## 5 Parametric Study
### Comprehensive Study of Vibration Isolation  
In the analysis of vibration isolation performance, the influence of vertical spring stiffness \(k_{v}\) and damping \(c_{v}\) on vibration isolation performance is mainly studied. Initially the conditions for achieving quasi-zero stiffness are disregarded, and only the vertical spring stiffness \(k_{v}\) is changed, allowing for exploration of the variation of the vibration transmissibility in relation to \(k_{v}\). Figure 5a shows the vibration transmissibility curve under different \(k_{v}\), and the values of other parameters are the same as those shown in Table 1. The base excitation amplitude is \(A=0.05\) m/s\({}^{2}\). It can be seen that when \(k_{v}\) changes from 1000 N/m to 2000 N/m, the peak of the vibration transmissibility curve gradually shifts to the left and the low orbit gradually decreases, indicating an improvement in vibration isolation performance. However, when \(k_{v}\) continues to increase, the peak of the vibration transmissibility curve moves to the right and the low orbit increases, indicating a deterioration in vibration isolation performance. Figure 4: The launch velocity \(v_{0}\) curve versus pre-compression length \(\delta\) of the linear design and the proposed design with (**a**) parameter set 1 and (**b**) parameter set 2. Figure 5: (**a**) The vibration transmissibility \(T\) and (**b**) the nonlinear restoring force with different \(k_{v}\) of the proposed design. To understand the essence of this changing pattern, the relation curve between nonlinear restoring force \(F\) and displacement \(x\) is drawn in Figure 5b. It can be seen that when \(k_{v}=1000,~{}1500~{}\text{N/m}\), the negative stiffness provided by the two vertical springs is insufficient to offset the positive stiffness of the transverse springs, thus exhibiting positive stiffness characteristics near \(x=0\). The smaller \(k_{v}\) is, the greater the positive stiffness, making the system tend towards a linear vibration isolation system. When \(k_{v}=2000~{}\text{N/m}\), according to Equation (7), the condition for quasi-zero stiffness is exactly met, showing zero stiffness characteristics near \(x=0\). Figure 5a indicates that under this condition, the peak is the smallest, the low-frequency range is the lowest, and the vibration isolation performance is the best. When \(k_{v}=2500,~{}3000~{}\text{N/m}\), the negative stiffness is too large, and the system exhibits negative stiffness near \(x=0\). The vibration transmissibility curve displays abnormal distortion, and the vibration isolation performance becomes worse. From the above analysis, it is clear that the best vibration isolation performance is achieved when the structural parameters satisfy the condition for quasi-zero stiffness. When considering quasi-zero stiffness characteristics, the stiffness ratio \(\alpha\) and the length ratio \(\lambda\) have a definite relationship: \(\alpha=1/\lambda-1\); thus, it can be said that the stiffness ratio and the length ratio have the same influence on the vibration isolation performance. Then, the influence of stiffness ratio \(\alpha\) on vibration isolation performance is studied by changing the parameter \(k_{v}\) while keeping \(k_{h}\) and \(d\) unchanged. Based on the deterministic relation \(\alpha=1/\lambda-1\), the length ratio \(\lambda\) changes with the stiffness ratio \(\alpha\), meaning that \(l\) changes with \(k_{v}\). Figure 6 shows the vibration transmissibility curve under different stiffness ratio \(\alpha\). The values of other parameters are the same as those shown in Table 1. The base excitation amplitude is \(A=0.05~{}\text{m/s}^{2}\). It can be seen that with the increase of stiffness ratio \(\alpha\), the peak frequency gradually increases, and the peak amplitude also gradually increases. However, the stiffness ratio \(\alpha\) has no effect on the low-orbit region. This indicates that a smaller stiffness ratio provides better vibration isolation performance.