# Nonlinear Vibration Isolation via a NiTiNOL Wire Rope
## 4 Results and Discussion
### Frequency Responses  
The frequency responses of the displacement transmissibility (\(T_{\text{D}}\)) were studied to demonstrate the overall isolation performance of the isolators. The peak transmissibility (dimensionless resonant amplitude) and the resonant frequency were calculated. Generally, lower peak transmissibility indicates a stronger damping effect, and a lower resonant frequency indicates a stronger low-frequency vibration isolation ability. The frequency responses of the isolators with the NiTiNOL and the steel wire ropes were obtained from the harmonic balance analysis introduced in Section 3.1. Figure 6 shows the frequency responses of the isolators with the parameters shown in Table 1. The responding displacement transmissibility with different excitation amplitudes is shown in Figure 6a. With \(A_{\text{b}}=0.7\) mm, both isolators demonstrated frequency responses with resonant peaks. The peak transmissibility and the resonant frequency of the isolator with the NiTiNOL wire rope were lower, indicating a better isolation performance. With \(A_{\text{b}}=1\) mm, the isolator with the NiTiNOL wire rope presented a resonant peak, while the isolator with the steel wire rope exhibited an unbounded response. This is a unique  
Figure 5: The generalized equivalent damping ratio of the isolator with different (**a**) Bouc-Wen model parameters (\(\beta/\gamma\)), (**b**) Bouc-Wen model parameters (\(\beta+\gamma\)), (**c**) pinching parameters \(\zeta\), and (**d**) pinching parameters \(y_{\text{b}}\). phenomenon for nonlinear isolators, and this is because the damping produced by the steel wire rope was not enough to suppress the resonance. The responses of the isolator with the NiTiNOL wire rope with different excitation amplitudes were compared. With \(A_{\text{b}}\) = 1 mm, the resonant frequency and the peak transmissibility are the lowest. This indicates that the relationships among the peak transmissibility, the resonant frequency, and the excitation amplitude are not monotonic, as shown in Figure 6b. With an increasing excitation amplitude, both the peak transmissibility and the resonant frequency decreased and then increased. With \(A_{\text{b-pmn}}\) = 1.08 mm, the peak transmissibility reached the lowest value (\(T_{\text{pmn}}\) = 2.65), and thus, \(A_{\text{b-pmn}}\) was identified as the most suitable excitation amplitude of the isolator for suppressing the resonance. With \(A_{\text{b-rmn}}\) = 1 mm, the resonant frequency reached the lowest value (\(\eta_{\text{rmn}}\) = 0.25), and thus, \(A_{\text{b-rmn}}\) was identified as the most suitable excitation amplitude of the isolator for low-frequency vibration. The isolator with the steel wire rope also demonstrated non-monotonic relationships among the peak transmissibility, the resonant frequency, and the excitation amplitude, as shown in Figure 6b. The non-monotonic relationships were due to the non-monotonic equivalent stiffness and damping ratio of the isolators, as shown in Figure 3.  
The frequency responses of the isolator with the NiTiNOL wire rope with \(A_{\text{b}}\) = 1 mm and different isolator parameters were studied. The frequency responses were also calculated from direct integration of Equations (7) and (8) based on the Runge-Kutta method. The results calculated from the harmonic balance analysis (denoted as \"HBM results\" in the following figures) coincide well with the Runge-Kutta method, based on the numerical results (denoted as \"RK results\" in the following figures). This verifies the effectiveness of the alternating frequency/time domain technique in dealing with the modified Bouc-Wen model, and that the third-order assumption of Equation (10) is reasonable. The frequency responses with different values of \(\beta/\gamma\) are shown in Figure 7. As discussed in Section 4.1, \(\beta/\gamma\) influenced the equivalent damping ratio, while it did not affect the equivalent stiffness. Thus, with a higher \(\beta/\gamma\) value, the peak transmissibility was lower, while the resonant frequency change was not obvious.