## 1 Introduction
For instance, a compact QZS Stewart isolator, which can be applied widely to isolate vibration for 1-6 degrees of freedom systems, has been studied by [PERSON] et al. [18], obtained the resonant frequency of the system. This can be reduced in 6 degrees of freedom and the load-bearing ability remains. [PERSON] et al. [19] analyzed and experimented with the negative stiffness magnetic spring, consisting of a pair of coaxing ring permanent magnetics in parallel with the mechanical spring to reduce the resonance frequency and achieve the desired isolation effectiveness. Based on Maxwell normal stress, [PERSON] et al. [20] designed and experimentally validated an innovative magnetic negative stiffness isolation system for enhancing isolation effectiveness in the low frequency range. A magnetic isolator, which shows the decreasing of the peak frequency and expanding of the isolation frequency towards the low frequency band, is proposed by [PERSON] et al. [21]. [PERSON] et al. [22] proposed an electromagnetic active negative stiffness generator which can produce high-static low-dynamic stiffness. The analysis and experimental results proved the natural frequency of optical platform to be reduced and its isolation performance was enhanced effectively towards the low frequency bandwidth. Besides, [PERSON] et al. [23] developed an isolator containing three coils with a ring magnet by controlling the current to the coils to improve the isolation performance under violent vibration conditions. The experimental results proved that this isolator is suitable for large excitation and can be tuned online. Except for the permanent magnetic method, isolators using the electromagnetic method can easily adjust the dynamic stiffness to obtain desirable isolation effectiveness, but the drawback of the magnetic spring is that it has low load ability. Accordingly, application limitation of the magnetic spring is inevitable. Another type of elastic element, which can overcome the issues mentioned above, is an air spring, due to easy control of the spring coefficient and high bearing capacity. Thus, in recent years, this type has been widely applied in vibration isolation platforms [24; 25; 26], etc. Therefore, it is interesting to combine the mechanical and pneumatic system to develop quasi-zero stiffness vibration isolation models which can extend the isolation region toward low frequency [27; 28]. A quasi-zero stiffness isolator based on a set of two double-acting pneumatic linear actuators is analyzed and demonstrated experimentally by [PERSON] et al. [29], obtaining a reduction in resonance frequency. [PERSON] et al. [30] analyzed an air spring vibration isolation system based on a negative stiffness structure, showing that the load capacity of the isolator can be adjusted easily via the gas pressure. Motivated by the interest of the scientific community toward quasi-zero stiffness vibration systems and promising application, authors developed and realized the static analysis of a quasi-zero stiffness pneumatic vibration isolator (QZSPVI) [31] in which a pneumatic cylinder with an auxiliary tank worked as an elastic element, revealing that the stiffness curve of the isolator is asymmetrical around the desirable equilibrium position. This asymmetry can effect the amplitude-frequency response as well as the dynamic behavior of the isolator. As is known, the isolation system mentioned above is the combination of components such as joints, sliding parts, and so on, which could involve self-locking or frictional effects as discussed in [32; 33]. Accordingly, the nonlinear dynamic response of the QZSPVI, which is inherited by the sliding friction between the piston and cylinder, will be studied comprehensively in the present paper. First, by using a multi-scale method, the primary frequency-amplitude response will be analyzed.