Method for grinding conical spiral groove through formed grinding wheel based on analytical optimization

Abstract

The invention discloses a method for grinding a conical spiral groove by a formed grinding wheel based on analytical optimization, which specifically comprises the following steps of: 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 rotary contour of the formed grinding wheel is calculated in an analytical optimization 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 analytical optimization. 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 guarantee 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 form 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 form grinding wheel under the specified grinding wheel grinding trajectory conditions. 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 form 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 form grinding wheels. Hsieh et al. [2] combined coordinate transformation with the conjugate surface theory, successfully established a kinematic model for the grinding of helical grooves, and determined the grinding wheel profile based on the conjugate contact theory, laying a foundation for the theoretical research on the helical 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 form 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 form 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 form 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 emerged. You Minglin et al. [7] proposed the pixel matrix method based on the helical 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 design of grinding wheels. Shen et al. [8] proposed a grinding wheel design method based on the envelope theory, which can machine different small helical grooves on the basis of the existing rake face of the helical groove, expanding the application range of form grinding wheel grinding. Wu Changzuo et al. [9] successfully obtained the corresponding profile of the form grinding wheel by solving the inverse envelope problem according to the given helical groove profile, further enriching the theoretical system of form grinding wheel design. In addition, Yang et al.

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

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

[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 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 grinding of a conical spiral groove while ensuring the full-profile accuracy of the cross-section of the spiral groove. To this end, the present invention provides a method for grinding a conical spiral groove with a formed grinding wheel based on analytical optimization.

[0017] A method for grinding a conical spiral groove with a formed grinding wheel based on analytical optimization 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 Completely coincides 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, used to describe the rotation axis of the grinding wheel; the coordinate plane X G Y G Is perpendicular to the grinding wheel axis, 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 performing envelope calculations on 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 let point P W be an arbitrary point on the rotary profile of the grinding wheel, with its rotary radius being R W , and the distance from it to the coordinate system plane X G Y G be h, and the angle between the line segment O G P W and the coordinate axis X G be the rotary 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 a kinematic model of grinding wheel grinding.

[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 it to 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 grinding pose of the formed grinding wheel 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 to rotate around the coordinate axis Z AThe rotation angle is the gyro angle ξ of the grinding wheel's helical motion, 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 helical motion with an equal lead relative to the cutting tool. Define the lead of the grinding wheel's helical motion as p h , and the taper of the helical motion as κ G , and its value is equal to the core thickness taper angle κ of the conical helical groove c . Then the gyro angle ξ of the grinding wheel's helical motion is expressed as:

[0046]

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

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

[0049]

[0050] Then the position of the profiling grinding wheel's grinding motion is expressed by the rotation matrix and the translation vector as follows:

[0051]

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

[0053]

[0054] 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 and the axis vector of the grinding wheel are (0, 0, 0) T and (0, 0, 1) T respectively. According to equation (8), the expressions of the origin O of the grinding wheel G and the grinding wheel axis vector Z G in the coordinate system ACS are established as:

[0055]

[0056] In the formula, (x OG_A , y,yOG_A , z OG_A ) is the origin O of the grinding wheel G is the spatial position coordinate, (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 W 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 and 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 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 exactly 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 ground conical spiral groove.

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

[0064]

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

[0066]

[0067] As can be seen from Equation (12), setting m = 0 gives the grinding curve of the grinding wheel under the reference axis section M, which is an expression related to the grinding wheel radius.

[0068] (2) Calculation of the rotational contour of the formed grinding wheel

[0069] Combined with the geometric constraint conditions of the axis section contour, the calculation method of using the grinding curve of the grinding wheel to envelope the axis section contour curve of the conical spiral groove is adopted to determine the coordinates of each point on the rotational contour of the formed grinding wheel, thereby obtaining its precise rotational contour; the specific solution for the contour of the formed grinding wheel is as follows:

[0070] 1) Tangency calculation between the grinding curve of the grinding wheel and the reference contour curve of the spiral groove axis section

[0071] According to the reference contour curve of the spiral groove axis section defined by Equation (1), by taking the derivative of the parameter u with respect to the point coordinates (x S_A , y S_A ) on the curve, the unit tangent vector τ S at point P S_A can be obtained, that is:

[0072]

[0073] Taking the partial derivative of Equation (12) with respect to the rotation angle of the grinding wheel, we get:

[0074]

[0075] In the formula,

[0076]

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

[0078]

[0079] Then the distance d between the grinding curve of the grinding wheel and the reference contour curve of the spiral groove axis section is expressed as:

[0080]

[0081] And the included angle θ between the vector τ G and the vector τ S is expressed as:

[0082]

[0083] When the grinding curve of the grinding wheel is tangent to the axis section contour curve of the spiral groove, its constraint conditions are that the coordinate values of the two curves at the common tangent point are equal and the tangent vectors are equal, that is, both the included angle θ and the distance d are 0, and its constraint conditions are expressed as:

[0084]

[0085] The problem of tangency solution is transformed into a numerical optimization problem. Taking the intersection distance d and the angle θ between the tangency vectors as the optimization objectives, a corresponding mathematical optimization model is established.

[0086] To improve the calculation efficiency, the coordinates are taken point by point on the reference profile curve S of the spiral groove axial section, and its tangent vector is calculated in advance. Substituting it into Equation (18) to eliminate the parameter u, so as to reduce the dimension of the variable; the mathematical optimization model after dimension reduction is shown as follows:

[0087]

[0088] Using an intelligent optimization algorithm to optimize the above formula as the objective function, the radius R W of the grinding wheel rotation profile point P W and its axial position h in the grinding wheel coordinate system GCS can be obtained.

[0089] 2) Direction judgment of the grinding wheel grinding curve and the groove profile curve

[0090] For the target optimization solution, Equation (19) can obtain two target solutions, that is, corresponding to two grinding wheel grinding curves tangent to the reference profile curve S of the spiral groove axial section. Therefore, it is necessary to constrain the solution range to select the grinding wheel grinding curve inside the reference profile curve of the spiral groove axial section.

[0091] Taking the direction angle of the intersection point of the grinding wheel grinding curve and the tool axial section circular profile as the judgment basis, the grinding wheel grinding curve is screened; establish an equation to solve the intersection points P G1 and P G2 .

[0092]

[0093] In the formula, r t_m is the radius of the outer circular profile of the tool axial section; when m = 0, r t_m is the tool circle radius under the reference axial section, and the tool circle radius under this section is defined as r t ; the tool profile taper is defined as k w , then the tool radius at any section is:

[0094] r t_m = r t_0 + m·tan(κ w ) (21)

[0095] Solving Equation (20), the intersection points P G1 and PG2 The coordinates (x G1 , y G1 ) and (x G2 , y G2 ) in the tool coordinate system ACS; define the intersection points P G1 and P G2 with the direction angle θ G , which is the angle between the line segment O A P G and the positive direction of the Y A axis. Then the corresponding direction angles θ G1 and θ G2 are calculated as follows:

[0096]

[0097] Compare the direction angles θ G1 and θ G2 of the axial section with the direction angles θ S1 and θ S2 at both ends of the reference profile curve of the helical groove axial section; if the direction angle of one of the grinding wheel curves is such that it represents that the grinding curve is outside the profile curve, then it is discarded, and another grinding wheel curve and the radius R W and the axial position h of any point P W corresponding to the grinding wheel rotation profile are selected, thereby obtaining the formed grinding wheel rotation profile.

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

[0099] The present invention proposes a calculation method for the formed grinding wheel rotation profile to achieve precise grinding of the given conical helical groove cross-section profile. First, a parametric geometric model of the formed grinding wheel and the conical helical 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 helical groove; secondly, based on the principle that the grinding wheel grinding curve is tangent to the cross-section profile curve, the formed grinding wheel rotation profile is calculated by means of analytical optimization; finally, the calculated grinding wheel profile is verified by actual processing. Through algorithm development and actual processing, a conical helical groove tool with a radius of 6 mm is successfully manufactured, and the machining accuracy error does not exceed 0.1 mm, verifying the correctness and effectiveness of the method. BRIEF DESCRIPTION OF THE DRAWINGS

[0100] Figure 1 It is a schematic diagram of the reference axial section coordinate system and the conical helical groove.

[0101] Figure 2 It is a schematic diagram of the formed grinding wheel coordinate system.

[0102] Figure 3Schematic diagram of the initial grinding pose of the formed grinding wheel.

[0103] Figure 4 Schematic diagram of the grinding motion of the formed grinding wheel.

[0104] Figure 5 Schematic diagram of the grinding curve of the grinding wheel in the axial section.

[0105] Figure 6 Schematic diagram of the grinding curve of the grinding wheel tangent to the reference contour curve of the axial section of the spiral groove.

[0106] Figure 7 Flow chart for calculating the rotation contour of the formed grinding wheel.

[0107] Figure 8 Shape of the cross-sectional contour.

[0108] Figure 9 Calculation results of the spiral groove envelope and the formed grinding wheel.

[0109] Figure 10 Direction diagram of the coordinate axes of the internal structure of the machine tool.

[0110] Figure 11 Cross-sectional contour after actual grinding. Specific implementation mode

[0111] The following further describes the present invention in detail with reference to the accompanying drawings and specific embodiments.

[0112] A method for grinding a conical spiral groove with a formed grinding wheel based on analytical optimization according to the present invention includes the following steps:

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

[0114] (1) Definition of the axial section

[0115] 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 brings great challenges 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 in the axial section, the complexity of the problem is significantly reduced, providing convenience for subsequent modeling and calculation.

[0116] To avoid the influence of the tool end edge structure on the geometric structure of the conical spiral groove and to 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.

[0117] (2) Definition of the axis cross-section coordinate system ACS

[0118] 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.

[0119] (3) Expression of the designed surface of the conical spiral groove

[0120] The designed surface of the conical spiral groove can be regarded as composed of multiple axis cross-section contours. The geometric shape of the designed surface of the cylindrical spiral groove is relatively simple, and usually the entire surface can be generated by the equal-profile sweeping of a single cross-section contour. However, due to the continuous change of the core thickness at the bottom of the groove of the conical spiral groove along the tool axis direction, the shape of each layer of cross-section is different, which increases the complexity of the expression of the designed surface.

[0121] 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:

[0122]

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

[0124] 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:

[0125]

[0126] In the formula, m is the axial position of other axis cross-sections relative to the reference axis cross-section.

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

[0128] (1) Definition of the formed grinding wheel coordinate system GCS

[0129] To accurately represent the rotational profile 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 profile 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 represent the rotational profile of the grinding wheel but also facilitate coordinate transformation with the axial section coordinate system ACS, thus realizing the motion control between the grinding wheel and the workpiece during the grinding process.

[0130] (2) Expression of the formed grinding wheel rotational profile

[0131] To facilitate the calculation of the formed grinding wheel rotational profile, the distances from the two end faces of the grinding wheel to the coordinate plane X G Y G are defined as h s , h e , respectively. That is, the effective width range of the formed grinding wheel rotational profile is [h s , h e . When constructing the grinding wheel rotational profile, the present invention adopts the method of discretizing the grinding wheel into slices along the grinding wheel axis, and constructs the complete rotational profile of the grinding wheel by enveloping and calculating the actual radii corresponding to each slice within the effective width range of the grinding wheel.

[0132] Define the formed grinding wheel rotational profile curve as W. Let point P W be any point on the grinding wheel rotational profile, with its rotational radius being R W , and the distance from it to the coordinate plane X G Y G being h. 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:

[0133]

[0134] Step 3: Establish a kinematic model of the grinding wheel.

[0135] (1) Definition of the initial grinding pose of the formed grinding wheel

[0136] 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 .

[0137] 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 .

[0138]

[0139] (2) Expression of the grinding motion of the formed grinding wheel

[0140] 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 .

[0141] 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 translation along the axis and 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 perform translation and rotation 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.

[0142] Define the coordinate system GCS to rotate around the axis Z AThe 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 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:

[0143]

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

[0145]

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

[0147]

[0148] 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:

[0149]

[0150] 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 expressions of the origin O of the grinding wheel G and the grinding wheel axis vector Z G in the coordinate system ACS as:

[0151]

[0152] 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:

[0153]

[0154] Step 4: Calculate the rotating contour of the profile grinding wheel.

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

[0156] 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 with 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.

[0157] 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 form the axial section contour of the ground conical spiral groove.

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

[0159]

[0160] Substitute Equation (11) into x W_A and y W_A in Equation (10), that is, the grinding curve G of the rotation surface of the grinding wheel on any axial section M m of the tool is obtained, and its coordinates are expressed in the coordinate system ACS as:

[0161]

[0162] 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.

[0163] (2) Calculation of the formed grinding wheel rotation contour

[0164] The formed grinding wheel contour and the grinding movement together determine the grinding accuracy of the conical spiral groove. In order to obtain the formed grinding wheel rotation contour, the present invention combines the geometric constraint conditions of the axial section contour and adopts a calculation method for the envelope of the grinding curve of the grinding wheel to the axial section contour curve of the conical spiral groove to determine the coordinates of each point on the formed grinding wheel rotation contour, thereby obtaining its accurate rotation contour. The calculation process of the formed grinding wheel rotation contour is as shown in Figure 7 shown, specifically as follows:

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

[0166] According to the reference profile curve of the spiral groove axial section defined by Equation (1), the unit tangent vector τ at point P can be obtained by differentiating with respect to the parameter u S That is: S_A

[0167]

[0168] Taking the partial derivative of Equation (12) with respect to the rotation angle of the grinding wheel, we get:

[0169]

[0170] Wherein,

[0171]

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

[0173]

[0174] Then the distance d between the grinding curve of the grinding wheel and the reference profile curve of the spiral groove axial section is expressed as:

[0175]

[0176] And the included angle θ between the vector τ G and the vector τ S is expressed as:

[0177]

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

[0179]

[0180] Due to the complexity of the non-linear solution problem, it is difficult to explicitly express the analytical solution formula of Equation (18). There are great difficulties in calculating the radius R at tangency by numerically solving Equation (18). In view of this, the present invention transforms the tangency solution problem into a numerical optimization problem, takes the intersection point spacing d and the included angle θ between the tangency vectors as the optimization objectives, and establishes a corresponding mathematical optimization model. W

[0181] ​To improve the computational efficiency, the coordinates of each point on the reference contour curve S of the spiral groove axial section are taken one by one, and its tangent vector is calculated in advance. Substitute it into Equation (18) to eliminate the parameter u, so as to reduce the dimension of the variable. The reduced mathematical optimization model is shown as follows:

[0182]

[0183] Using an intelligent optimization algorithm to optimize the above equation as the objective function, the radius R W of the grinding wheel rotation contour point P W and its axial position h in the grinding wheel coordinate system GCS can be obtained.

[0184] 2) Direction judgment of the grinding wheel grinding curve and the groove profile curve

[0185] For the target optimization solution, Equation (19) can obtain two target solutions, that is, corresponding to two grinding wheel grinding curves tangent to the reference contour curve S of the spiral groove axial section, as Figure 6 shown. Therefore, it is necessary to constrain the solution range to select the grinding wheel grinding curve inside the reference contour curve of the spiral groove axial section.

[0186] Taking the direction angle of the intersection point of the grinding wheel grinding curve and the tool axial section circular contour as the judgment basis, screen the grinding wheel grinding curve; establish an equation to solve the intersection points P G1 and P G2 of the two grinding wheel grinding curves and the outer circular contour of the tool axial section.

[0187]

[0188] In the formula, r t_m is the radius of the outer circular contour of the tool axial section; when m = 0, r t_m is the tool circle radius under the reference axial section, and the tool circle radius under this section is defined as r t ; define the tool profile taper as k w , then the tool radius at any section is:

[0189] r t_m = r t_0 + m·tan(κ w ) (21)

[0190] Solve Equation (20) to obtain the coordinates (x G1 , y G2 ) and (x G1 , y G1 ) of the intersection points P G2 and P G2 in the tool coordinate system ACS; define the direction angles of the intersection points P G1 and P G2 as θG , which is O A P G The included angle between the line segment and the positive direction of the Y A axis, then the corresponding direction angle θ G1 and θ G2 are calculated as follows:

[0191]

[0192] Compare the direction angles θ G1 and θ G2 of the axis section with the direction angles θ S1 and θ S2 at both ends of the reference contour curve of the helical groove axis section; if the direction angle of one of the grinding wheel grinding curves then it means that this grinding curve is outside the contour curve and is discarded. Select another grinding wheel grinding curve and the radius R W of any point P W on the corresponding grinding wheel rotation contour and the axial position h, so as to obtain the formed grinding wheel rotation contour.

[0193] Example:

[0194] 1. Section contour design

[0195] Taking a vertical milling cutter with a cylindrical core and a thick taper as an example, this embodiment illustrates the feasibility of the proposed analytical optimization calculation method for the grinding wheel contour of the conical helical groove through a set of examples. The given discrete point contours of the relevant helical grooves are shown in Table 1.

[0196] Table 1 Coordinates of some section contour points

[0197]

[0198]

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

[0200] Then the parameter values of the three arcs of groove I are shown in Table 2, and the range of δ in the table is expressed in angular measure.

[0201] Table 2 Parameter values of the three arcs of groove I

[0202]

[0203] The design of grinding process parameters needs to consider the overall tool parameters (r t , κ w ), grinding process parameters (m, d X , α, κ G , p h ) when grinding the spiral groove cross-section profile using the analytical optimization method. Based on the existing experimental conditions, the selected blank radius R of the grinding wheel in this embodiment g = 102.5 mm, and the installation height d X of the grinding wheel is set to be equal to the blank radius R g of the grinding wheel. The given position of the spiral groove cross-section profile is m = 0, that is, the tool end face. Then, all the settings of the grinding process parameters are summarized in Table 3.

[0204] Table 3 Overall tool and grinding process parameters of the grinding wheel

[0205]

[0206] 2. Analysis of calculation results

[0207] Through the setting of the grinding process parameters, the envelope calculation results of the spiral groove and the corresponding formed grinding wheel rotation profile are as Figure 9 shown. It can be seen that a smooth formed grinding wheel rotation profile is obtained, and some grinding wheel profile points are shown in Table 4.

[0208] Table 4 Some grinding wheel profile points

[0209]

[0210] In summary, through the comparison of the error results of the spiral groove envelope, the correctness and effectiveness of the formed grinding wheel profile calculation algorithm are verified. In order to further verify the correctness of the calculation of the cross-section profile algorithm for each layer, a simulation and actual machining verification are carried out here.

[0211] For the above-given spiral groove cross-section profile, an actual machining verification is carried out. The verification will be carried out on the MG05 CNC groove grinding machine, and the internal structure of this CNC grinding machine is as Figure 10 shown, corresponding to the 3D simulation model of the machine tool.

[0212] The spiral groove cross-section profile is actually ground, and the spiral groove cross-section profile is obtained. The PG1000 tool detector is used to measure and compare the tool spiral groove cross-section profile. The ground cross-section profile is as Figure 11 shown. The yellow line in the figure is the given cross-section profile, and it can be seen that it coincides highly with the ground cross-section profile.

[0213] In summary, based on the given spiral groove cross-sectional profile and grinding trajectory, the rotary profile of the formed grinding wheel is calculated, and then the spiral groove of the tool is ground using this formed grinding wheel, obtaining a result consistent with the given cross-sectional profile, verifying the correctness and engineering practicability of the calculation of the rotary profile of the formed grinding wheel and the calculation algorithm for any cross-sectional profile.

Claims

1. A method for grinding a conical spiral groove using a profiled grinding wheel based on analytical optimization, characterized in that: The following steps are involved: Step 1: Establish the geometric model of conical spiral groove; (1) Definition of axial section Define the plane perpendicular to the tool axis as the axis section, and select a tool axis section with a complete spiral groove intersection profile as the reference axis section, denoted as M; (2) Definition of axial section coordinate system ACS Coordinate origin O A Located at the center of the reference axis section M, coordinate axis Z A Coincident with the tool axis direction, the positive direction is from the small diameter end of the conical end mill to the large diameter end, the coordinate plane X A Y A Completely coincide with the reference axis section M; (3) Surface expression of conical spiral groove design Define the intersection profile of the conical spiral groove design surface and the reference shaft section M as the shaft section reference profile curve, denoted by S; define any point on the reference profile curve S as P S , its coordinates are expressed in the axial section coordinate system ACS as follows: Where, u is the spiral groove profile curve function variable; Similarly, define the other axis section profile curve L of the conical spiral groove m , then the design surface of the conical spiral groove is expressed as: Where 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 GCS coordinate system for forming grinding wheel Coordinate system origin O G Located on the grinding wheel axis, serving as the grinding wheel origin; coordinate axis Z G Coincident with the grinding wheel axis, used to describe the rotation axis 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) Expression of the rotation profile of the forming grinding wheel Define the two sides of the grinding wheel and the coordinate plane X G Y G The distances are h s 、h e , that is, the effective width range of the rotating profile of the forming grinding wheel is [h s ,h e ]; When constructing the grinding wheel rotation profile, the grinding wheel is discretized into slices along the grinding wheel axis, and the actual radius corresponding to each slice within the effective width of the grinding wheel is calculated by enveloping, so as to construct the complete rotation profile of the grinding wheel; Define the profile curve of the grinding wheel rotation as W, set point P W is any point on the grinding wheel's rotating profile, and its rotation radius is R W , and the coordinate plane X G Y G The distance between the line segment O and G P W With coordinate axis X G The angle is the grinding wheel rotation angle Then point P W The coordinates are expressed in the grinding wheel coordinate system as: Step 3: Establish the grinding wheel grinding kinematics model; (1) Definition of initial grinding posture of forming grinding wheel When the grinding wheel is in the initial grinding position, its origin O G Located in X A On the coordinate axis and with the origin O A The distance is d X , coordinate axis X G With X A The direction is consistent, the coordinate axis Z G With Z A The angle is the grinding wheel installation angle α; The geometric transformation relationship between the grinding wheel coordinate system GCS corresponding to the initial grinding wheel grinding posture and the reference axis section coordinate system ACS is expressed as the rotation matrix R x and the translation vector T x ; (2) Expression of grinding motion of profiled grinding wheel The grinding wheel grinding trajectory is defined as a conical spiral motion. The grinding process of the spiral groove is considered to be that the tool remains fixed, while the grinding wheel performs a conical spiral motion based on its initial grinding posture. This motion process is described as the grinding wheel origin O G Conical spiral motion is performed relative to the reference axis section coordinate system ACS; Define the coordinate system GCS around the coordinate axis Z A The 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: In the formula, z p O is the origin of the grinding wheel coordinate system G The axial movement distance of the grinding wheel 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 ; Set the grinding wheel origin O G The distances moved relative to the coordinate system ACS are Δx, Δy, and Δz, namely: The position of the grinding motion of the profile grinding wheel is expressed by the rotation matrix and the translation vector as shown below: 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: 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: In the formula, (x OG_A ,y OG_A ,z OG_A ) is the grinding wheel origin O G The spatial position coordinates, (i G_A ,j G_A ,k G_A ) is the grinding wheel axis vector; Furthermore, the combined equations (5)-(7) show that during the grinding process, any point P on the rotating surface of the forming grinding wheel W The coordinates of are expressed in the coordinate system ACS as: Step 4: Calculate the rotation profile of the forming grinding wheel; (1) Calculation of grinding wheel grinding curve based on shaft section The grinding wheel grinding curve refers to the trajectory curve formed by the intersection of the rotation circle of any point on the grinding wheel rotation profile and the shaft section during the grinding movement of the grinding wheel; the grinding wheel rotation surface is discretized according to the axial distance h, and it is divided into a series of grinding wheel profile rotation circles, and they are made to participate in the grinding movement of the grinding wheel; in this process, these grinding wheel profile rotation circles intersect with the shaft section, and their intersection trajectories constitute grinding curves corresponding to each grinding wheel profile rotation circle on the shaft section; multiple grinding curves formed by all grinding wheel profile rotation circles are collected together to form a complete grinding curve family; The process of grinding the conical spiral groove of the tool with a grinding wheel is regarded as a Boolean subtraction operation between two entities; the part of the grinding wheel grinding curve within the tool cross-section circle is the intuitive reflection of the tool part actually cut off by the grinding wheel during the grinding process; the grinding curve and its envelope together constitute the cross-section profile of the conical spiral groove shaft 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: From formula (12), we can see that by setting m = 0, we can get the grinding curve of the grinding wheel under the reference shaft section M, which is an expression of the grinding wheel radius; (2) Calculation of the rotation profile of the forming grinding wheel Combined with the geometric constraints of the shaft section profile, the calculation method of the grinding wheel grinding curve enveloping the conical spiral groove shaft section profile curve is adopted to determine the coordinates of each point on the forming grinding wheel rotation profile, so as to obtain its precise rotation profile; the specific solution of the forming grinding wheel profile is as follows: 1) Tangent calculation of grinding wheel grinding curve and reference profile curve of spiral groove shaft section According to the spiral groove shaft section reference profile curve defined by formula (1), the coordinates of the points on the curve (x S_A ,y S_A ) Taking the derivative of parameter u, we get point P S The unit tangent vector τ at S_A ,Right now: Apply formula (12) to the grinding wheel rotation angle Taking partial derivatives, we get: In the formula, Then any point P on the grinding curve G of the grinding wheel G The unit tangent vector at is expressed as τ G_A : Then the distance d between the grinding wheel grinding curve and the reference profile curve of the spiral groove shaft section is expressed as: And the vector τ G With the vector τ S The angle θ is expressed as: When the grinding wheel grinding curve is tangent to the axial section profile curve of the spiral groove, the constraint condition is that the coordinate values ​​of the two curves at the common tangent point are equal, and the tangent vectors are equal, that is, the angle θ and the distance d are both 0. The constraint condition is expressed as: The tangency solution problem is transformed into a numerical optimization problem, and the intersection distance d and the angle θ between the tangent vectors are taken as the optimization targets, and the corresponding mathematical optimization model is established; In order to improve the calculation efficiency, the coordinates of each point on the reference profile curve S of the spiral groove shaft section are taken and the tangent vector is calculated in advance. The coordinates are substituted into equation (18) to eliminate the parameter u to reduce the dimension of the variable. The mathematical optimization model after dimensionality reduction is shown as follows: The above formula is optimized as the objective function using the intelligent optimization algorithm, and the grinding wheel rotation contour point P is obtained. W The radius R W and its axial position h in the grinding wheel coordinate system GCS; 2) Determination of the direction of the grinding wheel grinding curve and the groove profile curve Solving equation (19) for target optimization can obtain two target solutions, which correspond to two grinding wheel grinding curves that are tangent to the reference profile curve S of the spiral groove shaft section. Therefore, it is necessary to constrain the solution range and select the grinding wheel grinding curve that is inside the reference profile curve of the spiral groove shaft section. The direction angle of the intersection of the grinding wheel grinding curve and the circular contour of the tool axis section is used as the judgment basis to screen the grinding wheel grinding curve; an equation is established to solve the intersection point P of the two grinding wheel grinding curves and the outer circular contour of the tool axis section. G1 and P G2 ; In the formula, r t_m is the outer circle contour radius of the tool axis section; when m=0, r t_m is the tool circle radius under the reference axis section, and the tool circle radius under this section is defined as r t ; Define the tool profile taper as k w , then the tool radius at any section is: r t_m =r t_0 +m·tan(κ w ) (21) Solve equation (20) to obtain the intersection point P G1 and P G2 The coordinates (x 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, then 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 with the two end points of the reference profile curve of the spiral groove shaft section S1 and θ S2 For comparison; if the direction angle θ of one of the grinding wheel grinding curves G1 , This means that the grinding curve outside the contour curve is discarded, and another grinding wheel grinding curve and the corresponding grinding wheel rotation contour arbitrary point P are selected. W The radius R W and axial position h, thereby obtaining the rotation profile of the forming grinding wheel.

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