4.1. 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.
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
k3 = 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
β +
γ mainly influenced the stiffness-softening characteristic with small displacements. The stiffness softening was more dramatic with a higher
β +
γ value. With
β +
γ = 0, the isolator exhibited a stiffness–hardening characteristic at all displacements. The generalized equivalent stiffness was independent of the individual values of
β and
γ, provided that
β +
γ was the same. As shown in
Figure 4c, the pinching parameter
ξ mainly influenced the initial stiffness. With a lower
ξ value, the initial stiffness was higher and the stiffness softening was more dramatic. With
ξ = 1, the initial stiffness was equal to
k1Kc, and it was independent with the Bouc-Wen model. As shown in
Figure 4d, the pinching parameter
yb influenced the stiffness hardening induced by the pinching effect. With a lower
yb value, this part of the stiffness-hardening effect was stronger.
The generalized equivalent damping ratios of the isolator were calculated with Equation (24) and are shown in
Figure 5. The black solid lines show the results calculated with the parameters shown in
Table 1. The generalized equivalent damping ratio was independent of the frequency, and it increased and then decreased with an increasing vibration amplitude. As shown in
Figure 5a, with a constant
β +
γ value, a higher
β value led to a higher damping ratio, and a lower amplitude occurred at the peak damping ratio. As shown in
Figure 5b, with a constant
β/
γ, a higher
β +
γ value led to a higher peak damping ratio and a lower corresponding amplitude. As shown in
Figure 5c, with a higher
ξ value, the damping ratio was smaller. As shown in
Figure 5d, with a higher
yb value, the damping ratio was smaller.
4.2. Frequency Responses
The frequency responses of the displacement transmissibility (TD) 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
Ab = 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
Ab = 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 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
Ab = 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
Ab-pmin = 1.08 mm, the peak transmissibility reached the lowest value (
Tpmin = 2.65), and thus,
Ab-pmin was identified as the most suitable excitation amplitude of the isolator for suppressing the resonance. With
Ab-rmin = 1 mm, the resonant frequency reached the lowest value (
ηrmin = 0.25), and thus,
Ab-rmin 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 Ab = 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 β/γ are shown in
Figure 7. As discussed in
Section 4.1, β/γ influenced the equivalent damping ratio, while it did not affect the equivalent stiffness. Thus, with a higher β/γ value, the peak transmissibility was lower, while the resonant frequency change was not obvious.
Table 2 shows the lowest peak transmissibility Tpmin, the lowest resonant frequency ηrmin, and the corre-sponding excitation amplitudes, Ab-pmin and Ab-rmin. Higher β/γ values led to lower Tpmin, higher Ab-pmin, and higher Ab-rmin values, while the ηrmin value stayed approximately the same.
The frequency responses of the isolator with
Ab = 1 mm and different values of
β +
γ are presented in
Figure 8. With higher
β +
γ 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
β +
γ values led to lower
Tpmin, lower
Ab-pmin, and lower
ηrmin values, while the relationship between
Ab-rmin and
β +
γ was not monotonic.
The frequency responses of the isolator with
Ab = 1 mm and different values of
ξ are shown in
Figure 9. With higher
ξ 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
ξ values led to higher
Tpmin, lower
Ab-pmin, lower
ηrmin, and lower
Ab-rmin values.
The influences of the linear spring stiffness (
Ks) on the responses were equivalent to the influences of the dimensionless parameter
k1. The frequency responses of the isolator with
Ab = 1 mm and different values of
k1 are presented in
Figure 10, including
k1 = 0.02 (
Ks = 2.26 N/mm),
k1 = 0.025 (
Ks = 13.33 N/mm), and
k1 = 0.03 (
Ks = 24.40 N/mm). With higher
k1 (
Ks) 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
k1 (
Ks) values led to higher
Tpmin, lower
Ab-pmin, higher
ηrmin, and lower
Ab-rmin values.
The isolator with strong nonlinearity exhibited diverse nonlinear phenomena, such as jump phenomena.
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
β/
γ value. With low
k3, high
β +
γ, and low
ξ 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
k1 = 0.02,
k3 = 0,
β = 0.1,
γ = 1.9,
ξ = 0, and
Ab = 1 mm, the resonance peak bends to the left and the jump-down frequency is lower than the resonant frequency. With high
k3 and low
β +
γ 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
k1 = 0.02,
k3 = 0.014,
β = 0.1,
γ = 0.4,
ξ = 0.5,
yb = 215.57, and
Ab = 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
where
Ub and
fb 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].
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
Ub = 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 super-harmonic resonances at about
fb = 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.