## 4 Performance Comparison
### Launch Velocity Comparison  
Considering the realization condition of quasi-zero stiffness (Equation (7)), two sets of different parameter values shown in Table 2 are selected for comparative analysis of the launch velocity \(v_{0}\). Figure 4 shows the launch velocity \(v_{0}\) curve versus pre-compression length \(\delta\) under the two parameter sets. The comparison of parameter Set 1 shows that for linear design, the launch velocity is strictly proportional to the pre-compression length, with a slope of 10. Since the original length \(l\) of the two vertical springs is greater than the vertical distance \(d\), there is pre-compression in the two vertical springs at the initial state. Thus, there is elastic potential energy at \(\delta=0\) for the proposed design, which can be converted into launch kinetic energy. At the beginning (\(\delta<0.098\) m), the compression amount of the two vertical springs decreases with the increase of the transverse spring pre-compression length \(\delta\), and the reduced elastic potential energy is greater than that increased by the transverse spring compressing. Therefore, the total elastic potential energy of the system is lower than that of the initial state, and the launch velocity is also lower than the initial launch velocity. In this process, when \(\delta<0.0663\) m, the launch velocity decreases with the increase of \(\delta\); when \(0.0663\) m \(<\delta<0.098\) m, the launch velocity increases with the increase of \(\delta\). After \(\delta>0.098\) m, the launch velocity approximately shows a linear increasing relationship with \(\delta\), with a slope of 21.55, which means that the launch velocity of the proposed design is increased 1.16-fold compared to the linear design. Since the stiffness ratio \(\alpha=k_{v}/k_{h}\) of the two parameter sets is the same, the results of parameter Set 2 also show the same trend as the launch velocity curve of parameter Set 1. The slope of launch velocity relative to \(\delta\) is approximately 26.38, which is 1.64 times higher than that of linear design.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Properties**} & **Set 1** & **Set 2** \\ \cline{2-3}  & **Proposed Design** & **Linear Design** & **Proposed Design** & **Linear Design** \\ \hline \(m\) & 10 kg & 10 kg & 10 kg \\ \cline{2-3} \(k_{h}\) & 1000 N/m & 1000 N/m & 1500 N/m \\ \hline \(c_{h}\) & 1 Ns/m & 1 Ns/m & 1 Ns/m \\ \hline \(k_{v}\) & 2000 N/m & & 3000 N/m \\ \hline \(c_{v}\) & 0.1 Ns/m & & 0.1 Ns/m \\ \hline \(l\) & 0.15 m & & 0.15 m \\ \hline \(d\) & 0.1 m & & 0.1 m \\ \hline \hline \end{tabular}
\end{table}
Table 2: The structure parameters for launch velocity comparison.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Properties**} & **Set 1** & **Set 2** \\ \cline{2-3}  & **Proposed Design** & **Linear Design** & **Proposed Design** & **Linear Design** \\ \hline \(m\) & 10 kg & 10 kg & 10 kg \\ \cline{2-3} \(k_{h}\) & 1000 N/m & 1000 N/m & 1500 N/m \\ \hline \(c_{h}\) & 1 Ns/m & 1 Ns/m & 1 Ns/m \\ \hline \(k_{v}\) & 2000 N/m & & 3000 N/m \\ \hline \(c_{v}\) & 0.1 Ns/m & & 0.1 Ns/m \\ \hline \(l\) & 0.15 m & & 0.15 m \\ \hline \(d\) & 0.1 m & & 0.1 m \\ \hline \hline \end{tabular}
\end{table}
Table 1: The structure parameters for vibration isolation performance comparison.