## 2 Model of Quasi-Zero Stiffness Pneumatic Vibration Isolator
### Stiffness Analysis
Furthermore, because of the asymmetry of the stiffness curve, the system can not achieve the lowest stiffness at the equilibrium position (\(u=0\)). As observed, for the 1 st  
dimensionless value of the tank volume, the position defined by \(\hat{u}_{ls}\) at which the stiffness achieving the lowest value is approximated 0.084 and the value of \(\hat{u}_{ls}\) can be reduced to 0.034 and 0.09 as the auxiliary tank volume is grown to 13.153 and 26.306, respectively. The lowest stiffness position can be predicted by differentiating Equation (9) to second order versus the displacement as expressed in Equation (11), and then it equals zero. It can be seen that this position will be asymptotic to the wanted equilibrium position when the auxiliary tank volume \(\hat{V}_{t}\) is large enough, as shown in Figure 5.  
\[2a_{2}+6a_{3}u+12a_{4}u^{2}+20a_{5}u^{3}=0 \tag{11}\]  
In addition, from Equation (8) for \(P_{e1}\) = 2 bar, by setting \(K_{s}\) = 0, the relationship between the pressure ratio and the auxiliary tank volume for which the pressure in cylinder of the SCM will be calculated is marked in Figure 6a. The pressure ratio curve is formed by cutting sections at the various values of the auxiliary tank volume \(\hat{V}_{t}\) as shown in Figure 6b, in which the values of \(\hat{V}_{t}\) and the notation for line types are noted in the middle of the panel. It can be observed that in the case of \(\hat{V}_{t}=34.198\), at the extremum point A corresponding to the pressure ratio \(\mu\) = 0.905, the oscillation system can obtain the quasi-zero stiffness at only one position \(\hat{u}_{A}=0.008\) but \(\mu\) > 0.905, there may exist two positions at which the stiffness is also nearly equal to zero such as at point A\({}_{1}\) (\(\hat{u}=0.381\)) and point A\({}_{2}\) (\(\hat{u}=-0.364\)) for \(\mu\) = 1.2. It is shown that lessening the volume \(\hat{V}_{t}\) leads to the increase of the pressure ratio. For instance, \(\hat{V}_{t}=21.045,\ 13.153,\) and 7.982, the lowest  
Figure 4: (**a**) The vertical stiffness surface in the space \((\hat{V}_{t},\hat{u},\hat{K}_{s})\) for pressure ratio \(\mu\) = 1.83; (**b**) The dynamic stiffness curves for different values of dimensionless tank volume given in the right top corner panel. Figure 3: (**a**) Comparison between of the original (solid line) and 5 th order approximated (dot line) curve of the elastic force; (**b**) The error percentage between the exact solution and approximation one. pressure ratios to obtain quasi-zero stiffness are 1.096, 1.379, and 1.834 are marked by the extremum point such as B, C, and D, respectively. The slope of the stiffness curve at points \(A(\hat{a}_{A}=0.008),B(\hat{a}_{B}=0.021),C(\hat{a}_{C}=0.046),D(\hat{a}_{D}=0.083)\) is nearly equal to zero as depicted in Figure 6c (all parameters are the same as in Figure 6b). Meanwhile, for other positions (A\({}_{1}\), A\({}_{2}\), etc.) these are nonzero, indicating that for the value of \(\mu\) at extreme points of the pressure ratio curve, the dynamic stiffness is equal to zero and obtains the minimum value in the expected working region. This result is also confirmed in Figure 7a, herein the values of \(\mu\) and annotation of the line types are given in the top right corner panel of the figure. Figure 5: The influence of auxiliary tank volume \(\hat{V}_{t}\) on the minimum stiffness position. Figure 6: (a). The quasi-zero stiffness surface in the space \((\hat{V}_{t},\hat{a},\mu)\); (b). The pressure ratio curve for various values of \(\hat{V}_{t}\); (c). Slope of stiffness curve in dimensionless form. As mentioned above, points may exist where the stiffness of the QZSPVI is quasi-zero but the slope of the stiffness curve is nonzero. For example, for \(\mu=1.2\),\(\tilde{V}_{t}=34.198\), the system has quasi-zero stiffness at the position A\({}_{1}\) (\(\hat{u}=0.381\)) and A\({}_{2}\) (\(\hat{u}=-0.364\)) (seen in Figure 6b), but at which the slope of the stiffness curve is larger than zero at A\({}_{2}\) and smaller than zero at A\({}_{1}\) (seen in Figure 6c).