## 2 Model of Quasi-Zero Stiffness Pneumatic Vibration Isolator  
### Description  
The QZSPVI, which is constructed by a positive stiffness structure connecting in parallel with a negative stiffness one, is described as in Figure 1a. The formerly named load-bearing mechanism (LBM), consisting of the wedges 7, rollers 5, and cylinder 1, is used to support the load plate 8, while the latter is used to correct the dynamic stiffness of the system and is called the stiffness correction mechanism (SCM), including semicircular cams 6, rollers 4, and cylinder 2. It is noted that the wedge 7 and cam 6 are fixed on the leg 14 which is fixed on the load plate 8 and only moves vertically through the guidance bars 11 and rollers 9, meanwhile the rollers 5 and 4 only realize horizontal motion via the rods and cylinders in which the cylinders can only slide horizontally without frictional phenomenon thanks to the sliding on fixed guidance bars 12 of sliders 13. An auxiliary tank 3 connects to the cylinder 1 of the LBM to improve symmetry of the dynamic stiffness curve, and the stiffness of the QZSPVI can offer desirable low values by adjusting the low pressure in cylinders 1 and 2. During the operation of the system, it can spend three states as shown in Figure 1b, including the undeformed, desirable equilibrium, and arbitrary position. For example, at the undeformed one, the air in the cylinders 1 and 2 is not compressed, meaning that no force is generated by the LBM and SCM. The vertical distance (\(H_{\text{o}}\)) between the centers of the cam 6 and roller 4 is called the static deformation of the system. The desirable equilibrium position (DEP), determined as the load plate, is loaded until the centers of cam and roller are on the same horizontal line. At this state, the vertical force which acts on the load plate is only caused by the load-bearing structure, whilst the arbitrary state is determined when the load plate moves away from the equilibrium position at distance \(u<H_{\text{o}}\), at which the load plate will be acted on by two vertical forces generated by both the LBM and SCMSupposing at an arbitrary position, the vertical forces generated by the elastic forces of the load-bearing mechanism (\(F_{\text{LBM}}\)) and the stiffness corrected mechanism (\(F_{\text{SCM}}\)) are determined as the following:  
\[F_{\text{s}}=F_{LBM}+F_{\text{SCM}} \tag{1}\]  
As analyzed in [28], the \(F_{\text{LBM}}\) and \(F_{\text{SCM}}\) are expressed as below  
\[F_{LBM}=2A_{1}\bigg{[}P_{so1}\bigg{(}\frac{A_{1}h_{1}+V_{t}}{A_{1}h_{1}+V_{t}- 2A_{1}H_{o}\tan\alpha+2A_{1}u\tan\alpha}\bigg{)}^{n}-P_{atm}\bigg{]}\tan\alpha \tag{2}\]  
\[F_{\text{SCM}}=2A_{2}\Bigg{[} \begin{array}{c}P_{so2}\bigg{(}\frac{A_{2}h_{2} }{A_{2}h_{2}+2A_{2}\sqrt{(R+r)^{2}-H_{o}^{2}}-2A_{2}\sqrt{(R+r)^{2}-u^{2}}} \bigg{)}^{n}\\ -P_{atm}\end{array} \Bigg{]}\frac{u}{\sqrt{\big{(}R+r\big{)}^{2}-u^{2}}} \tag{3}\]  
in which \(n\) is the ratio of specific heat capacity, \(R\) and \(r\) are the radii of the semicircular cam and roller in m, \(P_{atm}\) is the ambient pressure in Pa, while \(P_{so}\) is the pressure in cylinder at the undeformed position in Pa, the area of piston is denoted by \(A\) in m\({}^{2}\), \(h\) is the working stroke length of piston in m as denoted in Figure 1a, \(V_{t}\) is the volume of the auxiliary tank, \(u\) is the relative displacement between the base and load plate. It is noted that parameters that are representative for pneumatic springs 1 and 2 are noted by subscripts \"1\"and \"2\", respectively. As described above, the load plated is only supported through the pressure which is generated by the LBM. Hence, it is necessary to determine the relationship between the pressure in cylinder 1 and the weight of the isolated object as expressed by Equation (4), meaning that thanks to this relation, the system can obtain the desired equilibrium position. \[P_{ep1}=\frac{Mg}{2A_{1}\tan\phi}+P_{atm} \tag{4}\]  
Figure 1: (**a**) Schematic diagram of the QZSPVI composed by the LBM and SCM, (**b**) Specific states of the QZSPVI.