A partition type arc head rotary file grinding track calculation method

Abstract

The application discloses a kind of partition type arc head rotary file grinding track calculation methods, specifically: first, the structural characteristic parameters of partition type arc head rotary file and related coordinate system are defined, and the mathematical model of the peripheral edge line and end edge line of partition type arc head rotary file is established;Second, the reference grinding posture of grinding wheel is defined;On this basis, the motion mode of grinding wheel is described by using coordinate transformation matrix, and the calculation of grinding position and posture of grinding wheel is derived by means of kinematics principle.The method has the characteristics of good structural parameter adaptability, flexible partition adjustment, etc., and the edge line model of partition type arc head rotary file based on workpiece coordinate system can be obtained, which can meet the design and processing requirements of arc head rotary file.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of rotary file structure design and grinding manufacturing, and particularly relates to a grinding track calculation method for a partition type circular arc head rotary file. BACKGROUND

[0002] As a new type of cutter, the rotary file has the characteristics of high production efficiency, good machining quality, long service life, etc. [1] It is used for surface finishing of various parts, cleaning of burrs and welds of welding parts, and can also be used for machining of precise mold cavities and complex curved surfaces of impeller flow passage parts. It has been widely used in the manufacturing of ships, aircraft, molds and mechanical equipment. [2] [3] The overall size of the rotary file is relatively small, and there are various types and specifications, each with its own characteristics. The blade curve is a helix with smooth and same direction without intersection, and the number of teeth is large. The complexity of the blade curve increases the difficulty of grinding and manufacturing of the rotary file cutter. [4] [5] In the processing and manufacturing of rotary file cutters, the research on blade curve and grinding processing track planning and multi-axis numerical control processing technology is carried out, which provides important theoretical basis for improving the efficiency, cutting performance and machining quality of rotary file manufacturing.

[0003] Many scholars have carried out related research work on the planning of complex cutter blade lines including flat head rotary file, arc type rotary file and ball head rotary file. For example, Zhou Chongxiu et al. proposed the determination of the tooth groove forming principle and the grinding system motion function of the cutter for three different forms of plane curve type, equal helix angle and composite blade, which realized the diversity of the profile of the rotary surface cutter blade. [8] [9] Liu Huran carried out the forming theory research of partition type special rotary surface cutter.

[10] Tang Yifeng

[11] et al. studied the motion of each axis of the special numerical control machine tool for spiral file, realized the smooth transition of cylinder and ball head through the interpolation fitting of blade curve, and realized the motion track calculation through coordinate transformation. However, this method has large amount of calculation, and interference or overcutting is easy to occur in the processing process.

[0004] There is also a certain degree of research on the grinding manufacturing of rotary file. Liu Huran

[12]

[13] carried out simple modification on the tool grinder, and realized the semi-automatic processing of conical and arc rotary files through the forming method. Zhang Shichang and Li Guoqin

[14] et al. analyzed the geometric model of ball head rotary file under the condition of reducing the swing of grinding tool, and deduced the blade grinding motion model combined with the structure of four-coordinate machine tool, realized the blade grinding of rotary file on multi-coordinate linkage control machine tool, and increased the feasibility of four-coordinate numerical control blade grinding technology of rotary file.

[0005] However, there is little research on the circular arc head partition type rotary file, and there is even less literature on the partition method of circular arc head rotary file.

[0006] REFERENCES

[0007] [1] Kangjia. Research on programming technology of medical dental rotary file blade grinding [D]. Hubei University, 2018.

[0008] [2] Lin Xue, Tian Fengjie, Li Lun. Application of rotary file in complex curved surface processing [J]. Optoelectronic Technology Application, 2019, 34(05): 72-76.

[0009] [3] Song Qing. Research on automatic programming technology of multi-axis NC machining of partition type rotary file [D]. Huazhong University of Science and Technology, 2014.

[0010] [4] Zhao Jun. Development and research of rotary file NC grinding system based on IPC [D]. Tianjin University, 2004.

[0011] [5] Guo Weijun. Design and implementation of rotary file NC grinding machine control system [D]. Tianjin University, 2007.

[0012] [8] Zhou Changxiu. Principle of grinding special rotary surface cutter with equal helix angle blade [J]. Journal of Southeast University, 1992(01): 8-15.

[0013] [9] Zhou Changxiu. Forming principle of tooth slot of special rotary surface cutter with composite blade [J]. Journal of Southeast University, 1992(03): 68-74.

[0014]

[10] Liu Huran. Forming principle of special rotary surface cutter with partition type (staggered tooth) [J]. Mechanical Design and Manufacturing, 1995(05): 33-36.

[0015]

[11] Tang Yifeng, Chen Xinhu, Zhou Zhongwang, Zhao Junsheng. Research on NC machining method of rotary file combined with spherical surface and cylindrical surface [J]. Manufacturing Automation, 2010, 32(08): 188-190.

[0016]

[12] Liu Huran, Le Duiqian. Semi-automatic NC machining of arc rotary file [J]. Ordnance Automation, 2004(04): 42.

[0017]

[13] Liu Huran, Le Duiqian. Two coordinate machining of conical rotary file [J]. Ordnance Automation, 2004(04): 63.

[0018]

[14] Zhang Shichang, Li Guoqin. Research on four coordinate NC grinding technology of rotary file [J]. Tool Technology, 2001(03): 16-18. SUMMARY

[0019] In order to solve the problem of the poor cutting condition caused by the small top chip space and the intersection of the teeth of the common rotary file at the tool vertex, and improve the grinding processing quality of the arc head, the application provides a partition type arc head rotary file grinding track calculation method.

[0020] The partition type arc head rotary file grinding track calculation method of the application comprises the following steps:

[0021] Step 1: definition of the geometric parameters of the partition type arc head rotary file.

[0022] The tool starting rotation radius R w , that is, the tool radius at the starting position of the peripheral tooth helical edge;

[0023] The tool taper angle kappa: the included angle between the outer contour of the tool rotation body and the tool rotation center axis;

[0024] The peripheral edge length L w : the length of the peripheral tooth along the tool axis direction;

[0025] The peripheral edge helix angle beta: the included angle between the peripheral edge rotation contour generatrix and the peripheral edge edge line tangent vector;

[0026] The tooth angle alpha: the included angle of the center of the corresponding points of the two adjacent tool edges of the rotary file;

[0027] The cylindrical starting tooth depth d1: defined as the depth of the workpiece ground by the grinding wheel at the starting position of the cylindrical;

[0028] The end edge arc edge starting tooth depth d2: defined as the depth of the workpiece ground by the grinding wheel at the starting position of the end edge arc edge;

[0029] The end edge arc edge terminal tooth depth d3: defined as the depth of the workpiece ground by the grinding wheel at the terminal position of the end edge arc edge;

[0030] The tool top end tooth depth d4: defined as the depth of the workpiece ground by the grinding wheel at the top end position of the tool.

[0031] Step 2: definition of the coordinate system.

[0032] The workpiece coordinate system WCS

[0033] The workpiece coordinate system O W -X W Y W Z W , which takes the tool axis as the coordinate axis Z W , takes the origin O W , and takes the straight line pointing to the starting point of the helical edge as the coordinate axis X W .

[0034] The end edge coordinate system DCS

[0035] The end edge coordinate system OD -X D Y D Z D , the workpiece coordinate system rotates around the Z W axis by the rotation angle φ, and then moves along the Z W axis in the positive direction by L. w .

[0036] Step 3: Coordinate system transformation;

[0037] Define M D→W , T D→W as the rotation matrix and translation matrix from the end blade coordinate system to the workpiece coordinate system, respectively, then:

[0038] (1)

[0039] (2)

[0040] In the formula,

[0041] (3)

[0042] Step 4: Parametric modeling of the rotating blade line of the zoned circular arc head file, which divides the blade line into a peripheral blade and an end blade.

[0043] S4.1 Peripheral blade line model of zoned circular arc head file.

[0044] The peripheral blade line on the cylindrical surface is established in the workpiece coordinate system O W -X W Y W Z W , with the coordinate value z of the Z W axis as the independent variable, then the expression of the coordinates of any point P1 on the curve is as follows:

[0045] (4)

[0046] φ represents the rotation angle of the blade point P1 relative to the starting point of the blade line in the workpiece coordinate system, and its expression is:

[0047] (5)

[0048] In the formula, the independent variable z is the coordinate value of the Z W axis.

[0049] S4.2 Peripheral blade line model of zoned circular arc head file.

[0050] First, define the number of end-edge sections of the rotary file as m, and the number of teeth in each section as n. Then, use the method of finding the intersection line between a non-orthogonal helical rotation surface and a circular rotation surface. By introducing the latitude angle θ as the independent variable, establish its relationship with the rotation angle φ. r The relationship is used to obtain the expression for the arc-shaped cutting edge, which is then described in the end-edge coordinate system.

[0051] S4.2.1 Circular Arc Revolute Curve Section

[0052] This curve is obtained by the intersection of a circular arc surface of revolution and a non-orthogonal helical surface. The circular arc cutting edge curve is considered as a generalized helical line on the circular arc surface of revolution, and the circular arc cutting edge line is also considered as a curve formed by a moving point P moving on the circular arc surface of revolution according to a certain rotational law; then the coordinates of any cutting edge point P0 on this curve segment are expressed as:

[0053] (6)

[0054] In the formula, θ is the latitude angle, r is the radius of the end-edge arc, and R is the center distance of the arc. Its expression is:

[0055] (7)

[0056] φ r This represents the rotation angle at the cutting edge point P0 of the arc-shaped blade. Written in the form of latitude angle A function with parameters, where different cutting edges will produce different φ values ​​depending on the tool's partitioning. r The expression form is used to obtain φ. r The expression can then be used to obtain the circular arc revolution surface curve model.

[0057] (1) For the calculation of the rotation angle of the cutting edge point on the main cutting edge, since the cutting edge curve of the main cutting edge needs to pass through the center of the top of the rotating file, the rotation angle φ of the main cutting edge arc curve is obtained. r Expressed as:

[0058] (8)

[0059] (2) For the calculation of the rotation angle of the cutting edge point on the secondary cutting edge, the secondary cutting edge curve has an eccentricity, which does not pass through the center of the top of the rotating file. At this time, the rotation angle of the cutting edge point also needs to be added to the pitch angle between the cutting edges. Therefore, the rotation angle of the arc cutting edge curve of the secondary cutting edge is... Expressed as:

[0060] (9)

[0061] In the formula, h represents the eccentricity of the secondary cutting edge line, and n×m represents the total number of teeth on the rotary file.

[0062] S4.2.2 Planar curve section

[0063] The curve P3P4 is established on the plane M passing through the point P3 and tangent to the coordinate plane X D O D Y D The curve section can be regarded as a circular arc with the projection point O D O D Y D of the point P3 on the coordinate plane X r as the center; since the point P3 is the highest point of the circular arc rotary surface, the tangent vector F p1 of the curve P3P4 at the point P3 must be located on the plane M, and the point is located on the straight generatrix, which is also located on the plane, so that the connection from the curve section P2P3 to the curve section P3P4 is smooth, and the expression of the blade point P0 of the curve section is:

[0064] (10)

[0065] wherein η is the blade in-dip angle of the circular arc, and φ r (π / 2) represents the rotary angle at the blade point P3 when θ=π / 2, and φ h represents the included angle between the plane M and the X D axis, which is also divided into the expressions of the main blade and the secondary blade.

[0066] (1) The plane M of the main blade circular arc blade curve passes through the center of the top of the rotary file tool, and the included angle between the plane M and the X D axis, that is, the rotary angle φ h0 at the highest point P3 of the circular arc rotary surface, is:

[0067] (11)

[0068] Therefore, the expression of the planar curve of the main blade circular arc curve section is:

[0069] (12)

[0070] For the secondary blade blade line, since there is an eccentric amount, the plane M does not pass through the center of the top of the rotary file tool, the plane M passes through the point P3 and is tangent to the eccentric distance cylinder, and the curve P3P4 is established on the plane; therefore, the expression of the planar curve on the secondary blade curve is:

[0071] (13)

[0072] wherein φ h1 is the included angle between the plane M and the X D axis of the secondary blade circular arc blade curve, and the expression is:

[0073] (14)

[0074] S4.2.3 Straight blade portion

[0075] The straight blade portion is established on the plane M, in order to ensure that the straight blade and the end blade circular blade edge line are smoothly linked, the straight blade is along the vector direction of the end point of the circular blade, and the main blade and the auxiliary blade are also described separately. The straight blade is defined as a straight line segment P4P5 on the plane M. It is known that the coordinates of the end point P4 on the end blade plane curve are:

[0076] (15)

[0077] The tangent vector F at the end point P4 of the end blade plane curve is: t1_D

[0078] (16)

[0079] Introducing the independent variable L, the expression of the blade point P0 on the straight blade is:

[0080] (17)

[0081] In the formula, L h is the length of the end blade straight blade, and the mathematical expressions of the straight blade lengths of the main blade and the auxiliary blade are different. The main blade and the auxiliary blade are described separately.

[0082] (1) For the partition main blade, the blade curve has no eccentricity, and the plane M on which the straight blade lies passes through the top center of the circular arc head rotating tool, and also passes through the end point of the circular blade plane curve and the center of the end blade coordinate system. Through geometric relationship derivation, the expression of the straight blade length of the partition main blade is:

[0083] (18)

[0084] (2) For the auxiliary blade edge line, since there is eccentricity, the plane M does not pass through the rotating file tool top center, and the i-th auxiliary blade intersects with the next main blade of the partition at a point. The spatial distance between the intersection point P5 and the end point P4 of the auxiliary blade straight blade is the length of the auxiliary blade straight blade. Its expression can be derived from the geometric relationship as:

[0085] (19)

[0086] L i represents the projection segment of the partition auxiliary blade straight blade on the end blade coordinate system plane X D O D Y D length, which is expressed as: ​​

[0087] (20)

[0088] where k i is the straight line length of the main blade intersected by the sub-blade in the end-blade coordinate plane X D O D Y D The slope of the upper straight line projection can be obtained by the relative position between the sub-blade and the main blade and the indexing relationship of the rotary surface, which is expressed as:

[0089] (21)

[0090] where m represents the number of the rotary file tooth partition groups.

[0091] The straight line length of each sub-blade straight line can be obtained by simultaneously calculating formula (18), formula (19), formula (20) and formula (21). The expression of the straight line length of the sub-blade straight line is derived by geometric relationship, which is:

[0092] (22)

[0093] S4.2.4 Transformation to the workpiece coordinate system

[0094] The coordinates of any point on the partitioned blade line of the above end-blade part are transformed by the coordinate system transformation matrices (1) and (2), so as to obtain the complete expression of the main blade and each sub-blade blade line in the workpiece coordinate system within a partition. The blade lines of adjacent partitions can be rotated by an angle of 2π / m after returning to the initial machining position, so as to obtain the main blade and sub-blade blade lines of the next partition, and m is the number of rotary file partitions.

[0095] Step 5: Definition of the initial posture of the grinding wheel.

[0096] Reference grinding posture of the peripheral blade grinding wheel

[0097] The reference grinding posture of the grinding wheel is defined in the coordinate system WCS, and the center coordinates O g of the end face of the grinding wheel and the axial vector F g are used to describe the posture of the grinding wheel; the tangent vector F t at the P1 point is used as the tangent vector of the grinding wheel, which is expressed in the coordinate system WCS as:

[0098] (23)

[0099] The vector of the axis pointing to the coordinate Z W at the P1 point is used as the radial vector F b of the grinding wheel, which is expressed in the coordinate system WCS as:

[0100] (24)

[0101] wheel axis vector F g with tangent vector F t wheel radial vector F b are perpendicular to each other, which are expressed in the coordinate system WCS as:

[0102] (25)

[0103] reference grinding pose of the face wheel

[0104] grinding pose of the face wheel with respect to the face wheel arc curve, with the tangent vector F t as the tangent vector in the grinding process, which is expressed in the coordinate system DCS as:

[0105] (26)

[0106] where φ r represents the rotation angle at the blade line point P0, .

[0107] with the center O r of the arc pointing to the vector of the P2 point on the arc blade curve as the radial vector F b of the wheel, the coordinates of the midpoint O r of the arc are expressed as:

[0108] (27)

[0109] radial vector F b is expressed in the coordinate system DCS as:

[0110] (28)

[0111] where, .

[0112] wheel axis vector F g with tangent vector F t of the wheel and radial vector F b of the wheel, which are expressed in the coordinate system DCS as:

[0113] (29)

[0114] Step 6: Calculate the grinding pose of the wheel.

[0115] (a) Grinding pose of the wheel with peripheral blade rake angle

[0116] Define the wheel rake angle δ as the tangent vector F tis the angle of rotation of the rotating axis; let Rot(N, a) be the transformation matrix of rotating the vector N by an angle a, which is expressed as:

[0117] (30)

[0118] where , i N , j N , k N are the components of the vector N in the three directions, respectively, and a is the angle of rotation.

[0119] After introducing the wheel angle δ, the radial vector F b and the wheel axis vector F g are transformed into F′ b and F′ g , respectively, and the radial vector F b is expressed in the WCS coordinate system as:

[0120] (31)

[0121] The wheel axis vector F g is expressed in the WCS coordinate system as:

[0122] (32)

[0123] (b) End edge wheel grinding posture

[0124] Similarly, in the end edge wheel grinding posture, the wheel angle δ is also introduced. After introducing the wheel angle δ, the radial vector F b and the wheel axis vector F g are transformed into F′ b and F′ g , respectively, and are expressed in the DCS coordinate system as:

[0125] (33)

[0126] (34)

[0127] The wheel axis vector F′ g can be calculated as:

[0128] (35)

[0129] Step 7: Wheel grinding track calculation.

[0130] (1) Calculation of the peripheral edge wheel grinding track

[0131] According to the grinding posture definition, based on the workpiece coordinate system, with the grinding wheel end face center point O... g The coordinates describe the grinding position of the grinding wheel, with the constraint that the large end face of the grinding wheel is always in contact with the cutting edge during the grinding process; thus, the center point O of the grinding wheel end circle can be obtained. g The coordinates in the Zhouren coordinate system are expressed as follows:

[0132] (36)

[0133] Among them, R g Let d be the circumference radius of the large end face of the grinding wheel. 12 The grinding depth of the peripheral cutting edge of the rotary file, which is consistent with the tooth depth d1 of the peripheral cutting edge and the initial tooth depth d2 of the end cutting edge arc, is expressed as follows:

[0134] (37)

[0135] Where, k s The cone angle of the grinding wheel is represented by γ, the rake angle is represented by m×n, and the total number of teeth of the rotating file is represented by m×n.

[0136] (2) Calculation of grinding trajectory of end-edge grinding wheel

[0137] For the description of the grinding wheel trajectory of the end-edge portion, based on the definition of the end-edge cutting line and the grinding wheel posture, the grinding wheel center point O of this segment of the grinding wheel trajectory is... g The coordinates in the end-edge coordinate system are expressed as follows:

[0138] (38)

[0139] Among them, R g Let d be the circumference radius of the large end face of the grinding wheel. 24 Let d2 be the initial tooth depth of the rotary file's end cutting edge, and d3 be the final tooth depth of the arc-shaped cutting edge. Therefore, the grinding depth of the end cutting edge is expressed as:

[0140] (39)

[0141] The beneficial technical effects of this invention are as follows:

[0142] The method of this invention features good adaptability of structural parameters and flexible adjustment of partitions, and can obtain a partitioned circular arc head rotary file cutting edge model based on the workpiece coordinate system. It can meet the design and machining requirements of a circular arc head rotary file. Attached Figure Description

[0143] Figure 1 This is a schematic diagram of the parameters for a rotary file with an arc head.

[0144] Figure 2 This is a schematic diagram of the coordinate system.

[0145] Figure 3 End section view of the zoned circular head rotary file.

[0146] Figure 4 Modeling view of the peripheral edge edge line.

[0147] Figure 5 Modeling view of the end edge circular arc surface curve portion.

[0148] Figure 6 Modeling view of the main edge plane curve portion.

[0149] Figure 7 Modeling view of the auxiliary edge plane curve portion.

[0150] Figure 8 Modeling view of the main edge straight edge portion.

[0151] Figure 9 Modeling view of the auxiliary edge straight edge portion.

[0152] Figure 10 Reference grinding posture view of the peripheral edge of the circular arc type rotary file.

[0153] Figure 11 Reference grinding posture view of the end edge of the circular arc type rotary file.

[0154] Figure 12 Peripheral edge grinding posture view after adding the wheel rake angle.

[0155] Figure 13 End edge grinding posture view after adding the wheel rake angle.

[0156] Figure 14 VERICUT grinding simulation view of the zoned circular head rotary file.

[0157] Figure 15 Actual machining and tool measurement view. DETAILED DESCRIPTION

[0158] The application will be further described in detail below in combination with the drawings and specific embodiments.

[0159] A zoned circular head rotary file grinding track calculation method of the application comprises the following steps:

[0160] Step 1: Definition of the geometric parameters of the zoned circular head rotary file.

[0161] In order to more completely and accurately analyze and study the structure of the circular head rotary file, the following parameters are defined as shown in the drawings: Figure 1

[0162] ​Tool initial rotation radius R w ; i.e. tool radius at the start of the helical flute;

[0163] Tool taper angle k: the angle between the tool rotation body outer contour and the tool rotation center axis;

[0164] Flute length L w : the length of the helical flute along the tool axis direction;

[0165] Taper angle k: the angle between the helical flute rotation contour generatrix and the tool axis;

[0166] Flute helix angle b: the angle between the helical flute rotation contour generatrix and the helical flute blade tangent vector;

[0167] Inter-tooth angle a: the central angle between the corresponding points of two adjacent tool blade lines of the rotary file;

[0168] Cylindrical initial tooth depth d1: defined as the depth of the workpiece ground by the grinding wheel at the cylindrical initial position;

[0169] End blade circular arc blade initial tooth depth d2: defined as the depth of the workpiece ground by the grinding wheel at the end blade circular arc blade initial position;

[0170] End blade circular arc blade terminal tooth depth d3: defined as the depth of the workpiece ground by the grinding wheel at the end blade circular arc blade terminal position;

[0171] Tool tip tooth depth d4: defined as the depth of the workpiece ground by the grinding wheel at the tool tip position.

[0172] Step 2: Definition of coordinate system (as shown in Figure 2 ).

[0173] Workpiece coordinate system WCS

[0174] Definition of workpiece coordinate system O W -X W Y W Z W , with the tool axis as the coordinate axis Z W , with the origin O W , the straight line pointing to the start of the helical blade line as the coordinate axis X W . The tool position coordinates of the grinding wheel grinding track ultimately need to be described in the workpiece coordinate system.

[0175] End blade coordinate system DCS

[0176] Definition of end blade coordinate system O D -X D Y D Z D , rotating the workpiece coordinate system around the Z W axis by the rotation angle f, and translating along the positive direction of the Z W axis by Lw Obtained.

[0177] Step 3: Coordinate system transformation.

[0178] In order to facilitate the wheel tool and numerical control grinding post-processing, the following coordinate transformation matrix is constructed in this paper: define M D→W , T D→W , respectively, the rotation matrix and translation matrix from the end edge coordinate system to the workpiece coordinate system, then:

[0179] (1)

[0180] (2)

[0181] In the formula,

[0182] (3)

[0183] Step 4: Parametric modeling of partition type circular arc head rotary file edge line.

[0184] The partition type circular arc head rotary file is shown in Figure 3 In this partition type circular arc head rotary file edge line design method, the tool edge is divided into two kinds, one is the main edge defined as the tool edge converging on the center of the tool top, and the other is the secondary edge defined as the tool edge not converging on the center of the tool top. The tool edge is arranged in this way, which increases the chip space between the tool edges and can improve the cutting conditions. The partition type circular arc head rotary file edge curve is divided into cylindrical peripheral edge equal helical edge line and end edge part circular arc head edge curve. In order to ensure the smoothness of the edge curve and facilitate cutting, the circular arc edge curve and the peripheral edge line are smoothly connected, and the generalized helix angle of the circular arc head edge line and the peripheral edge equal helical edge line at the connection point needs to be the same. The edge line is divided into peripheral edge and end edge part for parametric modeling.

[0185] S4.1 Partition type circular arc head rotary file peripheral edge line model.

[0186] In the cylindrical part of the rotary file, the peripheral edge line is designed as an equal helix angle curve, that is, the helical motion direction and the profile generatrix have a constant included angle, in order to ensure that the chip flow direction of the rotary file remains consistent during grinding. The study of peripheral edge line has been described in many documents. In this invention, the peripheral edge line on the cylindrical surface is established in the workpiece coordinate system O W -X W Y W Z W , as shown in Figure 4 , taking the coordinate value z of Z W axis as the independent variable, then the expression of the coordinates of any point P1 on the curve is as follows:

[0187] (4)

[0188] φ represents the rotation angle of the blade point P1 relative to the start of the blade line in the workpiece coordinate system, and its expression is:

[0189] (5)

[0190] where the independent variable z is Z W axis coordinate value.

[0191] S4.2 Partition type arc head rotary file end blade blade line model.

[0192] In the description of the partition type arc head end blade line model, first define the number of end blade partitions of the rotary file tool as m, and the number of teeth of each partition as n. The end blade curve is divided into main blade and auxiliary blade according to the different cutting effect. In order to accurately and clearly describe the arc blade curve, this paper adopts the method of spiral surface of Cheng et al. to build the model of arc blade line part of end blade. By introducing the latitude angle θ as the independent variable, the relationship between θ and the rotation angle φ r is established, so as to obtain the expression of the arc head blade line, which is described in the end blade coordinate system respectively.

[0193] S4.2.1 Arc surface curve part

[0194] This curve part is obtained by the intersection of arc surface and non-orthogonal spiral surface. The arc blade curve is regarded as a generalized spiral line on the arc surface, and the arc blade line is also regarded as a curve formed by a certain rotation rule of a moving point P on the arc surface, as shown in Figure 5 , then the coordinates of any blade point P0 on the curve are:

[0195] (6)

[0196] where the independent variable θ is the latitude angle, r is the radius of the end blade arc, and R is the center distance of the arc, and its expression is:

[0197] (7)

[0198] φ r represents the rotation angle of the arc blade point P0, and the rotation angle is written as a function of the latitude angle as a parameter. According to the different partition of the tool blade, different φ r expressions will be produced, and the expression of φ r can be obtained by the expression of the arc surface curve model.

[0199] (1) For the calculation of the rotation angle of the main blade edge point, the main blade curve needs to pass through the center of the rotating file top, and the rotation angle φ of the main blade arc curve is obtained r is expressed as:

[0200] (8)

[0201] (2) For the calculation of the rotation angle of the sub-blade edge point, the sub-blade curve has an eccentric amount, which does not pass through the center of the rotating file top. At this time, the rotation angle of the blade line point also needs to add the division angle between the blade lines, and the rotation angle φ of the sub-blade arc curve is expressed as:

[0202] (9)

[0203] In the formula, h represents the eccentric amount of the sub-blade blade line, and n x m represents the total number of blade teeth of the rotating file.

[0204] S4.2.2 Planar curve part

[0205] The curve P3P4 is established on the plane M passing through the point P3 and perpendicular to the coordinate plane X D O D Y D . This segment of curve can be regarded as a circular arc with the projection point O r of the point P3 on the coordinate plane X D O D Y D as the center. Since the point P3 is the highest point of the arc rotation surface, the tangent vector F p1 of the curve P3P4 segment at the point P3 must be located on the plane M, and the point is located on the straight generatrix, which is also located on the plane. Therefore, the connection from the curve P2P3 segment to the curve P3P4 segment is smooth, and the expression of the blade point P0 of this segment of curve is:

[0206] (10)

[0207] where η is the inner inclination angle of the circular arc blade, and in the formula, φ r (π / 2) represents the rotation angle at the blade point P3 when θ=π / 2, and φ h represents the angle between the plane M and the X D axis, which is also divided into the expressions of the main blade and the sub-blade.

[0208] (1) The plane M on the main blade curve is passing through the center of the rotating file top, as shown in Figure 6 . The angle between the plane M and the X D axis on the main blade arc curve is the rotation angle φ h0 at the highest point P3 of the arc rotation surface, that is:

[0209] (11)

[0210] Therefore, the expression of the planar curve of the main blade circular arc curve portion is:

[0211] (12)

[0212] For the minor blade blade line, since there is eccentricity, the plane M will not pass through the center of the top of the file tool, as shown in Figure 7 The curve P3P4 is established on the plane M through point P3 and tangent to the eccentricity cylinder; therefore, the expression of the planar curve on the minor blade curve is:

[0213] (13)

[0214] In the formula, φ h1 is the angle between the plane M and the X D axis on the minor blade circular arc blade curve, and the expression is:

[0215] (14)

[0216] S4.2.3 Straight Blade Portion

[0217] The straight blade portion is established on the plane M, and in order to ensure that the straight blade and the end blade circular arc blade line are smoothly linked, the straight blade is along the vector direction of the end point of the circular arc blade, and the main blade and the minor blade are also described separately. The straight blade is defined as a straight line segment P4P5 on the plane M; it can be known that the coordinates of the end point P4 on the end blade planar curve are:

[0218] (15)

[0219] The tangent vector F t1_D at the end point P4 of the end blade planar curve is:

[0220] (16)

[0221] Introducing the independent variable L, the expression of the blade point P0 on the straight blade is:

[0222] (17)

[0223] In the formula, L h is the length of the straight blade of the end blade, and the mathematical expressions of the straight blade lengths of the main blade and the minor blade are different. The main blade and the minor blade are described separately.

[0224] (1) For the main blade of the partition, the blade curve has no eccentricity, and the straight blade lies in the plane M which passes through the vertex of the rotating tool of the circular-arc head, and also passes through the end point of the circular-arc blade plane curve and the center of the end blade coordinate system; as shown in FIG. 1. Through geometric relationship derivation, the expression of the length of the straight blade of the main blade of the partition is: Figure 8

[0225] (18)

[0226] (2) For the blade line of the secondary blade, since there is eccentricity, the plane M does not pass through the center of the top of the rotating file tool, and the i-th secondary blade intersects with the next main blade of the partition at a point, and the spatial distance between the intersection point P5 and the end point P4 of the straight blade of the secondary blade is the length of the straight blade of the secondary blade; as shown in FIG. 2. Through geometric relationship derivation, the expression is: Figure 9

[0227] (19)

[0228] L i represents the projection segment of the straight blade of the secondary blade of the partition on the end blade coordinate system plane X D O D Y D , and the expression is:

[0229] (20)

[0230] In the formula, k i is the slope of the straight line projection of the straight blade of the main blade intersecting with the secondary blade on the end blade coordinate plane X D O D Y D , which can be obtained through the relative position between the secondary blade and the main blade and the indexing relationship of the rotating surface, and the expression is:

[0231] (21)

[0232] In the formula, m represents the number of partition groups of the rotating file teeth;

[0233] Through simultaneous calculation of formula (18), formula (19), formula (20) and formula (21), the length of the straight blade of each secondary blade can be obtained, and through geometric relationship derivation, the expression of the length of the straight blade of the secondary blade is: the complete expression is:

[0234] (22)

[0235] S4.2.4 Transformation to the workpiece coordinate system

[0236] ​​​Since the cutting edge needs to be described in a unified coordinate system, the cutting edge expression calculated in the end-edge coordinate system needs to be transformed to the workpiece coordinate system. By transforming the coordinates of any point on the partitioned cutting edge of the end-edge part into coordinate system transformation matrices (1) and (2), the complete expression of the main cutting edge and each secondary cutting edge in a partition in the workpiece coordinate system can be obtained. The cutting edge of the adjacent partition can be rotated by 2π / m angles after returning to the initial machining position to obtain the main cutting edge and secondary cutting edge of the next partition, where m is the number of partitions of the rotary file.

[0237] Step 5: Define the initial attitude of the grinding wheel.

[0238] peripheral grinding wheel reference grinding posture

[0239] The grinding posture of the grinding wheel of the circular arc-head rotary file with circumferential edge used in this invention is as follows: Figure 10 As shown, the grinding wheel reference posture is defined in the WCS coordinate system. To facilitate the conversion of the grinding wheel posture into a CNC program, a conical grinding wheel is used in this paper. For a given grinding wheel, precise machining of the rake angle can be achieved. By reasonably selecting the bottom angle of the grinding wheel cone, not only can the rake angle specification be met, but also the design requirements for the groove depth can be satisfied. The coordinates of the grinding wheel end face center O are... g With grinding wheel axis vector F g Describe the grinding wheel's attitude; use the tangent vector at point P1 as the tangent vector F of the grinding wheel. t In the WCS coordinate system, it is expressed as:

[0240] (twenty three)

[0241] Point P1 to the plane coordinate Z where point P1 is located. W The axis vector is taken as the radial vector F of the grinding wheel. b In the WCS coordinate system, it is expressed as:

[0242] (twenty four)

[0243] Grinding wheel axis vector F g With tangent vector F t Radial vector F of the grinding wheel b Mutually perpendicular, expressed in the WCS coordinate system as:

[0244] (25)

[0245] End-edge grinding wheel reference grinding posture

[0246] The grinding posture of the grinding wheel on the end-edge portion used in this invention is as follows: Figure 11 As shown, taking the end-edge circular arc curve as the reference, and the tangent vector F of the curve... tAs the tangent vector of the grinding wheel in the grinding process, it is expressed as in the coordinate system DCS:

[0247] (26)

[0248] In the formula, φ r represents the rotation angle at the blade line point P0, .

[0249] The center O r of the circular arc points to the vector of the P2 point on the circular arc blade curve as the radial vector F b of the grinding wheel, r The coordinates of the midpoint O b of the circular arc are expressed as:

[0250] (27)

[0251] The radial vector F b is expressed as in the coordinate system DCS:

[0252] (28)

[0253] In the formula, .

[0254] The grinding wheel shaft vector F g is perpendicular to the tangent vector F t of the grinding wheel and the radial vector F b of the grinding wheel, which is expressed as in the coordinate system DCS:

[0255] (29)

[0256] Step 6: Grinding wheel grinding posture calculation.

[0257] (a) Peripheral blade rake angle grinding wheel grinding posture

[0258] In order to avoid the problem of interference between the rotating file teeth in the actual grinding process, the grinding wheel rake angle is defined as the rake angle δ that the grinding wheel rotates around the tangent vector F t of the rotating profile blade point; as Figure 12 shown. Let the transformation matrix of rotating the space by any unit vector N by an angle ε be Rot(N, ε), which is expressed as:

[0259] (30)

[0260] In the formula , i N , j N , k N represent the components of the vector N in three directions, and ε represents the rotation angle.

[0261] After the grinding wheel introduces a lifting angle δ, the radial vector F of the peripheral cutting edge... b With grinding wheel axis vector F g Transform into F′ respectively b and F′ g Radial vector F b Its expression in the WCS coordinate system is as follows:

[0262] (31)

[0263] Grinding wheel axis vector F g In the WCS coordinate system, this is expressed as:

[0264] (32)

[0265] (b) Grinding posture of the grinding wheel with the end edge raised.

[0266] Similarly, a wheel lift angle δ is also introduced under the grinding posture of the end-edge grinding wheel, such as... Figure 13 As shown. After introducing the grinding wheel lifting angle δ, the radial vector F of the end-edge portion... b With grinding wheel axis vector F g Transform into F′ b and F′ g In the DCS coordinate system, it is expressed as:

[0267] (33)

[0268] (34)

[0269] Grinding wheel axis vector F′ g It can be calculated using the following formula:

[0270] (35)

[0271] Step 7: Calculation of grinding wheel trajectory;

[0272] (1) Calculation of grinding trajectory of peripheral grinding wheel

[0273] According to the grinding posture definition, based on the workpiece coordinate system, with the grinding wheel end face center point O... g The coordinates describe the grinding position of the grinding wheel, with the constraint that the large end face of the grinding wheel is always in contact with the cutting edge during the grinding process; thus, the center point O of the grinding wheel end circle can be obtained. g The coordinates in the Zhouren coordinate system are expressed as follows:

[0274] (36)

[0275] Among them, R g Let d be the circumference radius of the large end face of the grinding wheel. 12The grinding depth of the peripheral edge portion of the rotary file is consistent with the tooth depth d1 of the peripheral edge portion and the initial tooth depth d2 of the end edge circular arc edge, and the expression is:

[0276] (37)

[0277] wherein k s represents the taper angle of the grinding wheel, γ represents the rake angle, and m x n represents the total number of teeth of the rotary file.

[0278] (2) End edge grinding wheel grinding track calculation

[0279] For the description of the end edge grinding wheel grinding track, according to the end edge edge line and the end edge grinding wheel grinding posture, the grinding wheel center point O g of the grinding wheel grinding track is defined.

[0280] (38)

[0281] wherein R g is the large end surface circumferential radius of the grinding wheel, d 24 is the grinding depth of the end edge portion of the rotary file, wherein the initial tooth depth d2 of the end edge circular arc edge and the terminal tooth depth d3 of the circular arc edge are equal, and thus the grinding depth of the end edge portion is expressed as:

[0282] (39)

[0283] Simulation verification:

[0284] In order to verify the grinding wheel grinding track algorithm, the present application develops an algorithm prototype using VC++ environment, inputs the related structure design parameters, outputs the tool position track file according to the grinding wheel grinding track algorithm, and performs three-dimensional grinding simulation in VERICUT8.0, and the used structure design parameters are shown in Table 1, and the process parameters are shown in Table 2.

[0285] Table 1 Structure design parameters of the circular arc head rotary file

[0286]

[0287] Table 2 Grinding process parameters of the partition type circular arc head rotary file

[0288]

[0289] The grinding wheel grinding track of the partition type circular arc head rotary file is shown in Table 3, and the simulation results are shown in Figure 14 . Then a G500T+ type five-axis numerical control tool grinder is used for actual grinding processing. The actual processing results are shown in Figure 15 .

[0290] Table 3 Parameter measurement values of the zoned circular head rotary file

[0291]

[0292] From the comparative analysis of the results in Table 3 and Table 1, it can be seen that the measurement values of the geometric parameters of the circular head rotary file after simulation processing are basically consistent with the design values, indicating that the grinding track algorithm of the circular head rotary file proposed in the application can meet the design and processing requirements of the circular head rotary file. In actual processing, due to the wear of the grinding wheel arc radius, there may be a small error in the actual grinding part parameters, and a compensation method can be used to improve the processing precision.

Claims

1. A method for calculating the grinding trajectory of a rotary file with a partitioned arc head, characterized in that, Includes the following steps: Step 1: Definition of geometric parameters for a partitioned, arc-shaped rotary file: Tool starting rotation radius R w That is, the tool radius at the beginning of the circumferential helical cutting edge; Tool taper angle к: The angle between the outer contour of the tool's rotating body and the axis of rotation of the tool's center; Circumferential cutting edge length L w : is the length of the tooth along the axis of the cutting tool; Circumferential helix angle β: is the angle between the generatrix of the circumferential cutting edge rotation profile and the tangent of the circumferential cutting edge line; Tooth angle α: The central angle between corresponding points of two adjacent cutting edges of a rotary file; The initial tooth depth d1 of the cylinder is defined as the depth to which the grinding wheel grinds the workpiece at the starting position of the cylinder. End-edge arc starting tooth depth d2: defined as the depth to which the grinding wheel grinds the workpiece at the starting position of the end-edge arc; End-edge arc edge tooth depth d3: Defined as the depth of grinding of the workpiece by the grinding wheel at the end position of the end-edge arc edge; Tool tip tooth depth d4: Defined as the depth to which the grinding wheel grinds the workpiece at the tool tip position; Step 2: Define the coordinate system; Workpiece Coordinate System (WCS) Define the workpiece coordinate system O W -X W Y W Z W Its coordinate axis is Z, with the tool axis as the coordinate axis. W With the origin O W The straight line pointing to the starting point of the helical cutting edge is the X-axis of the coordinate system. W ; End-edge coordinate system DCS Define the end-edge coordinate system O D -X D Y D Z D The workpiece coordinate system is rotated around Z W The axis rotates by a rotation angle φ, then along Z. W Translation L in the positive direction of the axis w get; Step 3: Coordinate system transformation; Define M D→W T D→W Let be the rotation and translation matrices from the end-edge coordinate system to the workpiece coordinate system, respectively. Then: （1） （2） In the formula, （3） Step 4: Parametric modeling of the cutting edge of the segmented arc-shaped rotary file, dividing the cutting edge into peripheral and end-edge sections for parametric modeling; S4.1 Partitioned Arc-Head Rotary File Peripheral Cutting Edge Model; The peripheral cutting edge line on the cylindrical surface is established in the workpiece coordinate system O. W -X W Y W Z W Below, with Z W If the coordinate value z of the axis is the independent variable, then the expression for the coordinates of any point P1 on the curve is as follows: （4） φ represents the rotation angle of the cutting edge point P1 relative to the starting point of the cutting edge in the workpiece coordinate system, and its expression is: （5） In the formula, the independent variable z is Z W Axis coordinate values; S4.2 Partitioned Arc-shaped Rotary File End Cutting Edge Model; First, define the number of end-edge sections of the rotary file as m, and the number of teeth in each section as n. Then, use the method of finding the intersection line between a non-orthogonal helical rotation surface and a circular rotation surface. By introducing the latitude angle θ as the independent variable, establish its relationship with the rotation angle φ. r The relationship is used to obtain the expression for the arc-shaped cutting edge, which is then described in the end-edge coordinate system. S4.2.1 Circular Arc Revolute Curve Section This curve is obtained by the intersection of a circular arc surface of revolution and a non-orthogonal helical surface. The circular arc cutting edge curve is considered as a generalized helical line on the circular arc surface of revolution, and the circular arc cutting edge line is also considered as a curve formed by a moving point P moving on the circular arc surface of revolution according to a certain rotational law; then the coordinates of any cutting edge point P0 on this curve segment are expressed as: （6） In the formula, θ is the latitude angle, r is the radius of the end-edge arc, and R is the center distance of the arc. Its expression is: （7） φ r This represents the rotation angle at the cutting edge point P0 of the arc-shaped blade. Written in the form of latitude angle A function with parameters, where different cutting edges will produce different φ values ​​depending on the tool's partitioning. r The expression form is used to obtain φ. r The expression can then be used to obtain the circular arc revolution surface curve model; (1) For the calculation of the rotation angle of the cutting edge point on the main cutting edge, since the cutting edge curve of the main cutting edge needs to pass through the center of the top of the rotating file, the rotation angle φ of the main cutting edge arc curve is obtained. r Expressed as: （8） (2) For the calculation of the rotation angle of the cutting edge point on the secondary cutting edge, the secondary cutting edge curve has an eccentricity, which does not pass through the center of the top of the rotating file. At this time, the rotation angle of the cutting edge point also needs to be added to the pitch angle between the cutting edges. Therefore, the rotation angle of the arc cutting edge curve of the secondary cutting edge is... Expressed as: （9） In the formula, h represents the eccentricity of the secondary cutting edge line, and n×m represents the total number of cutting teeth of the rotary file; S4.2.2 Planar Curve Section The curve P3P4 passes through point P3 and intersects the coordinate plane X. D O D Y D Established on a perpendicular plane M, this curve can be considered as having point P3 on the coordinate plane X. D O D Y D The projection point O r Let P3 be a segment of an arc centered at P3. Since P3 is the highest point of the arc's surface of revolution, the tangent vector F of the curve segment P3P4 at point P3 is... p1 The point must lie on plane M, and this point lies on the generatrix, which also lies on the plane. Therefore, the connection from curve segment P2P3 to curve segment P3P4 is smooth. Thus, the expression for the knife-edge point P0 of this curve segment is: （10） Where η is the inclination angle of the circular arc blade, and φ r (π / 2) represents the rotation angle at θ=π / 2, which is the blade tip point P3, φ h Representing planes M and X D The included angle of the shaft, as above, is also expressed as a main cutting edge and a secondary cutting edge; (1) The upper plane M of the main cutting edge curve passes through the center of the top of the rotary file, and the upper plane M of the main cutting edge arc curve is perpendicular to X. D The included angle of the axis is also the rotation angle φ at the highest point P3 of the circular arc surface of revolution. h0 ,Right now: （11） Therefore, the expression for the planar curve of the main cutting edge's circular arc section is: （12） For the secondary cutting edge line, due to its eccentricity, plane M will not pass through the center of the top of the rotary file. Plane M passes through point P3 and is tangent to the eccentricity cylinder. Curve P3P4 is established on this plane. Therefore, the expression for the plane curve on the secondary cutting edge curve is: （13） In the formula, φ h1 Then it is the plane M and X on the upper surface of the secondary cutting edge arc curve. D The included angle between the axes is expressed as: （14） S4.2.3 Straight-edge section The straight cutting edge is established on plane M. To ensure a smooth connection between the straight cutting edge and the arc-shaped cutting edge of the end edge, the straight cutting edge is defined as the vector direction along the endpoint of the arc-shaped cutting edge. The main cutting edge and the secondary cutting edge are also described separately. The straight cutting edge is defined as a straight line segment P4P5 on plane M. Therefore, the coordinates of the endpoint P4 on the plane curve of the end edge are: （15） The tangent vector F at the end point P4 of the end-edge plane curve t1_D for: （16） Introducing the independent variable L, the expression for the cutting edge point P0 on the straight cutting edge is: （17） In the formula, L h The mathematical expressions for the straight cutting length of the end edge, the main edge, and the secondary edge are different. The main edge and the secondary edge are described separately. (1) For the partitioned main cutting edge, its cutting edge curve has no eccentricity. The plane M containing the straight cutting edge passes through the vertex of the circular arc head rotating tool, and simultaneously passes through the end point of the circular arc cutting edge plane curve and the center of the end-edge coordinate system. Through the derivation of geometric relationships, the expression for the length of the straight cutting edge of the partitioned main cutting edge is: （18） (2) For the secondary cutting edge line, due to its eccentricity, the plane M will not pass through the center of the top of the rotary file. The i-th secondary cutting edge and the next main cutting edge of the partition must intersect at a point. The spatial distance between this intersection point P5 and the end point P4 of the secondary cutting edge is the length of the secondary cutting edge. Its expression can be derived from the geometric relationship: （19） L i This indicates that the partitioned secondary cutting edge is located on the X-plane of the end-edge coordinate system. D O D Y D Projection segment on Length, which is expressed as: （20） In the formula, k i The primary cutting edge, which is the intersecting secondary cutting edge, is located in the X-axis of the end-edge coordinate plane. D O D Y D The slope of the projection of the upper straight line can be obtained from the relative position between the secondary and primary cutting edges and the indexing relationship of the surface of revolution, and is expressed as: （21） In the formula, m represents the number of tooth groups in the rotary file; By combining equations (18), (19), (20), and (21) and performing simultaneous calculations, the length of the straight edge of each secondary cutting edge can be obtained. The expression for the length of the straight edge of the secondary cutting edge, derived through geometric relationships, is: Its complete expression is: （22） S4.2.4 Transform to workpiece coordinate system By transforming the coordinates of any point on the partitioned cutting edge of the above end-edge part through coordinate system transformation matrices (1) and (2), we can obtain the complete expression of the main cutting edge and each secondary cutting edge in the workpiece coordinate system within a partition. The cutting edge of the adjacent partition can be rotated by 2π / m angles after returning to the initial machining position to obtain the main cutting edge and secondary cutting edge of the next partition, where m is the number of partitions of the rotary file. Step 5: Define the initial attitude of the grinding wheel; peripheral grinding wheel reference grinding posture Define the grinding wheel reference grinding posture in the WCS coordinate system, with the coordinates of the grinding wheel end face center O. g With grinding wheel axis vector F g Describe the grinding wheel's attitude; use the tangent vector at point P1 as the tangent vector F of the grinding wheel. t In the WCS coordinate system, it is expressed as: （23） Point P1 to the plane coordinate Z where point P1 is located. W The axis vector is taken as the radial vector F of the grinding wheel. b In the WCS coordinate system, it is expressed as: （24） Grinding wheel axis vector F g With tangent vector F t Radial vector F of the grinding wheel b Mutually perpendicular, expressed in the WCS coordinate system as: （25） End-edge grinding wheel reference grinding posture The grinding posture of the grinding wheel on the end-edge portion is based on the end-edge arc curve, with the tangent vector F of the curve as the reference. t As the tangent vector in the grinding process, it is expressed in the DCS coordinate system as: （26） In the formula, φ r This indicates the rotation angle at point P0 on the cutting edge. ; With the center of the arc O r The vector pointing to point P2 on the arc-shaped cutting edge curve is taken as the radial vector F of the grinding wheel. b Midpoint O of the arc r The coordinates are expressed as: （27） Radial vector F b In the DCS coordinate system, this is expressed as: （28） In the formula, ; Grinding wheel axis vector F g Tangential vector F with the grinding wheel t and the radial vector F of the grinding wheel b Mutually perpendicular, expressed in the DCS coordinate system as: （29） Step 6: Calculate the grinding posture of the grinding wheel; (a) Grinding posture of the grinding wheel with the edge raised. Define the grinding wheel lift angle δ as the tangential vector F of the grinding wheel about the cutting edge point of the rotation profile. t Let α be the angle through which the axis of rotation passes; let Rot(N,ε) be the transformation matrix for rotating an angle α around any unit vector N in space, then it can be expressed as: （30） In the formula i N j N k N These represent the components of vector N in the three directions, respectively. Indicates the rotation angle; After introducing the grinding wheel lift angle δ, the radial vector F of the peripheral cutting edge... b With grinding wheel axis vector F g Transform into F′ respectively b and F′ g Radial vector F b Its expression in the WCS coordinate system is as follows: （31） Grinding wheel axis vector F g In the WCS coordinate system, this is expressed as: （32） (b) Grinding posture of the grinding wheel with the end edge raised. Similarly, a wheel lift angle δ is introduced under the end-edge grinding posture. After introducing the wheel lift angle δ, the radial vector F of the end-edge portion... b With grinding wheel axis vector F g Transform into F′ b and F′ g In the DCS coordinate system, it is expressed as: （33） （34） Grinding wheel axis vector F′ g It can be calculated using the following formula: （35） Step 7: Calculation of grinding wheel trajectory; (1) Calculation of grinding trajectory of peripheral grinding wheel According to the grinding posture definition, based on the workpiece coordinate system, with the grinding wheel end face center point O... g The coordinates describe the grinding position of the grinding wheel, with the constraint that the large end face of the grinding wheel is always in contact with the cutting edge during the grinding process; thus, the center point O of the grinding wheel end circle can be obtained. g The coordinates in the Zhouren coordinate system are expressed as follows: （36） Among them, R g Let d be the circumference radius of the large end face of the grinding wheel. 12 The grinding depth of the peripheral cutting edge of the rotary file, which is consistent with the tooth depth d1 of the peripheral cutting edge and the initial tooth depth d2 of the end cutting edge arc, is expressed as follows: （37） Where, k s γ represents the rake angle of the grinding wheel, m×n represents the total number of teeth on the rotating file; (2) Calculation of grinding trajectory of end-edge grinding wheel For the description of the grinding wheel trajectory of the end-edge portion, based on the definition of the end-edge cutting line and the grinding wheel posture, the grinding wheel center point O of this segment of the grinding wheel trajectory is... g The coordinates in the end-edge coordinate system are expressed as follows: （38） Among them, R g Let d be the circumference radius of the large end face of the grinding wheel. 24 Let d2 be the initial tooth depth of the rotary file's end cutting edge, and d3 be the final tooth depth of the arc-shaped cutting edge. Therefore, the grinding depth of the end cutting edge is expressed as: （39）。

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