# Nonlinear Vibration Isolation via a NiTiNOL Wire Rope
## 5 Conclusions  
In this study, we proposed a nonlinear isolator with a horizontal NiTiNOL wire rope, providing nonlinear stiffness and nonlinear damping, and a vertical linear spring, enhancing the load capacity. The nonlinearities were derived from geometric constraints, inner frictions, and phase transitions of the material. The restoring force model of the wire rope was established by a combined method of a beam constraint model and a Bouc-Wen model. The dynamic model of the nonlinear isolator was constructed and the frequency responses were analyzed through a harmonic balance method with an alternating frequency/time domain technique. The generalized equivalent stiffness and generalized equivalent damping ratio were defined considering the influences of the hysteretic damping and the pinching effect. Numerical simulations allowed us to draw the following conclusions. The isolator exhibited a stiffness-softening-hardening characteristic. In a dimensionless form, the initial stiffness, the stiffness softening at small displacements, and the stiffness hardening at large displacements mainly depended on the pinching parameter \(\zeta\), the Bouc-Wen parameter \(\beta+\gamma\), and the cubic stiffness \(k_{3}\), respectively. The generalized equivalent damping ratio increased and then decreased with increasing displacement. The Bouc-Wen \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\mathbf{M}\) & \(\mathbf{K_{\text{w1}}}\) & \(\mathbf{U_{\text{b}}}\) & \(\mathbf{K_{\text{d}}}\) & \(\mathbf{\xi}\) & \(\mathbf{Y_{\text{b}}}\) & \(\mathbf{G}\) & \(\mathbf{B}\) & \(\mathbf{n_{\text{b}}}\) \\ \hline
6.46 kg & 4.18 kN/m & 6.35 m/s\({}^{2}\) & 58.5 kN/m & 0.587 & 112 mm\({}^{2}\) & 0.544 kN\({}^{-n}\)\({}_{\text{b}}\)\({}^{+1}\)/mm & 0.823 kN\({}^{-n}\)\({}_{\text{b}}\)\({}^{+1}\)/mm & 1.2 \\
6.46 kg & 4.47 kN/m & 7.68 m/s\({}^{2}\) & 100 kN/m & 0.294 & 950 mm\({}^{2}\) & 2 kN\({}^{-n}\)\({}_{\text{b}}\)\({}^{+1}\)/mm & 1.831 kN\({}^{-n}\)\({}_{\text{b}}\)\({}^{+1}\)/mm & 1.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The identified parameters of the damper experiments [35]. Figure 12: The frequency responses of the damper under the excitation acceleration amplitudes of 6.35 m/s\({}^{2}\) and 7.68 m/s\({}^{2}\)[35]. “HBM”, “RK”, and “Exp” denote the results of the harmonic balance analysis, the Runge–Kutta-based numerical integration, and the experiments, respectively. parameters and the pinching parameter \(\zeta\) showed obvious influences on the damping ratio. Both the stiffness and the damping changes were non-monotonic. Compared to the isolator with a steel wire rope, the NiTiNOL wire rope decreased the initial equivalent stiffness and increased the damping. The isolator with the NiTiNOL wire rope enhanced the vibration isolation performance. The peak displacement transmissibility and the resonant frequency of the isolator were related to the excitation amplitude, and the relationships were non-monotonic. There were excitation amplitudes corresponding to the minimum peak transmissibility and the lowest resonant frequency, respectively, and they are referred to as the most suitable excitation amplitudes of the isolator. The isolator parameters affected these indexes in different ways depending on their influences on the stiffness and damping characteristics. With low cubic stiffness, high Bouc-Wen hysteresis, and a weak pinching effect, the isolator tended to demonstrate a stiffness-softening type of jump phenomenon, and the stiffness-hardening type of jump phenomenon tended to occur with contrary parameters. Conceptualization, L.-Q.C. and M.-Q.N. ; methodology, M.-Q.N. and L.-Q.C. ; software, M.-Q.N. ; validation, M.-Q.N. ; formal analysis, M.-Q.N. ; investigation, M.-Q.N. and L.-Q.C. ; resources, M.-Q.N. and L.-Q.C. ; data curation, M.-Q.N. ; writing--original draft preparation, M.-Q.N. ; writing--review and editing, L.-Q.C. ; visualization, M.-Q.N. ; supervision, L.-Q.C. ; project administration, L.-Q.C. ; funding acquisition, M.-Q.N. and L.-Q.C.