Method for calculating critical modification amount of face gear pair

Abstract

This invention discloses a method for calculating the critical profile modification amount of a face gear pair. First, the involute pinion undergoes bidirectional parabolic profile and tooth direction modification to establish a modified tooth surface model considering load deformation. Then, tooth surface contact analysis and load-bearing contact analysis are performed to obtain contact traces, contact stresses, and contact ellipse parameters. The core of this method lies in establishing an iterative calculation model based on the Newton-Raphson method. This involves solving for the geometric intersection points of the inner and outer diameter edge curves of the face gear and the pinion tooth surface, and using the convergence of this numerical iterative solution as the basis for determining the existence of edge contact. By incrementally increasing the pinion tooth direction modification amount parameter until the iterative solution changes from convergent to non-convergent, the minimum tooth direction modification amount required to eliminate load-bearing edge contact is accurately determined. This invention achieves quantitative calculation of the critical profile modification amount, overcomes the blindness of empirical design, effectively eliminates edge contact, improves tooth surface load distribution, and enhances the load-bearing capacity and service life of the face gear pair.

Description

Technical Field

[0001] This invention belongs to the field of gear design technology, specifically a method for calculating the critical modification amount of a face gear pair, mainly for the design and verification of face gear pair modification parameters. Background Technology

[0002] Face gear transmission is a unique form of gear transmission, in which the pinion is a cylindrical gear and the gear is a face gear with a conical tooth surface. This transmission boasts excellent characteristics such as no axial force on the pinion, compact structure, ability to achieve large transmission ratios, and multi-path splitting, making it promising for applications in high-power-density transmission fields such as helicopter main reducers and aero-engines.

[0003] However, when standard face gear pairs transmit heavy loads, due to factors such as tooth surface contact deformation, manufacturing and installation errors, and system deformation, contact easily occurs at the tooth tips, a phenomenon known as "edge contact." Edge contact leads to a sharp increase in contact stress in this area, resulting in severe stress concentration. This not only significantly reduces the contact fatigue strength and bending fatigue strength of the tooth surface but may also cause early failures such as tooth surface scuffing and spalling, severely restricting the improvement of the load-bearing capacity and service reliability of face gear transmissions.

[0004] To avoid edge contact, the tooth surface must be modified. Currently, extensive research has been conducted on gear modification, resulting in various technical solutions. For example, some methods optimize the tooth surface load distribution based on finite element simulation and trial-and-error adjustments, or find modification parameters with superior overall performance through parametric curves and intelligent algorithms. However, these methods generally suffer from limitations such as low computational efficiency, multiple optimization objectives, and reliance on iterative verification, making it difficult to accurately pinpoint the boundary values ​​of the modification amount required for the specific objective of "eliminating edge contact."

[0005] Specifically, in the design of face gear pairs, current technology and engineering practice still largely rely on empirical formulas and trial-and-error methods. Designers typically select modification parameters initially based on experience with similar operating conditions, and then check and evaluate the contact traces and contact stresses on the modified tooth surface through tooth contact analysis and load-bearing contact analysis. If edge contact is found, the modification parameters are adjusted based on experience and the analysis is repeated, and this process is repeated until the edge contact phenomenon disappears or weakens. This method lacks a quantitative calculation model for the modification amount required to precisely eliminate edge contact under different load and geometric parameter combinations, resulting in significant arbitrariness. Insufficient modification amount cannot completely avoid edge contact; excessive modification amount will lead to a reduction in effective contact area and an increase in contact stress in the middle of the tooth surface, which may weaken the overall load-bearing capacity of the gear. Therefore, how to scientifically and quantitatively determine the critical modification amount for avoiding edge contact in face gear pairs under specific operating conditions has become a key technical bottleneck for improving its design level and service performance. Summary of the Invention

[0006] In view of this, the purpose of this invention is to provide a method for calculating the critical tooth profile modification amount of a face gear pair. By establishing an iterative calculation model based on the Newton-Raphson method and using numerical convergence as the criterion, the critical tooth profile modification amount that just avoids edge contact is quantitatively calculated, thereby improving the load-bearing capacity and service life of the face gear pair.

[0007] To achieve the above objectives, the present invention provides the following technical solution: A method for calculating the critical modification amount of a face gear pair includes the following steps: S1. Tooth Surface Modeling: Based on the basic parameters of the pinion, a parabola is used to modify the tooth profile of the rack cutter, obtaining the tooth surface equation and normal vector of the rack cutter; based on the meshing principle, the tooth surface equation of the pinion after tooth profile modification is derived from the tooth surface of the rack cutter through coordinate transformation and relative motion velocity relationship; at the same time, a parabola is used to modify the tooth direction of the pinion, and the tooth surface equation of the pinion after tooth direction modification is obtained by offsetting the tooth surface point along the normal of the standard involute pinion tooth surface; S2. Tooth surface contact analysis: Based on the condition that the coordinates of the tooth surface points and the normal vector are equal at the meshing contact point between the face gear and the pinion, the contact point solution equation is established; the rotation angle of the pinion is used as the independent variable for iterative solution to obtain the complete contact trace of the face gear pair, and the single-tooth meshing area and double-tooth meshing area are determined. S3. Tooth surface bearing contact analysis: Based on the input torque and the distance from the contact point to the pinion axis, calculate the normal contact force in the single tooth meshing area; based on the principal curvature, principal direction and total deformation of the tooth surface at the contact point, calculate the major and minor axis dimensions of the contact ellipse under load conditions. S4. Iterative Calculation of Critical Modification Amount: Establish a set of equations to solve for the geometric intersection between the inner and outer diameter edge curves of the face gear tooth surface and the modified tooth surface of the pinion; use the Newton-Raphson method to iteratively solve the set of equations, and use the numerical convergence of the iterative solution as the basis for judging whether edge contact exists. The method is as follows: set a convergence threshold, and when the iterative solution meets the convergence condition, it is judged as convergent, indicating that edge contact exists; by gradually increasing the pinion tooth profile modification parameter, repeat the iterative calculation until the iterative solution cannot converge, then it is judged that the edge contact is eliminated, and the corresponding pinion tooth profile modification amount is the critical modification amount.

[0008] Furthermore, in step S1, the equation of the pinion tooth surface after profile modification is: in: The tooth surface of the pinion after tooth profile modification; Tool coordinate system To the pinion coordinate system The transformation matrix; Tool coordinate system The equation for the tooth surface of the rack cutter; This is the normal vector of the rack cutter tooth surface; The relative speed between the rack cutter and the pinion; , The two surface parameters for machining pinion rack and pinion tools correspond to the coordinate system respectively. The X coordinate below Coordinates; and: in: The coefficient of the parabolic curve for tooth profile modification; The pressure angle; coordinate system Below coordinate; This is the zero-point offset of the parabola; The distance from the vertex of the parabola to the origin. exist Distance in a direction.

[0009] Furthermore, in step S1, the equation of the pinion tooth surface after tooth profile modification is: in: The pinion tooth surface after double-shaped tooth profile modification; The tooth surface of the pinion after tooth profile modification; This is the unit normal vector of the pinion tooth surface after tooth profile modification; These are the profile modification parameters for the pinion teeth; , The two surface parameters for machining pinion rack and pinion tools correspond to the coordinate system respectively. The X coordinate below coordinate; This refers to the tooth width of the pinion.

[0010] Furthermore, in step S2, the contact point solution equation is established based on the tooth surface meshing condition, and is expressed as: in: The tooth surface of the pinion after tooth profile modification; This is the unit normal vector of the pinion tooth surface after tooth profile modification; The modified gear working tooth surface and tooth root transition surface formed by the envelope of the modified pinion; It is the unit normal vector of its corresponding tooth surface; From coordinate system To coordinate system The transformation matrix; , Let represent two parameters of the tooth surface of a face gear, where The X-coordinate in the coordinate system of the gear and rack tool for machining the surface. For the gear hobbing cutter's turning angle; and: in: This is the unit normal vector of the gear shaping cutter tooth surface; Here is the equation for the tooth surface of the gear shaper; The equation for the tooth surface of a face gear; For the gear shaping tool coordinate system To the gear coordinate system The transformation matrix; The relative speed between the face gear and the gear shaper cutter; For the gear hobbing cutter's angle; The X-coordinate is the coordinate system of the gear and rack tool used for machining the surface.

[0011] Furthermore, in step S2, the method for solving the single-tooth meshing region and double-tooth meshing region of the face gear pair is as follows: in: This is the critical point at which the double meshing zone transitions into the single meshing zone along the meshing direction; This is the critical point at which the single meshing zone enters the double meshing zone along the meshing direction; The engagement angle; This refers to the engagement angle; The degree of overlap; This represents the number of teeth on the pinion.

[0012] Furthermore, in step S3, the formula for calculating the normal contact force in the single-tooth meshing zone is: in: This is the normal contact force in the single-tooth meshing zone; This refers to the tangential contact force in the single-tooth meshing zone; Input torque; This is the distance from the contact point of the face gear pair to the axis of the pinion; and The contact point of the face gear pair in the follower coordinate system of the pinion The coordinates below; This is the pressure angle.

[0013] Furthermore, in step S3, the major axis of the contact ellipse is... With short axis The calculation formula is: in: This represents the total deformation of the two contacting tooth surfaces; The angle between the first principal directions of the two tooth surfaces; and These are the sum of the first and second principal curvatures of the two sides, respectively; and These are the differences between the first and second principal curvatures of the two surfaces, respectively; and: in: denoted by the normal curvature of the two surfaces; L, M, and N represent the first fundamental quantities of the tooth profile surface; E, F, and G represent the second fundamental quantities of the tooth profile surface. , is the independent variable of the surface equation.

[0014] Furthermore, in step S4, the set of equations used to determine edge contact is: in: The total deformation of the two contacting tooth surfaces; The angle of rotation of the face gear relative to the pinion caused by deformation; The pressure angle; This is the distance from the contact point to the axis of the face gear; This is an elliptic integral of the first kind; It is an elliptic integral of the second kind; Let be the eccentricity of the ellipse; and These are the sum of the first and second principal curvatures of the two sides, respectively; The elastic modulus of the pinion; The elastic modulus of the face gear; The Poisson's ratio for the pinion; For the face gear, Poisson's ratio is given. This represents the normal contact force in the single-tooth meshing zone.

[0015] Furthermore, the formula for iteratively solving the system of equations using the Newton-Raphson method is as follows: in: , and These are the two parameters of the pinion tooth surface equation after n iterations and the gear shaper rotation angle, respectively. It is a Jacobian matrix; The intersection point when the tooth surfaces of the face gear and the pinion are in edge contact. A 3rd order submatrix, and: in: This is the intersection point where the tooth surfaces of the face gear and the pinion are in edge contact. The pinion tooth surface after double-shaped tooth profile modification; coordinate system arrive The transformation matrix; , The two surface parameters for machining pinion rack and pinion tools correspond to the coordinate system respectively. The X coordinate below coordinate; For the gear hobbing cutter's angle; , and For the pinion tooth surface in The X, Y, and Z coordinates; , and The inner and outer diameter edge curves of the face gear in the coordinate system The X, Y, and Z coordinates; The inner and outer diameter edge curves of the face gear are represented as follows: in: The inner radius of the face gear; The outer radius of the face gear; If satisfied If the edge contact is found to be present, the solution is considered convergent; otherwise, the iterative solution fails to converge, and the edge contact is considered eliminated. This is the convergence threshold.

[0016] Furthermore, the Jacobian matrix calculated iteratively using the Newton-Raphson method is expressed as: .

[0017] The beneficial effects of this invention are as follows: The method for calculating the critical modification amount of face gear pairs in this invention has the following technical effects.

[0018] (1) Achieving precise quantitative calculation of critical profile modification amount: By establishing an iterative calculation model based on the Newton-Raphson method, and using numerical convergence as an objective criterion, the minimum tooth profile modification amount required to just eliminate load edge contact can be scientifically and quantitatively determined. This completely changes the traditional design mode that relies on experience-based trial and error and qualitative verification, and elevates the design of profile modification parameters from a cycle of "empirical estimation-repeated verification" to a precise process of "model calculation-one-time determination", significantly reducing design blindness and development cycle.

[0019] (2) Effective elimination of edge contact and optimization of load distribution: The critical modification amount determined by the method of this invention can ensure that the contact area of ​​the face gear pair is precisely controlled in the effective area in the middle of the tooth surface under the target working conditions, completely avoiding contact at the tooth tip or tooth root edge. This fundamentally eliminates the local stress concentration caused by edge contact, making the load distribution on the tooth surface more reasonable and uniform, laying a key foundation for improving the contact fatigue strength and bending fatigue strength of the gear.

[0020] (3) Maximizing load-bearing potential while ensuring performance: The method of this invention seeks a "critical" modification amount, that is, avoiding excessive modification while ensuring the elimination of edge contact. Excessive modification will reduce the effective contact area and may lead to increased stress in the middle of the tooth surface. By accurately locating the critical point, this invention achieves the elimination of harmful edge contact while preserving the complete contact load-bearing capacity of the tooth surface as much as possible, thereby optimizing and improving the load-bearing capacity and service life of the gear pair as a whole.

[0021] (4) Enhancing Design Reliability and Engineering Application Value: This invention provides a standardized and repeatable calculation process, the conclusions of which do not rely on personal experience, greatly improving the consistency and reliability of the design. It can be widely applied to the design of high-performance face gear transmissions in aerospace, high-end equipment and other fields, and has important engineering practical value for improving the power density, reliability and durability of transmission systems. Attached Figure Description

[0022] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration: Figure 1 This is a flowchart of the method for calculating the critical modification amount of the face gear pair according to the present invention; Figure 2 A schematic diagram of pinion tooth profile modification; Figure 3 A schematic diagram of pinion tooth profile modification; Figure 4 The contact trace of the gear tooth surface; Figure 5 For surface gear tooth surface bearing contact analysis; Figure 6 This refers to the relative position of the gear pair under no-load conditions; Figure 7 This refers to the relative position of the gear pair under heavy load conditions; Figure 8 This shows the contact condition of the unmodified gear pair. Figure 9 This describes the contact condition of the modified gear pair. Detailed Implementation

[0023] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0024] This embodiment proposes a method for calculating the critical modification amount of face gear pairs. This method addresses the technical problem that traditional face gear modification design relies on empirical trial and error, making it difficult to accurately determine the critical modification amount required to eliminate edge contact under different operating conditions. This improves the load-bearing capacity and service life of face gear pairs. Specifically, the method for calculating the critical modification amount of face gear pairs in this embodiment modifies the profile and tooth direction of the involute pinion in both directions, and introduces the relative rotation angle change caused by tooth surface contact deformation under load conditions to construct a modified tooth surface considering load deformation factors. An iterative calculation model based on Newton-Raphson is established to solve for the geometric intersection points of the inner and outer diameter edge curves of the face gear and the pinion tooth surface. Numerical convergence is used as the basis for judging whether the modification parameters meet the requirements for eliminating edge contact. By incrementally increasing the pinion tooth direction modification parameters when the iterative solution converges until the iterative solution becomes non-convergent, the critical tooth direction modification amount for reducing load edge contact is determined.

[0025] like Figure 1 As shown, the method for calculating the critical modification amount of the face gear pair in this embodiment includes the following steps.

[0026] S1. Tooth Surface Modeling: Based on the basic parameters of the pinion, a parabola is used to modify the tooth profile of the rack cutter, obtaining the tooth surface equation and normal vector of the rack cutter; based on the meshing principle, the tooth surface equation of the pinion after tooth profile modification is derived from the tooth surface of the rack cutter through coordinate transformation and relative motion velocity relationship; at the same time, a parabola is used to modify the tooth direction of the pinion, and the tooth surface equation of the pinion after tooth direction modification is obtained by offsetting the tooth surface point along the normal of the standard involute pinion tooth surface.

[0027] Existing technology shows that parabolic profile modification has a good vibration reduction effect. Therefore, in this embodiment, a parabolic profile is first used to modify the rack, and then the modified tooth surface of the pinion is calculated based on the meshing principle, such as... Figure 2 As shown.

[0028] coordinate system The equation for the tooth surface of the lower rack cutter is expressed as: in: The coefficient of the parabolic curve for tooth profile modification; The pressure angle; coordinate system Below coordinate; This is the zero-point offset of the parabola; The distance from the vertex of the parabola to the origin. exist Distance in direction. When When the value is zero, the pinion has a standard involute tooth profile.

[0029] rack cutter tooth surface normal vector It can be calculated using the following formula: Based on the meshing principle, the equation of the pinion tooth surface after tooth profile modification is... This can be expressed as: in: The tooth surface of the pinion after tooth profile modification; Tool coordinate system To the pinion coordinate system The transformation matrix; Tool coordinate system The equation for the tooth surface of the rack cutter; This is the normal vector of the rack cutter tooth surface; The relative speed between the rack cutter and the pinion.

[0030] Similarly, according to the meshing principle, the working tooth surface and the tooth root transition surface of the gear formed by the shaper cutter are... It can be calculated using the following formula: in: This is the unit normal vector of the gear shaping cutter tooth surface; Here is the equation for the tooth surface of the gear shaper; The equation for the tooth surface of a face gear; For the gear shaping tool coordinate system To the gear coordinate system The transformation matrix; The relative speed between the face gear and the gear shaper cutter; For the gear hobbing cutter's angle; The X-coordinate of the gear and rack tool in the coordinate system for machining the surface; At the same time, the pinion uses a parabolic profile to modify the tooth direction, such as... Figure 3 As shown. By offsetting the tooth surface points along the normal direction of the standard involute pinion tooth surface, the equation of the pinion tooth surface after tooth profile modification can be obtained, as shown in the following formula. The vertex of the tooth profile modification parabola is located at the midpoint of the tooth width, and the maximum tooth profile modification amount of the pinion is... .

[0031] in: The pinion tooth surface after double-shaped tooth profile modification; The tooth surface of the pinion after tooth profile modification; This is the unit normal vector of the pinion tooth surface after tooth profile modification; These are the profile modification parameters for the pinion teeth; , The two surface parameters for machining pinion rack and pinion tools correspond to the coordinate system respectively. The X coordinate below coordinate; This refers to the tooth width of the pinion.

[0032] S2. Tooth surface contact analysis: Based on the condition that the coordinates of the tooth surface points and the normal vectors at the meshing contact points of the face gear and the pinion are equal, the contact point solution equation is established; the rotation angle of the pinion is used as the independent variable for iterative solution to obtain the complete contact trace of the face gear pair, and the single-tooth meshing area and double-tooth meshing area are determined.

[0033] When a face gear and a pinion mesh, the coordinates of the tooth surface points at their contact points are equal to the normal vector. Based on this condition, the contact point of a face gear pair in a certain phase can be calculated using the following formula: in: The tooth surface of the pinion after tooth profile modification; This is the unit normal vector of the pinion tooth surface after tooth profile modification; The modified gear working tooth surface and tooth root transition surface formed by the envelope of the modified pinion; It is the unit normal vector of its corresponding tooth surface; From coordinate system To coordinate system The transformation matrix; , Let represent two parameters of the tooth surface of a face gear, where The X-coordinate in the coordinate system of the gear and rack tool for machining the surface. For the gear hobbing cutter's angle; Using the pinion's rotation angle as the independent variable, and rotating the pinion in steps of a certain size, the complete contact trace of the face gear pair can be solved according to the above formula, and the single-tooth meshing region and double-tooth meshing region of the face gear pair can be calculated. Specifically, the solution method for the single-tooth meshing region and double-tooth meshing region of the face gear pair is as follows: in: This is the critical point where the double-meshing zone transitions into the single-meshing zone along the meshing direction, i.e. Figure 4 Point DS in; This is the critical point where the single-meshing zone transitions into the double-meshing zone along the meshing direction, i.e. Figure 4 Point SD in the middle; The engagement angle; This refers to the engagement angle; The degree of overlap; This represents the number of teeth on the pinion.

[0034] like Figure 4 The figure shows the calculated contact traces of the gear teeth, where MI and MO are the engagement and disengagement points, respectively, and DS and SD are the critical points from the double-tooth meshing zone to the single-tooth meshing zone and from the single-tooth meshing zone to the double-tooth meshing zone, respectively.

[0035] S3. Tooth surface bearing contact analysis: Based on the input torque and the distance from the contact point to the pinion axis, calculate the normal contact force in the single tooth meshing area; based on the principal curvature, principal direction and total deformation of the tooth surface at the contact point, calculate the major and minor axis dimensions of the contact ellipse under load conditions.

[0036] During gear transmission, loads exist between face gear pairs, causing deformation of the tooth surfaces and transforming the gear pair from point contact to surface contact. With the pinion as the driving gear, the input torque... The distance from the contact point of the face gear pair to the axis of the pinion. Normal force in the single tooth meshing area of ​​a face gear pair : in: This is the normal contact force in the single-tooth meshing zone; This refers to the tangential contact force in the single-tooth meshing zone; Input torque; This is the distance from the contact point of the face gear pair to the axis of the pinion; and The contact point of the face gear pair in the follower coordinate system of the pinion The coordinates below; This is the pressure angle.

[0037] The principal curvature of the gear pair can be solved using the following formula, and the principal direction of the pinion can be derived from this equation. , Main direction of the face gear and This further determines the direction of the contact ellipse.

[0038] in: denoted by the normal curvature of the two surfaces; L, M, and N represent the first fundamental quantities of the tooth profile surface; E, F, and G represent the second fundamental quantities of the tooth profile surface. , is the independent variable of the surface equation.

[0039] like Figure 5 As shown, the major axis of the contact ellipse With short axis The calculation formula is: in: This represents the total deformation of the two contacting tooth surfaces; The angle between the first principal directions of the two tooth surfaces; and These are the sum of the first and second principal curvatures of the two sides, respectively; and These are the differences between the first and second principal curvatures of the two sides, respectively.

[0040] S4. Iterative Calculation of Critical Modification Amount: Establish a set of equations to solve for the geometric intersection between the inner and outer diameter edge curves of the face gear tooth surface and the modified tooth surface of the pinion; use the Newton-Raphson method to iteratively solve the set of equations, and use the numerical convergence of the iterative solution as the basis for judging whether edge contact exists. The method is as follows: set a convergence threshold, and when the iterative solution meets the convergence condition, it is judged as convergent, indicating that edge contact exists; by gradually increasing the pinion tooth profile modification parameter, repeat the iterative calculation until the iterative solution cannot converge, then it is judged that the edge contact is eliminated, and the corresponding pinion tooth profile modification amount is the critical modification amount.

[0041] like Figure 6 As shown, under no-load conditions, the included angle between adjacent tooth surfaces of the face gear is equal to the theoretical value. Under load, the tooth surface contact area deforms, with the maximum deformation being... The deformation causes a change in the angle of the face gear relative to the pinion. Under heavy loads, interference occurs in the face gear pair, leading to edge contact of the face gears, such as... Figure 7 As shown. In this embodiment, the equations used to determine edge contact are: in: The total deformation of the two contacting tooth surfaces; The angle of rotation of the face gear relative to the pinion caused by deformation; The pressure angle; This is the distance from the contact point to the axis of the face gear; This is an elliptic integral of the first kind; It is an elliptic integral of the second kind; Let be the eccentricity of the ellipse; and These are the sum of the first and second principal curvatures of the two sides, respectively; The elastic modulus of the pinion; The elastic modulus of the face gear; The Poisson's ratio for the pinion; For the face gear, Poisson's ratio is given. This represents the normal contact force in the single-tooth meshing zone.

[0042] Determining the pinion tooth profile modification parameters When considering factors such as the outer diameter and tooth tip strength of the face gear, in order to determine the tooth profile modification amount of the pinion without edge contact, the intersection points of the inner and outer diameter edge curves of the face gear tooth surface and the pinion tooth surface are determined based on the Newton-Raphson method to determine the critical tooth profile modification parameters of the pinion. The value of .

[0043] Specifically, the inner and outer diameter curves of the face gear Represented as: in: The inner radius of the face gear; Let be the outer radius of the face gear.

[0044] When it has edge contact with the pinion tooth surface, the intersection point satisfies: in: This is the intersection point where the tooth surfaces of the face gear and the pinion are in edge contact. The pinion tooth surface after double-shaped tooth profile modification; coordinate system arrive The transformation matrix; , The two surface parameters for machining pinion rack and pinion tools correspond to the coordinate system respectively. The X coordinate below coordinate; For the gear hobbing cutter's angle; , and For the pinion tooth surface in The X, Y, and Z coordinates; , and The inner and outer diameter edge curves of the face gear in the coordinate system The X, Y, and Z coordinates; The inner and outer diameter edge curves of the face gear.

[0045] The Newton-Raphson method is used for iterative solution. Specifically, the formula for iteratively solving the system of equations using the Newton-Raphson method is as follows: in: , and These are the two parameters of the pinion tooth surface equation after n iterations and the gear shaper rotation angle, respectively. It is a Jacobian matrix; The intersection point when the tooth surfaces of the face gear and the pinion are in edge contact. A 3rd order submatrix, and: The Jacobian matrix calculated iteratively using the Newton-Raphson method is expressed as: If satisfied If the edge contact is found to be present, the solution is considered convergent; otherwise, the iterative solution fails to converge, and the edge contact is considered eliminated. This is the convergence threshold.

[0046] Case Analysis This embodiment analyzes the above-mentioned method for calculating the critical modification amount of the face gear pair using a specific example. The face gear parameters selected in the example are shown in Table 1: Table 1. Parameter Table of Face Gear Pair The preset meshing performance of the face gear pair is: the parabolic coefficient of the pinion tooth profile modification. Given a value of -0.2, the critical profile modification parameter of the pinion is obtained by calculating the critical profile modification amount of the face gear pair using the method for calculating the critical profile modification amount of the face gear pair. As a parameter for modifying the tooth profile of the pinion.

[0047] The contact condition of the rear gear pair before and after modification was calculated. The calculation results are as follows: Figure 8 and Figure 9 As shown in the figure, the contact traces on the tooth surfaces of the unmodified face gear pair are distributed approximately perpendicularly along the tooth height direction, resulting in edge contact between the tip of the face gear and the tip of the pinion. After modification, no edge contact occurs in the face gear pair, and the contact area is confined to the middle of the tooth surface. The calculation results indicate that modifying the face gear pair can effectively avoid edge contact and improve the tooth surface contact.

[0048] This embodiment determines the critical profile adjustment amount required to eliminate edge contact through quantitative calculation, reducing the blind spots in the design process, effectively eliminating load concentration caused by edge contact, and improving the load-bearing capacity and service life of the gear teeth. The method of this embodiment can be applied to the tooth surface design of face gears, improving their meshing performance and increasing their load-bearing capacity and service life.

[0049] The above-described embodiments are merely preferred embodiments provided to fully illustrate the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A method for calculating the critical modification amount of a face gear pair, characterized in that: Includes the following steps: S1. Tooth profile modeling: Based on the basic parameters of the pinion, the tooth profile of the rack cutter is modified by parabola to obtain the tooth surface equation and normal vector of the rack cutter. Based on the meshing principle, the equation of the pinion tooth surface after profile modification is derived from the tooth surface of the rack cutter through coordinate transformation and relative motion velocity relationship; at the same time, the pinion tooth direction is modified by parabola, and the equation of the pinion tooth surface after tooth direction modification is obtained by offsetting the tooth surface point along the normal of the standard involute pinion tooth surface. S2. Tooth surface contact analysis: Based on the condition that the coordinates of the tooth surface points and the normal vector are equal at the meshing contact point between the face gear and the pinion, the contact point solution equation is established. The complete contact trace of the face gear pair is obtained by iteratively solving the problem with the rotation angle of the pinion as the independent variable, and the single-tooth meshing area and the double-tooth meshing area are determined. S3. Tooth surface bearing contact analysis: Based on the input torque and the distance from the contact point to the pinion axis, calculate the normal contact force in the single tooth meshing area; Based on the principal curvature and principal direction of the tooth surface at the contact point and the total deformation of the two contact tooth surfaces, calculate the major and minor axis dimensions of the contact ellipse under load conditions. S4. Iterative Calculation of Critical Modification Amount: Establish a set of equations to solve for the geometric intersection between the inner and outer diameter edge curves of the face gear tooth surface and the modified tooth surface of the pinion; use the Newton-Raphson method to iteratively solve the set of equations, and use the numerical convergence of the iterative solution as the basis for judging whether edge contact exists. The method is as follows: set a convergence threshold, and when the iterative solution meets the convergence condition, it is judged as convergent, indicating that edge contact exists; by gradually increasing the pinion tooth profile modification parameter, repeat the iterative calculation until the iterative solution cannot converge, then it is judged that the edge contact is eliminated, and the corresponding pinion tooth profile modification amount is the critical modification amount.

2. The method according to claim 1, characterized in that: In step S1, the equation of the pinion tooth surface after profile modification is: wherein: is the pinion tooth surface after tooth profile modification; is the tool coordinate system is the conversion matrix from the tool coordinate system to the pinion coordinate system; is the rack tool tooth surface equation in the tool coordinate system ; is the rack tool tooth surface normal vector; is the relative motion velocity between the rack tool and the pinion; , are two curved surface parameters of the rack tool for machining the pinion, corresponding to the X coordinate, the Y coordinate in the coordinate system ; and: in: The coefficient of the parabolic curve for tooth profile modification; The pressure angle; coordinate system Below coordinate; This is the zero-point offset of the parabola; The distance from the vertex of the parabola to the origin. exist Distance in a direction.

3. The method for calculating the critical modification amount of a face gear pair according to claim 1 or 2, characterized in that: In step S1, the equation of the pinion tooth surface after tooth profile modification is: in: The pinion tooth surface after double-shaped tooth profile modification; The tooth surface of the pinion after tooth profile modification; This is the unit normal vector of the pinion tooth surface after tooth profile modification; These are the profile modification parameters for the pinion teeth; , The two surface parameters for machining pinion rack and pinion tools correspond to the coordinate system respectively. The X coordinate below coordinate; This refers to the tooth width of the pinion.

4. The method for calculating the critical modification amount of a face gear pair according to claim 1, characterized in that: In step S2, the contact point solution equation is established based on the tooth surface meshing condition and is expressed as: in: The tooth surface of the pinion after tooth profile modification; This is the unit normal vector of the pinion tooth surface after tooth profile modification; The modified gear working tooth surface and tooth root transition surface formed by the envelope of the modified pinion; It is the unit normal vector of its corresponding tooth surface; From coordinate system To coordinate system The transformation matrix; , Let represent two parameters of the tooth surface of a face gear, where The X-coordinate in the coordinate system of the gear and rack tool for machining the surface. For the gear hobbing cutter's turning angle; and: in: This is the unit normal vector of the gear shaping cutter tooth surface; Here is the equation for the tooth surface of the gear shaper; The equation for the tooth surface of a face gear; For the gear shaping tool coordinate system To the gear coordinate system The transformation matrix; The relative speed between the face gear and the gear shaper cutter.

5. The method for calculating the critical modification amount of a face gear pair according to claim 1, characterized in that: In step S2, the method for solving the single-tooth meshing region and double-tooth meshing region of the face gear pair is as follows: in: This is the critical point at which the double meshing zone transitions into the single meshing zone along the meshing direction; This is the critical point at which the single meshing zone enters the double meshing zone along the meshing direction; The engagement angle; This refers to the engagement angle; The degree of overlap; This represents the number of teeth on the pinion.

6. The method for calculating the critical modification amount of a face gear pair according to claim 1, characterized in that: In step S3, the formula for calculating the normal contact force in the single-tooth meshing zone is: in: This is the normal contact force in the single-tooth meshing zone; This refers to the tangential contact force in the single-tooth meshing zone; Input torque; This is the distance from the contact point of the face gear pair to the axis of the pinion; and The contact point of the face gear pair in the follower coordinate system of the pinion The coordinates below; This is the pressure angle.

7. The method for calculating the critical modification amount of a face gear pair according to claim 1, characterized in that: In step S3, the major axis of the ellipse is contacted. With short axis The calculation formula is: in: This represents the total deformation of the two contacting tooth surfaces; The angle between the first principal directions of the two tooth surfaces; and These are the sum of the first and second principal curvatures of the two sides, respectively; and These are the differences between the first and second principal curvatures of the two surfaces, respectively; and: in: denoted by the normal curvature of the two surfaces; L, M, and N represent the first fundamental quantities of the tooth profile surface; E, F, and G represent the second fundamental quantities of the tooth profile surface. , is the independent variable of the surface equation.

8. The method for calculating the critical modification amount of a face gear pair according to claim 1, characterized in that: In step S4, the set of equations used to determine edge contact is as follows: in: The total deformation of the two contacting tooth surfaces; The angle of rotation of the face gear relative to the pinion caused by deformation; The pressure angle; This is the distance from the contact point to the axis of the face gear; It is an elliptic integral of the first kind; It is an elliptic integral of the second kind; Let be the eccentricity of the ellipse; and These are the sum of the first and second principal curvatures of the two sides, respectively; The elastic modulus of the pinion; The elastic modulus of the face gear; The Poisson's ratio for the pinion; For the face gear, Poisson's ratio is given. This represents the normal contact force in the single-tooth meshing zone.

9. The method for calculating the critical modification amount of a face gear pair according to claim 8, characterized in that: The formula for iteratively solving the system of equations using the Newton-Raphson method is as follows: in: , and These are the two parameters of the pinion tooth surface equation after n iterations and the gear shaper rotation angle, respectively. It is a Jacobian matrix; The intersection point when the tooth surfaces of the face gear and the pinion are in edge contact. A 3rd order submatrix, and: in: This is the intersection point where the tooth surfaces of the face gear and the pinion are in edge contact. The pinion tooth surface after double-shaped tooth profile modification; coordinate system arrive The transformation matrix; , The two surface parameters for machining pinion rack and pinion tools correspond to the coordinate system respectively. The X coordinate below coordinate; For the gear hobbing cutter's rotation angle; , and For the pinion tooth surface in The X, Y, and Z coordinates; , and The inner and outer diameter edge curves of the face gear in the coordinate system The X, Y, and Z coordinates; The inner and outer diameter edge curves of the face gear are represented as follows: in: The inner radius of the face gear; The outer radius of the face gear; If satisfied If the edge contact is found to be present, the solution is considered convergent; otherwise, the iterative solution fails to converge, and the edge contact is considered eliminated. This is the convergence threshold.

10. The method for calculating the critical modification amount of a face gear pair according to claim 9, characterized in that: The Jacobian matrix calculated iteratively using the Newton-Raphson method is expressed as: 。

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