Introduction

In modern engineering, cables play an indispensable role in power transmission, communication, and control systems. Among these, mining machine cables are responsible for providing power to the coal mining machine and transmitting control signals. Improving the comprehensive performance of mining machine cables is of great significance for ensuring safe coal mining, stabilizing coal supply, and improving mining efficiency. It is also a crucial guarantee for achieving intelligent fully mechanized coal mining under complex conditions.

Dynamic simulation of coal mining machine cable characteristics is crucial for performance evaluation and design optimization to enhance durability and reliability. Accurate cable model construction is key in this regard.

Cables are complex n-level helical structures, and some progress has been made in the field of modeling. Wang et al. analyzed the structure of round strand wire rope and proposed a general mathematical model to describe the paths of conductors with single and second-level helix forms1. Jiang et al. established a finite element model of seven-strand wire rope to analyze the termination effect2, and subsequently proposed an accurate and general finite element model for wire rope to analyze the tensile, shear, bending, torsion, contact, and frictional effects during loading3. Wang Guilan et al. established the spatial curve equations of wires in double-twist wire rope based on differential geometry and analyzed the influence of typical twist combinations on the curvature of wires in the rope4,5,6. Elata et al. proposed a new model to simulate the mechanical response of independent wire ropes7. Erdonmez et al. proposed a more realistic three-dimensional modeling method for independent wire ropes with a secondary helix structure8,9, and then they discussed the geometry of the tertiary helix based on the modeling of primary and secondary helix structures and extended it to the construction of n-level helix equations10. Wu Juan et al. derived the spatial vector expression of secondary helix lines through the Frenet-Serret frame, constructed a finite element analysis model of wire rope, and studied the stress distribution of wire rope11. Hu Yujiao established a submarine cable model by establishing a centerline using key points and studied its mechanical behavior12. Zhongbin modeled prefabricated spiral joints using parameter equations and studied the changes in gripping force using Abaqus13. Zhang et al. constructed a three-dimensional model of coal mining machine cables using the trajpar function in ProE and analyzed the dynamic characteristics of coal mining machines using homogenization theory and the volume averaging principle14.

Although some scholars have made certain achievements in the field of modeling n-level complex helical structures, traditional modeling methods have shortcomings in systematically and comprehensively constructing high-precision complex multi-layer helical structures and braided structures. This study aims to introduce a systematic, full-process, high-precision method to model complex multi-layer helical structure cables represented by coal mining machine cables, in order to overcome the limitations of existing methods and improve the accuracy and practicality of modeling complex multi-layer helical structures.

Modeling method of n-level helix curve based on parameter equations

Cables are typical complex n-level helical structures, and their accurate modeling is a complex and challenging task. N-level helical structures are formed by n-level helix curves as centerlines, and one of the key aspects is to establish accurate n-level helix curves.

Method for constructing first-level helix curve

In the Cartesian coordinate system, the first-level helix curve is swept by points moving linearly along the z-axis while undergoing circular motion in the x-y plane (the normal plane of the z-axis), as depicted in Fig. 1.

Figure 1
figure 1

Schematic diagram of the first-level helix curve and its Frenet-Serret frame.

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All points on this first-level helix curve can be represented by the parametric equation shown in Eq. (1).

$${}_{ }^{B} P_{1} = \left[ {\begin{array}{*{20}c} {{}_{ }^{B} x_{1} } \\ {{}_{ }^{B} y_{1} } \\ {{}_{ }^{B} z_{1} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {R_{1} cos\left( {2\pi \cdot \frac{{t_{1} }}{{L_{1} }} + \theta_{1} } \right)} \\ { - R_{1} sin\left( {2\pi \cdot \frac{{t_{1} }}{{L_{1} }} + \theta_{1} - flg_{1} \cdot \pi } \right)} \\ {t_{1} } \\ \end{array} } \right]$$
(1)

In Eq. (1), \(\theta_{1}\) represents the phase angle of the first-level helix curve, \(t_{1}\) denotes the distance traveled by the point along the z-direction of the base coordinate, \(R_{1}\) is the twisting radius of the first-level helix curve, \(L_{1}\) is the helical pitch of the first-level helix curve, and when \(flg_{1}\) equals 0, it indicates left-handed helicity, while when \(flg_{1}\) equals 1, it indicates right-handed helicity.

Revised method for constructing second-level helix curves

The second-level helix curve is swept by points undergoing circular motion in the normal plane of the first-level helix curve. Utilizing the theory of differential geometry, a set of Frenet-Serret frames is established for the first-level helix curve. The Frenet-Serret frame is an important mathematical tool for describing the geometric properties of spatial curves. It is based on three mutually perpendicular unit vectors: tangent, normal, and binormal vectors. Taking the tangent vector of the curve as the z-axis, the principal normal vector of the curve as the y-axis, and the binormal vector as the x-axis, a frame is constructed, as illustrated in Fig. 1.

Using the arc length of the first-level helix curve as the parameter, we establish the parametric equation. Here, \(t_{2}\) represents the distance traveled by the point in the direction of progression along the curve. \(R_{2}\) denotes the twisting radius of the second-level helix curve, and \(L_{2}\) represents the helical pitch of the second-level helix curve. \(t_{2}\) can be obtained by Eq. (2):

$$t_{2} = \frac{{t_{1} }}{{L_{1} }}\mathop \smallint \limits_{0}^{{t_{1} }} \sqrt {\left( {\frac{{2\pi R_{1} }}{{L_{1} }}} \right)^{2} + 1}$$
(2)

The direction vectors of the Frenet-Serret frame for the first-level helix curve are respectively given by Eqs. (3) and (4):

$$\left[ {\begin{array}{*{20}c} {\vec{x}_{2} } \\ {\vec{y}_{2} } \\ {\vec{z}_{2} } \\ \end{array} } \right]^{T} = \left[ {\begin{array}{*{20}c} {\vec{y}_{2} \times \vec{z}_{2} } \\ {\frac{{\ddot{P}_{1} }}{{\left| {\ddot{P}_{1} } \right|}}} \\ {\frac{{\dot{P}_{1} }}{{\left| {\dot{P}_{1} } \right|}}} \\ \end{array} } \right]^{T}$$
(3)
$$\begin{gathered} \vec{x}_{2} = \frac{1}{{\sqrt {\left( {\frac{{2\pi R_{1} }}{{L_{1} }}} \right)^{2} + 1} }}\left[ {\begin{array}{*{20}c} {sin\left( {\frac{{2\pi t_{1} }}{{L_{1} }} + \theta_{1} - flg_{1} \cdot \frac{pi}{2}} \right)} \\ {cos\left( {\frac{{2\pi t_{1} }}{{L_{1} }} + \theta_{1} + flg_{1} \cdot \frac{pi}{2}} \right)} \\ {\frac{{2\pi R_{1} }}{{L_{1} }}} \\ \end{array} } \right] \hfill \\ \vec{y}_{2} = \left[ {\begin{array}{*{20}c} { - cos\left( {\frac{{2\pi t_{1} }}{{L_{1} }} + \theta_{1} + flg_{1} \cdot \frac{pi}{2}} \right)} \\ {sin\left( {\frac{{2\pi t_{1} }}{L} + \theta_{1} - flg_{1} \cdot \frac{pi}{2}} \right)} \\ 0 \\ \end{array} } \right] \hfill \\ \vec{z}_{2} = \frac{1}{{\sqrt {\left( {\frac{{2\pi R_{1} }}{{L_{1} }}} \right)^{2} + 1} }}\left[ {\begin{array}{*{20}c} { - \frac{{2\pi R_{1} }}{{L_{1} }}sin\left( {\frac{{2\pi t_{1} }}{{L_{1} }} + \theta_{1} + fl{\text{g}}_{1} \cdot \frac{pi}{2}} \right)} \\ { - \frac{{2\pi R_{1} }}{{L_{1} }}cos\left( {\frac{{2\pi t_{1} }}{{L_{1} }} + \theta_{1} - fl{\text{g}}_{1} \cdot \frac{pi}{2}} \right)} \\ 1 \\ \end{array} } \right] \hfill \\ \end{gathered}$$
(4)

The coordinates of the second-level helix curve on the Frenet-Serret frame of the first-level helix curve are given by Eq. (5):

$${}_{ }^{{F_{1} }} P_{2} = \left[ {\begin{array}{*{20}c} {{}_{ }^{{F_{1} }} x_{2} } \\ {{}_{ }^{{F_{1} }} y_{2} } \\ {{}_{ }^{{F_{1} }} z_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {R_{2} \cdot cos\left( {2\pi \cdot \frac{{t_{2} }}{{L_{2} }} + \theta_{2} + flg_{2} \cdot \frac{pi}{2}} \right)} \\ { - R_{2} \cdot sin\left( {2\pi \cdot \frac{{t_{2} }}{{L_{2} }} + \theta_{2} - flg_{2} \cdot \frac{pi}{2}} \right)} \\ 0 \\ \end{array} } \right]$$
(5)

Using coordinate transformation, the coordinates of the first-level helix curve's Frenet-Serret frame are transformed to the base coordinate system. The coordinates of the second-level helix curve in the base coordinate system are given by Eq. (6):

$$\begin{gathered} {}_{ }^{B} P_{2} = \left[ {\begin{array}{*{20}c} {{}_{ }^{B} x_{2} } \\ {{}_{ }^{B} y_{2} } \\ {{}_{ }^{B} z_{2} } \\ \end{array} } \right] = {}_{B}^{{F_{1} }} R \cdot {}_{ }^{{F_{1} }} P_{2} + {}_{ }^{B} P_{1} \hfill \\ = \left[ {\begin{array}{*{20}c} {{\text{sin}}\left( {M_{12} + \theta_{1} } \right)} & { - {\text{cos}}\left( {M_{11} + \theta_{1} } \right)/N_{1} } & { - \frac{{2\pi R_{1} }}{{L_{1} }}\left( {M_{11} + \theta_{1} } \right)} \\ {{\text{cos}}\left( {M_{11} + \theta_{1} } \right)} & {{\text{sin}}\left( {M_{12} + \theta_{1} } \right)/N_{1} } & { - \frac{{2\pi R_{1} }}{{L_{1} }}\left( {M_{12} + \theta_{1} } \right)} \\ {\frac{{2\pi R_{1} }}{{L_{1} }}} & 0 & 1 \\ \end{array} } \right] \cdot N_{1} \hfill \\ \times \left[ {\begin{array}{*{20}c} {R_{2} \cdot cos\left( {M_{21} + \theta_{2} } \right)} \\ { - R_{2} \cdot sin\left( {M_{22} + \theta_{2} } \right)} \\ 0 \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {R_{1} cos\left( {M_{11} + \theta_{1} } \right)} \\ { - R_{1} sin\left( {M_{12} + \theta_{1} } \right)} \\ {t_{1} } \\ \end{array} } \right] \hfill \\ = \left[ {\begin{array}{*{20}c} {R_{2} \left( {{\text{cos}}\left( {M_{21} + \theta_{2} } \right) \cdot {\text{sin}}\left( {M_{12} + \theta_{1} } \right) \cdot N_{1} + {\text{sin}}\left( {M_{22} + \theta_{2} } \right) \cdot {\text{cos}}\left( {M_{11} + \theta_{1} } \right)} \right) + R_{1} cos\left( {M_{11} + \theta_{1} } \right)} \\ {R_{2} \left( {{\text{cos}}\left( {M_{21} + \theta_{2} } \right) \cdot {\text{cos}}\left( {M_{11} + \theta_{1} } \right) \cdot N_{1} - {\text{sin}}\left( {M_{22} + \theta_{2} } \right) \cdot {\text{sin}}\left( {M_{12} + \theta_{1} } \right)} \right) - R_{1} sin\left( {M_{12} + \theta_{1} } \right)} \\ {\frac{{2\pi R_{1} }}{{L_{1} }}N_{1} R_{2} \cdot cos\left( {M_{21} + \theta_{2} } \right) + t_{1} } \\ \end{array} } \right] \hfill \\ \end{gathered}$$
(6)

In Eq. (6):

$$M_{n1} = \frac{{2\pi t_{n} }}{{L_{n} }}, M_{n2} = \frac{{2\pi t_{n} }}{{L_{n} }} - fl{\text{g}}_{1} \cdot \pi , N_{n} = \frac{1}{{\sqrt {\left( {\frac{{2\pi R_{n} }}{{L_{n} }}} \right)^{2} + 1} }}$$
(7)

The arc length of the constructed second-level helix curve is given by Eq. (8):

$$LSUM_{2} = \mathop \smallint \limits_{0}^{{t_{1} }} \sqrt {\left( {\frac{{d{}_{ }^{B} x_{2} }}{{dt_{1} }}} \right)^{2} + \left( {\frac{{d{}_{ }^{B} y_{2} }}{{dt_{1} }}} \right)^{2} + \left( {\frac{{d{}_{ }^{B} z_{2} }}{{dt_{1} }}} \right)^{2} }$$
(8)

Using this conventional approach, the parameters are assigned according to Eq. (9) second-level helix curves are constructed separately for \(flg_{2} = 0\) and \(flg_{2} = 1\), and their lengths are calculated. The second-level helix curves are illustrated in Fig. 2. The arc length of the left-handed second-level helix curve is 65.813 mm, while the arc length of the right-handed second-level helix curve is 54.611 mm. Additionally, we can calculate the standard arc lengths of the first-level helix curve and the second-level helix curve as 37.242 mm and 59.809 mm, respectively, using Eq. (10).

$$\begin{gathered} LEN_{c} = 20\;{\text{mm}}\left( {0 \le t_{1} \le 20} \right) \hfill \\ L_{1} = 20\;{\text{mm}},R_{1} = 5\;{\text{mm}},f\lg_{1} = 0, \hfill \\ L_{2} = 10\;{\text{mm}},R_{2} = 2\;{\text{mm}} \hfill \\ \end{gathered}$$
(9)
$$\begin{gathered} LEN_{1} = \left( {LEN_{c} /L_{1} } \right)\sqrt {L_{1}^{2} + \left( {2\pi R_{1} } \right)^{2} } \hfill \\ LEN_{2} = \left( {LEN_{1} /L_{2} } \right)\sqrt {L_{2}^{2} + \left( {2\pi R_{2} } \right)^{2} } \hfill \\ \end{gathered}$$
(10)
Figure 2
figure 2

presents a comparison of the second-level helix curves before and after introducing the corrected pitch.

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Research has found that the lengths of the left-handed and right-handed second-level helix curves are not equal and do not match the theoretical arc length. The reason for this phenomenon is that the Frenet-Serret frame undergoes rotations in three directions as it progresses along the curve. These rotations are superimposed on the next-level helix, resulting in differences in arc length between the left-handed and right-handed next-level helix curves, and neither of them matches the theoretically calculated arc length. On the other hand, the first-level helix curve can be considered to exist on a set of frames along the central axis, which is a straight line. The axes of this set of frames are mutually parallel, without bending or twisting. Therefore, the arc lengths of the left-handed and right-handed first-level helix curves are both equal to the standard arc length.

Taking the coal mining machine cable as an example, according to its manufacturing process, all helical structures are twisted into the cable under the condition of first-level twisting. The method proposed in this paper, which adjusts the pitch in non-first-level helix curves to compensate for the lengths of double and higher helix curves, can effectively solve the distortion problem of the twisted structure model caused by using traditional methods to construct second-level helix curves and higher helix curves.

For the second-level helix curve, we replace \(L_{2}\) in Eq. (8) with the unknown variable \(L_{2}{\prime}\).We construct an integral equation using \(LEN_{2}\) and \(LSUM_{2}\), and solving this integral equation yields the corrected pitch \(L_{2}^{\prime}\).The constructed integral equation is shown in Eq. (11):

$$\left( {LEN_{1} /L_{2} } \right)\sqrt {L_{2}^{2} + \left( {2\pi R_{2} } \right)^{2} } = \mathop \smallint \limits_{0}^{{t_{1} }} \sqrt {\left( {\frac{{d{}_{ }^{B} x_{2} }}{{dt_{1} }}} \right)^{2} + \left( {\frac{{d{}_{ }^{B} y_{2} }}{{dt_{1} }}} \right)^{2} + \left( {\frac{{d{}_{ }^{B} z_{2} }}{{dt_{1} }}} \right)^{2} }$$
(11)

After solving for the corrected pitch, we can substitute it back into Eq. (6) to obtain the corrected second-level helix curve with compensated length, as shown in Fig. 2. Assigning values to each parameter according to Eq. (9), the numerical solutions for the corrected pitch of the left-handed and right-handed second-level helix curves are found to be 10.106 mm and 8.746 mm, respectively.

Generalization of corrected second-level helix curves to corrected n-level helix curves

Similarly, the coordinates of the n-level helix curve on the Frenet-Serret frame twisted at the (n-1)-level are given by Eq. (12):

$${}_{ }^{{F_{n - 1} }} P_{n} = \left[ {\begin{array}{*{20}c} {{}_{ }^{{F_{n - 1} }} x_{n} } \\ {{}_{ }^{{F_{n - 1} }} y_{n} } \\ {{}_{ }^{{F_{n - 1} }} z_{n} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {R_{n} \cdot cos\left( {2\pi \cdot t_{n} /L_{n}^{\prime} + \theta_{n} + flg_{n} \cdot \frac{pi}{2}} \right)} \\ { - R_{n} \cdot sin\left( {2\pi \cdot t_{n} /L_{n}^{\prime} + \theta_{n} - flg_{n} \cdot \frac{pi}{2}} \right)} \\ 0 \\ \end{array} } \right]$$
(12)

Using coordinate transformation, the coordinates of the n-level helix curve are transformed from the (n-1)-level Frenet-Serret frame to the base coordinate system. The equation for the n-level helix curve in the base coordinate system is given by Eq. (13):

$${}_{ }^{B} P_{n} = \left[ {\begin{array}{*{20}c} {{}_{ }^{B} x_{n} } \\ {{}_{ }^{B} y_{n} } \\ {{}_{ }^{B} z_{n} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{\ddot{P}_{n - 1} }}{{\left| {\ddot{P}_{n - 1} } \right|}} \times \frac{{\dot{P}_{n - 1} }}{{\left| {\dot{P}_{n - 1} } \right|}}} \\ {\frac{{\ddot{P}_{n - 1} }}{{\left| {\ddot{P}_{n - 1} } \right|}}} \\ {\frac{{\dot{P}_{n - 1} }}{{\left| {\dot{P}_{n - 1} } \right|}}} \\ \end{array} } \right]^{T} \left[ {\begin{array}{*{20}c} {R_{n} \cdot \cos \left( {M_{n1} + \theta_{2} } \right)} \\ { - R_{n} \cdot \sin \left( {M_{n2} + \theta_{2} } \right)} \\ 0 \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {{}_{ }^{B} x_{n - 1} } \\ {{}_{ }^{B} y_{n - 1} } \\ {{}_{ }^{B} z_{n - 1} } \\ \end{array} } \right]$$
(13)

As shown in Fig. 3, according to the cable manufacturing process, we can recursively calculate the standard length \(LEN_{n}\) of each level helix curve using Eq. (14).

$$LEN_{n} = \left( {LEN_{n - 1} /L_{n} } \right)\sqrt {L_{n}^{2} + \left( {2\pi R_{n} } \right)^{2} }$$
(14)
Figure 3
figure 3

The recursive calculation method for the standard length of the helix curve.

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Based on the equation of the n-level helix curve in the base coordinate system and the standard length of the n-level helix curve, we can construct an integral equation to solve for \(L_{n}^{\prime}\). The integral equation is given by Eq. (15):

$$LEN_{n} = \mathop \smallint \limits_{0}^{{t_{1} }} \sqrt {\left( {\frac{{d{}_{ }^{B} x_{n} }}{{dt_{1} }}} \right)^{2} + \left( {\frac{{d{}_{ }^{B} y_{n} }}{{dt_{1} }}} \right)^{2} + \left( {\frac{{d{}_{ }^{B} z_{n} }}{{dt_{1} }}} \right)^{2} }$$
(15)

Finally, bringing \(L_{n}^{\prime}\) back to \({}_{ }^{B} P_{n}\) yields the compensated length of the n-level helix curve. Taking the third-level helix curve as an example, assigning values to each parameter according to Eq. (16) and establishing the third-level helix curve and the corrected third-level helix curve, as shown in Fig. 4. The comparative data are presented in Table 1. The results indicate that after introducing the corrected pitch, the length error of the second-level helix curve decreases from 5.084106 to −0.98E−4 mm, and the length error of the third-level helix curve decreases from −3.5902 to 1.89E−4 mm.

$$\begin{gathered} LEN_{c} = 20\;{\text{mm}}\left( {0 \le t_{1} \le 20\;{\text{mm}}} \right); \hfill \\ L_{1} = 20\;{\text{mm}},R_{1} = 5\;{\text{mm}},fig_{1} = 0; \hfill \\ L_{2} = 10\;{\text{mm}},R_{2} = 2\;{\text{mm}},fig_{2} = 0; \hfill \\ L_{3} = 5\;{\text{mm}},R_{2} = 1\;{\text{mm}},fig_{3} = 0; \hfill \\ \end{gathered}$$
(16)
Figure 4
figure 4

Comparison diagram of the third-level helix curve.

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Table 1 Comparison of the lengths of the third-level helix curve.
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The establishment of the helix curve is a recursive process. A second-level helix curve is obtained recursively from the first-level helix curve using the aforementioned algorithm, and a third-level helix curve is obtained recursively from the second-level helix curve using the same algorithm. Similarly, an n-level helix curve is obtained recursively from the (n-1)-level helix curve using the aforementioned algorithm. Therefore, we can easily extend a third-level helix curve to an n-level helix curve recursively in the same manner. Due to space limitations, we only present detailed data for the first-level to third-level helix curves. To supplement this, we compared the line length accuracy of the fourth-level to tenth-level helix curves. The results show that the line length error of the tenth-level helix curve established using the method in this paper is still within the order of 1e−4 compared to the theoretical line length. This indicates that the helix curves of the third level and above established using the method in this paper can still ensure good accuracy.

In summary, by introducing the corrected pitch obtained through the integral equation, it is possible to effectively compensate for the length errors of multi-level helix curves caused by the rotation of the Frenet-Serret frame, providing an accurate and reliable basis for cable modeling. However, the aforementioned methods for modeling multi-level helix curves are all based on known values of the twisting radius and pitch. In practical engineering, it is often impossible to directly obtain the actual values of the twisting radius and pitch during the cable manufacturing process. Therefore, it is necessary to calculate the twisting radius \(R_{n}\) and pitch \(L_{n}\) of each level helix curve through geometric relationships and cable process parameters, and calculate the standard length \(LEN_{n}\) of each level helix curve based on the twisting radius and pitch.

Method for determining pitch and twisting radius of multi-layer helical structures

In multi-layer helical structures modeling, determining the pitch and twisting radius of each helix based on the actual object is indispensable. Taking a certain type of mining machine cable as the engineering object, its core line, power unit, and control unit are all relatively dense multi-layer helical structures. The core line is composed of multiple layers of fine wires, while the power unit and control unit are composed of multiple layers of core lines, as shown in Fig. 5.

Figure 5
figure 5

Cable structure diagram.

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Taking the core line as an example, the twisting radius of each layer of helical structure is related to the diameter of the wires and the number of wires in each layer. Traditional modeling methods generally calculate the twisting radius of each layer of helical structure through a circle tangent to each other15,16,17, while the pitch during modeling is often variable. Although this method is convenient, the cross-section of each wire on the overall plane of the core line is elliptical due to the different normal planes of each wire, as shown in Fig. 6. When tangential within the layer, if the twisting radius is calculated roughly as a circle according to the cross-sectional pattern, it will result in a smaller calculated twisting radius and interfere with the model, thereby reducing the accuracy of the model.

Figure 6
figure 6

Twisted wire section pattern.

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As shown in Fig. 7, there are mainly three distributions of twisted wires in multi-layer twisted structures: (1) meeting the condition of “in-layer tangency”, that is, the twisted wires within the layer are tangential to each other; (2) meeting the condition of “inter-layer tangency”, that is, each twisted wire in the layer is tangential to the outer circle of the previous layer; (3) meeting the condition of “compression deformation”, that is, the mutual compression of twisted wires within this layer leads to the deformation of the insulation part.

Figure 7
figure 7

Schematic diagram of the distribution of twisted wires.

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Calculation of twisting radius and pitch for first-level helix curve multi-layer twisted structure

First-level helix curve multi-layer twisted structure meeting the condition of “in-layer tangency”

Assuming that under the condition of “in-layer tangency”, the mth layer is composed of nm strands of twisted wires. The long axis of the ellipse is \(r_{p\_m}\), the short axis is \(r_{x\_m}\) (wire radius), the pitch is \(L_{m}\), and the twisting radius is \(R_{m}\). As shown in Fig. 6, when there is in-layer tangency, solving the twisting radius of the mth layer twist structure essentially involves finding the radius of the circle formed by the centers of \(n_{m}\) ellipses when they are tangent to each other. The shape of the ellipse determines the twisting radius \(R_{m}\), and this shape is also related to the pitch and twisting radius of the twisted wire, so it is necessary to establish a system of equations for solving.

As shown in Fig. 8a and b, we can flatten the cylinder surface where the wire centerline lies to determine the inclination angle \(\varphi_{m} = {\text{arctan}}\left( {L_{m} /2\pi R_{m} } \right)\) of the wire using geometric relationships. We establish a coordinate system \(f\) at the center of one ellipse, with the coordinates of a tangent point on the ellipse denoted as \(\left( {{}_{ }^{f} x_{m} ,{}_{ }^{f} y_{m} } \right)\), as shown in Fig. 8c. By using the relationship between \(r_{p\_m}\) and \(r_{x\_m}\), the ellipse equation, the coordinates of the ellipse tangent point, and the slope of the tangent line to the ellipse, we can construct a system of equations based on geometric relationships and relationships between variables to solve for \(L_{m}\) and \(R_{m}\). Additionally, the cable industry generally uses the outer diameter \(R_{o\_m}\) and the pitch-diameter ratio \(j_{m}\) to describe the twist structure. To better match engineering practice, we introduce the pitch- diameter ratio, a parameter widely used in the industry. With the introduction of the pitch-diameter ratio, \(L_{m}\) and \(R_{m}\) can be solved by Eq. (17).

$$\left\{ {\begin{array}{*{20}l} {r_{p\_m} = \frac{{r_{x\_m} }}{{{\text{sin}}\left( {{\text{arctan}}\left( {\frac{{L_{m} }}{{2\pi R_{m} }}} \right)} \right)}}} \hfill \\ {\frac{{{}_{ }^{f} x_{m} ^{2} }}{{r_{p\_m}^{2} }} + \frac{{{}_{ }^{f} y_{m} ^{2} }}{{r_{x\_m}^{2} }} = 1} \hfill \\ {{}_{ }^{f} y_{m} = {\text{tan}}\left( {\frac{\pi }{2} - \frac{\pi }{{n_{m} }}} \right){}_{ }^{f} x_{m} - R_{m} } \hfill \\ {\frac{{ - r_{x\_m}^{2} {}_{ }^{f} x_{m} }}{{r_{p\_m}^{2} {}_{ }^{f} y_{m} }} = {\text{tan}}\left( {\frac{\pi }{2} - \frac{\pi }{{n_{m} }}} \right)} \hfill \\ {j_{m} = \frac{{L_{m} }}{{2{*}\left( {R_{m} + r_{x\_m} } \right)}}} \hfill \\ {R_{o\_m} = R_{m} + r_{x\_m} } \hfill \\ {x_{m} > 0} \hfill \\ \end{array} } \right.$$
(17)
Figure 8
figure 8

Schematic diagram of the geometric relationship of stranded wires on the m layer.

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Using the above system of equations, we only need to input the pitch-diameter ratio, wire radius, and the number of wires in the mth layer to calculate the twisting radius and pitch of the mth layer in the multi-layer twisted structure when it satisfies the condition of “in-layer tangency” within the layer.

First-level helix curve multi-layer twisted structure meeting the condition of “inter-layer tangency”

When satisfying the condition of “inter-layer tangency”, i.e., when the mth layer is tangential to the (m-1)th layer, the twisting radius of the mth layer is equal to the radius of the outer envelope circle of the cross-section of the (m-1)th layer plus the wire radius of the mth layer. As shown in Fig. 9, which represents a particular case where \(R_{m} \ne R_{m - 1} + r_{x\_m} + r_{x\_m - 1}\), we can derive a general equation for calculating the twisting radius of the mth layer based on the twisting radius and pitch of the (m-1)th layer, as shown in Eq. (18).

$$\begin{gathered} R_{m} = \sqrt {\left( {a\cos \left( {\delta_{m - 1} } \right) + R_{m - 1} } \right)^{2} + \left( {b\sin \left( {\delta_{m - 1} } \right)} \right)^{2} } + r_{x\_m} \hfill \\ \frac{{d_{{R_{m} }} }}{{d_{{\delta_{m - 1} }} }} = \frac{{ - a^{2} \sin \left( {\delta_{m - 1} } \right) + b\left( {b\sin \left( {\delta_{m - 1} } \right) + R} \right)\cos \left( {\delta_{m - 1} } \right)}}{{\sqrt {\left( {a\cos \left( {\delta_{m - 1} } \right)} \right)^{2} + \left( {b\sin \left( {\delta_{m - 1} } \right) + R_{m - 1} } \right)^{2} } }} = 0 \hfill \\ a = r_{x\_m - 1} ,b = \frac{{r_{x\_m - 1} }}{{\sin \left( {\arctan \left( {\frac{{L_{m - 1} }}{{2\pi R_{m - 1} }}} \right)} \right)}},\delta_{m - 1} \in \left( {0,\pi } \right) \hfill \\ \end{gathered}$$
(18)
Figure 9
figure 9

Schematic diagram of the outer envelope circle of layer m-1.

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The method for solving the pitch in this case is shown in Eq. (19).

$$L_{m} = 2 \cdot j_{m} \cdot \left( {R_{m} + r_{x\_m} } \right)v$$
(19)

First-level helix curve multi-layer twisted structure meeting the condition of “compression deformation”

The method for solving the pitch and the twisting radius under the condition of “compression deformation”, that is, when the insulation undergoes deformation due to compression, is shown in Eq. (20).

$$\begin{gathered} L_{m} = j_{m} \cdot 2R_{o\_m} \hfill \\ R_{m} = R_{o\_m} - r_{x\_m} \hfill \\ \end{gathered}$$
(20)

Using the three methods mentioned above, we can calculate the twisting radius and pitch of first-level helix multi-layer twisted structures under conditions of “in-layer tangency”, “inter-layer tangency”, and “compression deformation”. When the sum of the twisting radius calculated using the “in-layer tangency” condition and the wire radius is greater than the difference between the outer radius and the wire radius, it indicates that “compression deformation” conditions are met. In cases where “compression deformation” conditions are not met, if the twisting radius calculated using the “in-layer tangency” condition is less than the twisting radius calculated using the “inter-layer tangency” condition, it indicates that the wires in the mth layer are not sufficient to completely cover the (m-1)th layer when each wire is in tangency. In this case, the data calculated using the “in-layer tangency” condition should be discarded, and the data calculated using the “inter-layer tangency” condition should be used instead. Conversely, if the twisting radius calculated using the “in-layer tangency” condition is greater than the twisting radius calculated using the “inter-layer tangency” condition, it indicates that the wires in the mth layer sufficiently cover the (m-1)th layer when each wire is in tangency. In this case, the data calculated using the “in-layer tangency ”condition should be used.

For the convenience of batch calculation of multi-layer twisted structures, we have developed a script to compute the required pitch and twisting radius for modeling purposes. This script takes as input the number of winding layers, arrays representing the number of wires per layer, arrays representing the wire radius per layer, arrays representing the pitch-diameter ratio per layer, and arrays representing the outer diameter per layer. The script can automatically calculate the twisting radius, pitch, and distribution condition of windings for each layer in the first-level helix multi-layer twisted structure. The program flowchart of the script is illustrated in Fig. 10.

Figure 10
figure 10

Flowchart of the calculation script of multi-layer twisted structure.

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Since the pitch-diameter ratio data is obtained before each level of winding is stranded into cable, utilizing the aforementioned script allows us to obtain the pitch and twisting radius of each layer's structure before cable-laying. Furthermore, by utilizing the twisting radius and pitch before stranding into cable, we can recursively solve for the standard wire length of each part of the twisted structure using formula (14).

Calculation of twisting radius and pitch for n-level helix curve multi-layer twisted structure

N-level helix curve multi-layer twisted structure is a winding structures which center line is (n-1)-level helix curve, where the center line of each wire in this structure is an n-level helix curve. Fig. 11 shows the first-level helix curve multi-layer twisted structure and the second-level helix curve multi-layer twisted structure. Sections "Revised method for constructing second-level helix curves" and "Generalization of corrected second-level helix curves to corrected n-level helix curves" mention that for second-level and higher-level helix curve, integral equations are needed to solve the corrected pitch. However, this corrected pitch is intended to compensate for errors caused by the twisting and bending of the Frenet-Serret frame. Therefore, after introducing the corrected pitch, the inclination of the wires in the n-level helix curve twisted structure relative to the overall center line (the pitch angle of the wires) will not change compared to before stranding. Therefore, the cross-sectional shape obtained by the normal plane of the overall center line before and after stranded into cable will not change. In summary, the calculation of the twisting radius of the multi-layer twisted structure after stranded to cable can still use the calculation script in Fig. 10, and the corrected pitch can be directly solved using formula 15.

Figure 11
figure 11

Fiist-level and second-level helix curve multi-layer twisted structure.

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Construction of the braid structure mode

The braided layer of the cable is a cross-woven structure made of metal wires or synthetic materials, covering the outer surface of its insulation layer. This layer is mainly used to provide electromagnetic shielding and mechanical protection, preventing electromagnetic interference from affecting the cable signal, and increasing the durability of the cable.

Traditional modeling of braided structures generally uses rose curves to describe the centerline of the braided structure. For general braided structures, the sine function in the rose curve can be replaced with a piecewise function of sine function and circular arc18, or the circle in the sine function of the rose curve can be replaced with an ellipse19 to transform the traditional rose curve into a modified rose curve, ensuring that the peaks of the rose curve have a certain width to accommodate multiple strands of braided wires arranged side by side. However, in engineering practice, we found that both methods have defects. The approach based on the sine function and circular arc needs to separately discuss single-strand braided structures and multi-strand braided structures, and interference may occur in the rising and falling segments of the braided wire when the braided wire is thick. As for the approach based on the modified rose curve, the braided structure it establishes will have a certain gap between the upper and lower layers at the intersection segment, and it is difficult to calculate a suitable major axis for the ellipse. To solve these problems, this paper proposes a piecewise function composed of a fifth-degree polynomial and circular arc to replace the rose curve in traditional methods. The fifth-degree polynomial allows direct control of the starting and ending points of the path, as well as their tangent directions and curvature, making precise adjustments to the path more convenient with such endpoint control.

The essence of the braided wire modeling method in this paper is to superimpose a piecewise function that controls oscillations on the base helix curve. Before establishing the braided wire, it is necessary to first solve for the pitch and twisting radius of the basic helix. The pitch of the base helix curve is denoted as \(L_{b}\) and the twisting radius as \(R_{b}\), which can be determined by Eq. (21). In Eq. (21), \(R_{o\_b}\) represents the outer diameter of the braided structure, and \(j_{b}\) represents the pitch-diameter ratio of the braided structure. The base helix curve and the braided wire are shown in Fig. 12.

$$\begin{gathered} L_{b} = j_{b} R_{o\_b} \hfill \\ R_{b} = R_{o\_b} - 2r_{x\_b} \hfill \\ \end{gathered}$$
(21)
Figure 12
figure 12

Braided structure base helix curve and braided wire.

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As shown in Fig. 13, the piecewise function controlling oscillations consists of 3 arc segments and 2 quintic polynomial segments. The braided structure includes both left-hand and right-hand braided wires, with the number of clusters in each direction denoted as p and the number of wires per cluster as q. The period of the piecewise function controlling oscillations is \(2\pi /p\). The cross-sectional shape of the braided wire arc segment on the centerline normal plane of the braided structure is an ellipse, with the major and minor axes corresponding to the radius of the braided wire \(r_{x\_b}\) and the major axis being \(r_{p\_b}\).

Figure 13
figure 13

Schematic diagram of piecewise function.

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Additionally, for the n-level helix curve braided structure (a braided structure with the centerline being an n-1 level helix curve), it is also necessary to introduce integral equations to solve for the corrected pitch \(L_{b} {^{\prime}}\). The corrected pitch of the basic helix curve of the n-level helix curve braided structure after stranding can be determined using Eq. (15).

Similar to Eq. (17), the central angle \(phase\_b\) occupied by each braided wire on the normal plane of the centerline in the braided structure can be determined using the equation in Eq. (22):

$$\left\{ {\begin{array}{*{20}l} {r_{p\_b} = r_{x\_b} /{\text{sin}}\left( {{\text{atan}}\left( {L_{b} /\left( {2\pi \left( {R_{b} - r_{x\_b} } \right)} \right)} \right)} \right)} \hfill \\ {y = {\text{tan}}\left( {\frac{phase\_b}{2}} \right)x - \left( {R_{b} - r_{x\_b} } \right)} \hfill \\ {\frac{{x^{2} }}{{r_{p\_b}^{2} }} + \frac{{y^{2} }}{{r_{x\_b}^{2} }} = 1} \hfill \\ { - \frac{{r_{p\_b}^{2} \cdot x}}{{r_{p\_x}^{2} \cdot y}} = {\text{tan}}\left( {\frac{phase\_b}{2}} \right)} \hfill \\ {x > 0,y < 0,phase\_b < \pi } \hfill \\ \end{array} } \right.$$
(22)

As shown in Fig. 14, after projecting all arc lengths onto the middle layer of the braid, the arc length \(r_{t\_b}\) occupied by the width of a single left-handed braided wire on the right-handed braided wire can be determined using Eq. (23), where \(\varphi_{b}\) represents the inclination angle of the braided wire.

$$\begin{gathered} r_{t\_b} = \frac{{R_{b} \cdot phase\_b}}{{2{\text{cos}}\left( {\varphi_{b} } \right)}} \hfill \\ \varphi_{b} = {\text{atan}}\left( {\frac{{L_{b} }}{{2\pi R_{b} }}} \right) \hfill \\ \end{gathered}$$
(23)
Figure 14
figure 14

Schematic diagram of braided wire occupying arc length.

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The angle \(phase\_x\) at which \(r_{t\_b}\) is converted into the rotation angle of the braided wire around the center line can be determined using Eq. (24):

$$phase\_x = \frac{{2\pi \cdot r_{t\_b} {\text{sin}}\left( {\varphi_{b} } \right)}}{{L_{b} }}$$
(24)

Referring to the method described in Eq. (12), the coordinates of the \(q_{x}\) braided wire in the \(p_{x}\) group on the Frenet-Serret frame of the center line can be represented as \({}_{ }^{{F_{b - 1} }} P_{{b\_p_{x} q_{x} }}\), as shown in Eq. (25). When \(flg_{b} = 0\), it represents a left-handed braided wire, while \(flg_{b} = 1\) represents a right-handed braided wire, where \(p_{x} = 0\sim p - 1\), \(q_{x} = 0\sim q - 1\).

$$\begin{gathered} {}_{ }^{{F_{b - 1} }} P_{{b\_p_{x} q_{x} }} = \left[ {\begin{array}{*{20}c} {{}_{ }^{{F_{b - 1} }} x_{{b\_p_{x} q_{x} }} } \\ {{}_{ }^{{F_{b - 1} }} y_{{b\_p_{x} q_{x} }} } \\ {{}_{ }^{{F_{b - 1} }} z_{{b\_p_{x} q_{x} }} } \\ \end{array} } \right] \hfill \\ = \left[ {\begin{array}{*{20}c} {\left( {R_{b} + r\_b\left( {p \cdot 2\pi \cdot t_{b} /L_{b}^{\prime} + \theta_{z} \left( {q_{x} } \right)} \right)} \right)\cos \left( {2\pi \cdot t_{b} /L_{b}^{\prime} + \theta_{b} \left( {p_{x} ,q_{x} } \right)} \right)} \\ { - \left( {R_{b} + r\_b\left( {p \cdot 2\pi \cdot t_{b} /L_{b}^{\prime} + \theta_{z} \left( {q_{x} } \right)} \right)} \right)\sin \left( {2\pi \cdot t_{b} /L_{b}^{\prime} - f\lg_{b} \cdot \pi + \theta_{b} \left( {p_{x} ,q_{x} } \right)} \right)} \\ 0 \\ \end{array} } \right] \hfill \\ \end{gathered}$$
(25)

In Eq. (25), \(r\_b\left( \alpha \right)\) is a segmented periodic function that controls the up-and-down fluctuations of the braided wire, and it can be calculated using Eq. (26).

$$r_{b\left( \alpha \right)} = \left\{ {\begin{array}{*{20}l} { - r_{x\_b} ,\left( {0 \le \alpha_{b} \le \frac{q}{2} \cdot phase\_x} \right);} \hfill \\ {a\left( {\alpha_{b} - \alpha_{b2} } \right)^{5} + b\left( {\alpha_{b} - \alpha_{b2} } \right)^{4} + c\left( {\alpha_{b} - \alpha_{b2} } \right)^{3} + d\left( {\alpha_{b} - \alpha_{b2} } \right)^{2} + e\left( {\alpha_{b} - \alpha_{b2} } \right) + f,} \hfill \\ {\left( {\frac{q}{2} \cdot phase\_x < \alpha_{b} < \frac{\pi }{p} - \frac{q}{2} \cdot phase\_x} \right);} \hfill \\ {r_{x\_b} ,\left( {\frac{\pi }{p} - \frac{q}{2} \cdot phase\_x \le \alpha_{b} \le \frac{\pi }{p} + \frac{q}{2} \cdot phase\_x} \right);} \hfill \\ { - a\left( {\alpha_{b} - \alpha_{b3} } \right)^{5} + b\left( {\alpha_{b} - \alpha_{b3} } \right)^{4} - c\left( {\alpha_{b} - \alpha_{b3} } \right)^{3} + d\left( {\alpha_{b} - \alpha_{b3} } \right)^{2} - e\left( {\alpha_{b} - \alpha_{b3} } \right) + f,} \hfill \\ {\left( {\begin{array}{*{20}c} {\frac{\pi }{p} + \frac{q}{2} \cdot phase\_x < \alpha_{b} < \frac{2\pi }{p} - \frac{q}{2} \cdot phase\_x} \\ \end{array} } \right);} \hfill \\ { - r_{x\_b} ,\left( {\frac{2\pi }{p} - \frac{q}{2} \cdot phase\_x \le \alpha_{b} \le \frac{2\pi }{p}} \right);} \hfill \\ \end{array} } \right.$$
(26)

where

$$\alpha_{b} = \left( {p \cdot 2\pi \cdot \frac{{t_{b - 1} }}{{L_{1} }}} \right){\text{\% }}\left( {\frac{2\pi }{p}} \right),$$
$$\alpha_{b2} = \frac{q}{2} \cdot phase\_x$$
$$\alpha_{b3} = \pi /p + q/2 \cdot phase\_x$$

In Eq. (25), \(a,b,c,d,e,f\) are the coefficients of the fifth-degree polynomial, which can be solved using the system of equations given in Eq. (27).

$$\left\{ {\begin{array}{*{20}l} {ax_{0}^{5} + bx_{0}^{4} + cx_{0}^{3} + dx_{0}^{2} + ex_{0} + f = y_{0} } \hfill \\ {ax_{1}^{5} + bx_{1}^{4} + cx_{1}^{3} + dx_{1}^{2} + ex_{1} + f = y_{1} } \hfill \\ {5ax_{0}^{4} + 4bx_{0}^{3} + 3cx_{0}^{2} + 2dx_{0} + e = 0} \hfill \\ {5ax_{1}^{4} + 4bx_{1}^{3} + 3cx_{1}^{2} + 2dx_{1} + e = 0} \hfill \\ {20ax_{0}^{3} + 12bx_{0}^{2} + 6cx_{0} + 2d = 0} \hfill \\ {20ax_{1}^{3} + 12bx_{1}^{2} + 6cx_{1} + 2d = 0} \hfill \\ \end{array} } \right.$$
(27)

where:

$$x_{0} = 0,y_{0} = - r_{x\_b} ,x_{1} = \pi /p - phase\_x,y_{1} = r_{x\_b}$$

In Eq. (25),\(\theta_{b} \left( {p_{x} ,q_{x} } \right)\) is a parameter that determines the position of the weaving line along the overall circumference of the structure and can be calculated using Eq. (28).

$$\begin{gathered} if\left( {\left( {q{\text{\% }}2} \right) = 1} \right): \hfill \\ \quad \quad \theta_{b} \left( {p_{x} ,q_{x} } \right) = \frac{{p_{x} \cdot 2\pi }}{p} + \left( {q_{x} - \frac{q}{2}} \right) \cdot phase\_b \hfill \\ if\left( {\left( {q{\text{\% }}2} \right) = 0} \right): \hfill \\ \quad \quad \theta_{b} \left( {p,q} \right) = \frac{{p_{x} \cdot 2\pi }}{p} + \left( {q_{x} - \left( {q + 1} \right)/2} \right) \cdot phase\_b \hfill \\ \end{gathered}$$
(28)

In Eq. (25), \(\theta_{z} \left( {q_{x} } \right)\) is a parameter controlling the phase of the weaving line's oscillation and can be calculated using Eq. (29).

$$\begin{gathered} if\left( {\left( {q{\text{\% }}2} \right) = 1} \right): \hfill \\ \quad \quad \theta_{z} \left( {q_{x} } \right) = \left( {q_{x} - \frac{q}{2}} \right) \cdot phase\_x \hfill \\ if\left( {\left( {q{\text{\% }}2} \right) = 0} \right): \hfill \\ \quad \quad \theta_{z} \left( {q_{x} } \right) = \left( {q_{x} - \left( {q + 1} \right)/2} \right) \cdot phase\_x \hfill \\ \end{gathered}$$
(29)

Finally, the coordinates in the Frenet-Serret frame need to be transformed into the base coordinate system using a coordinate transformation method. By applying the method described in Eq. (13), the coordinates of the weaving line in the base coordinate system can be obtained, as shown in Eq. (30).

$$\begin{gathered} {}_{ }^{B} P_{{b\_p_{x} q_{x} }} = \left[ {\begin{array}{*{20}c} {{}_{ }^{B} x_{{b\_p_{x} q_{x} }} } \\ {{}_{ }^{B} y_{{b\_p_{x} q_{x} }} } \\ {{}_{ }^{B} z_{{b\_p_{x} q_{x} }} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{\ddot{P}_{b - 1} }}{{\left| {\ddot{P}_{b - 1} } \right|}} \times \frac{{\dot{P}_{b - 1} }}{{\left| {\dot{P}_{b - 1} } \right|}}} \\ {\frac{{\ddot{P}_{b - 1} }}{{\left| {\ddot{P}_{b - 1} } \right|}}} \\ {\frac{{\dot{P}_{b - 1} }}{{\left| {\dot{P}_{b - 1} } \right|}}} \\ \end{array} } \right]^{T} \hfill \\ \left[ {\begin{array}{*{20}c} {{}_{ }^{{F_{b - 1} }} x_{{b\_p_{x} q_{x} }} } \\ {{}_{ }^{{F_{b - 1} }} y_{{b\_p_{x} q_{x} }} } \\ {{}_{ }^{{F_{b - 1} }} z_{{b\_p_{x} q_{x} }} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {{}_{ }^{B} x_{b - 1} } \\ {{}_{ }^{B} y_{b - 1} } \\ {{}_{ }^{B} z_{b - 1} } \\ \end{array} } \right] \hfill \\ \end{gathered}$$
(30)

Based on the above method, modeling of both single-strand and multi-strand weaving structures can be easily handled. As shown in Fig. 15, all parameters except for the process parameters are obtained through equations. In comparison to traditional methods, the modeling approach proposed in this paper does not differentiate between single-strand and multi-strand weaving structures. It is more unified and, at a sufficient density, avoids interference phenomena.

Figure 15
figure 15

Various braided structures.

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Comparison of modeling results

We utilized the algorithm proposed in this paper to develop a complete modeling toolset using C# script combined with Python script in Grasshopper. This toolset efficiently solves the equations in the algorithm using libraries such as numpy and scipy in Python script, and then builds the model efficiently using C# script, balancing functionality and modeling efficiency. Figure 16 illustrates the process flow of modeling a certain type of cable using this modeling toolset.

Figure 16
figure 16

Flowchart of modeling a certain type of cable.

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Specifically, this toolset primarily includes the "Twisting radius and pitch calculation tool", "Helix curve generation tool", "Braided structure generation tool" and "Insulation layer generation tool" as illustrated in Fig. 17. These tools are described as follows:

"Twisting radius and pitch calculation tool": This module leverages Python to solve the equations presented in Section "Method for determining pitch and twisting radius of multi-layer helical structures", enabling the calculation of twisting radius and pitch based on known parameters such as the "diameter-to-pitch ratio."

Figure 17
figure 17

Modules within the modeling toolset.

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“Helix curve generation tool”: Utilizing C# scripts, this module implements the corrected helix curve generation algorithm detailed in Section "Modeling method of n-level helix curve based on parameter equations". The script calculates points on the corrected helix curve and generates the helix curve through these discrete points, thereby creating an n+1 level corrected helix curve from an n-level corrected helix curve.

"Braided structure generation tool": This module employs both Python and C# scripts to realize the braiding line generation algorithm described in Section "Construction of the braid structure mode". Python is used to solve the parameters of the quintic polynomial and the central angle occupied by the braiding line (phase_b). C# is then used to generate discrete points on the modified braiding line, which are used to construct the braiding line.

"Insulation layer generation tool": This module utilizes built-in functions within Grasshopper to generate multiple insulation layers with non-standard cross-sections. It can batch-generate multiple hollow insulation layers that press against each other.

Using both the modeling method proposed in this paper and the traditional modeling method, we modeled the MCPT-1.9/3.3 3120 + 170 + 4 * 10 type of shearer cable with the folded yarns as the smallest unit, with a modeling length of 480mm (one pitch lenth). The process parameters of the cable are shown in Table 2, and the modeling parameters obtained using the method proposed in this paper are shown in Table 3. Thirteen different centerlines were selected (as shown in Fig. 18) to compare the errors of the two modeling methods, and the data are shown in Table 4. The error comparison between the traditional method (“Traditional error”) and the method proposed in this paper (“Paper error”) is shown in Fig. 19.

Table 2 MCPT-1.9/3.3 3 * 120 + 1 * 70 + 4 * 10 coal mining machine cable process parameters.
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Table 3 Modeling data obtained by this algorithm according to process parameters.
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Figure 18
figure 18

MCPT-1.9/3.3 3 * 120 + 1 * 70 + 4 * 10 coal mining machine cable rendering.

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Table 4 Data comparison between the two methods.
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Figure 19
figure 19

The traditional method and this paper’s method error comparison.

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From Table 4 and Fig. 19, it can be observed that the error between the length of the first-level helix curve and the standard wire length in the models established by both methods is within 1e−4 mm, indicating that there is little difference in accuracy between the two methods for the first-level helix curve, and both methods achieve high accuracy. For the second-level and higher-level helix curves, the length accuracy of the modeling method using corrected pitch and twisting radius introduced in this paper can still be maintained below 1e−4 mm. In contrast, for the traditional modeling method without introducing corrected pitch and twisting radius, the maximum error can reach −149.606188 mm, a difference in maximum error of 3.31E6 times. This demonstrates that the research method proposed in this paper can effectively reduce modeling errors and maintain a high level of accuracy in the model.

Alongside this, we cut a 1000 mm section of MCPT-1.9/3.3 3120 + 170 + 4 * 10 coal mining machine cable and dissected it. Each dissected part was measured five times, and the average value was taken to obtain the measured length. We organized the standard line length values (“Standard length”), measured line length values (“Measured length”), and the line length values obtained using the method in this paper (“Paper length”). Absolute and relative errors between the measured line length and the line length obtained using this paper's method (“Measure absolute error”, “Measure relative error”, “Paper absolute error” and “Paper relative error”) were calculated and summarized, as shown in Table 5. The comparison of the standard line length, measured line length, and the line length obtained using the method in this paper is shown in Fig. 20.

Table 5 Data comparison between standard length, measured length and paper length.
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Figure 20
figure 20

Comparison of Standard length, measured length and paper length.

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The data in Table 5 indicate that the "Measure relative error" is all below 0.195%, demonstrating that the measured lengths are relatively accurate. The "Paper relative error" is all below 0.22 parts per million, indicating that the cable model lengths established using the method in this paper have a high level of accuracy. As depicted in Fig. 20, comparing the values of "Standard length," "Measured length," and "Paper length," the relative errors between "Paper length" and both "Standard length" and "Measured length" are all less than 0.1941%. This indicates that the model established by the method in this paper exhibits good fidelity and high accuracy. This demonstrates that the method presented in this paper can effectively reduce the errors of traditional methods and significantly improve the authenticity of the model.

Summary

The paper first elaborates on the parametric modeling method of n-level helix curves and proposes compensating n-level helix curves with corrected pitches. The errors between the lengths of the second-level and third-level helical curves before and after introducing the corrected pitch are compared, and the comparative results show that the proposed research method in this paper can obtain accurate n-level helix curves, solving the problem of inaccurate center lines in n-level twisted structures. Next, equations were established for multi-layer twisted structures under conditions of “in-layer tangency”, “inter-layer tangency”, and “compression deformation”, respectively. The twisting radius and pitch under these three conditions were obtained by the pitch-diameter ratio, and an automatic calculation script was written to batch calculate the twisting radius and pitch of each layer structure in multi-layer twisted structures. Subsequently, we proposed a modeling method for braided structures based on a segmented function composed of fifth-degree polynomials and circular arcs, replacing the sine function in traditional braiding modeling methods. This method effectively avoids issues of insufficient contact between braided lines and interference between braided lines encountered in traditional methods. Finally, we developed a modeling tool using C# and Python in Grasshopper to implement the modeling algorithm. Taking the MCPT-1.9/3.3 3120 + 170 + 4 * 10 coal mining machine cable as an example, we modeled it using both the research method proposed in this paper and the traditional method. Comparative results show that the method proposed in this paper can reduce errors by up to 3.31E6 times in second-level and higher-level helical structures. Additionally, this paper compares the standard line length, measured line length, and the line length established by the model developed using the method proposed in this study. The results indicate that the relative errors between the line length established by the proposed method and the standard line length, as well as the measured line length, are both less than 0.1941%.This indicates that this paper provides a new, systematic, and high-precision modeling method for multi-level complex twisted structures, laying the foundation for high-precision modeling and simulation of cables. This method eliminates the manual adjustment of modeling parameters in traditional methods. Except for process parameters, all other parameters are automatically derived from equations, simplifying the modeling process while improving accuracy and efficiency. Moreover, the research results of this paper can be applied not only to the modeling of coal mining machine cables but also to the modeling of other complex multi-layer twisted structures.

The results of this study have been applied to the optimization of cable structures. Through mechanical performance simulation and electromagnetic performance simulation, we have successfully enhanced the overall performance of the cables. The optimized cables have demonstrated significantly improved electromagnetic characteristics and are expected to have a longer service life. Furthermore, the results of this study will be further extended to the field of fiber optic sensing cables, providing a theoretical foundation for constructing mathematical models and developing information calculation algorithms. This lays a solid foundation for the realization of intelligent sensing cables and contributes key technologies for the comprehensive intelligentization of underground operations. Therefore, this study is of profound theoretical and practical significance for enhancing the safety and stability of the coal industry and promoting its transformation towards high-quality development.