Ball-end milling blade cutting edge curve parametric modeling method

Abstract

The invention discloses a parametric modeling method for a cutting edge curve of a ball-end milling blade, and the method specifically comprises the steps: firstly, defining geometric feature parameters which meet the construction of a parametric geometric model of the ball-end milling blade, and guaranteeing the integrity of the parametric model; establishing a parameterized model of the blade line of each part based on each characteristic coordinate system, deducing an expression of blade point coordinates on the curve, and ensuring the smoothness of the connection between the sections of curves; a calculation formula of a blade point is given in the form of a formula, equation solving is avoided, blade curves with different geometric parameters are verified through CAD three-dimensional simulation, and the stability of a model calculation result and the adaptability of blade curve modeling under different geometric parameters are proved.

Description

Technical Field

[0001] The invention belongs to the field of tool grinding and manufacturing, and in particular relates to a parameterized modeling method for a ball-end milling insert blade curve. Background Art

[0002] Ball-end milling inserts are an important type of cutting tool for carbide tools. With the continuous development of the manufacturing industry, the requirements for carbide tools in precision manufacturing are becoming increasingly stringent, especially for the processing of difficult-to-machine materials such as high-temperature alloys and aerospace titanium alloys. The demand for their processing and manufacturing is also increasing [Ji Wei. Research on the integrated design and manufacturing of "form-property-use" of solid carbide ball-end milling cutters [D]. Harbin University of Science and Technology, 2015.]. The design and sharpening of ball-end milling inserts, as difficult issues in tool manufacturing, directly affect their cutting performance and tool life.

[0003] The parametric mathematical model of ball-end milling inserts is the foundation of their manufacturing. A good mathematical model helps improve the grinding accuracy and efficiency of ball-end milling inserts and helps fully utilize the excellent cutting performance of ball-end milling inserts. In the field of milling cutter grinding, Gong Zhihui et al., based on the processing characteristics of the tool, proposed to perform grinding through the cutting edge point, transforming the difficulties of milling cutter grinding manufacturing into cutting edge point solutions [Gong Zhihui, Bin Hongzan. Cutting edge point tracking CNC grinding method and its application on conical ball end mills [J]. Tool Technology, 1997(9):19-24.]. Therefore, it is very important to establish a parametric edge line model for ball-end milling inserts for their manufacturing.

[0004] At present, the research on the geometric model of ball-end milling inserts mainly focuses on the blade curve modeling. He et al. established a parametric model of the helical angle of the peripheral edge based on the geometric characteristics of the tapered end mill, and can be extended to the design of the helical edge line of other types of tapered milling cutters [He L, Wang X, Liu Z, et al, Mathematical Modeling and Parametric Design of Taper End Mills [J]. Advanced Materials Research, 2013, 2631 (797): 574-578.]. Wu Chunya et al. used the velocity analysis method to establish a helical angle edge line model of the rotary milling cutter, which provided a basis for the integrated design of tool cutting simulation parameters [Wu Chunya, Qi Biao, Wang Guangzhou, et al. Parametric modeling of end mills for titanium alloy processing [J]. Aviation Precision Manufacturing Technology, 2020, 56 (01): 1-4 + 8.]. Based on the structural characteristics of special types of milling cutters, Wang Jingping et al. established a general mathematical model for the spiral edge line of the peripheral edge of the end mill, and realized the modeling and design of special types of end mills with unequal tooth angles and helical angles [Wang Jingping, Li Rong, Cheng Xuefeng et al. Research on three-dimensional parametric design technology of special types of end mills [J]. Manufacturing Automation, 2014, 36 (12): 90-94.]. At present, in the study of the geometric model of ball-end milling inserts, Tsai et al. used the helical line analysis method to establish a continuous equal helical angle cutting edge line on the overall geometry of the cylinder and the ball head, providing a reference for the design of continuous edge lines on the surface of the ball-end rotating body [Tsai YC, Hsieh J MA Study of a Design and NC Manufacturing Model of Ball-End Cutters [J]. Journal of Materials Processing Technology, 2001, 117 (1): 183-192.]. However, its expression does not take the taper angle into account, so it cannot meet the requirements of the edge line modeling of conical ball-end milling inserts. Based on the universal geometric model of the uniform pitch rotary milling cutter, Dong Min et al. further designed a continuous uniform pitch blade curve model for ball-end milling cutters [Dong Min, Tang Yuyong, Dong Zengfu. On the uniform pitch continuous blade curve of ball-end milling cutters [J]. Journal of Harbin Institute of Technology, 2003, 35(3): 301-302]. However, this blade line model is only applicable to the manufacture of uniform pitch spiral blade line milling cutters.To ensure a smooth connection between the edge line of the ball-end cutting edge and the spiral edge line of the peripheral cutting edge, Lai et al. established a model for the spiral angle edge line on the ball-end cutting edge at a certain angle to the rotation axis [Lai HY, Chen W F. Precision Design and Numerical Control Machining of Tapered Ball-End Milling Cutters [J]. Proceedings of the Institution of Mechanical Engineers, Part B: Engineering Manufacture, 2002, 216(2): 183-197.]. However, this method is only applicable to solving equations under specific conditions and has many constraints, making it difficult to meet the requirements for edge curve modeling under different geometric parameters. Based on the mathematical model of the overall blade curve of a tapered ball-end end mill, Qiao Xiaofeng et al. established the constraint conditions for the smooth transition at the connection between the conical surface constant helix angle edge line and the spherical edge line [Qiao Xiaofeng, Pang Siqin, Wang Xibin, Liu Zhibing, He Lixun. Smooth transition of the blade curve of a tapered ball-end end mill [J]. Journal of Lanzhou University of Technology, 2013, 39(5): 37-41. Cheng et al. Considering the influence of tooth offset on the machining performance of ball-end milling cutters, they designed a mathematical model of the "S"-shaped edge line of a ball-end milling cutter considering tooth offset, realized the machining of the center-staggered tooth edge of the ball-end milling cutter, and provided a reference for the subsequent design of tools with tooth offset [Cheng X, Ding G, Li R, et al. A new design and grinding algorithm for ball-end milling cutter with tooth offset center [J]. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2014, 228(7): 687-697.]. Zeng Linlin et al. designed a new double-edged indexable ball-end insert and established a corresponding mathematical model of the cutting edge curve by combining the characteristics of the S-shaped cutting edge and the arc-shaped cutting edge [Zeng Linlin, Zhou Liping, Zhang Jingzhi. Design of a new double-edged indexable ball-end end mill and its processing simulation research [J]. Machine Tools and Hydraulics, 2016, 44(17): 71-75.]. Pang Siqin et al. applied the "single ball method" and "double ball method" to solve the problem of solving the complex cutting edge curve model of the ball-end milling cutter where the spherical surface intersects with two spatial planes at a certain angle [Pang Siqin, Qiao Xiaofeng, Wang Xibin, Peng Song. A mathematical modeling method for the "S"-shaped cutting edge curve of the ball-end milling cutter [J]. Journal of Hunan University (Natural Science Edition), 2014, 41(5): 44-49].However, it only targets the intersection of a sphere and multiple planes, and it is difficult to meet the requirements of ball-end milling insert edge line modeling under different geometric parameters. Summary of the Invention

[0005] In view of the above problems, the present invention provides a method for parameterized modeling of the blade curve of a ball-end milling insert.

[0006] A method for parameterizing a ball-end milling insert blade curve according to the present invention comprises the following steps:

[0007] Step 1: Define the blade curve geometry parameters.

[0008] The tool height is defined as the tool rotation blank height; the axial projection length of the ball end milling insert's overall edge line is defined as the tool overall length L t ; Among them, the axial projection length of the edge line of the peripheral edge is the peripheral edge length L W .

[0009] The tool turning radius at the starting point of the blade curve is defined as the tool starting turning radius R w .

[0010] The projected length of the end of the peripheral edge line on the blade axis is defined as the end edge radius R d .

[0011] The angle between the tangent vector of the helix of the peripheral edge and the generatrix of the tool's rotating body is defined as the helix angle β.

[0012] The angle between the outer contour of the tool rotation body and the tool rotation center axis is defined as the tool taper angle k.

[0013] The projection distance from the center point of the end edge to the tooth center edge line perpendicular to the axis direction is defined as the tooth eccentricity h.

[0014] The distance from the center point of the end tooth along the tangent vector direction of the end point of the non-orthogonal "S"-shaped edge line through the center point of the end face is defined as the tooth center distance l h .

[0015] The latitude angle corresponding to the non-orthogonal "S" shaped edge line on the sphere is defined as the latitude angle θ at the end of the non-orthogonal "S" shaped edge line. d .

[0016] Step 2: Modeling of the helical angle edge line of the peripheral edge.

[0017] Step 21: Define the workpiece coordinate system.

[0018] Define workpiece coordinate system O W -X W Y W Z W is WCS, with the center of the cross section corresponding to the starting point of the peripheral edge as the origin O W, coordinate axis Z W is the tool axis, coordinate axis Y W From the origin O W and the starting point of the peripheral blade, the coordinate axis Y W Determined by the right-hand rule.

[0019] Step 22: Create a spiral edge line model of the peripheral edge.

[0020] Define point P as any axial displacement z p Corresponding to the grinding points on the blade edge, the helical edge line of the peripheral edge is established as follows;

[0021]

[0022] Where, Indicates the rotation angle at the blade point P.

[0023] For cylindrical body of revolution blank, k=0:

[0024]

[0025] Where, Indicates the initial rotation angle.

[0026] For a conical body of revolution blank, k≠0:

[0027]

[0028] In this model, the axial displacement z p The maximum value of the circumferential blade length L W , whose expression is:

[0029] L W =L t -(1-sink)·R d (4)

[0030] Step 3: Modeling the ball end blade curve.

[0031] Step 31: Define the end blade coordinate system.

[0032] Establish coordinate system O d -X d Y d Z d , its Z d Axis and workpiece coordinate system Z W Axis coincidence, X d O d Z d The plane is located on the great circle of the spherical body of revolution and the origin is O d Determined by the center of the great circle, the coordinate axis X d The end cutting edges intersect.

[0033] Step 32: Create a ball head end blade curve model.

[0034] The ball head end blade edge line is modeled and described in the end blade coordinate system, and is described in sections based on the curvature variation characteristics of the blade points at different positions:

[0035] (1) Non-orthogonal “S” edge line modeling

[0036] First, define the tooth partial cylinder with radius h: the cylinder is in the workpiece coordinate system Z W The axis is the rotation axis, and the length of the rotation radius is the tooth eccentricity h; define P 1d P 2d The curve is a non-orthogonal "S"-shaped edge line. The coordinates of any point P0 on the curve in the end edge coordinate system are expressed as:

[0037]

[0038] In the formula, the independent variable θ represents the latitude angle, θ d Indicates the latitude angle of the end of the non-orthogonal "S" shaped edge line, and its value is affected by the tooth center amount l h The influence of the tooth eccentricity h is calculated as follows:

[0039] If the tooth over-center value is greater than 0, the expression is:

[0040]

[0041] If the tooth over-center value is less than or equal to 0, the expression is:

[0042]

[0043] in, Indicates the rotation angle at the position of P0. To ensure a smooth connection between the peripheral edge line and the end edge line, it is calculated as follows:

[0044]

[0045] (2) Modeling of the tooth passing through the center edge line.

[0046] First, define P 2d P' 3d The curve is a non-orthogonal "S" shaped edge line end point P 2d The introduced tangent segment, the introduction of the independent variable t represents the point on the tangent segment and the end point P of the non-orthogonal "S" shaped edge line 2d Distance; Get the end point P of the non-orthogonal "S" shaped edge line 2d (x1, y1, z1), θ = θ d Substituting into formula (5) we get:

[0047]

[0048] Derivative of equation (9) yields point P 2d The unit tangent vector F at P2_d for:

[0049]

[0050] Thus, from the end point P 2d The expression for any point on the derived tangent segment is:

[0051]

[0052] The tooth passes through the center edge line and is smoothly connected to the "S" shaped blade curve and passes through the end point P 2d , then the tooth passes through the center edge line from the end point P of the non-orthogonal "S" edge line 2d The derived tangent segment and the tool axis form a tangent plane that intersects with the rotating sphere. The expression of any point P0 on the curve in the end blade coordinate system is:

[0053]

[0054] Among them, t d The variable value t on the tangent segment corresponds to the latitude variable on the tooth passing through the center edge line, and its expression is:

[0055]

[0056] Step 33: Edge line constraint parameter analysis.

[0057] In the milling process of the ball end mill, the parts involved in the milling are the peripheral edge and the ball end edge; among them, the peripheral edge helix angle β i It is equal at each edge line point of the circumferential edge, and according to the definition of the generalized helix angle, the end edge helix angle is the angle between the generatrix tangent vector and the edge line tangent vector at the end edge line point.

[0058] Define the generatrix tangent vector at the end edge line point P0 as F e_m , whose expression is:

[0059]

[0060] Derivative of equation (5) yields the edge line tangent vector F at the edge line point P0: s for:

[0061]

[0062] Combining equations (14) and (15), we can get the edge helix angle β at any point on the end edge of the ball head: s Written as:

[0063]

[0064] In tool design, the ball head radius R d The milling cutter taper angle k is used as a known design value to determine the tool blank profile, and the peripheral edge helix angle β is designed i and tooth eccentricity h i To determine the "S" shape distribution of the end edge.

[0065] By changing the helix angle β of the peripheral edge line in equal increments i , we get the end edge line family (L l1 ,L l2 ,L l3 ); As the helix angle changes, the inclination angle of the cutting edge relative to the tool axis changes, which affects the shear force applied when the tool cuts into the workpiece.

[0066] From the relationship between the blade curve and the ball end blade rotation surface, it can be seen that the blade line is along the Y d The maximum tooth height in the direction must be within the tool blank contour, i.e., β i The value range must meet the following conditions:

[0067]

[0068] By changing l in equal steps i With h i , and obtain multiple tooth edge line families (L h1 ,L h2 ,L h3 ); solve the range of the edge offset of the edge line. According to formula (8), if the edge offset satisfies the following conditions: Then this formula has no solution, and thus the blade curve equation cannot be obtained; combined with the above requirements, solve the blade deflection h i The value range is as follows:

[0069] 0≤h i <R d cosk,k∈(-90,90) (18)

[0070] Step 34: Transform to the workpiece coordinate system.

[0071] In the workpiece coordinate system, the expression of the cutting edge point P' of the ball end cutting edge curve is:

[0072] P′=M d-W P+T d-W (19)

[0073] Where, P and P' are the coordinate vectors of points P and P'; M d-w Represents the rotation matrix from the ball head coordinate system to the workpiece coordinate system, T d-wRepresents the translation matrix from the ball head coordinate system to the workpiece coordinate system. The specific expression is as follows:

[0074]

[0075] Step 4: The ball-end milling insert edge curve is calculated based on the constructed ball-end milling insert edge curve parameterized model.

[0076] The beneficial technical effects of the present invention are:

[0077] By analyzing the existing modeling methods and restrictions of ball-end milling cutters, the present invention first defines the geometric feature parameters that meet the construction of the parametric geometric model of the ball-end milling insert, thereby ensuring the integrity of the parametric model; based on each characteristic coordinate system, a parametric model of each part of the edge line is established, and the expression of the coordinates of the blade point on the curve is derived, thereby ensuring the smoothness of the connection between each segment of the curve; and the calculation formula of the blade point is given in the form of a formula, thereby avoiding the solution of the equation. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] Figure 1 Schematic diagram of the workpiece coordinate system and the geometric characteristic parameters of the blade curve.

[0079] Figure 2 for Figure 1 Top view of .

[0080] Figure 3 Schematic diagram of the end blade coordinate system.

[0081] Figure 4 Schematic diagram for modeling the perimeter edge line.

[0082] Figure 5 Schematic diagram of the edge line of the ball head end blade.

[0083] Figure 6 Schematic diagram of latitude angles corresponding to different tooth offsets.

[0084] Figure 7 Schematic diagram of the influence of helix angle on the edge line distribution of the ball head end blade.

[0085] Figure 8 Schematic diagram of the effect of tooth offset on the edge line distribution of the ball head end cutting edge.

[0086] Figure 9 Schematic diagram of the cutting edge curve modeling of a two-tooth ball-end milling insert. (a) Stereoscopic view, (b) front view.

[0087] Figure 10 Schematic diagram of blade curve modeling corresponding to different helix angles; (a) β = 8°, (b) β = 12°, (c) β = 16°.

[0088] Figure 11Schematic diagram of blade curve modeling corresponding to different taper angles; (a): k = 0°; (b): k = -10°.

[0089] Figure 12 Schematic diagram of blade curve modeling corresponding to different tooth offsets; (a) h = 0 mm, (b) h = 0.1 mm, (c) h = 0.4 mm. DETAILED DESCRIPTION

[0090] The present invention will be further described in detail below with reference to the accompanying drawings and specific implementation methods.

[0091] A method for parameterizing a ball-end milling insert blade curve according to the present invention comprises the following steps:

[0092] Step 1: Define the blade curve geometry parameters.

[0093] like Figure 1 、 Figure 2 As shown in the figure, the cutting edge curve of the ball-end milling insert includes three parts: the peripheral cutting edge line with equal helical angles, the ball-end cutting edge curve and the non-orthogonal "S"-shaped cutting edge line.

[0094] In order to fully describe the structural characteristics of the ball-end milling insert, the geometric parameters of the blade curve model are first defined:

[0095] Blade length L: The tool height is defined as the tool blank height of the tool rotation body; the axial projection length of the ball end milling insert's overall edge line is defined as the overall tool length L t ; Among them, the axial projection length of the edge line of the peripheral edge is the peripheral edge length L W .

[0096] Tool turning radius R: The tool turning radius at the starting point of the blade curve is defined as the tool starting turning radius R w .

[0097] The projected length of the end of the peripheral edge line on the blade axis is defined as the end edge radius R d .

[0098] Helix angle β: The angle between the tangent vector of the circumferential blade helix and the generatrix of the tool's rotating body is defined as the helix angle β.

[0099] Taper angle k: The angle between the outer contour of the tool rotation body and the tool rotation center axis is defined as the tool taper angle k.

[0100] Tooth offset h: The projection distance from the center point of the end edge to the tooth center edge line perpendicular to the axis direction is defined as the tooth offset h.

[0101] Tooth over-center distance l h :The distance from the center point of the end tooth along the direction of the tangent vector of the end point of the non-orthogonal "S" shaped edge line through the center point of the end face is defined as the tooth center distance lh .

[0102] Tool width D: The vertical distance between the upper and lower surfaces of the tool is defined as the tool width D.

[0103] The latitude angle corresponding to the non-orthogonal "S" shaped edge line on the sphere is defined as the latitude angle θ at the end of the non-orthogonal "S" shaped edge line. d .

[0104] Step 2: Modeling of the helical angle edge line of the peripheral edge.

[0105] Step 21: Define the workpiece coordinate system.

[0106] like Figure 1 As shown, define the workpiece coordinate system O W -X W Y W Z W is WCS, with the center of the cross section corresponding to the starting point of the peripheral edge as the origin O W , coordinate axis Z W is the tool axis, coordinate axis Y W From the origin O W and the starting point of the peripheral blade, the coordinate axis Y W Determined by the right-hand rule.

[0107] Step 22: Create a spiral edge line model of the peripheral edge.

[0108] like Figure 4 As shown, the definition point P is any axial displacement z p Corresponding to the grinding points on the blade edge, the helical edge line of the peripheral edge is established as follows;

[0109]

[0110] Where, Indicates the rotation angle at the blade point P.

[0111] For cylindrical body of revolution blank, k=0:

[0112]

[0113] Where, Indicates the initial rotation angle.

[0114] For a conical body of revolution blank, k≠0:

[0115]

[0116] In this model, the axial displacement z p The maximum value of the circumferential blade length L W , whose expression is:

[0117] L W =L t -(1-sink)·R d (4)

[0118] Step 3: Modeling the ball end blade curve.

[0119] Step 31: Define the end blade coordinate system.

[0120] like Figure 3 As shown, establish the coordinate system O d -X d Y d Z d , its Z d Axis and workpiece coordinate system Z W Axis coincidence, X d O d Z d The plane is located on the great circle of the spherical body of revolution and the origin is O d Determined by the center of the great circle, the coordinate axis X d The end cutting edges intersect.

[0121] Step 32: Create a ball head end blade curve model.

[0122] Cheng et al. used the method of finding the intersection line between the non-orthogonal spiral rotation surface and the spherical surface to obtain the ball-end cutting edge curve in their research [Xuefeng Cheng, Guofu Ding, Rong Li, et al. A New Design and Grinding Algorithm for Ball-end Milling Cutter with Tooth Offset Center. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2014, 228 (7): 687-697.]. By introducing the latitude angle θ as an independent variable, the relationship between it and the rotation angle is established. , and thus obtain the expression of the ball head edge line.

[0123] The present invention adopts the above method to obtain the end tooth arc edge line by using the intersection line of the non-orthogonal spiral rotation surface and the ball head rotation surface. However, due to the existence of tooth offset, in order to achieve the connection with the other side of the blade, the non-orthogonal "S" edge line passes through the highest point P 2d After (when the independent variable reaches its maximum value), it extends in the direction of the tangent vector at the end point to reach P 3d Point, such as Figure 5 (a). If the curve P 2d P 3dThe segment is still obtained by the above method, then on the curve P 2d P 3d The actual tooth eccentricity h' obtained in the segment will be smaller than the theoretical value.

[0124] In order to ensure the accuracy of the tooth eccentricity h, the present invention improves the modeling method of the end tooth arc edge curve, describes it in a segmented form, and ensures its smooth connection. 1d P 2d The segment is obtained by finding the intersection line of the ball head rotation surface and the non-orthogonal spiral surface; the curve P1P2 is established on the plane passing through P1 and tangent to the eccentric cylinder.

[0125] The ball head end blade edge line is modeled and described in the end blade coordinate system, and is described in sections based on the curvature variation characteristics of the blade points at different positions:

[0126] (1) Non-orthogonal “S” edge line modeling

[0127] like Figure 5 As shown in (a), first define the tooth partial cylinder with radius h: the cylinder is in the workpiece coordinate system Z W The axis is the rotation axis, and the length of the rotation radius is the tooth eccentricity h; define P 1d P 2d The curve is a non-orthogonal "S"-shaped edge line. The coordinates of any point P0 on the curve in the end edge coordinate system are expressed as:

[0128]

[0129] In the formula, the independent variable θ represents the latitude angle, θ d Indicates the latitude angle of the end of the non-orthogonal "S" shaped edge line, and its value is affected by the tooth center amount l h The influence of the tooth eccentricity h is calculated as follows:

[0130] like Figure 6 As shown in (a), if the tooth over-center amount is greater than 0, its expression is:

[0131]

[0132] like Figure 6 As shown in (b), if the tooth over-center amount is less than or equal to 0, its expression is:

[0133]

[0134] in, Indicates the rotation angle at the position of P0. To ensure a smooth connection between the peripheral edge line and the end edge line, it is calculated as follows:

[0135]

[0136] (2) Modeling of the tooth passing through the center edge line.

[0137] like Figure 5 As shown in (b), first define P 2d P' 3d The curve is a non-orthogonal "S" shaped edge line end point P 2d The introduced tangent segment, the introduction of the independent variable t represents the point on the tangent segment and the end point P of the non-orthogonal "S" shaped edge line 2d Distance; Get the end point P of the non-orthogonal "S" shaped edge line 2d (x1, y1, z1), θ = θ d Substituting into formula (5) we get:

[0138]

[0139] Derivative of equation (9) yields point P 2d The unit tangent vector F at P2_d for:

[0140]

[0141] Thus, from the end point P 2d The expression for any point on the derived tangent segment is:

[0142]

[0143] The tooth passes through the center edge line and is smoothly connected to the "S" shaped blade curve and passes through the end point P 2d , then the tooth passes through the center edge line from the end point P of the non-orthogonal "S" edge line 2d The derived tangent segment and the tool axis form a tangent plane that intersects with the rotating sphere. The expression of any point P0 on the curve in the end blade coordinate system is:

[0144]

[0145] Among them, t d The variable value t on the tangent segment corresponds to the latitude variable on the tooth passing through the center edge line, and its expression is:

[0146]

[0147] Step 33: Edge line constraint parameter analysis.

[0148] In the milling process of the ball end mill, the parts involved in the milling are the peripheral edge and the ball end edge; among them, the peripheral edge helix angle β i It is equal at each edge line point of the circumferential edge, and according to the definition of the generalized helix angle, the end edge helix angle is the angle between the generatrix tangent vector and the edge line tangent vector at the end edge line point.

[0149] Define the generatrix tangent vector at the end edge line point P0 as F e_m , whose expression is:

[0150]

[0151] Derivative of equation (5) yields the edge line tangent vector F at the edge line point P0: s for:

[0152]

[0153] Combining equations (14) and (15), we can get the edge helix angle β at any point on the end edge of the ball head: s Written as:

[0154]

[0155] From formula (16), we can know that the helix angle β of the peripheral edge line is i and tooth eccentricity h i is the rotation angle of P0 The main influencing factors. Usually in tool design, the ball head radius R d The milling cutter taper angle k is used as a known design value to determine the tool blank profile, which can be determined by designing the peripheral edge helix angle β i and tooth eccentricity h i To determine the "S" shape distribution of the end edge.

[0156] By changing the helix angle β of the peripheral edge line in equal increments i , we get the end edge line family (L l1 ,L l2 ,L l3 ); As the helix angle changes, the inclination angle of the cutting edge relative to the tool axis changes, which affects the shear force applied when the tool cuts into the workpiece, such as Figure 7 shown.

[0157] From the relationship between the blade curve and the ball end blade rotation surface, it can be seen that the blade line is along the Y d The maximum tooth height in the direction must be within the tool blank contour, i.e., β i The value range must meet the following conditions:

[0158]

[0159] By changing l in equal steps i With h i , and obtain multiple tooth edge line families (L h1 ,L h2 ,L h3 ),like Figure 8As shown. Solve the range of the edge offset of the edge line. According to formula (8), if the edge offset satisfies the following conditions: Then this formula has no solution, and thus the blade curve equation cannot be obtained; combined with the above requirements, solve the blade deflection h i The value range is as follows:

[0160] 0≤h i <R d cosk,k∈(-90,90) (18)

[0161] Step 34: Transform to the workpiece coordinate system.

[0162] In the workpiece coordinate system, the expression of the cutting edge point P' of the ball end cutting edge curve is:

[0163] P′=M d-W P+T d-W (19)

[0164] Where, P and P' are the coordinate vectors of points P and P'; M d-w Represents the rotation matrix from the ball head coordinate system to the workpiece coordinate system, T d-w Represents the translation matrix from the ball head coordinate system to the workpiece coordinate system. The specific expression is as follows:

[0165]

[0166] Step 4: The ball-end milling insert edge curve is calculated based on the constructed ball-end milling insert edge curve parameterized model.

[0167] Experimental verification:

[0168] 1. Verification of blade curve calculation results.

[0169] The geometric parameters of the blade curve are set as shown in Table 1.

[0170] Table 1 Geometric parameters of the cutting edge curve of the two-tooth ball-end milling insert

[0171]

[0172] The calculated data of some cutting edge points are shown in Table 2.

[0173] Table 2 Partial blade point data

[0174]

[0175]

[0176] Draw the calculated blade point in CAD and get the following Figure 9As shown in the figure, a smooth transition is ensured between the peripheral edge line and the end edge line, realizing the scalable design of the ball end milling insert edge line model.

[0177] 2. Verification of blade curve with different parameters.

[0178] Change the input value of the helix angle β, and keep the other parameters the same as those in Table 1. The points drawn in CAD are as follows Figure 10 shown.

[0179] Change the input value of the taper angle k, and keep the other parameters the same as those in Table 1. The points drawn in CAD are as follows Figure 11 shown.

[0180] Change the input value of the tooth eccentricity h, and keep the other parameters the same as those in Table 1. The points drawn in CAD are as follows: Figure 12 shown.

[0181] According to the results of blade curve modeling under different parameters, it is found that the blade curve of the ball-end milling insert established in this paper has good adaptability and can be simultaneously applied to blade curves with different parameters of helix angle, taper angle and tooth eccentricity. The obtained blade curve achieves an overall smooth connection, which shows the reliability and applicability of the blade line model established in this paper.

Claims

1. A method for parameterizing the blade curve of a ball-end milling insert, characterized in that: The following steps are involved: Step 1: Define the blade curve geometric parameters; The tool height is defined as the tool rotation blank height; the axial projection length of the ball end milling insert's overall edge line is defined as the tool overall length L t ; Among them, the axial projection length of the edge line of the peripheral edge is the peripheral edge length L W ; The tool turning radius at the starting point of the blade curve is defined as the tool starting turning radius R w ; The projected length of the end of the peripheral edge line on the blade axis is defined as the end edge radius R d ; The angle between the tangent vector of the helix of the peripheral edge and the generatrix of the tool's rotating body is defined as the helix angle β; The included angle between the outer contour of the tool rotation body and the center axis of the tool rotation is defined as the tool taper angle k; The projection distance from the center point of the end edge to the tooth center edge line perpendicular to the axis direction is defined as the tooth eccentricity h; The distance from the center point of the end tooth along the tangent vector direction of the end point of the non-orthogonal "S"-shaped edge line through the center point of the end face is defined as the tooth center distance l h ; The latitude angle corresponding to the non-orthogonal "S" shaped edge line on the sphere is defined as the latitude angle θ at the end of the non-orthogonal "S" shaped edge line d ; Step 2: Modeling of the helical angle edge line of the peripheral edge; Step 21: Define the workpiece coordinate system; Define workpiece coordinate system O W -X W Y W Z W is WCS, with the center of the cross section corresponding to the starting point of the peripheral edge as the origin O W , coordinate axis Z W is the tool axis, coordinate axis Y W From the origin O W and the starting point of the peripheral blade, the coordinate axis Y W Determined by the right-hand rule; Step 22: Establish a spiral edge line model of the peripheral edge; Define point P as any axial displacement z p Corresponding to the grinding points on the blade edge, the helical edge line of the peripheral edge is established as follows; Where, represents the rotation angle at the blade point P; For cylindrical body of revolution blank, k=0: Where, represents the initial rotation angle; For a conical body of revolution blank, k≠0: In this model, the axial displacement z p The maximum value of the circumferential blade length L W , whose expression is: L W =L t -(1-sink)·R d (4) Step 3: Modeling the ball end edge curve; Step 31: Define the end blade coordinate system: Establish coordinate system O d -X d Y d Z d , its Z d Axis and workpiece coordinate system Z W Axis coincidence, X d O d Z d The plane is located on the great circle of the spherical body of revolution and the origin is O d Determined by the center of the great circle, the coordinate axis X d The end edge lines intersect; Step 32: Create the ball end blade curve model: The ball head end blade edge line is modeled and described in the end blade coordinate system, and is described in sections based on the curvature variation characteristics of the blade points at different positions: (1) Non-orthogonal "S" edge line modeling First, define the tooth partial cylinder with radius h: the cylinder is in the workpiece coordinate system Z W The axis is the rotation axis, and the length of the rotation radius is the tooth eccentricity h; define P 1d P 2d The curve is a non-orthogonal "S"-shaped edge line. The coordinates of any point P0 on the curve in the end edge coordinate system are expressed as: In the formula, the independent variable θ represents the latitude angle, θ d Indicates the latitude angle of the end of the non-orthogonal "S" shaped blade line, and its value is affected by the tooth center amount l h The influence of the tooth eccentricity h is calculated as follows: If the tooth over-center value is greater than 0, the expression is: If the tooth over-center value is less than or equal to 0, the expression is: in, Indicates the rotation angle at the position of P0. To ensure a smooth connection between the peripheral edge line and the end edge line, it is calculated as follows: (2) Modeling of the tooth passing through the center edge line; First, define P 2d P' 3d The curve is a non-orthogonal "S" shaped edge line end point P 2d The tangent segment is introduced, and the independent variable t is introduced to represent the point on the tangent segment and the end point P of the non-orthogonal "S" shaped blade line. 2d Get the end point P of the non-orthogonal "S" shaped edge line 2d (x1, y1, z1), θ = θ d Substituting into formula (5) we get: Derivative of equation (9) yields point P 2d The unit tangent vector F at P2d for: Thus, from the end point P 2d The expression for any point on the derived tangent segment is: The tooth passes through the center edge line and is smoothly connected to the "S" shaped blade curve and passes through the end point P 2d , then the tooth edge line through the center is from the end point P of the non-orthogonal "S" edge line 2d The derived tangent segment and the tool axis form a tangent plane that intersects with the rotating sphere. The expression of any point P0 on the curve in the end blade coordinate system is: Among them, t d The variable value t on the tangent segment corresponds to the latitude variable on the tooth passing through the center edge line, and its expression is: Step 33: Edge line constraint parameter analysis; In the milling process of the ball end mill, the parts involved in the milling are the peripheral edge and the ball end edge; among them, the peripheral edge helix angle β i It is equal at each edge line point of the circumferential edge, and according to the definition of the generalized helix angle, the end edge helix angle is the angle between the generatrix tangent vector and the edge line tangent vector at the end edge line point; Define the generatrix tangent vector at the end edge line point P0 as F em , whose expression is: Derivative of equation (5) yields the edge line tangent vector F at the edge line point P0: s for: Combining equations (14) and (15), we can get the edge helix angle β at any point on the end edge of the ball head: s Written as: In tool design, the ball head radius R d The milling cutter taper angle k is used as a known design value to determine the tool blank profile, and the peripheral edge helix angle β is designed i and tooth eccentricity h i To determine the "S" shape distribution of the end edge; By changing the helix angle β of the peripheral edge line in equal increments i , we get the end edge line family (L l1 ,L l2 ,L l3 ); As the helix angle changes, the inclination angle of the cutting edge relative to the tool axis changes, affecting the shear force applied by the tool when it cuts into the workpiece; From the relationship between the blade curve and the ball end blade rotation surface, it can be seen that the blade line is along the Y d The maximum tooth height in the direction must be within the tool blank contour, i.e., β i The value range must meet the following conditions: By changing l in equal steps i With h i , and obtain multiple tooth edge line families (L h1 ,L h2 ,L h3 ); solve the range of the edge offset of the edge line. According to formula (8), if the edge offset satisfies the following conditions: Then this formula has no solution, and thus the blade curve equation cannot be obtained; combined with the above requirements, solve the blade deflection h i The value range is as follows: 0≤h i <R d cosk,k∈(-90,90) (18) Step 34: Transform to the workpiece coordinate system; In the workpiece coordinate system, the expression of the cutting edge point P' of the ball end cutting edge curve is: P′=M d-W P+T d-W (19) Where, P and P' are the coordinate vectors of points P and P'; M d-w Represents the rotation matrix from the ball head coordinate system to the workpiece coordinate system, T d-w Represents the translation matrix from the ball head coordinate system to the workpiece coordinate system. The specific expression is as follows:

2. The method for parameterizing the blade curve of a ball-end milling insert according to claim 1, wherein: The method further includes step 4: calculating and obtaining the ball-end milling insert edge curve based on the constructed ball-end milling insert edge curve parameterized model.

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