Cutter outer curved surface compensation method based on differential geometry

Abstract

The invention relates to a tool outer curved surface compensation method based on differential geometry. Obtaining a tool outer blade curved surface equation expression; the generatrix of the curved surface of the cutter and the curved surface of the moving cutter track are projected to an XZ plane by adopting differential geometry Guass mapping, and Guass mapping is carried out on the generatrix of the outer boundary of the curved surface of the original cutter; extending and projecting geometric structure points of a boundary curve of the outer blade curved surface of the cutter in an Os-xsys plane, describing a contact relation between a cutter path curved surface and a material fracture curved surface, and determining a minimum axis inclination angle; differential compensation is carried out on the outer blade curved surface of the cutter, and a Gauss mapping curved surface compensation expression of the curved surface of the cutter and the material cutting section is listed; and performing verification and correction after performing differential compensation on the curved surface of the outer blade of the cutter. According to the method, the normal vectors of the outer blade curved surface of the cutter and the material contact curved surface are the same, inclined stubbles are reduced from the geometric root, accurate description and compensation of a dynamic contact track can be realized, and the consistency of cutting quality under different working conditions is ensured.

Description

Technical Field

[0001] This invention relates to the field of cutting tool outer surface compensation technology, specifically to the optimization technology of shredding tools in forage harvesting equipment, and particularly to a cutting tool outer surface compensation method based on differential geometry. This method is specifically designed for high-speed rolling cutting operations and is applicable to the geometric error correction of the outer cutting surface of shredding tools, precise compensation of dynamic contact trajectory, improvement of tool-material cutting synergy, and adaptation and optimization of biomimetic morphology and cutting performance. It can be widely integrated into agricultural machinery equipment such as forage harvesters and straw shredders, and belongs to the field of agricultural machinery technology. Background Technology

[0002] In modern large-scale forage harvesting operations, high-speed rolling is a core technology for ensuring material cutting quality and improving harvesting efficiency. The geometric characteristics of the outer curved surface of the cutting blade directly determine the cutting accuracy, operational stability, and equipment energy consumption. Currently, domestically produced forage harvesting equipment widely uses flat straight blades, which have relatively low overall cutting quality and service life. Imported equipment offers high cutting quality but is also more expensive and difficult to maintain. Existing blade optimization methods focus on static geometric fitting of the outer cutting surface, using traditional geometric methods or empirical corrections for error compensation. While this can reduce some static deviations, it does not fully consider the dynamic characteristics during high-speed rolling. Due to the uncertainty of the dynamic cutting contact point of the outer cutting surface of irregularly shaped rolling cutters, influenced by multiple factors such as material feed thickness, changes in the principal curvature of the blade surface, and slippage, existing methods struggle to achieve precise geometric coordination between the outer curved surface of the blade and the fracture surface of the material.

[0003] Therefore, this invention provides a tool outer surface compensation method based on differential geometry. By constructing a precise parametric model of the surface, realizing the geometric representation and compensation of the dynamic contact trajectory, and correcting multi-source errors, it fully explores the structural potential of irregular curved blades. This method ensures that the normal vector of the tool's outer cutting surface is the same as that of the material contact surface, reducing the generation of stubble from a geometrical perspective. It also achieves accurate description and compensation of the dynamic contact trajectory, ensuring consistent cutting quality under different working conditions. It reduces the concentration of contact pressure, lowers the risk of tool wear and material tearing, achieves refined correction of local surfaces, improves the safety of equipment operation, and fundamentally solves the bottlenecks of existing technologies. Ultimately, it effectively improves the stability of cutting quality and increases the harvesting efficiency of high-speed rolling cutting equipment such as forage harvesters. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies, this invention aims to provide a tool external curvature surface compensation method based on differential geometry. Specifically, considering the urgent need to improve tool cutting quality, and comprehensively taking into account factors such as the cutting quality, service life, and maintenance costs of typical harvesting equipment tools, a tool external curvature surface compensation method based on differential geometry is proposed. This method can overcome the shortcomings of traditional flat straight blade cutting quality, solve the problem of consistent cutting quality under different working conditions, reduce the risk of tool wear and material tearing, improve the stability of cutting quality, and ultimately improve the harvesting efficiency of forage harvesters.

[0005] The technical solution adopted in this invention is: a tool outer surface compensation method based on differential geometry, comprising the following steps:

[0006] S1. Combining the discrete geometric features of the tool surface and NUBRS theory, construct a geometric parameterized model of the tool's outer cutting edge surface, and list the expansion formula of the tool's outer cutting edge surface to make the actual cutting edge line close to the theoretical trajectory line.

[0007] S2. Based on spinor theory, the shredding process is divided into two motions. Through normal vector equivalence, position equivalence, geometric equivalence, and motion equivalence, the equation expression of the tool's outer surface is obtained.

[0008] S3. Use differential geometry Guass mapping to project the tool surface generatrix and the moving tool trajectory surface onto the XZ plane, and perform Guass mapping on the original tool surface outer boundary generatrix.

[0009] S4. Define a dynamic rotation coordinate system. Extend the projection of the geometric points of the boundary curve of the tool's outer cutting edge onto this coordinate system to describe the contact relationship between the tool trajectory surface and the material fracture surface, and determine the minimum axis tilt angle.

[0010] S5. Perform differential compensation on the outer cutting surface of the tool and list the Gaussian mapping surface compensation expression between the tool surface and the material cutting section.

[0011] S6. After performing differential compensation on the outer cutting edge surface of the tool, verification and correction are carried out.

[0012] Furthermore, in S1, the outer curved surface of the tool is a tension surface, and the arc boundary curve is the outer cutting edge. It is assumed that the coordinate system T of the dynamic rotation center is defined as {O}. S -x S y S z S} is a transition coordinate system used to characterize the dynamic position of the hob's revolution center, where the origin O is... s Fixed at the center of rotation of the hob, z S The axis is along the direction of the hob's revolution axis, x S The axis is fed radially along the reference direction of the hob, y S axis and x Sz S The axes are perpendicular to each other and the local coordinate system M {o of the outer edge m -x m y m z m A local coordinate system is used to characterize the geometry of the tool's outer cutting edge surface and the position of the cutting point. Integrating the discrete geometric features of the tool surface and NUBRS theory, a parameterized geometric model of the tool's outer cutting edge surface is constructed, and the expansion formula representing the tool's outer cutting edge surface is listed, making the actual cutting edge line close to the theoretical trajectory line. Step S1 also includes the following sub-steps:

[0013] S11. Define the coordinate system T{O} of the moving rotation center. S -x S y S z S} and the local coordinate system M{o of the outer edge m -x m y m z m} is used to characterize the local geometric features of the outer cutting edge surface of a cutting tool;

[0014] S12. List the tool surface expression in the coordinate system set in S11, extract the key geometric parameters of the tool's outer cutting edge surface, and simplify the cutting edge to a straight line;

[0015] In T{O S -x S y S z S The expression for the tool surface in the coordinate system is:

[0016]

[0017] Where l is the generatrix of the blade's curved surface, θ w It is a curved latitude line;

[0018] In M{o m -x m y m z m The approximate expression for the tool surface in the coordinate system is as follows: Where β is the tool axis tilt angle, The transformation matrix for the original surface deflected by β around the z-axis;

[0019] S13. Considering the assembly requirements of the inclined tool installation, γ1 and γ2 are defined as the boundary angles between the inner and outer cutting surfaces of the tool arc and the tool tip. These angles define the curvature and length of the inner and outer cutting surfaces of the tool, and the baseline of the tool arc is d. s The inner and outer cutting edge radius of the tool is R. C The tool axis inclination angle is β, the lowest point of the outer cutting edge generatrix is ​​C, and the tool thickness is limited by point D on the inner cutting edge generatrix.

[0020] S14. List the geometric expression of the tool generatrix and discretize it into multiple NUBRS circular arc transition curves.

[0021]

[0022] Where, l∈[R] c ·((cosβ)-cos(β-γ1)), Rc·((cosβ)-cos(β+γ2))], d s R is the baseline of the tool arc surface. c The radius of the inner and outer cutting edge arc surfaces of the cutting tool;

[0023] S15. Combining with S13, construct the parameterized equation of the generatrix as Q(l) to simplify the parameterized representation of the surface;

[0024] S16. Transform and correct the coordinate system, and list the extended expression for the external cutting edge surface of the tool.

[0025]

[0026] in, This is the parametric expression for the outer cutting edge surface of the tool in the coordinate system T of the moving rotation center. Z represents the latitude parameter of the arc surface, and Z represents the extended stretching surface.

[0027] Furthermore, in step S2, the shredding process is divided into two motions based on spinor theory. Through normal vector equivalence, position equivalence, geometric equivalence, and motion equivalence, the equation expression for the tool's outer surface is obtained. This also includes the following sub-steps:

[0028] S21. 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.

[0029] S22. Based on the principle of equivalence of normal vectors, list the expression for the normal vector at the cutting point of the tool;

[0030]

[0031] in, , ω1=[0,0,1] T ω2=[0,1,0] T p0 = [p1, 0, p3] T , n=[n x ,n y ,n z ,0] T This represents the normal vector of the tool surface at the cutting point in the T coordinate system. It is an interior angle. Let n0 be the exterior deflection angle, then n0 = [-1, 0, 0] T , The unit vector of the slip deflection axis. Let q be the unit vector of the sliding and throwing axis, q be the position vector of the sliding and throwing axis b-axis, p1 and p3 be the coordinate components of the throwing endpoint p0 in the coordinate system T of the dynamic rotation center, and n be the coordinate components of the throwing endpoint p0 in the coordinate system T of the dynamic rotation center. x ,n y ,n z These are the normal vector components of the tool surface;

[0032] S23. According to the principle of geometric equivalence, in order to ensure the geometric features and basic properties of the neighborhood of the cutting point p for the quality of the shredded segments, it is concluded that the principal curvature of the tool surface and the principal curvature of the material are the same.

[0033] S24. Based on the principles of motion equivalence and equivalent cutting, list information about x. s From the expressions for the axisymmetric distribution of the meridian and the meridian radius function, we can obtain the expression for the principal curvature R of the meridian on the tool surface.

[0034] Regarding x s The expression for an axisymmetric meridian is as follows:

[0035]

[0036] In the formula, c1 is the paraxial curvature of the tool surface meridian, c = 1 / R0, and R0 is the radius of curvature of the tool surface at the cutting edge; k n Let k be the eccentricity function of the tool surface about the cutting edge. n =-e 2 e is the eccentricity of the tool surface about the cutting edge, used to control the degree of surface opening; P i This is the deviation coefficient between the inner and outer curved surfaces of the cutting tool;

[0037] The expression for the meridian radius function is as follows:

[0038]

[0039] Combining these equations, we can obtain the expression for the principal curvature R of the tool surface meridian as follows:

[0040]

[0041] Where q = [q1, 0, q3] T q1 is the x-axis in coordinate system T. S The offset in the axial direction, q3 is its value in the z-axis. S Height in the axial direction, For tool generatrix parameters, Let l be the radial dimension of the generatrix at parameter l. The rotation angle around the sliding deflection axis a during the tool's compound motion;

[0042] S25. Determine the position q of the cutting point p corresponding to the b-axis, and adjust the parameters kn and higher-order terms B. i Define the equation expression for the meridian of the external curved surface;

[0043]

[0044] S26, combining with S25, list the equation expression for the outer surface of the tool;

[0045] .

[0046] Furthermore, differential geometry Guass mapping is used to project the tool surface generatrix and the moving tool trajectory surface onto the XZ plane, and Guass mapping is performed on the original tool surface outer boundary generatrix. S3 also includes the following sub-steps:

[0047] S31. Define the Gaussian mapping for the tool's own surface and the surface of the tool's motion trajectory, and determine the source and target of the Gaussian mapping;

[0048] S32. List the Gaussian mapping expression between the tool surface and the material fracture surface;

[0049]

[0050]

[0051] Where, n(l,θ) w ), n(t, ) is the surface normal vector, Q l Q θ R t , These are the tangent vectors of the surface along the parametric coordinates l, θ, t, and γ, respectively.

[0052] S33. Define the original surface of the tool as a regular surface, and the tool path and tool marks, i.e. the material fracture surface, as projected surfaces.

[0053] S34. Analyze the Gaussian mapping regions corresponding to the two endpoints A and B of the cutting tool and the boundary points A' and B' of the cutting edge trajectory surface during the cutting operation.

[0054] S35. Project the moving tool trajectory surface mapped by the tool surface generatrix and Gauss onto the XZ plane. When only considering the contact of the tool's outer cutting edge, perform Gauss mapping on the original tool surface outer boundary generatrix and confirm the new mapping point.

[0055] S36. Summarize the relationship between the inner and outer offset angles ψ1 and ψ2 after projection and the inner and outer edge boundary angles γ1 and γ2 as follows: γ2>ψ1, γ1>γ;

[0056] S37. Project the Gaussian mapping in the XZ plane onto O. s -x s y s In the plane, analyze the relationship between the axis tilt angle β and the outer edge boundary angle γ2.

[0057] Furthermore, in O s -x s y s The geometric points of the boundary curve of the tool's outer cutting edge are extended and projected onto the plane to describe the contact relationship between the tool trajectory surface and the material fracture surface, and to determine the minimum axial tilt angle β. S4 also includes the following sub-steps:

[0058] S41. Characterize the position and distribution of the normal vector at the cut point p after mapping in the local coordinate system {p-e1e2};

[0059] S42. Combining with S41, approximate the surface in the neighborhood of the cutting point p, list the approximate expression of the surface, and determine that the approximate surface of the tool surface near the cutting point must be a convex hyperbolic paraboloid.

[0060]

[0061] in, The principal curvature of the boundary line of the outer cutting edge of the tool;

[0062] S43. Based on the mathematical relationship between the normal curvature of the latitude and longitude lines and the envelope line on the tool surface, analyze the principal curvature constraint conditions and the approximate shape of the neighborhood surface corresponding to the different positions of the cutting point p on the tool curve.

[0063] S44, limiting the principal curvature along the meridian direction The minimum value should be greater than the maximum curvature of the material bending surface outside the fixed blade;

[0064] S45. Based on the conclusions obtained in S45, calculate the principal curvature of the boundary line of the tool's outer cutting surface using the deviation and induction method.

[0065]

[0066] in, Let p be the distance from the cutting point p to the axis of the tool surface;

[0067] S46. Based on the Gaussian mapping analysis of the unit spherical coverage area, determine the minimum axial tilt angle β = 25°;

[0068] Gaussian mapping analysis is used to analyze the unit spherical coverage area to ensure the tool normal space G. Q Includes the material cross-section normal vector space G R Given that γ1 < β, we take γ1 = 12°, sinβ > 0.2 × 10 × sin12.5° ≈ 0.4158, which means β > 24.56°. Taking β = 25° satisfies the constraint requirements.

[0069] Furthermore, differential compensation is performed on the outer cutting edge surface of the tool, and the Gaussian mapping surface compensation expression between the tool surface and the material cutting section is listed. S5 also includes the following sub-steps:

[0070] S51. Based on the corresponding neighborhoods D1 and D2 of the cutting point p on the material fracture surface and the cutting tool surface, derive the expression for calculating the local cutting ratio of point p.

[0071]

[0072] Where G is the local cutting ratio during tool operation; A1 and A2 are the areas corresponding to D1 and D2, respectively, in mm. 2 h1 is the thickness of the feed layer, mm; h2 is the material cutting thickness in the normal direction at point p, mm;

[0073] S52. Analyze the surface S containing the contact point p between the outer cutting edge of the tool and the cut material cross-section, and calculate the area A of the cutting domain D mapped onto a unit sphere. g The Gaussian curvature expression between A and B;

[0074]

[0075] Where K is the Gaussian curvature, K = , The principal curvature of the surface;

[0076] S53. Combining S52 with the material fracture surface and tool surface at the cutting point p to define the Guass curvature, we obtain the formula for the cuttable feed layer thickness at point p.

[0077]

[0078] Wherein, K1(p) and K2(p) are the curvature values ​​of the tool surface at the cutting point p along two mutually perpendicular principal directions;

[0079] S54. Considering the high-speed rolling motion of the tool, the cutting marks on the same weft line on the outer edge of the tool are evenly distributed along the main axis of the hob.

[0080] S55. List the Gaussian curvature expressions for the material fracture surface and the tool surface;

[0081]

[0082]

[0083] Where D1 and D2 are the corresponding neighborhoods of the cutting point p on the material fracture surface and the tool surface, respectively. Let be the Gaussian curvature of the fracture surface of the material. The principal curvature of the fracture surface of the material is denoted as . Let be the principal curvature of the tool surface. For the Gaussian mapping of a local region of the material fracture surface, Gaussian mapping for a local region of the tool surface;

[0084] S56. Combining S51 and S53, we obtain the expression for the local cutting ratio;

[0085]

[0086] in, Let be the Gaussian curvature of the fracture surface of the material. Let be the Gaussian curvature of the tool surface;

[0087] S57. List the Gaussian mapping surface compensation expression between the tool surface and the material cutting section after considering slip offset.

[0088]

[0089] in, This represents the normal vector mapping results for the tool surface region H and the fracture surface region H. , The normal vector mapping results for tool surface segment I and fracture surface segment I are shown. This represents the normal vector mapping results of the tool surface at point p and the fracture surface at point p. For the Gaussian mapping of a local region of the material fracture surface, For the Gaussian mapping of a local region of the tool surface, For material feeding speed, This represents the rotational speed of the hob.

[0090] Furthermore, in step S6, after performing differential compensation on the outer cutting edge surface of the tool and then verifying and correcting it, the following sub-steps are also included:

[0091] S61. Set the three-dimensional laser scanning parameters and scan the compensated outer cutting edge surface of the tool.

[0092] S62. Compare the surface shape obtained by the three-dimensional laser scanning system with the surface shape obtained by S61, select any cutting contact point in the neighborhood of the cutting edge on the outer edge surface of the tool, and project it into the extracted surface.

[0093] S63. Based on S62, obtain the expression for the actual normal error and the characterization diagram of the separation line of the tool's outer cutting surface.

[0094]

[0095] in, For the actual normal error, d p p is the actual coordinate offset at the cutting point p, p0 is the cutting and throwing endpoint, and n is the unit normal vector of the tool surface at the cutting point p.

[0096] S64. Based on the structural features of the beaver's lower incisors, the structural features of the outer cutting edge surface of the tool are further compensated and reconstructed.

[0097] S65. Based on the dynamic variation characteristics of the sliding angle of the beaver's lower incisors, select the range of the main sliding angle τ = 12°~55°;

[0098] S66. Based on the specific characteristics of the beaver lower incisor crown, the outer circle of the transition is compensated by the outer edge contour of the labial side of the tooth root, the outer edge tangent is compensated by the overall outer edge contour of the labial crown, the front main edge tangent is compensated by the inner ridge, the outer edge length is compensated by the connecting length of the outer ridge, and the rear main edge tangent is compensated by the outer ridge.

[0099] S67. By correcting the geometric parameters, the optimized feature points A'-D' are obtained, where A' is the top feature point of the outer cutting surface of the tool, which is obtained by compensating and correcting the original point A; B' is the middle feature point of the outer cutting surface of the tool, corresponding to the original point B; C' is the feature point where the outer cutting surface of the tool connects with the tool holder, corresponding to the original point C; and D' is the dividing point between the end of the outer cutting surface of the tool and the tool holder, corresponding to the original point D. Connect A'-D' in sequence and smoothly transition to form the outer cutting surface of the tool.

[0100] S68. Use the least squares method to solve for the correction amount and update the tool outer surface model.

[0101] The beneficial effects of this invention are:

[0102] This invention provides a tool outer surface compensation method based on differential geometry. Through Gauss mapping, it achieves precise matching between the tool outer surface and the material fracture surface normal space. Combined with principal curvature analysis and slip error compensation, it significantly improves material chopping quality. Based on constraint optimization of axis tilt angle and boundary angle, it geometrically eliminates the risk of interference between moving and fixed tools, improving the stability of continuous equipment operation. Utilizing NUBRS surface parametric modeling to optimize the blade surface curvature distribution reduces local stress concentration in the tool, lowering the frequency of sharpening and replacement costs. The multi-coordinate parametric model and 3D laser scanning error separation technology can flexibly adapt to different feed thicknesses, covering equipment such as forage harvesters. It ensures that the normal vectors of the tool outer cutting surface and the material contact surface are the same, reducing stubble formation from a geometrical perspective, and achieving precise description and compensation of the dynamic contact trajectory, ensuring consistent cutting quality under different working conditions. It reduces the concentration of contact pressure, lowers the risk of blade wear and material tearing, enables fine correction of local curved surfaces, improves the safety of equipment operation, and ultimately effectively improves the stability of cutting quality and increases the harvesting efficiency of high-speed rolling cutting equipment such as forage harvesters. Attached Figure Description

[0103] Figure 1 This is a schematic diagram illustrating the steps of the tool external surface compensation method based on differential geometry according to the present invention.

[0104] Figure 2 This is the present invention. Figure 1 A schematic diagram of the sub-steps in step S1;

[0105] Figure 3 This is the present invention. Figure 1 A schematic diagram of the sub-steps in step S2;

[0106] Figure 4 This is the present invention. Figure 1 A schematic diagram of the sub-steps in step S3;

[0107] Figure 5 This is the present invention. Figure 1 A schematic diagram of the sub-steps in step S4;

[0108] Figure 6 This is the present invention. Figure 1 A schematic diagram of the sub-steps in step S5;

[0109] Figure 7 This is the present invention. Figure 1 A schematic diagram of the sub-steps in step S6;

[0110] Figure 8 This is a schematic diagram illustrating the optimization of the outer curved surface boundary curve of the cutting tool according to the present invention;

[0111] Figure 9 This is a schematic diagram of the cutting of the outer curved surface of the tool in this invention;

[0112] Figure 10 This is a schematic diagram of the tool's outer curved surface before optimization in this invention;

[0113] Figure 11 This is a schematic diagram of the optimized compensation for the outer curved surface of the cutting tool in this invention. Detailed Implementation

[0114] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0115] Example 1: As Figures 1-9 As shown, this invention provides a tool outer surface compensation method based on differential geometry, comprising the following steps:

[0116] S1. Combining the discrete geometric features of the tool surface and NUBRS theory, construct a geometric parameterized model of the tool's outer cutting edge surface, and list the expansion formula of the tool's outer cutting edge surface to make the actual cutting edge line close to the theoretical trajectory line.

[0117] S2. Based on spinor theory, the shredding process is divided into two motions. Through normal vector equivalence, position equivalence, geometric equivalence, and motion equivalence, the equation expression of the tool's outer surface is obtained.

[0118] S3. Use differential geometry Guass mapping to project the tool surface generatrix and the moving tool trajectory surface onto the XZ plane, and perform Guass mapping on the original tool surface outer boundary generatrix.

[0119] S4. Define a dynamic rotation coordinate system. Extend the projection of the geometric points of the boundary curve of the tool's outer cutting edge onto this coordinate system to describe the contact relationship between the tool trajectory surface and the material fracture surface, and determine the minimum axis tilt angle.

[0120] S5. Perform differential compensation on the outer cutting surface of the tool and list the Gaussian mapping surface compensation expression between the tool surface and the material cutting section.

[0121] S6. After performing differential compensation on the outer cutting edge surface of the tool, verification and correction are carried out.

[0122] like Figure 2 As shown, further, in S1, the outer curved surface of the tool is a stretching surface, and the arc boundary curve is the outer cutting edge. It is assumed that the coordinate system T of the dynamic rotation center is defined as {O}. S -x S y S z S} is a transition coordinate system used to characterize the dynamic position of the hob's revolution center, where the origin O is... s Fixed at the center of rotation of the hob, z S The axis is along the direction of the hob's revolution axis, x SThe axis is fed radially along the reference direction of the hob, y S axis and x S z S The axes are perpendicular to each other and the local coordinate system M {o of the outer edge m -x m y m z m A local coordinate system is used to characterize the geometry of the tool's outer cutting edge surface and the position of the cutting point. Integrating the discrete geometric features of the tool surface and NUBRS theory, a parameterized geometric model of the tool's outer cutting edge surface is constructed, and the expansion formula representing the tool's outer cutting edge surface is listed, making the actual cutting edge line close to the theoretical trajectory line. Step S1 also includes the following sub-steps:

[0123] S11. Define the coordinate system T{O} of the moving rotation center. S -x S y S z S} and the local coordinate system M{o of the outer edge m -x m y m z m} is used to characterize the local geometric features of the outer cutting edge surface of a cutting tool;

[0124] S12. List the tool surface expression in the coordinate system set in S11, extract the key geometric parameters of the tool's outer cutting edge surface, and simplify the cutting edge to a straight line;

[0125] In T{O S -x S y S z S The expression for the tool surface in the coordinate system is:

[0126]

[0127] Where l is the generatrix of the blade's curved surface, θ w It is a curved latitude line;

[0128] In M{o m -x m y m z m The approximate expression for the tool surface in the coordinate system is as follows: Where β is the tool axis tilt angle, The transformation matrix for the original surface deflected by β around the z-axis;

[0129] S13. Considering the assembly requirements of the inclined tool installation, γ1 and γ2 are defined as the boundary angles between the inner and outer cutting surfaces of the tool arc and the tool tip. These angles define the curvature and length of the inner and outer cutting surfaces of the tool, and the baseline of the tool arc is d. s The inner and outer cutting edge radius of the tool is R. CThe tool axis inclination angle is β, the lowest point of the outer cutting edge generatrix is ​​C, and the tool thickness is limited by point D on the inner cutting edge generatrix.

[0130] S14. List the geometric expression of the tool generatrix and discretize it into multiple NUBRS circular arc transition curves.

[0131]

[0132] Where, l∈[R] c ·((cosβ)-cos(β-γ1)), Rc·((cosβ)-cos(β+γ2))], d s R is the baseline of the tool arc surface. c The radius of the inner and outer cutting edge arc surfaces of the cutting tool;

[0133] S15. Combining with S13, construct the parameterized equation of the generatrix as Q(l) to simplify the parameterized representation of the surface;

[0134] S16. Transform and correct the coordinate system, and list the extended expression for the external cutting edge surface of the tool.

[0135]

[0136] in, This is the parametric expression for the outer cutting edge surface of the tool in the coordinate system T of the moving rotation center. Z represents the latitude parameter of the arc surface, and Z represents the extended stretching surface.

[0137] like Figure 3 As shown, further, in step S2, the shredding process is divided into two motions based on spinor theory. The equation expression for the tool's outer surface is obtained through normal vector equivalence, position equivalence, geometric equivalence, and motion equivalence. The step also includes the following sub-steps:

[0138] S21. 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.

[0139] S22. Based on the principle of equivalence of normal vectors, list the expression for the normal vector at the cutting point of the tool;

[0140]

[0141] in, , ω1=[0,0,1] T ω2=[0,1,0] T p0 = [p1, 0, p3] T , n=[n x ,n y ,n z ,0]T This represents the normal vector of the tool surface at the cutting point in the T coordinate system. It is an interior angle. Let n0 be the exterior deflection angle, then n0 = [-1, 0, 0] T , The unit vector of the slip deflection axis. Let q be the unit vector of the sliding and throwing axis, q be the position vector of the sliding and throwing axis b-axis, p1 and p3 be the coordinate components of the throwing endpoint p0 in the coordinate system T of the dynamic rotation center, and n be the coordinate components of the throwing endpoint p0 in the coordinate system T of the dynamic rotation center. x ,n y ,n z These are the normal vector components of the tool surface;

[0142] S23. According to the principle of geometric equivalence, in order to ensure the geometric features and basic properties of the neighborhood of the cutting point p for the quality of the shredded segments, it is concluded that the principal curvature of the tool surface and the principal curvature of the material are the same.

[0143] S24. Based on the principles of motion equivalence and equivalent cutting, list information about x. s From the expressions for the axisymmetric distribution of the meridian and the meridian radius function, we can obtain the expression for the principal curvature R of the meridian on the tool surface.

[0144] Regarding x s The expression for an axisymmetric meridian is as follows:

[0145]

[0146] In the formula, c1 is the paraxial curvature of the tool surface meridian, c = 1 / R0, and R0 is the radius of curvature of the tool surface at the cutting edge; k n Let k be the eccentricity function of the tool surface about the cutting edge. n =-e 2 e is the eccentricity of the tool surface about the cutting edge, used to control the degree of surface opening; P i This is the deviation coefficient between the inner and outer curved surfaces of the cutting tool;

[0147] The expression for the meridian radius function is as follows:

[0148]

[0149] Combining these equations, we can obtain the expression for the principal curvature R of the tool surface meridian as follows:

[0150]

[0151] Where q = [q1, 0, q3] T q1 is the x-axis in coordinate system T. S The offset in the axial direction, q3 is its value in the z-axis. S Height in the axial direction, For tool generatrix parameters, Let l be the radial dimension of the generatrix at parameter l. The rotation angle around the sliding deflection axis a during the tool's compound motion;

[0152] S25. Determine the position q of the cutting point p corresponding to the b-axis, and adjust the parameters kn and higher-order terms B. i Define the equation expression for the meridian of the external curved surface;

[0153]

[0154] S26, combining with S25, list the equation expression for the outer surface of the tool;

[0155] .

[0156] like Figure 4 As shown, further, differential geometry Guass mapping is used to project the tool surface generatrix and the moving tool trajectory surface onto the XZ plane, and Guass mapping is performed on the original tool surface outer boundary generatrix. Step S3 also includes the following sub-steps:

[0157] S31. Define the Gaussian mapping for the tool's own surface and the surface of the tool's motion trajectory, and determine the source and target of the Gaussian mapping;

[0158] S32. List the Gaussian mapping expression between the tool surface and the material fracture surface;

[0159]

[0160]

[0161] Where, n(l,θ) w ), n(t, ) is the surface normal vector, Q l Q θ R t , These are the tangent vectors of the surface along the parametric coordinates l, θ, t, and γ, respectively.

[0162] S33. Define the original surface of the tool as a regular surface, and the tool path and tool marks, i.e. the material fracture surface, as projected surfaces.

[0163] S34. Analyze the Gaussian mapping regions corresponding to the two endpoints A and B of the cutting tool and the boundary points A' and B' of the cutting edge trajectory surface during the cutting operation.

[0164] S35. Project the moving tool trajectory surface mapped by the tool surface generatrix and Gauss onto the XZ plane. When only considering the contact of the tool's outer cutting edge, perform Gauss mapping on the original tool surface outer boundary generatrix and confirm the new mapping point.

[0165] S36. Summarize the relationship between the inner and outer offset angles ψ1 and ψ2 after projection and the inner and outer edge boundary angles γ1 and γ2 as follows: γ2>ψ1, γ1>γ;

[0166] S37. Project the Gaussian mapping in the XZ plane onto O. s -x s y s In the plane, analyze the relationship between the axis tilt angle β and the outer edge boundary angle γ2.

[0167] like Figure 5 As shown, further, in O s -x s y s The geometric points of the boundary curve of the tool's outer cutting edge are extended and projected onto the plane to describe the contact relationship between the tool trajectory surface and the material fracture surface, and to determine the minimum axial tilt angle β. S4 also includes the following sub-steps:

[0168] S41. Characterize the position and distribution of the normal vector at the cut point p after mapping in the local coordinate system {p-e1e2};

[0169] S42. Combining with S41, approximate the surface in the neighborhood of the cutting point p, list the approximate expression of the surface, and determine that the approximate surface of the tool surface near the cutting point must be a convex hyperbolic paraboloid.

[0170]

[0171] in, The principal curvature of the boundary line of the outer cutting edge of the tool;

[0172] S43. Based on the mathematical relationship between the normal curvature of the latitude and longitude lines and the envelope line on the tool surface, analyze the principal curvature constraint conditions and the approximate shape of the neighborhood surface corresponding to the different positions of the cutting point p on the tool curve.

[0173] S44, limiting the principal curvature along the meridian direction The minimum value should be greater than the maximum curvature of the material bending surface outside the fixed blade;

[0174] S45. Based on the conclusions obtained in S45, calculate the principal curvature of the boundary line of the tool's outer cutting surface using the deviation and induction method.

[0175]

[0176] in, Let p be the distance from the cutting point p to the axis of the tool surface;

[0177] S46. Based on the Gaussian mapping analysis of the unit spherical coverage area, determine the minimum axial tilt angle β = 25°;

[0178] Gaussian mapping analysis is used to analyze the unit spherical coverage area to ensure the tool normal space G. Q Includes the material cross-section normal vector space G R Given that γ1 < β, we take γ1 = 12°, sinβ > 0.2 × 10 × sin12.5° ≈ 0.4158, which means β > 24.56°. Taking β = 25° satisfies the constraint requirements.

[0179] like Figure 6 As shown, further, differential compensation is performed on the outer cutting edge surface of the tool, and the Gaussian mapping surface compensation expression between the tool surface and the material cutting section is listed. S5 also includes the following sub-steps:

[0180] S51. Based on the corresponding neighborhoods D1 and D2 of the cutting point p on the material fracture surface and the cutting tool surface, derive the expression for calculating the local cutting ratio of point p.

[0181]

[0182] Where G is the local cutting ratio during tool operation; A1 and A2 are the areas corresponding to D1 and D2, respectively, in mm. 2 h1 is the thickness of the feed layer, mm; h2 is the material cutting thickness in the normal direction at point p, mm;

[0183] S52. Analyze the surface S containing the contact point p between the outer cutting edge of the tool and the cut material cross-section, and calculate the area A of the cutting domain D mapped onto a unit sphere. g The Gaussian curvature expression between A and B;

[0184]

[0185] Where K is the Gaussian curvature, K = , The principal curvature of the surface;

[0186] S53. Combining S52 with the material fracture surface and tool surface at the cutting point p to define the Guass curvature, we obtain the formula for the cuttable feed layer thickness at point p.

[0187]

[0188] Wherein, K1(p) and K2(p) are the curvature values ​​of the tool surface at the cutting point p along two mutually perpendicular principal directions;

[0189] S54. Considering the high-speed rolling motion of the tool, the cutting marks on the same weft line on the outer edge of the tool are evenly distributed along the main axis of the hob.

[0190] S55. List the Gaussian curvature expressions for the material fracture surface and the tool surface;

[0191]

[0192]

[0193] Where D1 and D2 are the corresponding neighborhoods of the cutting point p on the material fracture surface and the tool surface, respectively. Let be the Gaussian curvature of the fracture surface of the material. The principal curvature of the fracture surface of the material is denoted as . Let be the principal curvature of the tool surface. For the Gaussian mapping of a local region of the material fracture surface, Gaussian mapping for a local region of the tool surface;

[0194] S56. Combining S51 and S53, we obtain the expression for the local cutting ratio;

[0195]

[0196] in, Let be the Gaussian curvature of the fracture surface of the material. Let be the Gaussian curvature of the tool surface;

[0197] S57. List the Gaussian mapping surface compensation expression between the tool surface and the material cutting section after considering slip offset.

[0198]

[0199] in, This represents the normal vector mapping results for the tool surface region H and the fracture surface region H. , The normal vector mapping results for tool surface segment I and fracture surface segment I are shown. This represents the normal vector mapping results of the tool surface at point p and the fracture surface at point p. For the Gaussian mapping of a local region of the material fracture surface, For the Gaussian mapping of a local region of the tool surface, For material feeding speed, This represents the rotational speed of the hob.

[0200] like Figure 7As shown, further, in step S6, after performing differential compensation on the outer cutting edge surface of the tool and then verifying and correcting it, the following sub-steps are also included:

[0201] S61. Set the three-dimensional laser scanning parameters and scan the compensated outer cutting edge surface of the tool.

[0202] S62. Compare the surface shape obtained by the three-dimensional laser scanning system with the surface shape obtained by S61, select any cutting contact point in the neighborhood of the cutting edge on the outer edge surface of the tool, and project it into the extracted surface.

[0203] S63. Based on S62, obtain the expression for the actual normal error and the characterization diagram of the separation line of the tool's outer cutting surface.

[0204]

[0205] in, For the actual normal error, d p p is the actual coordinate offset at the cutting point p, p0 is the cutting and throwing endpoint, and n is the unit normal vector of the tool surface at the cutting point p.

[0206] S64. Based on the structural features of the beaver's lower incisors, the structural features of the outer cutting edge surface of the tool are further compensated and reconstructed.

[0207] S65. Based on the dynamic variation characteristics of the sliding angle of the beaver's lower incisors, select the range of the main sliding angle τ = 12°~55°;

[0208] S66. Based on the specific characteristics of the beaver lower incisor crown, the outer circle of the transition is compensated by the outer edge contour of the labial side of the tooth root, the outer edge tangent is compensated by the overall outer edge contour of the labial crown, the front main edge tangent is compensated by the inner ridge, the outer edge length is compensated by the connecting length of the outer ridge, and the rear main edge tangent is compensated by the outer ridge.

[0209] S67. By correcting the geometric parameters, the optimized feature points A'-D' are obtained, where A' is the top feature point of the outer cutting surface of the tool, which is obtained by compensating and correcting the original point A; B' is the middle feature point of the outer cutting surface of the tool, corresponding to the original point B; C' is the feature point where the outer cutting surface of the tool connects with the tool holder, corresponding to the original point C; and D' is the dividing point between the end of the outer cutting surface of the tool and the tool holder, corresponding to the original point D. Connect A'-D' in sequence and smoothly transition to form the outer cutting surface of the tool.

[0210] S68. Use the least squares method to solve for the correction amount and update the tool outer surface model.

[0211] like Figure 8As shown, the tool path surface mapped by Gaussian mapping of the tool surface generatrix is ​​projected onto the XZ plane. The BA curve represents the ideal tool path. Considering only the contact of the tool's outer cutting edge, Gaussian mapping is performed on the original tool surface's inner and outer boundary generatrixes respectively. The outer cutting edge boundary point A is mapped to point A, the inner cutting edge boundary point D is Gaussian mapped to point E, and the outer cutting edge boundary point B is mapped to point F, resulting in a new point B. The g-axis diverges from the origin... Q (a) g Q (b) Forming inner edge boundary angle γ1 and outer edge boundary angle γ2 with the OZ coordinate axis respectively, g R (A'), g R (B') forms angles with the OZ coordinate axes, namely the inner offset angle ψ1 and the outer offset angle ψ2. There must exist a relationship between each projection vector: γ2>ψ1, γ1>γ. When the axis tilt angle β is fixed, the larger γ1 is, the larger the curvature of the tool surface generatrix, and the larger the material allowance that can be shredded. Adjusting the position of the geometric parameter points near the tool tip on the outer cutting surface can change the vector space distribution during curve segment projection. When γ1=γ2, the tool boundary and generatrix configuration corresponding to the axis tilt angle β are reasonable, and the space between the tool cutting normal vector and the projection normal vector remains unchanged. To minimize the slippage during dynamic cutting, a one-to-one mapping between the inner and outer cutting surfaces of the tool and the trajectory envelope surface is required. Simultaneously, a certain radial clearance must be maintained between all cutting points on the tool surface and the projection points on the trajectory surface to ensure no interference at the fixed tool location.

[0212] like Figure 9 As shown, as the cutting penetration depth of the tool changes, the cutting point p gradually moves from the tip of the tool to the curved surface of the cutting edge. Due to the high-speed rolling motion of the tool, the cutting marks on the same weft line on the outer edge of the tool are evenly distributed along the main axis of the hob.

[0213] like Figure 10 As shown, before optimization, the outer surface of the tool is a discontinuous surface spliced ​​with multiple circular arcs, connected by four feature points (marked with circles). The generatrix has an irregular broken line transition, the axis tilt angle β fluctuates randomly, and the overall surface is uneven and the transition is not smooth, with obvious geometric discreteness.

[0214] like Figure 11As shown, after optimization, a parametric curve is constructed, connecting A′B′, B′C′, and C′D′ sequentially and smoothly transitioning to form the outer cutting surface of the tool. The inner and outer cutting surfaces enclose and envelop the shredding curved blade head. The curve DB segment, after compensation, is simplified to a straight line D′B′. Since the tool is mounted obliquely on the tool holder by relying on the tool pad and fastening bolts, the compensated straight line D′B′ still has the function of clamping the material with positive pressure and providing positive pressure. Moreover, the simplified straight line shape is more suitable for the tool roller structure. The overall shape has a regular quadrilateral-like contour, and the continuity and smoothness of the curved surface are significantly improved. This reflects the results of the collaborative optimization of parameters such as curvature and normal vector of the tool's outer curved surface based on differential geometry methods, which can achieve the standardization of geometric parameters and the consistency of surface performance.

[0215] In this embodiment, based on the existing irregularly shaped roller cutter outer curved surface of forage harvesters, differential geometry theory is used as the core basis, and techniques such as Gauss mapping, principal curvature matching, and error separation are employed for surface optimization design. Based on the geometric characteristics of the dynamic contact between the cutter and material, the NUBRS surface parametric characterization method is introduced. By adjusting key parameters such as the axis tilt angle β, inner and outer edge boundary angles γ1 and γ2, and the arc surface contour radius Rc, the normal vector distribution and sliding trajectory of the cutter's outer curved surface are controlled. This achieves normal vector coordination between the cutter's outer cutting surface and the material's fracture surface, reducing cutting interference and slippage errors, making it easier to accurately cut high-toughness fiber materials, improving the operating efficiency of forage harvesting machinery, and simultaneously optimizing the cutter's cutting edge stress distribution, thus enhancing the cutter's wear resistance and service life.

[0216] This invention solves the problems of large cutting error, high stubble rate, easy interference between moving and fixed blades, high power consumption, and unstable cutting quality of existing forage harvesters with irregularly shaped roller blades. It improves the material cutting accuracy, extends the service life of the blades, and saves equipment operating costs.

[0217] This invention leverages the precision advantages of differential geometry in surface representation and optimization. Building upon existing irregularly shaped roller cutter outer surfaces, it introduces NUBRS surface discrete representation and Gaussian mapping compensation technology. Ultimately, after the chopped forage crop is fed in, the cutting blade achieves stable rolling and sliding cutting of the material through the optimized outer surface. Furthermore, differential compensation controls the principal curvature of the neighborhood at the cutting point, inducing the material's fracture cracks to propagate along an ideal direction, generating more uniform sliding cracks and making the material easier to cut. This increases the effective cutting contact area, reduces cutting resistance and stress concentration, significantly improves the blade's wear resistance, and enables one-time cutting of high-toughness forage stalks. It also reduces material slippage and stubble formation, improving the operational efficiency and quality stability of forage harvesting machinery. When dealing with forage materials of varying thickness and toughness, by adjusting parameters such as shaft tilt angle β, boundary angles γ1 and γ2, and arc surface radius Rc, combined with error separation results from three-dimensional laser scanning, a precise match between the outer curved surface of the cutter and the material characteristics can be achieved. This helps to minimize cutting resistance and reduce dynamic slippage error during chopping operations, ultimately achieving the goals of labor saving, low energy consumption, and uniform cutting. This reduces overall machine energy consumption, improves operating efficiency and material chopping quality, and provides a systematic technical approach for the high-precision design of the outer curved surface of irregularly shaped cutters in forage harvesters.

[0218] The specific embodiments 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 embodiments. Within the scope of knowledge possessed by those skilled in the art, various modifications and combinations can be made without departing from the essence of the present invention, and these modifications and combinations are still within the protection scope of the present invention.

Claims

1. A tool exterior surface compensation method based on differential geometry, characterized in that, Includes the following steps: S1. Combining the discrete geometric features of the tool surface and NUBRS theory, construct a geometric parameterized model of the tool's outer cutting edge surface, and list the expansion formula of the tool's outer cutting edge surface to make the actual cutting edge line close to the theoretical trajectory line. S2. Based on spinor theory, the shredding process is divided into two motions. Through normal vector equivalence, position equivalence, geometric equivalence, and motion equivalence, the equation expression of the tool's outer surface is obtained. S3. Use differential geometry Guass mapping to project the tool surface generatrix and the moving tool trajectory surface onto the XZ plane, and perform Guass mapping on the original tool surface outer boundary generatrix. S4. Define a dynamic rotation coordinate system. Extend the projection of the geometric points of the boundary curve of the tool's outer cutting edge onto this coordinate system to describe the contact relationship between the tool trajectory surface and the material fracture surface, and determine the minimum axis tilt angle. S5. Perform differential compensation on the outer cutting surface of the tool and list the Gaussian mapping surface compensation expression between the tool surface and the material cutting section. S6. After performing differential compensation on the outer cutting edge surface of the tool, verification and correction are carried out.

2. The tool outer surface compensation method based on differential geometry according to claim 1, characterized in that, In S1, the outer curved surface of the tool is a tension surface, and the arc boundary curve is the outer cutting edge. It is assumed that the coordinate system T of the dynamic rotation center is defined as {O}. S -x S y S z S } is a transition coordinate system used to characterize the dynamic position of the hob's revolution center, where the origin O is... s Fixed at the center of rotation of the hob, z S The axis is along the direction of the hob's revolution axis, x S The axis is fed radially along the reference direction of the hob, y S axis and x S z S The axes are perpendicular to each other and the local coordinate system of the outer edge M {o m -x m y m z m A local coordinate system is used to characterize the geometry of the tool's outer cutting edge surface and the position of the cutting point. Integrating the discrete geometric features of the tool surface and NUBRS theory, a parameterized geometric model of the tool's outer cutting edge surface is constructed, and the expansion formula representing the tool's outer cutting edge surface is listed, making the actual cutting edge line close to the theoretical trajectory line. Step S1 also includes the following sub-steps: S11. Define the coordinate system T{O} of the moving rotation center. S -x S y S z S } and the local coordinate system M{o of the outer edge m -x m y m z m } is used to characterize the local geometric features of the outer cutting edge surface of a cutting tool; S12. List the tool surface expression in the coordinate system set in S11, extract the key geometric parameters of the tool's outer cutting edge surface, and simplify the cutting edge to a straight line; In T{O S -x S y S z S The expression for the tool surface in the coordinate system is: ; Where l is the generatrix of the blade's curved surface, θ w It is a curved latitude line; In M{o m -x m y m z m The approximate expression for the tool surface in the coordinate system is as follows: Where β is the tool axis tilt angle, The transformation matrix for the original surface deflected by β around the z-axis; S13. Considering the assembly requirements of the inclined tool installation, γ1 and γ2 are defined as the boundary angles between the inner and outer cutting surfaces of the tool arc and the tool tip. These angles define the curvature and length of the inner and outer cutting surfaces of the tool, and the baseline of the tool arc is d. s The inner and outer cutting edge radius of the tool is R. C The tool axis inclination angle is β, the lowest point of the outer cutting edge generatrix is ​​C, and the tool thickness is limited by point D on the inner cutting edge generatrix. S14. List the geometric expression of the tool generatrix and discretize it into multiple NUBRS circular arc transition curves. ; Where, l∈[R] c ·((cosβ)-cos(β-γ1)), Rc·((cosβ)-cos(β+γ2))], d s R is the baseline of the tool arc surface. c The radius of the inner and outer cutting edge arc surfaces of the cutting tool; S15. Combining with S13, construct the parameterized equation of the generatrix as Q(l) to simplify the parameterized representation of the surface; S16. Transform and correct the coordinate system, and list the extended expression for the external cutting edge surface of the tool. ; in, This is the parametric expression for the outer cutting edge surface of the tool in the coordinate system T of the moving rotation center. Z represents the latitude parameter of the arc surface, and Z represents the extended stretching surface.

3. The tool outer surface compensation method based on differential geometry according to claim 1, characterized in that, In step S2, the shredding process is divided into two motions based on spinor theory. The equation expression for the outer surface of the tool is obtained through normal vector equivalence, position equivalence, geometric equivalence, and motion equivalence. The step also includes the following sub-steps: S21. 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. S22. Based on the principle of equivalence of normal vectors, list the expression of the normal vector at the cutting point of the tool; ; in, , ω1=[0,0,1] T ω2=[0,1,0] T p0 = [p1, 0, p3] T , n=[n x ,n y ,n z ,0] T This represents the normal vector of the tool surface at the cutting point in the T coordinate system. It is an interior angle. Let n0 be the exterior deflection angle, then n0 = [-1, 0, 0] T , The unit vector of the slip deflection axis. Let q be the unit vector of the sliding and throwing axis, q be the position vector of the sliding and throwing axis b-axis, p1 and p3 be the coordinate components of the throwing endpoint p0 in the coordinate system T of the moving rotation center, and n be the coordinate components of the throwing endpoint p0 in the coordinate system T of the moving rotation center. x ,n y ,n z These are the normal vector components of the tool surface; S23. According to the principle of geometric equivalence, in order to ensure the geometric features and basic properties of the neighborhood of the cutting point p for the quality of the shredded segments, it is concluded that the principal curvature of the tool surface and the principal curvature of the material are the same. S24. Based on the principles of motion equivalence and equivalent cutting, list information about x. s From the expressions for the axisymmetric distribution of the meridian and the meridian radius function, we can obtain the expression for the principal curvature R of the meridian on the tool surface. Regarding x s The expression for an axisymmetric meridian is as follows: ; In the formula, c1 is the paraxial curvature of the tool surface meridian, c = 1 / R0, and R0 is the radius of curvature of the tool surface at the cutting edge; k n Let k be the eccentricity function of the tool surface about the cutting edge. n =-e 2 e is the eccentricity of the tool surface about the cutting edge, used to control the degree of surface opening; P i This is the deviation coefficient between the inner and outer curved surfaces of the cutting tool; The expression for the meridian radius function is as follows: ; Combining these equations, we can obtain the expression for the principal curvature R of the tool surface meridian as follows: ; Where q = [q1, 0, q3] T q1 is the x-axis of the b-axis in coordinate system T. S The offset in the axial direction, q3 is its value in the z-axis. S Height in the axial direction, For tool generatrix parameters, Let l be the radial dimension of the generatrix at parameter l. The rotation angle around the sliding deflection axis a during the tool's compound motion; S25. Determine the position q of the cutting point p corresponding to the b-axis, and adjust the parameters kn and higher-order terms B. i Define the equation expression for the meridian of the external curved surface; ; S26, combining with S25, list the equation expression for the outer surface of the tool; 。 4. The tool outer surface compensation method based on differential geometry according to claim 1, characterized in that, Differential geometry Guass mapping is used to project the tool surface generatrix and the moving tool trajectory surface onto the XZ plane, and Guass mapping is performed on the original tool surface outer boundary generatrix. Step S3 also includes the following sub-steps: S31. Define the Gaussian mapping for the tool's own surface and the surface of the tool's motion trajectory, and determine the source and target of the Gaussian mapping; S32. List the Gaussian mapping expression between the tool surface and the material fracture surface; ; ; Where, n(l,θ) w ), n(t, Let Q be the surface normal vector. l Q θ R t , These are the tangent vectors of the surface along the parametric coordinates l, θ, t, and γ, respectively. S33. Define the original surface of the tool as a regular surface, and the tool path and tool marks, i.e. the material fracture surface, as projected surfaces. S34. Analyze the Gaussian mapping regions corresponding to the two endpoints A and B of the cutting tool and the boundary points A' and B' of the cutting edge trajectory surface during the cutting operation. S35. Project the moving tool trajectory surface mapped by the tool surface generatrix and Gauss onto the XZ plane. When only considering the contact of the tool's outer cutting edge, perform Gauss mapping on the original tool surface outer boundary generatrix and confirm the new mapping point. S36. Summarize the relationship between the inner and outer offset angles ψ1 and ψ2 after projection and the inner and outer edge boundary angles γ1 and γ2 as follows: γ2>ψ1, γ1>γ; S37. Project the Gaussian mapping in the XZ plane onto O. s -x s y s In the plane, analyze the relationship between the axis tilt angle β and the outer edge boundary angle γ2.

5. The tool outer surface compensation method based on differential geometry according to claim 1, characterized in that, In O s -x s y s The geometric points of the boundary curve of the tool's outer cutting edge are extended and projected onto the plane to describe the contact relationship between the tool trajectory surface and the material fracture surface, and to determine the minimum axial tilt angle β. S4 also includes the following sub-steps: S41. Characterize the position and distribution of the normal vector at the cut point p after mapping in the local coordinate system {p-e1e2}; S42. Combining with S41, approximate the surface in the neighborhood of the cutting point p, list the approximate expression of the surface, and determine that the approximate surface of the tool surface near the cutting point must be a convex hyperbolic paraboloid. ; in, The principal curvature of the boundary line of the outer cutting edge of the tool; S43. Based on the mathematical relationship between the normal curvature of the latitude and longitude lines and the envelope line on the tool surface, analyze the principal curvature constraint conditions and the approximate shape of the neighborhood surface corresponding to the different positions of the cutting point p on the tool curve. S44, limiting the principal curvature along the meridian direction The minimum value should be greater than the maximum curvature of the material bending surface outside the fixed blade; S45. Based on the conclusions obtained in S45, calculate the principal curvature of the boundary line of the tool's outer cutting surface using the deviation and induction method. ; in, Let p be the distance from the cutting point p to the axis of the tool surface; S46. Based on the Gaussian mapping analysis of the unit spherical coverage area, determine the minimum axial tilt angle β = 25°; Gaussian mapping analysis is used to analyze the unit spherical coverage area to ensure the tool normal space G. Q Includes the material cross-section normal vector space G R Given that γ1 < β, we take γ1 = 12°, sinβ > 0.2 × 10 × sin12.5° ≈ 0.4158, which means β > 24.56°. Taking β = 25° satisfies the constraint requirements.

6. The tool outer surface compensation method based on differential geometry according to claim 1, characterized in that, Differential compensation is performed on the outer cutting surface of the tool, and the Gaussian mapping surface compensation expression between the tool surface and the material cutting section is listed. S5 also includes the following sub-steps: S51. Based on the corresponding neighborhoods D1 and D2 of the cutting point p on the material fracture surface and the cutting tool surface, derive the expression for calculating the local cutting ratio of point p. ; Where G is the local cutting ratio during tool operation; A1 and A2 are the areas corresponding to D1 and D2, respectively, in mm. 2 h1 is the thickness of the feed layer, mm; h2 is the material cutting thickness in the normal direction at point p, mm; S52. Analyze the surface S containing the contact point p between the outer cutting edge of the tool and the cut material cross-section, and calculate the area A of the cutting domain D mapped onto a unit sphere. g The Gaussian curvature expression between A and B; ; Where K is the Gaussian curvature, K = , The principal curvature of the surface; S53. Combining S52 with the material fracture surface and tool surface at the cutting point p to define the Guass curvature, we obtain the formula for the thickness of the cuttable feed material layer at point p. ; Wherein, K1(p) and K2(p) are the curvature values ​​of the tool surface at the cutting point p along two mutually perpendicular principal directions; S54. Considering the high-speed rolling motion of the tool, the cutting marks on the same weft line on the outer edge of the tool are evenly distributed along the main axis of the hob. S55. List the Gaussian curvature expressions for the material fracture surface and the tool surface; ; ; Where D1 and D2 are the corresponding neighborhoods of the cutting point p on the material fracture surface and the tool surface, respectively. Let be the Gaussian curvature of the fracture surface of the material. The principal curvature of the fracture surface of the material is denoted as . Let be the principal curvature of the tool surface. For the Gaussian mapping of a local region of the material fracture surface, Gaussian mapping of a local region of the tool surface; S56. Combining S51 and S53, we obtain the expression for the local cutting ratio; ; in, Let be the Gaussian curvature of the fracture surface of the material. Let be the Gaussian curvature of the tool surface; S57. List the Gaussian mapping surface compensation expression between the tool surface and the material cutting section after considering slip offset. ; in, This represents the normal vector mapping results for the tool surface region H and the fracture surface region H. , The normal vector mapping results for tool surface segment I and fracture surface segment I are shown. This represents the normal vector mapping results of the tool surface at point p and the fracture surface at point p. For the Gaussian mapping of a local region of the material fracture surface, For the Gaussian mapping of a local region of the tool surface, For material feeding speed, This represents the rotational speed of the hob.

7. The tool outer surface compensation method based on differential geometry according to claim 1, characterized in that, In step S6, after performing differential compensation on the outer cutting edge surface of the tool and then verifying and correcting it, the following sub-steps are also included: S61. Set the three-dimensional laser scanning parameters and scan the compensated outer cutting edge surface of the tool. S62. Compare the surface shape obtained by the three-dimensional laser scanning system with the surface shape obtained by S61, select any cutting contact point in the neighborhood of the cutting edge on the outer edge surface of the tool, and project it into the extracted surface. S63. Based on S62, obtain the expression for the actual normal error and the characterization diagram of the separation line of the tool's outer cutting surface. ; in, For the actual normal error, d p p is the actual coordinate offset at the cutting point p, p0 is the cutting and throwing endpoint, and n is the unit normal vector of the tool surface at the cutting point p. S64. Based on the structural features of the beaver's lower incisors, the structural features of the outer cutting edge surface of the tool are further compensated and reconstructed. S65. Based on the dynamic change characteristics of the sliding angle of the beaver's lower incisors, select the range of the main sliding angle τ = 12°~55°; S66. Based on the specific characteristics of the beaver lower incisor crown, the outer circle of the transition is compensated by the outer edge contour of the labial side of the tooth root, the outer edge tangent is compensated by the overall outer edge contour of the labial crown, the front main edge tangent is compensated by the inner ridge, the outer edge length is compensated by the connecting length of the outer ridge, and the rear main edge tangent is compensated by the outer ridge. S67. By correcting the geometric parameters, the optimized feature points A'-D' are obtained, where A' is the top feature point of the outer cutting surface of the tool, which is obtained by compensating and correcting the original point A; B' is the middle feature point of the outer cutting surface of the tool, corresponding to the original point B; C' is the feature point where the outer cutting surface of the tool connects with the tool holder, corresponding to the original point C; and D' is the dividing point between the end of the outer cutting surface of the tool and the tool holder, corresponding to the original point D. Connect A'-D' in sequence and smoothly transition to form the outer cutting surface of the tool. S68. Use the least squares method to solve for the correction amount and update the tool outer surface model.

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