Shock and Vibration
Volume 2019 (2019), Article ID 4525798, 10 pages
https://doi.org/10.1155/2019/4525798

Research Article

Effect of Structure Parameters on Polycal Wire Rope Isolator Stiffness-Damping Characteristics

Shaowu Tu, Xiaofeng Lu, and Xiaolei Zhu

School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing 211800, China

Correspondence should be addressed to Xiaofeng Lu; xflu@njtech.edu.cn and Xiaolei Zhu; zhuxiaolei@njtech.edu.cn

Received 19 December 2018; Revised 3 March 2019; Accepted 5 May 2019; Published 2 June 2019

Academic Editor: Gilbert R. Gillich

Copyright © 2019 Shaowu Tu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Wire rope isolators are mainly used to isolate vibration and protect precise equipment. However, the issue of regulation of vibration isolators taking into account the nonlinearity of their characteristics was poorly understood in the modern literature. In this paper, the influence of structural parameters (diameter ratio and lay pitch of the single strand, and lay pitch and bending radius of the wire rope) on stiffness-damping characteristics of the Polycal WRI was investigated by the simplified finite element analysis method. The stiffness and damping prediction models including structural parameters and material properties were established. The results showed that the stiffness-damping characteristics were the best; when the diameter ratio of wire strand was 1.1, the inside layer wire pitch length was 6 times the diameter of the wire strand, the outside layer wire pitch length was 11 times the diameter of the wire strand, the pitch length of the wire rope was 7.5 times its diameter, and the bending radius was equal to 46.5 mm. The errors of the prediction for prestiffness and softened stiffness were within 5%, and the errors of prediction for the energy dissipation coefficient were within 10%.

1. Introduction

Vibration is common in our lives. Especially in many industries, vibration is caused by the equipments operation, fluid flushing in pipes, and aero engine. This is harmful for operation safety [1–3]. Many vibration reduction and vibration suppression methods have been studied. Vibration isolators are widely applicable to production and living. Particularly, it is widely used in equipments with high load and vibration reduction requirements. In recent years, new type of vibration isolators and design of vibration isolation systems are hot topic for scholars. Wire rope isolator (WRI) has excellent rigidity damping characteristics, especially with high bearing capacity. It is widely used in mechanical manufacturing and construction. Therefore, it is necessary and significant to study the characteristics of the WRI with different structural parameters.

The characteristics of the WRI are studied through the experimental and theoretical methods [4–11]. Chen et al. [12] investigated the contact statues of a steel wire rope from the perspective of theoretical analysis. The result shows that the effect of the lay angle on the stiffness of the wire rope is different under different loads. Tinker and Cutchins [13] obtained the data of stiffness and damping characteristics of the WRI through dynamic experiment. It is also found that the damping of the WRI is related to coulomb-type friction. Demetriades et al. [14] studied the response characteristics under different loads for different structures of the WRI. The result indicated that the WRI exhibits the same characteristics under shear and roll loads. Wang et al. [15] experimentally investigated the effects of load frequency, amplitude and structural parameters on the dynamic characteristics of O-type WRIs. He found that the loading amplitude and geometric parameters of the isolator directly affects the dynamic characteristics of the isolator, while the loading frequency has no effect on it. Gerges [9] investigated the tension-compression mode of the wire rope spring. He presented a semianalytical model for a wire rope vibration isolator through experiment. Rashidi and Ziaei-Rad [7] investigated the quasi-static and dynamic characteristics of the WRI. It is suggested that there is not obviously relationship between hysteresis loops and loading velocity under quasi-static load. The dynamic results indicated that by increasing the frequency of excitation, the area of the hysteresis loop starts decreasing. Finally, a hysteresis analytical prediction model with high coincidence degree was established.

The finite element analysis method can reduce the cost of the experiment, and thus has been widely used in studying the characteristics of the WRI and wire ropes. Jiang et al. [16] found effective simplified finite element analysis method of analyzing the contact statues of wire ropes. It was found that the local contact deformation affects the accuracy of the results. Wang et al. [17] investigated the mechanical behavior of the YS9-8 × 19 braided wire rope under tensile load. By comparing the results of finite element analysis and experiment, the error between them was small, and the accuracy of the model was verified. By the finite element method, Xiang et al. [18] obtained the elastic-plastic contact stresses under axial and torsion loads of wire ropes, and investigated the elastic-plastic behavior of it. The finite element analysis results have a good agreement with the experimental test results, and a new prediction model was proposed. Yu et al. [19] applied the beam finite element method to analyzing axial tensile properties of the 91-wire strand. By comparing with the experiment results, the beam FEM could be used to predict the tensile properties of the steel wire rope. Song et al. [20] analyzed distributions of stress and deformation in the braided wire rope subjected to torsional loading. He found that the wires in the strands have the tendency to be screwed tightly and are in a stretched state when the lay direction of the strand coincides with its torsion direction. Cao and Wu [21] established the finite element model of wire ropes with different structural parameters and analyzed the stress distribution and deformation under cantilever beam state. The accuracy of the results of finite element analysis was slightly lower than the theoretical calculation results. Du et al. [22] presented a simulation of the 6 × 36 + WS RHRL wire rope. It is found that the stress of the wire rope was uneven, and the maximum stress occurs at the side of the wire. Yong et al. [23] conducted a finite element analysis of the IWRC636WS wire rope, and the elastic behavior of the wire rope under tensile loads was simulated. It is reported that nonlinear relationship between the axial tension and the axial elongation of the wire rope. Cen et al. [24] found effective simplified finite element analysis method by combining finite element method with experimental test. This method can be used to analyze the characteristics of the Polycal WRI. The above studies have studied the characteristics of wire ropes and vibration isolators by experiments and finite element methods and obtained some results. Considering the complexity of the wire rope isolator structure, there is less research on the relationship between the structural parameters of the WRI and the stiffness and damping characteristics of the WRI. Therefore, it is necessary to study the relationship between them, and provide guidance for practical engineering applications.

In this paper, the stiffness and damping characteristics of WRIs with different structural parameters were investigated. These structural parameters were number of wire ropes, material of wire, rope diameter (D), rope lay pitch (), single wire rope diameter (d), single wire strand lay pitch (), and wire rope diameter ratio (nr). A stiffness-damping prediction model consists of structural parameters of the Polycal WRI were established, which aims to provide powerful help for the structural design and wire rope selection of the Polycal WRI.

2. Method

2.1. Energy Dissipation Coefficient

The energy dissipation coefficient is a key parameter evaluating the effective vibration isolation property of a WRI. It is an important reference for evaluating the damping characteristics of the WRI. Because of sliding friction between the wire strands and the internal friction of wires, the isolators exhibit nonlinear hysteretic behavior. Typical load-displacement curve of the WRI is shown in Figure 1. The damping characteristics of the WRI are related with the area which is enclosed by the loading and unloading curve of the WRI under compression. The energy dissipation coefficient C was calculated as follows:

where Aloop was the area of the hysteresis loop (N·mm), Fmax and Fmin were the maximum and minimum loads (N), respectively, of the WRI in the compression loading-unloading process. Xmax and Xmin were the maximum and minimum displacement (mm) in the loading-unloading process.

Figure 1: The typical load-displacement curve of the WRI.

In most of the wire rope isolators, during the load-bearing process, the upper and lower pallets are mainly supported by the curved steel wire rope, and the bending stiffness and deformation process of the steel wire rope play a decisive role. Therefore, this paper mainly uses the stiffness and energy dissipation coefficient as indicators to measure the effectiveness of the wire rope isolators.

2.2. Calculation Method and Finite Element Model

2.2.1. Calculation Method

As shown in Figure 2(a), the WRI was composed of two pallets and twelve 6 × 19 IWS wire ropes. In this paper, the simplified finite element method is used to obtain the WRI load-displacement hysteresis loop, and the WRI stiffness damping of different structural parameters is discussed. We have referenced the simplified finite element method which was established by Cen et al. [24]. This method mainly simplified the single strand into a single wire. The 6 × 19 IWS wire rope was simplified into the 1 × 7 wire rope, as presented in Figure 2(b). Based on the simplified method, this paper studies the stiffness and damping characteristics of the isolators with different structural parameters and establishes the prediction model of the stiffness damping of the wire rope isolator with structural parameters as variables.

(a)

(b)

Figure 2: WRI structure: (a) GGQ-99 WRI; (b) simplified model of 6 × 19 IWS wire ropes.

2.2.2. Definitions of Finite Element Model

This paper is mainly based on the GGQ-99 Polycal WRI. By changing structural parameters, the diameter ratio and lay pitch of the single strand and lay pitch and bending radius of the wire rope, different finite element models were set.

In this presented model, the data source of the simulation of the WRIs refers to papers of Cen et al. [24], Jiang et al. [25], and Erdonmez and Imrak [26]. The material properties of the center and side wires are defined by the bilinear elastic-plastic kinematic hardening model in the ABAQUS material library, as shown in Table 1. By compression of corresponding strands, the max equivalent compression stress of the strand and equivalent compression elastic modulus are equal to and Ep, respectively. Ee is equal to the equivalent tension elastic modulus which is measured by the tension of the strands. The density is ρ = 7850 kg·m⁻³, and the Poisson’s ratio is μ = 0.3 [24].

Table 1: Material parameters of the kinematic hardening model for the arc wire rope [24].

Arc strand
Ee (GPa)
Ep (GPa)
(MPa)

Center wire
64.25
10.66
16.80

Side wire
80.37
10.68
18.34

Reproduced from Cen et al. [24] (under the Creative Commons Attribution License/public domain).

In the general contact type, the All with self is selected to define the contact type between the center strand and lay strand. The tangential behaviour with the penalty friction coefficient is 0.115 and normal behaviour with hard contact [25, 26]. The mesh type is a 3D solid linear reduction integration unit (C3D8R). The down-pallet is fully constrained, and the up-pallet is constrained except for the displacement load applied in the vertical direction.

3. The Analysis of WRI Stiffness-Damping with Different Structural Wire Ropes

The stiffness of the WRI determines the load-bearing capacity of the vibration isolation system, regardless of whether the WRI is subjected to a static load or a strong impact load. The softening load of the WRI and the subsequent softening stiffness both affect the efficiency of vibration isolation and stability of the entire isolator system. The damping characteristic reflects the ability to absorb shock vibration energy of the isolator in the vibration isolation system. There are many factors that affect the static stiffness and damping characteristics of the WRI, including the selection of the wire rope, number of wire ropes, material of steel wire, wire rope diameter (D), rope lay pitch (), single wire rope diameter (d), single wire strand lay pitch (), wire rope diameter ratio (nr), and arc wire rope bending radius (R).

3.1. Wire Rope Diameter Ratio

As shown in Figure 3, the diameter ratio (nr) of the 1 + 6 + 12 center strand or the lay strand is defined as follows:

where rc and rs are the diameters of the center wire and lay wire in the single strand, respectively.

Figure 3: The 1 + 6 + 12 single-strand wire rope.

Figure 3 shows the 1 + 6 + 12 single-strand wire rope. It was stipulated that the 1 + 6 + 12 single-strand wire rope has the same other structural parameters; the first layer wire pitch length was 6 times the diameter of the single wire rope, and the second layer wire pitch length was 11 times the diameter of the single wire rope. The diameter of the wire rope was 8 mm in the study of the GGQ-99 Polycal WRI. The diameter of the center strand was 2.839 mm, and the diameter of the lay strand was 2.581 mm. Obviously, the former was 1.1 times more than the latter. The lay pitch length was 60 mm (7.5 times diameter of the wire rope), and the bending radius was 46.5 mm of the arc wire rope. The diameters of the center wire and lay wire in different strands with different diameter ratios of the GGQ-99 are listed in Table 2.

Table 2: Diameter of wire for single strands with different diameter ratios (mm).

Diameter of wire

1.1
1.2
1.3
1.4
1.5

Center strand

    Center wire
0.614
0.655
0.696
0.736
0.774

    Lay wire
0.556
0.546
0.536
0.526
0.516

Lay strand

    Center wire
0.557
0.595
0.633
0.669
0.704

    Lay wire
0.506
0.496
0.487
0.478
0.469

According to the simplified FEM model, the tensile and compression of the single strand with different diameter ratios were calculated. The elastic tensile and compression modulus and compression ultimate load of the single strand were obtained. These mechanical parameters were used to calculate the stiffness-damping characteristics of the WRI. It is presented in Table 3. It could be seen that the elastic tensile modulus significantly reduced with increased diameter ratio. But the elastic compression modulus increased with increased diameter ratio. The compression ultimate load is kept steady basically with increased diameter ratio. The reason is that the compression ultimate load indicated the friction properties between the center wire and lay wire. The friction between the center wire and lay wire was retained about the same during calculating the model with different diameter ratios. So, the compression ultimate loads showed little changes.

Table 3: The mechanical parameters of single strands with different diameter ratios.

FEM results of single strand

1.1
1.2
1.3
1.4
1.5

Center strand

     (GPa)
70.10
63.25
60.58
59.52
57.29

     (GPa)
11.52
12.38
13.48
14.80
15.94

     (MPa)
17.50
16.21
16.16
16.54
16.33

Lay strand

     (GPa)
77.61
69.16
67.05
65.73
61.94

     (GPa)
11.80
12.45
13.52
14.87
16.00

     (MPa)
17.78
16.41
16.15
16.93
16.77

According to the simplified FEM model, the compression loading-unloading processes of the GGQ-99 WRI with different diameter ratios were simulated. Different load-displacement hysteresis loop curves of the WRI are shown in Figure 4. The ratio between the prestiffness K₁ and softened stage stiffness K₂ indicated the impact resistance for the WRI. The smaller this value, the better the impact resistance. The energy dissipation coefficient was used to evaluate the damping characteristic. The higher this value, the better the damping characteristic. The results are shown in Figure 5.

Figure 4: Hysteresis curves comparison of the GGQ-99 WRI with different diameter ratios.

Figure 5: Relationship between stiffness ratio and energy absorption coefficient with different diameter ratios.

As shown in Figure 4, the compression load of the WRI increased with increasing of the diameter ratio of the wire strand. This result is related to the elastic compression modulus of the single strand. As shown in Figure 5, the ratio of K₁ to K₂ was increased with increasing of the diameter ratio of the wire strand. The ratio of K₁ to K₂ reached the minimum when the diameter ratio was equaled to 1.1. It means that the GGQ-99 WRI was easier to maintain the stability of the vibration isolation system through large deformation. The energy dissipation coefficient for the WRI decreased with increasing of the diameter ratio of the wire strand. When the ratio of the strand was equaled to 1.1, the energy dissipation coefficient reached maximum. These results indicate that the Polycal WRI has better damping characteristics, which could effectively consume the impact load and eliminate the vibration from the isolation system.

3.2. Lay Pitch of Single Strand

Based on the diameter ratio of the wire strand to 1.1, the main purpose of this section is to study the effect of the outer layer side wire pitch length () on the stiffness-damping of the Polycal WRI. During the process of creating the simulation model, the first layer side wire of the strand steel wire rope () was unchanged and equal to 6 times the diameter of the wire strand. The was 7 times, 8 times, 9 times, 10 times, and 11 times the diameter of the wire strand. The relation between the pitch length and diameter of the wire strand (d) is expressed in equation (3). The geometric dimensions of the single wire strands with different pitch lengths are shown in Table 4.

Table 4: The geometric dimensions of the single wire strands with different pitch length (mm).

Pitch length

t = 7
t = 8
t = 9
t = 10
t = 11

Center strand
17.034
19.873
22.712
25.551
28.39
31.229

Lay strand
15.486
18.067
20.648
23.229
25.81
28.391

As shown in Table 5, the equipment tensile modulus, compression modulus, and softened stress of the wire strands with different were obtained by axial tension and compression. The modulus of the tension was continuously increased with increasing of the outer layer side wire pitch. The modulus of the compression had a small variation with increasing of the outer layer side wire pitch basically, and the maximum load of the corresponding compression process decreased.

Table 5: The FEM results of single strands with different lay pitches.

FEM results of wire strands

t = 7
t = 8
t = 9
t = 10
t = 11

Center strand

     (GPa)
46.18
48.04
55.59
60.83
70.10

     (GPa)
11.82
11.80
11.76
11.41
11.52

     (MPa)
25.01
23.08
21.71
19.74
17.50

Lay strand

     (GPa)
52.90
55.48
61.13
66.61
77.61

     (GPa)
11.73
11.56
11.85
11.52
11.80

     (GPa)
24.49
23.23
21.75
19.80
17.78

The equipment tensile and compression performance of wire strands with different were used to define the bilinear elastic-plastic kinematic hardening material properties of the center and lay strand assembled in the GGQ-99 WRI. As shown in Figure 6, the load-displacement hysteresis loops of different lay pitches had a small variation. Besides, the modulus of the compression was not changing with increasing of the outer layer side wire pitch. Combined stiffness-damping characteristics of the GGQ-99 WRI with the different wire strand pout are shown in Figure 7. When the was 11 times the diameter of the wire strand, the ratio of K₁ to K₂ reached the minimum. At the same time, the Polycal WRI had large prestiffness and the most obvious softening characteristics, not only could withstand large loads but also could easily maintain the stability of the vibration isolation system through large deformation under the action of large loads. The energy dissipation coefficient of the WRI remained basically unchanged, when the ratio of the outside lay pitch and the diameter of the strand increased.

Figure 6: Hysteresis curves comparison of the GGQ-99 WRI with different lay pitches.

Figure 7: The stiffness-damping characteristics of the GGQ-99 WRI under the lay pitch.

3.3. Lay Pitch of Wire Rope

In the process of compression loading-unloading of the Polycal WRI, not only the slippage of the wires exists in the strand but also the overall slippage of the strands in the rope. Therefore, the influence rule of the pitch length on the wire rope is discussed in this section. The diameter ratio, the inside layer wire pitch length and outside layer wire pitch, was equaled to 1.1. It was 6 times the diameter of the wire strand and 11 times the diameter of the wire strand, respectively. The relation between the pitch length and diameter of the wire rope (D) is expressed in equation (4). Table 5 lists the geometric dimensions of the single wire strands with different pitch lengths. The load-displacement hysteresis loops of the GGQ-99 WRI with different pitch lengths are shown in Figure 8.

Figure 8: Hysteresis curves comparison of the GGQ-99 WRI with different arc rope lay pitches.

As shown in Figure 8, the compression load of the WRI increased with increasing of the pitch length of the wire rope. The increase of the pitch length contributed to the increase of the angle between the center strand and lay strand. The bearing axial load of the lay strand increased with angle between center strand and lay strand decreasing. So, the compression load of the WRI was increased with the increasing of the pitch length of the wire rope.

The stiffness-damping characteristics of the GGQ-99 WRI with different pitch lengths of wire ropes are shown in Figure 9. As shown in Figure 9, both the ratio of K₁ to K₂ and the energy dissipation coefficient of the WRI fluctuated with the increase of the pitch length of the arc rope. When the rope pitch was equaled to 7.5 times the diameter of rope, the ratio of K₁ to K₂ was the smallest and less than 0.2 and the energy dissipation coefficient was the biggest.

Figure 9: The stiffness-damping characteristics of the GGQ-99 WRI with different pitch lengths of wire ropes.

3.4. Bending Radius of Wire Rope

The characteristics of the Polycal WRI depend on the structure of the wire rope, including the diameter of the single strand, the pitch length of the single strand, and the bending radius of the wire rope. In this section, the influence of the bending radius (R) of the arc rope on the stiffness-damping characteristics of the Polycal WRI is studied. The diameter ratio of the wire strand was 1.1. The inside layer wire pitch length was 6 times the diameter of the wire strand. The outside layer wire pitch length was 11 times the diameter of the wire strand, and the pitch length of the wire rope was 7.5 times its diameter. The bending radius was equal to 46.5 mm, and the bending radius of the wire rope was 50 mm, 55 mm, 60 mm, and 65 mm, respectively. The FEM models of the GGQ-99 WRI with different bending radii are shown in Figure 10. The load-displacement hysteresis loops of the GGQ-99 WRI with different bending radii of arc ropes are shown in Figure 11.

(a)

(b)

(c)

(d)

(e)

Figure 10: GGQ-99 WRIs with different bending radii. (a) 46.5 mm. (b) 50 mm. (c) 55 mm. (d) 60 mm. (e) 65 mm.

Figure 11: Hysteresis curves comparison of the GGQ-99 WRI with different bending radii.

As shown in Figure 11, the compression load of the WRI increased with increasing bending radius of arc ropes. Because the compression load of the WRI was inversely proportional to the curvature of the wire rope, the curvature of the wire rope decreased with increasing bending radius of arc ropes. The compression load of the WRI increased with increasing bending radius of arc ropes.

The stiffness-damping characteristics of the GGQ-99 WRI with different bending radii of the wire rope are shown in Figure 12. The ratio of K₁ to K₂ increased with increasing bending radius. However, the energy dissipation coefficient of the WRI decreased with increasing bending. When the bending radius was equal to 46.5 mm, the ratio of K₁ to K₂ was the smallest and the energy dissipation coefficient was the biggest.

Figure 12: The stiffness-damping characteristics of the GGQ-99 WRI with different bending radii of the wire rope.

In conclusion, the stiffness-damping characteristics were the best; when the diameter ratio of the wire strand was 1.1, the inside layer wire pitch length was 6 times the diameter of the wire strand, the outside layer wire pitch length was 11 times the diameter of the wire strand, and the pitch length of the wire rope was 7.5 times its diameter. The bending radius was equal to 46.5 mm.

4. Stiffness-Damping Prediction Models of WRI

4.1. Prediction Model

In the previous section, the influence rules of the diameter ratio of the wire strand (nr), the pitch length of the wire strand (), the pitch length of the wire rope (P), and the bending radius (R) for the stiffness-damping performance of the Polycal WRI were discussed. The prediction model of the stiffness-damping characteristic of the WRI was established by using the dimensionless diameter ratio of the wire strand (nr), the pitch length of the wire strand (), the pitch length of the wire rope (P), and the bending radius (R). The theoretical basis of equations (5) and (6) is the theorem in dimensional analysis. According to the principle of dimensional analysis, all structural parameters are transformed into a dimensionless form, and the number of variables is reduced to obtain the corresponding calculation formula as follows:

where E₁ is the elastic modulus of the steel wire material (E₁ = 193000 MPa), H is the height of the GGQ-99 WRI (H = 99 mm), and D is the diameter of the arc rope (D = 8 mm). The x, k, and c are the undetermined coefficients in the stiffness and damping models, respectively.

As shown in Table 6, all values of the prestiffness K₁, softened stiffness K₂, and energy dissipation coefficient C of the GGQ-99 Polycal WRI with different structural parameters were obtained. The coefficient values are listed in Table 7. By using the MATLAB software, equations (5) and (6) were formulated fitting with the corresponding structural parameters, respectively.

Table 6: The stiffness-damping of the GGQ-99 WRI with different structural parameters.

Number
Structural parameters
Stiffness-damping

K₁ (N/mm)
K₂ (N/mm)
(%)

1#
1.1
0.545
7.5
5.8125
564.326
108.341
34.65

2#
1.2
0.545
7.5
5.8125
525.582
115.573
29.76

3#
1.3
0.545
7.5
5.8125
526.628
122.756
26.24

4#
1.4
0.545
7.5
5.8125
541.734
151.971
22.45

5#
1.5
0.545
7.5
5.8125
505.712
163.632
19.63

6#
1.1
0.857
7.5
5.8125
449.583
110.592
33.55

7#
1.1
0.75
7.5
5.8125
439.908
109.667
33.64

8#
1.1
0.667
7.5
5.8125
539.792
110.376
33.62

9#
1.1
0.6
7.5
5.8125
554.747
109.971
33.68

10#
1.1
0.545
5.5
5.8125
484.87
104.764
31.63

11#
1.1
0.545
6.5
5.8125
490.37
110.694
28.49

12#
1.1
0.545
8.5
5.8125
509.811
117.433
26.45

13#
1.1
0.545
9.5
5.8125
590.583
125.242
28.07

14#
1.1
0.545
7.5
6.25
571.113
153.811
22.30

15#
1.1
0.545
7.5
6.875
632.302
179.774
21.03

16#
1.1
0.545
7.5
7.5
569.738
176.736
22.90

17#
1.1
0.545
7.5
8.125
749.393
226.096
26.99

Table 7: The coefficient values after the formula was fitted.

Stiffness-damping
Coefficient

k/c

−0.1147
−0.3926
0.3223
0.7524
3.11e − 6

1.0194
−0.1066
0.3003
2.0788
7.15e − 8

−1.0778
0.4142
−0.2272
−0.7828
2.614

4.2. Error Analysis of Prediction Models

As shown in Table 8, the errors of the prestiffness and softened stiffness fitting formulas of the WRI were all within 10%, and more than half of the errors were within 5%. The error of the energy dissipation coefficient of the WRI with the formula was basically within 20%, and more than half of the errors were within 10%. It could be considered that the formula fitted in this study is accurate and reliable by analysing the error results of the overall data. In the design of the Polycal WRI, the stiffness-damping characteristics could be quickly obtained.

Table 8: The fitting error of the stiffness-damping of the WRI with different structural parameters.

Number
Fitted value
Error (%)

K₁ (N/mm)
K₂ (N/mm)
(%)

1#
536.828
114.165
29.27
−4.87
5.38
−15.54

2#
531.497
124.754
26.65
1.12
7.94
−10.47

3#
526.639
135.360
24.44
2.14e − 3
10.27
−6.85

4#
522.182
145.982
22.57
3.61
−3.94
0.52

5#
518.066
156.619
20.95
2.44
−4.29
6.72

6#
449.424
108.787
35.30
−0.04
−1.63
5.22

7#
473.512
110.345
33.40
7.65
−0.62
−0.70

8#
495.898
111.733
31.82
8.13
1.23
−5.36

9#
516.942
113.001
30.45
−6.81
2.76
−9.58

10#
485.760
104.012
31.40
0.18
−0.72
−0.72

11#
512.631
109.363
30.23
4.54
−1.2
6.12

12#
558.926
118.538
28.44
9.63
0.94
7.54

13#
579.326
122.564
27.73
−1.91
−2.14
−1.19

14#
566.955
132.755
27.65
−0.73
−13.69
23.99

15#
609.105
161.844
25.66
−3.67
−9.97
22.02

16#
650.316
193.933
23.97
14.14
9.73
4.68

17#
690.684
229.043
22.52
−7.83
1.30
−16.58

In order to evaluate the accuracy of the prediction model, the experimental results of Cen et al. [24] were referenced. In his paper, the static compression experiment of GGQ25-62 was studied.

The wire rope structural parameters of the GGQ25-62 Polycal WRI are given in Table 9. Besides, the values of H, D, and R are 62 mm, 4.68 mm, and 27.15 mm, respectively.

Table 9: The wire rope size parameters used in the GGQ25-62 Polycal WRI [24].

Arc strand
Parameters
Value

Center strand
The diameter of the rope
1.64

The diameter of core wire
0.36

The diameter of lay wire
0.32

The pitch length of the first layer
9.84

The pitch length of the second layer
8.52

Lay strand
The diameter of the rope
1.52

The diameter of lay wire
0.32

The diameter of lay wire
0.30

The pitch length of the first layer
9.12

The pitch length of the second layer
7.93

As shown in Table 10, the structural parameters of GGQ25-62 were processed in the form of the fitting formula.

Table 10: The structural parameters and stiffness-damping of the GGQ25-62 WRI.

K₁ (N/mm)
K₂ (N/mm)
(%)

Value
1.1
1.15
6.5
5.8125
199.98
58.57
35.86

Through the out-of-plane compression loading-unloading experiment of the WRI, its stiffness-damping characteristics were obtained. Figure 13 shows the load-displacement curve of the GGQ25-62WRI, and the values of K₁, K₂, and C could be calculated from this figure, and Table 10 also shows the specific values.

Figure 13: Experimental compression load-displacement results [24].

Figure 13 is reproduced from the work of Cen et al. [24] (under the Creative Commons Attribution License/public domain).

The fitted value was compared with experimental data. As shown in Table 11, the errors of the prestiffness and softened stiffness fitting formulas of the WRI the values were all within 20%. The reason for this phenomenon is that the finite element is an analysis under ideal conditions. During the test, the vibration isolator is in a complex state, and there will be some nonlinearity characteristics. Therefore, it can be seen that the predicted value given by the prediction model is in good agreement with the experimental value.

Table 11: Comparison between the fitted value and experimental data.

Result
K₁ (N/mm)
K₂ (N/mm)
(%)

Fitted value
239.11
62.99
41.25

Experimental
199.98
58.57
35.86

Error
19.57%
7.55%
15.04%

5. Conclusion

This paper firstly presented the results of the mechanical properties of the wire rope with different diameter ratios and lay pitches of the single strand. By using the corrected simplified finite element method of the Polycal WRI, the influence of the diameter ratio and lay pitch of the single strand and the lay pitch and bending radius of the wire rope on the stiffness and damping characteristics of the Polycal WRI were studied. Several key conclusions are summarized as follows:

(1)The stiffness-damping characteristics were best; when the diameter ratio of wire strand was 1.1, the inside layer wire pitch length was 6 times the diameter of the wire strand, the outside layer wire pitch length was 11 times the diameter of the wire strand, the pitch length of the wire rope was 7.5 times its diameter, and the bending radius was equal to 46.5 mm.(2)The effective prediction models of stiffness and damping about the structural parameters of the wire rope were established, which combined the properties of a steel wire material and the overall structure of the WRI. More than half of the errors of the prestiffness and softened stiffness fitting formulas of the Polycal WRI were within 5%. And more than half of the errors of the energy dissipation coefficient of the WRI were within 10%. The fitted value was compared with the experimental data, the errors were within 20%, and the prediction model was verified.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Financial supports from the National Natural Science Foundations of China (no. 11772147) and Natural Science Fund for Colleges and Universities in Jiangsu Province (16KJB130003) are gratefully acknowledged.

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