## 2 Fundamental Principles of Quasi-Zero Stiffness
### Several Types of Negative Stiffness Actuator
#### 2.3.3 Magnetic Spring as Negative Stiffness Actuator
The control theory for the magnetic-air hybrid QZS vibration isolation system is innovated with the utilization of a  
Figure 11: Structure of the double-ring permanent magnet type vibration isolator proposed by [PERSON] [56]. Figure 12: New passive vibration isolators with magnetic repulsion and magnetic absorption proposed by [PERSON] [57]. type-2 fuzzy controller. This ensures the vibration isolation system exhibits satisfactory QZS characteristics under a static load. [PERSON] [59] engineered a vibration isolator featuring high-static and low-dynamic stiffness. The isolator employs a hybrid structure consisting of an electromagnet and a permanent magnet to furnish negative stiffness. The study explores the optimal parameters to attain the optimal isolation effect and subsequently optimizes the system model. [PERSON] engineered a vibration isolator featuring high-static and low-dynamic stiffness. The isolator employs a hybrid structure consisting of an electromagnet and a permanent magnet to furnish negative stiffness. The polarity of the electromagnet can be controlled to regulate the stiffness of the structure by adjusting the current. The optimal parameters of the electromagnetic spring are investigated to achieve the optimal vibration isolation effect. The system model is optimized using the optimal parameters, and the mathematical relationship between the excitation force frequency and the displacement transfer rate is calculated. Ultimately, experiments demonstrate the excellent vibration isolation performance and real-time adjustment capability of this vibration isolator. [PERSON] [60,61] introduced a novel passive vibration isolator that furnishes both magnetic repulsion and suction. They formulated a nonlinear equation for calculating the magnetic force, deduced a fitting expression for the equivalent nonlinear magnetic stiffness, and experimentally validated that magnetic repulsion and suction fulfill the vibration isolation requirements across various bandwidths. [PERSON] [62] pioneered a new electromagnetic negative-stiffness spring employing a permanent ring magnet and a coil with a rectangular cross-section. They have established and experimentally corroborated this novel spring type utilizing the thin-wire method. An ENSS analytical model is formulated and experimentally corroborated employing the thin-wire method. The model quantitatively explores the factors influencing the electromagnetic force and stiffness characteristics. The correlation between the excitation frequency and the force transfer rate under varying currents is examined, and real-time adjustment of negative stiffness is accomplished. The magnetic-type negative stiffness mechanism boasts several advantages, encompassing a compact structure, rapid response, high static load-bearing capacity density, minimal loss, and minimal environmental requirements. Nevertheless, it can prove challenging to control during operation and is significantly influenced by the current and magnetism, with its nonlinear characteristics being unstable.