The Influence of Parameters and Load on the Spin Angle of Multilayer Strand Wire Rope

Abstract

The spin characteristics of multi-layer strand wire ropes can reduce the control accuracy of the crane lifting system, resulting in the crane being unable to meet the high-precision operation requirements. An object of this paper is to know about the influence of different parameters on the spin characteristics of multi-layer strand wire ropes. This article first theoretically analyzes the effects of wire rope twist angle, wire diameter, material, and load on the rotational characteristics of multi-layer strand wire ropes. In addition, we use CREO to establish a parameterized model for multi-layer strand wire ropes, and then set the material properties, contact parameters, and boundary conditions of the model in ABAQUS. Finally, the dynamic simulation of the multi-layer strand wire ropes model was completed. The results indicate that when the multi-layer strand wire rope is in a steady state, the twist angle has the greatest impact on the rotation angle, and under the same conditions, the rotation angle increases with the increase of the twist angle. The rotation angle at steady state is basically consistent with the theoretical calculation results, indicating that the theoretical influence of wire rope twist angle, wire diameter, material, and load on the rotation characteristics of multi-layer strand wire ropes is accurate.

1 Introduction

Multi-layer strand wire ropes can withstand large axial loads, with relatively low torsional and bending stiffness [1], and are an important component of cranes. With the improvement of the operational indicators of engineering projects, the lifting system of cranes needs to be precisely controlled to achieve higher operational accuracy. However, most existing control technologies overlook the flexible characteristics of steel wire ropes [2, 3], and do not consider the changes in the torsion angle of steel wire ropes under the combined effects of their own parameters and loads. As a result, existing control methods are difficult to meet higher accuracy requirements.

In previous studies, Nawrocki et al. [4] analyzed the mechanical theoretical model of steel wire ropes earlier, studying the mechanical characteristics of single steel wire rope and entire steel wire rope from both static and dynamic responses. Erdönme et al. [5] used nested spiral geometric parameter equations to define a nearly realistic analysis model for the core of steel wire rope. Zhang et al. [6] studied the dynamic torsional characteristics and internal rotational state of steel wire ropes by considering the changes in geometric parameters and mechanical properties of spiral elements in the axial tensile layer of the steel wire rope. Zhang et al. [7] conducted research on shallow well low load mine hoists and used dynamic theory and friction transmission theory to obtain the dynamic tension of steel wire ropes at different positions from the container. Fu et al. [8] applied the theory of differential geometry to derive the spatial curve parameter equation for compiling anti torsion steel wire ropes. Meng et al. [9, 10] proposed a parameterized model for the structure of circular steel wire ropes with arbitrary center lines, especially regarding the direction and type of strand wires, and derived a series of recursive formulas for the spatial winding equations of strand wires.

In response to the shortcomings of the above research content, this article starts from the basic parameters of steel wire rope, such as material, number of strands, wire diameter, twisting method, and twisting angle, and summarizes the variation law of the rotation angle of the steel wire rope in steady state based on the load borne by the steel wire rope.

2 Mathematical Modeling of Multi-layer Strand Wire Ropes

Multi-layer strand steel wire ropes have complex spatial spiral structures, and existing modeling methods use the Frenet standard framework to represent the parameter equations of the centerline of each layer of strand steel wire and complete three-dimensional modeling. This method mainly completes the mathematical modeling of multi-layer steel wire ropes through mathematical calculations, ignoring the role of steel wire rope structure in the modeling. Based on the parameters and structural characteristics of steel wire ropes, this article adopts a combination of numbers and shapes to complete the mathematical modeling of multi-layer strand steel wire ropes. Figure 1 shows the distribution of the basic elements that make up ropes and strands on the cross-section of the steel wire rope.

Fig. 1

18 × 7 + IWS multi-layer strand wire rope cross section strand

The centerline of the central strand core line is a straight line, and its parameter equation can be written as

$$ \left\{ {\begin{array}{*{20}l} {x = 0} \\ {y = 0} \\ {z = c \cdot t} \\ \end{array} } \right. $$

(1)

where c is the coefficient of rope length, t is changeable parameter.

The centerline of the strand side steel wire and the centerline of the strand side core wire are first-order spirals, and their parameter equations can be written as

$$ \left\{ {\begin{array}{*{20}l} {x = r_{1} \cdot \cos \left( {\frac{{z \cdot \tan \beta_{1} }}{{r_{1} }} \cdot \frac{360}{{2\pi }} + \theta_{1i} } \right)} \\ {y = r_{1} \cdot \sin \left( {\frac{{z \cdot \tan \beta_{1} }}{{r_{1} }} \cdot \frac{360}{{2\pi }} + \theta_{1i} } \right)} \\ {z = c \cdot t} \\ \end{array} } \right. $$

(2)

where ${{r}}_{{1}}$ is the distance between the centerline of the central strand side wire and the centerline of the central strand core wire, ${\beta }_{{1}}$ is the twist angle of the central strand, ${\theta }_{{{{1i}}}}$ is the starting position angle of the centerline of the central strand side wire or the centerline of the side strand core wire，$\theta_{1i} = 360 \times \frac{i}{m},$ ${{i}} = {1}\sim {{m}}$, m is the number of side strands or cores of side strands.

The centerline of the side strands is a quadratic helix, and its parameter equation can be written as

$$ \left\{ {\begin{array}{*{20}l} {x = r_{1} \cdot \cos \left( {\frac{{z \cdot \tan \beta_{1} }}{{r_{1} }} \cdot \frac{360}{{2\pi }} + \theta_{1i} } \right) + r_{2} \cdot \sin \left( {\frac{{\left( {\left( {z \cdot \tan \left( {90^{^\circ } - \beta_{1} } \right)} \right)^{2} + z^{2} } \right)^{\frac{1}{2}} \cdot \tan \left( {90^{^\circ } - \beta_{2} } \right)}}{{r_{2} }} \times \frac{360}{{2\pi }} + \theta_{2i} } \right)} \hfill \\ {y = r_{1} \cdot \sin \left( {\frac{{z \cdot \tan \beta_{1} }}{{r_{1} }} \cdot \frac{360}{{2\pi }} + \theta_{1i} } \right) + r_{2} \cdot \sin \left( {\frac{{\left( {\left( {z \cdot \tan \left( {90^{^\circ } - \beta_{1} } \right)} \right)^{2} + z^{2} } \right)^{\frac{1}{2}} \cdot \tan \left( {90^{^\circ } - \beta_{2} } \right)}}{{2\pi \cdot r_{2} }} \times \frac{360}{{2\pi }} + \theta_{2i} } \right)} \hfill \\ {z = c \cdot t} \hfill \\ \end{array} } \right. $$

(3)

where ${{r}}_{{1}}$ is the distance between the centerline of the strand side core wire and the centerline of the central strand core wire, ${\beta }_{{1}}$ is the twist angle of the central line of the strand side core wire, ${{r}}_{{2}}$ is the distance between the centerline of the side strand side wire and the centerline of the side strand core wire, ${\beta }_{{2}}$ is the twist angle of the centerline of the side strands, ${\theta }_{{{{2i}}}}$ is the starting position angle of the centerline of the lateral strands, $\theta_{1i} = 360 \times \frac{i}{n}$, ${{i}} = {1}\sim {{n}}$, n is the number of side strands and side wires.

The research object selected in this article is 18 × 7 + IWS multi-layer strand steel wire rope. Its central strand core wire has a radius of 1 mm, the central strand side wire has a radius of 0.8 mm, a twist angle of 13.46°, right-handed. The inner side strand core wire has a radius of 1 mm, a twist angle of 16.15°, left-handed, and the inner side strand side wire has a radius of 0.8 mm, a twist angle of 13.46°, left-handed. The outer core wire has a radius of 1 mm, a twist angle of 16.15°, right-handed, and the outer side wire has a radius of 0.8 mm, a twist angle of 13.46°, right-handed. We defined the parameters of the wire rope based on CREO 3D software and established a multi-layer strand wire rope model with a rope length of 54.25 mm. The established 3D model is shown in Fig. 2.

Fig. 2

3D model of multi-layer strand wire rope

3 Relationship Between Load and Deformation of Multi-layer Strand Steel Wire Ropes

3.1 Load Analysis of a Single Steel Wire

The spatial force situation of any object can be decomposed into rotation and motion along the coordinate axis. The spatial stress state of steel wire includes the load on the cross-section and the load per unit length. Therefore, the load on a single steel wire can be decomposed into 12 components. The load acting on a single steel wire is shown in Fig. 3. Where S is the arc length along the wire.

Fig. 3

Load acting on a single steel wire

In Fig. 3, we established a Frenet framework on a single wire cross-section. Where N and N′ respectively represent the shear force components in the x-axis and y-axis directions of the steel wire cross-section, T represents the axial tensile force on the steel wire, G and G′ respectively represent the bending moment components in the x-axis and y-axis directions of the cross-section, H represents the torsional moment on the steel wire, X, Y and Z respectively represent the line load components per unit length of the steel wire centerline in the x, y, and z directions, K, K′ and θ respectively represent the external torque per unit length of the steel wire centerline in the x, y, and z directions.

The steel wire is in equilibrium under the combined action of these forces and moments. We sum the forces and moments along the coordinate axis of the Frenet framework to obtain the mechanical equilibrium equation for a single metal wire. The equation can be written as

$$ \left\{ {\begin{array}{*{20}l} \begin{gathered} \frac{{dN}}{{ds}} - N^{\prime } \tau + T\kappa ^{\prime } + X = 0 \hfill \\ \hfill \\ \end{gathered} \hfill \\ \begin{gathered} \frac{{dN^{\prime } }}{{ds}} - T\kappa + N\tau + Y = 0 \hfill \\ \hfill \\ \end{gathered} \hfill \\ \begin{gathered} \frac{{dT}}{{ds}} - N^{\prime } \kappa ^{\prime } + N^{\prime } \kappa + Z = 0 \hfill \\ \hfill \\ \end{gathered} \hfill \\ \begin{gathered} \frac{{dG}}{{ds}} - G^{\prime } \tau + H\kappa ^{\prime } - N^{\prime } + K = 0 \hfill \\ \hfill \\ \end{gathered} \hfill \\ \begin{gathered} \frac{{dG^{\prime } }}{{ds}} - H\kappa + G\tau + N + K^{\prime } = 0 \hfill \\ \hfill \\ \end{gathered} \hfill \\ {\frac{{dH}}{{ds}} - G\kappa ^{\prime } + G^{\prime } \kappa + \theta = 0} \hfill \\ \end{array} } \right. $$

(4)

where $\kappa$ and $\kappa^{\prime }$ respectively represent the curvature components in the x-axis and y-axis directions on the Frenet frame; $\tau$ is the torsion angle per unit length of the steel wire.

4 The Relationship Between Load and Deformation of Single Strand Steel Wire Rope

The initial structure of a single outer wire before and after loading is shown in Fig. 4, and the force relationship between the wire before and after loading is shown in Fig. 5. Then, based on the geometric relationship, we can obtain the deformation parameters of the single strand steel wire rope after being loaded.

Fig. 4

Structure of single strand steel wire rope

Fig. 5

Force triangle of single strand steel wire rope steel wire

The axial strain of steel wire rope and its core wire can be written as

$$ \varepsilon = \varepsilon_{11} = \frac{{\overline{{h_{11} }} - h_{11} }}{{h_{11} }} $$

(5)

The axial strain of the force triangle of the wire rope core can be written as

$$ \varepsilon_{11}^{\prime } = \frac{{\overline{{h_{11}^{\prime } }} - h_{11}^{\prime } }}{{h_{11}^{\prime } }} = \frac{{h_{11} }}{{h_{11}^{\prime } }}\varepsilon_{11} $$

(6)

The triangular axial strain of the wire rope side force can be written as

$$ \varepsilon_{12}^{\prime } = \varepsilon_{11}^{\prime } + \frac{{\alpha_{12} - {{a sin}} \left( {\frac{{\varepsilon_{11}^{\prime } + 1}}{{\varepsilon_{11}^{\prime } + 1/\sin \alpha_{12} }}} \right)}}{{\tan \alpha_{12} }} $$

(7)

The torsion angle per unit length of the side wire of a steel wire rope can be written as

$$ \begin{aligned}\tau_{s12} &= \frac{{\left. {\overline{{\left( {\theta_{12} } \right.}} + 1} \right) - \left( {\theta_{12} + 1} \right)}}{h} = \frac{{\varepsilon_{12}^{\prime } }}{{r_{12} \tan \alpha_{12} }} - \frac{1}{{r_{12} }}{{a}} \sin \left( {\frac{{\varepsilon_{11}^{\prime } + 1}}{{\varepsilon_{12}^{\prime } + 1/\sin \alpha_{12} }}} \right) + \frac{{\alpha_{12} }}{{r_{12} }}\\& \quad + v\frac{{R_{11} \varepsilon_{11}^{\prime } + R_{12} \varepsilon_{12}^{\prime } }}{{{\text{r}}_{12}^{2} \tan \alpha_{12} }}\end{aligned} $$

(8)

The torsion angle per unit length of steel wire rope and its core wire can be written as

$$ \tau_{s} = \tau_{s11} = \tau_{s12} \cdot\frac{{6W_{t12} }}{{W_{t11} + 6W_{t12} }} $$

(9)

In work, one end of a steel wire rope can be considered fixed while the other end is in a free state. As shown in Fig. 6, the free end generates a component ${{p}}_{{{x}}}$ perpendicular to the axial direction of the wire rope under the influence of the weight and the structure of the wire rope, causing the wire rope to rotate. The turning angle of the steel wire rope increases with the increase of the twisting angle of the steel wire rope. The expression for ${{p}}_{{{x}}}$ can be written as

$$ {{p}}_{{{x}}} = {{p}}_{{{y}}} \cdot{{tan}}\upbeta $$

(10)

Fig. 6

Schematic diagram of wire rope rotation

Figure 7 shows the working state of the steel wire rope. Figures 8 and 9 show that the load of the wire rope can be decomposed into axial force F₁₁ and axial torque M₁₁ on the central wire, as well as axial force F₁₂ and axial torque M₁₂ on the external wire. According to Eq. (4), the torque balance equation of the steel wire rope can be written as

$$ \left\{ {\begin{array}{*{20}l} {M_{in} = - \left( {M_{11} + 6 \times M_{12} } \right)} \\ {M_{11} = \frac{E}{2(1 + v)}h_{11} \cdot \tau_{s11} \cdot W_{t11} } \\ {M_{12} = \left( {A_{M12} + B_{M12} R_{11} } \right)\varepsilon_{11}^{\prime } + \left( { - A_{M12} + B_{M12} R_{12} } \right)\varepsilon_{12}^{\prime } + \pi \frac{{r_{12} }}{{R_{12} }}\cos \alpha_{12} \varepsilon_{12}^{\prime } } \\ {A_{M12} = \frac{{\left[ {\left( {1 + \cos^{2} \alpha_{12} } \right)\cos 2\alpha_{12} - 2(1 + v)\left( {1 + \sin^{2} \alpha_{12} } \right)\cos^{2} \alpha_{12} } \right]\pi r_{12} \sin \alpha_{12} \tan \alpha_{12} }}{{4(1 + v)r_{12}^{2} /R_{12} }}ER_{12}^{3} } \\ {B_{M12} = \frac{{\left[ {\left( {1 + \cos^{2} \alpha_{12} } \right)\sin^{2} \alpha_{12} - (1 + v)\left( {1 + \sin^{2} \alpha_{12} } \right)\cos^{2} \alpha_{12} } \right]\pi v\cos \alpha_{12} }}{{4(1 + v)r_{12}^{2} /R_{12} }}ER_{12}^{3} } \\ \end{array} } \right. $$

(11)

Fig. 7

Steel wire rope for hanging heavy objects

Fig. 8

Cross section stress situation

Fig. 9

Section moment situation

The force balance equation of steel wire rope can be written as

$$ \left\{ {\begin{array}{*{20}l} {Q = F_{11} + 6 \times F_{12} } \\ {F_{11} = {\text{E}}\pi R_{11}^{2} \varepsilon_{11} } \\ {F_{12} = A_{F12} \varepsilon_{11}^{\prime } + B_{F12} \varepsilon_{12}^{\prime } } \\ {A_{F12} = \frac{{\left( {1 + 2v\sin^{2} \alpha_{12} } \right)\pi r_{12} \cos^{3} \alpha_{12} \tan \alpha_{12} - \pi v^{2} \sin \alpha_{12} \cos^{4} R_{11} }}{{4(1 + v)r_{12}^{3} /R_{12}^{2} }}ER_{12}^{2} } \\ {B_{F12} = \left[ {\pi \sin \alpha_{12} - \frac{{\left( {1 + 2v\sin^{2} \alpha_{12} } \right)\pi r_{12} \cos^{3} \alpha_{12} \tan \alpha_{12} - \pi v^{2} \sin \alpha_{12} \cos^{4} R_{12} }}{{4(1 + v)r_{12}^{3} /R_{12}^{2} }}} \right]ER_{12}^{2} } \\ \end{array} } \right. $$

(12)

4.1 Relationship Between Load and Deformation of Multilayer Strand Steel Wire Rope

Based on the torque balance equation of a single strand steel wire rope, we have obtained the torque balance equation of a multi-layer strand steel wire rope, which can be written as

$$ \left\{ {\begin{array}{*{20}l} \begin{gathered} M_{{in}} = - \left( {M_{{11}} + 6 \times M_{{12}} + 6 \times M_{{21}} + 36 \times M_{{22}} \cos \beta _{{22}} + 12 \times M_{{31}} } \right. \hfill \\ \left. {\quad \quad \quad + 72 \times M_{{32}} \cos \beta _{{32}} } \right) \hfill \\ \end{gathered} \hfill \\ {M_{{11}} = \frac{E}{{2(1 + v)}}h_{{11}} \cdot \tau _{{S11}} \cdot W_{{t11}} } \hfill \\ \begin{gathered} M_{{ij}} = \left( {A_{{Mij}} + B_{{Mij}} R_{{xin_{{ij}} }} } \right)\varepsilon _{{xin_{{ij}} }}^{\prime } + \left( { - A_{{Mij}} + B_{{Mij}} R_{{ij}} } \right)\varepsilon _{{12}}^{\prime } + \pi \frac{{r_{{ij}} }}{{R_{{ij}} }}\cos \alpha _{{ij}} \varepsilon _{{ij}}^{\prime } \hfill \\ \hfill \\ \end{gathered} \hfill \\ \begin{gathered} A_{{Mij}} = \frac{{\left[ {\left( {1 + \cos ^{2} \alpha _{{ij}} } \right)\cos 2\alpha _{{ij}} - 2(1 + v)\left( {1 + \sin ^{2} \alpha _{{ij}} } \right)\cos ^{2} \alpha _{{ij}} } \right]\pi r_{{ij}} \sin \alpha _{{ij}} \tan \alpha _{{ij}} }}{{4(1 + v)r_{{ij}}^{2} /R_{{ij}} }}ER_{{ij}}^{3} \hfill \\ \hfill \\ \end{gathered} \hfill \\ {B_{{Mij}} = \frac{{\left[ {\left( {1 + \cos ^{2} \alpha _{{ij}} } \right)\sin ^{2} \alpha _{{ij}} - (1 + v)\left( {1 + \sin ^{2} \alpha _{{ij}} } \right)\cos ^{2} \alpha _{{ij}} } \right]\pi v\cos \alpha _{{ij}} }}{{4(1 + v)r_{{ij}}^{2} /R_{{ij}} }}ER_{{ij}}^{3} } \hfill \\ \end{array} } \right. $$

(13)

The force balance equation of multi-layer wire rope can be written as

$$ \left\{ {\begin{array}{*{20}l} {Q = F_{{11}} + 6 \times F_{{12}} + 6 \times F_{{21}} + 36 \times F_{{22}} \cos \beta _{{22}} + 12 \times F_{{31}} + 72 \times F_{{32}} \cos \beta _{{32}} } \hfill \\ {F_{{11}} = E\pi R_{{11}}^{2} \varepsilon _{{11}} } \hfill \\ \begin{gathered} F_{{ij}} = A_{{Fij}} \varepsilon _{{xin_{{ij}} }}^{\prime } + B_{{Fij}} \varepsilon _{{ij}}^{\prime } \hfill \\ \hfill \\ \end{gathered} \hfill \\ \begin{gathered} A_{{Fij}} = \frac{{\left( {1 + 2v\sin ^{2} \alpha _{{ij}} } \right)\pi r_{{ij}} \cos ^{3} \alpha _{{ij}} \tan \alpha _{{ij}} - \pi v^{2} \sin \alpha _{{ij}} \cos ^{4} R_{{ij}} }}{{4(1 + v)r_{{ij}}^{3} /R_{{ij}}^{2} }}ER_{{ij}}^{2} \hfill \\ \hfill \\ \end{gathered} \hfill \\ {B_{{Fij}} = \left[ {\pi \sin \alpha _{{ij}} - \frac{{\left( {1 + 2v\sin ^{2} \alpha _{{ij}} } \right)\pi r_{{ij}} \cos ^{3} \alpha _{{ij}} \tan \alpha _{{ij}} - \pi v^{2} \sin \alpha _{{ij}} \cos ^{4} R_{{ij}} }}{{4(1 + v)r_{{ij}}^{3} /R_{{ij}}^{2} }}} \right]ER_{{ij}}^{2} } \hfill \\ \end{array} } \right. $$

(14)

5 Theoretical Calculation and ABAQUS Simulation

To verify the correctness of the relationship between load and deformation of multi-layer wire ropes, We calculated the deformation of 18 × 7 + IWS multi-layer strand steel wire rope under load. The parameters of the multi-layer wire rope are the same as those of the multi-layer wire rope in Fig. 2. The lifting weight is 70 kg, the elastic modulus is 202,000 Mpa, and the Poisson’s ratio is 0.3. The deformation results of the free end of the multi-layer strand steel wire rope under a load of 70 kg are shown in Table 1 through calculation.

Table 1 Deformation results of free end of multilayer strand steel wire rope

We established a three-dimensional model of multi-layer wire rope in CREO, and then imported the model into ABAQUS to add heavy loads and set material parameters. Finally, we conducted finite element analysis on the model. Figure 10 show the solid finite element model of the steel wire rope. Figure 11 show the finite element analysis of the solid finite element model of the steel wire rope with a heavy object. Figure 12 show the finite element displacement analysis results of the overall model of multi-layer stranded wires. Figure 13 show the deformation results of the free end of the central stranded wire in the z direction.

Fig. 10

Finite element model of multi-layer strand wire rope

Fig. 11

Finite element model of multi-layer strand steel wire rope with heavy objects

Fig. 12

Displacement results of the overall model

Fig. 13

z-direction deformation results of the free end of the central strand core wire

In the drawing function of ABAQUS, we have plotted the displacement curve of the top left corner vertex in the x and y directions in Fig. 12. The plotted curve is shown in Fig. 14, and the stable displacement value is shown in Fig. 15.

Fig. 14

Displacement curve of the top left corner vertex of the heavy object

Fig. 15

Displacement values of the top left corner vertex of the heavy object

The displacement of the top left corner vertex of the heavy object in the x direction is −0.310654 mm, and the displacement in the y direction is −0.311836 mm. After calculation, the torsion angle of the steel wire rope during balance is displayed as

$$ \tau_{s} = 0.178^{\circ } \, \;\left( {{\text{right}}\;{\text{rotation}}} \right) $$

The error between the theoretical calculation results and the simulation calculation results of the torsion angle during the balance of multi-layer wire ropes is written as

$$ \delta = \frac{{0.185^{ \circ } - 0.178^{ \circ } }}{{0.178^{ \circ } }} \times 100{{\% }} = 3.93{{\% }} $$

6 Conclusion

This article establishes a parameterized model of steel wire rope based on the relationship between its structure and parameters, providing an effective mathematical analysis model for studying new steel wire rope structures. Then we explored the relationship between load and deformation of single-layer and multi-layer steel wire ropes, and summarized the relationship between torsion angle and deformation of steel wire ropes under gravity. After verification, we found that the relative error between the deformation calculation results of multi-layer wire ropes under load and the finite element calculation results is 3.93%. The research results provide theoretical guidance for exploring the accuracy impact of steel wire ropes during crane operation.