## 2 Analytical Investigation of Electromagnetic Damper Model
### Magnetic Circuit and Operating Point  
The magnetic circuit was analyzed under the assumption of a quasi-static magnetic field system. The magnetic circuit is an adaption of the electric circuit approach for solving the magnetic field problem, and the electromagnetic field theory is interpreted in accordance with the circuit theory [13]. It is similar to an electric circuit, but a permanent magnet is used instead of a battery. To analyze the magnetic circuit, the operating point of the permanent magnet must be determined, as it defines the amount of energy produced by the permanent magnets in the circuit [11]. A coil-type electromagnetic damper with two activated coils is shown in Figure 1. Both coils are said to be \"activated,\" because they are constantly affected by the magnetic flux. The activated coils are affected by the magnetic field in the opposite direction, and the currents flow through the coils in opposite directions. The main dimensions that determine the damper structure and integration paths are also shown in Figure 1. The magnetic circuit flows in a closed-loop path denoted as C and passes through the surfaces from S1 to S3. The magnetic fields in the pole pieces and air gap represent the cross-section of the space distributed in the radial direction. The magnetization of the permanent magnet and the magnetic flow in the stator yoke are assumed to be in the axial direction, which is the transverse direction in Figure 1. Table 1 presents the main geometric parameters of the damper, which are shown in Figure 1.  
Equation (1) is can be derived by applying [PERSON]'s circuital law to the closed-loop magnetic flux path (C) passing through all enclosing surfaces (S1 to S3),  
\[\oint_{C}\stackrel{{\rightarrow}}{{H}}\cdot\stackrel{{ \rightarrow}}{{d}}\stackrel{{\rightarrow}}{{l}}=\int_{Magnet} \stackrel{{\rightarrow}}{{H}}_{m}\cdot\stackrel{{ \rightarrow}}{{d}}\stackrel{{\rightarrow}}{{l}}+2\int_{Pole} \stackrel{{\rightarrow}}{{H}}_{p}d\stackrel{{ \rightarrow}}{{l}}+2\int_{Air}\stackrel{{\rightarrow}}{{H}}_{a} d\stackrel{{\rightarrow}}{{l}}+2\int_{Call}\stackrel{{ \rightarrow}}{{H}}_{c}d\stackrel{{\rightarrow}}{{l}}+\int_{Shell} \stackrel{{\rightarrow}}{{H}}_{s}d\stackrel{{ \rightarrow}}{{l}}=0, \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Symbol** & **Description** \\ \hline Magnet length & \(l_{m}\) & Actual length of the magnet (smaller than the pole pitch) \\ Pole shoe thickness & \(l_{p}\) & Length excluding the magnetic length in the pole pitch \\ Coil length & \(l_{c}\) & Axial length of each armature coil \\ Mover radius & \(r_{m}\) & Outer radius of the magnet or pole pieces \\ Armature radius & \(r_{a}\) & Inside radius of the armature \\ Winding radius & \(r_{c}\) & Outside radius of the armature coil \\ Stator yoke radius & \(r_{s}\) & Outside radius of the armature shell \\ Air gap & \(g\) & Distance between the mover and armature windings \\ Coll height & \(l_{c}\) & Height of the armature coil section \\ Yoke thickness & \(h_{g}\) & Armature shell thickness \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the damper. Figure 1: Cross-section of a coil-type electromagnetic damper. where the subscripts of \(H\) indicate the magnetic-field intensity in each region. Equation (1) assumes that the magnetic fields \(\overset{\rightarrow}{H}\) and the path \(d\overset{\rightarrow}{l}\) are parallel along the path direction, and the magnetic leakage is ignored. It was assumed that magnetic saturation of the material did not occur. The relationship between the magnetic field and magnetic flux outside the magnet is given by Equation (2):  
\[H=\frac{B}{\mu_{0}\mu_{r}}\, \tag{2}\]  
where \(B\) represents the magnetic field strength, i.e., the magnetic flux density; \(\mu_{r}\) represents the relative magnetic permeability of the material; and \(\mu_{0}\) represents the vacuum magnetic permeability.