## 2 Force and Stiffness Characteristics of Circular Magnetic Springs
### DifferentConfigurations of Circular Magnetic Springs  
A pair of coaxial magnetic rings forms a circular magnetic spring (CMS), including the inner and outer magnetic rings, and is also called the single-layer CMS. Each magnetic ring can be radially or axially magnetized, so either the inner or the outer magnetic ring has two classes of main magnetizations. In total, there are four types of configurations for CMSs, including Type I (radial-radial magnetization), Type II (axial-axial magnetization), Type III (radially-axially perpendicular magnetization), and Type IV (axially-radially perpendicular magnetization). Furthermore, both the inner and the outer magnetic rings have two of their own magnetization directions, so each type of configuration has four structures. Here, one basic structure of each basic configuration is chosen optionally for a benchmark, and its denotation is listed in Table 1, where the arrows indicate the magnetization directions. Inside and outside radii of the inner magnetic ring are denoted as \(R_{1}\) and \(R_{2}\). Inside and outside radii of the outer magnetic ring are denoted as \(R_{3}\) and \(R_{4}\). Magnetic forces and stiffness of the four basic CMSs are denoted as \(F_{1}\sim F_{4}\) and \(K_{1}\sim K_{4}\), respectively.  
Next, all sixteen structures of CMS are summarized in Table 2. According to Table 1, the magnetic forces and stiffness of the other three structures in each type of CMS can be obtained by referring to the basic structure.  
Specifically, all denotations in Tables 1 and 2 are explained as follows: 'OUT\(\leftarrow\)' indicates that the magnetization direction of the outer magnetic ring is radially outward from the inside. 'OUT\(\rightarrow\)' indicates that the magnetization direction of the outer magnetic ring is radially inward from the outside. 'OUT\(\uparrow\)' indicates that the magnetization direction of the outer magnetic ring is axially from bottom to top. 'OUT\(\downarrow\)' indicates that the magnetization direction of the outer magnetic ring is axially from top to bottom. 'IN\(\rightarrow\)' indicates that the magnetization direction of the inner magnetic ring is radially outward from the inside. 'IN\(\leftarrow\)' indicates that the magnetization direction of the inner magnetic ring is radially inward from the outside. 'IN\(\uparrow\)' indicates that the magnetization direction of the inner magnetic ring is axially from bottom to top. 'IN\(\downarrow\)' indicates that the magnetization direction of the inner magnetic ring is axially from top to bottom.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Description** & **Type** & **Basic Structure** & **Denotation** & **Magnetic Force** & **Stiffness** \\ \hline Radial–radial magnetization & Type I & OUT \(\leftarrow\) IN \(\leftarrow\) & \(F_{1}\) & \(K_{1}\) \\ \hline Axial–axial magnetization & Type II & OUT \(\uparrow\) IN \(\uparrow\) & \(F_{2}\) & \(K_{2}\) \\ \hline Perpendicular magnetization & Type III & OUT \(\rightarrow\) IN \(\uparrow\) & \(F_{3}\) & \(K_{3}\) \\ (radial–axial) & Type IV & OUT \(\uparrow\) IN \(\rightarrow\) & \(F_{4}\) & \(K_{4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Four types of structural configurations for CMSs.