# Nonlinear Vibration Isolation via a NiTiNOL Wire Rope
## 4 Results and Discussion  
### Characteristic Indicators  
Numerical simulations were performed to study the characteristic indicators of the isolator, including the restoring force, the generalized equivalent stiffness, and the generalized equivalent damping ratio. The parameters of the wire ropes were derived from the identification results in [45], using a NiTiNOL wire rope and a steel wire rope for comparison. Table 1 shows the dimensional parameters of the isolators. The restoring forces of the isolators with different materials of the wire rope are shown in Figure 2. Compared to the steel wire rope, the NiTiNOL wire rope demonstrated a larger hysteretic loop, indicating a stronger energy dissipation ability. This is because NiTiNOL dissipates more energy due to the phase transition. Furthermore, the NiTiNOL wire rope presented a flatter curve around the balance location (\(Y=0\)). This phenomenon was a result of the pinching effect. For further comparison, the generalized equivalent stiffness and damping ratios of the isolators with steel and NiTiNOL wire ropes were calculated through Equations (20) and (24). As shown in Figure 3a, the initial equivalent stiffness at \(Y=0\) of the NiTiNOL wire rope was much lower than that of the steel wire rope due to the pinching effect. As shown in Figure 3b, the equivalent damping ratio of the NiTiNOL wire rope was higher than that of the steel wire rope due to the energy dissipation of the phase transition. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Material** & \(K_{\text{s}}\) **(N/mm)** & \(K_{\text{w1}}\) **(N/mm)** & \(K_{\text{w3}}\) **(N/mm\({}^{\text{s}}\))** & \(K_{\text{d}}\) **(N/mm)** & \(\xi\) & \(Y_{\text{b}}\) **(mm\({}^{\text{2}}\))** & \(G\) **(mm\({}^{\text{-1}}\))** & \(B\) **(mm\({}^{\text{-1}}\))** & \(n_{\text{b}}\) \\ \hline NiTiNOL & 2.26 & 2.17 & 0.53 & 217 & 0.9 & 215.57 & 0.591 & 0.501 & 1 \\ Steel & 2.26 & 5.66 & 0.53 & 21 & 0 & / & 0.500 & 0.544 & 1 \\ \hline \end{tabular}
\end{table} 
Table 1: Dimensional parameters of the isolators with a NiTiNOL wire rope and a steel wire rope. Figure 2: Restoring forces of the isolators with a steel wire rope and a NiTiNOL wire rope. The influences of the parameters on the characteristic indicators of the isolator with a NiTiNOL wire rope were studied. The dimensional generalized equivalent stiffness of the isolator is presented in Figure 4. The black solid lines show the results calculated from the parameters shown in Table 1. The equivalent stiffness was symmetric with the balance location. There was a positive initial stiffness induced by the Bouc-Wen hysteresis. The stiffness decreased dramatically and then increased with increasing displacement. Thus, the isolator exhibited a stiffness-softening-hardening characteristic. As shown in Figure 4a, with higher cubic stiffness, the equivalent stiffness increased more dramatically with large displacements. It should be noted that, with \(k_{3}=0\), there was still a weak stiffness-hardening characteristic due to the pinching effect, as shown with the red dashed line. As shown in Figure 4b, the Bouc-Wen model parameter \(\beta+\gamma\) mainly influenced the stiffness-softening characteristic with small displacements. The stiffness softening was more dramatic with a higher \(\beta+\gamma\) value. With \(\beta+\gamma=0\), the isolator exhibited a stiffness-hardening characteristic at all displacements. The generalized equivalent stiffness was independent of the individual values of \(\beta\) and \(\gamma\), provided that \(\beta+\gamma\) was the same. As shown in Figure 4c, the pinching parameter \(\xi\) mainly influenced the initial stiffness. With a lower \(\xi\) value, the initial stiffness was higher and the stiffness softening was more dramatic. With \(\xi=1\), the initial stiffness was equal to \(k_{1}K_{c}\), and it was independent with the Bouc-Wen model. As shown in Figure 4d, the pinching parameter \(y_{\text{b}}\) influenced the stiffness hardening induced by the pinching effect.