Sage Journals: Discover world-class research

Abstract

In order to design antirotation wire rope, while considering the control of the axial strain and flexibility, the requirements that make the wire rope antirotation while making the axial strain little and flexibility excellent were presented. The relationship between axial force and twisting moment of wire rope with axial and rotational strain was given according to the linear stiffness coefficient. It was concluded that the twisting moment and axial strain under tensile load only depend on wire rope's geometry parameters which consist of helical angle and wire diameter. And the expression of bending stiffness was given. Through the analysis of an example, we find that the outer strand has a stronger effect on the tensile, torsional, and bending stiffness. And the closer the strand's helical angle is to 90°, the bigger the tensile stiffness and the bending stiffness are. The helical angles that give consideration to all the three stiffness coefficients can be found. These can be used for reference when designing antirotation wire rope.

1. Introduction

Wire rope was widely used in the field of mine hoist system, crane, transport machinery, and so on due to its advantages of excellent flexibility and high tensile strength. Wire rope will generate a rotary motion or twisting moment when subjected to tensile load because of its special helical structure, and the tensile and bending stiffness vary with the change of twisting parameters. But we require the wire rope antirotation and axial elongation being not too big and the flexibility being excellent on occasion of mine hoist, crane lifting, and so forth. Some researches have been done into helical-structure rope [1 –4]. The physical model and force conditions of antirotation wire rope were studied by some scholars [5 –8]. The axial and torsion strain of wire ropes and a single strand under axial tensile load were studied by some researchers [9 –13]. The flexibility and bending stiffness of strand were also studied by some scholars [14 –16]. The linear stiffness coefficients of wire rope were deduced by Cao [4]. But only one parameter was considered when antirotation wire ropes were designed. There is little research on calculation for design of antirotation wire rope, and no researches were done to colligate these three parameters. The bending, torsion, and Poisson's effect of wire rope were ignored in traditional design of antirotation wire rope; therefore, the result was not precise. And the effect of design parameters on tensile and bending stiffness was not considered.

Therefore, coupling parameters were put forward, which consist of antirotation, tensile characteristics, and flexibility. After studying the relationship between twisting moment and axial strain with stiffness coefficient, the difficulty of solving the nonlinear equations was solved based on the linear stiffness coefficient that ignored the extrusion deformation of fiber-core. The bending stiffness of wire rope was also given. Wire rope could be designed conveniently using tensile, torsional, and bending stiffness coefficient. At last, the effect of helical angles for different layer on three stiffness coefficients was obtained. Wire ropes can be designed according to the conclusions.

2. Requirement of Antirotation and Little Axial Strain and Excellent Flexibility

Tensile and torsional movements will affect each other when external axial load is applied because of the helical structure of wire rope (as in Figure 1(a)); therefore, the relationship between force and strain can be formulated according to [2, 17]

\begin{matrix} T = Q_{1} ε + Q_{2} ϕ, \\ M = Q_{3} ε + Q_{4} ϕ, \end{matrix}

(1)

where T and Mdenote tensile force and twisting moment, ε and ϕ denote the axial strain and angle of twist per unit length, Q1 denotes tensile stiffness coefficient, Q₂ and Q₃ denote the interaction coefficient of tensile and torsional motion, and Q4 denotes torsional stiffness coefficient.

Figure 1:

The structure of wire rope 36 × 7 + FC.

In order to make wire rope antirotation under axial load, as long as make sure that twisting moment M do not vary with the change of axial strain ε. Only let $\partial M / \partial ε = 0$ according to (1); that is, Q₃ = 0. Q₂ and Q₃ are approximately equal according to [2, 4]. So Q₂ is approximately equal to zero when meeting the requirement of antirotation. Therefore, the relationship between axial strain and axial load can be formulated as ε = T/Q₁. The axial strain will become less only if Q₁ becomes larger. The expression of Q₁ and Q₃ which depends on size and location parameters of wire rope should be given at first in order to give the design method of antirotation wire rope.

Wire rope has excellent flexibility which means that the radius of curvature can be very small subject to a small bending moment. The relationship between them can be formulated as $M_{b} = B / ρ$ , where M_b is bending moment, B is bending stiffness, and ρ is radius of curvature. The expression of B should also be given to design wire rope with excellent flexibility.

3. Stiffness of Wire Rope

3.1. Linear Tensile and Torsional Stiffness

Figure 1(b) is the sectional view of wire rope; Figure 1(c) is the stretch-out view of strand centerline before and after deformation. In Figure 1(c), l_j, R_j, β_j, and θ_j denote the length, helical radius, helical angle, and angle swept out by helical strand, respectively, of the strand in jth layer, and h denotes the length of wire rope. ξ₀ is axial strain of wire rope and of the fiber-core; ξ_j, Δβ_j, ΔR_j, and ϕ_j are axial strain, increment of helical angle, reduction of helical radius, and angle of twist per unit length, respectively, of the strand in jth layer. “ $\bar{}$ ” denotes the variable under axial load in the following paragraphs. It is assumed that the fiber-core inside rope has been fully squeezed due to twisting or long-term service, so the extrusion deformation of fiber-core could be ignored.

According to [2, 4], the axial strain ε and angle of twist per unit length ϕ of wire rope can be written as

\begin{matrix} ε = ξ_{0} = \frac{\bar{h} - h}{h} = (1 + ξ_{j}) \cdot \frac{\sin {\bar{β}}_{j}}{\sin β_{j}} - 1, \\ ϕ = \frac{{\bar{θ}}_{j} - θ_{j}}{h} = \frac{cot {\bar{β}}_{j}}{{\bar{R}}_{j}} (1 + ξ_{0}) - \frac{cot β_{j}}{R_{j}} . \end{matrix}

(2)

By using Taylor Formula $\sin {\bar{β}}_{j} = \sin β_{j} + Δ β_{j} \cos β_{j}$ , $cot {\bar{β}}_{j} / {\bar{R}}_{j} = cot β_{j} / R_{j} - Δ β_{j} / R_{j} si n^{2} β_{j} + Δ R_{j} cot β_{j} / R_{j}^{2}$ , where higher-ordered terms are neglected. Hence, (2) can now be formulated as

ε = ξ_{0} = Δ β_{j} cot β_{j} + ξ_{j},

(3)

ϕ = \frac{1}{R_{j}} (ξ_{j} cot β_{j} + \frac{Δ R_{j}}{R_{j}} cot β_{j} - Δ β_{j}) .

(4)

The reduction of helical radius of strand in jth layer can be formulated as $Δ R_{j} = Δ r_{0} + 2 \sum_{k = 1}^{j - 1} Δ r_{k} + Δ r_{j}$ , where Δr_k is reduction of strand radius in kth layer, Δr₀ = ν_fr₀ξ₀ is reduction of fiber-core radius, and ν_f is Poisson's ratio for fiber-core. According to [4], Δr_j is believed to be expressed as Δr_j = ΔR_n′ + Δr_n′, where ΔR_n′ and Δr_n′ which are linearly formulated using ξ_j and ϕ_j denote the reduction of helical radius and wire radius for outmost wire. So Δr_j is linearly expressed as Δr_j = η_{Δr_j}ξ_j + λ_{Δr_j}ϕ_j, where η_{Δr_j} and λ_{Δr_j} are coefficients which depend on the geometric parameters of strand and Poisson's ratio for wire material only. By using the expression of Δr_j, ΔR_j can be given by the following expression:

Δ R_{j} = \sum_{k = 0}^{j} (Q_{Δ R a_j k} ξ_{k} + Q_{Δ R b_j k} ϕ_{k}),

(5)

where the coefficients Q_{ΔRa_jk} and Q_{ΔRb_jk} are given in Appendix (A.1).

By virtue of (3), Δβ_j can be expressed as

Δ β_{j} = (ξ_{0} - ξ_{j}) \tan β_{j} .

(6)

Angle of twist per unit length of strand inside wire rope under external load should be expressed by the change of torsion τ_j as $ϕ_{j} = {\bar{τ}}_{j} - τ_{j}$ , where ${\bar{τ}}_{j} = (\sin {\bar{β}}_{j} \cos {\bar{β}}_{j}) / {\bar{R}}_{j}$ . By using Taylor Formula and neglecting higher-ordered terms, ϕ_j could be converted into the form

ϕ_{j} = τ_{a j} Δ β_{j} + τ_{b j} Δ R_{j},

(7)

where τ_aj and τ_bj are given in Appendix (A.5).

After substituting (5) and (6) into (7), we have

ϕ_{j} = \sum_{k = 0}^{j} (Q_{ϕ a_j k} ξ_{k} + Q_{ϕ b_j k} ϕ_{k}),

(8)

where coefficients of Q_{ϕa_jk} and Q_{ϕb_jk} are given in Appendix (A.6).

Equation (8) can then be written in terms of matrix equation:

ϕ = Λ \cdot Z,

(9)

where $ϕ = {[\begin{bmatrix} ϕ_{1} & ϕ_{2} & \dots & ϕ_{n} \end{bmatrix}]}^{T}$ , $Ζ = {[\begin{bmatrix} ξ_{0} & ξ_{1} & \dots & ξ_{n} \end{bmatrix}]}^{T}$ , $Λ = M_{ϕ}^{- 1} \cdot M_{ε}$ , M_ϕ and M_ξ are given in Appendix (A.7), and n denotes the number of layers within wire rope.

Using (5) and (9), ΔR_j can be expressed as follows:

\begin{array}{l} Δ R_{j} = \sum_{k = 0}^{j} Q_{Δ R a_j k} ξ_{k} + \sum_{k = 1}^{j} (Q_{Δ R b_j k} \sum_{m = 0}^{n} Λ_{k (m + 1)} ξ_{m}) \\ = \sum_{k = 0}^{n} Q_{j_k} ξ_{k}, \end{array}

(10)

where Q_{j_k} is given in Appendix (A.8).

Combine (3), (4), and (10); the axial strain, variation of helical angle, and variation of helical radius for the strand in jth layer can be linearly expressed by axial strain and twist angle per unit length of wire rope as

ξ_{j} = η_{j} ξ_{0} + λ_{j} ϕ,

(11)

Δ β_{j} = η_{Δ β_{j}} ξ_{0} + λ_{Δ β_{j}} ϕ,

(12)

Δ R_{j} = η_{Δ R_{j}} ξ_{0} + λ_{Δ R_{j}} ϕ,

(13)

where the coefficients of η_j, λ_j, η_{Δβ_j}, λ_{Δβ_j}, η_{ΔR_j}, and λ_{ΔR_j} are given in Appendix (A.11)–(A.14).

The axial force and twisting moment of strand in jth layer could be formulated as T_j = E_jA_jξ_j, $H_{j} = C_{j} ({\bar{τ}}_{j} - τ_{j})$ , where E_j is Young's modulus for a strand in jth layer, A_j is sectional area of a strand in jth layer, and C_j is the torsional stiffness of strand in jth layer. It could be linearly expressed that T_j = Q_j1ξ_j + Q_j2ϕ_j and H_j = Q_j3ε_j + Q_j4ϕ_j according to [4, 17], where Q_j1, Q_j2, Q_j3, and Q_j4 are linear stiffness coefficient of strand in jth layer, which only depend on the geometric parameter of strand and wire material. The expressions of T_j and H_j could be further linearized with the use of (7) and ((11))–((13)) as the following expressions results:

T_{j} = η_{T_{j}} ξ_{0} + λ_{T_{j}} ϕ,

(14)

H_{j} = η_{H_{j}} ξ_{0} + λ_{H_{j}} ϕ,

(15)

where coefficients of η_{T
_j}, λ_{T
_j}, η_{H
_j}, and λ_{H
_j} are given in Appendix (A.15).

The bending moment of strand can be expressed by the variance of curvature κ_j as $G_{j} = B_{j} ({\bar{κ}}_{j} - κ_{j})$ according to [2, 18], where ${\bar{κ}}_{j} = \cos^{2} {\bar{β}}_{j} / {\bar{R}}_{j}$ , $B_{j} = (π E / 4) \cdot [r_{j 0}^{4} + \sum_{i = 1}^{n^{'}} (2 m_{j i} \sin α_{j i} / (2 + ν \cos^{2} α_{j i})) \cdot r_{j i}^{4}]$ is the bending stiffness of strands in jth layer, and ji denotes the ith layer of wire in jth layer of strand. By using Taylor Formula, G_j is linearly formulated as

G_{j} = η_{G_{j}} ξ_{0} + λ_{G_{j}} ϕ,

(16)

where η_{G
_j} and λ_{G
_j} are given in Appendix (A.16).

According to [2, 18], the shearing force of helical strand could be written as $N_{j} = H_{j} {\bar{κ}}_{j} - G_{j} {\bar{τ}}_{j}$ according to (15) and (16), the expression could be further linearized as

N_{j} = η_{N_{j}} ξ_{0} + λ_{N_{j}} ϕ,

(17)

where η_{N
_j} and λ_{N
_j} are given in Appendix (A.17).

According to [2, 4, 17], the total axial force and axial twisting moment can then be written as

\begin{matrix} T = E_{f} A_{0} ξ_{0} + \sum_{j = 1}^{n} m_{j} (T_{j} \sin β_{j} + N_{j} \cos β_{j}), \\ M = \sum_{j = 1}^{n} m_{j} [(H_{j} - N_{j} R_{j}) \sin β_{j} + (G_{j} + T_{j} R_{j}) \cos β_{j}], \end{matrix}

(18)

where E_f is Young's modulus for fiber-core, A₀ is sectional area of fiber-core, and m_j is the quantity of strands in jth layer.

Substitute ((14))–((17)) into (18), and by virtue of (1), Q₁ and Q₃ could be formulated as

Q_{1} = E_{f} A_{0} + \sum_{j = 1}^{n} m_{j} [η_{T_{j}} \sin β_{j} + η_{N_{j}} \cos β_{j}],

(19)

Q_{3} = \sum_{j = 1}^{n} m_{j} [(η_{H_{j}} - η_{N_{j}} R_{j}) \sin β_{j}) (+ (η_{G_{j}} + η_{T_{j}} R_{j}) \cos β_{j}] .

(20)

It should be stressed that tensile and torsional stiffness coefficients are constants for a given wire rope and are dependent on the rope geometry parameters and wire material properties only. The diameters of rope and wire are usually determined by the axial load that they have to bear. Therefore, we can do a little on diameter of wire to improve the performance of wire rope. The helical angle of strands and wires inside strands can be optimized to adapt to the special conditions.

3.2. Bending Stiffness

A strand inside wire rope can be regarded as a multiple helical spring; its bending stiffness can be defined by the following equation according to [2, 19, 20]:

B_{j} = \frac{π r_{j}^{4}}{4} E_{j} \cdot \frac{2 \sin β_{j}}{(2 + ν_{j} \cos^{2} β_{j})} .

(21)

The bending stiffness of fiber-core can be written as

B_{f} = \frac{π r_{0}^{4}}{4} E_{f} .

(22)

For a rope, the bending stiffness could be approximated by the bending stiffness of fiber-core and each strand in the rope because the friction has a small influence on it according to [2, 20]. So the bending stiffness of wire rope can be written as

B = \frac{π}{4} \cdot [E_{f} r_{0}^{4} + \sum_{j = 1}^{n} \frac{2 m_{j} E_{j} \sin β_{j}}{(2 + ν_{j} \cos^{2} β_{j})} \cdot r_{j}^{4}],

(23)

where the term ν_jcos⁡²β_j has much smaller effect and with negligible error ν_j may be set equal to Poisson's ratio of wire material ν. The Young modulus for strand in jth layer can be written as $E_{j} = Q_{j 1} / A_{j}$ Q_j1 is the tensile stiffness of jth strand. Hence, (23) could be converted into the following form:

B = \frac{π}{4} \cdot [E_{f} r_{0}^{4} + \sum_{j = 1}^{n} \frac{2 m_{j} Q_{j 1} \sin β_{j}}{A_{j} (2 + ν \cos^{2} β_{j})} r_{j}^{4}] .

(24)

4. Verification of Linear Stiffness Coefficient

4.1. Parameters

The 36 × 7 + FC wire rope is chosen, in which there is a fiber-core surrounded by thirty-six seven-wire strands. The innermost layer will be called strand 1; the middle and the outmost layer will be called strand 2 and strand 3, respectively. It is assuming that every strand is absolutely the same. The parameters are as follows: the diameter of fiber-core is 4.6 mm, the diameter of center wire inside strands is 1.316 mm, the diameter of outside wire inside strands is 1.25 mm, and the helical angle of wire inside strands is 75°. Young's modulus and Poisson's ratio of steel wire are 2 × 10¹¹ Pa and 0.3, respectively, while those of fiber-core are 3 × 10¹⁰ Pa and 0.4.

4.2. Result Analysis

The twisting moment could be solved by using both linear stiffness coefficient and simultaneous nonlinear equations, in case of rotational movements of two ends which are restricted and the value of axial strain is known. The twisting moment of wire rope under axial strain ε = 1.5 × 10⁻³ is depicted in Figures 2, 3, and 4; Figure 5 is the twisting moment of wire rope with axial strain ε varying from 1.0 × 10⁻³ to 2.0 × 10⁻³ when β₁ = 75^°, β₂ = 75^°, and β₃ = 105^°.

Figure 2:

Twisting moment with β₁ varying.

Figure 3:

Twisting moment with β₂ varying.

Figure 4:

Twisting moment with β₃ varying.

Figure 5:

Twisting moment with ε varying.

According to Figures 2 –5, the result of twist moment under external load that is obtained by linear and nonlinear method is basically the same, which proves the correctness of linear stiffness coefficient. Therefore, we can design antitwist wire rope using the linear stiffness coefficient in (19) and (20).

By virtue of Figures 2 –4, twisting moment has both positive and negative value; the reason is as follows. The twisting moment generated by strand 1 and strand 2 whose helical angles is below 90° is positive, while that generated by strand 3 is negative. Twisting moment is zero when moment of strand 1 and strand 2 is neutralized by that of strand 3. So there will be positive and negative twisting moment when the helical angles change.

5. Example Analysis

The parameters in Section 4.1 were used. In order to make twisting moment of all strands could cancel out each other, innermost layer and middle layer are Lang lay, and outermost layer is regular lay. Since the expression of Q₃ is very complex, solving another helical angle through Q₃ = 0 with two helical angles being known is very difficult. But it is convenient to solve the value of Q₃ when all helical angles are known. So we can draw the curve of Q₃ while two helical are fixed and another varies within a certain range. The curve will contain the point that makes Q₃ = 0. So the value of β₁, β₂, and β₃ that make wire rope antirotation could be found. The curves are depicted in Figures 6, 7, and 8.

Figure 6:

The curves of Q3 and Q1 with β₁ varying when β₂ = 73^° and β₃ = 101^°.

Figure 7:

The curves of Q3 and Q1 with β₂ varying when β₁ = 72^° and β₃ = 101.5^°.

Figure 8:

The curves of Q3 and Q1 with β₃ varying when β₁ = 72^° and β₂ = 70^°.

We can get three groups of helical angles that make wire rope antirotation according to Figures 6 –8: β₁ = 74.5^°, β₂ = 73^°, β₃ = 101^°; β₁ = 72^°, β₂ = 71.92^°, β₃ = 101.5^°; β₁ = 72^°, β₂ = 70^°, and β₃ = 102.008^°. And a lot of combinations will be obtained by changing the fixed angles. In addition, the closer the strands’ helical angle is to 90°, the larger the tensile stiffness Q₁ is; outer strands have stronger influence on Q₃ and Q₁ by observing the slope of curves.

The curves of bending stiffness with the helical angles varying are depicted in Figure 9, according to (24).

Figure 9:

Bending stiffness with helical angle varying.

According to Figure 9, the closer the strands’ helical angle is to 90°, the larger the bending stiffness is, that is, the worse the wire rope's flexibility is. In addition, the curves are symmetrical about β = 90^°. The outer strands have a stronger influence on bending stiffness according to the slope of curves.

Above all, when wire rope is designed, the tensile stiffness and torsional stiffness cannot reach the best at the same time while meeting the requirement of antirotation, so we should consider the actual working conditions to know how to choose the structure parameters reasonably.

6. Conclusion

The requirements that make the wire rope antirotation while with good performance in resisting axial elongation and excellent flexibility were given according to the force conditions of helical structure. The stiffness coefficients play the major role; the value of stiffness coefficients only depends on helical angle of every layer when the structure of wire rope and the radius of wire are known.

The difficulty of solving the nonlinear equations is solved by using the expressions of linear stiffness coefficient. We can get the helical angle of layers that make wire rope antirotation conveniently by making the curves of Q₃ and β. And its correctness was verified by comparing with the result of nonlinear calculation. So the linear stiffness coefficient can be used as theoretical basis of designing antirotation wire rope.

For the tensile and bending stiffness, when the helical angles approach 90 degrees, the response of stiffness to helical angles gets slower.

The outer layer has a stronger influence on stiffness than inner layers. The torsional, tensile, and bending stiffness can be regulated by slightly adjusting the helical angle of outer layer. These can be regarded as the theoretical basis of adjusting the structure parameters.

The tensile properties and flexibility of wire rope cannot reach the best at the same time. Therefore, appropriate value of helical angle should be chosen according to the needs of working conditions.

Footnotes

Appendix

The coefficient that is in expression of twisting radius variation ΔR_j is as follows:

where η_{Δr_k} and λ_{Δr_k} are coefficients which depend on the geometric parameter of strand and Poisson's ratio for wire material only.

According to [2], the curvature of strand can be formulated as

By using Taylor Formula ${\bar{κ}}_{j} = κ_{j} + (κ_{a j} Δ β_{j} + κ_{b j} Δ R_{j}) + \dots$ , neglect higher-ordered terms Δκ_j = κ_ajΔβ_j + κ_bjΔR_j, where

According to [2], the torsion of strand can be formulated as

By using Taylor Formula ${\bar{τ}}_{j} = τ_{j} + (τ_{a j} Δ β_{j} + τ_{b j} Δ R_{j}) + \dots$ , neglect higher-ordered terms Δτ_j = τ_ajΔβ_j + τ_bjΔR_j, where

The twist angle per unit length of strand in jth layer can be given as $ϕ_{j} = \sum_{k = 0}^{j} (Q_{ϕ a_j k} ξ_{k} + Q_{ϕ b_j k} ϕ_{k})$ , where coefficients are as follows:

The matrix of ϕ is formulated as ϕ = Λ·Z, where $Λ = M_{ϕ}^{- 1} \cdot M_{ε}$ the coefficient matrix is as follows:

The reduction of helical radius ΔR_j is written as $Δ R_{j} = \sum_{k = 0}^{j} Q_{Δ R a_j k} ξ_{k} + \sum_{k = 1}^{j} (Q_{Δ R b_{j k}} \sum_{m = 0}^{n} Λ_{k (m + 1)} ξ_{m})$ , and to simplify the expression, the two terms are combined, having $Δ R_{j} = \sum_{k = 0}^{n} Q_{j_k} ξ_{k}$ , where

Combine (3), (4), and (10) and obtain

When j varies from 1 to n,

Convert the n equations into matrix form, N_{ξ – ϕ}ξ = N_εε + N_ϕϕ, and after transposition of terms, it can be formulated as $ξ = N_{ξ - ϕ}^{- 1} N_{ε} ε + N_{ξ - ϕ}^{- 1} N_{ϕ} ϕ$ , where $ξ = {[\begin{bmatrix} ξ_{1} & ξ_{2} & \dots & ξ_{n} \end{bmatrix}]}^{T}$ . So ξ_j can be written as ξ_j = η_jε + λ_jϕ, where

where,

Substitute (11) into (3); the increment of helical angle for strand in jth layer is linearly written as Δβ_j = η_{Δβ_j}ξ₀ + λ_{Δβ_j}ϕ, where

Substitute (11) into (10); the reduction of helical radius for strand in jth layer is linearly written as ΔR_j = η_{ΔR_j}ξ₀ + λ_{ΔR_j}ϕ, where

The axial force and twisting moment of strand in jth layer could be formulated as T_j = Q_j1ξ_j + Q_j2ϕ_j and H_j = Q_j3ε_j + Q_j4ϕ_j. According to [4, 17], substitute the expression of ε_j and ϕ_j in (11) and (7) into it, having T_j = η_{T
_j}ξ₀ + λ_{T
_j}ϕ and H_j = η_{H
_j}ξ₀ + λ_{H
_j}ϕ, where

The bending moment of strand in jth strand can be expressed as $G_{j} = B_{j} ({\bar{κ}}_{j} - κ_{j})$ , according to [2, 18], by using Taylor Formula in (A.3) and the linear expression of Δβ_j and ΔR_j in (12) and (13) can be written as G_j = η_{G
_j}ξ₀ + λ_{G
_j}ϕ, where

The shearing force of helical strand in jth layer could be written as $N_{j} = H_{j} {\bar{κ}}_{j} - G_{j} {\bar{τ}}_{j}$ , according to [2, 18], by substituting the linear expression of H_j and G_j into it, having N_j = η_{N
_j}ξ₀ + λ_{N
_j}ϕ, where

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by “National Science Foundation for Young Scientists of China” (Grant no. 51005233), “National Key Basic Research Program of China (973 Program)” (Grant no. 2014CB049401), “Program for New Century Excellent Talents in University” (NCET-13-1017), “Program for Innovative Research Team in University” (IRT1292), and “a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions ‘PAPD'.”

References

Velinsky

S. A.

Anderson

G. L.

, and Costello

G. A.

, “Wire rope with complex cross sections,” Journal of Engineering Mechanics, vol. 110, no. 3, pp. 380–391, 1984.

Costello

G. A.

, Theory of Wire Rope, Springer, 1997.

Elata

Eshkenazy

, and Weiss

M. P.

, “The mechanical behavior of a wire rope with an independent wire rope core,” International Journal of Solids and Structures, vol. 41, no. 5–6, pp. 1157–1172, 2004.

Cao

G. H.

, Impact dynamic behaviors of mine hoisting rope during loading process [Ph.D. thesis], China University of Mining and Technology, 2009.

Utting

W. S.

and Jones

, “The response of wire rope strands to axial tensile loads—Part I. Experimental results and theoretical predictions,” International Journal of Mechanical Sciences, vol. 29, no. 9, pp. 605–619, 1987.

G. Z.

and Qin

W. X.

, “Exploring and analysis for special structure anti-rotation wire rope,” Steel Wire Products, vol. 34, no. 2, pp. 1–6, 2008.

Xie

X. P.

Jia

S. Y.

, and Niu

G. C.

, “Research of the mechanical model in non-rotating wire rope,” Coal Mine Machinery, vol. 31, no. 8, pp. 95–97, 2010.

H. J.

, Study on theory and life estimation of the non-rotating rope used in crane [Ph.D. thesis], South China University of Technology, 2012.

Machida

and Durelli

A. J.

, “Response of a strand to axial and torsional displacements,” Journal of Mechanical Engineering Science, vol. 15, no. 4, pp. 241–251, 1973.

10.

Lanteigne

, “Theoretical estimation of the response of helically armored cables to tension, torsion, and bending,” Journal of Applied Mechanics, vol. 52, no. 2, pp. 423–432, 1985.

11.

Erdem Imrak

and Erdönmez

, “On the problem of wire rope model generation with axial loading,” Mathematical and Computational Applications, vol. 15, no. 2, pp. 259–268, 2010.

12.

Erdonmez

and Imrak

C. E.

, “A finite element model for independent wire rope core with double helical geometry subjected to axial loads,” Sadhana, vol. 36, no. 6, pp. 995–1008, 2011.

13.

Beleznai

and Páczelt

, “Design curve determination for two-layered wire rope strand using p-version finite element code,” Engineering with Computers, vol. 29, no. 3, pp. 273–285, 2013.

14.

Costello

G. A.

and Butson

G. J.

, “Simplified bending theory for wire rope,” Journal of the Engineering Mechanics Division, vol. 108, no. 2, pp. 219–227, 1982.

15.

Jiang

, “A concise finite element model for pure bending analysis of simple wire strand,” International Journal of Mechanical Sciences, vol. 54, no. 1, pp. 69–73, 2012.

16.

Z. H.

and Hu

J. Q.

, “The fatigue and degradation mechanisms of wire ropes bending-over-sheaves,” Applied Mechanics and Materials, vol. 127, pp. 344–349, 2012.

17.

Velinsky

S. A.

, “Analysis of fiber-core wire rope,” Journal of Energy Resources Technology, vol. 107, no. 3, pp. 388–393, 1985.

18.

Love

A. E. H.

, A Treatise on the Mathematical Theory of Elasticity, Dover, 1944.

19.

Costello

G. A.

, “Large deflections of helical spring due to bending,” Journal of the Engineering Mechanics Division, vol. 103, no. 3, pp. 481–487, 1977.

20.

Costello

G. A.

and Miller

R. A.

, “Lay effect of wire rope,” Journal of the Engineering Mechanics Division, vol. 105, no. 4, pp. 597–608, 1979.

Study on Multicharacteristic of Antirotation Wire Rope Based on Linear Stiffness Coefficient

Abstract

1. Introduction

2. Requirement of Antirotation and Little Axial Strain and Excellent Flexibility

3. Stiffness of Wire Rope

3.1. Linear Tensile and Torsional Stiffness

3.2. Bending Stiffness

4. Verification of Linear Stiffness Coefficient

4.1. Parameters

4.2. Result Analysis

5. Example Analysis

6. Conclusion

Footnotes

Appendix

Conflict of Interests

Acknowledgments

References