Cutter inner curved surface compensation method based on principal curvature induction

Abstract

The invention relates to a main curvature induction-based tool inner curved surface compensation method, which comprises the following steps of: finishing NURBS (Non Uniform Rational B-Spline) curved surface separation of an inner blade curved surface of a prototype tool by utilizing Solid Works software, and carrying out geometric representation on a continuous curved surface generatrix; the curved surface circle center offset is substituted, and a tool inner edge curved surface equation is corrected; performing motion equivalence on the dynamic sliding of the curved surface, and defining an inner blade curved surface meridian equation to obtain an inner blade curved surface equation; according to the normal vector deviation and the dynamic offset of the cutting edge cutting track, the main curvature and the geodesic deflection of the inner edge curved surface are compensated and induced, and the inclined stubble height is calculated; cutter prototype curved surface slippage error calculation and cutter inner edge curved surface shape error calculation; and reconstructing normal and tangential compensation of the inner blade surface. According to the method, the shape of the inner curved surface of the cutter is optimized, an inclined stubble height calculation formula is deduced, curvature synergy of the inner blade curved surface of the cutter and the material fracture curved surface is guaranteed to the maximum extent, normal vector deviation is reduced, inclined stubbles are reduced, and the inner blade cutting efficiency can be effectively improved.

Description

Technical Field

[0001] This invention relates to a method for compensating the inner surface curvature of a cutting tool based on principal curvature induction, belonging to the field of agricultural machinery technology. Specifically, it relates to the optimization technology of chopping tools in forage harvesting equipment, and more particularly to a method for compensating the inner surface curvature of a cutting tool based on principal curvature induction adapted to high-speed rolling operations. This method is applicable to technical scenarios involving correction of geometric errors in the inner cutting surface of the cutting tool, biomimetic morphology matching, and improvement of cutting performance. Background Technology

[0002] In the field of forage harvesting in agricultural machinery, high-speed rolling cutters are the core operating method to ensure the quality of material cutting. Currently, the mainstream shredding blades are mainly divided into two categories: one is the domestically produced flat straight blade, which has a simple structure and low manufacturing cost. It cuts stalks through the rotational cutting of a straight blade and is widely used in small and medium-sized forage harvesting equipment. However, it suffers from poor cutting quality, numerous oblique burrs, and short service life. The other is the imported irregular curved blade, which adopts a multi-segment arc blade configuration and a herringbone installation method. It utilizes a dynamic sliding cutting trajectory to improve shredding ability, and the cutting quality is significantly better than that of the flat straight blade. However, it suffers from high cost, difficult maintenance, and insufficient adaptability. At the technical research level, existing patents mostly use differential geometry methods to optimize the blade surface. By matching the normal vector of the outer blade surface, cutting interference is reduced, which reduces static geometric errors to a certain extent. However, for the large-scale harvesting of high-toughness stalks, existing technologies still have key technical shortcomings and cannot meet the requirements for precise cutting.

[0003] Existing research largely focuses on optimizing the outer cutting surface, failing to fully consider the core geometric coordination requirements between the inner cutting surface and the material fracture surface, and neglecting to incorporate the biomimetic features of beaver incisors to optimize key parameters such as the cutting inclination angle and the cutting angle, resulting in poor effort-saving sliding cutting effects. Current technologies only compensate for surface shape errors, failing to quantify dynamic sliding errors during the cutting process. This causes the actual cutting trajectory to deviate from the theoretical biomimetic trajectory, leading to problems such as material slippage and jamming, and debris blockage, further exacerbating uneven cutting segments and tool wear.

[0004] Based on this, by achieving coordination of geodesic deflection and principal curvature between the inner cutting edge and the material fracture surface, integrating the comprehensive correction of dynamic slip error and multi-source static error, and combining the optimization of cutting edge inclination angle and cutting edge angle, while ensuring the curvature continuity of the multi-segment arc surface of the inner cutting edge, the stubble rate can be reduced, the wear resistance of the cutting tool can be improved, and the energy consumption of operation can be reduced, thus solving the adaptation bottleneck of existing technologies in the harvesting of high toughness materials. Summary of the Invention

[0005] This invention discloses a tool internal surface compensation method based on principal curvature-induced curves, aiming to solve the problems of insufficient accuracy and incomplete error correction in existing internal cutting surface compensation technologies. The method includes: 1. Establishing a dual-coordinate system and a parameterized model of the internal cutting surface based on the biomimetic features of beaver lower incisors, defining core parameters such as axis tilt angle, cutting edge tilt angle, and cutting edge angle; 2. Separating multiple sources of error, such as centerline offset, arc radius error, and slippage error, through NURBS surface reconstruction; 3. Prioritizing the reset of reference inflection points I and H, correcting the connection point G between the tool tip A and the tool holder, and establishing a biomimetic trajectory reference; 4. Adjusting auxiliary control points to ensure the continuity of curvature in the convex and concave arc segments, achieving synergy between the geodesic deflection and principal curvature of the internal cutting surface and the material fracture surface. This invention significantly reduces the stubble rate and cutting resistance, improves tool wear resistance and cutting quality, is compatible with existing harvesting equipment, and balances technological advancement with industrial practicality.

[0006] The technical solution of this invention is: a tool internal surface compensation method based on principal curvature-induced curves, comprising the following steps:

[0007] S1. Use SolidWorks software to separate the NURBS surface of the inner cutting edge of the prototype tool and perform geometric characterization of the generatrix of the continuous surface.

[0008] S2. Substitute the offset of the center of the curved surface to correct the equation of the inner cutting edge surface of the tool.

[0009] S3. Perform motion equivalents on the dynamic slip of the curved surface, define the meridian equation of the inner edge surface, and obtain the equation of the inner edge surface.

[0010] S4. Based on the normal vector deviation and dynamic offset of the blade cutting trajectory, compensate and induce the principal curvature and geodesic deflection of the inner blade surface, and calculate the slant height.

[0011] S5. Calculation of slippage error of the tool prototype surface;

[0012] S6. Calculation of surface shape error of the inner cutting edge of the tool;

[0013] S7, Reconstruction of normal and tangential directions of the inner cutting edge.

[0014] Furthermore, in S1, the inner curved surface of the tool is a tension surface, and the inner cutting edge is an arc-shaped boundary curve. First, two coordinate systems are set, namely, the coordinate system T of the moving rotation center is set as {O}. S -x S y S z S}, and then set the local coordinate system M as {o} when the hob rotates. m -x m y m z mSince the process of cutting materials by the tool is a continuous sliding cutting process, the motion of the cutting point p of the tool tip is divided into three parts: forward sliding cutting, reverse throwing and sliding deflection. Based on the characteristics of NURBS such as non-uniformity, rationality, local controllability and continuous controllability, SolidWorks software is used to complete the NURBS surface separation of the inner cutting surface of the prototype tool and to perform geometric characterization of the continuous surface generatrix.

[0015] Based on the principle of establishing coordinate systems using the Newton-Euler equations, the origin is set at the geometric center of the tool, i.e., the centroid. The specific process for establishing the two coordinate systems described above is as follows:

[0016] Moving rotation center coordinate system T:

[0017] (1) The z-axis is defined as being parallel to and in the same direction as the rotation axis of the tool;

[0018] (2) The x-axis points in the direction of the tool's movement, i.e., the forward direction;

[0019] (3) The y-axis is determined by the right-hand rule, that is, the four fingers of the right hand turn from the X-axis to the Y-axis, and the thumb points to the Z-axis;

[0020] Since the position, speed and force direction of each point on the cutting teeth are constantly changing when the hob is rotating and cutting, the origin O of M is set on the fixed cutting plane.

[0021] Local coordinate system M during hob rotation:

[0022] (1) The y-axis is perpendicular to the rotation direction of the tool's geometric center;

[0023] (2) The x-axis points in the direction of the tool's movement, i.e., the forward direction;

[0024] (3) The z-axis is determined by the right-hand rule.

[0025] Furthermore, step S1 also includes the following sub-steps:

[0026] S11. First, perform geometric abstraction and assumptions, and abstract the inner and outer cutting surfaces of the irregular curved knife as "stretched surfaces";

[0027] S12. Establish relevant coordinate systems, clarify the global coordinate system fixed to the frame and the local coordinate system fixed to the tool, denoted by {T} and {M} respectively, and clarify that their transformation relationship involves the axis tilt angle β transformation around the y-axis.

[0028] S13, in T{o s -x s y s z sThe expression of the tool surface is obtained in the coordinate system, a three-dimensional model of the prototype tool is established, the cutting edge is simplified to a straight line, and the complex tool generatrix is ​​discretized into multiple composite curves connected by NURBS curves.

[0029] S14, in M{o m -x m y m z m In the coordinate system, the geometric expression of the tool generatrix is ​​expressed;

[0030] S15. The generatrix is ​​parameterized as Q(l) in the initial coordinate system. In the static state, the generatrix lies in the plane o. m -x m y m Within, the inner cutting edge surface of the tool is obtained along z. s The characterization formula after directional extension;

[0031] S16. Expand the formula in S15, along the latitude θ respectively. w The partial derivatives of the direction and the z-direction of the extension line are calculated to obtain the expressions for the principal vectors e1 and e2;

[0032] S17. Based on the spinor theory, the overall cutting motion of the tool is divided into sliding and throwing motion around the b-axis and sliding and deflecting motion around the a'-axis that occurs with the material. The internal offset angle is solved according to the normal vector equivalence and position equivalence. The a'-axis is the tool sliding and deflecting axis.

[0033] S18. According to the kinematic equivalence, the distance from point p to the b axis and the principal curvature of the tool surface meridian are both R, so the formula for calculating R is obtained.

[0034] S19. To further obtain the specific position q of the cutting point p corresponding to the b-axis, the tool dynamically slides and propels the rotation axis, and the sliding deflection axis constructs the corresponding torsion angle to obtain x. s The expression for an axisymmetric meridian;

[0035] Solve by combining the expressions in S1-20, S18, and S19, as well as the expression for the meridian radius function;

[0036] S1-21. The formula for calculating the position q of the cutting point p corresponding to the b-axis is substituted into the compound rotation formula for twisting around the a' axis and then being thrown around the b-axis to obtain the information about the torsional coefficient k. n1 The expression;

[0037] S1-22. Correct the cutting point position and adjust the coefficient k. n1 and higher-order term A i The equation of the inner surface meridian is obtained, and finally the formula of the inner surface equation of the tool is obtained.

[0038] Furthermore, in step S2, substituting the offset of the curved surface center and correcting the equation of the inner cutting edge surface of the tool, the following sub-steps are also included:

[0039] S21. Based on the tool geometry characterization formula established in S1, perform internal cutting surface correction calculation;

[0040] The coordinate range before blade curvature compensation optimization is set to [285, -12];

[0041] S22. Draw a schematic diagram of the overall arc surface representation of the tool based on the positions of the two coordinate systems mentioned above;

[0042] S23, obtained in T{O s -x s y s z s The expression for the tool surface in the coordinate system;

[0043]

[0044] In the formula, the original surface of the tool is Q(l,θ) w ), where l is the generatrix of the curved surface of the cutting edge. Let θ be the geometric expression for the tool generatrix. w It is a curved latitude line;

[0045] S24. Establish a three-dimensional model of the prototype tool, and simplify the cutting edge to a straight line, and simplify the tool surface generatrix to a multi-segment curve formed by the connection of multiple NURBS circular arcs;

[0046] S25. Transform to the M coordinate system to obtain the approximate expression of the tool surface, and then derive the geometric expression of the tool generatrix.

[0047]

[0048] In the formula, d s R is the baseline of the tool arc surface. c β is the radius of the inner and outer cutting edge arc surfaces of the tool, and β is the tool axis inclination angle;

[0049] S26. Considering the herringbone configuration and the actual center offset, the equation in S25 is modified to obtain the inner cutting edge surface equation of the tool.

[0050]

[0051] In the formula, Z is the extended and stretched surface; Rot(y, β) is the transformation matrix of the original surface deflected by β about the z-axis;

[0052] S27, the formulas in S26 are respectively applied along the latitude θ wThe partial derivatives of the direction and the z-direction of the extension line are calculated to obtain the expressions for the principal vectors e1 and e2, and the two principal directions of the tool surface are defined.

[0053]

[0054] The above are the formulas for calculating partial derivatives in two directions;

[0055] In the formula, Rc is the radius of the inner and outer cutting edge arc surfaces of the tool, and θ w For the arc surface latitude line, β is the tool axis inclination angle, the entire system dynamically twists γ(Z) around the zs axis, and R is the principal curvature of the tool surface meridian;

[0056] S28. Based on spinor theory, the overall cutting motion of the tool is decomposed in the T coordinate system according to the equation of the inner cutting surface, and slip equivalence is performed.

[0057] Furthermore, in step S3, the dynamic sliding of the curved surface is kinematically equivalent, the meridian equation of the inner edge surface is defined, and the equation of the inner edge surface is obtained. This step also includes the following sub-steps:

[0058] S31. Based on the decomposed slip deflection motion, define the slip deflection axis as the a' axis in the T coordinate system. s x s The plane-directed shredding and throwing rotation axis is the b-axis, and the cutting point is point p;

[0059] S32. Set the distance from point p to the b axis and the principal curvature of the tool surface meridian to R, and obtain the expression for R;

[0060]

[0061] In the formula, Here is the formula for calculating the first-order partial derivative along the z-direction of the extended surface. The formula for calculating the second-order partial derivative along the z-direction of the extended surface;

[0062] S33. The tool dynamically slides and throws the rotating axis, and the sliding deflection axis constructs the corresponding torsion angle to obtain the specific position q of point p in the b axis;

[0063] S34. Let t be the projection height of the fixed tool normal onto the outer cutting edge surface of the tool. Define the inner cutting edge surface with respect to x. s The equation of an axisymmetric meridian;

[0064]

[0065] In the formula, c1 is the paraxial curvature of the tool surface meridian, and k n Let k be the eccentricity function of the tool surface about the cutting edge. n=-e2, where e2 is the eccentricity of the tool surface about the cutting edge, controlling the degree of surface opening; P i This is the deviation coefficient between the inner and outer curved surfaces of the cutting tool;

[0066] S35, combine S32, S34 and the meridian radius function formula to solve for R;

[0067]

[0068] In the formula, the meridian radius function formula is: c = 1 / R0, where R0 is the radius of curvature of the tool surface at the cutting edge, q is the position of the cutting point p on the b-axis, and q1 and q3 are the corresponding expressions in the two matrices. , q=[q1,0,q3] T ;

[0069] S36. Based on the above expression, we obtain a formula for q. Substituting this into the compound rotation, we obtain the torsional coefficient k. n1 The formula;

[0070]

[0071] In the formula, ψ1 is the interior offset angle, and n x n z Let n1 be the normal vector in each direction, and n1 = [n x ,n y ,n z ,0] T ;

[0072] S37. Combining equations S36 and correcting the formula for the cutting point p, we obtain the equivalent formula.

[0073] S38, Final Adjustment Coefficient k n1 and higher-order term A i Define the equation of the meridian of the inner curved surface;

[0074]

[0075] In the formula, Ai is incremented term by term;

[0076] S39. By combining all the equations, we obtain the equation for the inner blade surface.

[0077] .

[0078] Furthermore, in step S4, based on the deviation of the normal vector of the blade cutting trajectory and the dynamic offset, the principal curvature and geodesic deflection of the inner blade surface are compensated and induced, and the stubble height is calculated. This also includes the following sub-steps:

[0079] S41. Based on the formula of the inner surface equation, and the two principal vector expressions obtained in S27, further subdivide the material surface and the blade surface at the cutting point p into two principal directions and induced principal directions c1 and c2, respectively.

[0080] S42. Calculate the deflection of the material fracture surface and the tool surface in the c1 direction;

[0081]

[0082]

[0083] In the formula, , , , These are the two principal curvatures of the material fracture surface and the tool surface, respectively, c1 in {p;e 1 1;e 1 2}、{p;e 2 1;e 2 In 2}, the positions are ω1 and ω2, respectively; ω1 and ω2 represent position points; where point p is the cutting point; e 1 1, e 1 2 represents the two principal directions of the material surface, e 2 1, e 2 2 represents the two principal directions of the tool surface;

[0084] S43. Along the c1 direction, according to the geodesic torsion difference formula, the tangent of the principal curvatures of the two surfaces is obtained;

[0085]

[0086] S44. List the formulas for the induced principal curvature in the directions c1 and c2 respectively;

[0087]

[0088]

[0089] S45. Using the above formula, calculate the geodesic deflection of the material surface in the main material direction and the geodesic deflection of the inner cutting edge surface in the main tool direction.

[0090] S46. Make the geodesic torsion difference in the c1 direction zero, and process the obtained formula using the double-angle formula;

[0091]

[0092]

[0093]

[0094] S47. Express the formulas for the induced principal curvature in the principal directions c1 and c2 using Euler's formula;

[0095]

[0096]

[0097] S48. Substituting the slip distance formula into the formula for the plane where the oblique cutting is located, and solving the equations simultaneously, we can derive the formula for the tangent of the oblique cutting, i.e. ;

[0098] In the formula, Let c2 be any direction ω in the coordinate system {p,c1,c2}. c The induced curvature; ω c denoted as the sliding direction angle of the plane containing the slanted section; p' is a point within the slanted section area, and p is the cutting point; the length of the arc segment pp' is replaced by the line segment pp', denoted as 2s0, where s0 is half of the approximate line segment value.

[0099] Furthermore, in step S5, the calculation of the slippage error of the tool prototype surface also includes the following sub-steps:

[0100] S51. Using the static optimization parameters of the inner cutting surface as an ideal benchmark, extract the principal curvature and geodesic deflection parameters of the optimized inner cutting surface, and then... s The parameters Rc and β are associated with the principal curvature deviation and the geodesic torsion deviation, defining a bias-free reference state for slip error; where d s Rc is the baseline of the tool's arc surface, Rc is the radius of the tool's inner and outer cutting edge arc surface contours, and β is the tool axis tilt angle.

[0101] S52. Establish a new tool coordinate system {S1}, then rotate this coordinate system by dβ along the Y-axis and translate it along the Z-axis to obtain coordinate system {S2}.

[0102] The tool coordinate system is S, which is the same as the coordinate system of the tool's rotation center. {S1} is the cutting coordinate system corresponding to the cutting point p, i.e., corresponding to {O}. S1 ,X S1 ,Y S1 Z S1 In the new coordinate system formed by {S1} and {S2}, the corresponding point of point C is C1; S2 ,X S1 ,Y S1 Z S2 The coordinate system corresponding to {S1} is transformed from {S1};

[0103] S53. Based on the ideal state, re-establish the ideal position of the cutting point p in the new system, obtain the projection points of the lowest point C in the original curvature circle diagram in the two newly established coordinate systems, and derive the vector relationship between the origin O and the two projection points.

[0104]

[0105] {S2} is {S1} first rotated by dβ around the Y axis, then translated by R along the Z axis. c1 (cosβ0-cosβ) is obtained, R c1 The actual radius of the arc surface profile at the cutting point p;

[0106] S54. Based on the geometric relationship in the figure, derive the formula for the actual cutting point p's arc diameter based on the ideal cutting point p's corresponding tool arc diameter.

[0107]

[0108] In the formula, d s1 The cutting surface corresponding to the cutting point p under ideal conditions; The axial tilt angle at the cutting point;

[0109] S55. Obtain the formula for the slip error of the tool relative to the coordinate system {S1} in coordinate system {S2};

[0110]

[0111] S56. Perform partial differential linearization derivation on the formula in S55 to obtain the formulas for the actual coordinate offsets in the X, Y, and Z directions.

[0112]

[0113] In the process of calculating dp at any cutting point p, θ is assumed to remain unchanged. In the formula, θ1, θ2, and θ3 are the parametric coordinate directions on the corresponding surface; ψ1 is the inner offset angle; and ψ2 is the outer offset angle.

[0114] Furthermore, in step S6, the calculation of the surface shape error of the inner cutting edge of the tool also includes the following sub-steps:

[0115] S61. After machining the original tool model to a certain scale, perform a three-axis scan to separate the various geometric errors of the tool surface shape in reverse and perform geometric compensation to obtain the three-dimensional laser scanning separation analysis route and result diagram of the tool surface.

[0116] S62. Linearize the differential expression in S55.

[0117] S63. Compare the surface shape obtained by the 3D laser scanning system, select any cutting contact point in the vicinity of the cutting edge on the inner and outer cutting surfaces, and project it onto the extracted curved surface; calculate the actual normal error d. n The surface separation line type characterization diagram is obtained;

[0118] S64. For the tool arc surface error corresponding to the region near point p on the inner arc surface of the tool, construct the error evaluation point after surface discretization. Then, take 4 points evenly distributed in the neighborhood near the cutting point of each cutting parallel line to achieve full rank of the coefficient matrix of the surface error equation system.

[0119] S65. Select random area points accumulated on three adjacent inner cutting edges and use them as the overall error assessment. For any assessment point, express its actual coordinate error.

[0120]

[0121] In the formula, This indicates a small evaluation range, with micro-level values ​​taken for coordinate errors; x0、 y0 is the offset of the axis coordinate in the ideal geometric parameters of the inner arc surface of the tool. r is the radius error of the arc surface. The slant direction angle is any evaluation point;

[0122] S66. Take N evaluation points on three adjacent parallels of latitude and the generatrix, and express the joint equation formula for all points;

[0123] S67. Simplify the joint equation formula in S66 into matrix form and expand the matrix as a whole.

[0124] S68. Based on constrained least squares method, p o , o To find p from nearby evaluation points o Solve for the values ​​of each microvariable dd s By taking different values ​​of ψ1 and ψ2, dR at local areas on the tool generatrix or inner and outer arc surfaces can be obtained. c dd s , da and dβ, thereby correcting the tool geometry and configuration parameters;

[0125] p o , o p is the center point of the plane containing the simplified diagram of the laser scanning analysis results of the tool's curved surface; o To evaluate the center point O of the inner arc surface of the tool, the curve at the center point O is linearized; ψ1 is the inner offset angle; ψ2 is the outer offset angle; a' is the slip deflection axis; d a’This is the axial displacement error parameter caused by the slip deflection, used to quantify the linear displacement uncertainty of the tool along the slip deflection axis a'.

[0126] Furthermore, in S7, the normal and tangential directions of the inner cutting surface are compensated and reconstructed;

[0127] S71. Based on the surface error parameters and the results calculated from the equation system AX=B in S67, extract the three static error parameters: centerline offset, arc radius error, and assembly deviation.

[0128] S72. By deriving the actual normal error formula in reverse, the ideal radius after compensation is obtained, and then the compensation amount is derived to ensure the coordination of the principal curvature.

[0129] S73. Clarify that the range of the biomimetic blade inclination angle is 2°~10°, and obtain the blade inclination angle compensation amount by combining the geodesic deflection formula;

[0130] S74. According to the normal error formula, adjust the control vertices of the NURBS surface to make the compensated centerline return;

[0131] S75. Use a grinding machine to dress the inner cutting edge arc surface, and calculate the normal error d between the corrected measured coordinates and the ideal coordinates. n , ensure d n ≤0.01mm, completing the correction of the arc surface radius error;

[0132] S76. Take a smaller cutting edge angle of 2°~10°, and perform constrained geometric calculations on the region near the cutting tip using the least squares method.

[0133] S77. Further compensate and adjust the principal curvature of the cutting edge to obtain the coordinate range value after normal compensation optimization;

[0134] S78. Based on the original biomimetic irregular curved tool structure feature point AI, determine the tool ψ1 and ψ2 values, and modify and optimize the tool geometric parameters and configuration parameters by combining the biomimetic feature point A′-I′.

[0135] The beneficial effects of this invention are as follows: Based on the discrete geometric characteristics of the tool surface, a NURBS surface equation for the inner cutting surface is established, revealing the error coupling mechanism of dynamic contact at the tool-material interface and quantifying the influence of parameter errors on the chamfer angle of the cut surface. By compensating and optimizing the principal curvature and normal error of the inner cutting surface, the principal curvature of the inner cutting surface and the principal curvature of the material fracture surface are precisely coordinated, and the normal error is controlled within 0.002 mm, significantly reducing the chamfer generated during material cutting and greatly improving the flatness of the cut surface and the cutting quality. Finally, through compensation and reconstruction of the tool's arc surface structure, the cutting edge profile of the tool's outer edge is optimized. This invention is based on existing research and theory, requires no additional specialized equipment, and can greatly reduce the tool processing cost and manufacturing difficulty.

[0136] This invention provides a method for compensating the inner surface of a cutting tool based on principal curvature-induced cutting, which improves the operating efficiency of livestock machinery. Simultaneously, the cutting edge possesses a certain degree of self-sharpening capability, reducing the number of sharpening operations and extending tool life. This provides favorable conditions for the mechanized harvesting of stalks, significantly reducing costs and operating time, improving efficiency, and yielding good economic and social benefits, with a broad market prospect. Attached Figure Description

[0137] Figure 1 This is a schematic diagram of the overall steps of a tool internal surface compensation method based on principal curvature induction according to the present invention.

[0138] Figure 2 This is the present invention. Figure 1 A schematic diagram of the sub-steps in step S1;

[0139] Figure 3 This is the present invention. Figure 1 A schematic diagram of the sub-steps in step S2;

[0140] Figure 4 This is the present invention. Figure 1 A schematic diagram of the sub-steps in step S3;

[0141] Figure 5 This is the present invention. Figure 1 A schematic diagram of the sub-steps in step S4;

[0142] Figure 6 This is the present invention. Figure 1 A schematic diagram of the sub-steps in step S5;

[0143] Figure 7 This is the present invention. Figure 1 A schematic diagram of the sub-steps in step S6;

[0144] Figure 8 This is the present invention. Figure 1 A schematic diagram of the sub-steps in step S7;

[0145] Figure 9 This is a schematic diagram of a sub-step S78 of step S7 in the present invention;

[0146] Figure 10 This is a schematic diagram illustrating the specific system configuration of the internal cutting edge geometric parameters of the tool in this invention;

[0147] Figure 11 This is a schematic diagram of the complete geometric structure of the cutting tool in this invention;

[0148] Figure 12 This is a schematic diagram of the cutting of the inner cutting surface of the tool in this invention;

[0149] Figure 13This is a schematic diagram of the optimization of internal cutting surface compensation in this invention;

[0150] Figure 14 This is a schematic diagram of the complete curved surface of the cutting tool in this invention;

[0151] The irregular curved surface formed by the four points M, N, O, and S in the figure is the inner cutting surface of the tool, and the irregular curved surface formed by the four points Q, R1, P, and S is the outer cutting surface of the tool.

[0152] Figure 15 This is a schematic diagram showing the optimized features of the curved surface structure of the cutting tool in this invention;

[0153] Figure 16 This is a schematic diagram regarding tool tilting error in this invention;

[0154] Figure 17 This is a simplified diagram of the laser scanning analysis results of the tool's arc surface in this invention;

[0155] Figure 18 This is a schematic diagram illustrating the specific error assessment of the discrete line shape of the cutting edge in this invention;

[0156] Figure 19 This is an overall schematic diagram of the tool compensation optimization in this invention. Detailed Implementation

[0157] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0158] Example 1, as Figures 1-19 This invention provides a tool internal surface compensation method based on principal curvature-induced curves, comprising the following steps:

[0159] S1. Use SolidWorks software to separate the NURBS surface of the inner cutting edge of the prototype tool and perform geometric characterization of the generatrix of the continuous surface.

[0160] S2. Substitute the offset of the center of the curved surface to correct the equation of the inner cutting edge surface of the tool.

[0161] S3. Perform motion equivalents on the dynamic slip of the curved surface, define the meridian equation of the inner edge surface, and obtain the equation of the inner edge surface.

[0162] S4. Based on the normal vector deviation and dynamic offset of the blade cutting trajectory, compensate and induce the principal curvature and geodesic deflection of the inner blade surface, and calculate the slant height.

[0163] S5. Calculation of slippage error of the tool prototype surface;

[0164] S6. Calculation of surface shape error of the inner cutting edge of the tool;

[0165] S7, Reconstruction of normal and tangential directions of the inner cutting edge.

[0166] Furthermore, in S1, the inner curved surface of the tool is a tension surface, and the inner cutting edge is an arc-shaped boundary curve. First, two coordinate systems are set, namely, the coordinate system T of the moving rotation center is set as {O}. S -x S y S z S}, and then set the local coordinate system M as {o} when the hob rotates. m -x m y m z m Since the process of cutting materials by the tool is a continuous sliding cutting process, the motion of the cutting point p of the tool tip is divided into three parts: forward sliding cutting, reverse throwing and sliding deflection. Based on the characteristics of NURBS such as non-uniformity, rationality, local controllability and continuous controllability, SolidWorks software is used to complete the NURBS surface separation of the inner cutting surface of the prototype tool and to perform geometric characterization of the continuous surface generatrix.

[0167] Based on the principle of establishing coordinate systems using the Newton-Euler equations, the origin is set at the geometric center of the tool, i.e., the centroid. The specific process for establishing the two coordinate systems described above is as follows:

[0168] Moving rotation center coordinate system T:

[0169] (1) The z-axis is defined as being parallel to and in the same direction as the rotation axis of the tool;

[0170] (2) The x-axis points in the direction of the tool's movement, i.e., the forward direction;

[0171] (3) The y-axis is determined by the right-hand rule, that is, the four fingers of the right hand turn from the X-axis to the Y-axis, and the thumb points to the Z-axis;

[0172] Since the position, speed and force direction of each point on the cutting teeth are constantly changing when the hob is rotating and cutting, the origin O of M is set on the fixed cutting plane.

[0173] Local coordinate system M during hob rotation:

[0174] (1) The y-axis is perpendicular to the rotation direction of the tool's geometric center;

[0175] (2) The x-axis points in the direction of the tool's movement, i.e., the forward direction;

[0176] (3) The z-axis is determined by the right-hand rule.

[0177] Furthermore, step S1 also includes the following sub-steps:

[0178] S11. First, perform geometric abstraction and assumptions, and abstract the inner and outer cutting surfaces of the irregular curved knife as "stretched surfaces";

[0179] S12. Establish relevant coordinate systems, clarify the global coordinate system fixed to the frame and the local coordinate system fixed to the tool, denoted by {T} and {M} respectively, and clarify that their transformation relationship involves the axis tilt angle β transformation around the y-axis.

[0180] S13, in T{o s -x s y s z s The expression of the tool surface is obtained in the coordinate system, a three-dimensional model of the prototype tool is established, the cutting edge is simplified to a straight line, and the complex tool generatrix is ​​discretized into multiple composite curves connected by NURBS curves.

[0181] S14, in M{o m -x m y m z m In the coordinate system, the geometric expression of the tool generatrix is ​​expressed;

[0182] S15. The generatrix is ​​parameterized as Q(l) in the initial coordinate system. In the static state, the generatrix lies in the plane o. m -x m y m Within, the inner cutting edge surface of the tool is obtained along z. s The characterization formula after directional extension;

[0183] S16. Expand the formula in S15, along the latitude θ respectively. w The partial derivatives of the direction and the z-direction of the extension line are calculated to obtain the expressions for the principal vectors e1 and e2;

[0184] S17. Based on the spinor theory, the overall cutting motion of the tool is divided into sliding and throwing motion around the b-axis and sliding and deflecting motion around the a'-axis that occurs with the material. The internal offset angle is solved according to the normal vector equivalence and position equivalence. The a'-axis is the tool sliding and deflecting axis.

[0185] S18. According to the kinematic equivalence, the distance from point p to the b axis and the principal curvature of the tool surface meridian are both R, so the formula for calculating R is obtained.

[0186] S19. To further obtain the specific position q of the cutting point p corresponding to the b-axis, the tool dynamically slides and propels the rotation axis, and the sliding deflection axis constructs the corresponding torsion angle to obtain x. s The expression for an axisymmetric meridian;

[0187] Solve by combining the expressions in S1-20, S18, and S19, as well as the expression for the meridian radius function;

[0188] S1-21. The formula for calculating the position q of the cutting point p corresponding to the b-axis is substituted into the compound rotation formula for twisting around the a' axis and then being thrown around the b-axis to obtain the information about the torsional coefficient k. n1 The expression;

[0189] S1-22. Correct the cutting point position and adjust the coefficient k. n1 and higher-order term A i The equation of the inner surface meridian is obtained, and finally the formula of the inner surface equation of the tool is obtained.

[0190] Furthermore, in step S2, substituting the offset of the curved surface center and correcting the equation of the inner cutting edge surface of the tool, the following sub-steps are also included:

[0191] S21. Based on the tool geometry characterization formula established in S1, perform internal cutting surface correction calculation;

[0192] The coordinate range before blade curvature compensation optimization is set to [285, -12];

[0193] S22. Draw a schematic diagram of the overall arc surface representation of the tool based on the positions of the two coordinate systems mentioned above;

[0194] S23, obtained in T{O s -x s y s z s The expression for the tool surface in the coordinate system;

[0195]

[0196] In the formula, the original surface of the tool is Q(l,θ) w ), where l is the generatrix of the curved surface of the cutting edge. Let θ be the geometric expression for the tool generatrix. w It is a curved latitude line;

[0197] S24. Establish a three-dimensional model of the prototype tool, and simplify the cutting edge to a straight line, and simplify the tool surface generatrix to a multi-segment curve formed by the connection of multiple NURBS circular arcs;

[0198] S25. Transform to the M coordinate system to obtain the approximate expression of the tool surface, and then derive the geometric expression of the tool generatrix.

[0199]

[0200] In the formula, d s R is the baseline of the tool arc surface. c β is the radius of the inner and outer cutting edge arc surfaces of the tool, and β is the tool axis inclination angle;

[0201] S26. Considering the herringbone configuration and the actual center offset, the equation in S25 is modified to obtain the inner cutting edge surface equation of the tool.

[0202]

[0203] In the formula, Z is the extended and stretched surface; Rot(y, β) is the transformation matrix of the original surface deflected by β about the z-axis;

[0204] S27, the formulas in S26 are respectively applied along the latitude θ w The partial derivatives of the direction and the z-direction of the extension line are calculated to obtain the expressions for the principal vectors e1 and e2, and the two principal directions of the tool surface are defined.

[0205]

[0206] The above are the formulas for calculating partial derivatives in two directions;

[0207] In the formula, Rc is the radius of the inner and outer cutting edge arc surfaces of the tool, and θ w For the arc surface latitude line, β is the tool axis inclination angle, the entire system dynamically twists γ(Z) around the zs axis, and R is the principal curvature of the tool surface meridian;

[0208] S28. Based on spinor theory, the overall cutting motion of the tool is decomposed in the T coordinate system according to the equation of the inner cutting surface, and slip equivalence is performed.

[0209] Furthermore, in step S3, the dynamic sliding of the curved surface is kinematically equivalent, the meridian equation of the inner edge surface is defined, and the equation of the inner edge surface is obtained. This step also includes the following sub-steps:

[0210] S31. Based on the decomposed slip deflection motion, define the slip deflection axis as the a' axis in the T coordinate system. s x s The plane-directed shredding and throwing rotation axis is the b-axis, and the cutting point is point p;

[0211] S32. Set the distance from point p to the b axis and the principal curvature of the tool surface meridian to R, and obtain the expression for R;

[0212]

[0213] In the formula, Here is the formula for calculating the first-order partial derivative along the z-direction of the extended surface. The formula for calculating the second-order partial derivative along the z-direction of the extended surface;

[0214] S33. The tool dynamically slides and throws the rotating axis, and the sliding deflection axis constructs the corresponding torsion angle to obtain the specific position q of point p in the b axis;

[0215] S34. Let t be the projection height of the fixed tool normal onto the outer cutting edge surface of the tool. Define the inner cutting edge surface with respect to x. s The equation of an axisymmetric meridian;

[0216]

[0217] In the formula, c1 is the paraxial curvature of the tool surface meridian, and k n Let k be the eccentricity function of the tool surface about the cutting edge. n =-e2, where e2 is the eccentricity of the tool surface about the cutting edge, controlling the degree of surface opening; P i This is the deviation coefficient between the inner and outer curved surfaces of the cutting tool;

[0218] S35, combine S32, S34 and the meridian radius function formula to solve for R;

[0219]

[0220] In the formula, the meridian radius function formula is: c = 1 / R0, where R0 is the radius of curvature of the tool surface at the cutting edge, q is the position of the cutting point p on the b-axis, and q1 and q3 are the corresponding expressions in the two matrices. , q=[q1,0,q3] T ;

[0221] S36. Based on the above expression, we obtain a formula for q. Substituting this into the compound rotation, we obtain the torsional coefficient k. n1 The formula;

[0222]

[0223] In the formula, ψ1 is the interior offset angle, and n x n z Let n1 be the normal vector in each direction, and n1 = [n x ,n y ,n z ,0] T ;

[0224] S37. Combining equations S36 and correcting the formula for the cutting point p, we obtain the equivalent formula.

[0225] S38, Final Adjustment Coefficient k n1 and higher-order term A i Define the equation of the meridian of the inner curved surface;

[0226]

[0227] In the formula, Ai is incremented term by term;

[0228] S39. By combining all the equations, we obtain the equation for the inner blade surface.

[0229] .

[0230] Furthermore, in step S4, based on the deviation of the normal vector of the blade cutting trajectory and the dynamic offset, the principal curvature and geodesic deflection of the inner blade surface are compensated and induced, and the stubble height is calculated. This also includes the following sub-steps:

[0231] S41. Based on the formula of the inner surface equation, and the two principal vector expressions obtained in S27, further subdivide the material surface and the blade surface at the cutting point p into two principal directions and induced principal directions c1 and c2, respectively.

[0232] S42. Calculate the deflection of the material fracture surface and the tool surface in the c1 direction;

[0233]

[0234]

[0235] In the formula, , , , These are the two principal curvatures of the material fracture surface and the tool surface, respectively, c1 in {p;e 1 1;e 1 2}、{p;e 2 1;e 2 In 2}, the positions are ω1 and ω2, respectively; ω1 and ω2 represent position points; where point p is the cutting point; e 1 1, e 1 2 represents the two principal directions of the material surface, e 2 1, e 2 2 represents the two principal directions of the tool surface;

[0236] S43. Along the c1 direction, according to the geodesic torsion difference formula, the tangent of the principal curvatures of the two surfaces is obtained;

[0237]

[0238] S44. List the formulas for the induced principal curvature in the directions c1 and c2 respectively;

[0239]

[0240]

[0241] S45. Using the above formula, calculate the geodesic deflection of the material surface in the main material direction and the geodesic deflection of the inner cutting edge surface in the main tool direction.

[0242] S46. Make the geodesic torsion difference in the c1 direction zero, and process the obtained formula using the double-angle formula;

[0243]

[0244]

[0245]

[0246] S47. Express the formulas for the induced principal curvature in the principal directions c1 and c2 using Euler's formula;

[0247]

[0248]

[0249] S48. Substituting the slip distance formula into the formula for the plane where the oblique cutting is located, and solving the equations simultaneously, we can derive the formula for the tangent of the oblique cutting, i.e. ;

[0250] In the formula, Let c2 be any direction ω in the coordinate system {p,c1,c2}. c The induced curvature; ω c denoted as the sliding direction angle of the plane containing the slanted section; p' is a point within the slanted section area, and p is the cutting point; the length of the arc segment pp' is replaced by the line segment pp', denoted as 2s0, where s0 is half of the approximate line segment value.

[0251] Furthermore, in step S5, the calculation of the slippage error of the tool prototype surface also includes the following sub-steps:

[0252] S51. Using the static optimization parameters of the inner cutting surface as an ideal benchmark, extract the principal curvature and geodesic deflection parameters of the optimized inner cutting surface, and then... s The parameters Rc and β are associated with the principal curvature deviation and the geodesic torsion deviation, defining a bias-free reference state for slip error; where d s Rc is the baseline of the tool's arc surface, Rc is the radius of the tool's inner and outer cutting edge arc surface contours, and β is the tool axis tilt angle.

[0253] S52. Establish a new tool coordinate system {S1}, then rotate this coordinate system by dβ along the Y-axis and translate it along the Z-axis to obtain coordinate system {S2}.

[0254] The tool coordinate system is S, which is the same as the coordinate system of the tool's rotation center. {S1} is the cutting coordinate system corresponding to the cutting point p, i.e., corresponding to {O}. S1 ,X S1 ,Y S1 Z S1In the new coordinate system formed by {S1} and {S2}, the corresponding point of point C is C1; S2 ,X S1 ,Y S1 Z S2 The coordinate system corresponding to {S1} is shown below. Figure 16 ;

[0255] S53. Based on the ideal state, re-establish the ideal position of the cutting point p in the new system, obtain the projection points of the lowest point C in the original curvature circle diagram in the two newly established coordinate systems, and derive the vector relationship between the origin O and the two projection points.

[0256]

[0257] {S2} is {S1} first rotated by dβ around the Y axis, then translated by R along the Z axis. c1 (cosβ0-cosβ) is obtained, R c1 The actual radius of the arc surface profile at the cutting point p; Figure 16 In the diagram, the center of the circle of curvature at point p is O, and the lowest point of this circle of curvature at the position shown is C. The origin of the projection coordinates of C onto the tool rotation axis is O. s1 , {O S1 ,X S1 ,Y S1 Z S1 Let {S1} be the cutting coordinate system corresponding to the cutting point p, and let C1 be the corresponding point of point C; point C2 corresponds to C2 in the ideal state. ′ Point correspondence, C2 ′ C1 lies on the circle of curvature at the original point p, with a radius of curvature of R. c1 The projection of point C2 onto the tool axis during actual operation is O. S2 The corresponding new tool coordinate system {S2} is {O S2 ,X S1 ,Y S1 Z S2}; C2 ′ The projection of the point onto the ideal tool axis is O. S2 ′ ; The axial tilt angle at the cutting point.

[0258] S54. Based on the geometric relationship in the figure, derive the formula for the actual cutting point p's arc diameter based on the ideal cutting point p's corresponding tool arc diameter.

[0259]

[0260] In the formula, d s1 The cutting surface corresponding to the cutting point p under ideal conditions; The axial tilt angle at the cutting point;

[0261] S55. Obtain the formula for the slip error of the tool relative to the coordinate system {S1} in coordinate system {S2};

[0262]

[0263] S56. Perform partial differential linearization derivation on the formula in S55 to obtain the formulas for the actual coordinate offsets in the X, Y, and Z directions.

[0264]

[0265] In the process of calculating dp at any cutting point p, θ is assumed to remain unchanged. In the formula, θ1, θ2, and θ3 are the parametric coordinate directions on the corresponding surface; ψ1 is the inner offset angle; and ψ2 is the outer offset angle.

[0266] Furthermore, in step S6, the calculation of the surface shape error of the inner cutting edge of the tool also includes the following sub-steps:

[0267] S61. After machining the original tool model to a certain scale, perform a three-axis scan to separate the various geometric errors of the tool surface shape in reverse and perform geometric compensation to obtain the three-dimensional laser scanning separation analysis route and result diagram of the tool surface.

[0268] S62. Linearize the differential expression in S55.

[0269]

[0270] In the formula,

[0271] S63. Compare the surface shape obtained by the 3D laser scanning system, select any cutting contact point in the vicinity of the cutting edge on the inner and outer cutting surfaces, and project it onto the extracted curved surface; calculate the actual normal error d. n The surface separation line type characterization diagram is obtained;

[0272] S64. For the tool arc surface error corresponding to the region near point p on the inner arc surface of the tool, construct the error evaluation point after surface discretization. Then, take 4 points evenly distributed in the neighborhood near the cutting point of each cutting parallel line to achieve full rank of the coefficient matrix of the surface error equation system.

[0273] S65. Select random area points accumulated on three adjacent inner cutting edges and use them as the overall error assessment. For any assessment point, express its actual coordinate error.

[0274]

[0275] In the formula, This indicates a small evaluation range, with micro-level values ​​taken for coordinate errors; x0、 y0 is the offset of the axis coordinate in the ideal geometric parameters of the inner arc surface of the tool. r is the radius error of the arc surface. The slant direction angle is any evaluation point;

[0276] S66. Take N evaluation points on three adjacent parallels of latitude and the generatrix, and express the joint equation formula for all points;

[0277]

[0278] In the formula, The slant direction angle is from 1 to N evaluation points. These are the x and y coordinates of N arbitrary evaluation points, respectively.

[0279] S67. Simplify the joint equation formula in S66 into matrix form and expand the matrix as a whole.

[0280] That is, simplify the formula to A 2N×3 X 3×1 =B 2N×1 , where X=(A T A) -1 A T B, the matrix expansion operation is as follows:

[0281]

[0282]

[0283] In the formula, N represents any N evaluation points, and A and B are simplified letter symbols for matrix operations; This is the cumulative summation symbol;

[0284] S68. Based on constrained least squares method, p o , o To find p from nearby evaluation points o Solve for the values ​​of each microvariable dd s By taking different values ​​of ψ1 and ψ2, dR at local areas on the tool generatrix or inner and outer arc surfaces can be obtained. c dd s , da and dβ, thereby correcting the tool geometry and configuration parameters;

[0285] p o , o for Figure 17 The center point of the plane containing the simplified diagram of the laser scanning analysis results of the curved surface of the cutting tool; po To evaluate the center point O of the tool's inner arc surface, a linearized representation of the curve at center point O is used; ψ1 is the inner offset angle; ψ2 is the outer offset angle. In the formula, a' is the slip deflection axis; d a’ ψ1 and ψ2 are the axial displacement error parameters caused by the slip deflection, used to quantify the linear displacement uncertainty of the tool along the slip deflection axis a'; ψ1 and ψ2 are the inner offset angle and the outer offset angle, respectively.

[0286] Furthermore, in S7, the normal and tangential directions of the inner cutting surface are compensated and reconstructed;

[0287] S71. Based on the surface error parameters and the results calculated from the equation system AX=B in S67, extract the three static error parameters: centerline offset, arc radius error, and assembly deviation.

[0288] S72. By deriving the actual normal error formula in reverse, the ideal radius after compensation is obtained, and then the compensation amount is derived to ensure the coordination of the principal curvature.

[0289] S73. Clarify that the range of the biomimetic blade inclination angle is 2°~10°, and obtain the blade inclination angle compensation amount by combining the geodesic deflection formula;

[0290] S74. According to the normal error formula, adjust the control vertices of the NURBS surface to make the compensated centerline return;

[0291] S75. Use a grinding machine to dress the inner cutting edge arc surface, and calculate the normal error d between the corrected measured coordinates and the ideal coordinates. n , ensure d n ≤0.01mm, completing the correction of the arc surface radius error;

[0292] S76. Take a smaller cutting edge angle of 2°~10°, and perform constrained geometric calculations on the region near the cutting tip using the least squares method.

[0293] S77. Further compensate and adjust the principal curvature of the cutting edge to obtain the coordinate range value after normal compensation optimization;

[0294] S78. Based on the original biomimetic irregular curved tool structure feature point AI, determine the tool ψ1 and ψ2 values, and modify and optimize the tool geometric parameters and configuration parameters by combining the biomimetic feature point A′-I′.

[0295] For the optimized coordinate range in S77, compare it with the coordinate range set in S21 before the blade compensation optimization, which is [285, -12], and the optimized coordinate range is [290, 6].

[0296] The following are the detailed steps for modifying and optimizing the tool geometry and configuration parameters in S78:

[0297] S78-1. Optimize the reference boundary points I and H, and determine the convex and concave circular arc segments;

[0298] S78-2. Using the reset points I and H as references, shift point A in the opposite direction to offset the offset, and fine-tune point G to make the step size of segment HG meet the requirements, ensuring that the overall trajectory of A→I→H→G is smooth.

[0299] S78-3. Smoothly adjust auxiliary control points B, C, D, E, and F to correct local curvature deviations;

[0300] S78-4. Finally, refine the edge of the cutting edge at point A and the transition area between point G and the handle.

[0301] After correction in step S7, the compensated tool arc surface profile generatrix diagram is obtained.

[0302] The specific optimization path is as follows:

[0303] Point I (boundary between convex and concave arcs) → Point H (boundary between concave arc and guide section) → Point A (tool tip) → (Points B and C, auxiliary control points of convex arc segment) → (Points D, E, and F, auxiliary control points of concave arc segment) → Point G (tool holder connection point).

[0304] This invention, based on the discrete geometric characteristics of the tool surface, establishes the NURBS surface equation for the inner cutting surface, reveals the error coupling mechanism of dynamic contact at the tool-material interface, and quantifies the influence of parameter errors on the chamfer angle of the cut surface. By compensating and optimizing the principal curvature and normal error of the inner cutting surface, the principal curvature of the inner cutting surface is precisely coordinated with the principal curvature of the material fracture surface, and the normal error is controlled within 0.002 mm, significantly reducing the chamfer generated during material cutting and greatly improving the flatness of the cut surface and the cutting quality. Finally, through compensation and reconstruction of the tool's arc surface structure, the cutting edge profile of the tool's outer edge is optimized. Based on existing research and theory, this invention requires no additional specialized equipment and can greatly reduce tool processing costs and manufacturing difficulty.

[0305] The specific implementation steps of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above implementation steps. The optimization compensation method involved in the present invention is the core idea of ​​the present invention. The details need to be explained in detail with reference to the drawings.

Claims

1. A tool internal surface compensation method based on principal curvature-induced curves, characterized in that, Includes the following steps: S1. Use SolidWorks software to separate the NURBS surface of the inner cutting edge of the prototype tool and perform geometric characterization of the generatrix of the continuous surface. S2. Substitute the offset of the center of the curved surface to correct the equation of the inner cutting edge surface of the tool. S3. Perform motion equivalents on the dynamic slip of the curved surface, define the meridian equation of the inner edge surface, and obtain the equation of the inner edge surface; S4. Based on the normal vector deviation and dynamic offset of the blade cutting trajectory, compensate and induce the principal curvature and geodesic deflection of the inner blade surface, and calculate the slant height. S5. Calculation of slippage error of the tool prototype surface; S6. Calculation of surface shape error of the inner cutting edge of the tool; S7, Reconstruction of normal and tangential directions of the inner cutting edge.

2. The tool internal surface compensation method based on principal curvature-induced curves according to claim 1, characterized in that, In S1, the inner curved surface of the tool is a tension surface, and the inner cutting edge is an arc-shaped boundary curve. First, two coordinate systems are set, namely, the coordinate system T of the moving rotation center is set as {O}. S -x S y S z S }, and then set the local coordinate system M as {o} when the hob rotates. m -x m y m z m Since the process of the tool cutting material is a continuous sliding cutting process, the motion of the cutting point p is divided into three main parts: forward sliding cutting, reverse throwing, and sliding deflection. Based on the characteristics of NURBS, such as non-uniformity, rationality, local controllability, and continuous controllability, the following is utilized: SolidWorks software was used to separate the NURBS surface of the inner cutting edge of the prototype tool and to perform geometric characterization of the continuous surface generatrix. Based on the principle of establishing coordinate systems using the Newton-Euler equations, the origin is set at the geometric center of the tool, i.e., the centroid. The specific process for establishing the two coordinate systems described above is as follows: Moving rotation center coordinate system T: (1) The z-axis is defined as being parallel to and in the same direction as the rotation axis of the tool; (2) The x-axis points in the direction of the tool's movement, i.e., the forward direction; (3) The y-axis is determined by the right-hand rule, that is, the four fingers of the right hand turn from the X-axis to the Y-axis, and the thumb points to the Z-axis; Since the position, speed and force direction of each point on the cutting teeth are constantly changing when the hob is rotating and cutting, the origin O of M is set on the fixed cutting plane. Local coordinate system M during hob rotation: (1) The y-axis is perpendicular to the direction of rotation of the tool's geometric center; (2) The x-axis points in the direction of the tool's movement, i.e., the forward direction; (3) The z-axis is determined by the right-hand rule.

3. The tool internal surface compensation method based on principal curvature-induced curves according to claim 2, characterized in that, S1 further includes the following sub-steps: S11. First, perform geometric abstraction and assumptions, and abstract the inner and outer cutting surfaces of the irregular curved knife as "stretched surfaces"; S12. Establish relevant coordinate systems, clarify the global coordinate system fixed to the frame and the local coordinate system fixed to the tool, denoted by {T} and {M} respectively, and clarify that their transformation relationship involves the axis tilt angle β transformation around the y-axis. S13, in T{o s -x s y s z s The expression of the tool surface is obtained in the coordinate system, a three-dimensional model of the prototype tool is established, the cutting edge is simplified to a straight line, and the complex tool generatrix is ​​discretized into multiple composite curves connected by NURBS curves. S14, in M{o m -x m y m z m In the coordinate system, the geometric expression of the tool generatrix is ​​expressed; S15. The generatrix is ​​parameterized as Q(l) in the initial coordinate system. In the static state, the generatrix lies in the plane o. m -x m y m Within, the inner cutting edge surface of the tool is obtained along z. s The characterization formula after directional extension; S16. Expand the formula in S15, along the latitude θ respectively. w The partial derivatives of the direction and the z-direction of the extension line are calculated to obtain the expressions for the principal vectors e1 and e2; S17. Based on the spinor theory, the overall cutting motion of the tool is divided into sliding and throwing motion around the b-axis and sliding and deflecting motion around the a'-axis that occurs with the material. The internal offset angle is solved according to the normal vector equivalence and position equivalence. The a'-axis is the tool sliding and deflecting axis. S18. According to the kinematic equivalence, the distance from point p to the b axis and the principal curvature of the tool surface meridian are both R, so the formula for calculating R is obtained. S19. To further obtain the specific position q of the cutting point p corresponding to the b-axis, the tool dynamically slides and propels the rotation axis, and the sliding deflection axis constructs the corresponding torsion angle to obtain x. s The expression for an axisymmetric meridian; Solve by combining the expressions in S1-20, S18, and S19, as well as the expression for the meridian radius function; S1-21. The formula for calculating the position q of the cutting point p corresponding to the b-axis is substituted into the compound rotation formula for twisting around the a' axis and then being thrown around the b-axis to obtain the information about the torsional coefficient k. n1 The expression; S1-22. Correct the cutting point position and adjust the coefficient k. n1 and higher-order term A i We obtain the equation for the inner surface meridian and finally the formula for the inner surface equation of the tool.

4. The tool internal surface compensation method based on principal curvature-induced curves according to claim 1, characterized in that, In step S2, substituting the surface center offset and correcting the inner cutting edge surface equation of the tool, the following sub-steps are also included: S21. Based on the tool geometry characterization formula established in S1, perform internal cutting surface correction calculation; The coordinate range before blade curvature compensation optimization is set to [285, -12]; S22. Draw a schematic diagram of the overall arc surface representation of the tool based on the positions of the two coordinate systems mentioned above; S23, obtained in T{O s -x s y s z s The expression for the tool surface in the coordinate system; ； In the formula, the original surface of the tool is Q(l,θ) w ), where l is the generatrix of the curved surface of the cutting edge. Let θ be the geometric expression for the tool generatrix. w It is a curved latitude line; S24. Establish a three-dimensional model of the prototype tool, and simplify the cutting edge to a straight line, and simplify the tool surface generatrix to a multi-segment curve formed by connecting multiple NURBS circular arcs; S25. Transform to the M coordinate system to obtain the approximate expression of the tool surface, and then obtain the geometric expression of the tool generatrix. ； In the formula, d s R is the baseline of the tool arc surface. c β is the radius of the inner and outer cutting edge arc surfaces of the tool, and β is the tool axis inclination angle; S26. Considering the herringbone configuration and the actual center offset, the equation in S25 is modified to obtain the inner cutting edge surface equation of the tool. ； In the formula, Z is the extended and stretched surface; Rot(y, β) is the transformation matrix of the original surface deflected by β about the z-axis; S27, the formulas in S26 are respectively applied along the latitude θ w The partial derivatives of the direction and the z-direction of the extension line are calculated to obtain the expressions for the principal vectors e1 and e2, and the two principal directions of the tool surface are defined. ； The above are the formulas for calculating partial derivatives in two directions; In the formula, Rc is the radius of the inner and outer cutting edge arc surfaces of the tool, and θ w For the arc surface latitude line, β is the tool axis inclination angle, the entire system dynamically twists γ(Z) around the zs axis, and R is the principal curvature of the tool surface meridian; S28. Based on spinor theory, the overall cutting motion of the tool is decomposed in the T coordinate system according to the equation of the inner cutting surface, and slip equivalence is performed.

5. The tool internal surface compensation method based on principal curvature-induced curves according to claim 1, characterized in that, In step S3, the dynamic sliding of the curved surface is kinematically equivalent, the meridian equation of the inner edge surface is defined, and the equation of the inner edge surface is obtained. This step also includes the following sub-steps: S31. Based on the decomposed slip deflection motion, define the slip deflection axis as the a' axis in the T coordinate system. s x s The plane-directed shredding and throwing rotation axis is the b-axis, and the cutting point is point p; S32. Set the distance from point p to the b axis and the principal curvature of the tool surface meridian to R, and obtain the expression for R; ； In the formula, Here is the formula for calculating the first-order partial derivative along the z-direction of the extended surface. The formula for calculating the second-order partial derivative along the z-direction of the extended surface; S33. The tool dynamically slides and throws the rotating axis, and the sliding deflection axis constructs the corresponding torsion angle to obtain the specific position q of point p in the b axis; S34. Let t be the projection height of the fixed tool normal onto the outer cutting edge surface of the tool. Define the inner cutting edge surface with respect to x. s The equation of an axisymmetric meridian; ； In the formula, c1 is the paraxial curvature of the tool surface meridian, and k n Let k be the eccentricity function of the tool surface about the cutting edge. n =-e2, where e2 is the eccentricity of the tool surface about the cutting edge, controlling the degree of surface opening; P i This is the deviation coefficient between the inner and outer curved surfaces of the cutting tool; S35, combine S32, S34 and the meridian radius function formula to solve for R; ； In the formula, the meridian radius function formula is: c = 1 / R0, where R0 is the radius of curvature of the tool surface at the cutting edge, q is the position of the cutting point p on the b-axis, and q1 and q3 are the corresponding expressions in the two matrices. , q=[q1,0,q3] T ; S36. Based on the above expression, we obtain a formula for q. Substituting this into the compound rotation, we obtain the torsional coefficient k. n1 The formula; ； In the formula, ψ1 is the interior offset angle, and n x n z Let n1 be the normal vector in each direction, and n1 = [n x ,n y ,n z ,0] T ; S37. Combining equations S36 and correcting the formula for the cutting point p, we obtain the equivalent formula. S38, Final Adjustment Coefficient k n1 and higher-order term A i Define the equation of the meridian of the inner surface; ； In the formula, Ai is incremented term by term; S39. By combining all the equations, we obtain the equation for the inner blade surface. 。 6. The tool internal surface compensation method based on principal curvature-induced curves according to claim 1, characterized in that, In step S4, based on the normal vector deviation and dynamic offset of the cutting trajectory of the blade, the principal curvature and geodesic deflection of the inner blade surface are compensated and induced, and the slant height is calculated. This also includes the following sub-steps: S41. Based on the formula of the inner surface equation, and the two principal vector expressions obtained in S27, further subdivide the material surface and the blade surface at the cutting point p into two principal directions and induced principal directions c1 and c2, respectively. S42. Calculate the deflection of the material fracture surface and the tool surface in the c1 direction; ； ； In the formula, , , , These are the two principal curvatures of the material fracture surface and the tool surface, respectively, c1 in {p;e 1 1;e 1 2}、{p;e 2 1;e 2 In 2}, the positions are ω1 and ω2 respectively; ω1 and ω2 represent position points; where point p is the cutting point; e 1 1, e 1 2 represents the two principal directions of the material surface, e 2 1, e 2 2 represents the two principal directions of the tool surface; S43. Along the c1 direction, according to the geodesic torsion difference formula, the tangent of the principal curvatures of the two surfaces is obtained; ； S44. List the formulas for the induced principal curvature in the directions c1 and c2 respectively; ； ； S45. Using the above formula, calculate the geodesic deflection of the material surface in the main material direction and the geodesic deflection of the inner cutting edge surface in the main tool direction. S46. Make the geodesic torsion difference in the c1 direction zero, and process the obtained formula using the double-angle formula; ； ； ； S47. Express the formulas for the induced principal curvature in the principal directions c1 and c2 using Euler's formula; ； ； S48. Substituting the slip distance formula into the formula for the plane where the oblique cutting is located, and solving the equations simultaneously, we can derive the formula for the tangent of the oblique cutting, i.e. ; In the formula, Let c2 be any direction ω in the coordinate system {p,c1,c2}. c The induced curvature; ω c denoted as the sliding direction angle of the plane containing the slant; p' is a point within the slant area, and p is the cutting point; the length of the arc segment pp' is replaced by the line segment pp', denoted as 2s0, where s0 is half the approximate value of the line segment.

7. The tool internal surface compensation method based on principal curvature-induced curves according to claim 1, characterized in that, In step S5, the calculation of the slippage error of the tool prototype surface further includes the following sub-steps: S51. Using the static optimization parameters of the inner cutting surface as an ideal benchmark, extract the principal curvature and geodesic deflection parameters of the optimized inner cutting surface, and then... s The parameters Rc and β are associated with the principal curvature deviation and the geodesic torsion deviation, defining a bias-free reference state for slip error; where d s Rc is the baseline of the tool's arc surface, Rc is the radius of the tool's inner and outer cutting edge arc surface contours, and β is the tool axis tilt angle. S52. Establish a new tool coordinate system {S1}, then rotate this coordinate system by dβ along the Y-axis and translate it along the Z-axis to obtain coordinate system {S2}. The tool coordinate system is S, which is the same as the coordinate system of the tool's rotation center. {S1} is the cutting coordinate system corresponding to the cutting point p, i.e., corresponding to {O}. S1 ,X S1 ,Y S1 Z S1 In the new coordinate system formed by {S1} and {S2}, the corresponding point of point C is C1; S2 ,X S1 ,Y S1 Z S2 The coordinate system corresponding to {S1} is transformed from {S1}; S53. Based on the ideal state, re-establish the ideal position of the cutting point p in the new system, obtain the projection points of the lowest point C in the original curvature circle diagram in the two newly established coordinate systems, and derive the vector relationship between the origin O and the two projection points. ； {S2} is {S1} first rotated by dβ around the Y axis, then translated by R along the Z axis. c1 (cosβ0-cosβ) is obtained, R c1 The actual radius of the arc surface profile at the cutting point p; S54. Based on the geometric relationship in the figure, derive the formula for the actual diameter of the arc surface corresponding to the cutting point p based on the ideal diameter of the tool arc surface corresponding to the cutting point p. ； In the formula, d s1 The cutting surface corresponding to the cutting point p under ideal conditions; The axial tilt angle at the cutting point; S55. Obtain the formula for the slip error of the tool relative to the coordinate system {S1} in coordinate system {S2}; ； S56. Perform partial differential linearization derivation on the formula in S55 to obtain the formulas for the actual coordinate offsets in the X, Y, and Z directions. ； In the process of calculating dp at any cutting point p, θ is assumed to remain unchanged. In the formula, θ1, θ2, and θ3 are the parametric coordinate directions on the corresponding surface; ψ1 is the inner offset angle; and ψ2 is the outer offset angle.

8. The tool internal surface compensation method based on principal curvature-induced curves according to claim 7, characterized in that, In step S6, the calculation of the surface shape error of the inner cutting edge of the tool further includes the following sub-steps: S61. After machining the original tool model to a certain scale, perform a three-axis scan to separate the various geometric errors of the tool surface shape in reverse and perform geometric compensation to obtain the three-dimensional laser scanning separation analysis route and result diagram of the tool surface. S62. Linearize the differential expression in S55. S63. Compare the surface shape obtained by the 3D laser scanning system, select any cutting contact point in the vicinity of the cutting edge on the inner and outer cutting surfaces, and project it onto the extracted curved surface; calculate the actual normal error d. n The surface separation line type characterization diagram is obtained; S64. For the tool arc surface error corresponding to the region near point p on the inner arc surface of the tool, construct the error evaluation point after surface discretization. Then, take 4 points evenly distributed in the neighborhood near the cutting point of each cutting parallel line to achieve full rank of the coefficient matrix of the surface error equation system. S65. Select random area points accumulated on three adjacent inner cutting edges and use them as the overall error assessment. For any assessment point, express its actual coordinate error. ； In the formula, This indicates a small evaluation range, with micro-level values ​​taken for coordinate errors; x0、 y0 is the offset of the axis coordinate in the ideal geometric parameters of the inner arc surface of the tool. r is the radius error of the arc surface. The slant direction angle at any evaluation point; S66. Take N evaluation points on three adjacent parallels of latitude and the generatrix, and express the joint equation formula for all points; S67. Simplify the joint equation formula in S66 into matrix form and expand the matrix as a whole. S68. Based on constrained least squares method, p o , o To find p from nearby evaluation points o Solve for the values ​​of each microvariable dd s By taking different values ​​of ψ1 and ψ2, dR at local areas on the tool generatrix or inner and outer arc surfaces can be obtained. c dd s , da and dβ, thereby correcting the tool geometry and configuration parameters; p o , o p is the center point of the plane containing the simplified diagram of the laser scanning analysis results of the tool's curved surface; o To evaluate the center point O of the inner arc surface of the tool, the curve at the center point O is linearized; ψ1 is the inner offset angle; ψ2 is the outer offset angle; a' is the slip deflection axis; d a’ This is the axial displacement error parameter caused by the slip deflection, used to quantify the linear displacement uncertainty of the tool along the slip deflection axis a'.

9. The tool internal surface compensation method based on principal curvature-induced curves according to claim 8, characterized in that, In S7, the normal and tangential directions of the inner cutting surface are compensated and reconstructed; S71. Based on the surface error parameters and the results calculated from the equation system AX=B in S67, extract the three static error parameters: centerline offset, arc radius error, and assembly deviation. S72. By deriving the actual normal error formula in reverse, the ideal radius after compensation is obtained, and then the compensation amount is derived to ensure the coordination of the principal curvature. S73. Clarify that the range of the biomimetic blade inclination angle is 2°~10°, and obtain the blade inclination angle compensation amount by combining the geodesic deflection formula; S74. According to the normal error formula, adjust the control vertices of the NURBS surface to make the compensated centerline return; S75. Use a grinding machine to dress the inner cutting edge arc surface, and calculate the normal error d between the corrected measured coordinates and the ideal coordinates. n , ensure d n ≤0.01mm, completing the correction of the arc surface radius error; S76. Take a smaller cutting edge angle of 2°~10°, and perform constrained geometric calculations on the region near the cutting tip using the least squares method. S77. Further compensate and adjust the principal curvature of the cutting edge to obtain the coordinate range value after normal compensation optimization; S78. Based on the original biomimetic irregular curved tool structure feature point AI, determine the tool ψ1 and ψ2 values, and modify and optimize the tool geometric parameters and configuration parameters by combining the biomimetic feature point A′-I′.

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