## 1 Introduction  
Low and ultra-low-frequency vibration isolation systems are widely used to attenuate disturbances that affect device accuracy in the fields of precision/ultra-precision apparatus, navigation, and aerospace, among others. In general, a passive linear isolator can reduce vibration when the excitation frequency exceeds \(\sqrt{2}\) times the natural frequency of the isolation system, and the smaller stiffness the better isolation performance, but the weaker to anti-disturbance [1]. To achieve better vibration isolation, the effective isolation bandwidth can be enlarged using a relatively low stiffness system at the equilibrium position, or quasi-zero stiffness (QZS). Theoretically, all vibrations could be eliminated at a system stiffness of zero. However, such a system would readily cause relatively large system deflections and potential instability. To address this issue, non-linear, high-static-low-dynamic stiffness (HSLDS) springs could be used to achieve a stiffness of nearly zero and better vibration isolation effects. Furthermore, the simultaneous increase in system stiffness with increased deflection could help the system to quickly recover an equilibrium position, effectively enhancing the system's capacities for vibration isolation and stability. Non-linear passive vibration isolation has been intensively investigated [2]. Typical QZS isolation systems include the following: parallel positive and negative stiffness, non-linear Euler spring deflection, and single-pendulum principle-based isolations. The latter includes the use of folded, inverted, and conical pendulums to achieve high-static-low-dynamic QZS isolation. Additional options for isolation include a friction-rolling pendulum and viscoelastic materials. To briefly review this topic, [PERSON] et al. initially proposed the concept of a QZS isolator and analyzed the dynamic characteristics [3; 4; 5]. Subsequent research on QZS isolation systems developed rapidly following the demonstration of QZS isolator actualization using parallel positive and negative stiffness. Under low-frequency harmonic excitation, a QZS vibration isolation seat comprising vertical positive and negative stiffness linear helical springs was found to reduce the root mean square of deviation to 67.2% [6]. An experiment was designed in which the load was fixed at the middle of the two vertical springs, and adjacent magnets with like poles were arranged on both sides of the springs to form a HSLDS unit. [PERSON] proposed an integrated design approach of a QZS mechanism [7]. This unit exhibited better performance in response to a small vibration disturbance compared with a unit comprising linear springs [8]. The law of magnets, in which like poles repel and unlike poles attract, can be used to explain the similarities of like poles with a negative spring and of unlike poles with a positive spring. Negative and positive springs with rubber ligaments, among other elements, comprise the QZS isolation system. An experimental analysis of this model demonstrated the validity of the system [9; 10]. The characterization of a QZS isolation system composed of a Euler buckled beam as the negative stiffness corrector and a positive stiffness spring was analogous to the characterization of a soft spring [11]. Compared with a linear spring, the QZS isolation system performed better at a low frequency, with a decreased carrying capacity [12]. An experimental analysis of a QZS isolator composed of a cam-roller-spring as the negative stiffness corrector and a positive linear spring demonstrated an excellent capacity regarding either transmissibility or the initial isolation frequency, despite discontinuity of stiffness across the entire bandwidth [13]. In addition, the characteristic critical stability of the Euler column led to an ideal low-frequency vertical vibration isolation performance while experiencing elastic buckling under interference from mass load-induced pre-compression [14].