## 5 Conclusions  
A pneumatic vibration isolation model with quasi-zero stiffness characteristic was studied and analyzed in this paper. The dynamic stiffness of the QZSPVI was determined and numerically calculated. Analysis results indicated that the stiffness curve is asymmetrical around the equilibrium position and its asymmetry may be reduced according to the increase in the volume of the auxiliary tank. The condition for which the minimum stiffness obtains quasi-zero value was analyzed. Besides, the expected working region in which the pressure in the stiffness correction mechanism is always positive was obtained. Figure 27: Attraction basin for \(\omega=6\)rad/s, \(\mu=0.997\), and \(\hat{V}_{t}=26.30\), the same other parameters as in Figure 22.  
The spring force curve was approximated through expanding Taylor series to order-5. By using multi-scale methods, the force transmissibility of the QZSPVI was then analyzed in detail, showing that the transmitted curve may be bent to the left or the right depending on the auxiliary tank volume and the pressure ratio. Furthermore, the stability of the analysis response was also studied. In addition, parameter bifurcation analysis of the QZSPVI had been realized through numerical integration from the original dynamic equation. Simultaneously, fixed points had been also calculated by using a Poincare map. The result proved that the system could occur in period-1, period-2, or period-3 of oscillation depending on the initial conditions. Under given excited frequencies, the attractor-basin phase portrait had been detected. From these analysis results, we confirmed that the lower the stiffness, the larger the isolation region toward the low frequency. The bigger the volume of the tank is, the higher isolation effectiveness is. Additionally, the basin of initial conditions to occur in the non-resonance solution is extended according to reduction in the stiffness. This model can be applied effectively in the field of low frequency vibration isolation system, having a high isolated load such as platform for vibration sensitive instruments, an isolated seat for vehicles, mount of engine, and so forth. Conceptualization, N.Y.P.V. ; methodology, T.D.L. and N.Y.P.V. ; software, N.Y.P.V. ; validation, N.Y.P.V. ; formal analysis, N.Y.P.V. ; investigation, T.D.L. ; resources, N.Y.P.V. ; data curation, T.D.L. ; writing--original draft preparation, T.D.L. and N.Y.P.V. ; writing--review and editing, T.D.L. ; visualization, T.D.L. and N.Y.P.V. ; supervision, T.D.L. and N.Y.P.V. All authors have read and agreed to the published version of the manuscript. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2020.36. The authors declare no conflict of interest. * _[PERSON] (2021)_ _[PERSON] Optimizing elastic potential energy via geometric nonlinear stiffness._ Commun. Nonlinear Sci. Numer. Simul.__**2021**_, 103_, 106035. [CrossRef]
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