## 3 Analytical Stiffness of Halbach-Cylinder Magnetic Spring  
### Basic Configuration of an HCMS  
A Halbach array is a special arrangement of permanent magnets that enhances the magnetic field on one side of the array, while canceling the field to near zero on the other side. This is achieved by having a spatially rotating pattern of magnetization, which can be understood by observing the magnetic flux distribution shown in Figure 4. In Figure 4a, the pattern of magnetization rotates clockwise, so that the left magnetic field is enhanced while the right field is canceled. In Figure 4b, the pattern of magnetization rotates anticlockwise, so that the right magnetic field is enhanced while the left field is canceled. To date, two types of Halbach arrays have been proposed, namely the flat Halbach array [24] and the Halbach cylinder [26]. Figure 3: Structures of (**a**) Type III and (**b**)Type IV CMSs. Based on the mechanism in Figure 4, two coaxialHalbach cylinders are utilized to build an HCMS in this paper. That is to say, the HCMS is composed of the inner Halbach cylinder and the outer Halbach cylinder. In order to generate a larger magnetic force of the HCMS, the magnetization direction of the outer Halbach cylinder should change anticlockwise from top to bottom, while it should change clockwise for the inner Halbach cylinder. According to Figure 4, the magnetic field between the inner and outer Halbach cylinders can be greatly enhanced. In this paper, an HCMS is denoted as an '\(N\)-\(M\)' type (\(N,\ M\) should be odd), which means that the outer Halbach cylinderconsists of \(N\)-layer magnetic rings and the inner Halbach cylinderconsists of \(M\)-layer magnetic rings. Then, there are total \(NM\) CMSs in the HCMS. Therefore, the magnetic force and stiffness of the '\(N\)-\(M\)'-type HCMS can be written as  
\[ \begin{split} F_{H}(z)=\sum\limits_{i=1}^{NM}\widetilde{F}_{i}( z)\\ K_{H}(z)=-\partial F_{H}(z)/\partial z\end{split}  \tag{20}\]  
where \(\widetilde{F}_{i}(z)\) is the magnetic force of the \(i\)th CMS, which can refer to Section 2. As for each '\(N\)-\(M\)'-type HCMS, both the inner and the outer Halbach cylinders have four different configurations according to the magnetization directions, leading to sixteen structures. A representative structure of the 3-3-type HCMS is shown in Figure 5. The vertical center position of the outer Halbach cylinder is denoted as the horizontal axis, and the displacement between the inner and outer Halbach cylinders is denoted as \(z\). Figure 4: Formation mechanism of Halbach array: (**a**) left-side enhancement and (**b**) right-side enhancement. Figure 5: Schematic structure of the 3-3-type HCMS. negative value (attractive force) at \(z=0\). (ii) Magnetic force curves of Type I and Type II CMSs have asimilar sine-like shape, and the latter has the larger peak value. (iii) Magnetic force curves of Type III and Type IV CMSs are symmetrical in terms of the axis of \(z=0\), but the directions are reversed. Next, the stiffness curves of four basic CNSs are calculated and shown in Figure 7. The following can be seen: (i) All stiffness curves are nonlinear. (ii) Both Type I and Type II CMSs have symmetrical negative stiffness near \(z=0\). The stiffness of Type III CMSs is negative when \(z\in[-10,\ 0]\), and the stiffness of Type IV CMSs is negative when \(z\in[0,\ 10]\). (iii) Type II CMSs have the largest negative stiffness and the widest displacement range of negative stiffness, but Type IV CMSs have the best uniformity of negative stiffness. Therefore, it can be concluded that any CMS has negative stiffness with different displacement ranges. Considering the equilibrium position, Type II CMSs may be the most suitable for compact design. In real-world applications, the amplitude, the displacement range, and the uniformity of negative stiffness are three important metrics of QZS vibration isolators.