## 2 Force and Stiffness Characteristics of Circular Magnetic Springs
### Magnetic Force and Stiffness of Each Basic CMS
#### 2.2.2 Type II CMSs  
The detailed structure of Type II CMSs is shown in Figure 2a, where the geometric configuration is similar to that in Figure 1a.The inner and outer surfaces of the inner and outer magnetic rings are denoted as \(3\), \(3^{\prime}\), \(4\), and \(4^{\prime}\), respectively. The radial force between the two surfaces is also equal to zero due to the circular symmetry. The axial force between two surfaces is denoted as \(F_{ss^{\prime}}(z)\) (\(s,\
u\in\{3,\,4\}\)) and then the magnetic force of Type II CMSs can be calculated as  
\[F_{2}(z,2h_{1},2h_{2})=\sum\limits_{s=3}^{4}\sum\limits_{v=3}^{4}F_{ss^{ \prime}}(z) \tag{10}\]  
Next, the calculation of \(F_{33^{\prime}}(z)\) is selected as an example. The current directions of surface \(3\) and \(3^{\prime}\) are shown in Figure 2b by using the right-hand rule. \(h_{ss^{\prime}}(z)\) is the axial distance between the two surfaces. The axial magnetic force exerted on \(j_{1}R_{1}d\varphi dz_{1}\) at point \(N\) by \(j_{\rm O}R_{3}d\psi dz_{2}\) at point \(M\) is  
\[dF_{33^{\prime}}(z)=j_{1}R_{1}d\varphi dz_{1}\times d\mathbf{B}_{33^{\prime}} \tag{11}\]  
where \(d\mathbf{B}_{33^{\prime}}=\mu_{0}j_{\rm O}R_{3}d\psi z_{2}\times\mathbf{MN}_{3 3^{\prime}}/\left(4\pi|\mathbf{MN}_{33^{\prime}}|^{3}\right)\) is the magnetic flux density at point \(M\) produced by \(j_{\rm O}R_{3}d\psi dz_{2}\) at point \(N\); \(|\mathbf{MN}_{33^{\prime}}|=\sqrt{R_{1}^{2}+R_{2}^{3}+h_{33^{\prime}}^{2}-2R_{ 1}R_{3}\cos(\psi-\varphi)}\) is  
Figure 2: Schematic of Type II CMS: (**a**) structural configuration, (**b**) surface current directions, and (**c**) calculation of \(F_{33^{\prime}}(z)\).  
the vector from point \(M\) to point \(N\); and \(h_{33\prime}=z_{1}+z-z_{2}\) is the axial distance from surface 3 to surface \(3^{\prime}\).  
The axial magnetic force \(F_{33\prime}(z)\) can be obtained as  
\[F_{33\prime}(z)=-\frac{B_{r_{1}}B_{r_{2}}}{4\pi\mu_{0}}\int\limits_{0}^{2\pi} \int\limits_{0}^{2\pi}\int\limits_{-h_{1}}^{h_{1}}\int\limits_{-h_{2}}^{h_{2} }\frac{R_{1}R_{3}h_{33^{\prime}}\cos(\psi-\varphi)}{\left|\mathbf{MN}_{33\prime }\right|^{3}}dz_{1}dz_{2}d\varphi d\psi \tag{12}\]  
Similar to \(F_{33\prime}(z)\), axial magnetic forces \(F_{3\prime}(z)\), \(F_{43^{\prime}}(z)\), and \(F_{44^{\prime}}(z)\) can also be calculated as  
\[F_{s\
u^{\prime}}(z)=-(-1)^{s+v}\frac{B_{r_{1}}B_{r_{2}}}{4\pi\mu_{0}}\int \limits_{0}^{2\pi}\int\limits_{0}^{2\pi}\int\limits_{-h_{1}}^{h_{1}}\int \limits_{-h_{2}}^{h_{2}}\frac{R_{s-2}R_{v}h_{s\
u^{\prime}}\cos(\psi-\varphi )}{\left|\mathbf{MN}_{s\
u^{\prime}}\right|}dz_{1}dz_{2}d\varphi d\psi \tag{13}\]  
where \(\left|\mathbf{MN}_{s\
u^{\prime}}\right|=\sqrt{R_{s}^{2}+R_{v}^{2}+h_{s\
u^{ \prime}}^{2}-2R_{s-2}R_{v}\cos(\psi-\varphi)}\) and \(h_{s\
u^{\prime}}=z-(-1)^{s}z_{1}+(-1)^{v}z_{2}\), \(s,v\in\{3,4\}\).  
Combining Equations (10) and (11), \(F_{2}(z,2h_{1},2h_{2})\) can be rewritten as  
\[F_{2}(z,2h_{1},2h_{2})=-\frac{B_{r_{1}}B_{r_{2}}}{4\pi\mu_{0}}\sum\limits_{s= 3}^{4}\sum\limits_{p=3}^{4}\int\limits_{-h_{1}}^{h_{1}}\int\limits_{-h_{2}}^{ h_{2}}\int\limits_{0}^{2\pi}\int\limits_{0}^{2\pi}(-1)^{s+v}\frac{R_{s-2}R_{v}h_{s \
u^{\prime}}}{\left|\mathbf{MN}_{s\
u^{\prime}}\right|^{3}}\cos(\psi-\varphi )dz_{1}dz_{2}d\varphi d\psi \tag{14}\]  
Then, the stiffness \(K_{2}\) can be obtained as  
\[K_{2}(z,2h_{1},2h_{2})=-\frac{B_{r_{1}}B_{r_{2}}}{4\pi\mu_{0}}\sum\limits_{s= 3}^{4}\sum\limits_{p=3}^{2}\int\limits_{-h_{1}}^{h_{1}}\int\limits_{-h_{2}}^{ h_{2}}\int\limits_{0}^{2\pi}\int\limits_{0}^{2\pi}(-1)^{s+v}\frac{\left(\left| \mathbf{MN}_{s\
u^{\prime}}\right|^{2}-3h_{s\
u^{\prime}}^{2}\right)R_{s-2}R_{ v}}{\left|\mathbf{MN}_{s\
u^{\prime}}\right|^{3}}\cos(\psi-\varphi)dz_{1}dz_{2}d \varphi d\psi \tag{15}\]