## 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  
Figure 13 depicts the permanent magnet ring structure frequently employed in engineering applications encompassing magnetic bearings, gears, and couplings. The permanent magnet bearing harnesses the force between two permanent magnets as a supporting force. It can be categorized into radial and axial permanent magnet structures, contingent upon the magnetization orientation of the magnets and the distinct manifestations of the magnetic force. Figure 14 delineates the diverse configurations attainable contingent upon the varying types. Figure 14: Different configurations of permanent magnet bearings under radial and axial magnetization arrangements. (**a**) Two magnetic rings with unequal diameters, parallel heights, and axial polarization in the same direction; (**b**) Two magnetic rings with unequal diameters, parallel heights, wherein the outer ring exhibits radial polarization and the inner ring demonstrates reverse radial polarization; (**c**) Two magnetic rings with unequal diameters, parallel heights, both exhibiting mutually opposing axial polarization; (**d**) Two magnetic rings with unequal diameters, parallel heights, both  
Figure 13: Ring permanent magnet structure: (**a**) radial magnetization; and (**b**) axial magnetization. demonstrating reverse radial polarization; **(e)** Two magnetic rings with unequal diameters, non-parallel heights, wherein the outer ring exhibits axial polarization and the inner ring demonstrates reverse radial polarization; **(f)** Two magnetic rings with unequal diameters, non-parallel heights, wherein the outer ring displays radial polarization and the inner ring exhibits axial polarization; **(g)** Two magnetic rings with unequal diameters, non-parallel heights, wherein the outer ring demonstrates reverse axial polarization and the inner ring exhibits reverse radial polarization; **(h)** Two magnetic rings with unequal diameters, non-parallel heights, wherein the outer ring exhibits reverse radial polarization; **(i)** Two magnetic rings with unequal diameters, non-parallel heights, both exhibiting mutually opposing axial polarization; **(j)** Two magnetic rings with unequal diameters, non-parallel heights, both demonstrating radial polarization; **(k)** Two magnetic rings with unequal diameters, non-parallel heights, both exhibiting axial polarization; **(l)** Two magnetic rings with unequal diameters, non-parallel heights, wherein the outer ring exhibits reverse radial polarization and the inner ring exhibits radial polarization; **(m)** Two magnetic rings with unequal diameters, non-parallel heights, wherein the outer ring exhibits axial polarization and the inner ring displays radial polarization; **(n)** Two magnetic rings with unequal diameters, non-parallel heights, wherein the outer ring exhibits radial polarization and the inner ring displays radial polarization; **(n)** Two magnetic rings with unequal diameters, non-parallel heights, wherein the outer ring exhibits radial polarization and the inner ring demonstrates axial polarization; **(o)** Two magnetic rings with unequal diameters, non-parallel heights, wherein the outer ring exhibits reverse axial polarization and the inner ring exhibits radial polarization; **(p)** Two magnetic rings with unequal diameters, non-parallel heights, wherein the outer ring exhibits reverse radial polarization and the inner ring displays axial polarization; **(q)** Two magnetic rings with equal diameters, non-parallel heights, both exhibiting axial polarization; **(r)** Two magnetic rings with equal diameters, non-parallel heights, both demonstrating mutually opposing radial polarization; **(s)** Two magnetic rings with equal diameters, non-parallel mutually opposing axial polarization; **(t)** Two magnetic rings with equal diameters, non-parallel heights, both demonstrating reverse radial polarization.