## 3 Magnetic Negative Stiffness in QZS Isolators
### Mathematical Modeling of Permanent Magnet Negative Stiffness Elements
#### 3.2.1 Mechanical Modeling of Magnetic Springs with Negative Stiffness of Magnetic Rings
Also, based on the geometrical relationship between point P and point Q, we can obtain:  
\[r=\sqrt{x_{1}^{2}+\left(r_{3}\cos\beta-r_{2}\cos\alpha\right)^{2}+\left(r_{3} \sin\beta-r_{2}\sin\alpha\right)^{2}} \tag{12}\]  
By substituting Equations (11) and (12) into Equation (7) and integrating over the entire end face thereof, the magnitude of the magnetic force acting along the axial direction of the magnetic ring between the end face 2 of the inner ring and the end face 3 of the outer ring can be obtained as follows:  
\[F_{32}=C\int_{R_{1}}^{R_{2}}\int_{R_{3}}^{R_{4}}\int_{0}^{2\pi}\int_{0}^{2\pi} \frac{\left(r_{3}\sin\gamma-r_{2}\sin\varphi-x_{1}\right)r_{2}r_{3}dr_{2}dr_{ 3}dad\beta}{\left[x_{1}^{2}+\left(r_{3}\cos\beta-r_{2}\cos\alpha\right)^{2}+ \left(r_{3}\sin\beta-r_{2}\sin\alpha\right)^{2}\right]^{\frac{3}{2}}} \tag{13}\]  
Similarly, the magnitude of the magnetic force between the other end faces of the inner and outer rings along the axial direction of the ring can be obtained as the magnitude of the partial force:  
\[F_{31}=C\int_{R_{1}}^{R_{2}}\int_{R_{3}}^{R_{4}}\int_{0}^{2\pi}\int_{0}^{2\pi} \frac{\left(r_{3}\sin\gamma-r_{1}\sin\varphi-x_{1}\right)r_{1}r_{3}dr_{1}dr_{3 }dad\beta}{\left[\left(H_{1}+x_{1}\right)^{2}+\left(r_{3}\cos\beta-r_{1}\cos \alpha\right)^{2}+\left(r_{3}\sin\beta-r_{1}\sin\alpha\right)^{2}\right]^{ \frac{3}{2}}} \tag{14}\]  
\[F_{41}=C\int_{R_{1}}^{R_{2}}\int_{R_{3}}^{R_{4}}\int_{0}^{2\pi}\int_{0}^{2\pi} \frac{\left(r_{4}\sin\gamma-r_{1}\sin\varphi-x_{1}\right)r_{1}r_{4}dr_{1}dr_{ 4}dad\beta}{\left[x_{1}^{2}+\left(r_{4}\cos\beta-r_{1}\cos\alpha\right)^{2}+ \left(r_{4}\sin\beta-r_{1}\sin\alpha\right)^{2}\right]^{\frac{3}{2}}} \tag{15}\]  
\[F_{42}=C\int_{R_{1}}^{R_{2}}\int_{R_{3}}^{R_{4}}\int_{0}^{2\pi}\int_{0}^{2\pi} \frac{\left(r_{4}\sin\gamma-r_{2}\sin\varphi-x_{1}\right)r_{2}r_{4}dr_{2}dr_{ 4}dad\beta}{\left[\left(H_{2}-x_{1}\right)^{2}+\left(r_{4}\cos\beta-r_{2}\cos \alpha\right)^{2}+\left(r_{4}\sin\beta-r_{2}\sin\alpha\right)^{2}\right]^{ \frac{3}{2}}} \tag{16}\]  
where \(H_{1}\), \(H_{2}\) is the axial height of the inner and outer ring; \(r_{1}\), \(r_{2}\) is the inner and outer radius of the inner ring; \(r_{3}\), \(r_{4}\) is the inner and outer radius of the outer ring. The internal and external magnetic ring along the axial load-bearing force should be the superposition of the magnetic action between the end faces of the internal and external magnetic ring, having a positive and negative sign according to the two interacting end faces of the magnetic charge division, with the same number of magnetic charge is positive and the opposite number of magnetic charge is negative, that is:  
\[F=F_{41}+F_{32}-F_{31}-F_{42} \tag{17}\]  
From the definition of stiffness, the stiffness equation for a negative stiffness magnetic spring constructed from the inner and outer rings is obtained as:  
\[K_{N}=-\frac{dF}{dx_{1}} \tag{18}\]  
The negative sign in the formula indicates that the direction of the restoring force of this negative stiffness magnetic spring is opposite to the direction of the axial load-bearing force generated by the inner and outer magnetic rings.