A Parametric Design and Modeling Method for Cylindrical Gears with Locally Conjugate Circular Arc Tooth Lines

Abstract

This invention relates to the field of gear design technology and provides a parametric design and modeling method for cylindrical gears with locally conjugate circular arc tooth lines. The method involves obtaining and deriving the point cloud parameters of the gear based on the tooth profile equation derived by the applicant. Then, based on the point cloud parameters and the gear's design parameters, a tooth space or tooth solid is constructed. A three-dimensional model of the cylindrical gear with locally conjugate circular arc tooth lines is obtained by performing calculations on the gear blank using subtractive or additive manufacturing methods. During the calculation of the point cloud parameters, the gear rotation angle is converted into a tool rotation angle θ. i This simplifies the calculation of gear parameters by expressing them in this way; different tool rotation angles θ are used. i The point cloud data of the gear tooth surface is obtained through value calculation, which improves the efficiency of parameter calculation. The 3D modeling method of this application realizes parametric modeling, and the modeling process is consistent with the machining process, which can significantly improve modeling efficiency while ensuring modeling accuracy.

Description

Technical Field

[0001] This invention relates to the field of gear design technology, and in particular to a parametric design and modeling method for cylindrical gears with locally conjugate circular arc tooth lines. Background Technology

[0002] The locally conjugate circular arc tooth profile cylindrical gear is a novel type of gear with a unique tooth surface structure. The tooth profile is circularly arc-shaped, the mid-section tooth profile is involute, and the tooth profiles of other sections parallel to the mid-section are envelopes of a uniformly varying family of hyperbolas. Compared to traditional involute gears (spur, helical, and herringbone gears), it offers advantages such as better meshing performance, higher load-bearing capacity, greater overlap ratio, higher transmission efficiency, lower installation accuracy requirements, automatic centering, and no axial force component. Its overall performance surpasses the aforementioned three traditional gear types, demonstrating broad application prospects. Currently, parametric design for this type of gear has not been implemented, posing challenges to its 3D modeling and analysis. The parametric design modeling method proposed in this invention provides a model foundation for the machining and analysis of this gear, meeting industrial needs and laying the groundwork for its market application. Summary of the Invention

[0003] The purpose of this invention is to provide a parametric design and modeling method for locally conjugate circular arc toothed cylindrical gears. This application obtains the tooth profile equations of the working tooth surface and the transition tooth surface of the locally conjugate circular arc toothed cylindrical gear through theoretical derivation, and calculates point cloud data based on the equations, and constructs a three-dimensional model of the gear based on the calculated point cloud data.

[0004] Specifically, this application provides a parametric design method for cylindrical gears with locally conjugate circular arc tooth lines, comprising the following steps:

[0005] S10. Set gear parameters R T , m, z, B; where R T Where m is the nominal radius of the cutter head, z is the gear module, z is the number of teeth, and B is the tooth width.

[0006] S20. Calculate the tooth width parameter b, b = B / 2, and divide [-b, b] into K equal parts;

[0007] S30. Use the concave tooth surface equation and let t = 1, or use the convex tooth surface equation and let t = 2; let k = 1;

[0008] S40. Let z di =z dri =b k , k∈(1,K);

[0009] S50. Calculate the tool rotation angle θ at the tooth tip. i denoted as θ i刀 The tool rotation angle at the intersection of the working tooth profile and the tooth root transition curve is denoted as θ.i渡 , and the cutter rotation angle at the tooth root, denoted as θ i根 ;

[0010] S60. Take a number of discrete points between θ i刀 and θ i渡 , and solve the coordinates (x di , y di ) of the working tooth profile curve according to the tooth profile equation of the working tooth surface; take a number of discrete points between θ i渡 and θ i根 , and solve the coordinates (x dri , y dri ) of the root transition curve according to the tooth profile equation of the transition tooth surface;

[0011] S70. Store the (x di , y di , z di ) and (x dri , y dri , z dri ) obtained from the concave tooth surface equation into matrix A; store the (x di , y di , z di ) and (x dri , y dri , z dri ) obtained from the convex tooth surface equation into matrix B;

[0012] S80. Let k = k + 1. If b k < b, then return to step S40; otherwise, return to step S30 and change the value of t;

[0013] S90. When the equations of the concave and convex tooth surfaces are both calculated, output matrix A and B;

[0014] The tooth profile equation of the working tooth surface is:

[0015]

[0016] The tooth profile equation of the transition tooth surface is:​​​​​​​​​​​​​​​​​​​​​​​Let x be the pitch circle radius of the gear, and α be the pressure angle of the gear; dri y dri , z dri The tool rotation angle θ i The transition tooth surface at x d y d , z d coordinates on, (x d y d , z d () represents the moving coordinate system of the gear; r represents the radius of the tool tip fillet. This is the tooth height coefficient.

[0019] Preferably, step S90 further includes reversing the coordinate values ​​of matrix A around z. d Rotate the axis by 2π / z radians, store the resulting coordinates in matrix C, and output matrix C.

[0020] The second aspect of this application provides a method for modeling cylindrical gears with locally conjugate circular arc tooth lines, including the following steps:

[0021] L10. Calculate point cloud data using the design method described above, wherein the point cloud data includes matrices A and B;

[0022] L20. Use 3D modeling software to read gear parameters and point cloud data, and construct the tooth groove entity based on the point cloud data;

[0023] L30. By circumferentially arraying the tooth groove entity on the tooth blank according to the number of teeth, and subtracting it from the tooth blank entity by performing a Boolean operation, a three-dimensional digital model of the cylindrical gear with locally conjugate circular arc tooth lines can be obtained.

[0024] A third aspect of this application provides a method for modeling cylindrical gears with locally conjugate circular arc tooth lines, comprising the following steps:

[0025] L10'. Calculate point cloud data using the design method described above, wherein the point cloud data includes matrices C and B;

[0026] L20'. Use 3D modeling software to read gear parameters and point cloud data, and construct gear tooth entities based on the point cloud data;

[0027] L30'. By circumferentially arraying the gear teeth on the gear blank according to the number of teeth, and adding them to the gear blank entity through Boolean operation, a three-dimensional digital model of the cylindrical gear with locally conjugate circular arc tooth lines can be obtained.

[0028] The parametric design and modeling method for cylindrical gears with locally conjugate circular arc tooth lines proposed in this application has at least the following advantages compared to existing technologies:

[0029] (1) In the parameter design, this application converts the gear rotation angle into the tool rotation angle θ.i This simplifies the calculation of gear parameters by expressing them in this way; different tool rotation angles θ are used. i The point cloud data of the gear tooth surface can be obtained by value calculation, which can improve the efficiency of parameter calculation;

[0030] (2) In the calculation process, this application improves the method of selecting discrete points, which improves the non-smoothness of the calculated point cloud data, and does not increase the amount of calculation.

[0031] (3) This application uses the point cloud data obtained by calculation to perform three-dimensional modeling of locally conjugate circular arc toothed cylindrical gears by additive or subtractive methods, so as to realize parametric modeling. The modeling process is consistent with the machining process, which can greatly improve the modeling efficiency while ensuring the modeling accuracy. Attached Figure Description

[0032] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the description of the embodiments of the present invention or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0033] Figure 1 This is a flowchart illustrating a parametric design method for a cylindrical gear with locally conjugate circular arc tooth lines, as provided in Embodiment 1 of this application.

[0034] Figure 2 This is a schematic diagram of point cloud data obtained by the design method of Embodiment 1 of this application;

[0035] Figure 3 This is the point cloud data map before improvement in Embodiment 1 of this application;

[0036] Figure 4 This is an improved point cloud data map of Embodiment 1 of this application;

[0037] Figure 5 This is a schematic diagram of the three-dimensional modeling process of a cylindrical gear with locally conjugate circular arc tooth line through subtractive design in Embodiment 2 of this application;

[0038] Figure 6 This is a set of fitting curves for the subtractive design of a cylindrical gear with a locally conjugate circular arc tooth line in Embodiment 2 of this application;

[0039] Figure 7 This is a schematic diagram of the tooth groove structure of Embodiment 2 of this application;

[0040] Figure 8 This is a schematic diagram of the three-dimensional modeling process of a cylindrical gear with locally conjugate circular arc tooth lines through additive design in Embodiment 3 of this application;

[0041] Figure 9 This is a set of fitting curves for the additive design of locally conjugate circular arc toothed cylindrical gears in Embodiment 3 of this application;

[0042] Figure 10 This is a schematic diagram of the tooth groove structure of Embodiment 3 of this application;

[0043] Figure 11 This is a schematic diagram of a three-dimensional model of a cylindrical gear with a partially conjugate circular arc tooth line designed in Embodiment 2 or Embodiment 3 of this application. Detailed Implementation

[0044] The following description provides many different embodiments or examples for implementing various features of the invention. The elements and arrangements described in the specific examples below are only for concise expression of the invention and are merely examples, not intended to limit the invention.

[0045] Example 1

[0046] This embodiment 1 provides a parametric design method for cylindrical gears with locally conjugate circular arc tooth lines, such as... Figure 1 As shown, it includes the following steps:

[0047] S10. Set gear parameters R T , m, z, B; where R T Where m is the nominal radius of the cutter head, z is the gear module, z is the number of teeth, and B is the tooth width.

[0048] S20. Calculate the tooth width parameter b, b = B / 2, and divide [-b, b] into K equal parts;

[0049] S30. Use the concave tooth surface equation and let t = 1, or use the convex tooth surface equation and let t = 2; let k = 1; that is, in this step, the point cloud on the concave tooth surface can be calculated first using the concave tooth surface equation, or the point cloud on the convex tooth surface can be calculated first using the convex tooth surface equation. This application does not limit the order of these two calculation methods.

[0050] S40. Let z di =z dri =b k , k∈(1,K);

[0051] S50. Calculate the tool rotation angle θ at the tooth tip. i denoted as θ i刀 The tool rotation angle at the intersection of the working tooth profile and the tooth root transition curve is denoted as θ. i渡 And the tool rotation angle at the tooth root, denoted as θ i根 ;

[0052] In this embodiment, the tool rotation angle at the tooth tip, the tool rotation angle at the intersection of the working tooth profile and the tooth root transition curve, and the tool rotation angle at the tooth root are calculated according to the following formulas:

[0053] The tool rotation angle θ at the tooth tip is calculated using the following formula. i :

[0054]

[0055] The tool rotation angle θ at the root of the profile tooth is calculated using the following formula. i :

[0056]

[0057] The tool rotation angle θ at the intersection of the working tooth profile and the transition curve is calculated using the following formula. i :

[0058]

[0059] x di y di The tool rotation angle θ i The working tooth surface at x d y d The coordinates on R i Let x be the pitch circle radius of the gear. dri y dri The tool rotation angle θ i The transition tooth surface at x d y d coordinates on c is the tooth height coefficient. * is the tooth tip coefficient, and m is the module.

[0060] S60. In θ i刀 and θ i渡 Take several discrete points between them, and solve for the coordinates (x, y) of the working tooth profile curve based on the tooth profile equation of the working tooth surface. di y di );in θ i渡 and θ i根 Take several discrete points between them, and solve for the coordinates (x, y) of the tooth root transition curve based on the tooth profile equation of the transition tooth surface. dri ,y dri );

[0061] The tooth profile equation of the working tooth surface is:

[0062]

[0063] The tooth profile equation for the transition tooth surface is:

[0064]

[0065] In the formula, "±" and represent concave and convex tooth surfaces, where the upper sign is taken for the concave tooth surface and the lower sign for the convex tooth surface; x di , y di , z di are the coordinates of the working tooth surface at the cutter rotation angle θ i on x d , y d , z d ; R i is the pitch circle radius of the gear, and α is the pressure angle of the gear; x dri , y dri , z dri are the coordinates of the transitional tooth surface at the cutter rotation angle θ i on x d , y d , z d , (x d , y d , z d ) is the moving coordinate system of the gear; r is the radius of the tip fillet.

[0066] S70. Store the (x di , y di , z di ) and (x dri , y dri , z dri ) obtained from the concave tooth surface equation into matrix A; store the (x di , y di , z di ) and (x dri , y dri , z dri ) obtained from the convex tooth surface equation into matrix B; then matrices A and B are the point cloud matrices;

[0067] S80. Let k = k + 1. If b k < b, return to step S40; otherwise, return to step S30 and change the value of t, that is, if t = 1 in the first calculation, then take t = 2; if t = 2 in the first calculation, then take t = 1, so as to calculate both the convex tooth surface and the concave tooth surface once;

[0068] S90. When the equations of the concave tooth surface and the convex tooth surface are both calculated, output matrices A and B; thus, the gear tooth point cloud data of the locally conjugate circular arc tooth profile cylindrical gear is obtained.

[0069] Since matrices A and B are convenient for constructing the three-dimensional model of the tooth groove and are suitable for subtractive manufacturing of the three-dimensional model of the gear, see Embodiment 2 for details; matrix C can also be calculated from matrix A, such as Figure 2As shown, matrices B and C are suitable for constructing 3D models of gear teeth and for additively constructing 3D models of gears, as detailed in Example 3.

[0070] Therefore, step S90 further includes reversing the coordinate values ​​of matrix A around z. d Rotate the axis by 2π / z radians, store the resulting coordinates in matrix C, and output matrix C.

[0071] In this embodiment, the gear rotation angle is expressed as the tool rotation angle θ. i This simplifies the calculation of gear parameters; by taking different tool rotation angles θ i The point cloud data of the gear tooth surface is obtained by value calculation, which provides a data basis for the 3D modeling of the gear.

[0072] In step S60, where the tooth surface point cloud data is obtained by solving the tooth surface equation, the tool rotation angle θ is affected by the strong nonlinearity of the tooth surface equation. i If we divide its value range into equal parts, the resulting point cloud data is as follows: Figure 3 As shown, the data points along the tooth height are not evenly spaced, especially at the junction of the working tooth surface and the transition surface, where point cloud discontinuities may occur. This results in an uneven fit to the fitted tooth surface at that location. In such cases, this can generally be addressed by increasing θ. i The unevenness is improved by dividing the material into equal parts. However, increasing θ... i The number of equal divisions increases the amount of point cloud data, which lengthens the calculation cycle and makes tooth surface fitting more difficult.

[0073] In this application embodiment, a method is proposed to improve the above problem by introducing a nonlinear sequence, θ i The value of is calculated using the following formula. The calculated point cloud data is as follows: Figure 4 As shown, the point cloud data has been significantly improved, and the number of point cloud data has not increased.

[0074]

[0075] Where, θ i (j) is θ i The j-th value, θ imin For θ i The minimum value of θ imax For the maximum value, p is θ i The discrete number of θ, where a is a constant greater than 0. When a = 1, θ i (j) is an arithmetic sequence, and the value of a can be estimated using the following method:

[0076] The value of a corresponding to the working tooth profile curve is calculated using the following formula:

[0077]

[0078] The value of a corresponding to the tooth root transition curve is calculated using the following formula:

[0079]

[0080] After multiple experiments and explorations by the applicant, more preferably, when calculating the working tooth profile curve, the value range of a is [1.05, 1.55]; when calculating the tooth root transition curve, the value range of a is [1.8, 4.2].

[0081] Example 2

[0082] This embodiment provides a method for modeling cylindrical gears with locally conjugate circular arc tooth lines, based on point cloud data matrices A and B calculated in Embodiment 1. Specifically, as follows... Figure 5 As shown:

[0083] A method for modeling cylindrical gears with locally conjugate circular arc tooth lines includes the following steps:

[0084] L10. Calculate point cloud data using the design method described in Example 1, wherein the point cloud data includes matrices A and B;

[0085] L20. Use 3D modeling software to read gear parameters and point cloud data, and construct the tooth groove entity based on the point cloud data;

[0086] Specifically, step L20 includes the following steps:

[0087] L21. Fit the point cloud data into B-spline curves along the tooth width and tooth height directions respectively to obtain a set of curves that constitute the mesh surface;

[0088] like Figure 5 As described above, the number of rows l and columns n of the tooth surface in the point cloud data A and B are identified, and then the data of matrix A and matrix B are converted into spatial points respectively; B spline curves are fitted along the tooth width direction and tooth height direction respectively, and fitted into a crisscross tooth surface grid, which is then fitted into surface A1 and surface B1 respectively.

[0089] L22. Complete the top and bottom curves of the tooth groove;

[0090] L23. Construct the concave and convex tooth surfaces of a single tooth groove using mesh surfaces; construct the top and bottom surfaces of the tooth groove using the top and bottom curves of the tooth groove; select the edge lines of the two end faces of the tooth groove to construct a bounded plane, forming a complete surface of a single tooth groove.

[0091] The tooth tip boundary curve AK1 is circularly arrayed around the z-axis to obtain curves ABK2, ABK3, ABK4, and ABK5, with BK1 serving as the construction line for the tooth groove top surface. The tooth groove top curves KG1 and KG2 on both end faces are then completed as boundary lines. A mesh is constructed using curves AK1, ABK2, ABK3, ABK4, ABK5, BK1, KG1, and KG2 to fit the tooth tip surface D1. Finally, the tooth groove bottom curves KG3 and KG4 on both end faces are completed as boundary lines. Curves AK1, BK1, and KG2 are then used as boundary lines. l KG3 and KG4 are used to fit the boundary to obtain the bottom curved surface E1 of the tooth groove; as shown Figure 6 As shown.

[0092] L24. Stitch together the complete curved surfaces of a single tooth groove to form a tooth groove solid.

[0093] Construct a bounded plane F1 as the front face of the tooth groove, using curves AG1, BG1, KG1, and KG3 as boundaries; take curve AG n BG n KG2 and KG4 form a bounded plane G1 as the rear end face of the tooth groove; the curved surfaces A1, B1, D1, E1, F1, and G1 are stitched together to obtain a tooth groove solid I1; as shown Figure 7 As shown.

[0094] L30. By circumferentially arraying the tooth groove entity on the tooth blank according to the number of teeth, and subtracting it from the tooth blank entity by performing a Boolean operation, a three-dimensional digital model of the cylindrical gear with locally conjugate circular arc tooth lines can be obtained.

[0095] This embodiment describes a method for constructing a three-dimensional model on a gear blank using a subtractive process. Using matrices A and B calculated in Embodiment 1, a single tooth groove solid is constructed. The tooth groove solids are arranged circumferentially, and a Boolean operation is performed to subtract them from the gear blank solid to obtain the three-dimensional model of a cylindrical gear with locally conjugate circular arc tooth lines, as shown below. Figure 11 As shown.

[0096] Example 3

[0097] Compared with Example 2, Example 3 differs in that it uses additive manufacturing for 3D modeling. Specifically, it uses matrices C and B calculated in Example 1 to construct a single gear tooth entity. The gear tooth entities are then arranged circumferentially and added to the gear blank entity via Boolean operations to obtain the 3D model of the locally conjugate circular arc tooth line cylindrical gear. Specifically, as shown in Figure 8:

[0098] L10'. Point cloud data is calculated using the design method as described in Example 1, wherein the point cloud data includes matrices C and B;

[0099] L20'. Use 3D modeling software to read gear parameters and point cloud data, and construct gear tooth entities based on the point cloud data;

[0100] 1. In step S20', constructing the gear tooth entity based on point cloud data includes the following steps:

[0101] L21'. Fit the point cloud data into B-spline curves along the tooth width and tooth height directions respectively to obtain a set of curves that constitute the mesh surface;

[0102] like Figure 5 As described above, the number of rows l and columns n of the tooth surface in the C and B point cloud data are identified, and then the data of the C matrix and B matrix are converted into spatial points respectively; B spline curves are fitted along the tooth width direction and tooth height direction respectively, and fitted into a crisscross tooth surface mesh, which is fitted into surface C1 and surface B1 respectively.

[0103] L22'. Complete the top and bottom curves of the gear teeth;

[0104] L23'. Construct the concave and convex tooth surfaces of a single tooth groove using mesh surfaces; construct the top and bottom surfaces of the tooth using the top and bottom curves of the tooth; select the edge lines of the two end faces of the tooth to construct bounded planes, forming the complete surface of a single tooth;

[0105] The tooth tip boundary curve CK1 is circularly arrayed around the z-axis to obtain curves BCK2, BCK3, BCK4, and BCK5, with BK1 serving as the construction line for the tooth groove top surface. The tooth groove top curves KG1 and KG2 on both end faces are then completed as boundary lines. A mesh is constructed using curves CK1, BCK2, BCK3, BCK4, BCK5, BK1, KG1, and KG2 to fit the tooth tip surface D1. The tooth root boundary curve CK... l The curves BCK6, BCK7, BCK8, and BCK9 are obtained by circumferential arraying around the z-axis, and then compared with BK... l As the construction line of the bottom surface of the gear tooth;

[0106] Complete the bottom curves KG3 and KG4 of the tooth grooves on both ends as boundary lines, and take curve CK. l BCK6, BCK7, BCK8, BCK9, BK l KG3 and KG4 are used to construct a mesh to fit the bottom surface E1 of the gear teeth; as shown Figure 9 As shown.

[0107] L24'. Stitch together the complete curved surfaces of a single gear tooth to form a solid gear tooth;

[0108] Construct a bounded plane F1 as the front end face of the gear tooth, using curves AG1, BG1, KG1, and KG3 as boundaries; n BG nKG2 and KG4 form a bounded plane G1 as the rear end face of the gear tooth; the curved surfaces A1, B1, D1, E1, F1, and G1 are stitched together to obtain a tooth groove solid I1; as shown... Figure 10 As shown.

[0109] L30'. By circumferentially arraying the gear teeth on the gear blank according to the number of teeth, and adding them to the gear blank solid through Boolean operations, a three-dimensional digital model of the cylindrical gear with locally conjugate circular arc tooth lines can be obtained, such as... Figure 11 As shown.

[0110] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for parametric design of a local conjugate circular-arc tooth trace cylindrical gear, characterized in that, The method comprises the following steps: S10. Set gear parameters R T , m, z, B; wherein R T is the nominal radius of the cutter, m is the gear modulus, z is the number of teeth, and B is the tooth width. S20. Calculate the tooth width parameter b, b=B / 2, and divide [-b, b] into K parts; S30. Use the concave tooth surface equation, let t=1, or use the convex tooth surface equation, let t=2; let k=1; S40. Let z di =z dri =b k , (1,K); S50. respectively calculating the tool rotation angle at the addendum , denoted as , the tool rotation angle at the intersection of the working tooth profile and the dedendum transition curve, denoted as , and the tool rotation angle at the dedendum, denoted as ; S60. Between and , take several discrete points, according to the tooth profile equation of the working tooth surface, solve the coordinates (x di , y di ) of the working tooth profile curve; between and , take several discrete points, according to the tooth profile equation of the transition tooth surface, solve the coordinates (x dri , y dri ) of the root transition curve; In step S60, the discrete points are calculated according to the following formula: ， wherein is the th is the maximum value, p is the discrete number of a is a constant greater than 0; S70. Store the concave tooth surface equation found (x di , y di , z di ) and (x dri , y dri , z dri ) in matrix A; store the convex tooth surface equation found (x di , y di , z di ) and (x dri , y dri , z dri ) in matrix B; S80. Calculate the intersection of the two tooth surfaces by solving the equations of the two tooth surfaces simultaneously; S80. Let k = k + 1, if b k < b, return to step S40; otherwise, return to step S30, change the value of t; S90. When the concave tooth surface and the convex tooth surface equations are calculated, output the matrix A and B; The tooth profile equation of the working tooth surface is: ； The tooth profile equation of the transition tooth surface is: ； In the formula, " "and" " indicates concave and convex tooth surfaces, with the upper symbol for concave tooth surfaces and the lower symbol for convex tooth surfaces; x di y di , z di For tool rotation angle The working tooth surface at x d y d , z d Coordinates on; R i Let be the pitch circle radius of the gear. The pressure angle of the gear; x dri y dri , z dri For tool rotation angle The transition tooth surface at x d y d , z d The coordinates on the x-axis (x-axis) d y d , z d () represents the moving coordinate system of the gear; r is the radius of the tool tip fillet. This is the tooth height coefficient.

2. The method according to claim 1, wherein, Step S90 further includes reversing the coordinate values ​​of matrix A around z. d Rotate the axis by 2π / z radians, store the resulting coordinates in matrix C, and output matrix C.

3. The method according to claim 2, wherein, In step S50, the tool rotation angles at the tooth tip, the intersection of the working tooth profile and the tooth root transition curve, and the tool rotation angles at the tooth root are calculated according to the following formula respectively: The tool rotation angle at the tooth tip is calculated by the following formula : The tool rotation angle at the profile root is calculated by the following formula : The tool rotation angle at the intersection of the working profile and the transition curve is calculated by the following formula : ； wherein is the addendum.

4. The method according to claim 2, wherein, The value of a is estimated according to the following formula: The value of a corresponding to the working tooth profile curve is calculated by the following formula: The value of a corresponding to the tooth root transition curve is calculated by the following formula: 。 5. The method according to claim 4, wherein, The value of a when calculating the working tooth profile curve is in the range of [1.05, 1.55], and the value of a when calculating the tooth root transition curve is in the range of [1.8, 4.2].

6. A method of modeling a locally conjugate circular-arc tooth trace cylindrical gear, characterized in that, The method comprises the following steps: L10. Calculate the point cloud data using the design method of any one of claims 1-5, wherein the point cloud data comprises matrices A and B; L20. Read the gear parameters and the point cloud data using a three-dimensional modeling software, and construct a gear slot entity based on the point cloud data; L30. Perform a circular array of the gear slot entity on a blank according to the number of teeth, and perform a Boolean operation subtraction between the gear slot entity and a blank entity, so that a three-dimensional digital model of the local conjugate circular-arc tooth trace cylindrical gear is obtained.

7. The modeling method of a local-conjugate circular-arc tooth trace cylindrical gear according to claim 6, characterized in that, In step L20, constructing the gear slot entity based on the point cloud data comprises the following steps: L21. Fit the point cloud data into B-spline curves along the tooth width direction and the tooth height direction respectively to obtain a curve group constituting a grid surface; L22. Complete the top curve and the bottom curve of the gear slot; L23. Construct the concave and convex tooth surfaces of a single gear slot using the grid surface, construct the top surface and the bottom surface of the gear slot using the top curve and the bottom curve of the gear slot, and select the edge lines of the two end surfaces of the gear slot to construct a bounded plane to form a complete surface of the single gear slot; L24. Stitch the complete surface of the single gear slot to form a gear slot entity.

8. A method of modeling a locally conjugate circular-arc tooth trace cylindrical gear, characterized in that, The method comprises the following steps: L10'. Calculate the point cloud data using the design method of any one of claims 2-5, wherein the point cloud data comprises matrices C and B; L20'. Read the gear parameters and the point cloud data using a three-dimensional modeling software, and construct a gear tooth entity based on the point cloud data; L30'. Perform a circular array of the gear tooth entity on a blank according to the number of teeth, and perform a Boolean operation addition between the gear tooth entity and a blank entity, so that a three-dimensional digital model of the local conjugate circular-arc tooth trace cylindrical gear is obtained.

9. The modeling method of a local-conjugate circular-arc tooth trace cylindrical gear according to claim 8, characterized in that, In step S20', constructing the gear tooth entity based on the point cloud data comprises the following steps: L21'. Fit the point cloud data into B-spline curves along the tooth width direction and the tooth height direction respectively to obtain a curve group constituting a grid surface; L22'. Complete the top curve and the bottom curve of the gear tooth; L23'. Construct the concave and convex tooth surfaces of a single gear slot using the grid surface, construct the top surface and the bottom surface of the gear slot using the top curve and the bottom curve of the gear slot, and select the edge lines of the two end surfaces of the gear slot to construct a bounded plane to form a complete surface of the single gear slot; L24. Stitching the complete curved surface of a single tooth to form a tooth solid.

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