Method for grinding conical spiral groove through formed grinding wheel based on iterative approximation

Abstract

The invention discloses a method for grinding a conical spiral groove through a formed grinding wheel based on iterative approximation, and the method specifically comprises the steps: firstly, constructing a parameterized geometric model of the formed grinding wheel and the conical spiral groove; then, a formed grinding wheel grinding kinematics model is constructed and used for guiding the grinding process, and the grinding mode of the conical spiral groove is achieved; secondly, on the basis of the principle that a grinding wheel grinding curve is tangent to a section contour curve, the rotation contour of a formed grinding wheel is calculated in an iterative approximation mode; and finally, actual machining verification is carried out on the grinding wheel contour obtained through calculation. According to the method, the accuracy of the section contour of the spiral groove is comprehensively guaranteed, and the correctness and effectiveness of related theoretical models of the method are proved.

Description

Technical Field

[0001] The present invention belongs to the field of grinding and manufacturing of numerical control tools, and particularly relates to a method for grinding a conical spiral groove with a formed grinding wheel based on iterative approximation. Background Art

[0002] The spiral groove is a key component of a numerical control tool, which determines the cutting performance, stiffness and strength of the tool. The spiral groove with a linearly varying core thickness along the axial direction of the tool is called a conical spiral groove. With the continuous development of numerical control machining technology, due to the excellent performance of the conical spiral groove in chip evacuation ability, high stiffness and vibration resistance, the manufacturing industry increasingly tends to use numerical control tools with conical spiral grooves. However, the traditional standard grinding wheel grinding process can only ensure the basic geometric parameters of the conical spiral groove, but cannot ensure the design accuracy of the entire cross-sectional profile of the spiral groove; while the formed grinding wheel grinding process can only grind cylindrical spiral grooves, and the application range of the process grinding method is limited. Therefore, the present invention combines the formed grinding wheel grinding and the conical spiral groove to realize the grinding process method.

[0003] The research on the grinding process of formed grinding wheels is mainly based on the contact line theory and the conjugate theory of the grinding wheel and the groove surface, aiming to achieve the accurate calculation of the rotary profile of the formed grinding wheel under the specified grinding trajectory conditions of the grinding wheel. Many scholars have carried out in-depth research in this field and achieved remarkable results. Wasif et al. [1] constructed a simplified profile of the formed grinding wheel through line segments and arcs, skillfully combined the contact theory with the differential evolution algorithm, and accurately calculated and optimized the final profile of the grinding wheel, providing an effective numerical method for the design of formed grinding wheels. Hsieh et al. [2] combined coordinate transformation with the conjugate surface theory, successfully established the grinding kinematic model of the spiral groove, and determined the grinding wheel profile based on the conjugate contact theory, laying a foundation for the theoretical research of the spiral groove grinding process. Uhlmann et al. [3] focused on the complex contact conditions between the grinding wheel and the tool, and significantly improved the performance of the formed grinding wheel through optimized design. Jiang Lei et al. [4] calculated the grinding wheel profile based on the contact line theory, and further effectively improved the wear resistance of the grinding wheel and extended the service life of the formed grinding wheel by adjusting the posture of the grinding wheel. Tang Qian et al. [5] proposed a design method for the screw forming tool based on discrete points, namely the Form Position Geometry Method (FPGM), based on the contact line theory, providing a new idea for the tool design of complex workpieces. Shen Zhihuang et al. [6] deduced the contact line equation based on the conjugate theory, thus accurately obtaining the grinding wheel profile and successfully solving the problem of the smoothness of the formed grinding wheel profile. However, the contact theory often requires complex analytical solutions in practical applications, bringing certain difficulties to research and engineering practice. To overcome this problem, the envelope theory came into being. You Minglin et al. [7] proposed the pixel matrix method based on the spiral motion envelope method and mathematical morphology. By calculating the point cloud of the radial section of the grinding wheel and extracting the boundary of the point cloud, the grinding wheel profile was efficiently obtained, providing a simple and effective calculation method for the grinding wheel design. Shen et al. [8] proposed a grinding wheel design method based on the envelope theory, which can machine different small spiral grooves on the basis of the existing front rake face of the spiral groove, expanding the application range of the formed grinding wheel grinding. Wu Changzuo et al. [9] successfully obtained the corresponding profile of the formed grinding wheel by solving the inverse envelope problem according to the given spiral groove profile, further enriching the theoretical system of the formed grinding wheel design. In addition, Yang et al.

[10] solved the formed grinding wheel of the screw compressor rotor based on the graphic edge detection method of the alpha shape algorithm, providing a new technical approach for the design of the formed grinding wheel of complex workpieces. Wu Yuren et al.

[11] proposed a Radial Ray Shooting (RRS) method to obtain the formed grinding wheel profile corresponding to threaded cylindrical workpieces such as gears and screws. The research of the above researchers is all aimed at grinding cylindrical spiral grooves and has made certain progress, while the research on conical spiral grooves has not been publicly reported.

[0004] References:

[0005] [1] Wasif M, Iqbal SA, Ahmed A, et al. Optimization of simplified grinding wheel geometry for the accurate generation of end-mill cutters using the five-axis CNC grinding process[J]. The International Journal of Advanced Manufacturing Technology, 2019, 105(10): 4325 - 4344.

[0006] [2] Hsieh JF. Mathematical model and sensitivity analysis for helical groove machining[J]. International Journal of Machine Tools and Manufacture, 2006, 46(10): 1087 - 1096.

[0007] [3] Uhlmann E, Gülzow B, Muthulingam A. Optimising the grinding wheel design for flute grinding processes utilising numerical analysis of the complex contact conditions[J]. Journal of Machine Engineering, 2020.

[0008] [4] Jiang L, Yang Z, Li Y, et al. An optimized calculation method of the grinding wheel profile for the helical flute forming grinding[J]. The International Journal of Advanced Manufacturing Technology, 2024, 132(3): 1649 - 1664.

[0009] [5] Tang Q, Zhang Y, Jiang Z, et al. Design Method for Screw Forming Cutter Based on Tooth Profile Composed of Discrete Points[J]. Journal of Mechanical Design, 2015, 137(085002).

[0010] [6] Shen Z H, Lu R S, Zhang Z S, et al. Research on Calculation and Smoothing of Grinding Wheel Profile for Machining Screw Rotor[J]. Advanced Materials Research, 2011, 291 - 294: 2383 - 2387.

[0011] [7] You M, Yao B. Application of Mathematical Morphology in Solving the Profile of Forming Grinding Wheel[J]. Mathematical Problems in Engineering, 2022, 2022: e5735199.

[0012] [8] Shen C, Xiao Y, Xiong L. Grinding Wheel Parametric Design for Machining Arbitrary Grooves on the Helical Rake Face of the Tool[J]. International Journal of Precision Engineering and Manufacturing - Green Technology, 2022, 9(4): 997 - 1008.

[0013] [9]Wu C T,Chen C K.Manufacturing models for the design and NC grinding of a revolving tool with a circular arc generatrix[J].Journal of Materials Processing Technology,2001,116(2):114-123.

[0014]

[10] Yang J,Sun F H,Lu Z.Solving the screw compressor rotor-forming grinding wheel using the edge detection method based on the graphic method[J].Proceedings of the Institution of Mechanical Engineers,Part E:Journal of Process Mechanical Engineering,2019,233(5):967-979.

[0015]

[11] Wu Y R,Fong Z H,Zhang Z X.Simulation of a cylindrical form grinding process by the radial-ray shooting(RRS)method[J].Mechanism and Machine Theory,2010,45(2):261-272. Summary of the Invention

[0016] The present invention aims to achieve the full-profile accuracy of the spiral groove cross-section while grinding the conical spiral groove. To this end, the present invention provides a method for grinding a conical spiral groove with a formed grinding wheel based on iterative approximation.

[0017] A method for grinding a conical spiral groove with a formed grinding wheel based on iterative approximation according to the present invention includes the following steps:

[0018] Step 1: Establish a geometric model of the conical spiral groove.

[0019] (1) Definition of the axial section

[0020] Define the plane perpendicular to the tool axis as the axial section, and select a tool axial section with a complete intersecting contour of the spiral groove as the reference axial section, denoted as M.

[0021] (2) Definition of the axial section coordinate system ACS

[0022] Coordinate origin O A Located at the center of the reference axial section M, the coordinate axis Z A Coincides with the tool axis direction, and the positive direction points from the small-diameter end to the large-diameter end of the conical end mill. The coordinate plane X A Y A Coincides completely with the reference axial section M.

[0023] (3) Expression of the design surface of the conical spiral groove

[0024] Define the intersection contour of the design surface of the conical spiral groove and the reference axial section M as the axial section reference contour curve, denoted as S; define any point on the reference contour curve S as P S , and its coordinates are expressed in the axial section coordinate system ACS as:

[0025]

[0026] In the formula, u is the function variable of the spiral groove contour curve.

[0027] Similarly, define the other axial section contour curves L of the conical spiral groove m , then the expression of the design surface of the conical spiral groove is:

[0028]

[0029] In the formula, m is the axial position of other axial sections relative to the reference axial section.

[0030] Step 2: Establish the geometric model of the forming grinding wheel.

[0031] (1) Definition of the forming grinding wheel coordinate system GCS

[0032] Coordinate system origin O G Located on the grinding wheel axis, serving as the grinding wheel origin; the coordinate axis Z G Coincides with the grinding wheel axis and is used to describe the rotation axis of the grinding wheel; the coordinate plane X G Y G Is perpendicular to the grinding wheel axis and is used to define the rotary contour of the grinding wheel.

[0033] (2) Expression of the rotary contour of the forming grinding wheel

[0034] Define the distances between the two end faces of the grinding wheel and the coordinate planes X G Y G as h s 、h e , that is, the effective width range of the rotary contour of the forming grinding wheel is [h s ,h e .

[0035] When constructing the rotary profile of the grinding wheel, a method of discretizing the grinding wheel into slices along the axis of the grinding wheel is adopted. By envelope calculation of the actual radii corresponding to each slice within the effective width range of the grinding wheel, the complete rotary profile of the grinding wheel is constructed.

[0036] Define the rotary profile curve of the formed grinding wheel as W, and set point P W as an arbitrary point on the rotary profile of the grinding wheel, with its rotary radius being R W , and the distance from the coordinate plane X G Y G is h, and the angle between the line segment O G P W and the coordinate axis X G is the rotation angle of the grinding wheel Then the coordinates of point P W are expressed in the grinding wheel coordinate system as:

[0037]

[0038] Step 3: Establish the grinding kinematic model of the grinding wheel.

[0039] (1) Definition of the initial grinding pose of the formed grinding wheel

[0040] When the grinding wheel is in the initial grinding pose, its origin O G is located on the X A coordinate axis and the distance from the origin O A is d X , the coordinate axis X G is in the same direction as X A , and the angle between the coordinate axis Z G and Z A is the installation angle α of the grinding wheel.

[0041] The geometric transformation relationship between the grinding wheel coordinate system GCS corresponding to the initial formed grinding wheel grinding pose and the reference axis section coordinate system ACS is represented by the rotation matrix R x and the translation vector T x .

[0042]

[0043] (2) Expression of the grinding motion of the formed grinding wheel

[0044] Define the grinding trajectory of the grinding wheel as a conical spiral motion. The grinding process of the spiral groove is regarded as the tool remaining fixed, while the grinding wheel performs a conical spiral motion on the basis of its initial grinding pose; this motion process is described as the origin O G of the grinding wheel performing a conical spiral motion relative to the reference axis section coordinate system ACS.

[0045] Define the coordinate system GCS rotating around the coordinate axis Z AThe angle of rotation is the grinding wheel spiral motion rotation angle ξ, that is, line segment O A O G In the coordinate plane X A Y A The projection on the coordinate axis X A The angle between the grinding wheel and the tool is assuming that the grinding wheel performs a conical spiral motion with equal lead relative to the tool, and the lead of the spiral motion of the grinding wheel is defined as p h , the taper of the spiral motion is κ G , whose value is related to the cone angle κ of the conical spiral groove core thickness c If they are equal, the spiral motion rotation angle ξ of the grinding wheel can be expressed as:

[0046]

[0047] In the formula, z p O is the origin of the grinding wheel coordinate system G Assume that the grinding wheel moves along the coordinate axis Z T The unit rotation angle of the moving grinding wheel spiral motion is k ξ , then let k ξ =2π / p h

[0048] Set the grinding wheel origin O G The distances moved relative to the coordinate system ACS are Δx, Δy, and Δz, namely:

[0049]

[0050] The position of the grinding motion of the profile grinding wheel is expressed by the rotation matrix and the translation vector as shown below:

[0051]

[0052] Therefore, according to equations (4) and (7), the posture transformation of the profile grinding wheel grinding motion can be divided into the rotation matrix M = (p, n, v) and the translation matrix r, which can be expressed as:

[0053]

[0054] When the grinding wheel is at the initial position, the grinding wheel coordinate system coincides with the axis section coordinate system. At this time, the grinding wheel origin and the axis vector are (0,0,0) respectively. T and (0,0,1) T According to formula (8), establish the grinding wheel origin O G and the grinding wheel axis vector Z G The expression in the coordinate system ACS is:

[0055]

[0056] In the formula, O is the origin of the grinding wheelG The spatial position coordinates, (i G_A , j G_A , k G_A ) is the grinding wheel axis vector.

[0057] Furthermore, by combining equations (5)-(7), it is known that during the grinding motion, the coordinates of any point P on the rotating surface of the formed grinding wheel are expressed in the coordinate system ACS as: W The coordinates of any point P on the rotating surface of the formed grinding wheel are expressed in the coordinate system ACS as:

[0058]

[0059] Step 4: Calculate the rotating contour of the formed grinding wheel.

[0060] (1) Calculation of the grinding curve of the grinding wheel based on the axial section

[0061] The grinding curve of the grinding wheel refers to the trajectory curve formed by the intersection of the rotating circle of any point on the rotating contour of the grinding wheel with the axial section during the grinding motion of the grinding wheel; the rotating surface of the grinding wheel is discretized according to the axial distance h and divided into a series of rotating circles of the grinding wheel contour, and they are made to participate in the grinding motion of the grinding wheel; during this process, these rotating circles of the grinding wheel contour intersect with the axial section, and their intersection trajectories form the grinding curves corresponding to each rotating circle of the grinding wheel contour on the axial section; the set of multiple grinding curves formed by all the rotating circles of the grinding wheel contour together constitutes the complete grinding curve family.

[0062] The process of grinding the conical spiral groove of the cutting tool by the grinding wheel is regarded as a Boolean subtraction operation between two entities; among them, the part of the grinding curve of the grinding wheel within the cross-sectional circle of the cutting tool is exactly the intuitive manifestation of the part of the cutting tool actually removed by the grinding wheel during the grinding process; the grinding curve and its envelope together form the axial section contour of the conical spiral groove after grinding.

[0063] Let z in equation (10) W_A = m, then the axial movement distance Δz of the origin O of the grinding wheel is expressed as: G The axial movement distance Δz of the origin O of the grinding wheel is expressed as:

[0064]

[0065] Substitute equation (11) into x in equation (10) W_A and y W_A , that is, the grinding curve G of the rotating surface of the grinding wheel on any axial section M of the cutting tool is obtained, and its coordinates are expressed in the coordinate system ACS as: m The grinding curve G of the rotating surface of the grinding wheel on any axial section M of the cutting tool is obtained, and its coordinates are expressed in the coordinate system ACS as:

[0066]

[0067] It can be seen from equation (12) that by setting m = 0, the grinding curve of the grinding wheel under the reference axial section M can be obtained, which is an expression about the radius of the grinding wheel.

[0068] (2) Calculation of the Rotary Profile of the Molding Grinding Wheel

[0069] Iterative approximation: The self-parameters h and R of the grinding wheel profile are given in advance W And continuously iterate to calculate the number of intersection points between the grinding curve of the grinding wheel and the sectional profile during the iteration process. Through the judgment of the mutation of the number of intersection points, the approximate tangency between the grinding curve of the grinding wheel and the sectional profile is realized.

[0070] (1) Calculation of the Intersection Points between the Grinding Curve of the Grinding Wheel and the Sectional Profile

[0071] The sectional profile S is uniformly approximated by the profile broken line L composed of a large number of discrete points S , thus simplifying the calculation without affecting the calculation accuracy; during the iteration process of the self-parameters of the grinding wheel profile, when the grinding curve G of the grinding wheel and the profile broken line L S reach the approximate tangency state, the grinding wheel radius R W and the axial position h are determined; in order to accurately judge the approximate tangency state, the tangency calculation is transformed into the calculation of the number of intersection points K between the profile broken line L S and the grinding curve G of the grinding wheel, that is, calculate the number of intersection points K between the grinding curve of the grinding wheel corresponding to the adjacent grinding wheel radii R W_j-1 and R W_j , if there is a mutation in the number of intersection points, that is, K j-1 =0 mutates to K j >0 or K j-1 >0 mutates to K j =0, it indicates that the approximate tangency is achieved. At this time, the grinding wheel radius R j-1 ∈(R j , R W ), by reducing the iteration step R_step of the grinding wheel radius, the grinding wheel radius R with sufficient accuracy can be obtained W_j-1 , R W_j ). W .

[0072] In order to calculate the number of intersection points K, first define and calculate the intersection point P S between the profile broken line L K_j and the grinding curve G of the grinding wheel, where j = 0, 1,..., K; let the coordinates of the two endpoints of each small straight line segment in the profile broken line L S be (x s , y s ) and (x e , y e ), then the equation of the small straight line segment L S_k is expressed as the following formula:

[0073]

[0074] Assume that the axial position of the grinding wheel slice at this time is h and the radius of the grinding wheel is R W_j , and the corresponding grinding curve of the grinding wheel is G j , substitute x G_T and y G_T in Equation (12) into x and y in Equation (12) respectively, and the intersection point P of the small straight line segment and the grinding curve G of the grinding wheel can be solved K The corresponding rotational angle of the grinding wheel value, and substitute back into x G_T and y G_T to obtain the intersection point P K coordinates (x k_j , y k_j ). If x k_j satisfies x k_j ∈[x s , x e , it indicates that the intersection point is on the small straight line segment, and store this intersection point; otherwise, there is no intersection point between this small straight line segment and the grinding curve of the grinding wheel, that is, discard it; by traversing each small straight line segment, the contour broken line L S and all intersection point coordinates and the number K j-1 of the grinding curve of the grinding wheel can be obtained

[0075] Subsequently, continue to iterate the radius R of the grinding wheel W_j , and through the above steps, the corresponding number of intersection points K j is obtained. If there is a sudden change in the number of intersection points between adjacent grinding wheel radii during the iteration process, then the actual radius R W corresponding to the grinding wheel slice h at this time belongs to (R W_j-1 , R W_j ); without loss of generality, let R W =R W_j .

[0076] 2) Definition of the effective starting and ending grinding wheel slice ranges

[0077] The grinding curve of the grinding wheel needs to be constrained inside the opening of the cross-sectional contour. For the integrity of the grinding wheel contour calculation, the initially given axial position h init of the grinding wheel needs to satisfy the condition of h init <<h s . Therefore, the initially given values R W_init and h init are very likely to cause the grinding curve of the grinding wheel to be outside the cross-sectional contour. On the outside, continuously iterate the next grinding wheel with the given grinding wheel slice iteration step h_step until the two intersection points of the grinding curve of a certain grinding wheel and the cross-sectional circle are both inside the opening of the cross-sectional contour, that is, the corresponding direction angles θ G1 , θ G2 satisfy θ G1 , θG2 v(θ S1 , θ S2 ), at this time, the axial position of the grinding wheel is denoted as h s ; Subsequently, start iterating the radius of the grinding wheel until the number of intersection points K changes suddenly, and obtain the actual radius R of the grinding wheel W , and the axial position h of the grinding wheel at this time s and the radius R W are the effective starting grinding wheel {h s , R W}.

[0078] After determining the initial range h of the grinding wheel s , continue to calculate the radius and axial position of the subsequent grinding wheels in the above manner until the intersection points of the grinding curve of a certain grinding wheel and the section circle exceed the opening range of the section contour. At this time, the axial position h e of the previous grinding wheel is the effective termination grinding wheel, and the iteration process terminates, obtaining the rotary contour of the formed grinding wheel

[0079] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0080] The present invention proposes a calculation method for the rotary contour of a formed grinding wheel to achieve precise grinding of a given conical spiral groove section contour. First, a parametric geometric model of the formed grinding wheel and the conical spiral groove is constructed; subsequently, a kinematic model of the formed grinding wheel grinding is constructed to guide the grinding process and realize the grinding method of the conical spiral groove; secondly, based on the principle that the grinding curve of the grinding wheel is tangent to the section contour curve, an iterative approximation method is used to calculate the rotary contour of the formed grinding wheel; finally, the calculated grinding wheel contour is verified by actual processing. Through algorithm development and actual processing, a conical spiral groove tool with an end face radius of 3.65 mm and a blank radius of 5.1 mm is successfully manufactured, and the machining accuracy error does not exceed 0.01 mm, verifying the correctness and effectiveness of the method BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 is a schematic diagram of the reference axis section coordinate system and the conical spiral groove

[0082] Figure 2 is a schematic diagram of the formed grinding wheel coordinate system

[0083] Figure 3 is a schematic diagram of the initial grinding pose of the formed grinding wheel

[0084] Figure 4 is a schematic diagram of the grinding motion of the formed grinding wheel

[0085] Figure 5 is a schematic diagram of the grinding curve of the grinding wheel in the axis section

[0086] Figure 6 Schematic diagram of the mutation of the number of intersection points corresponding to adjacent grinding wheel radii.

[0087] Figure 7 Definition of the starting and ending ranges of the grinding wheel segment.

[0088] Figure 8 Calculation process of the iterative approximation method for the rotating contour of the formed grinding wheel.

[0089] Figure 9 Shape of the groove cross-section contour.

[0090] Figure 10 Calculation results of the spiral groove envelope and the formed grinding wheel by the iterative approximation method (where (a) is the calculation result of the groove II envelope, and (b) is the calculation result of the formed grinding wheel).

[0091] Figure 11 Internal structure of the machine tool (Figure a) and grinding wheel dressing diagram (Figure b).

[0092] Figure 12 Tool detector.

[0093] Figure 13 Cross-section contour after actual grinding. Specific implementation mode

[0094] The following further elaborates on the present invention in detail with reference to the accompanying drawings and specific embodiments.

[0095] A method for grinding a conical spiral groove with a formed grinding wheel based on iterative approximation according to the present invention includes the following steps:

[0096] Step 1: Establish a geometric model of the conical spiral groove.

[0097] (1) Definition of the axial section

[0098] Compared with the cylindrical spiral groove, the geometric shape complexity of the conical spiral groove increases significantly. Its helix not only extends axially but also changes with the radius, making the surface of the conical spiral groove present a gradually changing free-form surface feature, lacking a strict mathematical expression form, which poses a huge challenge to accurate modeling and grinding processes. To simplify the problem and facilitate analysis, the present invention introduces the concept of "axial section", and defines the plane perpendicular to the tool axis as the axial section. By transforming the conical spiral groove surface from three-dimensional space into a two-dimensional expression within the axial section, the complexity of the problem is significantly reduced, facilitating subsequent modeling and calculation.

[0099] To avoid the influence of the tool end-edge structure on the geometric structure of the conical spiral groove and meet the unified definition of the geometric structure of the conical spiral groove, the present invention selects a tool axis cross-section with a complete spiral groove intersection contour as the reference axis cross-section, denoted as M, providing a stable and unified reference benchmark for the geometric modeling and grinding process design of the conical spiral groove.

[0100] (2) Definition of the axis cross-section coordinate system ACS

[0101] To accurately express the geometric structure of the conical spiral groove through the reference axis cross-section M, an axis cross-section coordinate system (Axis Cross-Section Coordinate System, ACS) is defined. As Figure 1 shown, the coordinate origin O A is located at the center of the reference axis cross-section M, the coordinate axis Z A coincides with the tool axis direction, and the positive direction points from the small-diameter end to the large-diameter end of the conical end mill. The coordinate planes X A Y A completely coincide with the reference axis cross-section M.

[0102] (3) Expression of the conical spiral groove design surface

[0103] The conical spiral groove design surface can be regarded as composed of multiple axis cross-section contours. The geometric shape of the cylindrical spiral groove design surface is relatively simple and usually the entire surface can be generated by the equal-profile sweeping of a cross-section contour. However, for the conical spiral groove design surface, due to the continuous change of the core thickness at the bottom of the groove along the tool axis direction, the shape of each cross-section is different, which increases the complexity of the design surface expression.

[0104] To accurately express the geometric characteristics of the conical spiral groove design, the intersection contour of the conical spiral groove design surface and the reference axis cross-section M is defined as the axis cross-section reference contour curve, denoted as S; any point on the reference contour curve S is defined as P S , and its coordinates are expressed in the axis cross-section coordinate system ACS as:

[0105]

[0106] In the formula, u is the function variable of the spiral groove contour curve.

[0107] Similarly, other axis cross-section contour curves L of the conical spiral groove are defined m , then the expression of the conical spiral groove design surface is:

[0108]

[0109] In the formula, m is the axial position of other axis cross-sections relative to the reference axis cross-section.

[0110] Step 2: Establish the geometric model of the formed grinding wheel.

[0111] (1) Definition of the formed grinding wheel coordinate system GCS

[0112] To accurately express the rotational contour of the formed grinding wheel and establish the pose relationship between the formed grinding wheel and the axial section coordinate system (ACS) during the grinding process, the grinding wheel coordinate system (Grinding Coordinate System, GCS) is defined. As Figure 2 shown, the origin O of the coordinate system G is located on the grinding wheel axis and serves as the grinding wheel origin; the coordinate axis Z G coincides with the grinding wheel axis and is used to describe the rotation axis of the grinding wheel; the coordinate plane X G Y G is perpendicular to the grinding wheel axis and is used to define the rotational contour of the grinding wheel. Through this definition, the grinding wheel coordinate system GCS provides an accurate reference framework for describing the geometric characteristics and motion state of the formed grinding wheel. It can not only accurately express the rotational contour of the grinding wheel but also facilitate coordinate transformation with the axial section coordinate system ACS, thereby realizing the motion control between the grinding wheel and the workpiece during the grinding process.

[0113] (2) Expression of the formed grinding wheel rotational contour

[0114] To facilitate the calculation of the formed grinding wheel rotational contour, the distances from the two end faces of the grinding wheel to the coordinate plane X G Y G are defined as h s and h e , respectively. That is, the effective width range of the formed grinding wheel rotational contour is [h s , h e . When constructing the grinding wheel rotational contour, the present invention adopts the method of discretizing the grinding wheel into slices along the grinding wheel axis and calculates the actual radius corresponding to each slice within the effective width range of the grinding wheel through envelope calculation, thereby constructing the complete rotational contour of the grinding wheel.

[0115] Define the formed grinding wheel rotational contour curve as W. Let point P W be an arbitrary point on the grinding wheel rotational contour, its rotational radius is R W , the distance from it to the coordinate plane X G Y G is h, and the angle between the line segment O G P W and the coordinate axis X G is the grinding wheel rotation angle Then the coordinates of point P W are expressed in the grinding wheel coordinate system as:

[0116]

[0117] Step 3: Establish the kinematic model of the grinding wheel.

[0118] (1) Definition of the initial grinding pose of the formed grinding wheel

[0119] To facilitate the description of the position, orientation of the grinding wheel in the reference axis cross-section coordinate system ACS and the grinding motion relationship with the tool, the grinding pose of the formed grinding wheel is described by the geometric relationship between the grinding wheel coordinate system GCS and the reference axis cross-section coordinate system ACS. When the grinding wheel is in the initial grinding pose, its origin O G is located on the X A axis and the distance from the origin O A is d X , the axis X G is in the same direction as X A , and the included angle between the axis Z G and Z A is the grinding wheel installation angle α, as shown in Figure 3 .

[0120] The geometric transformation relationship between the grinding wheel coordinate system GCS corresponding to the initial formed grinding wheel grinding pose and the reference axis cross-section coordinate system ACS is represented by the rotation matrix R x and the translation vector T x .

[0121]

[0122] (2) Expression of the grinding motion of the formed grinding wheel

[0123] To systematically describe the grinding process of the formed grinding wheel for the conical spiral groove, the grinding trajectory of the grinding wheel is defined as a conical spiral motion in this paper. Specifically, the grinding process of the spiral groove can be regarded as the tool remaining fixed while the grinding wheel performs a conical spiral motion based on its initial grinding pose. This motion process can be described as the origin O of the grinding wheel G performing a conical spiral motion relative to the reference axis cross-section coordinate system ACS, as shown in Figure 4 .

[0124] During the grinding process of the conical spiral groove, its core characteristic parameters (such as core thickness, helix angle, etc.) are not constant but change dynamically with the grinding process. In the grinding of the cylindrical spiral groove, the motion of the grinding wheel mainly focuses on the translation along the axis and the rotation around the axis, while the grinding of the conical spiral groove is more complex. The origin O of the grinding wheel G not only needs to translate and rotate along the axis Z A but also needs to gradually move away from the axis Z A in the radial direction to adapt to the change in the geometry of the conical spiral groove.

[0125] Define the coordinate system GCS to rotate around the axis Z AThe rotation angle is the spiral motion rotation angle ξ of the grinding wheel, that is, the line segment O A O G The projection on the coordinate plane X A Y A and the included angle with the coordinate axis X A ; Assume that the grinding wheel performs a conical spiral motion with a constant lead relative to the tool, and define the lead of the grinding wheel spiral motion as p h , and the taper of the spiral motion is κ G , and its value is equal to the core thickness taper angle κ of the conical spiral groove c , then the grinding wheel spiral motion rotation angle ξ is expressed as:

[0126]

[0127] Let the origin O of the grinding wheel G The distances moved relative to the coordinate system ACS are Δx, Δy, and Δz respectively, that is:

[0128]

[0129] Then the position of the profile grinding wheel grinding motion is expressed by the rotation matrix and the translation vector, as shown below:

[0130]

[0131] Therefore, according to equations (4) and (7), the pose transformation of the profile grinding wheel grinding motion can be divided into the rotation matrix M=(p, n, v) and the translation matrix r, expressed as:

[0132]

[0133] When the grinding wheel is in the initial position, the grinding wheel coordinate system coincides with the axial section coordinate system. At this time, the origin of the grinding wheel and the axis vector are (0, 0, 0) T and (0, 0, 1) T , according to equation (8), establish the expression of the origin O of the grinding wheel G and the grinding wheel axis vector Z G in the coordinate system ACS as:

[0134]

[0135] Furthermore, by combining equations (5)-(7), it is known that during the grinding motion, the coordinates of any point P W on the rotating surface of the profile grinding wheel are expressed in the coordinate system ACS as:

[0136]

[0137] Step 4: Calculate the rotating contour of the profile grinding wheel.

[0138] (1) Calculation of Grinding Curve of Grinding Wheel Based on Axial Section

[0139] The grinding curve of the grinding wheel refers to the trajectory curve formed by the intersection of the rotation circle of any point on the rotation contour of the grinding wheel and the axial section during the grinding movement of the grinding wheel, as shown in Figure 5 the left figure. Specifically, the rotation surface of the grinding wheel is discretized according to the axial distance h, divided into a series of rotation circles of the grinding wheel contour, and made to participate in the grinding movement of the grinding wheel. During this process, these rotation circles of the grinding wheel contour intersect with the axial section, and their intersection trajectories form the grinding curves corresponding to each rotation circle of the grinding wheel contour on the axial section. The set of multiple grinding curves formed by all the rotation circles of the grinding wheel contour together constitutes the complete grinding curve family, as shown in Figure 5 the right figure.

[0140] The process of the grinding wheel grinding the conical spiral groove of the tool is regarded as a Boolean subtraction operation between two entities; among them, the part of the grinding curve of the grinding wheel within the circular section of the tool is exactly the intuitive manifestation of the part of the tool actually removed by the grinding wheel during the grinding process; the grinding curve and its envelope together constitute the axial section contour of the conical spiral groove after grinding.

[0141] Let z in formula (10) W_A = m, then the axial movement distance Δz of the origin O of the grinding wheel G is expressed as:

[0142]

[0143] Substitute formula (11) into x in formula (10) W_A and y W_A , that is, the grinding curve G of the grinding wheel on any axial section M of the tool of the rotation surface of the grinding wheel is obtained, and its coordinates are expressed in the coordinate system ACS as: m

[0144]

[0145] It can be seen from formula (12) that by setting m = 0, the grinding curve of the grinding wheel under the reference axial section M can be obtained, which is an expression about the radius of the grinding wheel.

[0146] (2) Calculation of Forming Rotation Contour of Grinding Wheel

[0147] Iterative approximation: The parameters h and R of the grinding wheel contour itself are given in advance W and iterated continuously. Calculate the number of intersection points between the grinding curves of adjacent radii and the section contour during the iteration process, and judge through the mutation of the number of intersection points, so as to realize the approximate tangency between the grinding curve of the grinding wheel and the section contour. The calculation process of the iterative approximation method for the forming rotation contour of the grinding wheel in the present invention is as shown in Figure 8 the figure.

[0148] 1) Calculation of the Intersection Points between the Grinding Curve of the Grinding Wheel and the Section Profile

[0149] Since the actual expression of the section profile may be relatively complex or composed of multiple expressions, it brings certain difficulties to the tangency calculation. Here, the section profile S is uniformly approximated by the profile broken line L composed of a large number of discrete points S , thus simplifying the calculation without affecting the calculation accuracy; during the iteration process of the parameters of the grinding wheel profile, when the grinding curve G of the grinding wheel and the profile broken line L S reach the approximate tangency state, the radius R of the grinding wheel is determined W and the axial position h; in order to accurately judge the approximate tangency state, the tangency calculation is transformed into the calculation of the number of intersection points K between the profile broken line L S and the grinding curve G of the grinding wheel, that is, calculating the number of intersection points K between the grinding curve of the grinding wheel corresponding to the adjacent grinding wheel radii R W_j-1 and R W_j and the profile broken line; if there is a sudden change in the number of intersection points, that is, K j-1 =0 mutates to K j >0 or K j-1 >0 mutates to K j =0, it indicates that the approximate tangency is achieved. At this time, the grinding wheel radius R j-1 v(R j ,R W ), as W_j-1 ,R W_j ) shown. By reducing the iteration step R_step of the grinding wheel radius, the grinding wheel radius R with sufficient accuracy can be obtained Figure 6 . W .

[0150] To calculate the number of intersection points K, first define and calculate the coordinates of the intersection point P S between the profile broken line L K_j and the grinding curve G of the grinding wheel, where j = 0, 1,..., K; let the coordinates of the two end points of each small straight line segment in the profile broken line L S be (x s ,y s ) and (x e ,y e ), then the equation of the small straight line segment L S_k is expressed as the following formula:

[0151]

[0152] As Figure 6 shown in the case, assume that the axial position of the grinding wheel slice during iteration is h and the grinding wheel radius is R W_j , and the corresponding grinding curve of the grinding wheel is G j , substitute x G_T and yG_T Substitute x and y into Equation (12) respectively, and the intersection point P between the small straight line segment and the grinding curve G of the grinding wheel can be solved K The corresponding rotation angle of the grinding wheel value, and substitute back into x G_T and y G_T to obtain the intersection point P K coordinates (x k_j , y k_j ). If x k_j satisfies x k_j v[x s , x e , it indicates that the intersection point is on the small straight line segment, and store this intersection point; otherwise, there is no intersection point between this small straight line segment and the grinding curve of the grinding wheel, that is, discard it; by traversing each small straight line segment, the contour broken line L S and all intersection point coordinates and the number K j-1 with the grinding curve of the grinding wheel are obtained. At this time, K j-1 = 0.

[0153] Subsequently, continue to iterate the grinding wheel radius R W_j , and through the above steps, the corresponding number of intersection points K j = 2 is obtained. During the iteration process, the number of intersection points has mutated for adjacent grinding wheel radii. Then, the actual grinding wheel radius R W v(R W_j-1 , R W_j ) corresponding to the grinding wheel slice h; without loss of generality, let R W = R W_j .

[0154] 2) Definition of the effective starting and ending grinding wheel slice ranges

[0155] The grinding curve of the grinding wheel needs to be constrained inside the opening of the cross-sectional contour. For the integrity of the grinding wheel contour calculation, the initially given axial position h init of the grinding wheel needs to satisfy the condition of h init << h s . Therefore, the initially given values of R W_init and h init are very likely to cause the grinding curve of the grinding wheel to be outside the cross-sectional contour. As shown in Figure 7 , when it is outside, continuously iterate the next grinding wheel with the given grinding wheel slice iteration step h_step until the two intersection points of the grinding curve of a certain grinding wheel and the cross-sectional circle are both inside the opening of the cross-sectional contour, that is, the corresponding direction angles θ G1 , θ G2 satisfy θ G1 , θ G2 ∈(θ S1 , θ S2 ). At this time, the axial position of the grinding wheel slice is denoted as hs ; Subsequently, start iterating the grinding wheel radius until the number of intersection points K changes abruptly, and obtain the actual grinding wheel radius R W , and the axial position h of the grinding wheel at this time s and the radius R W are the effective starting grinding wheel {h s , R W}.

[0156] After determining the initial grinding wheel range h s , continue to calculate the subsequent grinding wheel radius and axial position in the above manner until the intersection points of the grinding curve of a certain grinding wheel and the section circle exceed the opening range of the section contour. At this time, the axial position h of the previous grinding wheel e is the effective termination grinding wheel, and the iteration process terminates, obtaining the rotary contour of the formed grinding wheel.

[0157] Example:

[0158] 1. Section contour design

[0159] To verify the accuracy and generality of the present invention, a practical grinding case of the tool spiral groove section contour is given here. The tool end face radius r t = 3.65 mm, the tool outer contour radius r t_b = 5.1 mm, and the tool contour taper κ w = arctan(1.4 / 10) = 7.970°. The given contour of the spiral groove is as Figure 9 shown, which is obtained by reverse scanning with a scanner and consists of a large number of discrete points. Some section contour points are shown in Table 1.

[0160] Table 1 Coordinates of some section contour points

[0161]

[0162]

[0163] Grinding process parameter design What needs to be considered during grinding are the overall tool parameters (r t , κ w ), grinding process parameters (m, d X , α, κ G , p h ), and the self-parameters of the grinding wheel (h init , R W_init , h_step, R_step). To calculate an effective grinding wheel, let the axial position h of the initial grinding wheel init = -2·r t , and this value is much smaller than the effective starting grinding wheel h s , R W_init whose range should be within [dX –k t ·r t , d X , between which k t ∈(0, 1), where k t = 0.9, then R W_init = d X –0.9·r t . Based on the existing experimental conditions, the blank radius R selected in this study g = 100.0 mm, let the grinding wheel installation height d X be set to be equal to the respective grinding wheel blank radius R g , and the pitch p h is calculated according to the following formula.

[0164] p h = 2πr t tanα(14)

[0165] To simplify the experimental process, the grinding parameters of the two grooves are set to be the same. Given the spiral groove cross-section profile position m = 0 mm, the grinding wheel taper κ G = arctan(1 / 10) = 5.711°, then all the grinding process parameter settings are summarized in Tables 2 and 3.

[0166] Table 2 Overall tool parameters and grinding wheel self-parameters

[0167]

[0168] Table 3 Grinding wheel grinding process parameters

[0169]

[0170] 2. Analysis of calculation results

[0171] After calculation, the correct envelope of the spiral groove is achieved, as Figure 10 shown, and a relatively smooth formed grinding wheel rotation contour is obtained. Some grinding wheel contour points are shown in Table 4.

[0172] Table 4 Some grinding wheel contour points

[0173]

[0174] For the above-given spiral groove cross-section profile, actual machining verification is carried out. The verification will be carried out on a certain MG05 model CNC groove grinding machine, and the internal structure of the CNC grinding machine is as Figure 11 (a) shown. Before the actual machining test on the machine tool, first, according to the calculated formed grinding wheel rotation contour, a grinding wheel dressing program is generated by the supporting software of the CNC grinding machine, and the dressing wheel mounted above the grinding wheel blank is used to dress it, as Figure 11As shown in (b).

[0175] To verify the cross-sectional profile and core thickness calculation algorithms, cross-sections m = 0, 5, 10 mm were taken, and the tool was used for cutting. Each layer of cross-section is as Figure 13 shown, and the tool detector is as Figure 12 shown.

[0176] The structural parameters under the cross-section were measured and compared with the calculated values. The error values between the calculated profile and the actual profile, and the error of the measured core thickness of the cross-section are shown in Table 5 and Table 6 respectively.

[0177] Table 5 Cross-sectional profile error values

[0178]

[0179] Table 6 Core thickness error

[0180]

[0181] It can be seen that the maximum error value of the profile is 0.025 mm, and the maximum error of the core thickness is 0.068 mm. The profile error and core thickness error are within the allowable errors of 0.05 mm and 0.1 mm respectively, basically meeting the actual grinding requirements of the project. The reasons for this error may be the errors existing in the machine tool itself, the rapid wear of the grinding wheel during the grinding process, and the cross-section cutting error together.

[0182] In summary, the profile calculation algorithms of the formed grinding wheel have all been effectively verified through actual processing, verifying the correctness and engineering practicability of the algorithms.

Claims

1. A method for grinding a conical spiral groove with a formed grinding wheel based on iterative approximation, characterized in that, It includes the following steps: Step 1: Establish the geometric model of the conical spiral groove; (1) Definition of the axial section Define the plane perpendicular to the tool axis as the axial section, and select a tool axial section with a complete intersecting contour of the spiral groove as the reference axial section, denoted as M; (2) Definition of the axial section coordinate system ACS Coordinate origin O A Located at the center of the reference axis section M, coordinate axis Z A Coincides with the tool axis direction, and the positive direction points from the small diameter end to the large diameter end of the conical end mill, coordinate plane X A Y A Completely coincides with the reference axis section M; (3) Expression of the design surface of the conical spiral groove Define the intersection contour of the conical spiral groove design surface and the reference axis section M as the axis section reference contour curve, denoted as S; define any point on the reference contour curve S as P S , and its coordinates are expressed in the axis section coordinate system ACS as follows: In the formula, u is the function variable of the spiral groove contour curve; Similarly, the contour curves L of other axial sections of the conical spiral groove are defined m , and the design surface of the conical spiral groove is expressed as: In the formula, m is the axial position of other axial sections relative to the reference axial section; Step 2: Establish the geometric model of the forming grinding wheel; (1) Definition of the forming grinding wheel coordinate system GCS The origin O of the coordinate system G is located on the grinding wheel axis and serves as the origin of the grinding wheel; the coordinate axis Z G coincides with the grinding wheel axis and is used to describe the rotation axis of the grinding wheel; the coordinate plane X G Y G is perpendicular to the grinding wheel axis and is used to define the rotational contour of the grinding wheel; (2) Expression of the rotary contour of the forming grinding wheel Define the distances between the two end faces of the grinding wheel and the coordinate planes X G Y G as h s and h e respectively, that is, the effective width range of the formed grinding wheel rotation contour is [h s , h e ; When constructing the rotary contour of the grinding wheel, the method of discretizing the grinding wheel into slices along the grinding wheel axis is adopted. By enveloping and calculating the actual radii corresponding to each slice within the effective width range of the grinding wheel, the complete rotary contour of the grinding wheel is constructed; Define the rotational contour curve of the molding sand wheel as W, and set point P W as an arbitrary point on the rotational contour of the sand wheel, with its rotational radius being R W , and the distance from the coordinate plane X G Y G is h. The angle between the line segment O G P W and the coordinate axis X G is the rotational angle of the sand wheel Then the coordinates of point P W are expressed in the sand wheel coordinate system as: Step 3: Establish the kinematic model of the grinding wheel grinding; (1) Definition of the initial grinding pose of the forming grinding wheel Define the origin O of the grinding wheel when it is in the initial grinding pose G is located on the X A axis and the distance from the origin O A is d X , the X axis G is in the same direction as X A , and the included angle between the Z axis G and Z A is the grinding wheel installation angle α; The geometric transformation relationship between the grinding wheel coordinate system GCS corresponding to the initial forming grinding wheel pose and the reference axis cross-section coordinate system ACS is represented by the rotation matrix R x and the translation vector T x ; (2) Expression of the grinding motion of the forming grinding wheel Define the grinding wheel grinding trajectory as a conical spiral motion. The grinding process of the spiral groove is regarded as the tool remaining fixed, while the grinding wheel performs a conical spiral motion based on its initial grinding pose; this motion process is described as the origin O of the grinding wheel G performing a conical spiral motion with respect to the reference axis section coordinate system ACS; Define that the coordinate system GCS rotates around the Z-axis A The rotation angle is the grinding wheel spiral motion rotation angle ξ, that is, the line segment O A O G The projection on the coordinate plane X A Y A And the included angle with the X-axis; Assume that the grinding wheel performs a constant-lead conical spiral motion relative to the tool, and define the lead of the grinding wheel spiral motion as p A , the taper of the spiral motion is κ h , and its value is equal to the core thickness taper angle κ of the conical spiral groove G c , then the grinding wheel spiral motion rotation angle ξ is expressed as:​ where z p is the axial movement distance of the origin O G of the grinding wheel coordinate system; assuming that the unit rotation angle of the spiral movement of the grinding wheel moving along the coordinate axis Z T is k ξ , then let k ξ = 2π / p h ; Set the origin O of the grinding wheel G The distances moved relative to the coordinate system ACS are Δx, Δy, and Δz respectively, that is: Then the position of the forming grinding wheel grinding motion is expressed by the rotation matrix and the translation vector, as shown below: Therefore, according to Equation (4) and Equation (7), the pose transformation of the forming grinding wheel grinding motion can be divided into the rotation matrix M=(p, n, v) and the translation matrix r, and is expressed as: When the grinding wheel is in the initial position, the coordinate system of the grinding wheel coincides with the coordinate system of the axial section. At this time, the origin and axis vector of the grinding wheel are (0, 0, 0) T and (0, 0, 1) T , according to Equation (8), the origin O of the grinding wheel G and the axis vector Z of the grinding wheel G are expressed in the coordinate system ACS as follows: In the formula, is the spatial position coordinate of the grinding wheel origin O G , and (i G_A , j G_A , k G_A ) is the grinding wheel axis vector; Furthermore, from the combined equations (5)-(7), it is known that during the grinding motion, the coordinates of any point P on the rotating surface of the formed grinding wheel are expressed in the coordinate system ACS as follows: W ​ Step 4: Calculate the rotary contour of the forming grinding wheel; (1) Calculation of the grinding curve of the grinding wheel based on the axial section The grinding curve of the grinding wheel refers to the trajectory curve formed by the intersection of the rotary circle of any point on the rotary contour of the grinding wheel and the axial section during the grinding motion of the grinding wheel; the rotary surface of the grinding wheel is discretized according to the axial distance h, divided into a series of rotary circles of the grinding wheel contour, and made to participate in the grinding motion of the grinding wheel; during this process, these rotary circles of the grinding wheel contour intersect with the axial section, and their intersection trajectories form the grinding curves corresponding to each rotary circle of the grinding wheel contour on the axial section; the set of multiple grinding curves formed by all the rotary circles of the grinding wheel contour together constitutes the complete grinding curve family; The process of the grinding wheel grinding the conical spiral groove of the tool is regarded as a Boolean subtraction operation between two entities; among them, the part of the grinding curve of the grinding wheel within the tool section circle is the intuitive manifestation of the part of the tool actually cut by the grinding wheel during the grinding process; the grinding curve and its envelope together form the axial section contour of the conical spiral groove after grinding; Let z in formula (10) W_A =m, then the origin of the grinding wheel is O G The axial movement distance Δz is expressed as: Substituting equation (11) into equation (10) W_A and W_A , that is, the grinding wheel rotating surface is obtained at any axial section M of the tool m The grinding wheel grinding curve G, its coordinates are expressed in the coordinate system ACS as follows: It can be seen from Equation (12) that by setting m = 0, the grinding curve of the grinding wheel under the reference axial section M can be obtained, which is an expression about the radius of the grinding wheel; (2) Calculation of the rotary contour of the forming grinding wheel Iterative approximation: Grinding wheel profile parameters h and R are given in advance W And iterates continuously, calculates the number of intersections between the grinding wheel grinding curve and the cross-sectional profile of adjacent radii during the iteration process, and determines the sudden change in the number of intersections to achieve approximate tangency between the grinding wheel grinding curve and the cross-sectional profile; 1) Calculation of the intersection points between the grinding curve of the grinding wheel and the section contour The cross-sectional profile S is uniformly approximated by the contour polyline L composed of a large number of discrete points, so as to simplify the calculation without affecting the calculation accuracy. During the iteration process of the parameters of the grinding wheel profile, when the grinding curve G of the grinding wheel and the contour polyline L reach an approximately tangent state, the grinding wheel radius R and the axial position h are determined. In order to accurately judge the approximately tangent state, the tangency calculation is transformed into the calculation of the number of intersections K between the contour polyline L and the grinding curve G of the grinding wheel, that is, calculate the number of intersections K between the grinding curve of the grinding wheel corresponding to the adjacent grinding wheel radii R and R. If there is a sudden change in the number of intersections, that is, K changes from 0 to K>0 or K>0 changes to K = 0, it indicates that the approximate tangency is achieved. At this time, the grinding wheel radius R ∈ (R, R). By reducing the iteration step size R_step of the grinding wheel radius, the grinding wheel radius R with sufficient accuracy can be obtained. S , thus simplifying the calculation without affecting the calculation accuracy; during the iteration process of the parameters of the grinding wheel profile itself, when the grinding curve G of the grinding wheel and the contour polyline L S reach an approximately tangent state, the grinding wheel radius R W and the axial position h are determined; in order to accurately judge the approximately tangent state, the tangency calculation is transformed into the calculation of the number of intersections K between the contour polyline L S and the grinding curve G of the grinding wheel, that is, calculate the number of intersections K between the grinding curve of the grinding wheel corresponding to the adjacent grinding wheel radii R W_j-1 and R W_j , and the number of intersections K j-1 and K j between the corresponding grinding curve of the grinding wheel and the contour polyline; if there is a sudden change in the number of intersections, that is, K j-1 = 0 changes to K j >0 or K j-1 >0 changes to K j = 0, it indicates that the approximate tangency is achieved. At this time, the grinding wheel radius R W ∈ (R W_j-1 , R W_j ), and by reducing the iteration step size R_step of the grinding wheel radius, the grinding wheel radius R with sufficient accuracy can be obtained W ; To calculate the number of intersection points K, first define and calculate the contour polyline L S The intersection point P K_j Coordinates with the grinding wheel grinding curve G, where j = 0, 1, …, K; Let the coordinates of the two endpoints of each small straight line segment in the contour polyline L S Be (x s , y s ) and (x e , y e ), then the equation of the small straight line segment L S_k Is expressed as the following formula: Assume that the axial position of the grinding wheel during iteration is h and the radius of the grinding wheel is R W_j , and the corresponding grinding curve of the grinding wheel is G j , substitute x G_T and y G_T in Equation (12) into x and y in Equation (12) respectively, and then solve for the intersection point P of the small straight line segment and the grinding curve G of the grinding wheel K The corresponding rotational angle value of the grinding wheel , and substitute back into x G_T and y G_T to obtain the coordinates (x K , y k_j ) of the intersection point P k_j . If x k_j satisfies x k_j ∈ [x s , x e , it indicates that the intersection point is on the small straight line segment, and store this intersection point; otherwise, there is no intersection point between this small straight line segment and the grinding curve of the grinding wheel, that is, discard it; by traversing each small straight line segment, the contour broken line L S and all the intersection point coordinates and the number K j-1 of the grinding curve of the grinding wheel can be obtained; Subsequently, continue to iterate the grinding wheel radius R W_j , and through the above steps, obtain the corresponding number of intersection points K j . During the iteration process, if the number of intersection points has mutated for adjacent grinding wheel radii, then the actual grinding wheel radius R corresponding to the grinding wheel h at this time W ∈(R W_j-1 , R W_j ); Without loss of generality, let R W = R W_j ; 2) Definition of the range of the effective starting and ending grinding wheel slices The grinding curve of the grinding wheel needs to be constrained inside the opening of the cross-sectional profile. For the completeness of the grinding wheel profile calculation, the initially given axial position h of the grinding wheel init needs to satisfy h init << h s condition. Therefore, the initially given values of R W_init and h init are very likely to cause the grinding curve of the grinding wheel to be outside the cross-sectional profile. On the outside, the iterative step h_step of the grinding wheel disc is used to iterate the next grinding wheel until the two intersection points of the grinding curve of a certain grinding wheel with the cross-sectional circle are both inside the opening of the cross-sectional profile, that is, the corresponding direction angles θ G1 , θ G2 satisfy θ G1 , θ G2 ∈(θ S1 , θ S2 ). At this time, the axial position of the grinding wheel disc is denoted as h s ; Subsequently, the radius of the grinding wheel is iterated until the number of intersection points K changes suddenly, and the actual radius R of the grinding wheel disc W is obtained. At this time, the axial position h s of the grinding wheel disc and the radius R W are the effective starting grinding wheel disc {h s , R W}; After determining the initial grinding wheel slice range h s continue to calculate the subsequent grinding wheel radius and axial position in the above manner until the intersection point of the grinding curve of a certain grinding wheel and the section circle exceeds the opening range of the section contour. At this time, the axial position h e of the previous grinding wheel is the effective termination grinding wheel slice, the iteration process terminates, and the formed grinding wheel rotation contour is obtained.

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