## 4 Numerical Simulation
### Influence of Parameters on the Force Transmitted Curve
In this case, the pressure ratio is calculated through Equation (8) as shown in Figure 12 for which in the working region, the minimum stiffness of the QZSPVI will be nearly equal to zero, as depicted in Figure 13. It is noteworthy to see that the asymmetry of the system around the equilibrium position is reduced according to the increase in the volume of the auxiliary tank, indicating that the position at which the dynamic stiffness is quasi-zero moves towards the equilibrium position. Figure 11: Effect of the pressure ratio on the peak frequency \(\sigma_{p}\) of the QZSPVI for the same parameters as in Figure 8.  
Figure 10: Comparing force transmissibility of the QZSPVI and EPVI for various values of \(\mu\) and the same other parameters as in Figure 8 (the details of types of lines are presented in panel). In this case, the peak frequency and amplitude curve, with respect to the tank volume of the QZSPVI having the dynamic stiffness curves shown in Figure 13, is presented in Figure 14. It can be seen that if the dimensionless volume of the auxiliary tank is larger than \(21.24\) and smaller than \(43.85\) the value of \(\sigma_{\rm p}\) is positive, but this value is negative because the volume of auxiliary tank is out of this range, as shown in Figure 14a. Besides, increasing the tank volume will lead to a lessening in the peak amplitude as observed in Figure 14b, while this may have no appearance in the peak frequency. For instance, with \(\hat{V}_{t}=13.5\) the value of \(\sigma_{\rm p}\) is around \(-10.5\), but if \(\hat{V}_{t}=13.6\), this value is nearly equal to \(-21\). Furthermore, as the dimensionless volume of the auxiliary tank is changed from \(43.83\) (at point A) to \(43.88\) (at point B), the value of \(\sigma_{p}\) is changed suddenly from positive to negative. Simultaneously, the peak amplitude is decreased remarkably. In addition, it is essential to note that increasing the auxiliary tank volume will extend the region isolation toward low frequency, as shown in Figure 10.0 This indicates that improving the symmetry of the stiffness curve will increase the effectiveness of suppressing the force transmissibility from the load plate to the base. Figure 10.1 The relation of pressure ratio versus the auxiliary tank volume, the same other parameters as in the first case. Figure 10.2 The dynamic stiffness curve of the QZSPVI for the different values of the auxiliary tank volume as annotated in Figure. Meanwhile, the pressure ratio is calculated as in Figure 10.3  
Figure 10.4 The peak frequency curve (**a**) and peak amplitude curve (**b**) of the QZSPVI for the same parameters as in Figure 10.5  
Figure 10.6 Force transmissibility of the QZSPVI for various values of \(\hat{V}_{t}\) including 7.89; 14.46, 18.41, 24.99, 39.45, and the same other parameters as in Figure 14 (The annotation of line types indicated in panel of figure). The effect of nonlinearity in dynamic characteristic, the amplitude-frequency curve, can be bent as analyzed above, which indicates the appearance of the down (\(J_{D}\)) and up (\(J_{U}\)) jump frequency as expressed in Figure 10.7 Herein, the values of \(\mu\) and \(\hat{V}_{t}\) are given in the right-top corner panel, and the other parameters are the same as in Figure 10.8 The points on the response curve lying between \(J_{D}\) and \(J_{U}\) are unstable solutions because these points are in the unstable region calculated by Equation (34). When the down jump frequency which is also the peak frequency \(\sigma_{p}\) moves closely to zero, the phenomenon of the frequency jump may be neglected, meaning that all points on the amplitude-frequency curve are stable as numerically simulated by the dashed line in Figure 10.9 Likewise, the stable branches of the proposed model with various values of \(\mu\) and \(\hat{V}_{t}\) are denoted in Figures 10 and 15. The detailed annotation for line types is presented in the lower-left and upper-left corner of Figures 10 and 15, respectively.