## 3 QZS Characteristics of QVIP
### Parametric Study  
The initial model parameters are set as \(L=0.1\), \(\varepsilon=1\), \(\lambda=1\), \(\mu=0.68\), \(\
u=1\), \(\xi=1\), \(\theta=\pi/6\), \(k=2000\). At first, using different rod length ratios (\(\lambda\)), the results are shown in Figure 2a; as \(\lambda\) gradually increases from 0 (\(\lambda<1\)), the equivalent nonlinear stiffness force (\(F_{k}\)) significantly increases; when \(\lambda=1\), the QZS range appears; as \(\lambda\) further increases beyond 1, \(F_{k}\) decreases, and the QZS range shortens. By adjusting \(\lambda\), the \(F_{k}\) curves near \(\lambda=1\) are shown in Figure 2b. It is observed that an extended QZS range occurs only when \(\lambda=1\). If \(\lambda\
eq 1\), negative stiffness appears after \(F_{k}\) reaches its maximum value. To facilitate comparison of QZS state in the QVIP, the system is considered to achieve the QZS state when the slope of the equivalent nonlinear stiffness force curve reaches \(0<k_{\text{FK}}<0.1\). \(\eta\) is the ratio of the QZS width to the maximum compression displacement. The influence of rod length \(L\) is shown in Figure 3. As \(L\) increases, \(F_{k}\) increases significantly, while the \(\eta\) remains invariant. The influence of stiffness ratio \(\mu\) is shown in Figure 4. As \(\mu\) increases, the positive stiffness also increases, it is necessary to increase \(\mu\) to a certain value to achieve QZS for QVIP, e.g., \(\mu=0.68\). More than this value, the stiffness is always positive as the compression continues and QZS will not appear. As shown in Figure 5, the equivalent nonlinear stiffness force of the QVIP is related to the stiffness ratio \(\
u\). Increasing \(\
u\) enhances the maximum equivalent nonlinear stiffness force under the same compression while reducing the slope \(k_{\text{FK}}\) of the equivalent nonlinear stiffness force. By adjusting the ratio \(\mu\) between the stiffness of the inclined springs \(k_{p}\) and \(k\), a wide QZS range can be achieved for the QVIP under different stiffness ratios \(\
u\). As \(\
u\) increases, the required compression for the QIP platform to reach the QZS state also increases, leading to a decrease in the QZS range ratio \(\eta\). Consequently, the QVIP becomes less stable under ultra-low frequency vibrations. Figure 3: Different rod length \(L\) influence on static stiffness force (parameters are set as \(\varepsilon=1\), \(\lambda=1\), \(\mu=0.68\), \(\
u=1\), \(\xi=1\), \(k=2000\)). Figure 2: Parametric influence on static stiffness force (parameters are set as \(L=0.1\), \(\varepsilon=1\), \(\mu=0.68\), \(\
u=1\), \(\xi=1\), \(k=2000\)): (**a**) using different rod length ratio \(\lambda\), and (**b**) when length ratios near \(\lambda=1\). Figure 4: The influence of stiffness ratios \(\
u\) and \(\mu\) on static stiffness force (parameters are set as \(L=0.1\), \(\varepsilon=1\), \(\lambda=1\), \(\xi=1\), \(k=2000\)): (**a**) when stiffness ratio \(\
u=1\), (**b**) when stiffness ratio \(\
u=3.18\), (**c**) when stiffness ratio \(\
u=5.28\), and (**d**) comparison of the optimal QZS effect in (**a–c**). Figure 5: The influence of stiffness ratios \(\mu\) and \(\
u\) on static stiffness force (parameters are set as \(L=0.1\), \(\varepsilon=1\), \(\lambda=1\), \(\xi=1\), \(k=2000\)): (**a**) when stiffness ratio \(\mu=0.68\), (**b**) when stiffness ratio \(\mu=0.82\), (**c**) when stiffness ratio \(\mu=1.1\), and (**d**) comparison of the optimal QZS effect in (**a–c**). As can be seen from Figure 6, the equivalent nonlinear stiffness force of the QVIP is related to the initial angle \(\theta\). Increasing the initial angle \(\theta\) will increase the slope \(k_{Fk}\) of the system's equivalent nonlinear stiffness force while reducing the maximum equivalent nonlinear stiffness force under the same compression amount.