## 1 Introduction  
Vibration exists widely in nature, which is often defined as the mechanical movement of an object oscillating around an equilibrium position. However, vibration is undesirable in many domains, and unwanted vibrations can cause serious harm to mechanical systems, human health, and so on. For example, vibrations of vehicle seats often reduce ride comfort [1], vibrations of machine tools will reduce the manufacturing precision [2], and earthquake-induced vibrations will damage buildings [3]. In particular, mechanical vibrations are often a byproduct of mechanical systems, and it is impossible to eliminate vibration sources in practice. Vibration isolation can prevent vibration transmission between different components, so it is important to implement vibration isolators. Nowadays, linear isolators are always employed to reduce mechanical vibrations. According to the linear isolation theory, however, vibrations cannot be suppressed when the excitation frequency is less than \(\sqrt{2}\omega_{n}\) (\(\omega_{n}\) is the natural frequency of the mechanical system) [4]. This is a significant drawback of linear isolators since many systems require low-frequency vibration isolation. One possible strategy for extending isolation ability to low-frequency bands is to reduce the stiffness of the system, but this approach will lead to low loading capacity and system instability. Therefore, innovative structures are needed to realize low-frequency or ultra-low-frequency vibration isolation. In recent years, quasi-zero stiffness (QZS) vibration isolators have been proven to be an effective and feasible means to solve low-frequency vibration problems. The concept of QZS was first proposed by [PERSON] [5] and is described as positive stiffness and negative stiffness. QZS has the unique feature of high static and low dynamic (HSLD) stiffness near the static equilibrium position, which can greatly reduce the dynamic stiffness without reducing the static stiffness. Therefore, QZS can be utilized to realize low-frequency vibration control and solve the contradiction between low natural frequency and small static deformation. In particular, the key point of QZS is to design proper negative-stiffness structures [6; 7]. According to the literature, there are some ways of achieving negative stiffness. The first way is to use oblique springs. [PERSON] et al. [8] built a QZS-based vibration isolation system consisting of a vertical spring and two oblique springs in parallel. The second way is to use a spring and a rod. [PERSON] et al. [9] proposed a QZS vibration isolator based on the'spring-rod' structure. The third way is to use buckling beams. [PERSON] et al. [10] built a large negative stiffness under small displacement by utilizing the Euler buckling beam. The fourth way is to use cams. [PERSON] et al. [11] built a QZS vibration isolator by using a cam structure whose profile was designed to follow the force-displacement relation of QZS. The fifth way is to use magnets. [PERSON] et al. [12] used a pair of magnetic rings to generate QZS for low-frequency simultaneous vibration isolation and energy harvesting. To date, QZS has shown great potential in the field of low-frequency vibration isolation. In recent years, the implementation of permanent magnets in QZS vibration isolators has attracted significant attention from many scholars. Among different kinds of negative-stiffness structures, the magnetic spring is commonly built by using a mobile magnet and a stationary magnet, which utilizes the repulsive or attractive forces between two permanent magnets to achieve negative stiffness. [PERSON] and [PERSON] [13] gave a comprehensive review on magnetic negative stiffness in vibration isolation systems from theoretical models to engineering applications. Generally speaking, the shapes of magnetic springs can be rectangular [14] or circular [15] according to the shapes of the permanent magnets.