## 4 Numerical Simulation  
### Influence of Parameters on the Force Transmitted Curve  
In this section, the effects of the pressure ratio and volume of auxiliary tank on the force transmissibility of the QZSPVI are investigated. Firstly, the pressure ratio will be studied. Meanwhile, a set of parameters of the isolation system comprising of \(\xi=0.1;\;\hat{A}=0.16;\;\hat{V}_{t}=13.153\); \(\hat{H}_{o}=0.53\) is given, simultaneously at the equilibrium position, the pressure \(P_{\rm{ep1}}\) of the LBM is set at the value of 1.85 bar, for which the weight of the isolated load is 122.41 kg. As observed in Figure 8, the dynamic stiffness (denoted by the solid line) of the QZSPVI for various values of \(\mu\) within 1.350 to 1.377 is always smaller than that of the EPVI (exhibited by the dashed line) when the position of the load plate is in the working region (\(|a|<\hat{H}_{o}\)). Especially, when \(\mu\) = 1.379, the minimum stiffness of the QZSPVI is nearly equal to zero and it will be increased according to reduction in the pressure ratio. The result is the force transmitted curves shown in Figure 9 (the detailed annotation of the type of lines and the chosen values of \(\mu\) are presented in the left-upper corner panel of the figure). It can be argued that the nonlinearity of the proposed model bends the force transmitted curves to the left or the right depending on the pressure ratio. In the case of \(\mu\) = 1.379, the curve is bent to the left, indicating that this is the case of the soft nonlinear system. Reducing the pressure ratio will lead to reducing the soft nonlinearity to become the hard nonlinearity. As observed, the amplitude-frequency curve is bent to the right for \(\mu\) = 1.370. However, the bending level of the response curve to the right may be lessened as there is a lessening in the pressure ratio such as \(\mu\) = 1.360, \(\mu\) = 1.350, and even trends no bending for \(\mu\) = 0, as shown in Figure 9. This is evident because the pressure of the stiffness corrected mechanism will be reduced according to the reduction in the value of \(\mu\), meaning that the effects of this mechanism on the dynamic stiffness of the QZSPVI become light, and hence, the peak amplitude is increased. Furthermore, the increase of the pressure ratio will broaden the isolation region of the hardening system toward the low frequency as plotted in Figure 10. On the other hand, the comparison between the proposed system with the EPVI in which the SCM is removed is shown in Figure 10. It is noteworthy to observe that the suppression of the force transmissibility from the load plate to the base of the QZSPVI is better than the EPVI, meaning that isolation region and vibration attenuated ratio of the former is larger than those of the latter. As analyzed above, the amplitude-frequency response of the proposed system can be soft or hard depending on the peak frequency \(\sigma_{p}\). Indeed, this value is negative corresponding to the response curve bending to the left. Contrastingly, the bending to the right appeared for the positive value of \(\sigma_{p}\). Figure 11 shows the influence of the pressure ratio on the peak frequency of the QZSPVI for the same parameters as in Figure 8. It is interesting to see that if the pressure ratio is larger than 1.376, the value of \(\sigma_{p}\) is negative, indicating that the response curve is bent to the left as confirmed in Figure 9. Meanwhile, the bending of the curve to the right occurred for \(\mu<1.376\) due to \(\sigma_{p}>0\). Secondly, the dimensionless volume of the auxiliary tank of the load-bearing mechanism is set at the values 7.892; 14.460; 18.814; 24.992; 39.453; 47.358, while the other values for the pressure ratio are the same as in the first case.