## 3 Sensitivity Analysis and Genetic Algorithms for Design Optimization of the MRE-Based ATDVA
### Derivation of Equations for the Magnetic Field and Mass of the MRE-Based ATDVA  
The shear modulus of MREs varies with the application of a magnetic field to the MRE-based ATDVA. For such shear modulus to be used as stiffness, the installation of MREs was designed to keep the plane of MRP vibration perpendicular to the direction of MRP orientation. The overall structure of the MRE-based ATDVA is as follows (Figure 12):  
The mass part of the MRE-based ATDVA is shown in Figure 13a. The mass part consisted of a coil and a core (Figure 13a). The MRE was made installable inside so that a magnetic field could flow through the MRE when applied to the coil, and a magnetic conductor was configured as a closed circuit for the flow of the magnetic field (Figure 13b). In addition, a joint was created for connection between the magnetic conductor and the mass part. The portion connected with the MRE was shaped to have a circular cross section so the flow of the magnetic field could be facilitated, and other magnetic conductors were formed into quadrangular shapes for the ease of fabrication. Magnetic conductors were made of SC10, which is characterized by high permeability as one of the lowest-carbon steels for structural applications in mechanical parts. They were configured as closed circuits to maximize the performance of applying a magnetic field, and their edges were filleted for prevention of fringing flux. Figure 11: Variation rate of shear modulus isotropic/anisotropic MRE. (**a**) Isotropic MRE, (**b**) Anisotropic MRE. Non-magnetic brass was used for the core of the mass part. The joint between the mass part and magnetic conductor was made of the carbon steel SC10 as for the magnetic conductors. The MRE was designed in a solenoid shape in combination with the coil so that it could be positioned in the center of the core where the magnetic field was strongest. The materials used for the MRE-based ATDVA are listed in Table 3.  
The magnetic field strength for application to the MRE can be derived theoretically as follows:  
Magnetomotive force is measured in ampere-turns (A-t) of coils and represents the force that produces a magnetic field in a magnetic circuit, as given by Equation (10) (Figure 14). \[F=N\cdot I \tag{10}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Coil** & **Core** & **Connector** & **Magnetic Conductor** \\ \hline Copper & Brass & SC10 & SC10 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Materials of ATMVA. Figure 12: Cross sectional-view of adaptive tuned MRE vibration absorber. Figure 13: Components of adaptive tuned MRE vibration absorber. **(a)** coil part, **(b)** magnetic conductor and connector. where \(F\) is the magnitude of magnetomotive force, which is expressed in ampere-turns (A-t); \(N\) is the number of coil turns; and \(I\) is the current (A) in the coil. The magnetic field strength is defined as the magnetomotive force per unit length in a magnetic field. The average magnetic field strength in the magnetic circuit section made up of homogeneous material is determined by dividing the magnetomotive force applied to the section by the section's effective length (I), as expressed by Equation (11). \[H=\frac{F}{I}=\frac{N\cdot I}{I} \tag{11}\]  
If the magnetic circuit has a uniform cross-sectional area, the magnetic field strength is equal throughout the magnetic circuit. If different cross-sectional areas or materials are used, the magnetic field strength varies from section to section. In Figure 14, the respective sides of the core can be considered as serially connected magnetic resistances, so if they are defined as \(1_{1}\), \(l_{2}\), \(l_{3}\) and \(l_{4}\) in the clockwise direction from the side on which the coil is wound, the magnetic resistance of the core can be expressed as shown in equation (12). \[R_{eq}=\frac{l_{1}}{\mu\pi R_{S}^{2}}+\frac{2l_{2}}{\mu bc}+\frac{l_{3}}{\mu b (h-d)} \tag{12}\]  
Therefore, the total magnetic flux passing through the magnetic circuit is given by Equation (13).