## 4 Electromagnetic Analysis of the Proposed Damper
### Analysis Results  
The magnetic-field distribution in the air gap by the driver was examined [33]. With consideration of the magnetic saturation, the thickness of the pole pieces was adjusted so that the maximum magnetic flux density outside the air gap was distributed in the range of 1.5-2.0 T. Figure 11 shows the magnetic flux density along the axial length inside and outside the air gap. The potential was maximized when the thickness of the inner and outer pole pieces were 7 and 5 mm, respectively. For the mesh of the transient analysis model, the conditions for magnetostatic analysis were applied to all the components of the damper and the band of the stroke space. In the magnetostatic analysis, the mesh was applied considering the density of the pole pieces and the air gap [34; 35]. The maximum length of each element was 0.004 m, and the maximum number of elements was 1000. The faces, edges, etc. of each object were all divided equally. Copper modeled as a coil has high conductivity, but has little effect on penetration depth; thus, a skin-depth mesh was not considered. The mesh configuration applied to the damper model is shown in Figure 12a. Figure 12b shows the distribution of the magnetic flux density at the maximum motion force, when the periodic motion of the driving part was stabilized and neutral. The magnetic flux density was concentrated on the teeth outside the pole pieces and on the magnetic path and exhibited a vertically symmetrical distribution. The maximum magnetic flux density appeared along the interface of the teeth facing the pole pieces. Low magnetic flux densities of \(\leq\)1 T were observed in the back iron of the magnetic path passing through the activated coils. The distribution of the magnetic flux lines is shown in Figure 12c, confirming the formation of magnetic circuits. The magnetic flux lines of the permanent magnets and pole pieces exhibited a dense distribution as they approached the stator. The strongest and most dense magnetic flux distribution was observed in the teeth located on the magnetic path, which faced the pole pieces as the driver moved. Conversely, the weakest magnetic flux distribution was observed on the inner paths of the activated coils and flux lines in the back iron. Additionally, magnetic flux lines appeared on a short path through the slot in which the insulating material was disposed between the coils. Figure 11: Magnetic flux density of the ferromagnetic pole pieces. The current-density distribution is shown in Figure 12d. The damper windings were assumed to be stranded models. Therefore, since there was no skin depth and solid loss did not occur, a uniform current flowed through the coil cross-section. The coils exhibited a uniform density distribution and a concentrated distribution in the conductor, where each pole piece was located. To examine the changes in the damping force of the proposed damper, the motion force acting on the driver during a period of two cycles was investigated. The position, speed, and force of the driver over time are shown in Figure 13. The driving force changed periodically with the changes in the position and speed. As the speed increased, the force increased and acted in the opposite direction to the movement of the driver, because the force was generated in a direction to resist changes in the magnetic flux. The maximum damping force exhibited a delay of \(\geq\)10 ms. A weak damping force was observed at the beginning of the driving, but the damping force stabilized rapidly from the second cycle onwards. After a delay of 2 ms, a constant period and damping force magnitude appeared. The maximum damping force was 1.01 kN when the speed of the damper was 376.99 mm/s, which was 88% stronger than that of the coil-type electromagnetic damper and 20% stronger than that of the composite-type electromagnetic damper. The electrical characteristics of the damper model affected the energy-harvesting performance.