Circular-section steel wire ring closed-loop winding parameter design method based on closing criterion

Abstract

The invention provides a circular-section steel wire ring closed-loop winding parameter design method based on a closing criterion, and belongs to the technical field of metal product machining. Aiming at the problem that an existing design method cannot completely meet parameter combination of a closed condition, the method comprises the following steps: firstly, constructing a circle core curve equation of a closed ring, and establishing a Freen-Serret frame on a core line of the closed ring to obtain a mathematical model of a first-order spiral line; and constructing a Freen-Serret frame on the first-order spiral line to derive a second-order spiral line mathematical model, and establishing a Freen-Serret frame based on the second-order spiral line to generate a third-order spiral line mathematical model. And establishing an outer layer spiral winding model of the steel wire ring with the circular section through geometrical relationship association of the circular core curve and the first-order spiral line to the third-order spiral line. And establishing a closing criterion of the first-order to third-order spiral steel wires based on the model, finally inputting the mathematical model into modeling software, selecting parameters according to the closing criterion, establishing all core wires, and generating each spiral structure model.

Description

Technical Field

[0001] The invention relates to a closed-loop winding parameter design method for a circular-section steel wire ring based on a closure criterion, and belongs to the technical field of metal product processing. Background Art

[0002] The bead ring is a key load-bearing component of the tire, providing a secure seal between the tire and the rim, and transmitting braking, steering, and impact loads. Its reliability is crucial to the safe operation of the tire. Currently, the cross-sectional shapes of bead rings used in tires primarily include circular, hexagonal, and parallelogram shapes. Compared to bead rings with other cross-sectional shapes, circular bead rings, due to their unique geometric structure, exhibit significant mechanical advantages: highly uniform force distribution, high specific strength, and strong impact resistance. They are primarily used in aircraft tires.

[0003] A circular wire traveler consists of two structural components: an inner ring of wire and an outer ring of wire (strands, ropes) spirally wound around the ring's surface. Existing research on mathematical models for circular wire travelers is limited and primarily relies on production experience. There has been no systematic study of mathematical models, and no research has been reported on the closure of wound wires. Research on mathematical models and closure criteria for closed wire ropes, which have a similar structure to circular travelers, has been conducted. Although these models and closure criteria are not universal due to different winding methods, they are still useful as references. However, research has revealed that the current closure criteria for closed wire ropes are insufficient. These criteria cannot cover all parameter combinations that meet the closure requirements, resulting in the omission of optional parameters. Although manual intervention and adjustment can achieve end-to-end closure in the actual production of circular travelers, uneven winding trajectories can lead to poor product consistency and dynamic balance issues for the traveler. Furthermore, a lack of systematic research on mathematical models has prevented the development of three-dimensional models for mechanical research using finite element methods. Currently, the mechanical properties of circular cross-section wire rings are mainly verified through experimental methods, which have the disadvantages of high cost, low efficiency, and inability to cover all design parameters. Summary of the Invention

[0004] The present invention aims to solve the problem that the accuracy of the design parameters selected in the existing circular cross-section wire ring design is deviated, resulting in the wire ring being unable to be completely closed. A closed-loop winding parameter design method for a circular cross-section wire ring based on a closure criterion is proposed.

[0005] The technical solution adopted by the present invention to solve the above problems is: the present invention comprises the following steps: Step 1: Based on the core radius of the wire ring and the angle of the core curve, the curve equation of the closed circular core line is constructed. A Frenet-Serret frame is established on the core line of the closed circular core line to obtain the mathematical model of the first-order helix. The Frenet-Serret frame is then constructed on the first-order helix to derive the mathematical model of the second-order helix. Finally, a Frenet-Serret frame is established based on the second-order helix to generate the mathematical model of the third-order helix. Step 2: Based on the geometric relationship between the core curve, first-order helix, second-order helix, and third-order helix, establish the closure criteria for the first-order to third-order helical wires of the circular cross-section wire ring; Step 3: Import the established mathematical model into the modeling software, select the geometric parameters that meet the spiral closure criteria based on the first-order to third-order spiral closure criteria of the circular cross-section wire ring, establish all core wires based on the mathematical model, and generate each spiral structure model.

[0006] Furthermore, step 1 specifically includes: Step 1.1: Construct the curve equation of the closed circular core line based on the core radius of the wire ring and the turning angle of the core curve; Step 1.2: Establish a Frenet-Serret frame on the core line of the closed loop and solve the Frenet-Serret frame. n θ - b θ - t θ The three unit vectors of , based on the solution results, construct the Frenet-Serret frame n θ - b θ - t θ The transformation matrix from the mid-direction vector to the global coordinate system is n θ - b θ - t θ Coordinate system to establish circular motion formula q' s , and converted into a direction vector in the global coordinate system q s , and the direction vector h Combined to obtain the direction vector of the first-order circular spiral h s , we get the equation of the first-order circular spiral curve; Step 1.3: Establish a Frenet-Serret frame on the first-order circular spiral curve and calculate the unit normal vector of the Frenet-Serret frame.n s , unit binormal vector b s and the unit tangent vector t s , construct the Frenet-Serret frame based on the solution results n s - b s - t s The transformation matrix from the mid-direction vector to the global coordinate system is n s - b s - t s Coordinate system establishes the direction vector of circular motion q' d , and converted into a direction vector in the global coordinate system q d , and the direction vector h s The direction vector of the second-order circular spiral is obtained by combining h d , we get the equation of the second-order circular spiral curve; Step 1.4: Establish a Frenet-Serret frame on the obtained second-order circular spiral curve, and repeat the steps in step 1.3 to solve the unit normal vector, unit binormal vector and unit tangent vector and the Frenet-Serret frame to global coordinate system transformation matrix to solve the circular motion direction vector q t , and with the direction vector h d The direction vector of the third-order circular spiral is obtained by combining h t , we get the equation of the third-order circular spiral curve; The curve equation of the closed circular core line is expressed as: (1); In formula (1), R is the radius of the ring core, θ is the ring core turning angle, “±” determines the forming direction of the closed ring core curve, and also determines the winding direction of the outer layer of winding wire. When “+” is taken, it is counterclockwise winding, and when “-” is taken, it is clockwise winding.

[0007] Furthermore, step 1.2 specifically includes: Step 1.2.1: Create a Frenet-Serret frame on the curve of the closed circle core n θ -b θ - t θ ; Step 1.2.2: Solve the Frenet-Serret moving frame n θ - b θ - t θ The unit normal vector in n θ , unit binormal vector b θ and the unit tangent vector t θ , based on the unit normal vector n θ , unit binormal vector b θ and the unit tangent vector t θ Calculate the Frenet-Serret frame n θ - b θ - t θ The transformation matrix from the inner direction vector to the global coordinate system; Step 1.2.3: n θ - b θ - t θ Establishing the circular motion formula in the coordinate system q' s ,Will q' s Convert to the direction vector in the global coordinate system q s , and the direction vector h Combined to obtain the direction vector of the first-order circular spiral h s , we get the equation of the first-order circular spiral curve; Unit normal vector n θ , unit binormal vector b θ and the unit tangent vector t θ The calculation formula is: (2); In formula (2), h' and h'' They are direction vectors h The first and second derivatives of ; Frenet-Serret frame n θ - b θ - t θ The calculation formula of the transformation matrix from the inner direction vector to the global coordinate system is: (3); Establishing the circular motion formula q' s The expression is: (4); In formula (4), r s is the spiral radius of the first-order circular spiral, φ s The angle of rotation around the core of the closed circle is determined by “±”, which means the spiral direction is right-handed and “-” means left-handed. q' s The expression in the global coordinate system is: (5); Direction vector of first-order circular spiral h s The calculation formula is: (6).

[0008] Furthermore, step 1.3 specifically includes: Step 1.3.1: Establish a Frenet-Serret frame on the obtained first-order circular spiral curve; Step 1.3.2: Place the core of the first-order circular spiral in the A Tangent vector at point t s Tangent vector to the toroidal core around which the first-order circular helix is ​​wound t θ The angle between , the core line of the first-order circular spiral A The tangent vector of the circular motion at point is recorded as t φ s , based on the tangent vector t φ s Sum and tangent vector t θ Calculate the principal normal vector of the first-order circular spiral curve n s , based on the tangent vector t θ , winding angle and the tangent vector t φ s Calculate the unit tangent vector t s , based on the principal normal vector n s and the unit tangent vector t s Calculate the unit binormal vector b s ; Step 1.3.3: Create the Frenet-Serret frame based on the solution results n s - b s - t s The transformation matrix from the direction vector to the global coordinate system is n s - b s - t s Coordinate system establishes the direction vector of the circular spiral q' d ,Will q' d Convert to the direction vector in the global coordinate system q d and with the direction vector h s The direction vector of the second-order circular spiral is obtained by combining h d , we get the equation of the second-order circular spiral curve; The calculation formula of the transformation matrix is: (7); Normal vector t θ , winding angle and the tangent vector t φ s The calculation formula is: (8); Tangent vector t φ s The calculation formula is: (9); Ring spiral direction vector q' d The expression is: (10); In formula (10), rd is the spiral radius of the second-order circular spiral, φ d is the rotation angle of the second-order circular spiral winding around the first-order circular spiral; Direction vector in the global coordinate system q d The expression is: (11); In formula (11), is the winding angle of the first-order helix; direction vector of the second-order circular spiral h d The expression is: (12).

[0009] Furthermore, the direction vector of the third-order circular spiral in step 1.4 q t The expression is: (13); In formula (13), r t is the spiral radius of the third-order spiral, α d is the winding angle of the second-order helix, φ t is the circumferential motion angle of the third-order helix around the second-order helix; direction vector of the third-order circular spiral h t The calculation formula is: (14).

[0010] Furthermore, step 2 specifically includes: Calculation of the winding angle of the first-order helix of a circular cross-section wire ring based on the geometric relationship among the core curve, first-order helix, second-order helix and third-order helix α s , the winding angle of the second-order helix α d and the winding angle of the third-order helix α t Expressions of According to the winding characteristics of the circular cross-section wire ring and the geometric relationship between the ring core curve and the spiral of each order, the ring core angle is determined θ and the rotation angle relationship between the first to third order spiral wires of circular cross-section wire rings; Based on the rotation angle relationship of the first-order to third-order spiral steel wire of the circular cross-section steel wire ring, the expression of the closure criterion of the first-order to third-order spiral steel wire of the circular cross-section steel wire ring is obtained. In the closure criterion, the closure of the circular spiral must satisfy that the normal rotation angle is an integer multiple of the rotation angle of the core wire around it. The expression of the rotation angle relationship of the first to third order spiral wire of the circular cross-section wire ring is: (15); (16); (17); In formula (15), m The number of turns of the winding wire around the coil core in the circumferential direction indicates the number of steel wires in the cross section of the finished product; The expression of the closure criterion for the first to third order spiral steel wire of circular cross-section traveler is: (18); (19); (20); Furthermore, step 3 specifically includes: Input the established mathematical model into the modeling software; Input the corresponding core curve, the annular rotation angle relationship of the first-order to third-order circular spiral, and the curve equations corresponding to steps 1.2-1.4. Obtain other design parameters except the winding angle according to the size and load requirements, and re-substitute the parameters into the corresponding closure criterion to obtain the winding angle that satisfies the closure relationship. Input all parameters into the software, use the established core curve and spiral lines of each order as guide lines, and establish circular curves of corresponding diameters as generatrixes in the normal planes of the core curve and spiral lines of each order to generate a three-dimensional model of each spiral structure.

[0011] The beneficial effects of the present invention are: This invention begins with parameter design, solving the Frenet-Serret frame based on geometric relationships and establishing a mathematical model. It then devises closure criteria for first- to third-order circular spirals, accurately obtaining the winding angles required for modeling. This ensures that all modeling parameters meet the perfect closure conditions for the spiral, further ensuring precise closure at both ends of the wire winding. Furthermore, the invention optimizes the solution method, reducing computational effort and improving solution efficiency. The established mathematical model and modeling method provide an accurate three-dimensional model for finite element analysis of circular cross-section wire travelers. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 A schematic flow chart of a closed-loop winding parameter design method for a circular cross-section wire ring based on a closure criterion provided by the present invention; Figure 2 A schematic diagram of the second-order spiral derivation principle provided by the present invention; Figure 3 Schematic diagram of solving the unit vector of the Frenet-Serret frame provided by the present invention; Figure 4 This is a schematic diagram of the geometric relationship between the coil core curve, first-order helix, second-order helix and third-order helix provided by the present invention. DETAILED DESCRIPTION

[0013] Combine Figure 1-4 This embodiment is described as follows. Figure 1 As shown, the method for designing closed-loop winding parameters of a circular cross-section wire ring based on the closure criterion in this embodiment includes the following steps: S101: Construct a curve equation of the core line of the closed circle, establish a Frenet-Serret frame on the core line of the closed circle, and obtain a mathematical model; This embodiment constructs the curve equation of the closed circular core line based on the core radius of the wire ring and the rotation angle of the circular core line: (1); In formula (1), R is the radius of the ring core, θ is the ring core turning angle, “±” determines the forming direction of the closed ring core curve, and also determines the winding direction of the outer layer of winding wire. When “+” is taken, it is counterclockwise winding, and when “-” is taken, it is clockwise winding.

[0014] S102: Establishment of the first-order circular spiral curve equation: This embodiment establishes a Frenet-Serret frame on the core line of the closed loop n θ - b θ - t θ , calculate the Frenet-Serret moving frame n θ - b θ - t θ The unit normal vector in n θ , unit binormal vector b θ and the unit tangent vector t θ : (2); In formula (2), h' and h''They are direction vectors h The first and second derivatives of ; Based on unit normal vector n θ , unit binormal vector b θ and the unit tangent vector t θ Calculate the Frenet-Serret frame n θ - b θ - t θ The transformation matrix from the mid-direction vector to the global coordinate system: (3); Since the circular spiral motion is the composite motion of two sub-motions: the motion along the core curve and the circular motion around the core. The motion around the core curve can be expressed by formula (1). n θ - b θ - t θ The circular motion formula can be established in the coordinate system. The expression of circular motion is shown in formula (4): (4); In formula (4), r s is the spiral radius of the first-order circular spiral, φ s The angle of rotation around the core of the closed circle is determined by “±”, which means the spiral direction is right-handed and “-” means left-handed. Will q' s Convert to the direction vector in the global coordinate system q s : (5); Will q s and direction vector h Combined to obtain the direction vector of the first-order circular spiral h s , we get the equation of the first-order circular spiral curve; Direction vector of first-order circular spiral h s The calculation formula is: (6).

[0015] The first-order circular spiral is realized by combining different winding directions and spiral directionsq s The operators are shown in Table 1: Table 1

[0016] S103: Establish the equation of the second-order circular spiral curve: The derivation principle of the second-order circular spiral curve equation is the same as that of the first-order circular spiral curve. Figure 2 As shown, the following steps are included: S10301: Establish a Frenet-Serret frame on the obtained first-order circular spiral curve; S10302: If Figure 3 As shown, the core line of the first-order circular spiral is A Normal vector at point t θ The angle in the opposite direction to the circular motion of the first-order spiral winding core curve is recorded as the winding angle , the core line of the first-order circular spiral A The tangent vector of the circular motion at point is recorded as t φ s , based on the tangent vector t φ s and the tangent vector to the center of the ring t θ Calculate the principal normal vector n s , based on the tangent vector t θ , winding angle and the tangent vector t φ s Calculate the unit tangent vector t s , based on the principal normal vector n s and the unit tangent vector t s Calculate the unit binormal vector b s , the Frenet-Serret frame will be established based on the solution results n s - b s - t s The transformation matrix of the internal direction vector to the global coordinate system, and converts it into the direction vector in the global coordinate system; like Figure 3 As shown, according to the kinematic formation principle of the circular spiral, the tangent vector of any point on the first-order circular spiral ist s It can be decomposed into two components: (1) the tangent vector of the center line of the basic core line t θ (2) Tangent vector related to the circular winding motion in the normal plane of the core centerline t φ s Since the tangent vectors of both components of motion lie in the tangent plane, the principal normal vector must be orthogonal to these tangential directions. Therefore, the principal normal vector n s Available through t θ and t φ s The cross product of is obtained. The tangent vector t s , normal vector n s With binormal vector b s The equation is shown in formula (8). t φ s The calculation formula is shown in formula (9).

[0017] The calculation formula of the transformation matrix is: (7); Normal vector t θ , winding angle and the tangent vector t φ s The calculation formula is: (8); Tangent vector t φ s The calculation formula is: (9); S20203: In n s - b s - t s Coordinate system establishes circular motion direction vector q' d : (10); Will q' d Convert to the direction vector in the global coordinate system q d : (11); In formula (11), is the winding angle of the first-order helix; In formula (10), r d is the spiral radius of the second-order circular spiral, φ d is the rotation angle of the second-order circular spiral winding around the first-order circular spiral; The direction vector in the global coordinate system q d With direction vector h The direction vector of the second-order circular spiral is obtained by combining h d , we get the equation of the second-order circular spiral curve: (12).

[0018] The combination of different winding directions and spiral directions of the second-order circular spiral is realized q d The operators are shown in Table 2: Table 2

[0019] S104: Establish the equation of the third-order circular spiral curve: The derivation method of the third-order circular spiral curve equation is the same as that of the second-order spiral curve. The only difference is that the Frenet-Serret frame needs to be established on the second-order spiral curve. Based on this derivation method, the equation of the third-order circular spiral curve can be established. n The derived third-order spiral curve equation is the direction vector of the circular motion of the third-order spiral in the global coordinate system. q t As shown in formula (13), the direction vector of the third-order circular spiral curve is shown in formula (14). The different winding directions of the third-order circular spiral are combined with the spiral direction to achieve q t The operators are shown in Table 3.

[0020] (13); In formula (13), r t is the spiral radius of the third-order spiral, α d is the winding angle of the second-order helix, φ t is the circumferential motion angle of the third-order helix around the second-order helix; (14).

[0021] Table 3

[0022] In summary, the present invention uses geometric relationships to establish a mathematical model for the Frenet-Serret frame, optimizes the solution method, reduces the amount of calculation, and improves the solution efficiency.

[0023] S2: Based on the geometric relationship between the core curve, first-order helix, second-order helix and third-order helix, the closure criteria of the first-order to third-order helix steel wire of the circular cross-section wire ring are established; In this embodiment, the geometric relationship between the core curve, the first-order helix, the second-order helix and the third-order helix is ​​as follows: Figure 4 As shown, Figure 4 middle, l c 、 l s 、 l d 、 l t are the lengths of the core curve, first-order helix, second-order helix and third-order helix respectively, α s is the winding angle of the first-order helix, α d is the winding angle of the second-order helix, α t is the winding angle of the third-order helix.

[0024] according to Figure 4 The rotation angle relationship between spirals of different orders can be derived, and the rotation angle relationship is shown in formulas (15)-(17): (15); (16); (17); In formula (15), m The number of turns of the winding wire around the coil core in the circumferential direction indicates the number of steel wires in the cross section of the finished product; When it is necessary to establish a mathematical model of a circular cross-section wire ring, the parameter relationship of formulas (15)-(17) is substituted into the corresponding curve equation to realize the model of a single-layer single-wire reciprocating winding structure.

[0025] For circular cross-section wire rings, the main structural parameters are set according to the tire specifications and load capacity, and the ring core radius can be obtained. R 、Spiral radius( r s 、 r d 、 r t ), number of lapsm , number of leads N The closing criterion of the present invention takes the winding angle as the final design parameter, so that the value of the winding angle satisfies a certain relationship with other parameters, thereby ensuring the precise closing of the product.

[0026] From formulas (15)-(17), the closure criteria for the first-order to third-order spiral steel wires of circular cross-section steel rings can be obtained as shown in formulas (18)-(20): (18); (19); (20); When designing the parameters of a circular cross-section wire ring, other parameters can be designed according to the size and load requirements. Finally, the various parameters must be substituted into formulas (18)-(20) to solve the winding angle that satisfies the closed relationship, and this winding angle is used for three-dimensional modeling or production processing.

[0027] S3: Input the established core curve, mathematical expressions of the circular spiral curves of each order, the angle relations of formulas (15)-(17), and the parameters designed according to the closure criterion into the simulation software, and define the relevant parameter values ​​to generate each spiral structure model; S301: Inputting the established mathematical expressions of the core curve, the circular spiral curves of each order, the relationship between the corresponding curve angles, and the relevant parameters designed according to the closure criterion into the modeling software NX UG; S302: Create all core wires and create a circular sketch with the same diameter as the corresponding steel wire in the normal plane of each core wire; S304: Use the established center curve as the guide line and the circular curve of the corresponding diameter as the generatrix to sweep and generate each spiral structure model In summary, the present invention designs the closure criteria for first-order to third-order circular spirals, accurately obtains the winding angle and rotation angle relationship required for modeling, ensures that all parameters used in modeling meet the perfect closure conditions of the spiral, and further ensures the precise closure of the winding ends of the steel wire.

[0028] The above description is merely a preferred embodiment of the present invention and does not constitute any form of limitation to the present invention. Although the present invention has been disclosed as a preferred embodiment as above, it is not intended to limit the present invention. Any technician familiar with the present profession can make some changes or modifications to equivalent embodiments of equivalent changes using the technical content disclosed above without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent replacement and improvement of the above embodiments made according to the technical essence of the present invention, within the spirit and principles of the present invention, without departing from the content of the technical solution of the present invention, shall still fall within the scope of protection of the technical solution of the present invention.

Claims

1. A closed loop winding parameter design method for circular cross-section wire rings based on the closure criterion, characterized in that: include: Step 1: Based on the core radius of the wire ring and the angle of the core curve, the curve equation of the closed circular core line is constructed. A Frenet-Serret frame is established on the core line of the closed circular core line to obtain the mathematical model of the first-order helix. The Frenet-Serret frame is then constructed on the first-order helix to derive the mathematical model of the second-order helix. Finally, a Frenet-Serret frame is established based on the second-order helix to generate the mathematical model of the third-order helix. Step 2: Based on the geometric relationship between the core curve, first-order helix, second-order helix, and third-order helix, establish the closure criteria for the first-order to third-order helical wires of the circular cross-section wire ring; Step 3: Input the established mathematical model into the modeling software, select the geometric parameters that meet the spiral closure criteria based on the first-order to third-order spiral closure criteria of the circular cross-section wire ring, establish all core wires based on the mathematical model, and generate each spiral structure model.

2. The closed-loop winding parameter design method for a circular cross-section steel wire ring based on the closure criterion according to claim 1 is characterized in that: Step 1 specifically includes: Step 1.1: Construct the curve equation of the closed circular core line based on the core radius of the wire ring and the turning angle of the core curve; Step 1.2: Establish a Frenet-Serret frame on the core line of the closed loop and solve the Frenet-Serret frame. n θ - b θ - t θ The three unit vectors of , based on the solution results, construct the Frenet-Serret frame n θ - b θ - t θ The transformation matrix from the mid-direction vector to the global coordinate system is n θ - b θ - t θ Coordinate system to establish circular motion formula q' s , and converted into a direction vector in the global coordinate system q s , and the direction vector h Combined to obtain the direction vector of the first-order circular spiral h s , we get the equation of the first-order circular spiral curve; Step 1.3: Establish a Frenet-Serret frame on the first-order circular spiral curve and calculate the unit normal vector of the Frenet-Serret frame. n s , unit binormal vector b s and the unit tangent vector t s , construct the Frenet-Serret frame based on the solution results n s - b s - t s The transformation matrix from the mid-direction vector to the global coordinate system is n s - b s - t s Coordinate system establishes the direction vector of circular motion q' d , and converted into a direction vector in the global coordinate system q d , and the direction vector h s The direction vector of the second-order circular spiral is obtained by combining h d , we get the equation of the second-order circular spiral curve; Step 1.4: Establish a Frenet-Serret frame on the obtained second-order circular spiral curve, and repeat the steps in step 1.3 to solve the unit normal vector, unit binormal vector and unit tangent vector and the Frenet-Serret frame to global coordinate system transformation matrix to solve the circular motion direction vector q t , and with the direction vector h d The direction vector of the third-order circular spiral is obtained by combining h t , we get the equation of the third-order circular spiral curve; The curve equation of the closed circular core line is expressed as: （1）； In formula (1), R is the radius of the ring core, θ is the ring core rotation angle, "±" determines the formation direction of the closed ring core curve, and also determines the winding direction of the outer layer of winding wire. When "+" is taken, it is counterclockwise winding, and when "-" is taken, it is clockwise winding.

3. The closed-loop winding parameter design method for a circular cross-section steel wire ring based on the closure criterion according to claim 2 is characterized in that: Step 1.2 specifically includes: Step 1.2.1: Create a Frenet-Serret frame on the curve of the closed circle core n θ - b θ - t θ ; Step 1.2.2: Solve the Frenet-Serret moving frame n θ - b θ - t θ The unit normal vector in n θ , unit binormal vector b θ and the unit tangent vector t θ , based on the unit normal vector n θ , unit binormal vector b θ and the unit tangent vector t θ Calculate the Frenet-Serret frame n θ - b θ - t θ The transformation matrix from the inner direction vector to the global coordinate system; Step 1.2.3: n θ - b θ - t θ Establishing the circular motion formula in the coordinate system q' s ,Will q' s Convert to the direction vector in the global coordinate system q s , and the direction vector h Combined to obtain the direction vector of the first-order circular spiral h s , we get the equation of the first-order circular spiral curve; Unit normal vector n θ , unit binormal vector b θ and the unit tangent vector t θ The calculation formula is: （2）； In formula (2), h' and h'' They are direction vectors h The first and second derivatives of ; Frenet-Serret frame n θ - b θ - t θ The calculation formula of the transformation matrix from the inner direction vector to the global coordinate system is: （3）； Establishing the circular motion formula q' s The expression is: （4）； In formula (4), r s is the spiral radius of the first-order circular spiral, φ s The angle of rotation around the core of the closed loop, "±" determines the direction of the spiral, "+" is right-handed, and "-" is left-handed; q' s The expression in the global coordinate system is: （5）； Direction vector of first-order circular spiral h s The calculation formula is: （6）。 4. The closed-loop winding parameter design method for a circular cross-section steel wire ring based on the closure criterion according to claim 2 is characterized in that: Step 1.3 specifically includes: Step 1.3.1: Establish a Frenet-Serret frame on the obtained first-order circular spiral curve; Step 1.3.2: Place the core of the first-order circular spiral in the A Tangent vector at point t s Tangent vector to the toroidal core around which the first-order circular helix is ​​wound t θ The angle between , the core line of the first-order circular spiral A The tangent vector of the circular motion at point is recorded as t φ s , based on the tangent vector t φ s Sum and tangent vector t θ Calculate the principal normal vector of the first-order circular spiral curve n s , based on the tangent vector t θ , winding angle and the tangent vector t φ s Calculate the unit tangent vector t s , based on the principal normal vector n s and the unit tangent vector t s Calculate the unit binormal vector b s ; Step 1.3.3: Create the Frenet-Serret frame based on the solution results n s - b s - t s The transformation matrix from the direction vector to the global coordinate system is n s - b s - t s Coordinate system establishes the direction vector of the circular spiral q' d ,Will q' d Convert to the direction vector in the global coordinate system q d and with the direction vector h s The direction vector of the second-order circular spiral is obtained by combining h d , we get the equation of the second-order circular spiral curve; The calculation formula of the transformation matrix is: （7）； Normal vector t θ , winding angle and the tangent vector t φ s The calculation formula is: （8）； Tangent vector t φ s The calculation formula is: （9）； Ring spiral direction vector q' d The expression is: （10）； In formula (10), r d is the spiral radius of the second-order circular spiral, φ d is the rotation angle of the second-order circular spiral winding around the first-order circular spiral; Direction vector in the global coordinate system q d The expression is: （11）； In formula (11), is the winding angle of the first-order helix; direction vector of the second-order circular spiral h d The expression is: （12）。 5. The closed-loop winding parameter design method for a circular cross-section steel wire ring based on the closure criterion according to claim 2 is characterized in that: The direction vector of the third-order circular spiral in step 1.4 q t The expression is: （13）； In formula (13), r t is the spiral radius of the third-order spiral, α d is the winding angle of the second-order helix, φ t is the circumferential motion angle of the third-order helix around the second-order helix; direction vector of the third-order circular spiral h t The calculation formula is: （14）。 6. The closed loop winding parameter design method for a circular cross-section steel wire ring based on the closure criterion according to claim 1 is characterized in that: Step 2 specifically includes: Calculation of the rotation angle of the first-order helix of a circular cross-section wire ring based on the geometric relationship among the core curve, first-order helix, second-order helix, and third-order helix φ s , the turning angle of the second-order helix φ d and the turning angle of the third-order helix φ t Expressions of According to the winding characteristics of the circular cross-section wire ring and the geometric relationship between the ring core curve and the spiral of each order, the ring core angle is determined θ and the rotation angle relationship between the first to third order spiral wires of circular cross-section wire rings; Based on the rotation angle relationship of the first-order to third-order spiral steel wire of the circular cross-section steel wire ring, the expression of the closure criterion of the first-order to third-order spiral steel wire of the circular cross-section steel wire ring is obtained. In the closure criterion, the closure of the circular spiral must satisfy that the normal rotation angle is an integer multiple of the rotation angle of the core wire around it. The expression of the rotation angle relationship of the first to third order spiral wire of the circular cross-section wire ring is: （15）； （16）； （17）； In formula (15), m The number of turns of the winding wire around the coil core in the circumferential direction indicates the number of steel wires in the cross section of the finished product; The expression of the closure criterion for the first to third order spiral steel wire of circular cross-section traveler is: （18）； （19）； （20）。 7. The closed-loop winding parameter design method for a circular cross-section steel wire ring based on the closure criterion according to claim 6, characterized in that: Step 3 specifically includes: Input the established mathematical model into the modeling software; Input the corresponding core curve, the annular rotation angle relationship of the first-order to third-order circular spiral, and the curve equations corresponding to steps 1.2-1.

4. Obtain other design parameters except the winding angle according to the size and load requirements, and re-substitute the parameters into the corresponding closure criterion to obtain the winding angle that satisfies the closure relationship. Input all parameters into the software, use the established core curve and spiral lines of each order as guide lines, and establish circular curves of corresponding diameters as generatrixes in the normal planes of the core curve and spiral lines of each order to generate a three-dimensional model of each spiral structure.

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