A method for three-section modification of a gear

Abstract

This invention relates to the field of gear machining technology and discloses a method for three-section shaping of gears. First, a mathematical model of the gear tooth surface and the profile of the grinding wheel is established. Then, the gear is shaped along the tooth width and tooth direction. The shaped gear cross-section is obtained, and the shaped tooth surface is calculated based on the cross-section of a portion of the gear. The gear cross-section is then meshed, and the shaping amount at each mesh coordinate is obtained. Based on the shaping amount and the motion equation of the servo axis, the grinding wheel is controlled to shape the tooth surface according to the shaped tooth surface. The beneficial effect is that, in this invention, the gear cross-section after tooth width and tooth direction shaping is divided into a nine-grid system. By detecting the gear cross-sections in regions 1, 3, and 7, the tooth shape and tooth direction of each region's dividing line are controlled to be consistent to obtain the shaped tooth surface. Then, the shaping amount at each mesh coordinate is calculated based on the shaped tooth surface, so that a specified shaping amount is obtained at a specified location, allowing the tooth surface to be reverse-torsed as required.

Description

Technical Field

[0001] This invention relates to the field of gear machining technology, and in particular to a method for modifying the three-section profile of a gear. Background Technology

[0002] Tooth surface modification, also known as gear tooth profile modification, refers to the deliberate, minute alteration of the gear tooth surface to deviate from the theoretical tooth surface. Depending on the modification location, gear tooth modification can be divided into profile modification and tooth direction modification. Profile modification includes edge trimming, root trimming, and root hollowing. Its function is to reduce engagement and disengagement impacts, alleviate dynamic loads, eliminate positive base pitch errors during meshing, improve load distribution, and reduce impact vibrations. Tooth direction modification includes tooth tip thinning, helix angle adjustment, drum-shaped trimming, and curved surface trimming. Its function is to ensure uniform load distribution along the tooth width direction under load; that is, the modification curve must compensate for meshing tooth direction errors, avoid edge point contact, and reduce edge contact.

[0003] Profile grinding is a grinding process for gears that achieves higher standards. When the profile of the gear end section is twisted along the tooth direction, forming a distorted surface, the actual machined modified tooth surface will not coincide with the theoretical modified tooth surface at each end section profile position. The offset error shows opposite trends in the upper and lower end faces of the gear and in the tooth height direction. Tooth surface distortion is mainly caused by the uneven removal of tooth surface by the forming grinding wheel during the machining process. The existence of tooth surface distortion not only seriously affects the transmission accuracy of the gear but also generates significant noise. Simply improving the accuracy of the machine tool cannot effectively eliminate tooth surface distortion; optimization of the modification methods, including design and manufacturing processes, is also necessary.

[0004] However, in the existing technology, there is a problem that it is impossible to perform effective and high-precision gear three-section modification. Summary of the Invention

[0005] To overcome the problems existing in the prior art, this invention proposes a method for modifying the three-section shape of gears, the detailed technical solution of which is as follows:

[0006] A method for three-section shaping of gears, implemented by a multi-axis linkage CNC system controlling a grinding wheel and a gear, wherein the multi-axis linkage CNC system includes at least three linear coordinate axes (X, Y, Z) and two rotary coordinate axes (A, C); the grinding wheel performs helical motion along the gear axis and parabolic motion along the gear radial direction; the method includes the following steps:

[0007] S01, Establish a mathematical model of the gear tooth surface and the sand profile; Based on the intersection of the normal on the helical gear tooth surface and the grinding wheel axis, find all the contact points between the helical gear and the grinding wheel, and then obtain the contact line. Then, rotate the contact line around the grinding wheel axis to obtain the sand profile.

[0008] S02, the gear tooth surface is meshed, and the gear is indexed while rotating along the C-axis. The grinding wheel is fed along the radial X-axis of the gear. The geometric center of the grinding wheel is kept aligned with the center of the gear tooth groove. The C-axis, X-axis, and Z-axis are linked to detect the tooth direction and tooth profile of the gear, obtaining the tooth surface area. The grinding profile is obtained based on the tooth surface area and the mathematical model in S01. The motion equations of the workpiece rotation angle and the servo axis are established using the grinding profile and gear parameters. The unit normal vector of the tooth surface is calculated using the grinding profile, and the tooth surface equation and the unit normal vector equation are solved to obtain the meshing multi-point topological surface of the tooth surface and the grinding profile. By fine-tuning the motion parameters of the servo axis, the tooth width and tooth direction are modified according to the meshing multi-point topological surface.

[0009] S03: Obtain the gear tooth surface after processing in S02, and divide the gear cross-section into 9 regions of equal area. Region 1 is designated as the central region, Region 2 as the upper left corner region, and Regions 3 through 9 are sequentially marked by rotating Region 2 clockwise around Region 1. The grid lines separating Regions 1 through 9 are labeled a, b, c, d, e, f, g, and h. Obtain the gear tooth surfaces corresponding to Regions 1, 3, and 7, and obtain the corresponding sand profile based on the mathematical model of the gear tooth surface and the sand profile. Calculate the entire gear tooth surface corresponding to Regions 1, 3, and 7 based on the sand profile, and denote it as the virtual tooth surface. Place this virtual tooth surface in the tool coordinate system Y... w The axis is translated by a preset value in the direction of the gear to obtain the modified tooth surface in the workpiece coordinate system. The gear tooth surface after S02 machining and the modified tooth surface are compared to obtain the modification amount of each grid coordinate in the workpiece coordinate system. Based on the modification amount and the motion equation of the servo axis, the grinding wheel is controlled to modify the tooth surface.

[0010] Furthermore, in S01, the grinding wheel and the gear are absolutely offset, and an O-shaped gap is established on the grinding wheel. c X c Y c Z c The tool coordinate system is established on the gear, and the OX coordinate system is created. w Y w Z w Establish the workpiece coordinate system and the transformation formula between the tool coordinate system and the workpiece coordinate system; based on the intersection of the normals of the grinding wheel axis and the gear helical tooth surface, solve to obtain the contact line equation, and then rotate the contact line around the grinding wheel axis to obtain the grinding profile.

[0011] Furthermore, in S01, O c X c Y c Z c To OX w Y w Z w The transformation relationship formula is as follows

[0012] x=α-X;y=-Ycos∑-Zsin∑, formula (1);

[0013] z = -Ysin∑ - Zcos∑.

[0014] In equation (1): x, y and z are the coordinates of a point in the workpiece coordinate system; X, Y and Z are the coordinates of a point in the tool coordinate system;

[0015] Establishing the workpiece coordinate system OX w Y w Z w Then, assume X w OY w There is a smooth curve r(t) on the coordinate plane.

[0016] r(t) = {x(t), y(t)}, t ∈ [t, t 2 Equation (2);

[0017] In the formula, x(t) and y(t) represent the coordinate components of the curve on the x-axis and y-axis of the workpiece coordinate system, respectively;

[0018] Let the curve be along Z w If a helical motion with a lead of h is performed in the positive direction of the axis, a helical surface R(θ,t) can be obtained in the workpiece coordinate system.

[0019] R(θ,t)={x(t)cosθ-y(t)sinθ,x(t)sinθ+y(t)cosθ,pθ}, equation (3)

[0020] In the formula: θ is the curve rotation angle parameter, and p is the helical parameter; the normal n at any point on the helical surface can be obtained according to the formula;

[0021] Establish the forming tool coordinate system O c X c Y c Z c If a radial vector R is drawn from the origin of the tool coordinate system to a point on the helical surface, then the contact line condition on the helical surface can be expressed as follows:

[0022] (k'×R)·n=0, Equation (4);

[0023] In the formula, k' is the tool axis;

[0024] Considering the relationship between the workpiece coordinate system and the forming tool coordinate system, the simplified formula is...

[0025] -(ycos∑+xsin∑)n1+n2(xa)cos∑+n3(ax)sin∑=0, equation (5)

[0026] In the formula, n1, n2, and n3 are the coordinate components of the normal to any point on the helical surface, respectively;

[0027] By combining equations (3) and (5), we can obtain the contact line equation of the helical surface in the workpiece coordinate system.

[0028] -(ycos ∑+xsin∑)[p(x'(t)sinθ+y'(t)cosθ)]-

[0029] [p(x'(t)cosθ-y'(t)sinθ)](xa)cos ∑+

[0030] {[p(x'(t)sinθ+y'(t)cosθ)]y(t)-[p(x'(t)

[0031] cos0-y'(t)sinθ)]}x(t) / p(ax)sinx=0, equation (6);

[0032] In the formula, θ and t satisfy equation (3);

[0033] By rotating the contact line around the forming tool axis, the tool rotation surface can be obtained; the contact line can be transformed to the tool coordinate system O using coordinate transformation formula (1). c X c Y c Z c Then the axial section of the forming tool's rotating surface can be expressed as:

[0034] ,

[0035] Z=Z, equation (7).

[0036] Furthermore, in S02, the motion equation of the workpiece rotation angle and the servo axis is as follows:

[0037]

[0038] In the formula, φ1 is the workpiece rotation angle, γ m d is a constant; f and d w δ, u1, and σ1 are the root circle diameter and pitch circle diameter of the gear, respectively; d1 is the grinding wheel diameter; β is the gear helix angle; δ, u1, and σ1 are gear parameters.

[0039] Furthermore, in S02, based on the sand profile equation, the modified tooth surface equation and the tooth surface unit normal vector equation are obtained through the positional relationship and meshing equation between the grinding wheel and the gear, thereby obtaining the meshing multi-point topological surface of the tooth surface and the sand profile.

[0040] According to any of the above embodiments, the present invention has at least the following beneficial effects: In the present invention, the gear cross-section after tooth width and tooth direction modification is divided into a nine-grid system. By detecting the gear cross-sections in regions 1, 3, and 7, the tooth shape and tooth direction of each region's dividing line are controlled to be consistent to obtain the modified tooth surface. Then, the modification amount of each grid coordinate is calculated based on the modified tooth surface.

[0041] This invention enables a specified amount of shaping at a designated location, allowing the tooth surface to be reversed as required. The method for three-section shaping of gears according to this invention can be used in five-axis CNC gear grinding machines, effectively reducing tooth surface distortion, improving gear transmission accuracy, and reducing noise. Attached Figure Description

[0042] Figure 1 This is a coordinate diagram of the workpiece coordinate system and the grinding wheel coordinate system of the present invention.

[0043] Figure 2 This is a schematic diagram of the tooth profile modification of the present invention.

[0044] Figure 3 This is a schematic diagram of the gear tooth surface grid distribution of the present invention.

[0045] Figure 4 This is a schematic diagram of the tooth surface mesh distribution of the three-section modification of the present invention. Detailed Implementation

[0046] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0047] A method for modifying the three-section profile of a gear specifically includes the following steps:

[0048] 1. Establish a mathematical model of the workpiece tooth surface and the sand profile. Based on the intersection of the normal on the helical gear tooth surface and the grinding wheel axis, find all the contact points between the helical gear and the grinding wheel, and then obtain the contact line. Then, rotate the contact line around the grinding wheel axis to obtain the sand profile.

[0049] When the tool surface is a rotating surface, there exists a contact line between the workpiece helical tooth surface and the tool surface conjugate to this surface. The normal to a point on this contact line intersects the tool axis. Based on this principle, by using the condition that the normal on the helical tooth surface of the helical gear intersects the grinding wheel axis, all contact points between the helical gear and the grinding wheel can be determined, thus obtaining the contact line. Then, by rotating this contact line around the grinding wheel axis, the grinding profile can be obtained.

[0050] The grinding wheel used as the cutting tool is absolutely offset from the gear to be machined to ensure the stability of continuous machining of the helical surface. The positional relationship between the forming tool and the workpiece is as follows: Figure 1 As shown, in Figure 1 Chinese: OX w Y w Z w For the workpiece coordinate system, O c X c Y c Z c Let X be the tool coordinate system; ∑ be the angle between the tool axis and the workpiece axis; α be the center moment between the forming tool and the workpiece; X be the workpiece coordinate system. w Shaft and forming tool X c The axes coincide, but the directions are opposite.

[0051] Based on the coordinate position relationship diagram, we obtain the coordinates from O. c X c Y c Z c To OX w Y w Z w The transformation relationship formula is: x = α - X; y = -Ycos∑ - Zsin∑, Equation (1);

[0052] z = -Ysin∑ - Zcos∑.

[0053] In equation (1): x, y and z are the coordinates of points in the workpiece coordinate system; X, Y and Z are the coordinates of points in the tool coordinate system.

[0054] Establishing the workpiece coordinate system OX w Y w Z w Then, assume X w OY w There is a smooth curve r(t) on the coordinate plane.

[0055] r(t)={x(t),y(t)},t∈[t,t2] , formula (2)

[0056] In the formula, x(t) and y(t) represent the coordinate components of the curve on the x-axis and y-axis of the workpiece coordinate system, respectively. Assuming the curve undergoes helical motion with a lead of h along the positive Zw axis, a helical surface R(θ,t) can be obtained in the workpiece coordinate system.

[0057] R(θ,t)={x(t)cosθ-y(t)sinθ,x(t)sinθ+y(t)cosθ,pθ}, equation (3)

[0058] In the formula: θ is the curve rotation angle parameter, and p is the helical parameter. Here, only the right-hand helical case is considered. The normal n at any point on the helical surface can be obtained using the formula.

[0059] Establish the forming tool coordinate system O c X c Y c Z c If a radial vector R is drawn from the origin of the tool coordinate system to a point on the helical surface, then the contact line condition on the helical surface can be expressed as follows:

[0060] (k'×R)·n=0, Equation (4);

[0061] In the formula, k' is the tool axis.

[0062] Considering the relationship between the workpiece coordinate system and the forming tool coordinate system, the simplified formula is...

[0063] -(ycos∑+xsin∑)n1+n2(xa)cos∑+n3(ax)sin∑=0, equation (5);

[0064] In the formula, n1, n2, and n3 are the coordinate components of the normal to any point on the helical surface, respectively.

[0065] By combining equations (3) and (5), we can obtain the contact line equation of the helical surface in the workpiece coordinate system.

[0066] -(ycos ∑+xsin∑)[p(x'(t)sinθ+y'(t)cosθ)]-

[0067] [p(x'(t)cosθ-y'(t)sinθ)](xa)cos ∑+

[0068] {[p(x'(t)sinθ+y'(t)cosθ)]y(t)-[p(x'(t)

[0069] cos0-y'(t)sinθ)]}x(t) / p(ax)sinx=0, equation (6);

[0070] In the formula, θ and t satisfy equation (3).

[0071] By rotating the contact line around the forming tool axis, the tool rotation surface can be obtained. The contact line can then be transformed to the tool coordinate system O using coordinate transformation formula (1). c X c Y c Z c Then the axial section of the forming tool's rotating surface can be expressed as:

[0072] ,

[0073] Z=Z, equation (7).

[0074] 2. Machining of meshing multi-point topological surfaces.

[0075] When using form grinding, the grinding wheel spirals upwards along the workpiece's axial direction while simultaneously undergoing a parabolic motion along its radial direction, thus shaping the workpiece in the tooth width and tooth direction. Given the gear end section profile, the grinding profile can be solved using coordinate transformations and the contact equation between the grinding wheel and the workpiece. During form grinding, a family of contact lines exists on the workpiece's tooth surface. This family of contact lines is the collection of contact lines between the grinding wheel and the workpiece at every instant, forming the workpiece's tooth surface. The grinding profile can be solved using appropriate coordinate transformations. Conversely, when the grinding profile equation is known, the equation of the shaped tooth surface can also be obtained through the positional relationship and meshing equation between the grinding wheel and the workpiece.

[0076] like Figure 2 As shown, the workpiece gear is indexed while rotating along the C-axis, and the grinding wheel is fed along the radial X-axis of the workpiece. The C, X, and Z axes are linked to flexibly control the positional relationship between the gear and the grinding wheel in space, thus completing the tooth profile machining. During grinding wheel dressing, the Y and Z axes are linked to maintain a certain accuracy in the grinding profile. During machine measurement, the CNC system controls the linkage of the C, X, and Z axes to detect the tooth direction and profile, and obtain the tooth surface area. Based on the mathematical model and the tooth surface area, the grinding profile is obtained, denoted as r(u,θ), and the grinding profile is transformed from the workpiece coordinate system to the tool coordinate system. The corresponding equation is:

[0077]

[0078]

[0079]

[0080]

[0081] During form grinding, the geometric center of the control grinding wheel and the center of the gear tooth groove are always aligned. Therefore, the C-axis rotation angle is opposite to the workpiece rotation angle, and similarly, the A-axis rotation angle is opposite to the workpiece helix angle. The tooth surface equation can be obtained from the sand profile through coordinate transformation, and similarly, the sand profile can be obtained from the tooth surface equation through coordinate transformation. Since the corresponding angle and position parameters in the above equations can be obtained, the motion equations of each servo axis during CNC form grinding of the workpiece can be obtained, specifically:

[0082]

[0083] In the formula, φ1 is the workpiece rotation angle, and γ m It is a constant;

[0084] d f and d w δ, u1, and σ1 are the root circle diameter and pitch circle diameter of the gear, respectively; d1 is the grinding wheel diameter; β is the gear helix angle; δ, u1, and σ1 are gear parameters.

[0085] After coordinate transformation, the modified tooth surface vector (u, θ, q) is obtained from the sand profile, where u and θ are tooth surface parameters, and q is a motion parameter. The unit normal vector of the tooth surface in the coordinate system allows for the calculation of the tooth surface equation and the unit normal vector equation, thus yielding the meshing multi-point topological surface of the tooth surface and the sand profile. By fine-tuning the motion parameters of the servo axis, the machining of the meshing multi-point topological surface is carried out.

[0086] 3. Three-section shaping.

[0087] Three-section profile modification of gears is an internationally recognized and demanding modification method, primarily requiring that the tooth profile be consistent across multiple points on the tooth surface after profile modification. For example... Figure 4 As shown, the solid line represents the tooth profile before modification, while the dashed line represents the tooth profile obtained after machining according to the multi-point meshing topology. It can be seen that tooth surface distortion causes significant damage to the upper and lower end faces, as well as the left and right end faces. The key to three-section modification lies in the grid coordinate allocation of the tooth surface modification amount, ensuring that a specified modification amount is obtained at a specified location, thus allowing the tooth surface to reverse the distortion as required. Therefore, establishing a three-section mathematical model algorithm and enabling the CNC grinding system to acquire vector point trajectory data are crucial to the research and development.

[0088] In this invention, when using the form grinding method to grind the workpiece, the grinding wheel spirals upward along the gear axis while simultaneously undergoing a parabolic motion along the gear radial direction, completing the gear's profile modification in the tooth width and tooth direction. Then, the gear's cross-sectional profile is inspected to obtain the profile, and the grinding profile is calculated based on the cross-sectional profile and the contact line equation. During form grinding, a family of contact lines exists on the workpiece tooth surface. This family of contact lines is the collection of contact lines between the grinding wheel and the workpiece at every instant, thus forming the workpiece tooth surface. After appropriate coordinate transformation, the grinding profile can be solved. Conversely, when the grinding profile equation is known, the modified tooth surface equation can also be obtained through the positional relationship between the grinding wheel and the workpiece and the meshing equation.

[0089] The coordinate system for the relative positional relationship between the grinding wheel and the workpiece is as follows: Figure 2As shown, the workpiece coordinate system and the tool coordinate system are fixed to the gear and the grinding wheel, respectively. φ is the grinding wheel mounting angle, which can be obtained from the gear's helix angle β. α is the center distance between the grinding wheel and the workpiece, and its value is related to the grinding wheel radius and workpiece parameters. In this invention, the tooth surface is first divided into a grid. More specifically, the tooth surface is divided into nine equal-area nine-square grid regions. The central region is designated as region 1, and the upper left corner region is designated as region 2. The upper left corner region is rotated clockwise and its region number is incremented by 1, sequentially labeling regions 2 through 9. The grid lines separating regions 1 through 9 are denoted as a, b, c, d, e, f, g, and h. In regions 1, 3, and 7, only the tooth profile is modified because there is no tooth profile modification. The tooth surface vector (u, θ, q) in the workpiece coordinate system for these regions is known, where q is a motion parameter. Since there is no tooth profile modification, q is 0. The tooth surfaces of regions 1, 3, and 7 are inspected. The corresponding tooth surface vectors for regions 1, 3, and 7 in the workpiece coordinate system are obtained. Virtual tooth surfaces are established with tooth directions and profiles consistent for a, b, c, and d, and tooth directions and profiles consistent for e, f, g, and h. These virtual tooth surfaces are then mapped to the tool coordinate system Y. w The axis is translated towards the gear direction by a set distance to obtain the modified tooth surface in the workpiece coordinate system. The actual tooth surface of the gear is detected, and the actual tooth surface and the modified tooth surface are compared to obtain the modification amount of each grid coordinate in the workpiece coordinate system. Based on the modification amount and the motion equation of each servo axis, the grinding wheel is controlled to modify the tooth surface.

[0090] In this invention, the core of the three-section shaping is to calculate the sand profile using coordinate conversion through the tooth surfaces of regions 1, 3 and 7, then calculate the virtual tooth surface based on the sand profile, and then translate the virtual tooth surface in the direction of the gear to obtain the shaped tooth surface. Then, the grinding wheel is controlled to shape accordingly. The purpose is to ensure that the tooth direction and tooth shape of the curves at a, b, c, d, e, f, g and h are consistent.

[0091] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for three-section shaping of gears, implemented by a multi-axis linkage CNC system controlling a grinding wheel and a gear, characterized in that, The multi-axis linkage CNC system includes at least three linear coordinate axes (X, Y, Z) and two rotary coordinate axes (A, C); the grinding wheel moves helically along the gear axis and parabolically along the gear radial direction. Includes the following steps: S01, Establish a mathematical model of the gear tooth surface and the sand profile; Based on the intersection of the normal on the gear helical tooth surface and the grinding wheel axis, find all the contact points between the gear and the grinding wheel, and then obtain the contact line. Then, rotate the contact line around the grinding wheel axis to obtain the sand profile. S02, the gear tooth surface is meshed, and the gear is indexed while rotating along the C-axis. The grinding wheel is fed along the radial X-axis of the gear. The geometric center of the grinding wheel is kept aligned with the center of the gear tooth groove. The C-axis, X-axis, and Z-axis are linked to detect the tooth direction and tooth profile of the gear, obtaining the tooth surface area. The grinding profile is obtained based on the tooth surface area and the mathematical model in S01. The motion equations of the workpiece rotation angle and the servo axis are established using the grinding profile and gear parameters. The unit normal vector of the tooth surface is calculated using the grinding profile, and the tooth surface equation and the unit normal vector equation are solved to obtain the meshing multi-point topological surface of the tooth surface and the grinding profile. By fine-tuning the motion parameters of the servo axis, the tooth width and tooth direction are modified according to the meshing multi-point topological surface. S03: Obtain the gear tooth surface after processing in S02, and divide the gear cross-section into 9 regions of equal area. Region 1 is designated as the central region, Region 2 as the upper left corner region, and Regions 3 through 9 are sequentially marked by rotating Region 2 clockwise around Region 1. The grid lines separating Regions 1 through 9 are labeled a, b, c, d, e, f, g, and h. Obtain the gear tooth surfaces corresponding to Regions 1, 3, and 7, and obtain the corresponding sand profile based on the mathematical model of the gear tooth surface and the sand profile. Calculate the entire gear tooth surface corresponding to Regions 1, 3, and 7 based on the sand profile, and denote it as the virtual tooth surface. Place this virtual tooth surface in the tool coordinate system Y... w The axis is translated by a preset value in the direction of the gear to obtain the modified tooth surface in the workpiece coordinate system. The gear tooth surface after S02 machining and the modified tooth surface are compared to obtain the modification amount of each grid coordinate in the workpiece coordinate system. Based on the modification amount and the motion equation of the servo axis, the grinding wheel is controlled to modify the tooth surface.

2. The method for modifying the three sections of a gear according to claim 1, characterized in that, In S01, the grinding wheel and the gear are absolutely offset, and an O is established on the grinding wheel. c X c Y c Z c The tool coordinate system is established on the gear, and the OX coordinate system is created. w Y w Z w Establish the workpiece coordinate system and the transformation formula between the tool coordinate system and the workpiece coordinate system; based on the intersection of the normals of the grinding wheel axis and the gear helical tooth surface, solve to obtain the contact line equation, and then rotate the contact line around the grinding wheel axis to obtain the grinding profile.

3. The method for modifying the three sections of a gear according to claim 1, characterized in that, In S02, the motion equation of the workpiece rotation angle and the servo axis is: In the formula, φ1 is the workpiece rotation angle, γ m d is a constant; f and d w δ, u1, and σ1 are the root circle diameter and pitch circle diameter of the gear, respectively; d1 is the grinding wheel diameter; β is the gear helix angle; δ, u1, and σ1 are gear parameters.

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