## 2 Model of Quasi-Zero Stiffness Pneumatic Vibration Isolator
### Stiffness Analysis  
Taking a derivation of Equation (1) versus dimensionless displacement \(\hat{u}\), the vertical stiffness is expressed in the form of dimensionless as following:  
\[ \begin{array}{ll}\hat{K}_{s}=&4n\frac{\hat{V}^{\alpha\pi}_{\sigma\rho}\left( \hat{V}_{\sigma\rho 1}+\hat{V}_{t}\right)^{n}\tan^{2}\alpha}{\left(\hat{V}_{\sigma 1}+\hat{V}_{t}-2\hat{H}_{o}\tan \alpha+2\delta\tan\alpha\right)^{n+1}}+\\ &\frac{2\hat{A}}{\sqrt{1-\hat{u}^{2}}}\left[\hat{P}_{atm}-\hat{\
u}_{ep2}^{ \alpha}\left(\hat{\hat{V}}_{\sigma\rho 2+2\hat{A}\sqrt{1-\hat{H}_{o}^{2}}-2\hat{A} \sqrt{1-\hat{u}^{2}}}\right)^{n}\right]\left(\frac{1}{1-\hat{u}^{2}}\right) \\ &+\frac{\hat{u}^{2}}{1-\hat{u}^{2}}\frac{4n\mu A^{2}\hat{V}_{ep2}^{\alpha}\hat {V}_{\sigma\rho 2}^{\alpha}}{\left(\hat{V}_{\sigma\rho 2}+2\hat{A}\sqrt{1-\hat{u}^{2}}-2 \hat{A}\sqrt{1-\hat{u}^{2}}\right)^{n+1}}\end{array}  \tag{8}\]  
in which  
\[ \begin{array}{ll}\hat{K}_{S}=\frac{K_{S}b}{A_{1}P_{A1}}\hat{A}=\frac{A_{2}}{A _{1}};\;\hat{H}_{o}=\frac{H_{o}}{R+\hat{r}};\;\hat{P}_{atm}=\frac{P_{atm}}{P_{ \hat{H}_{1}}};\;\hat{u}=\frac{u}{R+\hat{r}};\\ \hat{V}_{cyl}=\frac{V_{cyl}}{A_{1}(R+\hat{r})};\;V_{cyl}=A_{1}h_{1};\;\hat{V}_ {cyl2}=\frac{V_{cyl2}}{A_{1}(R+\hat{r})};\;V_{cyl2}=A_{2}h_{2};\\ \hat{V}_{t}=\frac{V_{t}}{A_{1}(R+\hat{r})};\;\hat{V}_{cyl1}=\frac{V_{cyl1}}{V_ {cyl}+\hat{V}_{t}};\;\hat{V}_{cyl2}=\frac{V_{cyl2}}{V_{cyl2}};\hat{F}_{LBM}= \frac{F_{LBM}}{A_{1}P_{\sigma\rho 1}};\;\mu=\frac{P_{cyl2}}{P_{cyl}}.\end{array} \]  
In order to simplify the dynamic analysis in the next section, the restoring force given by Equation (1) can be expressed approximately (\(F_{ap}\)) by expanding the Taylor series around the wanted equilibrium position \(u=0\) as below:  
\[F_{ap}=F_{o}+a_{1}u+a_{2}u^{2}+a_{3}u^{3}+a_{4}u^{4}+a_{5}u^{5}+O(u^{6}) \tag{9}\]  
herein,  
\[ \begin{array}{ll}F_{o}=2A_{1}\left[P_{S01}\left(\frac{A_{1}h_{1}+V_{t}}{A_{ 1}h_{1}+V_{t}-2A_{1}H_{2}\tan\alpha}\right)^{n}-P_{atm}\right]\tan\alpha\\ a_{n}=\frac{1}{n}\left(\frac{dF_{LBM}{}^{n}}{du^{n}}+\frac{dF_{SCM}{}^{n}}{du^{n }}\right)\Big{|}_{u=0}n=1,2,3,4,5\end{array}  \tag{10}\]  
The original elastic force given by Equation (1) and its approximated form expressed by Equation (9) are compared as in Figure 2(a). The former is denoted by the solid line, while the latter is drawn by the dot line. It is interesting to see that the curve of Equation (1) is approximated to 5 th order through expanding Taylor series, which is in good agreement with the other. Furthermore, the error percentage of the approximated curve is lower than 4% compared with the exact solution, as shown in Figure 2(b). As observed, at the equilibrium position, the error is equal to zero, but when moving away from this position, error is increased. Indeed, within \(\hat{u}=-0.5\div 0.5\), the slope of the force curve is very low. Meanwhile, out of this region, the slope is increased significantly as shown in Figure 2(a), indicating that the rate of changing error within \(\hat{u}=-0.5\div 0.5\) is also very low, outside it is developed strongly. This means that the 5 th order polynomial is suitable for small amplitude oscillation. This approximated solution will be used for analyzing the primary resonance response of the proposed system. Figure 3(a) shows the dynamic stiffness surface in the space \(\hat{V}_{t},\hat{u},\hat{K}_{s}\) in which the pressure ratio \(\mu\) is 1.83, and other parameters include \(\hat{V}_{cyl}=1.875;\;\hat{V}_{cyl2}=0.248\). It revealed that the stiffness of the QZSPVI is strongly affected by the auxiliary tank volume \(V_{t}\) of the load-bearing mechanism. Reducing the value of \(V_{t}\) will increase the value of the system stiffness. Simultaneously, the asymmetric level of the stiffness curve around the equilibrium position is also developed. This is seen clearly in Figure 3(b), and created by cutting sections at \(\hat{V}_{t}=7.892,\;13.153,\;26.306\) (types of lines are presented in the top-right corner panel).