## 4 Dynamics and Isolation Performance of the Vibration Isolator
### Advantage of QCRM in Calculation Accuracy  
Unlike the classic NSC whose complex stiffness needs to be approximated by the Taylor series expansion in dynamic analysis, the NSC composed of QCRMs can directly provide the negative stiffness in the form of a quadratic polynomial. In this subsection, the NSC with a semicircular cam trajectory is exemplified and compared to show the harmful effects of stiffness approximation on response prediction and parameter selection. Except for its positive stiffness element constructed by the same arrangement in Figure 6, the structure and derivation of its negative stiffness was detailed in [23]. Hence, the non-dimensional stiffness of the NSC with a semicircular cam trajectory is obtained as:  
\[\overline{K}_{\text{cir}}=1-\frac{1}{\overline{\delta}_{\text{cir}}}\Bigg{[}1 +\frac{\overline{\delta}_{\text{cir}}-1}{\left(1-\overline{u}^{2}\right)^{3/2} }\Bigg{]} \tag{25}\]  
Its approximate stiffness from a second order Taylor expansion at the equilibrium point is:  
\[\overline{K}_{\text{cir}}=\frac{3(1-\overline{\delta}_{\text{cir}})}{2 \overline{\delta}_{\text{cir}}}\overline{u}^{2} \tag{26}\]  
where \(\overline{\delta}_{\text{cir}}\) is the non-dimensional pre-compressed length of the horizontal spring in the NSC with a semicircular cam trajectory, and equal to \(2k_{\text{v}}/k_{\text{n}}\) under the QZS condition.  
Figure 9 shows the comparisons between the non-dimensional original stiffness (OS) and the non-dimensional approximate stiffness (AS) with different \(\overline{\delta}_{\text{cir}}\). It can be found that the approximation accuracy is determined by both the structure parameter and the displacement. When \(\overline{\delta}_{\text{cir}}=0.4\), the approximation error is large, and only a very small part of the displacement around \(\overline{u}=0\) can meet the calculation requirement. When \(\overline{\delta}_{\text{cir}}=0.9\) (the choice of [23]), the approximation accuracy around small displacement is improved, but the error is still obvious at large displacement. The error caused by the stiffness approximation will reduce the reliability of the dynamic analysis of the vibration isolation system. To make matters worse, the choice of structure parameters and the working range of isolation displacement are limited to ensure that calculation results are acceptable. Higher order Taylor approximation can improve this issue, but it is rarely used because of the increased computational cost. By contrast, the stiffness provided by the QCRM-NSC is a quadratic function directly. This means that analysis under arbitrary displacements and parameters is reliable and easy to implement. The simple stiffness expression also facilitates the study of the NSC in isolators with nonlinear supporting or multi-layer nested structure.  
Figure 9: Comparisons between original and approximate stiffness curves of cam with semicircular trajectory.  
Figure 10 demonstrates the influence of stiffness approximation from the perspective of dynamic response. The numerical response solutions of the dynamic equations with the original and the approximate stiffness, calculated by Runge-Kutta method (RKM), are plotted as the pink dashed line and the blue dash-dotted line, respectively. Compared with the FCR of the approximate analytical solution obtained by HBM, plotted as the black solid and dotted line, it can be seen that different calculation methods have little effect on calculation accuracy, which has been verified in literature. However, the approximation of stiffness makes the FRCs quite different. In the approximate FRC, not only does the large tilt on the original FRC disappear, but the correct peak amplitude and jump frequencies are not obtained. Moreover, the unbounded response is incorrectly given as bounded.