A method for grinding a conical helical groove of a forming wheel based on analytical optimization

Abstract

The application discloses a method for grinding a conical helical groove by using a formed grinding wheel based on analytical optimization, and particularly relates to the following steps: firstly, constructing a parameterized geometric model of the formed grinding wheel and the conical helical groove; then, constructing a grinding kinematic model of the formed grinding wheel, which is used for guiding the grinding process and realizing the grinding mode of the conical helical groove; secondly, based on the principle that the grinding curve of the grinding wheel is tangent to the cross-sectional profile curve, the rotation profile of the formed grinding wheel is calculated by using an analytical optimization mode; and finally, the calculated grinding wheel profile is verified by actual processing. The application comprehensively guarantees the precision of the helical groove cross-sectional profile, and proves the correctness and effectiveness of the related theoretical model of the method.

Description

Technical Field

[0001] This invention belongs to the field of CNC tool grinding and manufacturing, and particularly relates to a method for grinding conical spiral grooves with a shaped grinding wheel based on analytical optimization. Background Technology

[0002] Spiral grooves are a key component of CNC cutting tools, determining their cutting performance, rigidity, and strength. Spiral grooves with a core thickness that linearly varies along the tool's axis are called conical spiral grooves. With the continuous development of CNC machining technology, the manufacturing industry increasingly favors CNC cutting tools with conical spiral grooves due to their superior chip removal capability, high rigidity, and vibration resistance. However, traditional standard grinding wheel processes can only guarantee the basic geometric parameters of conical spiral grooves, but cannot ensure the design accuracy of the entire cross-sectional profile of the groove; while profile grinding wheel processes can only grind cylindrical spiral grooves, limiting the application range of these grinding methods. Therefore, this invention combines profile grinding wheel processes and conical spiral groove grinding to achieve this grinding process method.

[0003] The research on the grinding process of the forming grinding wheel is mainly based on the contact line theory and the conjugate theory of the grinding wheel and the groove surface. It is committed to the accurate calculation of the rotational profile of the forming grinding wheel under the specified grinding wheel grinding trajectory. 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 forming grinding wheel by line segments and arcs, and cleverly combined the contact theory and differential evolution algorithm to accurately calculate and optimize the final profile of the grinding wheel, providing an effective numerical method for the design of the forming grinding wheel. 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 the 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 forming grinding wheel through optimized design. Jiang Lei et al. [4] calculated the profile of the grinding wheel based on the contact line theory, and further improved the wear resistance of the grinding wheel and extended the service life of the forming grinding wheel by adjusting the grinding wheel grinding posture. Tang Qian et al. [5] proposed a screw forming tool design method based on discrete points, namely the Form Position Geometry Method (FPGM), based on the contact line theory, which provides a new idea for tool design of complex workpieces. Shen Zhihuang et al. [6] derived the contact line equation based on the conjugate theory, thus accurately obtaining the grinding wheel profile and successfully solving the smoothing problem of the forming grinding wheel profile. However, the contact theory often requires complex analytical solutions in practical applications, which brings certain difficulties to research and engineering practice. To overcome this problem, the envelope theory has emerged. 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 point cloud boundary, the grinding wheel profile was obtained efficiently, providing a simple and effective calculation method for grinding wheel design. Shen et al. [8] proposed a grinding wheel design method based on the envelope theory that can process different small spiral grooves on the existing spiral groove rake face, expanding the application range of forming grinding wheel grinding. Wu Changzuo et al. [9] successfully obtained the corresponding profile of the forming grinding wheel by solving the inverse envelope problem based on the given spiral groove profile, further enriching the theoretical system of forming grinding wheel design. In addition, Yang et al.

[10] solved the forming 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 forming grinding wheels for complex workpieces. Wu Yuren et al.

[11] proposed a radial ray shooting (RRS) method to obtain the profile of the forming grinding wheel corresponding to the 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 some progress, while the research on conical spiral grooves has not yet been published.

[0004] References:

[0005] [1]Wasif M, Iqbal S A, 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 J F. 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 FormingCutter Based on Tooth Profile Composed of Discrete Points[J].Journal ofMechanical Design,2015,137(085002).

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

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

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

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

[0014]

[10] Yang J, Sun FH, Lu Z. Solving the screw compressor rotor-forminggrinding 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 YR, Fong ZH, Zhang Z Summary of the Invention

[0016] This invention aims to achieve full contour accuracy of the conical helical groove cross-section while grinding it. To this end, this invention provides a method for grinding conical helical grooves using a profiled grinding wheel based on analytical optimization.

[0017] The present invention provides a method for grinding conical helical grooves using a profiled grinding wheel based on analytical optimization, comprising the following steps:

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

[0019] (1) Definition of 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 profile of the helical grooves as the reference axial section, denoted as M.

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

[0022] Origin of coordinates A The center of the circle located at the reference axial section M, coordinate axis Z A Aligned with the tool axis, the positive direction extends from the small diameter end of the tapered end mill to the large diameter end, in the coordinate plane X. A Y A It completely coincides with the reference axis section M.

[0023] (3) Surface representation of conical spiral groove design

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

[0025]

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

[0027] Similarly, the profile curves L of other axial sections of the conical helical groove are defined. m The surface design of the conical spiral groove is expressed as follows:

[0028]

[0029] In the formula, m represents 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 Grinding Wheel Coordinate System (GCS)

[0032] Origin of coordinate system G Located on the axis of the grinding wheel, serving as the origin of the grinding wheel; coordinate axis Z G Aligned with the axis of rotation of the grinding wheel, used to describe the axis of rotation of the grinding wheel; coordinate plane X G Y G Perpendicular to the grinding wheel axis, used to define the grinding wheel rotation profile.

[0033] (2) Representation of the rotating contour of the forming grinding wheel

[0034] Define the two end faces of the grinding wheel and the coordinate plane X. G Y G The distances are h respectively s h e That is, the effective width range of the forming grinding wheel's rotating profile is [h] s ,h e ].

[0035] When constructing the rotation profile of the grinding wheel, the grinding wheel is discretized into pieces along the axis of the grinding wheel. The actual radius corresponding to each piece within the effective width range of the grinding wheel is calculated by the envelope, thereby constructing the complete rotation profile of the grinding wheel.

[0036] Define the profile curve of the forming grinding wheel as W, and set point P. W Let R be any point on the rotational profile of the grinding wheel. W , and the coordinate system plane X G Y G The distance is h, and the line segment O G P W With coordinate axis X G The included angle is the grinding wheel rotation angle. Then point P W The coordinates in the grinding wheel coordinate system are expressed as:

[0037]

[0038] Step 3: Establish a kinematic model for grinding with a grinding wheel.

[0039] (1) Definition of the initial grinding posture of the forming grinding wheel

[0040] When the grinding wheel is in its initial grinding position, its origin O is defined. G Located in X A On the coordinate axis and at the origin O A The distance is d X coordinate axis X G With X A Consistent direction, coordinate axis Z G With Z A The included angle is the grinding wheel installation angle α.

[0041] The geometric transformation relationship between the grinding wheel coordinate system GCS and the reference axis section coordinate system ACS corresponding to the initial grinding wheel posture is expressed as the rotation matrix R. x Translation vector T x .

[0042]

[0043] (2) Expression of grinding motion of forming grinding wheel

[0044] The grinding trajectory of the grinding wheel is defined as a conical helical motion. The grinding process of the helical groove is considered as the tool remaining fixed, while the grinding wheel performs conical helical motion based on its initial grinding posture; this motion process is described by the grinding wheel origin O. G It performs conical helical motion relative to the reference axis section coordinate system ACS.

[0045] Define coordinate system GCS around coordinate axis Z AThe angle of rotation is the rotation angle ξ of the grinding wheel's helical motion, i.e., the angle of rotation of line segment O. A O G In the coordinate plane X A Y A Projection onto the coordinate axis X A The included angle; assuming the grinding wheel performs a constant-lead conical helical motion relative to the cutting tool, the lead of the grinding wheel's helical motion is defined as p. h The cone angle of the spiral motion is κ. G Its value is related to the cone angle κ of the conical spiral groove core. c If they are equal, then the rotation angle ξ of the grinding wheel's helical motion can be expressed as:

[0046]

[0047] In the formula, z p O is the origin of the grinding wheel coordinate system. G The axial movement distance of the grinding wheel. Assume the grinding wheel moves along the coordinate axis Z. T The unit rotation angle of the moving grinding wheel's helical motion is k. ξ Let k ξ =2π / p h .

[0048] Let the origin of the grinding wheel be O. G The distances traveled relative to the coordinate system ACS are Δx, Δy, and Δz, respectively, that is:

[0049]

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

[0051]

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

[0053]

[0054] When the grinding wheel is in its initial position, the grinding wheel coordinate system coincides with the axial section coordinate system, and the origin and axis vector of the grinding wheel are (0,0,0) at this time. T and (0,0,1) T According to equation (8), the origin O of the grinding wheel is established. G and grinding wheel axis vector Z G In the ACS coordinate system, it is expressed as:

[0055]

[0056] In the formula, (x OG_A ,yOG_A ,z OG_A ) represents the origin O of the grinding wheel. G Spatial location coordinates, (i G_A ,j G_A ,k G_A ) is the grinding wheel axis vector.

[0057] Furthermore, by combining equations (5)-(7), we know that during the grinding process, any point P on the rotating surface of the forming grinding wheel... W The coordinates of the coordinates are expressed in the ACS coordinate system as:

[0058]

[0059] Step 4: Calculate the rotation profile of the forming grinding wheel.

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

[0061] The grinding curve of a grinding wheel refers to the trajectory curve formed by the intersection of the circle of revolution at any point on the grinding wheel's rotational profile and the axial section during the grinding process. The grinding wheel's rotational surface is discretized according to the axial distance h, dividing it into a series of grinding wheel profile rotational circles, which participate in the grinding motion of the grinding wheel. During this process, these grinding wheel profile rotational circles intersect with the axial section, and their intersection trajectories form grinding curves corresponding to each grinding wheel profile rotational circle on the axial section. The collection of multiple grinding curves formed by all the grinding wheel profile rotational circles constitutes a complete family of grinding curves.

[0062] The process of grinding a conical spiral groove into a cutting tool with a grinding wheel is regarded as a Boolean subtraction operation between two entities. The part of the grinding wheel curve inside the tool cross-section circle is a direct representation of the portion of the tool actually removed by the grinding wheel during the grinding process. The grinding curve and its envelope together form the profile of the axial cross-section of the ground conical spiral groove.

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

[0064]

[0065] Substituting equation (11) into equation (10) for x W_A and y W_A That is, the rotating surface of the grinding wheel is obtained at any axial section M of the tool. m The grinding curve G of the grinding wheel, whose coordinates are expressed in the coordinate system ACS, is as follows:

[0066]

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

[0068] (2) Calculation of the rotating profile of the forming grinding wheel

[0069] Based on the geometric constraints of the shaft section profile, the calculation method of the shaft section profile curve of the conical spiral groove enveloped by the grinding wheel grinding curve was adopted to determine the coordinates of each point on the rotational profile of the forming grinding wheel, thereby obtaining its accurate rotational profile; the specific solution of the forming grinding wheel profile is as follows:

[0070] 1) Tangency calculation of the grinding curve of the grinding wheel and the reference profile curve of the spiral groove axial section

[0071] Based on the reference profile curve of the spiral groove axial section defined by equation (1), the coordinates of the points on the curve (x) are obtained. S_A ,y S_A Taking the derivative with respect to parameter u, we can obtain point P. S The unit tangent vector τ at the location S_A ,Right now:

[0072]

[0073] Equation (12) is applied to the grinding wheel rotation angle. Taking the partial derivative, we get:

[0074]

[0075] In the formula,

[0076]

[0077] Then any point P on the grinding curve G of the grinding wheel G The unit tangent vector at point τ is expressed as τ G_A :

[0078]

[0079] The distance d between the grinding wheel curve and the reference profile curve of the helical groove shaft section is expressed as:

[0080]

[0081] And vector τ G With vector τ S The included angle θ is expressed as:

[0082]

[0083] When the grinding curve of the grinding wheel is tangent to the axial profile curve of the spiral groove, the constraint condition is that the coordinate values ​​of the two curves at the common tangency point are equal, and the tangent vectors are equal, that is, the included angle θ and the distance d are both 0. The constraint condition is expressed as:

[0084]

[0085] The problem of finding the tangency vectors is transformed into a numerical optimization problem. The optimization objectives are the distance between the intersection points (d) and the angle between the tangent vectors (θ). A corresponding mathematical optimization model is then established.

[0086] To improve computational efficiency, the coordinates of each point on the reference profile curve S of the spiral groove shaft section are taken and its tangent vector is pre-calculated. The vector is then substituted into equation (18) to eliminate the parameter u, thereby reducing the dimensionality of the variables. The mathematical optimization model after dimensionality reduction is shown in the following equation:

[0087]

[0088] The above equation is optimized using an intelligent optimization algorithm as the objective function, thus obtaining the grinding wheel rotation profile point P. W radius R W And its axial position h in the grinding wheel coordinate system GCS.

[0089] 2) Determining the direction of the grinding wheel curve and the groove profile curve

[0090] The optimization solution of equation (19) for the target can yield two target solutions, namely two grinding curves that are tangent to the reference profile curve S of the spiral groove shaft section. Therefore, it is necessary to constrain the range of the solution and select the grinding curve that is inside the reference profile curve of the spiral groove shaft section.

[0091] The grinding curves are selected based on the direction angle of the intersection point between the grinding wheel curve and the outer circular profile of the tool's axial section. An equation is then established to solve for the intersection point P of the two grinding wheel curves and the outer circular profile of the tool's axial section. G1 and P G2 .

[0092]

[0093] In the formula, r t_m The radius of the outer circle of the tool's axial section; when m = 0, r t_m Let r be the radius of the tool circle under the reference axial section. t Define the tool profile taper as k. w Then the tool radius under any cross section is:

[0094] r t_m =r t_0 +m·tan(κ w ) (twenty one)

[0095] Solving equation (20) yields the intersection point P. G1 and PG2 Coordinates (x) in the tool coordinate system ACS G1 ,y G1 ) and (x G2 ,y G2 Define the intersection point P. G1 and P G2 The direction angle is θ G , which is O A P G line segment and Y A The angle between the positive directions of the axes is the corresponding direction angle θ. G1 and θ G2 The calculation is as follows:

[0096]

[0097] The direction angle θ of the axial section G1 and θ G2 The direction angle θ between the two endpoints of the reference profile curve of the spiral groove axial section S1 and θ S2 Compare; if the direction angle of one of the grinding wheel curves is... This means that the grinding curve is outside the profile curve and is therefore discarded. Instead, another grinding curve and any point P on the corresponding grinding wheel rotation profile are selected. W radius R W The axial position h is used to obtain the rotating profile of the forming grinding wheel.

[0098] Compared with the prior art, the beneficial effects of the present invention are:

[0099] This invention proposes a method for calculating the rotational profile of a forming grinding wheel to achieve precise grinding of a given conical helical groove cross-sectional profile. First, a parametric geometric model of the forming grinding wheel and the conical helical groove is constructed. Then, a kinematic model of the forming grinding wheel grinding is constructed to guide the grinding process and realize the grinding method of the conical helical groove. Second, based on the principle that the grinding wheel grinding curve is tangent to the cross-sectional profile curve, an analytical optimization method is used to calculate the rotational profile of the forming grinding wheel. Finally, the calculated grinding wheel profile is verified through actual machining. Through algorithm development and actual machining, a conical helical groove tool with a radius of 6mm was successfully manufactured with a machining accuracy error of no more than 0.1mm, verifying the correctness and effectiveness of the method. Attached Figure Description

[0100] Figure 1 This is a schematic diagram of the reference axis section coordinate system and the conical spiral groove.

[0101] Figure 2 This is a schematic diagram of the coordinate system for forming the grinding wheel.

[0102] Figure 3This is a schematic diagram of the initial grinding posture of the forming grinding wheel.

[0103] Figure 4 This is a schematic diagram of the grinding motion of a forming grinding wheel.

[0104] Figure 5 This is a schematic diagram of the grinding curve of the grinding wheel with a axial section.

[0105] Figure 6 A schematic diagram of the grinding curve of the grinding wheel that is tangent to the reference profile curve of the spiral groove axial section.

[0106] Figure 7 This is a flowchart for calculating the rotational profile of the forming grinding wheel.

[0107] Figure 8 The cross-sectional profile shape.

[0108] Figure 9 The calculation results are for the spiral groove envelope and the shaped grinding wheel.

[0109] Figure 10 This is a coordinate axis diagram of the internal structure of the machine tool.

[0110] Figure 11 This is the actual cross-sectional profile after grinding. Detailed Implementation

[0111] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0112] The present invention provides a method for grinding conical helical grooves using a profiled grinding wheel based on analytical optimization, comprising the following steps:

[0113] Step 1: Establish the geometric model of the conical spiral groove.

[0114] (1) Definition of axial section

[0115] Compared to cylindrical helical grooves, conical helical grooves exhibit significantly increased geometric complexity. Their helix extends not only axially but also varies with the radius, resulting in a gradually changing free-form surface characteristic. This lack of a rigorous mathematical expression poses a significant challenge to accurate modeling and grinding processes. To simplify the problem and facilitate analysis, this invention introduces the concept of a "axial section," defining a plane perpendicular to the tool axis as the axial section. By transforming the conical helical groove surface from three-dimensional space into a two-dimensional representation within the axial section, the complexity of the problem is significantly reduced, facilitating subsequent modeling and calculations.

[0116] To avoid the influence of the tool end-cutting structure on the geometry of the conical helical groove and to meet the unified definition of the geometry of the conical helical groove, this invention selects a tool axial section with a complete intersecting profile of the helical groove as the reference axial section, denoted as M, which provides a stable and unified reference for the geometric modeling and grinding process design of the conical helical groove.

[0117] (2) Definition of the axial section coordinate system ACS

[0118] To accurately represent the geometry of the conical helical groove using a reference axial section M, an Axis Cross-Section Coordinate System (ACS) is defined. For example... Figure 1 As shown, the origin O of the coordinate system A The center of the circle located at the reference axial section M, coordinate axis Z A Aligned with the tool axis, the positive direction extends from the small diameter end of the tapered end mill to the large diameter end, in the coordinate plane X. A Y A It completely coincides with the reference axis section M.

[0119] (3) Surface representation of conical spiral groove design

[0120] The design surface of a conical helical groove can be considered as being composed of multiple layers of axial cross-sectional profiles. The geometry of a cylindrical helical groove design surface is relatively simple, and the entire surface can usually be generated by sweeping an equal cross-sectional profile. However, the conical helical groove design surface has a different shape for each cross-section because the core thickness at the bottom of the groove changes continuously along the tool axis, which increases the complexity of the design surface representation.

[0121] To accurately represent the geometric characteristics of the conical helical groove design, the intersection profile of the conical helical groove design surface and the reference axial section M is defined as the reference profile curve of the axial section, denoted as S; any point on the reference profile curve S is defined as P. S Its coordinates, expressed in the axial section coordinate system ACS, are as follows:

[0122]

[0123] In the formula, u is the function variable of the spiral groove profile curve.

[0124] Similarly, the profile curves L of other axial sections of the conical helical groove are defined. m The surface design of the conical spiral groove is expressed as follows:

[0125]

[0126] In the formula, m represents the axial position of other axial sections relative to the reference axial section.

[0127] Step 2: Establish the geometric model of the forming grinding wheel.

[0128] (1) Definition of the Grinding Wheel Coordinate System (GCS)

[0129] To accurately represent the rotational profile of the forming grinding wheel and establish the positional relationship between the forming grinding wheel and the axial section coordinate system (ACS) during grinding, a grinding wheel coordinate system (GCS) is defined. For example... Figure 2 As shown, the origin O of the coordinate system G Located on the axis of the grinding wheel, serving as the origin of the grinding wheel; coordinate axis Z G Aligned with the axis of rotation of the grinding wheel, used to describe the axis of rotation of the grinding wheel; coordinate plane X G Y G Perpendicular to the grinding wheel axis, it is used to define the grinding wheel's rotational profile. Through this definition, the grinding wheel coordinate system (GCS) provides a precise reference frame for describing the geometric features and motion state of the forming grinding wheel. It can not only accurately express the grinding wheel's rotational profile, but also easily perform coordinate transformation with the axial section coordinate system (ACS), thereby realizing motion control between the grinding wheel and the workpiece during the grinding process.

[0130] (2) Representation of the rotating contour of the forming grinding wheel

[0131] To facilitate the calculation of the rotating profile of the forming grinding wheel, the two end faces of the grinding wheel and the coordinate plane X are defined. G Y G The distances are h respectively s h e That is, the effective width range of the forming grinding wheel's rotating profile is [h] s ,h e In constructing the rotational profile of the grinding wheel, this invention employs a method of discretizing the grinding wheel into pieces along its axis, and calculating the actual radius corresponding to each piece within the effective width range of the grinding wheel through envelope calculation, thereby constructing the complete rotational profile of the grinding wheel.

[0132] Define the profile curve of the forming grinding wheel as W, and set point P. W Let R be any point on the rotational profile of the grinding wheel. W , and the coordinate system plane X G Y G The distance is h, and the line segment O G P W With coordinate axis X G The included angle is the grinding wheel rotation angle. Then point P W The coordinates in the grinding wheel coordinate system are expressed as:

[0133]

[0134] Step 3: Establish a kinematic model for grinding with a grinding wheel.

[0135] (1) Definition of the initial grinding posture of the forming grinding wheel

[0136] To facilitate the description of the position and orientation of the grinding wheel in the reference axisymmetric coordinate system ACS, as well as its grinding motion relationship with the cutting tool, the grinding posture of the forming grinding wheel is described by the geometric relationship between the grinding wheel coordinate system GCS and the reference axisymmetric coordinate system ACS. The origin O of the grinding wheel is defined as follows when it is in its initial grinding posture: G Located in X A On the coordinate axis and at the origin O A The distance is d X coordinate axis X G With X A Consistent direction, coordinate axis Z G With Z A The included angle is the grinding wheel mounting angle α, such as Figure 3 As shown.

[0137] The geometric transformation relationship between the grinding wheel coordinate system GCS and the reference axis section coordinate system ACS corresponding to the initial grinding wheel posture is expressed as the rotation matrix R. x Translation vector T x .

[0138]

[0139] (2) Expression of grinding motion of forming grinding wheel

[0140] To systematically describe the forming grinding process of conical helical grooves, this paper defines the grinding trajectory of the grinding wheel as conical helical motion. Specifically, the grinding process of the helical groove can be regarded as the tool remaining fixed, while the grinding wheel performs conical helical motion based on its initial grinding posture. This motion process can be described as the grinding wheel origin O... G Performing conical helical motion relative to the reference axis section coordinate system ACS, such as... Figure 4 As shown.

[0141] In the grinding process of conical helical grooves, the core characteristic parameters (such as core thickness and helix angle) are not constant, but change dynamically with the grinding process. In the grinding of cylindrical helical grooves, the movement of the grinding wheel mainly focuses on axial translation and rotation around the axis, while the grinding of conical helical grooves is more complex. Grinding wheel origin O G Not only along the Z-axis A To perform translation and rotation, you also need to gradually move away from the Z-axis in the radial direction. A To adapt to changes in the geometry of the conical spiral groove.

[0142] Define coordinate system GCS around coordinate axis Z AThe angle of rotation is the rotation angle ξ of the grinding wheel's helical motion, i.e., the angle of rotation of line segment O. A O G In the coordinate plane X A Y A Projection onto the coordinate axis X A The included angle; assuming the grinding wheel performs a constant-lead conical helical motion relative to the cutting tool, the lead of the grinding wheel's helical motion is defined as p. h The cone angle of the spiral motion is κ. G Its value is related to the cone angle κ of the conical spiral groove core. c If they are equal, then the rotation angle ξ of the grinding wheel's helical motion can be expressed as:

[0143]

[0144] Let the origin of the grinding wheel be O. G The distances traveled relative to the coordinate system ACS are Δx, Δy, and Δz, respectively, that is:

[0145]

[0146] The position of the grinding motion of the forming grinding wheel is expressed by a rotation matrix and a translation vector, as shown below:

[0147]

[0148] Therefore, according to equations (4) and (7), the pose transformation of the grinding motion of the forming wheel can be divided into a rotation matrix M = (p,n,v) and a translation matrix r, expressed as:

[0149]

[0150] When the grinding wheel is in its initial position, the grinding wheel coordinate system coincides with the axial section coordinate system, and the origin and axis vector of the grinding wheel are (0,0,0) at this time. T and (0,0,1) T According to equation (8), the origin O of the grinding wheel is established. G and grinding wheel axis vector Z G In the ACS coordinate system, it is expressed as:

[0151]

[0152] Furthermore, by combining equations (5)-(7), we know that during the grinding process, any point P on the rotating surface of the forming grinding wheel... W The coordinates of the coordinates are expressed in the ACS coordinate system as:

[0153]

[0154] Step 4: Calculate the rotation profile of the forming grinding wheel.

[0155] (1) Calculation of grinding curve of grinding wheel based on axial section

[0156] The grinding curve of a grinding wheel refers to the trajectory curve formed by the intersection of the circle of revolution at any point on the grinding wheel's rotational profile and the axial section during the grinding process. Figure 5 Left figure. Specifically, the grinding wheel's rotating surface is discretized according to the axial distance h, dividing it into a series of grinding wheel profile rotation circles, which then participate in the grinding motion of the grinding wheel. During this process, these grinding wheel profile rotation circles intersect with the axial section, and their intersection trajectories form grinding curves corresponding to each grinding wheel profile rotation circle on the axial section. The multiple grinding curves formed by all the grinding wheel profile rotation circles are collected together to form a complete family of grinding curves, such as... Figure 5 The image on the right.

[0157] The process of grinding a conical spiral groove into a cutting tool with a grinding wheel is regarded as a Boolean subtraction operation between two entities. The part of the grinding wheel curve inside the tool cross-section circle is a direct representation of the portion of the tool actually removed by the grinding wheel during the grinding process. The grinding curve and its envelope together form the profile of the axial cross-section of the ground conical spiral groove.

[0158] Let z in equation (10) W_A =m, then the origin of the grinding wheel is O G The axial movement distance Δz is expressed as:

[0159]

[0160] Substituting equation (11) into equation (10) for x W_A and y W_A That is, the rotating surface of the grinding wheel is obtained at any axial section M of the tool. m The grinding curve G of the grinding wheel, whose coordinates are expressed in the coordinate system ACS, is as follows:

[0161]

[0162] As can be seen from equation (12), by setting m = 0, the grinding curve of the grinding wheel under the reference axial section M can be obtained, which is an expression for the grinding wheel radius.

[0163] (2) Calculation of the rotating profile of the forming grinding wheel

[0164] The profile of the forming grinding wheel and the grinding motion together determine the grinding accuracy of the conical helical groove. To obtain the rotational profile of the forming grinding wheel, this invention combines the geometric constraints of the shaft section profile and employs a calculation method that uses the grinding wheel grinding curve to envelop the shaft section profile curve of the conical helical groove. This method determines the coordinates of each point on the rotational profile of the forming grinding wheel, thereby obtaining its accurate rotational profile. The calculation process for the rotational profile of the forming grinding wheel is as follows: Figure 7 As shown, the details are as follows:

[0165] 1) Tangency calculation of the grinding curve of the grinding wheel and the reference profile curve of the spiral groove axial section

[0166] Based on the reference profile curve of the spiral groove axial section defined by equation (1), point P can be obtained by differentiating the parameter u. S The unit tangent vector τ at the location S_A ,Right now:

[0167]

[0168] Equation (12) is applied to the grinding wheel rotation angle. Taking the partial derivative, we get:

[0169]

[0170] In the formula,

[0171]

[0172] Then any point P on the grinding curve G of the grinding wheel G The unit tangent vector at point τ is expressed as τ G_A :

[0173]

[0174] The distance d between the grinding wheel curve and the reference profile curve of the helical groove shaft section is expressed as:

[0175]

[0176] And vector τ G With vector τ S The included angle θ is expressed as:

[0177]

[0178] When the grinding curve of the grinding wheel is tangent to the axial profile curve of the spiral groove, the constraint condition is that the coordinate values ​​of the two curves at the common tangency point are equal, and the tangent vectors are equal, that is, the included angle θ and the distance d are both 0. The constraint condition is expressed as:

[0179]

[0180] Due to the complexity of nonlinear problem-solving, the analytical solution formula of equation (18) is difficult to express explicitly. Therefore, equation (18) is solved numerically to calculate the radius R at tangency. W This presents significant challenges. Therefore, this invention transforms the tangency problem into a numerical optimization problem, using the intersection distance d and the angle θ between the tangent vectors as optimization objectives, and establishes a corresponding mathematical optimization model.

[0181] To improve computational efficiency, the coordinates of each point on the reference profile curve S of the spiral groove shaft section are taken and its tangent vector is pre-calculated. The vector is then substituted into equation (18) to eliminate the parameter u, thereby reducing the dimensionality of the variables. The mathematical optimization model after dimensionality reduction is shown in the following equation:

[0182]

[0183] The above equation is optimized using an intelligent optimization algorithm as the objective function, thus obtaining the grinding wheel rotation profile point P. W radius R W And its axial position h in the grinding wheel coordinate system GCS.

[0184] 2) Determining the direction of the grinding wheel curve and the groove profile curve

[0185] Solving equation (19) for the target optimization yields two target solutions, namely, two grinding curves corresponding to the grinding wheel grinding curves tangent to the reference profile curve S of the spiral groove axial section, such as... Figure 6 As shown. Therefore, it is necessary to constrain the solution range and select the grinding wheel curve that lies inside the reference profile curve of the helical groove axial section.

[0186] The grinding curves are selected based on the direction angle of the intersection point between the grinding wheel curve and the outer circular profile of the tool's axial section. An equation is then established to solve for the intersection point P of the two grinding wheel curves and the outer circular profile of the tool's axial section. G1 and P G2 .

[0187]

[0188] In the formula, r t_m The radius of the outer circle of the tool's axial section; when m = 0, r t_m Let r be the radius of the tool circle under the reference axial section. t Define the tool profile taper as k. w Then the tool radius under any cross section is:

[0189] r t_m =r t_0 +m·tan(κ w ) (twenty one)

[0190] Solving equation (20) yields the intersection point P. G1 and P G2 Coordinates (x) in the tool coordinate system ACS G1 ,y G1 ) and (x G2 ,y G2 Define the intersection point P. G1 and P G2 The direction angle is θG , which is O A P G line segment and Y A The angle between the positive directions of the axes is the corresponding direction angle θ. G1 and θ G2 The calculation is as follows:

[0191]

[0192] The direction angle θ of the axial section G1 and θ G2 The direction angle θ between the two endpoints of the reference profile curve of the spiral groove axial section S1 and θ S2 Compare; if the direction angle of one of the grinding wheel curves is... This means that the grinding curve is outside the profile curve and is therefore discarded. Instead, another grinding curve and any point P on the corresponding grinding wheel rotation profile are selected. W radius R W The axial position h is used to obtain the rotating profile of the forming grinding wheel.

[0193] Example:

[0194] 1. Cross-sectional profile design

[0195] Taking a cylindrical end mill with a thick taper and a cylindrical core as an example, this embodiment illustrates the feasibility of the proposed analytical optimization calculation method for the grinding wheel profile of a conical helical groove through a series of examples. The relevant discrete point profiles of the helical groove are shown in Table 1.

[0196] Table 1 Coordinates of some cross-sectional profile points

[0197]

[0198]

[0199] In the formula, x A and y A Let x0 and y0 be the component coordinates of any point on the arc segment, respectively, and let ρ be the radius of the arc segment. Let δ be the line segment connecting a point on the arc and the center point, and let X be the component coordinates of the arc segment. T The angle between the positive directions of the axis.

[0200] The parameter values ​​of the three circular arcs of groove I are shown in Table 2. The range of δ in the table is expressed in degrees.

[0201] Table 2. Parameter values ​​of the three-segment circular arc of groove I.

[0202]

[0203] When designing grinding process parameters for grinding the profile of a helical groove using analytical optimization methods, the overall tool parameters (r) need to be considered. t ,κ w Grinding process parameters (m,d) X ,α,κ G ,p h Based on the existing experimental conditions, the radius R of the grinding wheel blank selected in this embodiment is... g =102.5mm, the grinding wheel installation height d X Let it be equal to the radius R of the grinding wheel blank. g Given that the position of the helical groove cross-section profile is m=0, i.e., the tool end face, all grinding process parameter settings are summarized in Table 3.

[0204] Table 3 Overall Tool and Grinding Process Parameters

[0205]

[0206] 2. Analysis of Calculation Results

[0207] By setting the grinding process parameters, the calculation results of the spiral groove envelope and the corresponding forming grinding wheel rotation profile are as follows: Figure 9 As shown in Table 4, a smooth shaped grinding wheel rotation profile was obtained through calculation.

[0208] Table 4 shows some grinding wheel profile points.

[0209]

[0210] In summary, the correctness and effectiveness of the forming wheel profile calculation algorithm were verified by comparing the error results of the spiral groove envelope. To further verify the correctness of the algorithm calculation for each layer of cross-section profile, simulation and actual machining verification were performed.

[0211] The actual machining verification of the given spiral groove cross-sectional profile will be carried out on an MG05 CNC groove grinding machine, the internal structure of which is as follows: Figure 10 As shown, it corresponds to the three-dimensional simulation model of the machine tool.

[0212] The cross-sectional profile of the helical groove was actually ground, and the profile was obtained. A PG1000 tool measuring instrument was used to measure and compare the profile. The ground profile is shown below. Figure 11 As shown in the figure, the yellow line represents the given cross-sectional profile, which can be seen to highly coincide with the grinding cross-sectional profile.

[0213] In summary, based on the given spiral groove cross-sectional profile and grinding trajectory, the rotational profile of the forming grinding wheel was calculated. Then, the forming grinding wheel was used to grind the spiral groove of the tool, and the result was consistent with the given cross-sectional profile. This verifies the correctness and engineering practicality of the algorithms for calculating the rotational profile of the forming grinding wheel and the arbitrary cross-sectional profile.

Claims

1. A method for grinding conical helical grooves with a forming grinding wheel based on analytical optimization, characterized in that, Includes the following steps: Step 1: Establish the geometric model of the conical spiral groove; (1) Definition of axial section Define the plane perpendicular to the tool axis as the axial section, and select a tool axial section with a complete intersecting profile of the helical grooves as the reference axial section, denoted as M; (2) Definition of the axial section coordinate system ACS Origin of coordinates A The center of the circle located at the reference axial section M, coordinate axis Z A Aligned with the tool axis, the positive direction extends from the small diameter end of the tapered end mill to the large diameter end, in the coordinate plane X. A Y A Completely coincides with the reference axial section M; (3) Surface representation of conical spiral groove design Define the intersection profile of the conical helical groove design surface and the reference axial section M as the reference profile curve of the axial section, denoted as S; define any point on the reference profile curve S as P. S Its coordinates, expressed in the axial section coordinate system ACS, are as follows: In the formula, u is the function variable of the spiral groove profile curve; Similarly, the profile curves L of other axial sections of the conical helical groove are defined. m The surface design of the conical spiral groove is expressed as follows: In the formula, m represents 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 Grinding Wheel Coordinate System (GCS) Origin of coordinate system G Located on the axis of the grinding wheel, serving as the origin of the grinding wheel; coordinate axis Z G Aligned with the axis of rotation of the grinding wheel, used to describe the axis of rotation of the grinding wheel; coordinate plane X G Y G Perpendicular to the grinding wheel axis, used to define the grinding wheel rotation profile; (2) Representation of the rotating contour of the forming grinding wheel Define the two end faces of the grinding wheel and the coordinate plane X. G Y G The distances are h respectively s h e That is, the effective width range of the forming grinding wheel's rotating profile is [h] s ,h e ]; When constructing the rotation profile of the grinding wheel, the grinding wheel is discretized into pieces along the axis of the grinding wheel. The actual radius of each piece within the effective width range of the grinding wheel is calculated by envelope calculation, thereby constructing the complete rotation profile of the grinding wheel. Define the profile curve of the forming grinding wheel as W, and set point P. W Let R be any point on the rotational profile of the grinding wheel. W , and the coordinate system plane X G Y G The distance is h, and the line segment O G P W With coordinate axis X G The included angle is the grinding wheel rotation angle. Then point P W The coordinates in the grinding wheel coordinate system are expressed as: Step 3: Establish a kinematic model for grinding with a grinding wheel; (1) Definition of the initial grinding posture of the forming grinding wheel When the grinding wheel is in its initial grinding position, its origin O is defined. G Located in X A On the coordinate axis and at the origin O A The distance is d X coordinate axis X G With X A Consistent direction, coordinate axis Z G With Z A The included angle is the grinding wheel installation angle α; The geometric transformation relationship between the grinding wheel coordinate system GCS and the reference axis section coordinate system ACS corresponding to the initial grinding wheel posture is expressed as the rotation matrix R. x Translation vector T x ; (2) Expression of grinding motion of forming grinding wheel The grinding trajectory of the grinding wheel is defined as a conical helical motion. The grinding process of the helical groove is considered as the tool remaining fixed, while the grinding wheel performs conical helical motion based on its initial grinding posture; this motion process is described by the grinding wheel origin O. G Perform conical-helical motion relative to the reference axis section coordinate system ACS; Define coordinate system GCS around coordinate axis Z A The angle of rotation is the rotation angle ξ of the grinding wheel's helical motion, i.e., the angle of rotation of line segment O. A O G In the coordinate plane X A Y A Projection onto the coordinate axis X A The included angle; assuming the grinding wheel performs a constant-lead conical helical motion relative to the cutting tool, the lead of the grinding wheel's helical motion is defined as p. h The cone angle of the spiral motion is κ. G Its value is related to the cone angle κ of the conical spiral groove core. c If they are equal, then the rotation angle ξ of the grinding wheel's helical motion can be expressed as: In the formula, z p O is the origin of the grinding wheel coordinate system. G The axial movement distance of the grinding wheel; assuming the grinding wheel moves along the coordinate axis Z. T The unit rotation angle of the moving grinding wheel's helical motion is k. ξ Let k ξ =2π / p h ; Let the origin of the grinding wheel be O. G The distances traveled relative to the coordinate system ACS are Δx, Δy, and Δz, respectively, that is: The position of the grinding motion of the forming grinding wheel is expressed by a rotation matrix and a translation vector, as shown below: Therefore, according to equations (4) and (7), the pose transformation of the grinding motion of the forming wheel can be divided into a rotation matrix M = (p,n,v) and a translation matrix r, expressed as: When the grinding wheel is in its initial position, the grinding wheel coordinate system coincides with the axial section coordinate system, and the origin and axis vector of the grinding wheel are (0,0,0) at this time. T and (0,0,1) T According to equation (8), the origin O of the grinding wheel is established. G and grinding wheel axis vector Z G In the ACS coordinate system, it is expressed as: In the formula, (x OG_A ,y OG_A ,z OG_A ) represents the origin O of the grinding wheel. G Spatial location coordinates, (i G_A ,j G_A ,k G_A ) represents the grinding wheel axis vector; Furthermore, by combining equations (5)-(7), we know that during the grinding process, any point P on the rotating surface of the forming grinding wheel... W The coordinates of the coordinates are expressed in the ACS coordinate system as: Step 4: Calculate the rotational profile of the forming grinding wheel; (1) Calculation of grinding curve of grinding wheel based on axial section A grinding curve is a trajectory curve formed by the intersection of the circle of revolution at any point on the grinding wheel's rotational profile and the axial section during the grinding process. The grinding wheel's rotational surface is discretized according to an axial distance h, dividing it into a series of grinding wheel profile rotational circles, which then participate in the grinding motion. During this process, these grinding wheel profile rotational circles intersect with the axial section, and their intersection trajectories form grinding curves corresponding to each grinding wheel profile rotational circle on the axial section. The collection of multiple grinding curves formed by all the grinding wheel profile rotational circles constitutes a complete family of grinding curves. The process of grinding a conical spiral groove into a cutting tool with a grinding wheel is regarded as a Boolean subtraction operation between two entities; the part of the grinding wheel curve inside the tool section circle is a direct representation of the actual cutting tool portion removed by the grinding wheel during the grinding process; the grinding curve and its envelope together form the profile of the axial section of the ground conical spiral groove. Let z in equation (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) for x W_A and y W_A That is, the rotating surface of the grinding wheel is obtained at any axial section M of the tool. m The grinding curve G of the grinding wheel, whose coordinates are expressed in the coordinate system ACS, is as follows: As can be seen from equation (12), 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 grinding wheel radius; (2) Calculation of the rotating profile of the forming grinding wheel Based on the geometric constraints of the shaft section profile, the calculation method of the shaft section profile curve of the conical spiral groove enveloped by the grinding wheel grinding curve was adopted to determine the coordinates of each point on the rotational profile of the forming grinding wheel, thereby obtaining its accurate rotational profile; the specific solution of the forming grinding wheel profile is as follows: 1) Tangency calculation of the grinding curve of the grinding wheel and the reference profile curve of the spiral groove axial section Based on the reference profile curve of the spiral groove axial section defined by equation (1), the coordinates of the points on the curve (x) are obtained. S_A ,y S_A Taking the derivative with respect to parameter u, we can obtain point P. S The unit tangent vector τ at the location S_A ,Right now: Equation (12) is applied to the grinding wheel rotation angle. Taking the partial derivative, we get: In the formula, Then any point P on the grinding curve G of the grinding wheel G The unit tangent vector at point τ is expressed as τ G_A : The distance d between the grinding wheel curve and the reference profile curve of the helical groove shaft section is expressed as: And vector τ G With vector τ S The included angle θ is expressed as: When the grinding curve of the grinding wheel is tangent to the axial profile curve of the spiral groove, the constraint condition is that the coordinate values ​​of the two curves at the common tangency point are equal, and the tangent vectors are equal, that is, the included angle θ and the distance d are both 0. The constraint condition is expressed as: The problem of finding the tangency is transformed into a numerical optimization problem. The optimization objectives are the distance d between the intersection points and the angle θ between the tangent vectors. A corresponding mathematical optimization model is then established. To improve computational efficiency, the coordinates of each point on the reference profile curve S of the spiral groove shaft section are taken and its tangent vector is pre-calculated. The vector is then substituted into equation (18) to eliminate the parameter u, thereby reducing the dimensionality of the variables. The mathematical optimization model after dimensionality reduction is shown in the following equation: The above equation is optimized using an intelligent optimization algorithm as the objective function, thus obtaining the grinding wheel rotation profile point P. W radius R W And its axial position h in the grinding wheel coordinate system GCS; 2) Determining the direction of the grinding wheel curve and the groove profile curve The optimization solution of equation (19) for the target can yield two target solutions, namely two grinding wheel curves that are tangent to the reference profile curve S of the spiral groove shaft section. Therefore, it is necessary to constrain the range of the solution and select the grinding wheel curve that is inside the reference profile curve of the spiral groove shaft section. The grinding curves are selected based on the direction angle of the intersection point between the grinding wheel curve and the outer circular profile of the tool's axial section. An equation is then established to solve for the intersection point P of the two grinding wheel curves and the outer circular profile of the tool's axial section. G1 and P G2 ; In the formula, r t_m The radius of the outer circle of the tool's axial section; when m = 0, r t_m Let r be the radius of the tool circle under the reference axial section. t Define the tool profile taper as k. w Then the tool radius under any cross section is: r t_m =r t_0 +m·tan(κ w ) (21) Solving equation (20) yields the intersection point P. G1 and P G2 Coordinates (x) in the tool coordinate system ACS G1 ,y G1 ) and (x G2 ,y G2 Define the intersection point P. G1 and P G2 The direction angle is θ G , which is O A P G line segment and Y A The angle between the positive directions of the axes is the corresponding direction angle θ. G1 and θ G2 The calculation is as follows: The direction angle θ of the axial section G1 and θ G2 The direction angle θ between the two endpoints of the reference profile curve of the spiral groove axial section S1 and θ S2 Compare; if the direction angle θ of one of the grinding wheel curves... G1 , This means that the grinding curve is outside the profile curve and is therefore discarded. Instead, another grinding curve and any point P on the corresponding grinding wheel rotation profile are selected. W radius R W The axial position h is used to obtain the rotating profile of the forming grinding wheel.

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