## 4 Electromagnetic Analysis of the Proposed Damper
### Analysis Conditions  
The drive part of the proposed damper performs translational motion, and the non-linear material properties of the permanent magnet should be considered. Additionally, the magnetic field due to the effect of the induced current on the surface should be considered. Therefore, a transient analysis was  
Figure 9: Regenerative hybrid electrodynamic damper (RHED) model: (**a**) damper shape; (**b**) finite element analysis model of the damper; (**c**) connected circuit.  
performed. The geometry of the damper's finite-element model was modeled in two dimensions and rotated 360\({}^{\circ}\) around the \(z\)-axis.  
The neodymium used as the permanent magnet of the proposed damper was N45M-20C grade and from Arnold Magnetic Technologies Corp. (Rochester, NY, USA). The demagnetization curve indicating the non-linear relative permeability is shown in Figure 10. Other important non-linear attributes are presented in Table 4.  
By placing pole pieces made of pure iron at the boundary between the magnets and at both ends of the driver, the magnetic path loss was minimized, and the magnetic flux was guided in the radial direction. The relative permeability of pure iron used in the analysis was 4000, and the bulk conductivity was 10,300,000 S/m.  
It was assumed that the solenoid coils consisted of individually insulated conductors such that the eddy current was negligible. Therefore, the eddy current and displacement current inside the conductor were not calculated, and the current density was assumed to be uniform throughout the conductor. The diameter of each solenoid coil was 0.7 mm, and the number of turns was 625. According to the actual measurements, the series resistance of the coil was 6.5 (\(\pm\)0.1) \(\Omega\).  
Both the coils and the oxygen-free copper tube were assumed to be made of copper. The relative permeability used in the analysis was 0.999991, and the bulk conductivity was 58,000,000 S/m. The intermeshed teeth structures were assumed to be made of copper and pure iron, respectively. In the tooth structure, the part made of pure iron was combined with the outermost shell, which was assumed to be made of the same material.  
To manufacture an actual damper prototype and conduct an experiment with it, a rod had to be fixed to a permanent magnet. Additionally, perforated covers and bearings at both ends were needed to constrain the rod stroke. These components were assumed to be made of aluminum and stainless steel. Because these components are non-magnetic, they were not considered in the electromagnetic analysis. Additionally, it was assumed that each component of the damper had no deformation due to the magnetic field. Therefore, the Young's modulus, Poisson's ratio, and magnetostriction were not considered.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Characteristics** & **Description** & **Values** & **Units** \\ \hline B\({}_{\text{r}}\) & Residual induction & 1.35 & T \\ H\({}_{\text{c}}\) & Coercivity & \(-\)1,074,295.87 & A/m \\ BH\({}_{\text{max}}\) & Maximum energy product & 36,2574.85 & J/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Properties of the non-linear permanent magnet.  
Figure 10: BH curve of the applied neodymium magnet.  
The analysis region was set to 100% offset of the damper dimension in each axial direction considering the magnetic material of the model. Vacuum conditions were assumed for the space outside the damper, the clearance for the stroke of the drive, and the air gap. To implement translational motion in the driving part, permanent magnets and pole pieces were wrapped to restrain them from moving integrally under external excitation. The tied part was made to move in the \(z\)-axis inside the band that realized the stroke space of the driving part. The excitation was a sinusoidal wave with an amplitude of 12 mm and a frequency of 5 Hz. A transient analysis was performed in an adaptive time step with a time interval of at least 0.001 s. The analysis was performed for two cycles of excitation.