## 3 Magnetic Negative Stiffness in QZS Isolators
### Mathematical Modeling of Permanent Magnet Negative Stiffness Elements
#### 3.2.2 Mechanical Modeling of Rectangular Magnet Negative Stiffness Springs  
Accurately calculating the magnetic force of permanent magnets is crucial for establishing a magneto-negative stiffness model for rectangular permanent magnets. Extensive research has been conducted in this area. In 1984, [PERSON] [67] proposed an analytical model based on the surface charge model for calculating the magnetic force of a three-dimensionalrectangular permanent magnet with parallel magnetization directions. This model was derived by considering the differential of the magnetic interaction energy concerning displacements. It accurately computes the interaction force and moment between two permanent magnets with parallel magnetization directions, facilitating the design and optimization of magnetic parameters. Additionally, [PERSON] derived a three-dimensional force and moment model for two rectangular permanent magnets with perpendicular magnetization directions. [PERSON] [68] introduced the concept of a magnetic node and developed a highly accurate three-dimensional analytical model for the magnetic field and force of a rectangular permanent magnet based on this concept, suitable for designing and optimizing magnetic parameters. However, recent years have witnessed an in-depth exploration of the negative stiffness structure. The magnetic negative stiffness structure offers a promising alternative to conventional mechanical structures, characterized by its compact design, efficient space utilization, and absence of mechanical friction, rendering it particularly suitable for precision equipment. Rectangular or ring-shaped permanent magnetic materials serve as elastic support structures to provide gravitational or repulsive forces in mechanical vibration applications, with effective arrangements implemented to mitigate the influence of nonlinear permanent magnetic materials. The magnetic force exerted by the rectangular permanent magnetic materials below is computed. Figure 16 illustrates a structure featuring magnetically repulsive negative stiffness. Three magnets are vertically arranged, with the upper and lower magnets firmly affixed to the ground. The magnetization direction of all three magnets is aligned. When the middle magnet undergoes horizontal displacement from its equilibrium position, it experiences a combined force from the upper and lower magnets in the same direction, propelling it further from the equilibrium position, thus demonstrating negative stiffness characteristics. Similarly, the vertical movement of the middle magnet results in its restoration to the equilibrium position due to the repulsive force, leading to the manifestation of positive stiffness characteristics within the system. The aforementioned analysis elucidates that both magnetic suction and magnetic repulsion arrangements can yield negative stiffness characteristics, contingent upon the control of the intermediate magnet's movement direction. The magnetic force between permanent magnets is dictated by the magnetic field they generate. This field is influenced by multiple factors, encompassing the dimensions of the magnets, the magnetic material employed, the degree of magnetization, and the spatial configuration of the magnets. Consequently, when modeling magnetic force, it is imperative to account for distinct parameters such as the magnet size, geometry, and inter-magnet positioning. Figure 16: Structure of negative stiffness with magnetic attraction and magnetic repulsion: (**a**) magnetic suction negative stiffness structure; and (**b**) magnetically repulsive negative stiffness structure. The analysis of the magnetic force expression between two rectangular permanent magnets is conducted. Additionally, the mathematical model for negative stiffness in a three-magnet structure is formulated. As illustrated in Figure 17, two rectangular magnets, denoted as M1 and M2, exhibit rigidity and homogeneity.