# Nonlinear Vibration Isolation via a NiTiNOL Wire Rope
## 4 Results and Discussion
### Frequency Responses
Table 2 shows the lowest peak transmissibility Tpmin, the lowest resonant frequency nrmn, and the core-sponding excitation amplitudes, Ab-pmn and Ab-rmn. Higher \(\beta/\gamma\) values led to lower Tpmin, higher Ab-pmn, and higher Ab-rmn values, while the nrmn value stayed approximately the same. Figure 6: The frequency responses of the isolators. (**a**) The displacement transmissibility with different excitation amplitudes, and (**b**) the relationships among the peak transmissibility, the resonant frequency, and the excitation amplitude. The frequency responses of the isolator with \(A_{\text{b}}=1\) mm and different values of \(\beta+\gamma\) are presented in Figure 8. With higher \(\beta+\gamma\) values, the peak transmissibility was lower due to a higher equivalent damping ratio, as shown in Figure 5b. The resonant frequency was lower due to the lower equivalent stiffness, as shown in Figure 4b. As shown in Table 3, higher \(\beta+\gamma\) values led to lower \(T_{\text{pmin}}\), lower \(A_{\text{b-pmin}}\), and lower \(\eta_{\text{rmin}}\) values, while the relationship between \(A_{\text{b-rmin}}\) and \(\beta+\gamma\) was not monotonic. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\mathbf{T_{\text{pmin}}}\) & \(A_{\text{b-pmin}}\) & \(\eta_{\text{rmin}}\) & \(A_{\text{b-rmin}}\) \\ \hline \(\beta=0.25\), \(\gamma=0.842\) & 3.29 & 1.00 mm & 0.26 & 0.99 mm \\ \(\beta=0.501\), \(\gamma=0.591\) & 2.65 & 1.08 mm & 0.25 & 1.00 mm \\ \(\beta=0.75\), \(\gamma=0.342\) & 2.37 & 1.16 mm & 0.25 & 1.16 mm \\ \hline \end{tabular}
\end{table} 
Table 2: The lowest peak transmissibility/resonant frequency and the corresponding excitation amplitudes with different values of \(\beta/\gamma\). Figure 8: The frequency responses of the isolator with different values of \(\beta+\gamma\). Figure 7: The frequency responses with \(A_{\text{b}}=1\) mm and different values of \(\beta/\gamma\). The frequency responses of the isolator with \(A_{\text{b}}=1\) mm and different values of \(\xi\) are shown in Figure 9. With higher \(\xi\) values, the peak transmissibility was higher due to a smaller equivalent damping ratio, as shown in Figure 5c. The resonant frequency was lower due to the lower equivalent stiffness, as shown in Figure 4c. As shown in Table 4, higher \(\xi\) values led to higher \(T_{\text{pmin}}\), lower \(A_{\text{b-pmin}}\), lower \(\eta_{\text{rmin}}\), and lower \(A_{\text{b-rmin}}\) values. The influences of the linear spring stiffness (\(K_{\text{s}}\)) on the responses were equivalent to the influences of the dimensionless parameter \(k_{1}\). The frequency responses of the isolator with \(A_{\text{b}}=1\) mm and different values of \(k_{1}\) are presented in Figure 10, including \(k_{1}=0.02\) (\(K_{\text{s}}=2.26\) N/mm), \(k_{1}=0.025\) (\(K_{\text{s}}=13.33\) N/mm), and \(k_{1}=0.03\) (\(K_{\text{s}}=24.40\) N/mm). With higher \(k_{1}\) (\(K_{\text{s}}\)) values, both the peak transmissibility and the resonant frequency became higher, while the load capacity of the isolator was enhanced. As shown in Table 5, higher \(k_{1}\) (\(K_{\text{s}}\)) values led to higher \(T_{\text{pmin}}\), lower \(A_{\text{b-pmin}}\), higher \(\eta_{\text{rmin}}\), and lower \(A_{\text{b-rmin}}\) values. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(T_{\text{pmin}}\) & \(A_{\text{b-pmin}}\) & \(\eta_{\text{rmin}}\) & \(A_{\text{b-rmin}}\) \\ \hline \(\xi=0.7\) & 1.81 & 2.20 mm & 0.33 & 2.10 mm \\ \(\xi=0.8\) & 2.04 & 1.74 mm & 0.30 & 1.72 mm \\ \(\xi=0.9\) & 2.65 & 1.08 mm & 0.25 & 1.00 mm \\ \hline \end{tabular}
\end{table} 
Table 4: The lowest peak transmissibility/resonant frequency and the corresponding excitation amplitudes with different values of \(\xi\). Figure 10: The frequency responses of the isolator with different values of \(k_{1}\). Figure 9: The frequency responses of the isolator with different values of \(\xi\). The isolator with strong nonlinearity exhibited diverse nonlinear phenomena, such as jump phenomena.