A method for calculating the CNC grinding trajectory of a ball-head rotary file

Abstract

This invention discloses a CNC grinding trajectory calculation method for a ball-end rotary file. Specifically, the method involves: first, defining the modeling coordinate systems for the ball-end rotary file—the workpiece coordinate system and the end-edge coordinate system—and establishing the machining coordinate system—the grinding wheel coordinate system; second, using kinematic theory, defining the grinding process of the ball-end rotary file, decomposing the grinding process into process parameter expressions, initial grinding pose definitions, and grinding trajectory definitions, thereby simplifying the definition of the grinding process and enabling flexible machining of the ball-end rotary file. This invention offers good tooling flexibility and machining accuracy.

Description

Technical Field

[0001] This invention belongs to the technical field of complex rotary tool structure design, and particularly relates to a method for calculating the CNC grinding trajectory of a ball-head rotary file. Background Technology

[0002] He et al. proposed a new tool machining trajectory optimization method, applying the surface matching method to the optimization process of the initial tool machining trajectory, and using a sequential quadratic programming algorithm to solve the surface matching problem [He et al., Pang Kairui, Sang Yicun et al. Application of surface matching method in tool machining trajectory optimization [J]. Journal of Engineering Design, 2019, 26(2):190-196.]. Chen et al. proposed an iterative tool helical groove parameter control algorithm and implemented the algorithm in C++ [Zhan Ji Chen, Wei He, Genghuang Liu et al. Iteration based calculation of position and orientation of grinding wheel for solid cutting tool flutegrinding [J]. Journal of Manufacturing Processes, 2018, 36:209-215.]. B. Karpuschewski proposed an automatic wheel position search method for a given wheel profile and flute [Karpuschewski B, Jandecka K, Mourek D etc. Automatic search for wheelposition in flute grinding of cutting tools[J].Cirp Annals-ManufacturingTechnology,2011,60(1):347-350.]. Lei Ren proposed a five-axis grinding method based on standard 1V1 / 1A1 grinding wheels to address the problem that given wheel geometry cannot fully guarantee flute parameters [Lei Ren, Shilong Wang, Lili Yi etc. An accurate method for five-axis flute grinding in cylindrical end-mills using standard 1V1 / 1A1 grinding wheels,Precision Engineering,2016,43:387-394.].Zhao et al. proposed a method for generating the guide curve tool path of an elliptical torus tool based on a minimum distance computation algorithm [Zhao Mingbo, Cai Yonglin, Wang Haitong etc. Elliptical torus cuttertool path generation based on minimum distance computation[J]. Proceedings Of The Institution Of Mechanical Engineers, Part C: Journal Of Mechanical Engineering Science, 2021, 235(24):7672-7684]. Xi addressed the efficiency problem of calculating the tool position point using a torus tool based on the projection method in CNC machining of curved surfaces, proposing a new algorithm to directly calculate the tool position point by approximating the surface to be machined with a torus [Xiaolin Xi, Yonglin Cai, Fenglei Wang etc. An efficient algorithm for calculating the cutter location point based on projection method[J]. International Journal Of Production Research, 2018, 56(4):1722-1731.]. Wu used the principles of differential geometry to give the relevant model and simulation results of the cutting edge of the ball end mill with a constant helical angle. Based on this, he gave a solution method for the inverse problem of the manufacturing model [CT Wu, CK Chen, YY Tang etc. Modelling and computer simulation of grinding of the ball end type rotating cutter with a constant helical angle[J]. Proceedings Of The Institution Of Mechanical Engineers, Part B: Journal Of Engineering Manufacture, 2001, 215(11): 1581-1594.]. Summary of the Invention

[0003] Currently, there is no fully disclosed parameterized definition or parameterized CNC grinding trajectory algorithm for the machining of ball-end rotary files. This invention provides a method for calculating the CNC grinding trajectory of ball-end rotary files.

[0004] The present invention provides a method for calculating the CNC grinding trajectory of a ball-head rotary file, comprising the following steps:

[0005] Step 1: Define the key coordinate system.

[0006] Workpiece coordinate system

[0007] Define the workpiece coordinate system O w -X w Y w Z w As the absolute coordinate system in the coordinate system; define the workpiece coordinate system X. w O w Y w The plane is the plane where the starting point of the circumferential spiral cutting edge is located, and the origin of the coordinate system is X. w O w Y w Intersection of the plane and the tool axis; Z-axis w Coaxial with the tool rotation axis; X-axis w Intersecting at the starting point of the initial spiral cutting edge; Y w It is determined by the right-hand rule.

[0008] End-edge coordinate system

[0009] Define the end-edge coordinate system O d -X d Y d Z d Z-axis d Coincident with the tool axis, axis X d Parallel to X w Y w The plane intersects at the endpoint of the peripheral cutting edge. Considering the continuity of the cutting edge and the simplification of the modeling process, the end-edge coordinate system has a phase angle transformation relationship with the workpiece coordinate system, with the X-axis as the reference point. d With X w The angle between The functional relationship is expressed as follows:

[0010]

[0011] In the formula, L z O is the origin of the end-edge coordinate system. d With respect to the workpiece coordinate system plane X w Y w The distance, R w For X w Y w In-plane blank radius, β w κ is the helix angle of the tooth. w It is the angle between the generatrix of the rotating body of the tool and the axis.

[0012] As required by the coordinate system, the end-cutting line representation in the end-cutting coordinate system needs to be transformed to the workpiece coordinate system. Therefore, there exists a transformation relationship from the end-cutting coordinate system to the workpiece coordinate system, such as matrix M. d→w As shown:

[0013]

[0014] In the formula, L is the helix angle at the end of the peripheral cutting edge. w The length of the circumferential blade.

[0015] Grinding wheel coordinate system

[0016] Define the origin O of the grinding wheel coordinate system G The cutting edge is along the workpiece coordinate system Z. w A moving point X moving in the positive direction G The axis intersects the tool axis; Y G Tangent to the spiral blade line.

[0017] Step 2: Establish the perimeter cutting edge model.

[0018] First, define the motion trajectory of a moving point P on the tool blank in a helical forward motion as the cutting edge line, and establish the circumferential cutting edge line of the rotary file with axial displacement z. p The parameterized equation expression with respect to the independent variable is as follows:

[0019]

[0020] In the formula, Let z be the independent variable p The corresponding circumferential turning angle.

[0021] Define the number of rotary file cutting edges as N, and define the angle between two adjacent cutting edges as the division angle. It depends on the defined number of cutting edge groups N g Number of blades N in the group b And N = N g ·N b The specified number of a partitioned rotary file, N, is not a prime number; the graduation angle... The function expression is as follows:

[0022]

[0023] Establish the trajectory P of point P on the circumferential cutting edge of the rotary file using equations (3) and (4) in combination. tracks The expression is shown in equation (5):

[0024]

[0025] In the formula, n represents the coordinate system X of the workpiece. w The nth cutting edge line in the counterclockwise direction starting from the axis.

[0026] Step 3: Establish the end-edge cutting line model;

[0027] The ball-end cutting edge is parametrically modeled using spherical coordinates, with the origin of the end-edge coordinate system placed at the center of the end-edge structure, and X... d The axis serves as the reference for the independent variable and rotation angle in spherical coordinates. To unify the coordinate representation, and considering that the tool position point is expressed in three-dimensional coordinates, the polar coordinate expression in the spherical coordinate system is finally transformed into the expression in the three-dimensional coordinate system. The Z-axis of the end-edge coordinate system is constructed using projection. d The trajectories formed by the intersection of the perpendicular line and the tangent line of the cylinder with tooth deviation h are the cutting edges without tooth deviation and with tooth deviation h, respectively. The cutting edge equations are uniformly constructed in spherical coordinates. As shown below:

[0028]

[0029] In the formula, θ is the end-edge helix angle. s The latitude angle is 0, and the tooth deviation is 0. <h<R s cosκ w R s Let be the radius of the sphere head.

[0030] The independent variable is affected by the tooth deviation and the peripheral taper, so it is rotated back to plane M. xz The independent variable θ can be obtained from this. s The range of values ​​is as follows:

[0031] θ s ∈(κ,arccos(h / R s (7)

[0032] The cutting edge of the ball head in the spherical coordinate system is expressed in three-dimensional coordinates as shown in equation (8):

[0033]

[0034] Step 4: Establish the blade line partition control model.

[0035] The ball-end cutting lines are divided into groups, each group consisting of a main cutting edge passing through the vertex of the ball end and several secondary cutting edges; the length of the endpoint of the i-th secondary cutting edge from the origin is defined as the cutting edge excess l. i The distance from the center is h. i The density of the cutting edge lines near the tip of the ball head is mainly determined by the cutting edge deflection h. i The impact.

[0036] Since the i-th ball head, i = 1, 2, ..., N b -1 Initial rotation angle of the secondary cutting edge The existence of any set of primary and secondary cutting edge endpoint extensions on the projection plane X d O d Y d The inner edges must intersect. It is easy to see that the distance from the i-th set of cutting edges to the origin O is... d The length is the distance l through the center of the blade. i The intersection point is P. u_i To establish the functional relationship between the relative positions of the primary and secondary cutting edges, it is necessary to first determine the included angle ∠P formed by the intersection of the endpoints of the primary and secondary cutting edges. u_i Typically, the end point of the main cutting edge is located at the vertex of the rotary file. The tangent vector at the end point of the main cutting edge in the end-edge coordinate system is obtained by differentiating equation (8), which is the θ vector under the condition of no tooth deviation. s The tangent vector of the cutting edge at 90° is projected onto the coordinate plane X. d O d Y d The unit vector F in P-end As shown in equation (9):

[0037]

[0038] The vector at the end of the i-th cutting edge is obtained from equation (10). For a partitioned rotary file, a set of angle parameters within the cutting edge line can be obtained.

[0039]

[0040] In the formula, θ(h) i ) represents the independent variable θ corresponding to the i-th cutting edge. s Upper limit, and θ(h) i =arccos(h i / R s ), h i Let N be the cutting edge deviation of the i-th secondary cutting edge, 0≤i≤N b .

[0041] By combining equations (9) and (10), we can further obtain the angle ∠P between the tangent vectors at the last point of the primary cutting edge and the i-th secondary cutting edge within the cutting edge line in the projection plane. u_i for:

[0042]

[0043] As can be seen from the method of forming the edge deflection line, its end point is tangent to the origin O of the edge coordinate system. d Tooth deflection h centered on the circle i The ring can form a right triangle ΔP. h_end_i P u_i P end Based on the right triangle tangent relationship, the following functional relationship can be constructed:

[0044]

[0045] In the formula, h i For the i-th secondary cutting edge offset within the same cutting edge group, l i Let P be the intersection point. u_i With the main cutting edge P end Distance between endpoints.

[0046] Solving equations (9) to (12) together yields the result regarding the variable θ(h) i ), h i and l i The mathematical model is shown in equation (13):

[0047]

[0048] In the formula, θ(h i =arccos(h i / R s Therefore, this formula is only related to h. i and l i The relevant functional relationships.

[0049] Further selection l i h is selected as the control variable for the blade line partition. i As a dependent parameter for the density of the cutting edges, in order to achieve unified control over the zoning of each secondary cutting edge, it is stipulated that all secondary cutting edges within a group intersect the main cutting edge at the same point P in the end-edge coordinate system plane. m and with the main cutting edge tooth excess l m As a zoning control variable, the ball head edge line is zoned; therefore, sequentially in X... d O d Y d Calculate the cutting edge deviation h of the i-th set of cutting edges in the same group of cutting edges in the coordinate plane. i This causes the extensions of the endpoints of each secondary cutting edge to intersect the extension of the primary cutting edge at point P. m Point.

[0050] Solve for the edge deviation h i The range of values ​​for is shown in equation (14):

[0051] 0≤h i <R s cosκ w ,κ w ∈(-90,90) (14)

[0052] Step 5: Solve for the cutting edge parameters.

[0053] First, set the partition control parameters lm Substitute the cutting edge deviation parameter h of the i-th secondary cutting edge in equation (12) to solve the partition model. i , i = 1, 2, 3…, and the function equation of the i-th secondary cutting edge of the ball end cutting edge is further determined by equation (6). As shown in equation (15):

[0054]

[0055] Number of sections of ball head rotating file line group N g Number of blades N in the group b It can be seen that the phase angle between any two main cutting edges is Phase angle between any two adjacent cutting edges and the turning angle on the main chip cutting edge Find the upper limit θ of the independent variable interval of the secondary cutting edge corresponding to the intersecting cutting edge points in spherical coordinates. max-i Solve equations (6) and (15) simultaneously to find the intersection point P in spherical coordinates. max-i i = 1, 2…N b-1 As shown in equation (16):

[0056]

[0057] In the formula, The longitude angle of the helix of the main cutting edge; then the independent variable θ for the grinding of the zoned ball-head rotating file cutting edge is... s-i The interval is:

[0058] κ w ≤θ s-i ≤θ max-i (17)

[0059] Step 6: Grinding motion model of grinding wheel.

[0060] (1) Definition of grinding motion and model of grinding wheel rotation surface:

[0061] The rotary file chip groove is machined using 1V1 / 1A1 parallel grinding wheel. The grinding process can be regarded as follows: the workpiece is fixed and the grinding wheel moves along the spiral cutting edge in an initial grinding posture.

[0062] The mathematical model of the grinding wheel's rotation profile is established in the grinding wheel coordinate system, with the center of the large end face of the grinding wheel coinciding with the origin of the grinding wheel coordinate system, and the axis of rotation of the grinding wheel aligned with the Z-axis of the grinding wheel coordinate system. G Overlap, along the Z-axis of the grinding wheel coordinate system G A discrete mathematical model of the grinding wheel solid is established in the positive direction of the axis, and the grinding wheel rotation profile equation is established as shown in equation (18):

[0063]

[0064] In the formula, h G Let ψ be the independent variable in the direction of the grinding wheel height. G For grinding wheel h G The independent variable of the cross-sectional rotation angle, κ G This refers to the taper of the grinding wheel.

[0065] (2) Grinding wheel process parameter transformation matrix:

[0066] Define the chip flute rake angle as γ and the depth of cut parameter as cd. To avoid mutual interference between parameters during the initial grinding wheel pose setting process, the transformation process of the grinding wheel parameter definition is specified as follows: First, the grinding wheel mathematical model is moved along the grinding wheel coordinate system X. G Translation R in the positive direction of the axis G -cd; makes the origin O of the coordinate system... G Coincident at grinding point P, then rotate the grinding wheel around the grinding wheel coordinate system Y. G Rotate the axis clockwise by an angle γ, and the parameter transformation matrix M G-para (γ,cd) is shown in equation (19):

[0067]

[0068] (3) Initial grinding position of the cylindrical peripheral cutting edge in the workpiece coordinate system:

[0069] The grinding wheel coordinate system under the initial grinding posture should satisfy the following conditions: the origin of the grinding wheel coordinate system O G Always moving along the blade trajectory; coordinate axis Y G Parallel to the helical blade tangent vector F t ; coordinate axis X G With respect to the workpiece coordinate system axis X w Overlap, that is, when grinding a conical rotary file, X G Only with the coordinate axis X w Intersecting; furthermore, the initial position of the grinding wheel coordinate system is also determined by the cutting edge helix angle β. w Workpiece radius R w and taper κ w Decide.

[0070] Define the transformation matrix M of the initial posture of the cutting edge grinding wheel in the workpiece coordinate system. Cylinder As shown in equation (20):

[0071]

[0072] (4) Initial grinding position of the conical peripheral cutting edge in the workpiece coordinate system:

[0073] When setting the initial orientation of a tapered blank, the taper κ needs to be taken into account. wTo avoid the influence of the rotation matrices, a rotation matrix Rot(N,ξ) around any vector N is introduced in equation (20) to avoid parameter errors caused by the mutual influence of the rotation matrices. Rot(N,ξ) is shown in equation (21):

[0074]

[0075] In the formula, ξ is the rotation angle about vector N, and its direction is determined by the right-hand rule.

[0076] After introducing the rotation matrix Rot(N,ξ), it is further necessary to determine the arbitrary rotation vector N, where N is the initial tangent plane X. G O G Y G normal vector F N The rotation angle ξ represents the grinding wheel coordinate system Y under the initial attitude of the grinding wheel without considering the taper. G The extension line in the opposite direction intersects the tangent vector F of the cone's circumferential edge. t The included angle; first, differentiate the formula for the spiral cutting edge to obtain the tangent vector F at any point P on the cutting edge. t As shown in equation (22):

[0077]

[0078] In the formula,

[0079]

[0080] Vector F N Through the grinding wheel tangent vector F t |z p＝0 With respect to the workpiece coordinate axis X w Direction vector Cross product calculation yields:

[0081]

[0082] The rotation angle ξ is the circumferential cutting vector F t |z p＝0 The Y-axis of the grinding wheel coordinate system is relative to the initial pose without considering the workpiece taper. G Let the angle between the opposite direction vectors be... F YG-neg The workpiece coordinate system can be obtained using the following formula:

[0083]

[0084] The expression for the rotation angle ξ is:

[0085]

[0086] To ensure that the rotation transformation does not affect the coordinate system position, the displacement can be performed after introducing the rotation matrix. Therefore, the general transformation matrix M for the conical peripheral cutting edge from the initial grinding wheel coordinate system to the workpiece coordinate system is... Cone As shown in equation (27):

[0087]

[0088] (5) Peripheral grinding motion model:

[0089] As can be seen from the definition of the peripheral helical cutting edge, it has an equal helical cutting edge structure. Since the motion of the grinding wheel can be regarded as "fixed initial posture + rotational displacement motion", the aforementioned initial grinding wheel posture is z. p When the grinding wheel posture is 0, during the peripheral grinding process, the movement of the grinding wheel is only the rotational movement of the workpiece axis, the axial displacement movement, and the movement of the grinding wheel at the workpiece taper κ. w Radial displacement motion when ≠ 0; thus establishing the grinding motion matrix M during the peripheral cutting process. blade (z p ,κ w As shown in the following formula:

[0090]

[0091] In the formula, R(z) p ) = R w -R w ·tan(κ w ), z p z is the independent variable p ∈(0,L w ).

[0092] (6) End-edge grinding motion model

[0093] Define the tangent vector of the grinding point P on the spherical profile in the generatrix direction as F. m Define the tangent vector of the grinding point P along the direction of the helical cutting edge as F. ts From the definition of the helix angle, we know that the vector F m With F ts The included angle between them is the helix angle β. s It is easy to see that vector F m With F ts The plane formed lies within the tangent plane of the spherical profile at the grinding point P. Therefore, the tangent vector F of the helical cutting edge at any point on the ball head is... ts It can be regarded as the generatrix tangent vector F at the grinding point. m The grinding point normal vector F, which points from the center of the sphere to the grinding point P. NP The helix angle β at the rotary grinding point P s get.

[0094] During the grinding process, the attitude of the computational coordinate system needs to be adjusted in real time. This attitude adjustment requires the real-time helix angle β at the grinding point P. s The calculation begins by determining the tangent F at any point on the cutting edge of the ball head, based on the mathematical expression (8) for the cutting edge line. ts The equation is shown below:

[0095]

[0096] In the formula, Regarding θ s derivative As shown in the following formula:

[0097]

[0098] In the formula, β s Let P be the helix angle at the grinding point.

[0099] To determine the helix angle β s It is also necessary to determine the generatrix vector F at the grinding point P. m F m Considered as being based on the X plane of the grinding wheel coordinate system d O d Z d Inner generatrix vector F′ m Around the coordinate axis Z d Rotation angle It is easy to see that F′ NP Perpendicular to F′ m And F′ NP The unit vector is expressed as F′ NP (0,cosθ s sinθ s After discarding the opposite values ​​that do not meet the requirements, take F′. m The value is (0, -sinθ) s cosθ s ,0)T;thenF m It can be obtained from the following formula:

[0100]

[0101] In the formula, θ s ∈(-90,90).

[0102] Given the formulas for the two tangent vectors on the cutting edge, the helix angle β at any point on the cutting edge of the end blade is... s It can be further obtained from equation (32):

[0103]

[0104] The motion of the end-edge grinding trajectory is defined as: around the Z-axis of the end-edge coordinate system. d Y d Rotational motion in the direction of grinding, and the normal vector F around the grinding point P. NP The helix angle transformation; similarly considering the mutual influence of parameters in the rotation transformation process, the grinding motion trajectory matrix in the end-edge coordinate system is obtained as shown in equation (33):

[0105]

[0106] Represent the coordinates of the center point O of the large end face of the grinding wheel in matrix form. g-st and axis vector F g-st The initial values ​​are [0,0,0,1]T and [0,0,1,0]T, respectively. By combining formulas (19), (20), (28) and (2), (19), (27), (33), the grinding wheel motion transformation matrix M for the grinding process of the peripheral and end edges in the workpiece coordinate system is obtained. body and M head As shown in the formula:

[0107]

[0108] The beneficial technical effects of this invention are as follows:

[0109] This invention enables flexible machining of spherical rotary files, offering good tooling flexibility and machining accuracy. Attached Figure Description

[0110] Figure 1 A schematic diagram defining the coordinate system.

[0111] Figure 2 This is a schematic diagram of the circumferential cutting edge model.

[0112] Figure 3 This is a schematic diagram of the end-edge cutting line model.

[0113] Figure 4 A comparison chart of zoned and clustered blade lines.

[0114] Figure 5 Modeling diagram for the relationship between partitioned blade lines.

[0115] Figure 6 This is a schematic diagram of the parameter control for a zoned cutting edge.

[0116] Figure 7 Solve for the interval of parameters for the partitioned cutting edge.

[0117] Figure 8 This is a schematic diagram of a grinding wheel model.

[0118] Figure 9 A grinding wheel model for incorporating process parameters.

[0119] Figure 10 This is a schematic diagram of the initial grinding position of the cylindrical peripheral cutting edge.

[0120] Figure 11 This is a schematic diagram of the initial grinding posture of the conical circumferential cutting edge.

[0121] Figure 12 This is a schematic diagram of the peripheral cutting motion of a grinding wheel.

[0122] Figure 13 This is a schematic diagram of the grinding motion of a ball-head rotating file tip.

[0123] Figure 14 The simulation results are for a cylindrical ball-head rotary file.

[0124] Figure 15 This is the actual machining result of a cylindrical ball-head rotary file. Detailed Implementation

[0125] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0126] The present invention provides a method for calculating the CNC grinding trajectory of a ball-head rotary file, comprising the following steps:

[0127] Step 1: Define the key coordinate system.

[0128] Workpiece coordinate system

[0129] like Figure 1 As shown, the workpiece coordinate system O is defined. w -X w Y w Z w As the absolute coordinate system in the coordinate system; define the workpiece coordinate system X. w O w Y w The plane is the plane where the starting point of the circumferential spiral cutting edge is located, and the origin of the coordinate system is X. w O w Y w Intersection of the plane and the tool axis; Z-axis w Coaxial with the tool rotation axis; X-axis w Intersecting at the starting point of the initial spiral cutting edge; Y w It is determined by the right-hand rule.

[0130] End-edge coordinate system

[0131] like Figure 1 As shown, define the end-edge coordinate system O. d -X d Y d Z d Z-axis d Coincident with the tool axis, axis X dParallel to X w Y w The plane intersects at the endpoint of the peripheral cutting edge. Considering the continuity of the cutting edge and the simplification of the modeling process, the end-edge coordinate system has a phase angle transformation relationship with the workpiece coordinate system, with the X-axis as the reference point. d With X w The angle between The functional relationship is expressed as follows:

[0132]

[0133] In the formula, L z O is the origin of the end-edge coordinate system. d With respect to the workpiece coordinate system plane X w Y w The distance, R w For X w Y w In-plane blank radius, β w κ is the helix angle of the tooth. w It is the angle between the generatrix of the rotating body of the tool and the axis.

[0134] As required by the coordinate system, the end-cutting line representation in the end-cutting coordinate system needs to be transformed to the workpiece coordinate system. Therefore, there exists a transformation relationship from the end-cutting coordinate system to the workpiece coordinate system, such as matrix M. d→w As shown:

[0135]

[0136] In the formula, L is the helix angle at the end of the peripheral cutting edge. w The length of the circumferential blade.

[0137] Grinding wheel coordinate system

[0138] Define the origin O of the grinding wheel coordinate system G The cutting edge is along the workpiece coordinate system Z. w A moving point X moving in the positive direction G The axis intersects the tool axis; Y G Tangent to the spiral blade line.

[0139] Step 2: Establish the perimeter cutting edge model.

[0140] like Figure 2 As shown, this invention selects the equal helix angle cutting edge curve, which is widely used in actual engineering, as the peripheral cutting edge structure of the rotary file, and the principle of this peripheral cutting edge modeling structure is already very mature. First, the motion trajectory of a moving point P on the tool blank in a helical forward manner is defined as the cutting edge. The peripheral cutting edge of the rotary file is then established with an axial displacement z. pThe parameterized equation expression for the independent variable is as follows (for ease of coordinate transformation, all point coordinates and vector coordinates are expressed in the form of a single-column four-row matrix in this paper, and this expression does not affect the final calculation result):

[0141]

[0142] In the formula, Let z be the independent variable p The corresponding circumferential turning angle.

[0143] The number of rotary file cutting edges is defined as N. Unlike general rotary tools, the number of rotary file cutting edges is generally between 10 and 24. For the peripheral cutting edge structure, all cutting edges have the same structure, but they are evenly distributed on the peripheral edge of the tool in a central array. The angle between two adjacent cutting edges is defined as the pitch angle. It depends on the defined number of cutting edge groups N g Number of blades N in the group b And N = N g ·N b The specified number of a partitioned rotary file, N, is not a prime number; the graduation angle... The function expression is as follows:

[0144]

[0145] Establish the trajectory P of point P on the circumferential cutting edge of the rotary file using equations (3) and (4) in combination. tracks The expression is shown in equation (5):

[0146]

[0147] In the formula, n represents the coordinate system X of the workpiece. w The nth cutting edge line in the counterclockwise direction starting from the axis.

[0148] Step 3: Establish the end-edge cutting line model;

[0149] The ball-end cutting edge is parametrically modeled using spherical coordinates, with the origin of the end-edge coordinate system placed at the center of the end-edge structure, and X... d The axis serves as the reference for the independent variable and rotation angle in spherical coordinates. To unify the coordinate representation, and considering that the tool position point is expressed in three-dimensional coordinates, the polar coordinate expression in the spherical coordinate system is finally transformed into an expression in the three-dimensional coordinate system. For example... Figure 3 As shown, the Z-axis of the end-edge coordinate system is constructed using projection. d The trajectories formed by the intersection of the perpendicular line and the tangent line of the cylinder with tooth deviation h are the cutting edges without tooth deviation and with tooth deviation h, respectively. The cutting edge equations are uniformly constructed in spherical coordinates. As shown below:

[0150]

[0151] In the formula, θ is the end-edge helix angle. s The latitude angle is 0, and the tooth deviation is 0. <h<R s cosκ w R s Let be the radius of the sphere head.

[0152] The independent variable is affected by the tooth deviation and the peripheral taper, so it is rotated back to plane M. xz The independent variable θ can be obtained from this. s The range of values ​​is as follows:

[0153] θ s ∈κ( w arccos(h / R) s (7)

[0154] To facilitate the expression of the cutting edge in the workpiece coordinate system, the cutting edge of the ball end in the spherical coordinate system is expressed in three-dimensional coordinates as shown in equation (8):

[0155]

[0156] Step 4: Establish the blade line partition control model.

[0157] According to He Yaoxiong's understanding [He Yaoxiong, Zhou Yunfei, Zhou Ji. Ball end tool cutting edge modeling and transition edge design [J]. Journal of Mechanical Engineering, 2001, 37(9): 101-104.], the essence of ball end tool cutting edge partitioning design is a node interpolation problem on the ball end profile surface.

[0158] like Figure 4 As shown, since the cutting edge lines are spatial curves distributed on a sphere, the calculation process for direct zoning design is quite complex. To simplify the calculation, we can use a projection method to transform the three-dimensional calculation problem into a two-dimensional calculation problem, projecting the cutting edge lines onto the X plane of the end-edge coordinate system. d O d Y d The partitioning method used is as follows: the ball-end cutting lines are divided into groups, and each group consists of a main cutting edge passing through the vertex of the ball and several secondary cutting edges; the length of the endpoint of the i-th secondary cutting edge from the origin is defined as the cutting edge excess l. i The distance from the center is h. i The density of the cutting edge lines near the tip of the ball head is mainly determined by the cutting edge deflection h. i The impact.

[0159] like Figure 5 As shown, since the i-th ball head, i = 1, 2…Nb -1 Initial rotation angle of the secondary cutting edge The existence of any set of primary and secondary cutting edge endpoint extensions on the projection plane X d O d Y d The inner edges must intersect. It is easy to see that the distance from the i-th set of cutting edges to the origin O is... d The length is the distance l through the center of the blade. i The intersection point is P. u_i To establish the functional relationship between the relative positions of the primary and secondary cutting edges, it is necessary to first determine the included angle ∠P formed by the intersection of the endpoints of the primary and secondary cutting edges. u_i Typically, the end point of the main cutting edge is located at the vertex of the rotary file. The tangent vector at the end point of the main cutting edge in the end-edge coordinate system is obtained by differentiating equation (8), which is the θ vector under the condition of no tooth deviation. s The tangent vector of the cutting edge at 90° is projected onto the coordinate plane X. d O d Y d The unit vector F in P-end As shown in equation (9):

[0160]

[0161] The vector at the end of the i-th cutting edge is obtained from equation (10). For a partitioned rotary file, a set of angle parameters within the cutting edge line can be obtained.

[0162]

[0163] In the formula, θ(h) i ) represents the independent variable θ corresponding to the i-th cutting edge. s Upper limit, and θ(h) i =arccos(h i / R s ), h i Let N be the cutting edge deviation of the i-th secondary cutting edge, 0≤i≤N b .

[0164] By combining equations (9) and (10), we can further obtain the angle ∠P between the tangent vectors at the last point of the primary cutting edge and the i-th secondary cutting edge within the cutting edge line in the projection plane. u_i for:

[0165]

[0166] As can be seen from the method of forming the edge deflection line, its end point is tangent to the origin O of the edge coordinate system. d Tooth deflection h centered on the circle i The ring can form a right triangle ΔP. h_end_i P u_i P endBased on the right triangle tangent relationship, the following functional relationship can be constructed:

[0167]

[0168] In the formula, h i For the i-th secondary cutting edge offset within the same cutting edge group, l i Let P be the intersection point. u_i With the main cutting edge P end Distance between endpoints.

[0169] Solving equations (9) to (12) together yields the result regarding the variable θ(h) i ), h i and l i The mathematical model is shown in equation (13):

[0170]

[0171] In the formula, θ(h i =arccos(h i / R s Therefore, this formula is only related to h. i and l i The relevant functional relationships.

[0172] Further selection l i h is selected as the control variable for the blade line partition. i As a dependent parameter for the density of the cutting edges, in order to achieve unified control over the zoning of each secondary cutting edge, it is stipulated that all secondary cutting edges within a group intersect the main cutting edge at the same point P in the end-edge coordinate system plane. m and with the main cutting edge tooth excess l m As a zoning control variable, the ball head edge line is used for zoning control, such as Figure 6 As shown. Therefore, sequentially in X d O d Y d Calculate the cutting edge deviation h of the i-th set of cutting edges in the same group of cutting edges in the coordinate plane. i This causes the extensions of the endpoints of each secondary cutting edge to intersect the extension of the primary cutting edge at point P. m Point.

[0173] To determine the range of tooth deviation along the cutting edge, the tooth deviation intervals of both the primary and secondary cutting edges should be considered. Furthermore, as shown in equation (6), if the tooth deviation satisfies the following condition: Therefore, this equation has no solution, and thus the equation for the secondary cutting edge cannot be obtained. Based on the above requirements, the cutting edge deviation h needs to be solved. i The range of values ​​for is shown in equation (14):

[0174] 0≤h i <Rs cosκ w ,κ w ∈(-90,90) (14)

[0175] For a focused ball-end rotary file, its cutting edge structure can be considered as a special case where the offset of all its secondary cutting edges is zero. In summary, treating the focused ball-end rotary file as a special case of a partitioned ball-end rotary file simplifies the cutting edge structure model and the partitioning process, thereby enabling parameterized partitioning adjustment of the ball-end rotary file.

[0176] Step 5: Solve for the cutting edge parameters.

[0177] As defined in step 4, the distribution of each group of cutting edges follows the same pattern, and it is necessary to ensure that the distribution of each group of rotary file cutting edges is within the region of two adjacent main cutting edges, and that the cutting edges do not intersect. Therefore, it is necessary to further solve for the range of values ​​for the rotary file cutting edge parameters. The boundary of each group of ball end cutting edges is the next adjacent main cutting edge. For example... Figure 7 As shown, due to the characteristics of spherical coordinates, every point on the sphere can be represented by latitude and longitude coordinates. Therefore, under the requirement of non-intersection, the endpoint of each set of cutting edges and the intersection point of the intersecting main cutting edges have the same latitude and longitude coordinates. The corresponding latitude independent variable value θ is then used. s This serves as the upper limit of the independent variable interval when the blade lines do not intersect.

[0178] First, set the partition control parameters l m Substitute the cutting edge deviation parameter h of the i-th secondary cutting edge in equation (12) to solve the partition model. i , i = 1, 2, 3…, and the function equation of the i-th secondary cutting edge of the ball end cutting edge is further determined by equation (6). As shown in equation (15):

[0179]

[0180] Number of sections of ball head rotating file line group N g Number of blades N in the group b It can be seen that the phase angle between any two main cutting edges is Phase angle between any two adjacent cutting edges and the turning angle on the main chip cutting edge Find the upper limit θ of the independent variable interval of the secondary cutting edge corresponding to the intersecting cutting edge points in spherical coordinates. max-i Solve equations (6) and (15) simultaneously to find the intersection point P in spherical coordinates. max-i i = 1, 2…N b-1 As shown in equation (16):

[0181]

[0182] In the formula, The longitude angle of the helix of the main cutting edge; then the independent variable θ for the grinding of the zoned ball-head rotating file cutting edge is... s-i The interval is:

[0183] κ w ≤θ s-i ≤θ max-i (17)

[0184] Step 6: Grinding motion model of grinding wheel.

[0185] (1) Definition of grinding motion and model of grinding wheel rotation surface:

[0186] The rotary file chip groove is machined using 1V1 / 1A1 parallel grinding wheel. The grinding process can be regarded as follows: the workpiece is fixed and the grinding wheel moves along the spiral cutting edge in an initial grinding posture.

[0187] like Figure 8 As shown, the mathematical model of the grinding wheel's rotation profile is established in the grinding wheel coordinate system. The center of the large end face of the grinding wheel is aligned with the origin of the grinding wheel coordinate system, and the axis of rotation of the grinding wheel is aligned with the Z-axis of the grinding wheel coordinate system. G Overlap, along the Z-axis of the grinding wheel coordinate system G A discrete mathematical model of the grinding wheel solid is established in the positive direction of the axis, and the grinding wheel rotation profile equation is established as shown in equation (18):

[0188]

[0189] In the formula, h G Let ψ be the independent variable in the direction of the grinding wheel height. G For grinding wheel h G The independent variable of the cross-sectional rotation angle, κ G This refers to the taper of the grinding wheel.

[0190] (2) Grinding wheel process parameter transformation matrix:

[0191] Define the chip flute rake angle as γ and the depth of cut parameter as cd. To avoid mutual interference between parameters during the initial grinding wheel pose setting process, the transformation process of the grinding wheel parameter definition is specified as follows: First, the grinding wheel mathematical model is moved along the grinding wheel coordinate system X. G Translation R in the positive direction of the axis G -cd. For example... Figure 9 As shown, this makes the origin O of the coordinate system... G Coincident at grinding point P, then rotate the grinding wheel around the grinding wheel coordinate system Y. G Rotate the axis clockwise by an angle γ, and the parameter transformation matrix M G-para (γ,cd) is shown in equation (19):

[0192]

[0193] (3) Initial grinding position of the cylindrical peripheral cutting edge in the workpiece coordinate system:

[0194] like Figure 10 As shown, the grinding wheel coordinate system under the initial grinding posture should satisfy the following condition: the origin O of the grinding wheel coordinate system G Always moving along the blade trajectory; coordinate axis Y G Parallel to the helical blade tangent vector F t ; coordinate axis X G With respect to the workpiece coordinate system axis X w Overlap, that is, when grinding a conical rotary file, X G Only with the coordinate axis X w Intersecting; furthermore, the initial position of the grinding wheel coordinate system is also determined by the cutting edge helix angle β. w Workpiece radius R w and taper κ w Decide.

[0195] Define the transformation matrix M of the initial posture of the cutting edge grinding wheel in the workpiece coordinate system. Cylinder As shown in equation (20):

[0196]

[0197] (4) Initial grinding position of the conical peripheral cutting edge in the workpiece coordinate system:

[0198] like Figure 11 As shown, the initial attitude setting for the conical blank needs to take into account the taper κ. w Therefore, in equation (20), an additional rotation matrix is ​​introduced. In the above transformation matrix expression, if multiple matrices rotating around the coordinate axis are introduced, the matrices will affect each other, which will lead to the inability to guarantee parameters such as the chip flute rake angle during grinding. Therefore, only one rotation matrix around the coordinate axis that guarantees the helix angle is retained in the above matrix, and on this basis, an additional matrix Rot(N,ξ) rotating around any vector N is introduced to avoid parameter errors caused by the mutual influence of the rotation matrices. Rot(N,ξ) is shown in equation (21):

[0199]

[0200] In the formula, ξ is the rotation angle about vector N, and its direction is determined by the right-hand rule.

[0201] After introducing the rotation matrix Rot(N,ξ), it is further necessary to determine the arbitrary rotation vector N, where N is the initial tangent plane X. G O G Y G normal vector F N The rotation angle ξ represents the grinding wheel coordinate system Y under the initial attitude of the grinding wheel without considering the taper.G The extension line in the opposite direction intersects the tangent vector F of the cone's circumferential edge. t The included angle; first, differentiate the formula for the spiral cutting edge to obtain the tangent vector F at any point P on the cutting edge. t As shown in equation (22):

[0202]

[0203] In the formula,

[0204]

[0205] Vector F N Through the grinding wheel tangent vector F t |z p＝0 With respect to the workpiece coordinate axis X w Direction vector Cross product calculation yields:

[0206]

[0207] The rotation angle ξ is the circumferential cutting vector F t |z p＝0 The Y-axis of the grinding wheel coordinate system is relative to the initial pose without considering the workpiece taper. G Let the angle (acute angle) between the opposite direction vectors be... F YG-neg The workpiece coordinate system can be obtained using the following formula:

[0208]

[0209] The expression for the rotation angle ξ is:

[0210]

[0211] To ensure that the rotation transformation does not affect the coordinate system position, the displacement can be performed after introducing the rotation matrix. Therefore, the general transformation matrix M for the conical peripheral cutting edge from the initial grinding wheel coordinate system to the workpiece coordinate system is... Cone As shown in equation (27):

[0212]

[0213] (5) Peripheral grinding motion model:

[0214] The study of the motion model for peripheral grinding is further divided into cylindrical and conical cutting edges. The cylindrical cutting edge can be regarded as the workpiece taper κ. w For the special case of 0, this paper will establish a unified law of motion transformation for ease of description. For example... Figure 12As shown, based on the definition of the peripheral helical cutting edge, it is a constant helical cutting edge structure. Since the grinding wheel motion can be considered as "fixed initial posture + rotational displacement motion", the aforementioned initial grinding wheel posture is z. p When the grinding wheel posture is 0, during the peripheral grinding process, the movement of the grinding wheel is only the rotational movement of the workpiece axis, the axial displacement movement, and the movement of the grinding wheel at the workpiece taper κ. w Radial displacement motion when ≠ 0; thus establishing the grinding motion matrix M during the peripheral cutting process. blade (z p ,κ w As shown in the following formula:

[0215]

[0216] In the formula, R(z) p ) = R w -R w ·tan(κ w ), z p z is the independent variable p ∈(0,L w ).

[0217] (6) End-edge grinding motion model

[0218] like Figure 13 As shown, the tangent vector of the grinding point P on the spherical profile along the generatrix direction is defined as F. m Define the tangent vector of the grinding point P along the direction of the helical cutting edge as F. ts From the definition of the helix angle, we know that the vector F m With F ts The included angle between them is the helix angle β. s It is easy to see that vector F m With F ts The plane formed lies within the tangent plane of the spherical profile at the grinding point P. Therefore, the tangent vector F of the helical cutting edge at any point on the ball head is... ts It can be regarded as the generatrix tangent vector F at the grinding point. m The grinding point normal vector F, which points from the center of the sphere to the grinding point P. NP The helix angle β at the rotary grinding point P s get.

[0219] To ensure the grinding wheel coordinate system remains relatively stationary, the attitude of the computational coordinate system needs to be adjusted in real time during the grinding process. This attitude adjustment requires considering the real-time helix angle β at the grinding point P. s The calculation begins by determining the tangent F at any point on the cutting edge of the ball head, based on the mathematical expression (8) for the cutting edge line. ts The equation is shown below:

[0220]

[0221] In the formula, Regarding θ s derivative As shown in the following formula:

[0222]

[0223] In the formula, β s Let P be the helix angle at the grinding point.

[0224] To determine the helix angle β s It is also necessary to determine the generatrix vector F at the grinding point P. m F m Considered as being based on the X plane of the grinding wheel coordinate system d O d Z d Inner generatrix vector F′ m Around the coordinate axis Z d Rotation angle It is easy to see that F′ NP Perpendicular to F′ m And F′ NP The unit vector is expressed as After discarding the inverse values ​​that do not meet the requirements, take F′. m Value Then F m It can be obtained from the following formula:

[0225]

[0226] In the formula, θ s ∈(-90,90).

[0227] Given the formulas for the two tangent vectors on the cutting edge, the helix angle β at any point on the cutting edge of the end blade is... s It can be further obtained from equation (32):

[0228]

[0229] To ensure the continuity of the cutting edge, the helix angle β at the beginning of the end cutting edge is... s It should be consistent with the helix angle β of the peripheral edge. Therefore, during the grinding process from the peripheral edge to the end edge, the endpoint position of the end edge coincides with the starting position of the peripheral edge in the grinding wheel coordinate system, meaning the grinding wheel posture is consistent. This facilitates a one-step forming process in actual machining, and eliminates the need to redefine the transformation matrix from the end edge grinding wheel coordinate system to the workpiece coordinate system. In summary, the motion of the end edge grinding trajectory is defined as: around the Z-axis of the end edge coordinate system. d Y d Rotational motion in the direction of grinding, and the normal vector F around the grinding point P. NPThe helix angle transformation; similarly considering the mutual influence of parameters in the rotation transformation process, the grinding motion trajectory matrix in the end-edge coordinate system is obtained as shown in equation (33):

[0230]

[0231] By superimposing the grinding wheel grinding parameter model, the initial posture transformation matrix, and the grinding motion trajectory, a complete motion pose model of the grinding wheel process can be obtained. This mathematical model includes the coordinate trajectory of the grinding wheel center and the motion axis vector trajectory. The coordinates of the center point O of the large end face of the grinding wheel are represented in matrix form. g-st and axis vector F g-st The initial values ​​are [0,0,0,1]T and [0,0,1,0]T, respectively. By combining formulas (19), (20), (28) and (2), (19), (27), (33), the grinding wheel motion transformation matrix M for the grinding process of the peripheral and end edges in the workpiece coordinate system is obtained. body and M head As shown in the formula:

[0232]

[0233] Simulation example:

[0234] The grinding solution was developed using the Visual Studio 2019 environment. Based on the reliability requirements of the process scheme and algorithm module, simulation verification and actual machining verification of the calculation results of the grinding parameter model and algorithm module were implemented using Vericut 8.0. The machining parameters are shown in Table 1 below:

[0235] Table 1 Process Parameters

[0236]

[0237]

[0238] Simulation verification, such as Figure 14 As shown, after completing the corresponding grinding process, the overall structural dimensions and local dimensions of the cylindrical ball-head rotary file were measured using an optical measuring instrument PG1000. The relevant measured parameters of the cylindrical ball-head rotary file are shown in Table 2.

[0239] Table 2 Measured Parameters

[0240]

[0241] Actual processing such as Figure 15 As shown in the figure, the actual measurement results show that although the accuracy of the measurement parameters in the actual machining is lower than that in the simulation machining, and the measurement allowance between the cutting edges is 0.03mm, the measurement error ε is still significant.m The error is still within the set range of ε∈(-0.025mm, +0.025mm). Since errors are unavoidable in actual processing, considering the combined effects of multiple factors such as grinding wheel wear, blank accuracy, and machine tool error, the error is still within a reasonable range.

Claims

1. A method for calculating the CNC grinding trajectory of a ball-head rotary file, characterized in that, Includes the following steps: Step 1: Define the key coordinate system; Workpiece coordinate system Define the workpiece coordinate system O w -X w Y w Z w As the absolute coordinate system in the coordinate system; define the workpiece coordinate system X. w O w Y w The plane is the plane where the starting point of the circumferential spiral cutting edge is located, and the origin of the coordinate system is X. w O w Y w Intersection of the plane and the tool axis; coordinate axis Z w Coaxial with the tool rotation axis; X-axis w Intersecting at the starting point of the initial spiral cutting edge; Y w Determined by the right-hand rule; End-edge coordinate system Define the end-edge coordinate system O d -X d Y d Z d Z-axis d Coincident with the tool axis, axis X d Parallel to X w Y w The plane intersects at the endpoint of the peripheral cutting edge. Considering the continuity of the cutting edge and the simplification of the modeling process, the end-edge coordinate system has a phase angle transformation relationship with the workpiece coordinate system, with the X-axis as the reference point. d With X w The angle between The functional relationship is expressed as follows: In the formula, L z O is the origin of the end-edge coordinate system. d With respect to the workpiece coordinate system plane X w Y w The distance, R w For X w Y w In-plane blank radius, β w κ is the helix angle of the tooth. w The angle between the generatrix of the tool's rotating body and the axis; As required by the coordinate system, the end-cutting line representation in the end-cutting coordinate system needs to be transformed to the workpiece coordinate system. Therefore, there exists a transformation relationship from the end-cutting coordinate system to the workpiece coordinate system, such as matrix M. d→w As shown: In the formula, L is the helix angle at the end of the peripheral cutting edge. w The length of the circumferential cutting edge; Grinding wheel coordinate system Define the origin O of the grinding wheel coordinate system G The cutting edge is along the workpiece coordinate system Z. w A moving point X moving in the positive direction G The axis intersects the tool axis; Y G Tangent to the spiral cutting edge; Step 2: Establish the perimeter cutting edge model; First, define the motion trajectory of a moving point P on the tool blank in a helical forward motion as the cutting edge line, and establish the circumferential cutting edge line of the rotary file with axial displacement z. p The parameterized equation expression with respect to the independent variable is as follows: In the formula, Let z be the independent variable p The corresponding circumferential blade rotation angle; Define the number of rotary file cutting edges as N, and define the angle between two adjacent cutting edges as the division angle. It depends on the defined number of cutting edge groups N g Number of blades N in the group b And N = N g ·N b The specified number of a partitioned rotary file, N, is not a prime number; the graduation angle... The function expression is as follows: Establish the trajectory P of point P on the circumferential cutting edge of the rotary file using equations (3) and (4) in combination. tracks The expression is shown in equation (5): In the formula, n represents the coordinate system X of the workpiece. w The nth cutting edge in the counterclockwise direction starting from the axis; Step 3: Establish the end-edge cutting line model; The ball-end cutting edge is parametrically modeled using spherical coordinates, with the origin of the end-edge coordinate system placed at the center of the end-edge structure, and X... d The axis serves as the reference for the independent variable and rotation angle in spherical coordinates. To unify the coordinate representation, and considering that the tool position point is expressed in three-dimensional coordinates, the polar coordinate expression in the spherical coordinate system is finally transformed into the expression in the three-dimensional coordinate system. The Z-axis of the end-edge coordinate system is constructed using projection. d The trajectories formed by the intersection of the perpendicular line and the tangent line of the cylinder with tooth deviation h are the cutting edges without tooth deviation and with tooth deviation h, respectively. The cutting edge equations are uniformly constructed in spherical coordinates. As shown below: In the formula, θ is the end-edge helix angle. s The latitude angle is 0, and the tooth deviation is 0. <h<R s cosκ w R s The radius of the ball head; The independent variable is affected by the tooth deviation and the peripheral taper, so it is rotated back to plane M. xz The independent variable θ can be obtained from this. s The range of values ​​is as follows: θ s ∈(κ w arccos(h / R) s (7) The cutting edge line of the ball head in the spherical coordinate system is expressed in three-dimensional coordinates as shown in equation (8): Step 4: Establishment of the blade line zoning control model; The ball-end cutting lines are divided into groups, and each group of cutting lines consists of a main cutting edge passing through the vertex of the ball and several secondary cutting edges; Define the distance from the end point of the i-th cutting edge to the origin as the cutting edge excess l. i The distance from the center is h. i The density of the cutting edge lines near the tip of the ball head is mainly determined by the cutting edge deflection h. i The impact; Since the i-th ball head, i = 1, 2, ..., N b -1 Initial rotation angle of the secondary cutting edge The existence of any set of primary and secondary cutting edge endpoint extensions on the projection plane X d O d Y d The inner edges must intersect. It is easy to see that the distance from the i-th set of cutting edges to the origin O is... d The length is the distance l through the center of the blade. i The intersection point is P. u_i To establish the functional relationship between the relative positions of the primary and secondary cutting edges, it is necessary to first determine the included angle ∠P formed by the intersection of the endpoints of the primary and secondary cutting edges. u_i Typically, the end point of the main cutting edge is located at the vertex of the rotary file. The tangent vector at the end point of the main cutting edge in the end-edge coordinate system is obtained by differentiating equation (8), which is the θ vector under the condition of no tooth deviation. s The tangent vector of the cutting edge at 90° is projected onto the coordinate plane X. d O d Y d The unit vector F in P-end As shown in equation (9): The vector at the end of the i-th cutting edge is obtained from equation (10). For a partitioned rotary file, a set of angle parameters within the cutting edge line can be obtained. In the formula, θ(h) i ) represents the independent variable θ corresponding to the i-th cutting edge. s Upper limit, and θ(h) i =arccos(h i / R s ), h i Let N be the cutting edge deviation of the i-th secondary cutting edge, 0≤i≤N b ; Alliance Equations (9) and (10) further yield the angle ∠P between the tangent vectors at the last point of the primary cutting edge and the i-th secondary cutting edge within the cutting edge line in the projection plane. u_i for: As can be seen from the method of forming the edge deflection line, its end point is tangent to the origin O of the edge coordinate system. d Tooth deflection h centered on the circle i The ring can form a right triangle ΔP. h_end_i P u_i P end Based on the right triangle tangent relationship, the following functional relationship can be constructed: In the formula, h i For the i-th secondary cutting edge offset within the same cutting edge group, l i Let P be the intersection point. u_i With the main cutting edge P end Distance between endpoints; Solving equations (9) to (12) together yields the result regarding the variable θ(h) i ), h i and l i The mathematical model is shown in equation (13): In the formula, θ(h i =arccos(h i / R s Therefore, this formula is only related to h. i and l i The relevant functional relationship; Further selection l i h is selected as the control variable for the blade line partition. i As a dependent parameter for the density of the cutting edges, in order to achieve unified control over the zoning of each secondary cutting edge, it is stipulated that all secondary cutting edges within a group intersect the main cutting edge at the same point P in the end-edge coordinate system plane. m and with the main cutting edge tooth excess l m As a zoning control variable, the ball head edge line is zoned; therefore, sequentially in X... d O d Y d Calculate the cutting edge deviation h of the i-th set of cutting edges in the same group of cutting edges in the coordinate plane. i This causes the extensions of the endpoints of each secondary cutting edge to intersect the extension of the primary cutting edge at point P. m Point; Solve for the edge deviation h i The range of values ​​for is shown in equation (14): 0≤h i <R s cost w ,k w ∈(-90,90) (14) Step 5: Solve for the cutting edge parameters; First, set the partition control parameters l m Substitute the cutting edge deviation parameter h of the i-th secondary cutting edge in equation (12) to solve the partition model. i , i = 1, 2, 3…, and the function equation of the i-th secondary cutting edge of the ball end cutting edge is further determined by equation (6). As shown in equation (15): Number of sections of ball head rotating file line group N g Number of blades N in the group b It can be seen that the phase angle between any two main cutting edges is Phase angle between any two adjacent cutting edges and the turning angle on the main chip cutting edge Find the upper limit θ of the independent variable interval of the secondary cutting edge corresponding to the intersecting cutting edge points in spherical coordinates. max-i Solve equations (6) and (15) simultaneously to find the intersection point P in spherical coordinates. max-i i = 1, 2…N b-1 As shown in equation (16): In the formula, The longitude angle of the helix of the main cutting edge; then the independent variable θ for the grinding of the zoned ball-head rotating file cutting edge is... s-i The interval is: k w ≤θ s-i ≤θ max-i (17) Step 6: Grinding motion model of grinding wheel; (1) Definition of grinding motion and model of grinding wheel rotation surface: The rotary file chip groove is machined using 1V1 / 1A1 parallel grinding wheel grinding. The grinding process can be regarded as: the workpiece is fixed and the grinding wheel moves along the spiral cutting edge in an initial grinding posture. The mathematical model of the grinding wheel's rotation profile is established in the grinding wheel coordinate system, with the center of the large end face of the grinding wheel coinciding with the origin of the grinding wheel coordinate system, and the axis of rotation of the grinding wheel aligned with the Z-axis of the grinding wheel coordinate system. G Overlap, along the Z-axis of the grinding wheel coordinate system G A discrete mathematical model of the grinding wheel solid is established in the positive direction of the axis, and the grinding wheel rotation profile equation is established as shown in equation (18): In the formula, h G Let ψ be the independent variable in the direction of the grinding wheel height. G For grinding wheel h G The independent variable of the cross-sectional rotation angle, κ G For the taper of the grinding wheel; (2) Grinding wheel process parameter transformation matrix: Define the chip flute rake angle as γ and the depth of cut parameter as cd. To avoid mutual interference between parameters during the initial grinding wheel pose setting process, the transformation process of the grinding wheel parameter definition is specified as follows: First, the grinding wheel mathematical model is moved along the grinding wheel coordinate system X. G Translation R in the positive direction of the axis G -cd; makes the origin O of the coordinate system... G Coincident at grinding point P, then rotate the grinding wheel around the grinding wheel coordinate system Y. G Rotate the axis clockwise by an angle γ, and the parameter transformation matrix M G-para (γ,cd) is shown in equation (19): (3) Initial grinding position of the cylindrical peripheral cutting edge in the workpiece coordinate system: The grinding wheel coordinate system under the initial grinding posture should satisfy the following conditions: the origin of the grinding wheel coordinate system O G Always moving along the blade trajectory; coordinate axis Y G Parallel to the helical blade tangent vector F t ; coordinate axis X G With respect to the workpiece coordinate system axis X w Overlap, that is, when grinding a conical rotary file, X G Only with the coordinate axis X w Intersecting; furthermore, the initial position of the grinding wheel coordinate system is also determined by the cutting edge helix angle β. w Workpiece radius R w and taper κ w Decide; Define the transformation matrix M of the initial posture of the cutting edge grinding wheel in the workpiece coordinate system. Cylinder As shown in equation (20): (4) Initial grinding position of the conical peripheral cutting edge in the workpiece coordinate system: When setting the initial orientation of a tapered blank, the taper κ needs to be taken into account. w To avoid the influence of the rotation matrices, a rotation matrix Rot(N,ξ) around any vector N is introduced in equation (20) to avoid parameter errors caused by the mutual influence of the rotation matrices. Rot(N,ξ) is shown in equation (21): In the formula, ξ is the rotation angle about vector N, and the direction is determined by the right-hand rule; After introducing the rotation matrix Rot(N,ξ), it is further necessary to determine the arbitrary rotation vector N, where N is the initial tangent plane X. G O G Y G normal vector F N The rotation angle ξ represents the grinding wheel coordinate system Y under the initial attitude of the grinding wheel without considering the taper. G The extension line in the opposite direction intersects the tangent vector F of the cone's circumferential edge. t The included angle; first, differentiate the formula for the spiral cutting edge to obtain the tangent vector F at any point P on the cutting edge. t As shown in equation (22): In the formula, Vector F N Through the grinding wheel tangent vector F t |z p＝0 With respect to the workpiece coordinate axis X w Direction vector F x_w =[1,0,0,0] T Cross product calculation yields: The rotation angle ξ is the circumferential cutting vector F t |z p＝0 The Y-axis of the grinding wheel coordinate system is relative to the initial pose without considering the workpiece taper. G Let F be the angle between the opposite direction vectors. YG-neg =[0,-1,0,0] T F YG-neg The workpiece coordinate system can be obtained using the following formula: The expression for the rotation angle ξ is: To ensure that the rotation transformation does not affect the coordinate system position, the displacement can be performed after introducing the rotation matrix. Therefore, the general transformation matrix M for the conical peripheral cutting edge from the initial grinding wheel coordinate system to the workpiece coordinate system is... Cone As shown in equation (27): (5) Peripheral grinding motion model: As can be seen from the definition of the peripheral helical cutting edge, it has an equal helical cutting edge structure. Since the motion of the grinding wheel can be regarded as "fixed initial posture + rotational displacement motion", the aforementioned initial grinding wheel posture is z. p When the grinding wheel posture is 0, during the peripheral grinding process, the movement of the grinding wheel is only the rotational movement of the workpiece axis, the axial displacement movement, and the movement of the grinding wheel at the workpiece taper κ. w Radial displacement motion when ≠ 0; thus establishing the grinding motion matrix M during the peripheral cutting process. blade (z p ,κ w As shown in the following formula: In the formula, R(z) p ) = R w -R w ·tan(κ w ), z p z is the independent variable p ∈(0,L w ); (6) End-edge grinding motion model Define the tangent vector of the grinding point P on the spherical profile in the generatrix direction as F. m Define the tangent vector of the grinding point P along the direction of the helical cutting edge as F. ts From the definition of the helix angle, we know that the vector F m With F ts The included angle between them is the helix angle β. s It is easy to see that vector F m With F ts The plane formed lies within the tangent plane of the spherical profile at the grinding point P. Therefore, the tangent vector F of the helical cutting edge at any point on the ball head is... ts It can be regarded as the generatrix tangent vector F at the grinding point. m The grinding point normal vector F, which points from the center of the sphere to the grinding point P. NP The helix angle β at the rotary grinding point P s get; During the grinding process, the attitude of the computational coordinate system needs to be adjusted in real time. This attitude adjustment requires the real-time helix angle β at the grinding point P. s The calculation begins by determining the tangent F at any point on the cutting edge of the ball head, based on the mathematical expression (8) for the cutting edge line. ts The equation is shown below: In the formula, Regarding θ s derivative As shown in the following formula: In the formula, β s The helix angle at grinding point P; To determine the helix angle β s It is also necessary to determine the generatrix vector F at the grinding point P. m F m Considered as being based on the X plane of the grinding wheel coordinate system d O d Z d Inner generatrix vector F′ m Around the coordinate axis Z d Rotation angle It is easy to see that F′ NP Perpendicular to F′ m And F′ NP The unit vector is expressed as F′ NP (0,cosθ s sinθ s ,0) T After discarding the opposite values ​​that do not meet the requirements, take F′. m The value is (0, -sinθ) s cosθ s ,0) T Then F m It can be obtained from the following formula: In the formula, θ s ∈(-90,90); Given the formulas for the two tangent vectors on the cutting edge, the helix angle β at any point on the cutting edge of the end blade is... s It can be further obtained from equation (32): The motion of the end-edge grinding trajectory is defined as: around the Z-axis of the end-edge coordinate system. d Y d Rotational motion in the direction of grinding, and the normal vector F around the grinding point P. NP The helix angle transformation; similarly considering the mutual influence of parameters in the rotation transformation process, the grinding motion trajectory matrix in the end-edge coordinate system is obtained as shown in equation (33): Represent the coordinates of the center point O of the large end face of the grinding wheel in matrix form. g-st and axis vector F g-st The initial values ​​are [0,0,0,1]. T [0,0,1,0] T By combining formulas (19), (20), (28) and (2), (19), (27), (33), the grinding wheel motion transformation matrix M for the grinding process of the peripheral and end edges in the workpiece coordinate system is obtained. body and M head As shown in the formula:

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