## 2 Fundamental Principles of Quasi-Zero Stiffness
### Basic Types of QZS Vibration Isolators
#### 2.2.1 Linear Spring as Negative Stiffness Mechanism  
This mechanism commonly employs inclined mechanical springs as the negative stiffness component. It is renowned for its high elastic potential energy yet exhibits a limited load-carrying capacity. [PERSON] [35] optimized the three-spring configuration and devised a lateral-adjusting mechanism. This allows the vibration isolator to encompass a broader spectrum of vibration isolation frequencies without compromising the dimensional constraints. Consequently, it can fulfill the vibration isolation requirements across varying loading conditions. [PERSON] [36] proposed a six-degree-of-freedom QZS isolation platform constructed from six modules utilizing compressed-spring structures, as depicted in Figure 5a. The underlying QZS principles in both the translational and torsional directions are expounded upon. Upon establishment of the static model of the platform and its linearization at the static equilibrium position, it was found that linear cross-coupling effects emerge  
Figure 4: Schematic diagram of the high static and low dynamic stiffness vibration isolator containing magnets proposed by [PERSON] et al. [33; 34]. Figure 3: Schematic diagram of the three-spring QZS vibration isolator proposed by [PERSON] et al. [33; 34]. in the presence of static loads. Subsequent linearized dynamic analysis of the isolation platform revealed that through spring compression, the isolation frequency band can be extended to encompass a lower frequency range in all six directions. A novel linkage antivibration structure comprised a symmetric polygonal configuration and a vertical spring [37], as depicted in Figure 5b. Initial investigation into the static stiffness behavior unveiled the inherent positive and negative stiffness compensation mechanisms of linear springs within the symmetric polygonal structure. By fine-tuning the structural parameters, the linear spring can effectively offset the linear negative stiffness of the symmetric polygonal structure, thereby achieving enhanced quasi-zero stiffness characteristics over a broader stroke. To mitigate the spring-back behavior of conventional flexure hinges, a positive and negative stiffness matching method was employed [38]. The spring four-bar linkage (4 BSL) was engineered as a negative stiffness rotation mechanism, while the inner and outer rings of the flexure hinge (IORFH) served as positive stiffness mechanisms. Through coupling the 4 BSL and the IORFH, a novel zero-stiffness flexure hinge was devised. [PERSON] [27] proposed a two-degree-of-freedom seismic system model based on a smooth discontinuous oscillator. The model comprised two horizontally and orthogonally oriented vibration isolators, each exhibiting stable QZS characteristics, thereby reducing the initial isolation frequency to below 0.5 Hz. [PERSON] [39] investigated a vibration isolator comprising a vertical linear spring and two nonlinear inclined springs. The authors analyzed the evolution of the system from a doubly periodic bifurcation to chaotic motion through dynamics analysis. They ascertained the frequency of the initial doubly periodic bifurcation through both approximate analysis and numerical simulation, alongside calculating the damping and other system parameters. [PERSON] [40] proposed a quasi-zero stiffness vibration isolator featuring a branch cam-roller mechanism comprising three springs to mitigate the challenge of transmitting the large-amplitude excitation induced by circular cams. The dynamic response of the vibration isolator was scrutinized and validated under harmonic and shock excitation utilizing the fourth-order Runge-Kutta method. While this vibration isolator was tailored for isolating vibrations at specific frequencies rather than random vibrations, it significantly reduced the maximum stiffness by a factor of 10 compared to its predecessor.