# Theory and Implementation of No-Joint Wire Rope Spatial Geometry Modeling

Authors: Wu Zhang; Xinqian Zhang; Chuanwei Zhang; Jiafu Zhang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper takes a cross section of 6 × 7 + IWS wire rope as an example to complete the studies on a method of geometric modeling for no-joint wire rope and implementation of a 3D solid model. Frenet–Serret frame theory establishes three types of Frenet–Serret frame, which are double-helix winding straight line, single-helix winding arc, and double-helix winding arc, and it allows derivation of eight types of center line equation of wire. This paper has ascertained tangent condition of center lines of steel wires with the same name between arc and straight-line wire rope and closed condition of no-joint wire rope. It has been concluded that the length of steel wire must be an integer multiple of its lay pitch either in arc wire rope or in straight-line wire rope, and the name of wires in each wire rope must be consistent at the same time. Finally, using Pro/ENGINEER Wildfire 5.0 software generates a 3D solid model of no-joint wire rope.


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## Body

## 1. Introduction

No-joint wire rope (Figure1) has unique advantages in engineering applications. It is convenient to connect the weight to hoisting equipment, it does not easily slide or fracture, and it has more carrying capacity than ordinary wire rope (Figure 2).Figure 1
No-joint wire.Figure 2
Ordinary wire rope.For no-joint wire rope that is closed into a ring, its production process is more complex and the cost is higher than for ordinary wire rope. No-joint wire rope is often used in transmission machinery; for example, the wire rope-continuously variable transmission (WR-CVT) [1], which uses no-joint wire ropes instead of steel bands, is a typical application. A transmission system using no-joint wire rope is shown in Figure 3.Figure 3
Transmission system using no-joint wire rope.No-joint wire rope is different from the ordinary wire rope. Its closed condition of each steel wire should be considered in the process of geometric modeling. Thus, it is affected by lay angle, helix angle, lay pitch, and so on, and its modeling is difficult. At present, there are three main theories for wire rope modeling: matrix transformation theory [2], Frenet–Serret frame theory [3, 4], and enveloping surface theory [5]. The model shape mainly concentrates in the straight wire rope and arc wire rope, but there are a few studies on the modeling of no-joint wire rope. In 2000, Nawrocki and Labrosse [6] established a simple straight-stranded wire rope model and analyzed the reliability under axial loads and bending compound loads. Later, Judge et al. [7] established the multilayer single-twist wire rope model to study the complex mechanical behavior among steel wires. Erdonmez and Imrak established the model of straight wire rope with independent wire rope core [8], and three years later, the recurrence formula of n-tuple helix of the center line of straight round wire rope [9] was deduced. In 2012, Eva Stanová [10] used Pro/ENGINEER Wildfire 5.0 (Pro/E) software to generate the n0 + (2n0 + 4) + n2 straight wire rope model with approximate ellipse section. Two years later, the author worked with Fedorko et al. [11] to deduce a (3 + 9) and (3 + 9 + 15) mathematical model of wire rope with a triangular section and accurately calculate the solid model of double-layered triangular steel wire rope. Both Ma et al. [12] and Wu et al. [13] completed the modeling of arc wire rope based on Frenet–Serret frame theory. In order to correctly analyze the mechanical characteristics of multiple superposition of wire rope in groove in mine hoisting, Peng et al. [14] simplified the multiple superposition of wire rope into the superposition of no-joint wire ropes.The main purpose of wire rope geometric modeling is to facilitate the simulation of the mechanical properties and friction-wear characteristics of wire ropes under different conditions. Zhou and Tian [15] has established a finite-element model of linear single-twist wire rope to calculate its pitch length according to Timoshenko beam theory. The calculation results show that both tension and bending deformation are very close to calculation results of the Costello theory, which verifies the correctness and accuracy of the model. Stanova et al. [16] took the linear single-twist wire rope model with multiple layers as the research object and used the finite-element method to study its tensile mechanical properties. Elata et al. [17] estimated the overall mechanical properties of the wire rope by constructing a wire rope model with independent wire rope core. Páczelt and Beleznai [18] have developed a finite-element program that takes nonlinear contact, Poisson effect, friction, and wear into account, and the program can analyze the small displacement and deformation of wire ropes under a variety of loads, including stretching, torsion, and bending. Cruzado et al. [19], based on finite-element code, performed three studies on steel wire, which investigated the area of wear scar, depth of wear scar, and volume of wear scar, and the results showed that the error could be controlled within 10%. Two years later, Cruzado et al. [20] also carried out the wear scar prediction with a cross-angle for 15°, 45°, and 60°, and the simulation wear scar size and experimental results are basically consistent. At the same time, Wang et al. [21], according to Archard wear theory, analyzed the influence of different intersection angles on the fretting wear depth of steel wire, and the predicted fretting wear depth is in good agreement with the experimental results. Leonard et al. [22] used microscopes and high-speed cameras to record videos and images of contact position, and the experimental device can study the fretting wear among various contact surfaces.Based on the aforementioned research, the geometric modeling of no-joint wire rope will be carried out in this paper. The focus will be on analyzing and implementing the closed condition of no-joint wire rope and the tangent condition between straight wire rope and arc wire rope. Using Frenet–Serret frame theory, we can deduce the center line equation of straight wire rope and arc wire rope and also set up the coordinate system on different rope segments to solve the wire tangent problem. Finally, Pro/E software is used to generate a 3D solid model of no-joint wire rope of transmission. This study not only lays the foundation for the analysis of the dynamic performance of no-joint wire rope but also provides the theory and method for the wire rope modeling of other shapes.


## 2. Basic Structure of No-Joint Wire Rope

### 2.1. Basic Assumptions

This paper takes a cross section of 6 × 7 + IWS wire rope on right-hand Lang’s lay as an example and establishes a geometric model of no-joint wire rope, collectively referred to as NJ-6 × 7 + IWS. The establishment of this model is also based on the following assumptions:(a)
Ignoring the slight deformation of no-joint wire rope in the process of winding and bending, it is assumed that the cross section of each wire is circular(b)
Ignoring the small changes in helix angle of strand and helix angle of wire rope, which are caused by the production process, it is assumed that these helix angles remain constant in the wire rope(c)
It is assumed that the straight-line wire rope and the arc wire rope are in the same plane in the transmission process of the no-joint wire rope


### 2.2. Structure Analysis of NJ-6 × 7 + IWS

The structure of NJ-6 × 7 + IWS is shown in Figure4. Helix angle and lay pitch of strand are α and d1, helix angle, and lay pitch of wire rope are β and d1′, and all wires are round wires with radius r. Wire exists in three shapes in the rope. The shape of CC is straight line or arc. The shapes of CS and OC are single-helix winding CC, and their helical radiuses are r1 and r2. The shape of OS is double-helix winding OC and its helical radius is also r1. The wire is evenly distributed around each winding center. “ηi = (i − 1)60°” indicates OCi initial phase, and “i” is strand number, i = 1, 2, …, 6. “ξj = (j − 1)60°” indicates CSj initial phase, and “j” is side wire number, j = 1, 2, …, 6. “ξij = (j − 1)60°” indicates OSij initial phase, and “i” and “j” have the same meaning as stated previously.Figure 4
Structure of NJ-6 × 7 + IWS.


## 2.1. Basic Assumptions

This paper takes a cross section of 6 × 7 + IWS wire rope on right-hand Lang’s lay as an example and establishes a geometric model of no-joint wire rope, collectively referred to as NJ-6 × 7 + IWS. The establishment of this model is also based on the following assumptions:(a)
Ignoring the slight deformation of no-joint wire rope in the process of winding and bending, it is assumed that the cross section of each wire is circular(b)
Ignoring the small changes in helix angle of strand and helix angle of wire rope, which are caused by the production process, it is assumed that these helix angles remain constant in the wire rope(c)
It is assumed that the straight-line wire rope and the arc wire rope are in the same plane in the transmission process of the no-joint wire rope


## 2.2. Structure Analysis of NJ-6 × 7 + IWS

The structure of NJ-6 × 7 + IWS is shown in Figure4. Helix angle and lay pitch of strand are α and d1, helix angle, and lay pitch of wire rope are β and d1′, and all wires are round wires with radius r. Wire exists in three shapes in the rope. The shape of CC is straight line or arc. The shapes of CS and OC are single-helix winding CC, and their helical radiuses are r1 and r2. The shape of OS is double-helix winding OC and its helical radius is also r1. The wire is evenly distributed around each winding center. “ηi = (i − 1)60°” indicates OCi initial phase, and “i” is strand number, i = 1, 2, …, 6. “ξj = (j − 1)60°” indicates CSj initial phase, and “j” is side wire number, j = 1, 2, …, 6. “ξij = (j − 1)60°” indicates OSij initial phase, and “i” and “j” have the same meaning as stated previously.Figure 4
Structure of NJ-6 × 7 + IWS.


## 3. Frenet–Serret Frame Theory in No-Joint Wire Rope

### 3.1. Frenet–Serret Frame

As shown in Figure5, there is a curve hγ=xγ,yγ,zγ in a Cartesian coordinate system (O: x, y, z), and γ is a function argument. There is a moving point P on the curve, and N, B, and T are the point’s normal vector, binormal vector, and tangent vector. The local coordinate system n–b–t composed of the unit vectors n, b, and t corresponding to the N, B, and T is called a Frenet–Serret frame. The calculation formulas are as follows:(1)n=hγ¨hγ¨,b=t×n,t=hγ˙hγ˙.Figure 5
Frenet–Serret frame.


### 3.2. Application of the Frame in the Calculation of Arc Wire Rope

The no-joint wire rope is divided into arc segment and straight-line segment in the transmission process, so it is necessary to calculate the frame of each segment separately, and the calculation method of arc segment is shown in Figure6.Figure 6
Use of the frame in the calculation of helix winding arc.The single-helix winding arc is calculated according to the following method, taking CS as an example. Firstly, the framen1–b1–t1 is established on the center line of as the local coordinate, and h1, the vector form of CS center line in the local coordinate, is written out, secondly through the relationship between the coordinate (O: x, y, z), which is at the center of the arc, and the frame n1–b1–t1 transform vector h1 into (O: x, y, z) to get the vector expression of h1 in (O: x, y, z). Finally, take H0, the vector form of CC, add the result of the preceding transformation, and H1, the vector form of CS centerline in coordinate (O: x, y, z), is obtained:(2)H1=H0+h1n1b1t1T.The double-helix winding arc is OS, and the calculation method is as follows: OS winds OC, son2–b2–t2 is established on the center line of OC as a local coordinate system. However, because the shape of this wire rope is an arc, in order to ensure that the OS always winds the center line of the OC, it is necessary to use matrix T to transform the local coordinate system. The latter step is similar to calculation of a single helix, so H2, the vector expression of the OS centerline in the coordinate (O: x, y, z), is obtained:(3)H2=H1′+h2Tn2b2t2T,where H1′ is the vector form of center line of OC in coordinate (O: x, y, z), and h2 is the vector form of center line of OS in the frame n2–b2–t2.


### 3.3. Application of the Frame in the Calculation of Straight Wire Rope

The single-helix winding straight line is OS and OC, which can be written directly. The double helix is OS and needs to be deduced by using the frame, as shown in Figure7.Figure 7
Use of the frame in the calculation of helix winding straight line.The coordinate (O: x, y, z) is established at the starting position of the center line of CC, and the frame nb–bb–tb is established on the center line of OC as the local coordinate system. The vector of center line of OS in the nb–bb–tb is l2, and then transform l2 into the coordinate (O: x, y, z). Finally, by adding L1′, the vector form of center line of OC in the coordinate system (O: x, y, z), to the preceding transformation result for l2, L2, the vector center line of OS in the coordinate (O: x, y, z), is obtained:(4)L2=L1′+l2nbbbtbT.


## 3.1. Frenet–Serret Frame

As shown in Figure5, there is a curve hγ=xγ,yγ,zγ in a Cartesian coordinate system (O: x, y, z), and γ is a function argument. There is a moving point P on the curve, and N, B, and T are the point’s normal vector, binormal vector, and tangent vector. The local coordinate system n–b–t composed of the unit vectors n, b, and t corresponding to the N, B, and T is called a Frenet–Serret frame. The calculation formulas are as follows:(1)n=hγ¨hγ¨,b=t×n,t=hγ˙hγ˙.Figure 5
Frenet–Serret frame.


## 3.2. Application of the Frame in the Calculation of Arc Wire Rope

The no-joint wire rope is divided into arc segment and straight-line segment in the transmission process, so it is necessary to calculate the frame of each segment separately, and the calculation method of arc segment is shown in Figure6.Figure 6
Use of the frame in the calculation of helix winding arc.The single-helix winding arc is calculated according to the following method, taking CS as an example. Firstly, the framen1–b1–t1 is established on the center line of as the local coordinate, and h1, the vector form of CS center line in the local coordinate, is written out, secondly through the relationship between the coordinate (O: x, y, z), which is at the center of the arc, and the frame n1–b1–t1 transform vector h1 into (O: x, y, z) to get the vector expression of h1 in (O: x, y, z). Finally, take H0, the vector form of CC, add the result of the preceding transformation, and H1, the vector form of CS centerline in coordinate (O: x, y, z), is obtained:(2)H1=H0+h1n1b1t1T.The double-helix winding arc is OS, and the calculation method is as follows: OS winds OC, son2–b2–t2 is established on the center line of OC as a local coordinate system. However, because the shape of this wire rope is an arc, in order to ensure that the OS always winds the center line of the OC, it is necessary to use matrix T to transform the local coordinate system. The latter step is similar to calculation of a single helix, so H2, the vector expression of the OS centerline in the coordinate (O: x, y, z), is obtained:(3)H2=H1′+h2Tn2b2t2T,where H1′ is the vector form of center line of OC in coordinate (O: x, y, z), and h2 is the vector form of center line of OS in the frame n2–b2–t2.


## 3.3. Application of the Frame in the Calculation of Straight Wire Rope

The single-helix winding straight line is OS and OC, which can be written directly. The double helix is OS and needs to be deduced by using the frame, as shown in Figure7.Figure 7
Use of the frame in the calculation of helix winding straight line.The coordinate (O: x, y, z) is established at the starting position of the center line of CC, and the frame nb–bb–tb is established on the center line of OC as the local coordinate system. The vector of center line of OS in the nb–bb–tb is l2, and then transform l2 into the coordinate (O: x, y, z). Finally, by adding L1′, the vector form of center line of OC in the coordinate system (O: x, y, z), to the preceding transformation result for l2, L2, the vector center line of OS in the coordinate (O: x, y, z), is obtained:(4)L2=L1′+l2nbbbtbT.


## 4. Center Line Equations of Wire in NJ-6 × 7 + IWS

### 4.1. Tangent and Closed Conditions of Wire

The arc and the straight line are tangent in the transmission process of no-joint wire rope. Therefore, it is necessary to consider the tangential problem between steel wires in each rope segment when deducing the center line equations of wire. When the end faces of the corresponding wires in the arc and the straight line coincide and their tangents are the same, the arc and the straight-line wire rope are tangent. Based on this condition, set up the coordinate system as shown in Figure8.Figure 8
Coordinate systems at rope segments of no-joint wire rope.The global coordinate systems of Arc I and Arc II is (O1: x1, y1, z1) and (O3: x3, y3, z3), located at the center of each arc. The global coordinate systems of Line I and Line II are (O2: x2, y2, z2) and (O4: x4, y4, z4), which are, respectively, located at the tangent points of Arc I and Line I and Line II, and the direction of the y-axis is the same as the tangent direction of Arc I at the endpoint. Rn and ωn (n = 1, 2) are, respectively, the radius of arc and the rotation angle.The closed conditions, in essence, are to control the position of the end face of the wire and make the end section of each wire coincide with the starting section of the next connected wire, so as to achieve the purpose of closing the wire rope.According to the structural analysis, there are two types of wire winding ways, as shown in Figures9 and 10. Let the length of CC be Smn (m indicates shape of wire rope when “m = 1” is the arc and “m = 2” is the straight line; n is the number of wire rope segment, n = 1, 2). S1n = Rnωn, S2n = LcosΔ, and derive the winding angles θmn of CS, φmn of OC, and ψmn of OS as follows:(5)θmn=Smnd1×2π,ϕmn=Smnd1′×2π,ψmn=Smnd1cosβ×2π.Figure 9
Winding ofα.Figure 10
Winding ofβ.When the coefficient before 2π is rounded, the wires will be winded around an integer multiple of their lay pitch, and all wire rope segments are closed.


### 4.2. Equations of Center Line of CC

The shapes of CC center line in arc wire rope and straight-line wire rope are arc and straight line, listing their equations directly. The equations of Arc I and Arc II are as follows:(6)xωn=0,yωn=Rncosωn,zωn=Rnsinωn,where ω1∈π/2+Δ,3π/2−Δ and ω2∈−π/2−Δ,π/2+Δ.The equations of Line I and Line II are as follows:(7)
x
t
=
0
,
y
t
=
S
2
n
t
,
t
∈
0,1
,
z
t
=
0.


### 4.3. Equations of Single Helix

Both CS and OC center lines are single helix. Because the equation of single-helix winding arc is different from that of winding line, it is deduced separately.


#### 4.3.1. Equations of Single-Helix Winding Arc I and Arc II

Firstly, the framen1–b1–t1 is established on the center line of CC in arc, calculated by equations (1) and (6).For instance, equation (1) is used to calculate a response surface as follows:(8)n1=0,−cosωn,−sinωn,b1=1,0,0,t1=0,−sinωn,cosωn.Put the starting point of the center line of on theb1 axis, and rotate counterclockwise around t1 by θ1n degrees with radius r1. Then, the vector h1 in the frame is obtained:(9)h1=−r1sinθ1n+ξj,r1cosθ1n+ξj,0.TakingH0=xωn,yωn,zωn, equations (8) and (9) into equation (2), equations of center line of CS in Arc I and Arc II are obtained:(10)xθ1n=xωn+r1cosθ1n+ξj,yθ1n=yωn+r1sinθ1n+ξjcosωn,zθ1n=zωn+r1sinθ1n+ξjsinωn.The equation derivation process of central line of OC in arc is similar to CS, so the equation is directly listed as follows:(11)xφ1n=xωn+r2cosφ1n+ηi,yφ1n=yωn+r2sinφ1n+ηicosωn,zφ1n=zωn+r2sinφ1n+ηisinωn.


#### 4.3.2. Equations of Single-Helix Winding Line I and Line II

Single-helix winding line includes CS and OC, and their equations are simple and listed as follows:(12)xθ2n=r1cosθ2n+ξj,yθ2n=r1tanαθ2n,zθ2n=−r1sinθ2n+ξj,(13)xφ2n=r2cosφ2n+ηi,yφ2n=r2tanβφ2n,zφ2n=−r2sinφ2n+ηi.


### 4.4. Equations of Double Helix

Only the center line of OS is double helix in the no-joint wire rope, because double helix equations of arc and straight line are not the same, so it needs to be deduced.


#### 4.4.1. Double-Helix Equations of Arc I and Arc II

Establish the framen2–b2–t2 on the central line of OC in the arc; this frame is calculated by equations (1) and (11):(14)n2=−cosφ1n+ηi,−sinφ1n+ηicosωn,−sinφ1n+ηisinωn,b2=0,−sinωn,cosωn,t2=−sinφ1n+ηi,cosφ1n+ηicosωn,cosφ1n+ηisinωn.The matrixT is used to make the angle between tangent of center line of OS and OC equal to β:(15)T=1000sinβ−cosβ0cosβsinβ.The starting point of center line of OS is located atn2 and rotates counterclockwise around t2 by ψ1n degrees with radius r1. Then, the vector h2 in the frame is obtained:(16)h2=−r1cosψ1n+ξij,r1sinψ1n+ξij,0.TakingH1=xφ1n,yφ1n,zφ1n and substituting equations (14), (15), and (16) into equation (3), equations of center line of OS in Arc I and Arc II are obtained:(17)xψ1n=xφ1n+r2cosψ1n+ξijcosφ1n+ηi−sinψ1n+ξijsinφ1n+ηicosβ,yψ1n=yφ1n+r2sinψ1n+ξijsinβsinωn+cosφ1n+ηicosβcosωn+r2cosψ1n+ξijsinφ1n+ηicosωn,zψ1n=zφ1n+r2sinψ1n+ξij−sinβsinωn+cosφ1n+ηicosβsinωn+r2cosψ1n+ξijsinφ1n+ηisinωn.


#### 4.4.2. Double-Helix Equations of Line I and Line II

Establish the framenb–bb–tb, which is calculated by equations (1) and (13), on the center line of OC in the Line I and Line II:(18)nb=−cosφ2n+ηi,0,sinφ2n+ηi,bb=cosβsinφ2n+ηi,sinβ,cosβcosφ2n+ηi,tb=−sinβsinφ2n+ηi,cosβ,−sinβcosφ2n+ηi.The starting point of the center line of OS is located atnb, and it rotates clockwise around tb by ψ2n degrees with radius r1. Then, the vector l2 in the frame is obtained:(19)l2=−r1cosψ2n+ξij,r1sinψ2n+ξij,0.TakingL1′=xφ2n,yφ2n,zφ2n, equations (18) and (19) into equation (4), equations of center line of OS in Line I and Line II are obtained:(20)xψ2n=xφ2n+r1cosψ2n+ξijcosφ2n+ηi+sinψ2n+ξijsinφ2n+ηicosβ,yψ2n=yφ2n+r1sinψ2n+ξijsinβ,zψ2n=zφ2n+r1−cosψ2n+ξijsinφ2n+ηi+sinψ2n+ξijcosφ2n+ηicosβ.


## 4.1. Tangent and Closed Conditions of Wire

The arc and the straight line are tangent in the transmission process of no-joint wire rope. Therefore, it is necessary to consider the tangential problem between steel wires in each rope segment when deducing the center line equations of wire. When the end faces of the corresponding wires in the arc and the straight line coincide and their tangents are the same, the arc and the straight-line wire rope are tangent. Based on this condition, set up the coordinate system as shown in Figure8.Figure 8
Coordinate systems at rope segments of no-joint wire rope.The global coordinate systems of Arc I and Arc II is (O1: x1, y1, z1) and (O3: x3, y3, z3), located at the center of each arc. The global coordinate systems of Line I and Line II are (O2: x2, y2, z2) and (O4: x4, y4, z4), which are, respectively, located at the tangent points of Arc I and Line I and Line II, and the direction of the y-axis is the same as the tangent direction of Arc I at the endpoint. Rn and ωn (n = 1, 2) are, respectively, the radius of arc and the rotation angle.The closed conditions, in essence, are to control the position of the end face of the wire and make the end section of each wire coincide with the starting section of the next connected wire, so as to achieve the purpose of closing the wire rope.According to the structural analysis, there are two types of wire winding ways, as shown in Figures9 and 10. Let the length of CC be Smn (m indicates shape of wire rope when “m = 1” is the arc and “m = 2” is the straight line; n is the number of wire rope segment, n = 1, 2). S1n = Rnωn, S2n = LcosΔ, and derive the winding angles θmn of CS, φmn of OC, and ψmn of OS as follows:(5)θmn=Smnd1×2π,ϕmn=Smnd1′×2π,ψmn=Smnd1cosβ×2π.Figure 9
Winding ofα.Figure 10
Winding ofβ.When the coefficient before 2π is rounded, the wires will be winded around an integer multiple of their lay pitch, and all wire rope segments are closed.


## 4.2. Equations of Center Line of CC

The shapes of CC center line in arc wire rope and straight-line wire rope are arc and straight line, listing their equations directly. The equations of Arc I and Arc II are as follows:(6)xωn=0,yωn=Rncosωn,zωn=Rnsinωn,where ω1∈π/2+Δ,3π/2−Δ and ω2∈−π/2−Δ,π/2+Δ.The equations of Line I and Line II are as follows:(7)
x
t
=
0
,
y
t
=
S
2
n
t
,
t
∈
0,1
,
z
t
=
0.


## 4.3. Equations of Single Helix

Both CS and OC center lines are single helix. Because the equation of single-helix winding arc is different from that of winding line, it is deduced separately.


### 4.3.1. Equations of Single-Helix Winding Arc I and Arc II

Firstly, the framen1–b1–t1 is established on the center line of CC in arc, calculated by equations (1) and (6).For instance, equation (1) is used to calculate a response surface as follows:(8)n1=0,−cosωn,−sinωn,b1=1,0,0,t1=0,−sinωn,cosωn.Put the starting point of the center line of on theb1 axis, and rotate counterclockwise around t1 by θ1n degrees with radius r1. Then, the vector h1 in the frame is obtained:(9)h1=−r1sinθ1n+ξj,r1cosθ1n+ξj,0.TakingH0=xωn,yωn,zωn, equations (8) and (9) into equation (2), equations of center line of CS in Arc I and Arc II are obtained:(10)xθ1n=xωn+r1cosθ1n+ξj,yθ1n=yωn+r1sinθ1n+ξjcosωn,zθ1n=zωn+r1sinθ1n+ξjsinωn.The equation derivation process of central line of OC in arc is similar to CS, so the equation is directly listed as follows:(11)xφ1n=xωn+r2cosφ1n+ηi,yφ1n=yωn+r2sinφ1n+ηicosωn,zφ1n=zωn+r2sinφ1n+ηisinωn.


### 4.3.2. Equations of Single-Helix Winding Line I and Line II

Single-helix winding line includes CS and OC, and their equations are simple and listed as follows:(12)xθ2n=r1cosθ2n+ξj,yθ2n=r1tanαθ2n,zθ2n=−r1sinθ2n+ξj,(13)xφ2n=r2cosφ2n+ηi,yφ2n=r2tanβφ2n,zφ2n=−r2sinφ2n+ηi.


## 4.3.1. Equations of Single-Helix Winding Arc I and Arc II

Firstly, the framen1–b1–t1 is established on the center line of CC in arc, calculated by equations (1) and (6).For instance, equation (1) is used to calculate a response surface as follows:(8)n1=0,−cosωn,−sinωn,b1=1,0,0,t1=0,−sinωn,cosωn.Put the starting point of the center line of on theb1 axis, and rotate counterclockwise around t1 by θ1n degrees with radius r1. Then, the vector h1 in the frame is obtained:(9)h1=−r1sinθ1n+ξj,r1cosθ1n+ξj,0.TakingH0=xωn,yωn,zωn, equations (8) and (9) into equation (2), equations of center line of CS in Arc I and Arc II are obtained:(10)xθ1n=xωn+r1cosθ1n+ξj,yθ1n=yωn+r1sinθ1n+ξjcosωn,zθ1n=zωn+r1sinθ1n+ξjsinωn.The equation derivation process of central line of OC in arc is similar to CS, so the equation is directly listed as follows:(11)xφ1n=xωn+r2cosφ1n+ηi,yφ1n=yωn+r2sinφ1n+ηicosωn,zφ1n=zωn+r2sinφ1n+ηisinωn.


## 4.3.2. Equations of Single-Helix Winding Line I and Line II

Single-helix winding line includes CS and OC, and their equations are simple and listed as follows:(12)xθ2n=r1cosθ2n+ξj,yθ2n=r1tanαθ2n,zθ2n=−r1sinθ2n+ξj,(13)xφ2n=r2cosφ2n+ηi,yφ2n=r2tanβφ2n,zφ2n=−r2sinφ2n+ηi.


## 4.4. Equations of Double Helix

Only the center line of OS is double helix in the no-joint wire rope, because double helix equations of arc and straight line are not the same, so it needs to be deduced.


### 4.4.1. Double-Helix Equations of Arc I and Arc II

Establish the framen2–b2–t2 on the central line of OC in the arc; this frame is calculated by equations (1) and (11):(14)n2=−cosφ1n+ηi,−sinφ1n+ηicosωn,−sinφ1n+ηisinωn,b2=0,−sinωn,cosωn,t2=−sinφ1n+ηi,cosφ1n+ηicosωn,cosφ1n+ηisinωn.The matrixT is used to make the angle between tangent of center line of OS and OC equal to β:(15)T=1000sinβ−cosβ0cosβsinβ.The starting point of center line of OS is located atn2 and rotates counterclockwise around t2 by ψ1n degrees with radius r1. Then, the vector h2 in the frame is obtained:(16)h2=−r1cosψ1n+ξij,r1sinψ1n+ξij,0.TakingH1=xφ1n,yφ1n,zφ1n and substituting equations (14), (15), and (16) into equation (3), equations of center line of OS in Arc I and Arc II are obtained:(17)xψ1n=xφ1n+r2cosψ1n+ξijcosφ1n+ηi−sinψ1n+ξijsinφ1n+ηicosβ,yψ1n=yφ1n+r2sinψ1n+ξijsinβsinωn+cosφ1n+ηicosβcosωn+r2cosψ1n+ξijsinφ1n+ηicosωn,zψ1n=zφ1n+r2sinψ1n+ξij−sinβsinωn+cosφ1n+ηicosβsinωn+r2cosψ1n+ξijsinφ1n+ηisinωn.


### 4.4.2. Double-Helix Equations of Line I and Line II

Establish the framenb–bb–tb, which is calculated by equations (1) and (13), on the center line of OC in the Line I and Line II:(18)nb=−cosφ2n+ηi,0,sinφ2n+ηi,bb=cosβsinφ2n+ηi,sinβ,cosβcosφ2n+ηi,tb=−sinβsinφ2n+ηi,cosβ,−sinβcosφ2n+ηi.The starting point of the center line of OS is located atnb, and it rotates clockwise around tb by ψ2n degrees with radius r1. Then, the vector l2 in the frame is obtained:(19)l2=−r1cosψ2n+ξij,r1sinψ2n+ξij,0.TakingL1′=xφ2n,yφ2n,zφ2n, equations (18) and (19) into equation (4), equations of center line of OS in Line I and Line II are obtained:(20)xψ2n=xφ2n+r1cosψ2n+ξijcosφ2n+ηi+sinψ2n+ξijsinφ2n+ηicosβ,yψ2n=yφ2n+r1sinψ2n+ξijsinβ,zψ2n=zφ2n+r1−cosψ2n+ξijsinφ2n+ηi+sinψ2n+ξijcosφ2n+ηicosβ.


## 4.4.1. Double-Helix Equations of Arc I and Arc II

Establish the framen2–b2–t2 on the central line of OC in the arc; this frame is calculated by equations (1) and (11):(14)n2=−cosφ1n+ηi,−sinφ1n+ηicosωn,−sinφ1n+ηisinωn,b2=0,−sinωn,cosωn,t2=−sinφ1n+ηi,cosφ1n+ηicosωn,cosφ1n+ηisinωn.The matrixT is used to make the angle between tangent of center line of OS and OC equal to β:(15)T=1000sinβ−cosβ0cosβsinβ.The starting point of center line of OS is located atn2 and rotates counterclockwise around t2 by ψ1n degrees with radius r1. Then, the vector h2 in the frame is obtained:(16)h2=−r1cosψ1n+ξij,r1sinψ1n+ξij,0.TakingH1=xφ1n,yφ1n,zφ1n and substituting equations (14), (15), and (16) into equation (3), equations of center line of OS in Arc I and Arc II are obtained:(17)xψ1n=xφ1n+r2cosψ1n+ξijcosφ1n+ηi−sinψ1n+ξijsinφ1n+ηicosβ,yψ1n=yφ1n+r2sinψ1n+ξijsinβsinωn+cosφ1n+ηicosβcosωn+r2cosψ1n+ξijsinφ1n+ηicosωn,zψ1n=zφ1n+r2sinψ1n+ξij−sinβsinωn+cosφ1n+ηicosβsinωn+r2cosψ1n+ξijsinφ1n+ηisinωn.


## 4.4.2. Double-Helix Equations of Line I and Line II

Establish the framenb–bb–tb, which is calculated by equations (1) and (13), on the center line of OC in the Line I and Line II:(18)nb=−cosφ2n+ηi,0,sinφ2n+ηi,bb=cosβsinφ2n+ηi,sinβ,cosβcosφ2n+ηi,tb=−sinβsinφ2n+ηi,cosβ,−sinβcosφ2n+ηi.The starting point of the center line of OS is located atnb, and it rotates clockwise around tb by ψ2n degrees with radius r1. Then, the vector l2 in the frame is obtained:(19)l2=−r1cosψ2n+ξij,r1sinψ2n+ξij,0.TakingL1′=xφ2n,yφ2n,zφ2n, equations (18) and (19) into equation (4), equations of center line of OS in Line I and Line II are obtained:(20)xψ2n=xφ2n+r1cosψ2n+ξijcosφ2n+ηi+sinψ2n+ξijsinφ2n+ηicosβ,yψ2n=yφ2n+r1sinψ2n+ξijsinβ,zψ2n=zφ2n+r1−cosψ2n+ξijsinφ2n+ηi+sinψ2n+ξijcosφ2n+ηicosβ.


## 5. Generate 3D Solid Model of NJ-6 × 7 + IWS

### 5.1. Parameters of NJ-6 × 7 + IWS

Table1 shows the parameters of right-hand Lang’s lay NJ-6 × 7 + IWS, on which 3D modeling is mainly based.Table 1
Parameters of right-hand Lang’s lay NJ-6 × 7 + IWS.
Parameter symbol
Value
Unit
R
1
10
mm
R
2
20
mm
L
50
mm
r
0.1
mm
r
1
0.22
mm
r
2
0.66
mm
α
12.41
°
β
18.26
°


### 5.2. Generation of 3D Solid Model

The modeling software used in this paper is Pro/ENGINEER Wildfire 5.0, and the solid model of the no-joint wire rope is established according to the preceding equations.Firstly, the coordinate systems are established according to the tangent condition, and the center line of CC in Arc I is used as the reference. The result is obtained by using reference coordinate, datum, reference axis, and more, as shown in Figure11.Figure 11
Setup for the coordinate systems.Secondly, generate the solid model of the center strand, which is composed of CC solid model and CS solid model. The generation process of CC solid model is shown in Figure12, in which Panels ① to ③ show the generation process of center line of CC. Panel ④ is the CC solid model, and small graphs in Panel ④ are the magnification of junction between line and arc. This expression method in the subsequent diagram is the same as here.Figure 12
Generation process of the CC solid model.The generation step of the CS solid model is shown in Figure13; Panels ① to ④ show the generation process of the center line of CS1. Other generation processes of center line of CS are similar to CS1, and their generation results are shown in Panel ⑤. Then, generate the solid model of CS, in which Panel ⑥ is CS1 for the solid model. Generation processes from CS2 to CS6 are similar to Panel ⑥, and the results are shown in Panel ⑦. At the same time, the solid model of the center strand is also completed.Figure 13
Generation process of the solid model of CS.Finally, generate the solid models of side strand as shown in Figure14. Take one side strand as an example to show. Panels ① to ④ show the generation process of the center line of OC11, and Panel ⑤ is the OC11 solid model. Panel ⑥ shows the generation result of the center line from OS11 to OS16, Panel ⑦ is the solid model from OS11 to OS16, and a solid model of one side strand is also established at the same time. The generation process of other side strands is similar to one side strand, so we only show the generation results. Panels ⑧ to ⑫ show the solid model from two side strands to six side strands, and the solid model of no-joint wire rope is also established at the same time.Figure 14
Generation process of the solid model of side strand.If the wire rope parameter value is changed, the no-joint wire rope, which is used in WR-CVT, can be built when the transmission ratio is 1, as shown in Figure15.Figure 15
WR-CVT and no-joint wire rope when the transmission ratio is 1.


## 5.1. Parameters of NJ-6 × 7 + IWS

Table1 shows the parameters of right-hand Lang’s lay NJ-6 × 7 + IWS, on which 3D modeling is mainly based.Table 1
Parameters of right-hand Lang’s lay NJ-6 × 7 + IWS.
Parameter symbol
Value
Unit
R
1
10
mm
R
2
20
mm
L
50
mm
r
0.1
mm
r
1
0.22
mm
r
2
0.66
mm
α
12.41
°
β
18.26
°


## 5.2. Generation of 3D Solid Model

The modeling software used in this paper is Pro/ENGINEER Wildfire 5.0, and the solid model of the no-joint wire rope is established according to the preceding equations.Firstly, the coordinate systems are established according to the tangent condition, and the center line of CC in Arc I is used as the reference. The result is obtained by using reference coordinate, datum, reference axis, and more, as shown in Figure11.Figure 11
Setup for the coordinate systems.Secondly, generate the solid model of the center strand, which is composed of CC solid model and CS solid model. The generation process of CC solid model is shown in Figure12, in which Panels ① to ③ show the generation process of center line of CC. Panel ④ is the CC solid model, and small graphs in Panel ④ are the magnification of junction between line and arc. This expression method in the subsequent diagram is the same as here.Figure 12
Generation process of the CC solid model.The generation step of the CS solid model is shown in Figure13; Panels ① to ④ show the generation process of the center line of CS1. Other generation processes of center line of CS are similar to CS1, and their generation results are shown in Panel ⑤. Then, generate the solid model of CS, in which Panel ⑥ is CS1 for the solid model. Generation processes from CS2 to CS6 are similar to Panel ⑥, and the results are shown in Panel ⑦. At the same time, the solid model of the center strand is also completed.Figure 13
Generation process of the solid model of CS.Finally, generate the solid models of side strand as shown in Figure14. Take one side strand as an example to show. Panels ① to ④ show the generation process of the center line of OC11, and Panel ⑤ is the OC11 solid model. Panel ⑥ shows the generation result of the center line from OS11 to OS16, Panel ⑦ is the solid model from OS11 to OS16, and a solid model of one side strand is also established at the same time. The generation process of other side strands is similar to one side strand, so we only show the generation results. Panels ⑧ to ⑫ show the solid model from two side strands to six side strands, and the solid model of no-joint wire rope is also established at the same time.Figure 14
Generation process of the solid model of side strand.If the wire rope parameter value is changed, the no-joint wire rope, which is used in WR-CVT, can be built when the transmission ratio is 1, as shown in Figure15.Figure 15
WR-CVT and no-joint wire rope when the transmission ratio is 1.


## 6. Conclusions

This paper, taking a basic form of belt transmission as the background, establishes a no-joint wire rope model and mainly studies the geometric modeling method of no-joint wire rope, which consists of two straight lines and two arcs, as well as the implementation of a 3D solid model. The main conclusions are as follows.Based on the cross section of 6 × 7 + IWS wire rope, this study analyzes the geometrical relationship among various kinds of wires and set up three kinds of Frenet–Serret frame, which are on the double-helix winding straight line, single-helix winding arc, and double-helix winding arc, combined with Frenet–Serret frame theory, and derives four types of equations when wire rope segments are straight lines and arcs. The tangent condition has been obtained where the direction of the center line of same numbered wire must be tangent and the closed condition where the length of wire must be an integer multiple of its lay pitch, and the wire’s number must be the same. Finally, using Pro/E software generates two typical shapes of the no-joint wire rope 3D solid model when the transmission ratio of WR-CVT is 1 and is not 1. This paper further improves the geometric modeling system of wire rope and lays a theoretical foundation for the mechanical and dynamic analyses of no-joint wire rope.



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*Source: 8760578-2019-10-21.xml*