## 2 Fundamental Principles of Quasi-Zero Stiffness
### Several Types of Negative Stiffness Actuator
#### 2.3.3 Magnetic Spring as Negative Stiffness Actuator  
This mechanism employs permanent magnets or electromagnets as actuators for negative stiffness. Negative stiffness characteristics are attained by incorporating cams or other mechanical devices into the structural configuration. Its notable characteristic lies in the requirement to counterbalance or diminish the system response force by utilizing self-generated force. This imposes greater demands on the materials and structure. [PERSON] employed three permanent magnets with opposite poles to achieve negative stiffness, resulting in a reduction in the static equilibrium position at the intrinsic frequency. Magnetic springs utilize the repulsive or attractive forces between magnets to achieve negative stiffness. They are often used in applications where non-contact force transmission is desirable. For example, [PERSON] et al. [32] studied the dynamic behavior of magnetic springs and their potential in vibration isolation systems. Magnetic springs offer the advantage of tunable stiffness characteristics by varying the distance or orientation of the magnets. This tunability allows for precise control of the system's dynamic response, making them suitable for a wide range of vibration isolation applications. Additionally, magnetic springs can be combined with other mechanical elements to create hybrid systems that further enhance their performance. Nevertheless, the load on the vibration isolation system significantly affects the vibration isolation effectiveness, rendering it insufficient for low-frequency overload situations. Figure 11 illustrates a novel semi-active electromagnetic quasi-zero stiffness (SEQZS) vibration isolator proposed by [PERSON] [56]. The vibration isolator is designed to broaden the bandwidth of low-frequency or ultra-low-frequency vibration isolation. The article meticulously derives the theoretical equations for both static and dynamic analyses of the SEQZS vibration isolator under harmonic excitation and conducts parametric analyses to offer design insights. Furthermore, a prototype is fabricated, and the design concept is experimentally validated. The experimental results  
Figure 10: Air spring structures of a QZS isolator: (**a**) disc spring structures of the QZS isolator model proposed by [PERSON] [54]; and (**b**) schematic diagram of the nonlinear stiffness-modulated anti-vibration structure proposed by [PERSON] [55]. validate the theoretical formulations and illustrate that the vibration isolation performance of the proposed SEQZS vibration isolator significantly surpasses that of the corresponding linear isolator. Figure 12 illustrates the proposed concept for an adjustable negative stiffness spring (SMNS) based on Maxwell normal stress, as introduced by [PERSON] [57]. This spring enhances the adjustable stiffness range and energy utilization efficiency through a newly devised magnetic circuit. Additionally, this electromagnetic negative stiffness device boasts the advantages of being frictionless, springback-free, compact, and easily controllable. The paper introduces an electromagnetic force analysis model based on magnetic circuit analysis and conducts parametric analysis. A high-static, low-dynamic-stiffness isolator is fabricated by connecting the SMNS in parallel with a linear vibration isolator. The experimental results demonstrate that the SMNS yields real-time adjustable negative stiffness, broadens the isolation bandwidth, and markedly enhances the vibration isolation performance. This study proposes an innovative solution to enhance the performance of low-frequency vibration isolation and expand the isolation bandwidth. This is of considerable significance in the realm of vibration isolation technology. [PERSON] [58] devised a hybrid magnetic-air QZS vibration isolation system. The system employs an electromagnetic spring to supply negative stiffness and a vertically positioned air spring as a positive stiffness mechanism.