# Nonlinear Vibration Isolation via a NiTiNOL Wire Rope
## 4 Results and Discussion
### Frequency Responses
Figure 11 presents two types of jump phenomena, with the stable branches shown as black solid lines, and the unstable branches shown as red dashed lines. Generally, the jump phenomenon tends to exist with weak damping, namely a low \(\beta/\gamma\) value. With low \(k_{3}\), high \(\beta+\gamma\), and low \(\xi\) values, the equivalent stiffness decreased dramatically at small displacements, and the stiffness-softening type of jump phenomenon could occur. As shown in Figure 11a, with \(k_{1}=0.02\), \(k_{3}=0\), \(\beta=0.1\), \(\gamma=1.9\), \(\xi=0\), and \(A_{\text{b}}=1\) mm, the resonance peak bends to the left and the jump-down frequency is lower than the resonant frequency. With high \(k_{3}\) and low \(\beta+\gamma\) values, the equivalent stiffness increased dramatically at large displacements, and the stiffness-hardening type of jump phenomenon could occur. As shown in Figure 11b, with \(k_{1}=0.02\), \(k_{3}=0.014\), \(\beta=0.1\), \(\gamma=0.4\), \(\xi=0.5\), \(y_{\text{b}}=215.57\), and \(A_{\text{b}}=1\) mm, the resonance peak bends to the right and the jump-down frequency is higher than the resonant frequency. To verify the analysis method of the frequency responses, the calculation results were compared with the experimental and numerical results of a damper with NiTiNOL wire ropes used in [35]. The wire ropes are horizontal and symmetrically arranged, and the structure is similar to the proposed isolator, as shown in Figure 1, without the linear spring. Therefore, the proposed modeling and analysis methods were applied to the damper. The damper was tested under swept-frequency experiments with a constant acceleration amplitude, and the responding displacement amplitude was measured. Similar to Equation (6), the dynamic model of the damper in the experiment can be expressed as  
\[MY^{w}+K_{\text{w1}}Y+Z=MU_{\text{b}}\cos(2\pi f_{\text{b}}T) \tag{25}\]  
where \(U_{\text{b}}\) and \(f_{\text{b}}\) are the amplitude and the frequency of the excitation acceleration, respectively, and \(Z\) is expressed in Equation (4). The damper was tested in two conditions, and the identified parameters are shown in Table 6[35]. Figure 11: The frequency responses with jump phenomena. (**a**) Stiffness softening and (**b**) stiffness hardening. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(T_{\text{pmin}}\) & \(A_{\text{b-pmin}}\) & \(\eta_{\text{rmin}}\) & \(A_{\text{b-rmin}}\) \\ \hline \(k_{1}=0.2\) & 2.65 & 1.08 mm & 0.25 & 1.00 mm \\ \(k_{1}=0.25\) & 2.84 & 1.00 mm & 0.26 & 0.95 mm \\ \(k_{1}=0.3\) & 3.03 & 0.92 mm & 0.27 & 0.92 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The lowest peak transmissibility/resonant frequency and the corresponding excitation amplitudes with different values of \(k_{1}\). The frequency responses of the damper with NiTiNOL wire ropes under two excitation acceleration amplitudes are shown in Figure 12. The results calculated from the proposed harmonic balance analysis essentially coincide with the Runge-Kutta-based numerical results and the experimental results demonstrated in [35]. The main differences appeared in two ways. The experimental results with \(U_{\text{b}}=7.68\) m/s\({}^{2}\) exhibited the stiffness-softening phenomenon, while both the proposed method and the numerical integration failed to characterize the phenomenon. The harmonic balance analysis failed to present the superharmonic resonances at about \(f_{\text{b}}=2.4\) Hz, as demonstrated in the numerical results due to the truncation error of Equations (10) and (11). The numerical results and the experimental results verify the effectiveness of the proposed harmonic balance analysis.