## 2 Force and Stiffness Characteristics of Circular Magnetic Springs
### Magnetic Force and Stiffness of Each Basic CMS
#### 2.2.1 Type I CMSs  
The detailed structure of Type I CMSs is shown in Figure 1a, where the inner magnetic ring is unconstrained and the outer magnetic ring is fixed. The heights of the inner and outer magnetic rings are denoted as \(2h_{1},\ 2h_{2}\) and the residual flux densities of the inner and outer magnetic rings are denoted as \(B_{r_{1}},\ B_{r_{2}}\). The middle position of the outer magnetic ring is defined as the horizontal axis, and the distance between the two magnetic rings is denoted as \(z\). According to the Amperian current model [27], the magnetic force of two magnets is equivalent to the interaction between equivalent currents distributed on the magnets. For \begin{table}
\begin{tabular}{c c c c} \hline
**Type** & **All Denotations** & **Magnetic Force** & **Stiffness** \\ \hline \multirow{4}{*}{Type I} & OUT \(\leftarrow\) IN \(\leftarrow\) & \(F_{1}\) & \(K_{1}\) \\  & OUT \(\leftarrow\) IN \(\rightarrow\) & \(-F_{1}\) & \(-K_{1}\) \\  & OUT \(\rightarrow\) IN \(\leftarrow\) & \(-F_{1}\) & \(-K_{1}\) \\  & OUT \(\rightarrow\) IN \(\rightarrow\) & \(F_{1}\) & \(K_{1}\) \\ \hline \multirow{4}{*}{Type II} & OUT \(\uparrow\) IN \(\uparrow\) & \(F_{2}\) & \(K_{2}\) \\  & OUT \(\uparrow\) IN \(\downarrow\) & \(-F_{2}\) & \(-K_{2}\) \\  & OUT \(\downarrow\) IN \(\uparrow\) & \(-F_{2}\) & \(-K_{2}\) \\  & OUT \(\downarrow\) IN \(\downarrow\) & \(F_{2}\) & \(K_{2}\) \\ \hline \multirow{4}{*}{Type III} & OUT \(\rightarrow\) IN \(\uparrow\) & \(F_{3}\) & \(K_{3}\) \\  & OUT \(\rightarrow\) IN \(\downarrow\) & \(-F_{3}\) & \(-K_{3}\) \\  & OUT \(\leftarrow\) IN \(\downarrow\) & \(F_{3}\) & \(K_{3}\) \\ \hline \multirow{4}{*}{Type IV} & OUT \(\uparrow\) IN \(\rightarrow\) & \(F_{4}\) & \(K_{4}\) \\  & OUT \(\uparrow\) IN \(\leftarrow\) & \(-F_{4}\) & \(-K_{4}\) \\ \cline{1-1}  & OUT \(\downarrow\) IN \(\leftarrow\) & \(F_{4}\) & \(K_{4}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Magnetic force and stiffness of each configuration of CMSs. Figure 1: Schematic of Type I CMS: (**a**) structural configuration, (**b**) surface current directions, and (**c**) calculation of \(F_{11r}(z)\). the magnetic ring with radial magnetization, the equivalent current only distributes on its upper and lower surfaces, and the surface current density can be written as  
\[\textbf{J}=-\textbf{n}\times\textbf{B}/\mu_{0} \tag{1}\]  
where **n** is the unit vector normal to the surface, **B** is the magnetic-field vector, \(\mu_{0}=4\pi\times 10^{-7}\) V\(\cdot\)s/(A\(\cdot\)m) is the permeability of the vacuum, and '\(\times\)' denotes the vector product. Furthermore, the upper and lowersurfaces of the inner magnetic ring are denoted as 1 and 2, and the counterparts of the outer magnetic ring are denoted as 1\({}^{\prime}\) and 2\({}^{\prime}\), respectively. The radial force between the two surfaces is equal to zero due to the circular symmetry. The axial force between two surfaces is denoted as \(F_{s\psi^{\prime}}(z)(s,v\in\{1,\ 2\})\), and \(h_{s\psi^{\prime}}(z)\) is the axial distance between two surfaces. Then, the magnetic force of Type I CMSs can be represented as the sum of the four axial forces. \[F_{1}(z,2h_{1},2h_{2})=\sum\limits_{s=1}^{2}\sum\limits_{v=1}^{2}F_{s\psi^{ \prime}}(z) \tag{2}\]  
Next, the calculation of \(F_{11}(z)\) is selected as an example. The current directions of surface 1 and 1\({}^{\prime}\) are shown in Figure 1b by using the right-hand rule.