## 1 Introduction  
Along with technological developments in engineering sciences and micro-fabricate systems, ultra-precision measurements have been studied deeply [1, 2]. However, the accuracy as well as the lifespan of the manufacturing systems or measurement depends on many causes, in which vibration is one of the main reasons. Hence, minimizing the transmissibility of unpleasant vibrations from source to isolated object is a significant demand. In recent years, nonlinear structures with quasi-zero stiffness characteristic have offered promising effectiveness for preventing low frequency vibrations [3, 4]. Hence, studying dynamic characteristics of these nonlinear systems or structures has drawn increasing interest from researchers. [PERSON] et al. [5] analyzed the dynamic response of quasi-zero stiffness isolator, as well as the jump-up and jump-down phenomenon of the amplitude-frequency curve. The nonlinear dynamic response of an isolator using the negative stiffness structure is experimentally investigated by [PERSON] et al. [6]. The result obtained that the amplitude-frequency curve moves towards the low frequency. [PERSON] et al. [7] built and analyzed the nonlinear dynamic model of a prototype of quasi-zero stiffness (QZS) vibration isolation model with cam-roller-spring, obtaining an effective isolation response at low frequency bands. A useful and compact prototype of an oscillator, including two oblique springs and one vertical spring to get the QZS characteristic, was designed and experimented by [PERSON] et al. [8]. A novel study of compact isolators, combining a symmetrically scissor-like structure in horizontal direction to a spring-mass-damper system in order to improve the isolation performance in three directions with high-static and quasi-zero dynamic stiffness characteristics, is proposed by [PERSON] et al. [9]. Instead of using the approximate technique, the dynamic equation with original irrational nonlinearity was built and analyzed by [PERSON] et al. [10], which means that the prediction of a large displacement can obtain high accuracy. [PERSON] et al. [11] studied the vibration transmissibility of a novel two-stage quasi-zero stiffness nonlinear oscillator subjected to harmonic force excitation. Vibration characteristics of a fluid-conveying pipe supported by two nonlinear isolators featuring quasi-zero stiffness is numerically simulated by [PERSON] et al. [12]. The simulated results revealed that the natural vibration frequency of the system can be shifted to the low frequency region. The static and dynamic of a one degree of freedom quasi-zero stiffness oscillator for large deflection was analyzed by [PERSON] [13]. Unlike the above system, another perspective for designing low frequency vibration isolation models was studied and experimented by [PERSON] et al. [14] in which a novel compact isolation mount is proposed based on the X-shaped structure with a small size, aiming to adjust both the quasi-zero stiffness and loading capacity. Through a specific nonlinear mechanical mode, the overall approach was generalized by [PERSON] [15] in which the parameters of the system are optimized to enhance the elastic potential energy, which obtains a quasi-zero stiffness behavior achieving at maximum displacement. Additionally, [PERSON] et al. [16] presented comprehensive methodology of low frequency isolation which can be classified into three mechanisms, including stiffness adjustment, auxiliary mass, and damping mechanism. Oscillators with the quasi-zero stiffness characteristic mentioned above are to use the mechanical elastic elements such as coil spring. It is simple and easy to apply in practice. However, a very important aspect of using these models for vibration isolation is that changing the isolated load can result in a reduction in isolation effectiveness because of fixed elastic parameters, meaning that the isolated load influences the frequency-amplitude response as demonstrated by [PERSON] et al. [17]. Besides, magnetic springs have been used to study the low frequency vibration isolator.