm, so there was no magnetic field leakage at the time of magnetic-field excitation. For easy fabrication, the MRE was prepared in thicknesses of 2 mm, 4 mm, 6 mm, 8 mm and 10 mm, and the rate of variation of its shear modulus was measured for each thickness. Natural rubber was used as the base matrix, which was prepared by mixing MRPs and additives. The volume ratio of MRPs was set to 40 vol.% at which the rate of variation of shear modulus was highest [21]. Additives necessary for MRE mixing are listed in Table 2.  
Isotropic and anisotropic MREs were prepared through the following process (Figure 10):  
For fabrication of MREs, the natural rubber and different additives (Figure 10a) listed in Table 2 were mixed on a roll-mill. During the repeated mixing process on the roll-mill (Figure 10b), the primary compound of homogeneous MREs was prepared by mixing the natural rubber base matrix, MRPs and \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Matrix**} & **ZnO** & **Stearic Acid** & **CZ** & **Sulfur (Crosslink** \\  & **(Activator)** & **(Activator)** & **(Accelerator)** & **agent)** \\ \hline Natural Rubber & 5 phr & 2 phr & 0.8 phr & 2.5 phr \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Composition of MRE materials. Figure 9: Numerical simulation results of adaptive tuned vibration absorber. (**a**) Frequency response, (**b**) Time response (acceleration). additives. For its molecular chain to relax, the prepared primary compound was stabilized for 24 h at room temperature. After complete stabilization at room temperature, the isotropic MRE did not undergo any orientation process, and MRPs in the anisotropic MRE were oriented by application of a magnetic field (Figure 10c). Specimens were prepared by vulcanizing the MREs in a press for approximately 10 min under the conditions of 160 \({}^{\circ}\)C and 2000 psi (14.8 kPa) (Figure 10d). The following is how the shear modulus of an MRE is measured:  
The lower-middle part of a beam with both ends fixed is excited by 20-kHz white noise with a shaker, and the exciting force is measured using a force transducer. The MRE is placed between a magnetic-field application device (magnetic-flux generator) and an oscillator so that it is movable in the shear direction. With a current of 0-3 A applied to the magnetic flux generator, the transfer function is measured through an accelerator and the exciting force. The MRE's shear modulus is calculated by substituting the frequency of the measured transfer function's secondary eigenmode into Equation (9) as presented by [PERSON] et al. \[G_{mre}=\frac{2\pi^{2}f^{2}mh}{A} \tag{9}\]  
where \(f\) denotes the frequency of the measured secondary eigenmode, \(m\) is the mass of the oscillator, \(h\) is the thickness of the specimen, and \(A\) is the area of the specimen. The rate of variation of the prepared MRE's shear modulus was measured as follows (Figure 11):  
The rate of variation of shear modulus was greater in the anisotropic MRE than in the isotropic MRE and when their thickness was smaller. The anisotropic MRE is thought to offer greater resistance in the shear direction when a magnetic field is applied with MRPs clustered. During the fabrication of MREs, as their thickness increases, the magnetic field becomes more unlikely to pass through the MREs, so clusters are not formed properly.