A general parametric geometric modeling method for round and irregularly shaped wire ropes

Abstract

To overcome the difficulty of accurately modeling the three-dimensional geometry of irregularly shaped wire ropes when analyzing the mechanical properties of different types and parameters, this invention proposes a general parametric geometric modeling method for round and irregularly shaped wire ropes. First, this invention derives the first-order helical spatial structure parametric equations for the centroid lines of each wire in round, triangular, and elliptical wire ropes. Second, through coordinate transformation and geometric relationship analysis, it derives the second-order helical spatial structure parametric equations for the centroid lines of each wire applicable to these three types of wire ropes. Third, combining the dimensions and cross-sectional shapes of each wire in the wire rope, it sweeps along the obtained centroid lines of each wire to establish a three-dimensional solid model of the entire wire rope for different types. After deriving the parametric equations using this invention, substituting the parameters into the output equation yields the centroid lines. To obtain different types of wire ropes, simply adjust the parameters to generate the centroid lines of each wire for subsequent modeling, greatly improving modeling efficiency.

Description

Technical Field

[0001] This invention belongs to the field of three-dimensional geometric modeling of wire ropes, specifically relating to a three-dimensional geometric modeling method for round strand wire ropes and irregular strand wire ropes. Background Technology

[0002] Steel wire rope is a helical bundle of multiple steel wires with required mechanical properties and geometric dimensions twisted together according to certain rules. It is an ideal engineering load-bearing component for withstanding tensile loads, possessing characteristics of high strength, light weight, and good flexibility. Its high tensile strength and fatigue resistance ensure stable and reliable operation, preventing sudden breakage of the entire rope. These excellent properties make steel wire rope widely used in various industries. Steel wire rope can be classified by strand shape into round strands and shaped strands, with shaped strands further divided into triangular strands and elliptical strands. Shaped strand steel wire ropes have more contact points between strands, resulting in higher strength, longer service life, and better compressive strength and fatigue resistance. They also have a larger effective cross-sectional area, increasing the breaking tensile force by 20%-25% compared to round strand steel wire ropes at the same diameter and strength. Furthermore, they have a compact structure and a smooth surface. Today, modern industry continuously demands higher mechanical performance from steel wire ropes, and these characteristics further highlight the advantages of shaped strand steel wire ropes.

[0003] Due to the complex structure of irregularly shaped wire ropes, domestic and international research has generally focused on simulation studies of round wire ropes, with relatively little research on irregularly shaped wire ropes. Current research on wire ropes is divided into two types: physical testing and simulation analysis. Because physical testing is costly, numerical simulation is widely used to study the load-bearing characteristics of wire ropes as simulation technology becomes more accurate and efficient. Among these, an accurate three-dimensional geometric model of the wire rope is the prerequisite and foundation for mechanical performance analysis and optimization design. The literature "Gao Wenjing. Wire Rope Structural Selection and Design Software Development [D]. Yanshan University, 2013" proposes a modeling method for round wire ropes; however, it does not provide a modeling method for irregularly shaped wire ropes, such as elliptical and triangular strands. Therefore, it is necessary to propose a general three-dimensional geometric modeling method for round and irregularly shaped wire ropes to provide an accurate geometric model for subsequent mechanical analysis of wire ropes. Summary of the Invention

[0004] To overcome the difficulty of accurate three-dimensional geometric modeling of irregularly shaped wire ropes when performing mechanical property analysis on wire ropes of different types and parameters, this invention proposes a general parametric geometric modeling method for round and irregularly shaped wire ropes.

[0005] The concept of this invention is:

[0006] First, the relationships between key geometric parameters in the wire rope are accurately expressed. Then, a coordinate system for the whole rope and a coordinate system for the strands describing the centroids of the wires are established, and the geometric parameter equations of the first-order helical spatial structure of the centroids of the wires in the round strand, triangular strand, and elliptical strand are derived respectively. Second, through coordinate transformation and geometric relationship analysis, the geometric parameter equations of the second-order helical spatial structure of the centroids of the wires applicable to the above three types of wire ropes are derived. Third, by combining the size and cross-sectional shape of each wire in the wire rope and sweeping it along the obtained centroids of each wire, a precise three-dimensional solid model of the whole rope of different types of wire ropes can be established.

[0007] The technical solution adopted by this invention to solve its technical problem is as follows:

[0008] A general parametric geometric modeling method for round strand and irregular strand steel wire ropes, wherein the irregular strand steel wire ropes include triangular strand and elliptical strand steel wire ropes; its special feature is that it includes the following steps:

[0009] Step 1: Derive the mathematical relationships between key geometric parameters in the wire rope and establish the whole rope coordinate system and strand rope coordinate system to describe the centroid of the wire.

[0010] Step 2: Derive the parametric equations of the centroid lines of each wire in the circular, triangular, and elliptical wire ropes for the first-order helical model in the overall rope coordinate system;

[0011] Step 3: Based on the parametric equations of the first spiral model and the mathematical relationship between the key geometric parameters obtained in Step 1, the parametric equations of the second spiral model of each wire centroid line applicable to circular strands, triangular strands and elliptical strands in the whole rope coordinate system are derived through coordinate transformation and geometric relationship analysis.

[0012] Step 4: Combining the dimensions and cross-sectional shape of each wire in the round, triangular, and elliptical wire ropes, sweep along the obtained centroid lines of each wire to obtain a three-dimensional solid model of the entire wire rope of the corresponding type.

[0013] Furthermore, the mathematical relationship between the key geometric parameters mentioned in step 1 is the helical polar angle θ of the steel wire. w And the helical polar angle θ of the strand s The relationship between them satisfies the following equation:

[0014]

[0015] In the formula: f(θ) w When the helix polar angle of the steel wire is θ w The arc length r traced by the projection curve of the centroid of the steel wire in the strand onto the end face perpendicular to the entire strand; s The radius of the spiral is given.

[0016] Furthermore, in step 1:

[0017] The origin O of the whole rope coordinate system r Let z be the center point of the bottom surface of the wire rope core; coordinate axis z r Aligned with the axis of the wire rope and pointing upwards; y r The axis is horizontal to the right; x r The axis is determined by the right-hand rule.

[0018] The origin O of the rope coordinate system w Located on the centroid line of the strand; coordinate axis z w The direction and point O on the center line of the strand w The tangents at z are in the same direction, and z w The positive direction is the direction in which the spiral ascends; x w The axis is simultaneously intersecting with the coordinate plane x of the whole rope coordinate system. r O r y r and line segment P1O r Parallel, P1 is point O w In the coordinate plane x r O r y r Projection onto the coordinate axis; x-axis w The positive direction points outward from the wire rope; the coordinate axis y r It is determined by the right-hand rule.

[0019] Furthermore, in step 2:

[0020] (1) The parametric equations of the first-order helical model of each wire centroid of the round strand steel wire rope in the whole rope coordinate system are as follows:

[0021]

[0022] In the formula: (x s , y s , z s Let R be any point on the centroid of the steel wire in the whole rope coordinate system CSO. r The coordinates in r; s Let be the helix radius of the steel wire. The helix angle of the steel wire. The helix angle of the steel wire;

[0023] (2) The parametric equations of the first-order helical model of the centroid of each wire in the elliptical strand wire rope in the whole rope coordinate system are as follows:

[0024] (2.1) Parametric equation S1 of the first helical model of the centroid line of the side strand core wire of the elliptical strand wire rope i (x s1i ,y s1 i , z s1 i )for:

[0025]

[0026] In the formula: i is the i-th core wire; b2 = b1 / 2, b1 is the vertical distance between the center of the first core wire and the center of the third core wire; L s The twist pitch of the elliptical strand rope; The angle at which the side strands of an elliptical wire rope twist around the core strand is called the helix polar angle, which starts from 0°. r A starts counting counterclockwise;

[0027] (2.2) The parametric equations of the first-order helical model of the centroid lines of the inner and outer layers of the side strands of the elliptical strand steel wire rope are as follows:

[0028] When the helix polar angle When ∈[0,2θ2], the parametric equation S1 i (x s1 i , y s1 i , z s1 i )for:

[0029]

[0030] In the formula: θ2 = arctan(b2 / l1);

[0031] When the helix polar angle When ∈[2θ2,π], the parametric equation S2 2i (x s2 2i , y s2 2i , z s2 2i )for:

[0032] ;

[0033] When the helix polar angle When ∈[π,π+2θ2], the parametric equation S2 3i (x s2 3i , y s2 3i , z s2 3i )for:

[0034] ;

[0035] When the helix polar angle When ∈[π+2θ2,2π], the parametric equation S2 4i (x s2 4i , y s2 4i , z s2 4i ) for:

[0036] ;

[0037] (3) The parametric equations of the first-order helical model of the centroid lines of each wire in the triangular strand wire rope in the whole rope coordinate system are as follows:

[0038] (3.1) Parametric equation S1 of the first helical model of the centroid line of the side strand core wire of the triangular strand wire rope i (x s1 i ,y s1 i , z s1 i )for:

[0039]

[0040]

[0041]

[0042] In the formula: L s The twist pitch of the triangular strand rope; The angle at which the side strand of the triangular wire rope twists around the core wire is the helical polar angle, where i is the i-th core wire.

[0043] (3.2) The parametric equations of the first-order helical model of the centroid lines of the inner, middle, and outer layers of the side strands of the triangular strand steel wire rope are as follows:

[0044] When the helix polar angle When ∈[0,2θ1], the parametric equation S2 1i (x s2 1i , y s2 1i , z s2 1i )for:

[0045]

[0046] In the formula: =arctan( ); From Or A t Start counting counterclockwise;

[0047] When the helix polar angle ∈ At that time, parametric equation S2 2i (x s2 2i , y s2 2i , z s2 2i )for:

[0048] ;

[0049] When the helix polar angle ∈ At that time, parametric equation S2 3i (x s2 3i , y s2 3i , z s2 3i )for:

[0050] ;

[0051] When the helix polar angle ∈ At that time, parametric equation S2 4i (x s2 4i , y s2 4i , z s2 4i )for:

[0052] ;

[0053] When the helix polar angle ∈ At that time, parametric equation S2 5i (x s2 5i , y s2 5i , z s2 5i )for:

[0054] ;

[0055] When the helix polar angle ∈ Parametric equation S2 6i (x s2 6i , y s2 6i, z s2 6i )for:

[0056] .

[0057] Furthermore, the secondary helical model of the centroid lines of the various wires applicable to round, triangular, and elliptical strands in step 3 is as follows:

[0058] ;

[0059] ;

[0060] ;

[0061] in: .

[0062] Further, step 4 specifically involves: first, based on the given parameter dimensions and the parametric equation of the quadratic helix obtained in step 3, solving for and exporting the coordinate data of the points on the centroid lines of each wire; then, importing the coordinate data of the points on the centroid lines of the wires into 3D modeling software and fitting a curve; by sweeping along the corresponding centroid lines of the wires with a given wire cross-section, a 3D solid model of the specified wire can be generated. Repeating the above operations will generate 3D solid models of all the wires constituting the wire rope.

[0063] The present invention also provides a non-volatile computer-readable storage medium storing a computer program thereon; its special feature is that the computer program implements the above-mentioned general parametric geometric modeling method for round and irregular strand steel wire ropes when it is executed.

[0064] The present invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable thereon; its special feature is that the computer program implements the above-mentioned general parametric geometric modeling method for round and irregularly shaped steel wire ropes when it runs.

[0065] The beneficial effects of this invention are:

[0066] 1. This invention first studies and analyzes the relationships between various parameters of a wire rope; then, it establishes a coordinate system for the whole rope and strands describing the positional relationships of the wires; next, it solves the parametric equations for the centroidal axis models of three types of stranded wire ropes using first-order and second-order helical patterns; finally, combining the wire rope dimensional parameters, it sweeps a given wire cross-section along the centroidal axis to establish accurate three-dimensional geometric models of round strand and irregular strand wire ropes. After deriving the parametric equations using the method of this invention, the centroidal axis can be obtained by substituting the parameters into programming software. To obtain different types of wire ropes, only the parameters need to be adjusted to generate the centroidal axis of each wire for subsequent modeling. Therefore, this invention can effectively avoid the inconvenience caused by the complexity of three-dimensional geometric modeling of wire ropes with different parameters and types when performing load simulation analysis, greatly improving the modeling efficiency of complex steel cables.

[0067] 2. This invention provides a general parametric geometric modeling method for geometric models of various complex steel wire rope structures. Attached Figure Description

[0068] Figure 1 This is a schematic diagram of the triangular plane of the unfolded steel wire rope.

[0069] Figure 2 (a) is a schematic diagram of the coordinate system of the whole rope, and (b) is a schematic diagram of the coordinate system of the elliptical strand rope.

[0070] Figure 3 This is a schematic diagram of the cross-sectional geometry of a typical elliptical strand steel wire rope.

[0071] Figure 4 This is a geometric diagram of the cross-section of the steel wire in a triangular strand steel wire rope.

[0072] Figure 5 It explains the relationship between coordinate systems and the calculation of projection vectors.

[0073] Figure 6 This is a schematic diagram of a two-dimensional cross-section of a round strand steel wire rope.

[0074] Figure 7 It is the centroid of the steel wire in the first spiral model of a round strand steel wire rope.

[0075] Figure 8 It is the centroid line of the steel wire in the first spiral model of an elliptical strand steel wire rope.

[0076] Figure 9 This is a schematic diagram of a two-dimensional cross-section of a triangular strand steel wire rope.

[0077] Figure 10 It is the centroid of the steel wire in the first spiral model of a triangular strand steel wire rope.

[0078] Figure 11It is the centroid line of the entire steel wire in the secondary spiral model of a round strand steel wire rope.

[0079] Figure 12 It is the centroid line of the entire steel wire in the double spiral model of elliptical strand steel wire rope.

[0080] Figure 13 It is the centroid line of the entire steel wire in the secondary spiral model of triangular strand steel wire rope.

[0081] Figure 14 It is a three-dimensional geometric model of a single spiral and a double spiral of a round wire rope.

[0082] Figure 15 (a) is a three-dimensional geometric model of a triangular strand wire rope, and (b) is a three-dimensional geometric model of an elliptical strand wire rope. Detailed Implementation

[0083] The present invention will be further described in detail below with reference to the accompanying drawings.

[0084] This invention provides a general three-dimensional geometric modeling method for round and irregularly shaped steel wire ropes, specifically including the following steps:

[0085] Step 1: Accurately express the relationship between key geometric parameters in the wire rope

[0086] First, the key parameters describing the precise geometry of the wire rope are given. For example... Figure 1 The diagram shows the unfolded shape of the wire rope along a plane, where: S r For the length of the steel wire rope, S s S is the length of the strand. w θ is the length of the spiral wire in the strand. s Let α be the polar angle of the strand's helix. s r is the helix angle of the strand. s Let θ be the radius of the spiral. w Let α be the polar angle of the steel wire's helix. w Let f(θ) be the helix angle of the steel wire. w When the helix polar angle of the steel wire is θ w The arc length traced by the projection curve of the centroid of the steel wire in the strand onto the end face perpendicular to the whole strand.

[0087] Steel wire rope length S r The polar angle θ of the strand s The helix angle α of the strand s and the spiral radius r s The relationship between them can be represented as:

[0088] (1-1)

[0089] Length S of the strand sIt can be represented as:

[0090] (1-2)

[0091] When the helix polar angle θ of the strand s When = 2π, the length of the strand is equal to the strand lay length L. s .

[0092] Length S of the strand s The helix angle α of the steel wire w The relationship between them can be represented as:

[0093] (1-3)

[0094] For a round strand steel wire rope, the helix radius of the steel wire is r. w Therefore, in equation (1-3), the function f(θ) w ) can be used with r w ·θ w Equation (1-3) can be expressed as:

[0095] (1-4)

[0096] For irregularly shaped stranded steel wire ropes, the function f(θ) w (Keep the above definition.)

[0097] The length S of the spiral wire in the strand w It can be represented as:

[0098] (1-5)

[0099] When the helix angle θ of the steel wire w When = 2π, the length S of the strand s Equal to wire lay length L w Due to the length S of the strand... s It can be expressed by two sets of geometric parameters, that is

[0100] (1-6)

[0101] Based on the above analysis, the helix polar angle θ of the steel wire can be derived. w And the helical polar angle θ of the strand s The relationship between them:

[0102] (1-7)

[0103] Step 2: Establish the whole rope coordinate system (CSO) to describe the positional relationships of the steel wires. r and the CSO coordinate system w

[0104] Step 2.1 To clearly describe the geometric relationship of each wire in round strand and shaped strand wire ropes, establish the whole rope coordinate system CSO. r like Figure 2 As shown in (a). The whole rope coordinate system CSO r O of the coordinate system r (Fixed) is the center point of the bottom surface of the wire rope core; coordinate axis z r Aligned with the axis of the wire rope and pointing upwards; y r The axis is horizontal to the right; x r The axis is determined by the right-hand rule.

[0105] Step 2.2 In order to clearly describe the geometric relationship of each wire in the strands that make up the whole rope, in the whole rope coordinate system CSO r Based on this, establish the strand coordinate system CSO w Taking the elliptical strand steel wire rope coordinate system as an example, such as Figure 2 As shown in (b). The CSO (Cyclic Strut) coordinate system. w O of the coordinate system w The moving point lies on the center of mass line of the strand; the coordinate axis z w The direction and point O on the center line of the strand w The tangents at z are in the same direction, and z w The positive direction is the direction in which the spiral ascends; x w The axis is simultaneously with the coordinate plane x r O r y r and line segment P1O r Parallel, P1 is point O w In the coordinate plane x r O r y r Projection on, x w The positive direction points outward from the wire rope; the coordinate axis y r It is determined by the right-hand rule.

[0106] The order of steps 1 and 2 above can be interchanged.

[0107] Step 3: Solve the parametric equations of the first-order helical model of the centroid lines of each wire in the round strand steel wire rope.

[0108] A single-strand helical wire rope is a type of wire rope constructed by spirally winding a single wire around a core wire. Solving the parametric equations of the single-strand helical wire rope model is fundamental to deriving the double-strand helical model. The following section solves the parametric equations of the single-strand helical model of a round strand wire rope along the centroid of the wire.

[0109] In the whole rope coordinate system CSO rIn the linear spiral model of the wire's centroid, the tangent at any point R makes an angle equal to the centerline of the wire rope. The vector describing the centroid of the wire is... It can be expressed by the following formula:

[0110] (3-1)

[0111] in: , , The coordinate system is CSO. r The unit vectors of the x, y, and z axes, (x s , y s , z s Let R be a point in the whole rope coordinate system CSO. r The coordinates in the coordinate system are expressed by the following parameter:

[0112] (3-2)

[0113] Where: r s The radius of the strand helix. The helix angle of the strand. The polar angle of the strand is the helix angle.

[0114] Step 4: Solve the parametric equations of the first-order helical model of the centroid lines of each wire in the elliptical strand steel wire rope.

[0115] The single-helix model of an elliptical strand steel wire rope is obtained by spirally winding a single steel wire around a core steel wire along an "elliptical track." For example... Figure 3 The diagram shows a two-dimensional cross-sectional geometry of a typical elliptical strand steel wire rope. The "elliptical track" is formed by connecting a straight line segment AB, a semicircular arc segment BC, a straight line segment CD, and a semicircular arc segment DA. The length of the straight line segment AB is b1, and in the y... r To the right of the axis, with y r The axis is parallel and the distance is y r The distance between the axes is l1; the semicircular arc BC has O' as its center and a radius of l1; the straight line segment CD has a length of b1, and in y r To the left of the axis, with y r The axis is parallel and the distance is y r The distance between the axes is l1; the semicircular arc DA has O'' as its center and l1 as its radius. There are a total of 3 core strands, along the y... r The core wires are distributed sequentially along the positive axis, with the bottommost being the first wire. b1 is the vertical distance between the center of the first core wire and the center of the third core wire. O' is in the whole rope coordinate system CSO. r The coordinates in the coordinate system are (0, b2, 0), and O'' is in the whole rope coordinate system CSO. rThe coordinates in the diagram are (0, -b2, 0), and l1 is the horizontal distance between the center of the inner layer wire and the center of the first core wire at point A.

[0116] The positions of the core steel wire and the inner layer steel wire wound around the core are analyzed below.

[0117] Step 4.1 Position analysis of the side strand core wire:

[0118] The geometry of the centroid of the side strand core wire is perpendicular to x. r O r y r A straight line in a plane, a first-order spiral model of the centroid line of the side strand core wire in the whole rope coordinate system CSO r Parametric equation S1 i (x s1 i , y s1 i , z s1 i It can be expressed by the following formula:

[0119] (4-1)

[0120] Where i is the i-th core wire; b2 = b1 / 2; L s The twist pitch of the elliptical strand rope; The angle at which the side strands of an elliptical wire rope twist around the core strand is called the helix polar angle, which starts from 0°. r A starts counting counterclockwise.

[0121] It should be noted that if there are more than three core wires, the above formula (4-1) still applies and does not affect the subsequent analysis method for the positions of the inner and outer layer wires, because it is only necessary to know that the center of each core wire is in the whole rope coordinate system CSO. r The position in the middle is fine.

[0122] Step 4.2 Analysis of the position of the inner layer of steel wires in the side strands wound around the core:

[0123] The geometry of the centroid of the inner layer of steel wire in the side strand is a spatial curve spiraling upward along the "elliptical track". The "elliptical track" needs to be studied in segments, and then the parametric equations of each segment need to be solved.

[0124] (1) When the helix polar angle ∈[0,2θ2], that is, the endpoint of the centroid of the steel wire lies in the plane x r O r y r When the projection of is on the straight segment AB of the "elliptical track"

[0125] The first-order helical model of the centroid of the inner layer steel wire wound around the core in the whole rope coordinate system CSO r Parametric equation S2 1i (x s2 1i , y s2 1i , z s2 1i It can be expressed by the following formula:

[0126] (4-2)

[0127] Where θ2 = arctan(b2 / l1), From O r A is counted counterclockwise.

[0128] (2) When the helix polar angle ∈[2θ2,π], that is, the endpoint of the center of mass of the steel wire lies in the plane x r O r y r When the projection is on the arc segment BC in the "elliptical track"

[0129] For ease of description, a local coordinate system CSO' is established with point O' as the origin. The coordinate axes of the local coordinate system CSO' are intersected with the full-length coordinate system CSO. r The corresponding axes are parallel and in the same direction.

[0130] Parametric equations of the first-order helical model of the centroid of the inner layer steel wire wound around the core in the local coordinate system CSO' It can be expressed by the following formula:

[0131] (4-3)

[0132] Where, θ h The parameter θ refers to the angular change of the arc segment BC, calculated counterclockwise from the x' axis of the local coordinate system CSO'. h ∈[0,π].

[0133] Then, parametric equations In the whole rope coordinate system CSO r It can be derived from the parametric equation S2 2i (x s2 2i , y s2 2i , z s2 2i )express:

[0134] (4-4)

[0135] To standardize the expression of parametric equations, the following solution is performed. and The relationship between them.

[0136] The side length O can be obtained using the Law of Cosines. r E:

[0137] (4-5)

[0138] Then, using the law of sines, ∠EO can be obtained. r O':

[0139] (4-6)

[0140] but With ∠EO r The relationship of O':

[0141] (4-7)

[0142] Substituting into the above formula, we can obtain the whole rope coordinate system CSO. r The corresponding angular coordinates of the same point in the local coordinate system CSO'. and The relationship between them:

[0143] (4-8)

[0144] (3) When the helix polar angle ∈[π,π+2θ2], that is, the endpoint of the center of mass of the steel wire lies in the plane x r O r y r When the projection is on the straight segment CD of the "elliptical track"

[0145] The first-order helical model of the centroid of the inner layer steel wire wound around the core in the whole rope coordinate system CSO r Parametric equation S2 3i (x s2 3i , y s2 3i , z s2 3i It can be expressed by the following formula:

[0146] (4-9)

[0147] in, From O r A is counted counterclockwise.

[0148] (4) When the helix polar angle ∈[π+2θ²,2π], that is, the endpoint of the center of mass of the steel wire lies in the plane x r O r y r When the projection is on the arc segment DA

[0149] For ease of description, a local coordinate system CSO'' is established with point O'' as the origin. The local coordinate system CSO'' and the whole rope coordinate system CSO r The corresponding axes are parallel and in the same direction.

[0150] Parametric equations of the centroid of the inner layer of steel wire wound around the core in the local coordinate system CSO'' It can be expressed by the following formula:

[0151] (4-10)

[0152] in, The parameter is used to describe the angular change of the elliptical arc segment DA, calculated from the x'' axis in a counterclockwise direction. ∈[π,2π].

[0153] Then, parametric equations In the whole rope coordinate system CSO r It can be derived from the parametric equation S2 4i (x s2 4i , y s2 4i , z s2 4i )express:

[0154] (4-11)

[0155] To standardize the expression of parametric equations, the following solution is performed. and The relationship between them.

[0156] The side length O can be obtained using the Law of Cosines. r F:

[0157] (4-12)

[0158] Then, using the law of sines, ∠FO can be obtained. r O'':

[0159] (4-13)

[0160] but With ∠FO r The relationship of O'':

[0161] (4-14)

[0162] Substituting into the above formula, we can obtain the whole rope coordinate system CSO. r The corresponding angular coordinates of the same point in the local coordinate system CSO''. and The relationship between them:

[0163] (4-15)

[0164] The positional analysis of the outer layer steel wires of the side strands wound around the core follows the same principle and method as the positional analysis of the inner layer steel wires. The only difference is that the value of l1 needs to be changed based on the parametric equation of the centroid line of the inner layer steel wire obtained by the positional analysis method of the inner layer steel wires to obtain the parametric equation of the centroid line of the outer layer steel wires.

[0165] Step 5: Solve the parametric equations of the first-order helical model of the centroid of each wire in the triangular strand steel wire rope.

[0166] The single-strand spiral model of a triangular strand steel wire rope is obtained by spirally winding a single steel wire along a triangular trajectory with the core steel wire as the axis. For example... Figure 4 The diagram shows a two-dimensional cross-sectional geometry of a triangular strand steel wire rope. The triangular trajectory is formed by straight line segment A. t B t Arc segment B t C t Line segment C t D t Circular segment D t E t Line segment E t F t and circular arc segment F t A t It is formed by connecting the wires. Assume the diameter of the core wire is r1, and the diameter of the outer wire wound around the core is r2; straight segment A t B t The length is r1, in y r To the right of the axis, with y r The axes are parallel and the distance is R a ; Arc segment B t C t With point O' as the center, Q a Let B be the radius, starting from point B t Along O'B t Rotate counterclockwise We obtain line segment C. t D t The length is r1, and the arc segment B t C t Tangent at point C t; Circular segment D t E t With O'' as the center, Q a Let the radius be , from point D t Along O''D t Rotate counterclockwise We obtain: line segment E t F t The length is r1, and the arc segment D t E t Tangent at point E t ; Circular arc segment F t A t With O''' as the center, Q a Let F be the radius, from point F t Along O'''F t Rotate counterclockwise We obtain R. a For the whole rope coordinate system CSO r O of the coordinate system r to line segment A t B t The vertical distance, Q a Let O' be the line segment from point O' to line A. t B t The vertical distance between point O' and point O' in the whole rope coordinate system CSO r The coordinates in are ( , Point O'' lies in the whole rope coordinate system CSO. r The coordinates in are ( Point O''' lies in the whole rope coordinate system CSO (0, 0). r The coordinates in are ( , , 0).

[0167] The positions of the core wire and the wire wound around the core are analyzed below.

[0168] Step 5.1 Analysis of the position of the core wire:

[0169] The geometric shape of the centroid of the three core steel wires is perpendicular to x. r O r y r A straight line in a plane, the centroid of the core wire in the whole rope coordinate system CSO r Parametric equation S1 i (x s1 i , y s1 i , z s1 i It can be expressed by the following formula:

[0170] (5-1)

[0171] (5-2)

[0172] (5-3)

[0173] Among them, L s The twist pitch of the triangular strand rope; The angle at which the side strands of a triangular wire rope twist around the core strand, i.e., the helix polar angle, starts from 0°. r A t Starting from the counterclockwise direction; i is the i-th core wire.

[0174] Step 5.2 Positional analysis of the steel wire wound around the core:

[0175] The geometric shape of the centroid of the steel wire wound around the core is a spatial curve spiraling upward along a triangular trajectory. The triangular trajectory needs to be studied in segments, and then the parametric equations of each segment need to be solved.

[0176] (1) When the helix polar angle ∈[0,2θ1], that is, the endpoint of the center of mass of the steel wire lies in the plane x r O r y r The projection of on line segment A t B t Shangshi

[0177] The first-order helical model of the centroid of the steel wire wound around the core in the whole rope coordinate system CSO r Parametric equation S2 1i (x s2 1i , y s2 1i , z s2 1i It can be expressed by the following formula:

[0178] (5-4)

[0179] in, =arctan( ); As a parameter, it originates from O r A t Start counting counterclockwise.

[0180] (2) When the helix polar angle ∈ That is, the endpoint of the center of mass of the steel wire lies in the plane x. r O r yr The projection on arc segment B t C t Shangshi

[0181] For ease of description, a local coordinate system CSO' is established with point O' as the origin. The local coordinate system CSO' is parallel to the overall coordinate system CSO. r The corresponding axes are parallel and in the same direction.

[0182] The parametric equation S2 of the first-order helical model of the centroidal wire wound around the core. 2i (x s2 2i , y s2 2i , z s2 2i In the whole rope coordinate system CSO r The middle can be represented by the following formula:

[0183] (5-5)

[0184] Where, θ l The parameter is used to describe the arc segment B. t C t The trajectory change, starting from the x' axis in a counterclockwise direction, θ l ∈[0, π].

[0185] To standardize the expression of parametric equations, we will now solve for θ. l and The quantitative relationship between them.

[0186] The side length O can be obtained using the Law of Cosines. r E:

[0187] (5-6)

[0188] Then, using the law of sines, we can obtain... :

[0189] (5-7)

[0190] but and Relationship:

[0191] (5-8)

[0192] Substituting into the above formula, we can obtain the whole rope coordinate system CSO. r The corresponding angular coordinate θ of the same point in the local coordinate system CSO'. l and The relationship between them:

[0193] (5-9)

[0194] (3) When the helix polar angle ∈ That is, the endpoint of the center of mass of the steel wire lies in the plane x. r O r y r The projection of on line segment C t D t Shangshi

[0195] The first-order helical model of the centroid of the steel wire wound around the core in the whole rope coordinate system CSO r Parametric equation S2 3i (x s2 3i , y s2 3i , z s2 3i It can be expressed by the following formula:

[0196] (5-10)

[0197] in, As a parameter, from O r A t Counting from counterclockwise.

[0198] (4) When the helix polar angle ∈ That is, the endpoint of the center of mass of the steel wire lies in the plane x. r O r y r The projection on the arc segment D t E t Shangshi

[0199] For ease of description, a local coordinate system CSO'' is established with point O'' as the origin. The coordinate axis x'' in the local coordinate system CSO'' is parallel to the vector O''D. t With the same positive direction, the coordinate axis y'' and the vector O''D t Vertical, pointing outwards from the strand.

[0200] The parametric equation S2 of the centroid of the steel wire wound around the core. 4i (x s2 4i , y s2 4i , z s2 4i In the whole rope coordinate system CSO r The middle can be represented by the following formula:

[0201] (5-11)

[0202] in, The parameter is used to describe the arc segment D. t E t The trajectory change, starting from the x'' axis in a counterclockwise direction. ∈[0, ].

[0203] To standardize the expression of parametric equations, the following solution is performed. and The quantitative relationship between them.

[0204] The side length O can be obtained using the Law of Cosines. r F:

[0205] (5-12)

[0206] Then, using the law of sines, ∠FO can be obtained. r O'':

[0207] (5-13)

[0208] but and Relationship:

[0209] (5-14)

[0210] Substituting into the above formula, we can obtain the whole rope coordinate system CSO. r The corresponding angular coordinates of the same point in the local coordinate system CSO''. and The relationship between them:

[0211] (5-15)

[0212] (5) When the helix polar angle ∈ That is, the endpoint of the center of mass of the steel wire lies in the plane x. r O r y r The projection of on line segment E t F t Shangshi

[0213] The parametric equation S2 of the centroid of the steel wire wound around the core. 5i (x s2 5i , y s2 5i , z s2 5i In the whole rope coordinate system CSOr The middle can be represented by the following formula:

[0214] (5-16)

[0215] in, As a parameter, from O r A t Counting from counterclockwise.

[0216] (6) When the helix polar angle ∈ That is, the endpoint of the center of mass of the steel wire lies in the plane x. r O r y r The projection on the arc segment F t A t Shangshi

[0217] For ease of description, a local coordinate system CSO''' is established with point O''' as the origin. The coordinate axes x''' of the local coordinate system CSO''' are parallel to the vector O'''F. t With the same positive direction, the coordinate axis y''' and the vector O'''F t Vertical, pointing outwards from the strand.

[0218] The parametric equation S2 of the centroid of the steel wire wound around the core. 6i (x s2 6i , y s2 6i , z s2 6i In the whole rope coordinate system CSO r The middle can be represented by the following formula:

[0219] (5-17)

[0220] in, The parameter is used to describe the arc segment F. t A t The trajectory change, starting from the x''' axis in a counterclockwise direction. ∈[0, ].

[0221] To standardize the expression of parametric equations, the following solution is performed. and The quantitative relationship between them.

[0222] The side length O can be obtained using the Law of Cosines. r H:

[0223] (5-18)

[0224] Then, using the law of sines, we can obtain... :

[0225] (5-19)

[0226] but and Relationship:

[0227] (5-20)

[0228] Substituting into the above formula, we can obtain the whole rope coordinate system CSO. r The corresponding angular coordinates of the same point in the local coordinate system CSO'''. and The relationship between them:

[0229] (5-21)

[0230] The positional analysis of the middle and outer layer wires of the side strands wound around the core follows the same principle and method as the positional analysis of the inner layer wires described above. The only difference is that the parametric equation of the centroid line of the inner layer wire, obtained through the inner layer wire positional analysis method, needs to be modified by changing R. a By obtaining the numerical value, the parametric equation of the centroid of the outer steel wire can be obtained.

[0231] The order of steps 3-5 above can be interchanged arbitrarily.

[0232] Step 6: Solve the parametric equations for the universal quadratic helix of the centroid of the steel wires in round and irregularly shaped steel wire ropes.

[0233] A double-helix wire rope refers to a wire rope in which the strands are wound along a helix with the core wire as the axis, and each strand is composed of multiple wires. The double-helix model of the wire's center of mass is defined in the whole rope coordinate system CSO. r The vector R and the vector defined in the strand coordinate system CSO w coordinate plane x w O w y w The vector q on the vector together describes the vector, specifically as follows: Figure 5 As shown.

[0234] Assume that the coordinates are defined in the whole rope coordinate system CSO. r A position vector P, whose direction is defined in the rope coordinate system CSO w The point (x) in w1 , y w1 , z w1 ), where point (x) w1 , y w1 , z w1If a vector P lies on the centroid of the described wire, then the vector P can be represented as:

[0235] (6-1)

[0236] in, Let vector P be in the whole rope coordinate system CSO r Coordinates in; , The coordinate system is CSO. r The unit vectors along the x, y, and z axes.

[0237] Vector q in the rope coordinate system CSO w The middle can be represented as:

[0238] (6-2)

[0239] in, Let vector P be in the rope coordinate system CSO w The coordinates in the equation, and z w1 =0; , The CSO coordinate system is the same as the strand coordinate system. w The unit vectors along the x, y, and z axes.

[0240] Since vector R points to the tail of vector q, vector P can be obtained by summing vectors R and q, i.e.

[0241] (6-3)

[0242] Vector R represents the parametric expression of the first-order helical model of the centroid of the steel wire in various types of steel wire ropes, that is, vector R in the whole rope coordinate system CSO. r The parameters in the equation have already been obtained from steps 3 to 5. Therefore, the next step is to determine the vector q in the whole rope coordinate system CSO. r The parameters in the diagram can be used to determine the centroid of the steel wires constituting the whole rope in the whole rope coordinate system CSO. r The coordinates in the diagram.

[0243] The following solutions are provided for the CSO coordinate system defined in the strand coordinate system. w The two components x of the vector q w1 and y w1 In the whole rope coordinate system CSO r Projected coordinates (x) r1 , y r1 , z r1 ) and (x r2 , y r2 , z r2 ).

[0244] x of vector q w1 Components in the whole rope coordinate system CSO r The projection in can be represented as:

[0245] (6-4)

[0246] Where, θ s The polar angle of the strand is the helix angle.

[0247] According to the right-angle projection theorem, if two perpendicularly intersecting lines, and one of them is parallel to a projection plane, then the projections of both lines onto that projection plane are also right angles. Therefore, the y-axis of vector q... w1 Components in the whole rope coordinate system CSO r The projection in can be represented as:

[0248] (6-5)

[0249] Where, α s It is the helix angle of the strand.

[0250] Then, vector q in the whole rope coordinate system CSO r The middle can be represented as

[0251] (6-6)

[0252] Similarly, vector P in the whole rope coordinate system CSO r The middle can be represented as

[0253] (6-7)

[0254] Steps 3 to 5 have yielded the parameter expressions for the single-strand helical model of the steel wire centroid. Since all three types of steel wire ropes are wound in the manner of round strand steel wire rope, the only difference being the type of strand, all parameters in equation (6-7), i.e., x, are used. s x r1 x r2 y s y r1 y r2 z s z r1 z r2 By substituting the parameters of the first-order spiral model of the wire centroid in steps 3 to 5, the centroid of the wire in the whole rope coordinate system CSO in round and irregular strand wire ropes can be obtained. r The quadratic spiral model in the text uses vector P(x) r3 , y r3 , z r3 )express:

[0255] x of vector P r3 coordinate:

[0256] (6-8)

[0257] y of vector P r3 coordinate:

[0258] (6-9)

[0259] z of vector P r3 coordinate:

[0260] (6-10)

[0261] The above analysis yielded the vector P representing the double helix model of the wire's centroid in the whole rope coordinate system CSO. r Regarding the polar angle θ of the strand s The parameter expression. Combining with formula (1-7) in step 1, the helix polar angle θ of the strand. s The polar angle θ of the steel wire's helix can be used. w Represented as:

[0262] (6-11)

[0263] Then vector P in the whole rope coordinate system CSO r Regarding the helix polar angle θ of the steel wire w It can be represented as:

[0264] x of vector P r3 coordinate:

[0265] (6-12)

[0266] y of vector P r3 coordinate:

[0267] (6-13)

[0268] z of vector P r3 coordinate:

[0269] (6-14)

[0270] The above formulas (6-12)-(6-14) are the parameter expressions of the secondary helix model applicable to describing the centroid lines of each layer of steel wires in round, elliptical, and triangular steel wire ropes, that is, the parametric equations of the secondary helix of the centroid lines of the steel wires.

[0271] Step 7: Based on the derived parametric equation of the quadratic helix of the wire centroid, realize the three-dimensional geometric model of round strand and irregular strand wire ropes.

[0272] Based on the quadratic helical parameter equation of the wire centroid obtained in step 6, the cross-sectional geometry of the wire can be swept along the wire centroid to achieve accurate three-dimensional geometric modeling of round and irregularly shaped wire ropes.

[0273] The specific steps are as follows:

[0274] Step 7.1 First, based on the given parameter dimensions and the parametric equations of the secondary helix used to describe the centroid lines of each wire in the round and irregular strand wire ropes obtained in Step 6 above, solve the equations and export the coordinate data of the points on the centroid lines of each wire.

[0275] Step 7.2 Import the coordinate data of the points on the centroid line of the steel wire into the 3D modeling software and fit the curve. Then, sweep along the corresponding centroid line of the steel wire with the given steel wire cross section to generate a 3D solid model of the specified steel wire.

[0276] By repeating the above steps, a three-dimensional solid model of all the steel wires that make up the steel wire rope can be generated.

[0277] Example:

[0278] The following section describes the three-dimensional geometric modeling of the primary and secondary helical models of round and irregularly shaped steel wire ropes:

[0279] Step 1: Accurately describe the relationship between the key geometric parameters of the wire rope, determine the wire rope dimensions and the parameters for generating the centroid of the wire.

[0280] Take the 6×K31WS-FC model, right-hand twisted round strand steel wire rope as an example.

[0281] The lay length P of the main strand of the wire rope is 250mm, and the lay length of the side strand is... =258.7mm, the diameter of the wire rope is 43mm. The side strand has 31 wires, divided into the side strand core wire (the central wire of the side strand rope), the inner layer wires of the side bone, the middle layer wires of the side bone, and the outer layer wires of the side bone. There is 1 side strand core wire, 6 inner layer wires, 12 middle layer wires, and 12 outer layer wires. The diameter of the rope core (the central wire of the entire wire rope) is r=15mm, the diameter of the side strand core is r1=2.0mm; the diameter of the inner layer wire is r2=1.7mm; the middle layer wires consist of two types of wires, 6 of each, with diameters of r3=2.0mm and r4=1.52mm respectively; the diameter of the outer layer wire is r5=2.6mm. A two-dimensional cross-sectional geometric diagram of the side strand of this type of wire rope is shown below. Figure 6 As shown.

[0282] The remaining parameters required to generate the centroid of the steel wire can be calculated, including the helical radius r of the inner steel wire of the side strand around the side strand core. s1 =r1 / 2+r2 / 2=1.85mm; the helical radius r of the middle layer steel wire of the side strand rope around the side strand core. s2 =r1 / 2+r2+r4 / 2=3.46mm; the helical radius r of the outer steel wire of the side strand rope around the side strand core. s3 =r1 / 2+r2+r4 / 2+r5 / 2=4.76mm; the helical radius r of the side strand core wound around the rope core. s =r / 2 + r1 / 2 + r2 + r4 + r5 = 13.23 mm; Helix angle of the strand .

[0283] Because the wire rope is a line contact wire rope, and the lay pitch of each layer of wire in a line contact wire rope is equal, the lay pitch of the known side strands can be used to determine its properties. Calculate the helix angle of each layer of wire in the side strand rope.

[0284] Helix angle of the inner layer of steel wire in the side strand: .

[0285] Helix angle of the middle layer steel wire in the side strand: .

[0286] Helix angle of the outer layer of steel wire in the side strand: .

[0287] Step 2: Establish a coordinate system for the whole rope and strands to describe the positional relationship of the steel wires.

[0288] First, the geometric relationship of the wires in round strand and irregular strand wire ropes is described, and the whole rope coordinate system CSO is established. r Then, the geometric relationships of the steel wires in the strands that make up the whole rope are described in the whole rope coordinate system CSO. r Based on this, establish the strand coordinate system CSO w ,like Figure 2 As shown, it accurately describes the positional relationship of the wires within various types of stranded wire ropes, facilitating geometric modeling of round and irregularly shaped stranded wire ropes.

[0289] Step 3: Solve the helical model of the centroid lines of each wire in the circular wire rope. parametric equations

[0290] The first helical model of the centroid of a single helical circular strand wire rope in the whole rope coordinate system CSO is calculated by the following formula. r Parametric equations in:

[0291]

[0292] In the formula, θ sLet α be the polar angle of the strand's helix. s r is the helix angle of the strand. s The radius of the spiral is given.

[0293] Substitute the strand parameter r of the round wire rope from step 1 into the upward formula. s1 r s2 r s3 , , and This yields the centroid of the steel wire in the first-order helical model of the round steel wire rope, such as... Figure 7 As shown.

[0294] Step 4: Solve the first-order spiral model of the centroid lines of each wire in the elliptical strand steel wire rope. parametric equations

[0295] Take the 6Q×27-FC right-handed, clockwise twisted elliptical strand steel wire rope as an example.

[0296] The lay length P of the main strand of the wire rope is 250mm, and the lay length of the side strand is... =245mm, the diameter of the wire rope is 24mm. For example... Figure 3 The diagram shows a two-dimensional cross-sectional geometry of the side strand of an elliptical wire rope. The side strand has 27 wires, divided into core wires, inner layer wires, and outer layer wires. There are 3 core wires, 9 inner layer wires, and 15 outer layer wires. The core diameter is r = 12 mm. All side strand wires are identical, with a diameter d = 1.2 mm for each wire. Parameters used to calculate the centroid orientation include: b1 = 2.46 mm, b2 = 1.23 mm, l1 = 1.18 mm, l2 = 1.18 mm, θ1 = 27.5279°, and θ2 = 46.1885°.

[0297] Based on the above parameters, the remaining parameters required to generate the centroid of the steel wire can be calculated, including the helical radius r of the side strand core around the rope core. s =r / 2 + 5d / 2 = 8.875mm, the helix angle of the strand is... .

[0298] The first-order spiral model of the centroid line of the strand is calculated using the following formula in the whole rope coordinate system CSO. r Parametric equation S1 i (x s1 i , y s1 i , z s1 i ):

[0299]

[0300] Calculating the parametric equations of a single-helix model of the centroid line of a stranded wire (wire wound around the core of the strand) requires solving in segments (taking the centroid line of the inner layer of the stranded wire rope as an example):

[0301] (1) When the helix polar angle ∈[0,2θ2], that is, the endpoint of the centroid of the steel wire lies in the plane x r O r y r When the projection of is on line segment AB

[0302] The first-order helical model of the centroid of the inner layer steel wire wound around the core in the whole rope coordinate system CSO r The parametric equations in S2 are derived from... 1i (x s2 1i , y s2 1i , z s2 1i )express:

[0303]

[0304] Where θ2 = arctan(b2 / l1);

[0305] (2) When the helix polar angle ∈[2θ2,π], that is, the endpoint of the center of mass of the steel wire lies in the plane x r O r y r When the projection is on the arc segment BC

[0306] The first-order helical model of the centroid of the inner layer steel wire wound around the core in the whole rope coordinate system CSO r The parametric equation is derived from S2 i (x s2 2i , y s2 2i , z s2 2i )express:

[0307]

[0308] in With θ h Relationship:

[0309]

[0310] (3) When the helix polar angle ∈[π,π+2θ2], that is, the endpoint of the center of mass of the steel wire lies in the plane x r O r yr When the projection is on the line segment CD

[0311] The first-order helical model of the centroid of the inner layer steel wire wound around the core in the whole rope coordinate system CSO r The parametric equations in S2 are derived from... 3i (x s2 3i , y s2 3i , z s2 3i )express:

[0312]

[0313] (4) When the helix polar angle ∈[π+2θ²,2π], that is, the endpoint of the center of mass of the steel wire lies in the plane x r O r y r When the projection is on the arc segment DA

[0314] The first-order helical model of the centroid of the inner layer steel wire wound around the core in the whole rope coordinate system CSO r The parametric equations in S2 are derived from... 4i (x s2 4i , y s2 4i , z s2 4i )express:

[0315]

[0316] in and Relationship:

[0317]

[0318] Substituting the strand parameters of the elliptical strand wire rope into the parametric equations solved in this step, the centroid line of the first-order helical model of the elliptical strand wire rope can be obtained, as shown below. Figure 8 As shown.

[0319] Step 5: Solve the parametric equations of the first-order helical model of the centroid lines of each wire in the triangular strand wire rope.

[0320] Take the 6V×34-FC right-hand twisted triangular strand steel wire rope as an example.

[0321] The lay length P of the solid strand of the wire rope is 255mm, and the lay length of the side strand is... =250mm, the wire rope diameter is 36mm. There are 34 wires in the side strands, divided into core wires, inner layer wires, middle layer wires, and outer layer wires. Specifically, there is 1 core wire, 9 inner layer wires, 12 middle layer wires, and 12 outer layer wires. The core diameter is r=14mm, the core diameter of the side strand is r1=1.25mm; the inner layer wire diameter is r2=1.25mm; the middle layer wire diameter is r3=1.6mm; and the outer layer wire diameter is r4=2.3mm. Parameters used to calculate the centroid line also include: θ1=38.2678°, θ2=28.654°. For example... Figure 9 The figure shown is a two-dimensional cross-sectional geometric diagram of the side strand of a triangular strand steel wire rope.

[0322] Based on the above parameters, the remaining parameters required to generate the centroid of the steel wire can be calculated, including the helical radius r of the side strand core around the rope core. s =r / 2 + 5r1 / 2 + r2 + r4 = 14.575mm, the helix angle of the strand. .

[0323] The first-order spiral model of the centroid line of the strand is calculated using the following formula in the whole rope coordinate system CSO. r Parametric equation S1 i (x s1 i , y s1 i , z s1 i ):

[0324]

[0325]

[0326]

[0327] Calculating the centroidal parametric equation of the strand wire requires solving it piecewise (taking the inner layer of the strand wire as an example):

[0328] (1) When the helix polar angle ∈[0,2θ1], that is, the endpoint of the center of mass of the steel wire lies in the plane x r O r y r The projection of on line segment A t B t Shangshi

[0329] The first-order helical model of the centroid of the inner layer steel wire wound around the core in the whole rope coordinate system CSO r Parametric equation S2 1i (x s21i , y s2 1i , z s2 1i ):

[0330]

[0331] (2) When the helix polar angle ∈ That is, the endpoint of the center of mass of the steel wire lies in the plane x. r O r y r The projection on arc segment B t C t Shangshi

[0332] The first-order helical model of the centroid of the inner layer steel wire wound around the core in the whole rope coordinate system CSO r Parametric equation S2 i (x s2 2i , y s2 2i , z s2 2i ):

[0333]

[0334] in With θ l Relationship:

[0335]

[0336] (3) When the helix polar angle ∈ That is, the endpoint of the center of mass of the steel wire lies in the plane x. r O r y r The projection of on line segment C t D t Shangshi

[0337] The first-order helical model of the centroid of the inner layer steel wire wound around the core in the whole rope coordinate system CSO r Parametric equation S2 3i (x s2 3i , y s2 3i , z s2 3i ):

[0338]

[0339] (4) When the helix polar angle ∈ That is, the endpoint of the center of mass of the steel wire lies in the plane x. r O r y r The projection on the arc segment D t E t Shangshi

[0340] The first-order helical model of the centroid of the inner layer steel wire wound around the core in the whole rope coordinate system CSO r Parametric equation S2 4i (x s2 4i , y s2 4i , z s2 4i ):

[0341]

[0342] in and Relationship:

[0343]

[0344] (5) When the helix polar angle ∈ That is, the endpoint of the center of mass of the steel wire lies in the plane x. r O r y r The projection of on line segment E t F t Shangshi

[0345] The first-order helical model of the centroid of the inner layer steel wire wound around the core in the whole rope coordinate system CSO r Parametric equation S2 5i (x s2 5i , y s2 5i , z s2 5i ):

[0346]

[0347] (6) When the helix polar angle ∈ That is, the endpoint of the center of mass of the steel wire lies in the plane x. r O r y r The projection on the arc segment F t A t Shangshi

[0348] The first-order helical model of the centroid of the inner layer steel wire wound around the core in the whole rope coordinate system CSOr Parametric equation S2 6i (x s2 6i , y s2 6i , z s2 6i ):

[0349]

[0350] in and Relationship:

[0351]

[0352] Substituting the strand parameters of the triangular strand wire rope into the parametric equations solved in this step, the centroid line of the first-order helical model of the triangular strand wire rope can be obtained, as shown below. Figure 10 As shown.

[0353] Step 6: Solve the secondary helical model of the centroid lines of each wire in round and irregularly shaped wire ropes. parametric equations

[0354] Calculate the centroid of the steel wire in the whole rope coordinate system CSO r Parametric equations for the quadratic spiral model:

[0355] x of vector P r3 coordinate:

[0356]

[0357] y of vector P r3 coordinate:

[0358]

[0359] z of vector P r3 coordinate:

[0360]

[0361] Substituting the strand parameters of the round and irregularly shaped wire ropes into the parametric equations obtained in this step, the centroid lines of the secondary helical models of the round, elliptical, and triangular wire ropes can be solved respectively. For example... Figure 11 As shown, this is the centroid of the wire in the double helix model of a circular strand steel wire rope; similarly, the centroid of the wire in the double helix model of an elliptical strand steel wire rope is obtained, as shown below. Figure 12 As shown; the centroid line of the triangular strand steel wire rope secondary helix model is obtained, as follows. Figure 13 As shown.

[0362] Step 7: Using the obtained centroid of the steel wire in the secondary helix model, achieve three-dimensional geometric modeling of round, elliptical, and triangular secondary helix steel wire ropes.

[0363] The centroids of the first and second spirals of the whole rope wire are obtained through programming. The discrete interval of the independent variable, the spiral polar angle, is set to 0.005, and the coordinate data of the discrete points are exported in .dat format.

[0364] The centroid lines of each wire constituting the entire rope are generated using UGNX's spline curve function. A reference plane is created at the endpoints of each wire's centroid line. Using these endpoints as the origin, the cross-section of each wire is constructed, and a 3D solid model of the wire rope is generated by "sweeping along the guide line." Figure 14 In diagram (a), the centroids of the core wire and the outermost single wire of the round wire rope are shown. Figure 14 (b) is a three-dimensional geometric model of a single outer steel wire, as shown in Figure 1. Figure 14 Image (c) shows a three-dimensional geometric model of a single-strand spiral of a round wire rope, as follows: Figure 14 Figure (d) shows the three-dimensional geometric model of a double helix of a circular wire rope; as shown in Figure (d). Figure 15 As shown in (a), this is the three-dimensional geometric model of the triangular strand steel wire rope obtained based on the aforementioned steps and known parameters; Figure 15 As shown in (b), this is the three-dimensional geometric model of the elliptical strand steel wire rope obtained based on the aforementioned steps and known parameters.

[0365] As can be seen, this example obtained the whole three-dimensional geometric model of steel wire ropes with round strands, elliptical strands, and triangular strands, verifying the effectiveness of the general three-dimensional geometric modeling method for round strands and irregularly shaped strands of steel wire ropes in this invention.

[0366] In addition to the modeling methods described above, this invention also provides a non-volatile computer-readable storage medium storing a computer program that, when executed, implements the general parametric geometric modeling method for round and irregularly shaped wire ropes of this invention.

[0367] In addition, the present invention provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable thereon, wherein the computer program implements the general parametric geometric modeling method for round and irregularly shaped wire ropes of the present invention when it is executed.

Claims

1. A general parametric geometric modeling method for round strand and irregular strand steel wire ropes, wherein the irregular strand steel wire ropes include triangular strand and elliptical strand steel wire ropes; characterized in that, Includes the following steps: Step 1: Derive the mathematical relationships between key geometric parameters in the wire rope and establish the whole rope coordinate system and strand rope coordinate system to describe the centroid of the wire. Step 2: Derive the parametric equations of the first-order spiral model of each wire centroid line of the round, triangular, and elliptical wire ropes in the overall rope coordinate system, respectively. The parametric equations of the first-order spiral model of each wire centroid line of the triangular wire rope include the parametric equations of the first-order spiral model of the centroid line of the core wire of the side strand and the centroid lines of the inner, middle, and outer layers of the side strand. The parametric equations of the first-order spiral model of the centroid lines of the inner, middle, and outer layers of the side strand are established based on the straight line segments and circular arc segments of the triangular trajectory. The parametric equations of the first-order spiral model of each wire centroid line of the elliptical wire rope include the parametric equations of the first-order spiral model of the centroid line of the core wire of the side strand and the centroid lines of the inner and outer layers of the side strand. The parametric equations of the first-order spiral model of the centroid lines of the inner and outer layers of the side strand are established based on the straight line segments and circular arc segments of the elliptical trajectory. Step 3: Based on the parametric equations of the first-order spiral model and the mathematical relationships between the key geometric parameters obtained in Step 1, the parametric equations of the second-order spiral model for each wire centroid line applicable to circular, triangular, and elliptical strands in the whole rope coordinate system are derived through coordinate transformation and geometric relationship analysis: ； ； ； in: Let be the coordinates of vector P in the whole rope coordinate system, used to represent the quadratic helical model of the wire's center of mass; vector P is jointly described by vector R defined in the whole rope coordinate system and vector q defined in the strand coordinate system. Vector R represents the parametric representation of a single-stage spiral model of the steel wire's centroid; x w1 and y w1 These are the two components of vector q; r s α is the radius of the spiral strand; s The helix angle of the strand; = f(θ) w When the helix polar angle of the steel wire is θ w The arc length α traced by the projection curve of the centroid of the steel wire in the strand onto the end face perpendicular to the whole strand; w The helix angle of the steel wire; Step 4: Combining the dimensions and cross-sectional shape of each wire in the round, triangular, and elliptical wire ropes, sweep along the obtained centroid lines of each wire to obtain a three-dimensional solid model of the entire wire rope of the corresponding type.

2. The general parametric geometric modeling method for round and irregularly shaped wire ropes according to claim 1, characterized in that, In step 1: The origin O of the whole rope coordinate system r Let z be the center point of the bottom surface of the wire rope core; coordinate axis z r Aligned with the axis of the wire rope and pointing upwards; y r The axis is horizontal to the right; x r The axis is determined by the right-hand rule; The origin O of the rope coordinate system w Located on the centroid line of the strand; coordinate axis z w The direction and point O on the center line of the strand w The tangents at z are in the same direction, and z w The positive direction is the direction in which the spiral ascends; x w The axis is simultaneously intersecting with the coordinate plane x of the whole rope coordinate system. r O r y r and line segment P1O r Parallel, P1 is point O w In the coordinate plane x r O r y r Projection onto the coordinate axis; x w The positive direction points outward from the wire rope; the coordinate axis y r It is determined by the right-hand rule.

3. The general parametric geometric modeling method for round and irregularly shaped wire ropes according to claim 2, characterized in that, In step 2: (1) The parametric equations of the first-order helical model of each wire centroid of the round strand steel wire rope in the whole rope coordinate system are as follows: In the formula: (x s , y s , z s Let R be any point on the centroid of the steel wire in the whole rope coordinate system CSO. r The coordinates in r; s Let be the helix radius of the strand. The helix angle of the strand. The polar angle of the helix of the strand; (2) The elliptical trajectory is formed by connecting the straight line segment AB, the semicircular arc segment BC, the straight line segment CD, and the semicircular arc segment DA in sequence. Among them, the straight line segment AB is in the whole rope coordinate system. r To the right of the axis, and with y r The axes are parallel; the semicircular arcs BC and DA are centered at O' and O'' respectively, with radii of l1; the coordinates of O' and O'' in the whole rope coordinate system are (0, b2, 0) and (0, -b2, 0) respectively; then the parametric equations of the first-order helical model of the centroid lines of each wire of the established elliptical strand steel wire rope in the whole rope coordinate system are: (2.1) Parametric equation S1 of the first helical model of the centroid line of the side strand core wire of the elliptical strand wire rope i (x s1 i , y s1 i ,z s1 i )for: In the formula: i is the i-th side strand core wire; b2 = b1 / 2, b1 is the vertical distance between the center of the first side strand core wire and the center of the third side strand core wire, and the length of each straight line segment in the elliptical trajectory is equal to b1; L s The twist pitch of the strand; The helical polar angle of the side strand steel wire, which starts from O r A starts counting counterclockwise; (2.2) The parametric equations of the first-order helical model of the centroid lines of the inner and outer layers of the side strands of the elliptical strand steel wire rope are as follows: When the helix polar angle When ∈[0,2θ2], the parametric equation S1 i (x s1 i , y s1 i , z s1 i )for: In the formula: Let θ2 be the polar angle of the helix of the side strand of steel wire; θ2 = arctan(b2 / l1); The horizontal distance between the center of the inner layer wire at point A and the center of the first side strand core wire is given. The radii of the semicircular arcs BC and DA in the elliptical trajectory are both equal to... ; When the helix polar angle When ∈[2θ2,π], the parametric equation S2 2i (x s2 2i , y s2 2i , z s2 2i )for: ； In the formula, θ h The parameter refers to the change in angle of the semicircular arc segment BC in the elliptical trajectory; When the helix polar angle When ∈[π,π+2θ2], the parametric equation S2 3i (x s2 3i , y s2 3i , z s2 3i )for: ； When the helix polar angle When ∈[π+2θ2,2π], the parametric equation S2 4i (x s2 4i , y s2 4i , z s2 4i ) for: ； In the formula, The parameter is used to describe the angular change of the semicircular arc segment DA in the elliptical trajectory; (3) The triangular trajectory is formed by the straight line segment A t B t Arc segment B t C t Line segment C t D t Circular segment D t E t Line segment E t F t and circular arc segment F t A t It is formed by connecting them sequentially, where the straight line segment A t B t In the whole rope coordinate system y r To the right of the axis, and with y r Axis parallel; arc segment B t C t D t E t With F t A t With points O', O'', and O''' as centers, and each circle having radius Q, the circles are as follows: a The coordinates of O', O'', and O''' in the whole rope coordinate system are respectively ( , , 0), ( , 0, 0) and ( , , 0); r1 is the diameter of the side strand core wire; then the parametric equations of the first-order helical model of the centroid lines of each wire of the established triangular strand wire rope in the whole rope coordinate system are: (3.1) Parametric equation S1 of the first helical model of the centroid line of the side strand core wire of the triangular strand wire rope i (x s1 i , y s1 i ,z s1 i )for: In the formula: L s The twist pitch of the strand; Let be the helix polar angle of the side strand steel wire, and i be the i-th side strand core steel wire; (3.2) The parametric equations of the first-order helical model of the centroid lines of the inner, middle, and outer layers of the side strands of the triangular strand steel wire rope are as follows: When the helix polar angle When ∈[0,2θ1], the parametric equation S2 1i (x s2 1i , y s2 1i , z s2 1i )for: In the formula: The diameter of the side strand core wire is given, and the length of each straight segment in the triangular trajectory is equal to... ; The diameter of the steel wire wound around the core; Let A be the straight line segment in the triangular trajectory. t B t With y r Distance between axes; The helical polar angle of the side strand steel wire; =arctan( ); From O r A t Start counting counterclockwise; When the helix polar angle ∈ At that time, parametric equation S2 2i (x s2 2i , y s2 2i , z s2 2i )for: ； In the formula, The parameter is used to describe the circular arc segment B in the triangular trajectory. t C t The trajectory change of Q; a For arc segment B t C t From the center point O' to the line segment A t B t The perpendicular distance, and the radius of each arc segment in the triangular trajectory is equal to Q. a ; When the helix polar angle ∈ At that time, parametric equation S2 3i (x s2 3i , y s2 3i , z s2 3i )for: ； When the helix polar angle ∈ At that time, parametric equation S2 4i (x s2 4i , y s2 4i , z s2 4i )for: ； In the formula, The parameter is used to describe the circular arc segment D in the triangular trajectory. t E t The trajectory change; When the helix polar angle ∈ At that time, parametric equation S2 5i (x s2 5i , y s2 5i , z s2 5i )for: ； When the helix polar angle ∈ Parametric equation S2 6i (x s2 6i , y s2 6i , z s2 6i )for: In the formula, The parameter is used to describe the circular arc segment F in the triangular trajectory. t A t The trajectory changes.

4. The general parametric geometric modeling method for round and irregularly shaped wire ropes according to claim 3, characterized in that, Step 4 is as follows: First, based on the given parameter dimensions and the parametric equation of the quadratic helix obtained in step 3, solve the problem and export the coordinate data of the points on the centroid of each wire. Then, the coordinate data of the points on the centroid line of the steel wire are imported into the 3D modeling software and the curve is fitted. The given steel wire cross section is swept along the corresponding centroid line of the steel wire to generate a 3D solid model of the specified steel wire. By repeating the above steps, a three-dimensional solid model of all the steel wires that make up the steel wire rope can be generated.

5. A non-volatile computer-readable storage medium having a computer program stored thereon; characterized in that: When the computer program is run, it implements the general parametric geometric modeling method for round and irregularly shaped wire ropes as described in any one of claims 1-4.

6. An electronic device comprising a memory, a processor, and a computer program stored in and executable thereon; characterized in that: When the computer program is run, it implements the general parametric geometric modeling method for round and irregularly shaped wire ropes as described in any one of claims 1-4.

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