Face gear tooth surface discretization method, medium and equipment

Abstract

The invention discloses a face gear tooth surface discretization method, medium and equipment, and the method comprises the steps: firstly building a tooth surface equation of a cylindrical gear of an involute and equiangular helix combined tooth profile and a conjugate face gear, and generating a three-dimensional model for morphology verification; secondly, determining a contact boundary by solving key feature points such as a tooth crest engaging-in point and a tooth root engaging-out point, dividing a tooth surface into a plurality of meshing areas and performing high-precision discretization to obtain coordinates of a large number of discrete points and corresponding meshing angles; and finally, based on the discretized data, accurately calculating the total length, the sliding rate and the main curvature distribution of a tooth surface contact line, and further quantitatively evaluating the bearing capacity and the meshing transmission performance of the face gear pair. According to the method, refined and digital analysis of the tooth surface contact performance of the face gear is realized, and reliable data support and theoretical basis are provided for design and optimization of the high-performance face gear.

Description

Technical Field

[0001] This invention relates to the field of gear manufacturing, and more specifically to a method, medium, and device for discretizing the tooth surface of a face gear. Background Technology

[0002] Currently, high-end mechanical equipment is developing towards high power density, high efficiency, and high reliability, which poses unprecedented challenges to the performance of core transmission components in high-end mechanical equipment. Face gear transmission, as a new type of transmission that uses the meshing of cylindrical gears and bevel gears to achieve motion and power transmission between intersecting shafts, is gradually being applied in key areas such as helicopter main gearboxes due to its significant advantages such as compact structure, smooth transmission, high overlap ratio, and no axial force.

[0003] However, face gears have complex tooth profiles, and their meshing performance and load-bearing capacity are highly dependent on the tooth surface contact characteristics. Under high-speed, heavy-load conditions, the distribution of parameters such as tooth surface contact line length, slip ratio, and curvature directly affects their transmission efficiency, wear life, and anti-galling ability. Traditional design methods often rely on empirical formulas or simplified models, making it difficult to accurately describe and quantify the tooth surface contact morphology digitally. This results in the inability to accurately predict and optimize the performance bottlenecks of face gear pairs during actual meshing.

[0004] Therefore, developing a method that can achieve accurate discretization modeling of the tooth surface and perform efficient and accurate calculation and analysis of key performance parameters such as contact line, slip ratio, and curvature based on discrete data is of vital importance for fundamentally improving the design level of face gear transmissions, tapping their load-bearing potential, and ensuring their service reliability. Summary of the Invention

[0005] In view of the above problems, the present invention provides a method, medium and device for discretizing the tooth surface of face gears, aiming to provide a solution for accurately quantifying and evaluating the meshing performance of face gears with arbitrary tooth profiles, so as to solve the technical problem that the existing technology lacks effective means to accurately predict and compare the advantages and disadvantages of different tooth profile designs (such as involute, equiangular helix or their combination).

[0006] To achieve the above objectives, in a first aspect, this application provides a method for discretizing the tooth surface of a face gear, the method comprising the following steps:

[0007] S1: Establish tooth surface equation: Based on the basic parameters of the gear, calculate the tooth surface equation of the cylindrical gear with the combined tooth profile of involute and equiangular helix. Based on the meshing principle of face gears and coordinate transformation theory, solve to obtain the tooth surface equation of the face gear conjugate with the cylindrical gear. The cylindrical gear and the face gear constitute a face gear pair.

[0008] S2: Generate and visualize 3D models: Calculate the coordinates of the tooth surface points according to the tooth surface equation, generate 3D digital models of cylindrical gears and face gears, and render and display them for verification of tooth profile morphology.

[0009] S3: Feature point solution and tooth surface discretization: Based on the tooth surface equation of the face gear, solve for the coordinates of multiple feature points on the tooth surface of the face gear and their corresponding meshing angles. The multiple feature points include the tooth tip engagement point, the tooth root engagement point, the tooth vertex of the internal tooth profile and the tooth root point of the external tooth profile. Based on the multiple feature points, determine the contact boundary. According to the contact boundary, divide the tooth surface of the face gear into multiple meshing regions. In each of the meshing regions, calculate the coordinates of multiple discrete points and the meshing angles corresponding to each discrete point.

[0010] S4: Performance Analysis: Based on the coordinates of the discrete points obtained in step S3 and the meshing angles corresponding to each discrete point, calculate the total length of the tooth surface contact line, the sliding ratio, and the principal curvature of the face gear pair, and generate the corresponding distribution map.

[0011] S5: Result Evaluation: Based on the analysis results of the total contact line length, sliding rate distribution and principal curvature distribution obtained in step S4, the load-bearing capacity and meshing transmission performance of the face gear pair are quantitatively evaluated.

[0012] Furthermore, based on the basic parameters of the gear, the calculation formula for the equation of the cylindrical gear tooth surface with a combined involute and equiangular helix tooth profile is as follows:

[0013]

[0014] Where k is the constant of the equiangular helix, r0 is the initial radius of the equiangular helix, and r is the pitch circle radius of the cylindrical gear; bs μ is the base circle radius of the cylindrical gear. s For the tooth width parameter, θ s Let θ be the tooth profile expansion angle. os This is the angle parameter from the vertical axis of symmetry to the starting point of the tooth profile.

[0015] Furthermore, in step S1, based on the meshing principle of face gears and coordinate transformation theory, the equation of the face gear tooth surface conjugate with the cylindrical gear is obtained by solving the following:

[0016] S11: Based on the principle of face gear meshing and coordinate transformation theory, the coordinate transformation matrix M is obtained. 2s and the relative velocity v at the meshing point s2 ;

[0017] The coordinate transformation matrix M 2s The calculation formula is as follows:

[0018]

[0019] The relative velocity v at the engagement point s2 The calculation formula is as follows:

[0020]

[0021] Where, φ s φ2 is the rotation angle of the cylindrical gear, ω is the rotation angle of the face gear. s The rotational speed of the cylindrical gear is m. 2s The gear ratio is the face gear pair transmission ratio.

[0022] S12: Based on the coordinate transformation matrix M 2s and the relative velocity v at the engagement point s2 The equation of the tooth surface of the face gear, which is conjugate to the cylindrical gear, is obtained by solving the following formula:

[0023]

[0024] Where, φ θ =θ s +θ os ±φ s m is the modulus, h a h is the tooth tip height. f The tooth root height is given by z, which is the z-coordinate of the face gear.

[0025] Furthermore, in step S2, calculating the coordinates of the tooth surface points based on the tooth surface equation to generate a three-dimensional digital model of the cylindrical gear and the face gear includes:

[0026] The left and right tooth profile curves are generated from the coordinate points of the left and right tooth profiles, then stretched into the left and right tooth surfaces, and then the tooth surfaces are arrayed to obtain the tooth surfaces of all the teeth of the cylindrical gear, thus generating a three-dimensional digital model of the cylindrical gear.

[0027] In a computer simulation environment, the three-dimensional digital model of the cylindrical gear is set as a machining tool. Boolean subtraction is used to simulate the cutting process of the machining tool on the face gear blank to obtain a face gear single tooth with machining marks. The face gear single tooth is then surface-finished to obtain a smooth single tooth. All the smooth single teeth are arranged in an array to obtain the three-dimensional digital model of the face gear.

[0028] The three-dimensional digital model of the cylindrical gear is assembled with the three-dimensional digital model of the face gear to obtain the face gear pair transmission model.

[0029] Furthermore, in step S2, the verification of the tooth profile includes:

[0030] Compare the tooth surface morphology of the rendered face gear transmission model with the expected tooth surface morphology to check whether there is tooth profile distortion or undercutting phenomenon in the tooth surface morphology of the face gear transmission model.

[0031] Furthermore, based on the multiple feature points, a contact boundary is determined, and the tooth surface of the face gear is divided into multiple meshing regions according to the contact boundary. The coordinates of multiple discrete points are calculated within each meshing region, specifically including:

[0032] Step S3, which involves determining the contact boundary based on four feature points and calculating discrete points within each engagement region, specifically includes:

[0033] Based on four characteristic points—the tooth tip engagement point, the tooth root engagement point, the apex of the internal tooth profile, and the tooth root point of the external tooth profile—and their corresponding meshing angles, two contact boundary lines are determined. These two contact boundary lines divide the tooth surface of the face gear into three meshing regions. The two contact boundary lines include a first contact boundary line and a second contact boundary line. The three meshing regions include:

[0034] The first region, located between the tooth tip engagement point and the first contact boundary line, is used to complete the discretization of the upper half of the external tooth profile and the tooth tip of the face gear.

[0035] The second region, located between the first contact boundary line and the second contact boundary line, is used to complete the discretization of the lower half of the outer tooth profile and the upper half of the inner tooth profile of the face gear.

[0036] The third region, located between the second contact boundary line and the transition curve, is used to complete the discretization of the face gear transition curve and the lower half of the internal tooth profile.

[0037] Furthermore, the engagement angle corresponding to the tooth tip engagement point, the engagement angle corresponding to the tooth vertex of the internal tooth profile, the engagement angle corresponding to the tooth root engagement point, and the engagement angle corresponding to the tooth root point of the external tooth profile are obtained by substituting the inner diameter, outer diameter, tooth height parameters of the face gear and the tooth tip development angle parameter of the cylindrical gear into the face gear tooth surface equation for solving.

[0038] Furthermore, in step S4, the total length of the tooth surface contact line of the face gear pair is obtained by iteratively calculating the spatial coordinates of adjacent discrete points on the contact line, as shown in the following formula:

[0039]

[0040] Where L represents the total length of the tooth surface contact line, n represents the number of discrete points, and x i y i z i Let x and x represent the spatial coordinates of the i-th discrete point, respectively. i+1 y i+1 zi+1 Let l represent the spatial coordinates of the (i+1)th discrete point, respectively. i This represents the distance between the (i+1)th discrete point and the ith discrete point.

[0041] The formula for calculating the slip ratio is as follows:

[0042]

[0043] Where ΔS1 and ΔS2 are the moving arc lengths, and dS1 and dS2 are the differential arc lengths of the two conjugate tooth profiles. The formulas for calculating dS1 and dS2 are as follows:

[0044]

[0045] The formula for calculating the principal curvature is as follows:

[0046]

[0047] Where L, M, and N represent the second fundamental quantities of the surface, and E, F, and G represent the first fundamental quantities of the surface.

[0048] In a second aspect, this application provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the face gear tooth surface discretization method as described in the first aspect of this application.

[0049] In a third aspect, this application provides an electronic device having a computer program stored thereon, including a processor and a storage medium, wherein the computer program is stored on the storage medium, and when executed by the processor, the computer program implements the face gear tooth surface discretization method as described in the first aspect of this application.

[0050] Unlike existing technologies, the above-mentioned technical solution provides a method, medium, and device for discretizing the tooth surface of face gears. This method first establishes the tooth surface equations of a cylindrical gear with a combined involute and equiangular helical tooth profile and its conjugate face gear, generating a three-dimensional model for morphological verification. Secondly, by solving key feature points such as the tooth tip engagement point and tooth root disengagement point, the contact boundary is determined, dividing the tooth surface into multiple meshing regions and performing high-precision discretization to obtain the coordinates and corresponding meshing angles of a large number of discrete points. Finally, based on this discretized data, the total length of the tooth surface contact line, the slip ratio, and the principal curvature distribution are accurately calculated, thereby quantitatively evaluating the load-bearing capacity and meshing transmission performance of the face gear pair. This invention achieves refined and digital analysis of the tooth surface contact performance of face gears, providing reliable data support and theoretical basis for the design and optimization of high-performance face gears.

[0051] The above description of the invention is merely an overview of the technical solution of the present invention. In order to enable those skilled in the art to better understand the technical solution of the present invention and to implement it based on the description and drawings, and to make the above-mentioned objectives and other objectives, features and advantages of the present invention easier to understand, the following description is provided in conjunction with the specific embodiments and drawings of the present invention. Attached Figure Description

[0052] The accompanying drawings are only used to illustrate the principles, implementation methods, applications, features, and effects of specific embodiments of the present invention and other related contents, and should not be considered as limitations on the present invention.

[0053] In the accompanying drawings of the instruction manual:

[0054] Figure 1 This is a first flowchart of the face gear tooth surface discretization method involved in a specific implementation;

[0055] Figure 2 This is a schematic diagram of the combined tooth profile cylindrical gear according to an exemplary embodiment of this application;

[0056] Figure 3 This is a three-dimensional model of a combined tooth profile cylindrical gear according to an exemplary embodiment of this application;

[0057] Figure 4 This is a three-dimensional model of a combined tooth profile gear according to an exemplary embodiment of this application;

[0058] Figure 5 This is an exemplary embodiment of the combined tooth profile gear pair transmission model according to this application;

[0059] Figure 6 This is a visualization of a combined tooth profile cylindrical gear according to an exemplary embodiment of this application;

[0060] Figure 7 This is a visualization of a combined tooth profile gear according to an exemplary embodiment of this application;

[0061] Figure 8 This is a discretized diagram of the tooth surface of a face gear according to an exemplary embodiment of this application;

[0062] Figure 9 This is a discretized diagram of the contact line of the tooth surface of a face gear according to an exemplary embodiment of this application;

[0063] Figure 10 This is an analysis diagram of the meshing state of a face gear teeth according to an exemplary embodiment of this application;

[0064] Figure 11A schematic diagram illustrating the variation law of the sliding rate of the combined tooth profile gear pair along the tooth profile direction obtained by solving for an exemplary embodiment of this application;

[0065] Figure 12 A schematic diagram illustrating the variation law of the principal curvature of the combined tooth profile gear pair along the tooth height direction obtained by solving for an exemplary embodiment of this application;

[0066] Figure 13 A schematic diagram showing the variation law of the principal curvature of the combined tooth profile gear pair along the tooth width direction obtained by solving for an exemplary embodiment of this application;

[0067] Figure 14 A schematic diagram of the modules of the electronic device described in a specific embodiment;

[0068] The reference numerals used in the above figures are explained as follows:

[0069] 10. Electronic devices;

[0070] 101. Processor;

[0071] 102. Storage medium. Detailed Implementation

[0072] To illustrate the possible application scenarios, technical principles, implementable specific solutions, and achievable objectives and effects of this invention in detail, the following description, in conjunction with the listed specific embodiments and accompanying drawings, provides a detailed explanation. The embodiments described herein are merely illustrative of the technical solutions of this invention and are therefore intended only as examples, not as limiting the scope of protection of this invention.

[0073] In this document, the term "embodiment" means that a specific feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of the invention. The term "embodiment" appearing in various places throughout the specification does not necessarily refer to the same embodiment, nor does it specifically limit its independence or connection with other embodiments. In principle, in this invention, as long as there are no technical contradictions or conflicts, the technical features mentioned in each embodiment can be combined in any way to form corresponding implementable technical solutions.

[0074] Unless otherwise defined, the technical terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains; the use of related terms herein is merely for the purpose of describing particular embodiments and is not intended to limit the invention.

[0075] In the description of this invention, the term "and / or" is used to describe the logical relationship between objects, indicating that three relationships can exist. For example, A and / or B means: A exists, B exists, and A and B exist simultaneously. Additionally, the character " / " generally indicates that the preceding and following objects have an "or" logical relationship.

[0076] In this invention, terms such as “first” and “second” are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any actual quantity, hierarchy, or order between these entities or operations.

[0077] Without further limitations, the use of terms such as “comprising,” “including,” “having,” or other similar expressions in this invention is intended to cover non-exclusive inclusion, which does not exclude the presence of additional elements in a process, method, or product that includes the stated elements, such that a process, method, or product that includes a list of elements may include not only those defined elements but also other elements not expressly listed, or elements inherent to such a process, method, or product.

[0078] In this invention, expressions such as "greater than", "less than", and "exceeding" are understood to exclude the stated number; expressions such as "above", "below", and "within" are understood to include the stated number. Furthermore, in the description of the embodiments of this invention, "multiple" means two or more (including two), and similar expressions related to "multiple" are also understood in this way, such as "multiple groups" and "multiple times", unless otherwise explicitly specified.

[0079] In the description of the embodiments of the present invention, the spatial related expressions used, such as "center," "longitudinal," "lateral," "total length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "vertical," "top," "bottom," "inner," "outer," "clockwise," "counterclockwise," "axial," "radial," "circumferential," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the specific embodiments or drawings. They are only for the purpose of describing the specific embodiments of the present invention or for the reader's understanding, and do not indicate or imply that the device or component referred to must have a specific position, a specific orientation, or be constructed or operated in a specific orientation. Therefore, they should not be construed as limitations on the embodiments of the present invention.

[0080] Unless otherwise explicitly stated or limited, the terms "installation," "connection," "linking," "fixing," and "setting," as used in the description of the embodiments of this invention, should be interpreted broadly. For example, "connection" can be a fixed connection, a detachable connection, or an integral arrangement; it can be a mechanical connection, an electrical connection, or a communication connection; it can be a direct connection or an indirect connection through an intermediate medium; it can be the internal connection of two components or the interaction between two components. Those skilled in the art can understand the specific meaning of the above terms in the embodiments of this invention according to the specific circumstances.

[0081] like Figure 1 As shown, in a first aspect, this application provides a method for discretizing the tooth surface of a face gear, the method comprising the following steps:

[0082] S1: Establish the tooth surface equation: Based on the basic parameters of the gear, calculate the tooth surface equation of the cylindrical gear with the combined tooth profile of the involute and the equiangular helix. Based on the meshing principle of face gears and the coordinate transformation theory, solve for the tooth surface equation of the face gear that is conjugate to the cylindrical gear. The cylindrical gear and the face gear constitute a face gear pair.

[0083] In step S1, the basic gear parameters include module, number of teeth, tooth width, tooth height, pitch circle radius, and base circle radius, which are the foundational data for deriving the tooth surface equation and generating the three-dimensional model. In this invention, a short-tooth system is selected, with the following specific parameters: the cylindrical gear has a module of 3, 23 teeth, a tooth width of 12mm, and a tooth height of 5.7mm; the face gear has a module of 3, 59 teeth, a tooth width of 9mm, and a tooth height of 5.7mm, and the inner and outer radii of the face gear are 86-95mm.

[0084] like Figure 2 As shown, the combined involute and equiangular helix tooth profile is defined by the pitch circle of the cylindrical gear. The tooth profile from the root circle to the pitch circle is an equiangular helix, while the tooth profile from the pitch circle to the addendum circle is an involute. This design combines the advantages of both tooth profiles, balancing load-bearing capacity with the avoidance of undercut.

[0085] Conjugate refers to the fact that the tooth profile curves of a cylindrical gear and a face gear satisfy the meshing condition. That is, during the meshing process, the common normal at the contact point of a pair of tooth profiles always passes through the node, ensuring the smoothness and accuracy of the transmission and realizing the effective transmission of motion and power between intersecting shafts.

[0086] Step S1 first derives the gear tooth surface equation of the combined involute and equiangular helix based on the determined basic gear parameters and the mathematical properties of the involute and equiangular helix. The involute part is based on the formation principle of the involute, i.e., when a straight line rolls purely on the base circle, the trajectory of a point on the line is the involute, and its equation can be determined by parameters such as the base circle radius and the tooth profile development angle. The equiangular helix part is derived based on the polar coordinate equation of the equiangular helix, combined with tooth width parameters, to obtain the combined gear tooth surface equation.

[0087] Next, based on the meshing principle of face gears, when a face gear meshes with a cylindrical gear, there is a specific transmission ratio between their motions. By establishing coordinate systems for the cylindrical gear and the face gear, and using coordinate transformation theory, points on the cylindrical gear tooth surface are transformed into the face gear coordinate system. Simultaneously, considering the relative velocity condition at the meshing point—that is, the component of the relative velocity of the two tooth profiles at the meshing point in the direction of the common normal is zero—the equations for the face gear tooth surface, conjugate to the cylindrical gear, are solved simultaneously, thus determining the tooth profile shape and geometric characteristics of the face gear.

[0088] S2: Generate and visualize 3D models: Calculate the coordinates of the tooth surface points based on the tooth surface equation, generate 3D digital models of cylindrical gears and face gears, and render and display them for verification of tooth profile morphology.

[0089] In step S2, the spatial coordinates (x, y, z coordinates) of a large number of points on the tooth surface are calculated by substituting the basic gear parameters into the equations of the cylindrical gear and face gear obtained in S1. For the cylindrical gear, the left and right tooth profile curves are generated from the coordinate points of the left and right tooth profiles respectively. Then, the tooth profile curves are stretched along the tooth width direction to form the left and right tooth surfaces. Then, the tooth surfaces are arrayed according to the number of teeth to obtain the tooth surfaces of all the teeth of the cylindrical gear. Finally, a three-dimensional digital model of the cylindrical gear is generated through a solidification operation, such as... Figure 3 As shown.

[0090] For face gears, in a computer simulation environment, the generated 3D digital model of the cylindrical gear is used as a machining tool to simulate the actual gear machining process. Through Boolean subtraction, the "tool gear" cuts the face gear blank, removing excess material to obtain a single tooth of the face gear with machining marks. Then, the tooth surface of the single tooth is smoothed to eliminate machining marks and make the tooth surface smooth. Finally, the smooth single teeth are arranged in an array according to the number of teeth of the face gear to generate a 3D digital model of the face gear, such as... Figure 4 As shown.

[0091] Finally, the 3D digital models of the cylindrical gear and the face gear are assembled according to the actual meshing relationship, ensuring that the center distance, transmission ratio, etc., meet the design requirements, resulting in a face gear pair transmission model. The 3D model is then rendered and displayed using computer graphics technology to visually present the tooth profile morphology, such as... Figure 5 As shown.

[0092] S3: Feature point solution and tooth surface discretization: Based on the tooth surface equation of the face gear, solve for the coordinates of multiple feature points on the tooth surface of the face gear and their corresponding meshing angles. The multiple feature points include the tooth tip engagement point, the tooth root engagement point, the tooth vertex of the internal tooth profile, and the tooth root point of the external tooth profile. Based on the multiple feature points, determine the contact boundary. According to the contact boundary, divide the tooth surface of the face gear into multiple meshing regions. In each of the meshing regions, calculate the coordinates of multiple discrete points and the meshing angles corresponding to each discrete point.

[0093] In step S3, tooth surface discretization refers to dividing the continuous tooth surface of a face gear into multiple discrete regions according to certain rules, and selecting multiple discrete points in each region. The meshing characteristics and performance parameters of the tooth surface are studied by analyzing the discrete points, which is a key step in subsequent performance analysis.

[0094] Based on the equation of the combined tooth profile cylindrical gear tooth surface, the xyz coordinate values ​​of the cylindrical gear tooth surface can be obtained, thus enabling the visualization of the combined tooth profile cylindrical gear tooth surface, such as... Figure 6 As shown. Similarly, based on the tooth surface equation of the combined tooth profile gear, a visual representation of the combined tooth profile gear can be achieved, such as... Figure 7 As shown. By visualizing the tooth surface, the shape of the combined tooth profile can be intuitively displayed, and the correctness of the derived equation for the combined tooth profile gear tooth surface can also be verified.

[0095] Based on the tooth surface equation of the face gear established in S1, and combined with the geometric characteristics and meshing conditions of the face gear, the coordinates of the feature points on the tooth surface and their corresponding meshing angles are solved. The tooth tip engagement point is the contact point at the upper right corner of the outer tooth profile when meshing begins, which can be solved by substituting the outer diameter and tooth tip height parameters of the face gear into the tooth surface equation; the tooth root disengagement point is the contact point at the lower left corner of the inner tooth profile when meshing ends, which is calculated using the inner diameter and tooth root height parameters of the face gear; the tooth vertex of the inner tooth profile is the highest point at the upper left corner of the inner tooth profile, and the tooth root point of the outer tooth profile is the lowest point at the lower right corner of the outer tooth profile, both of which are obtained by solving the corresponding geometric parameters and tooth surface equation.

[0096] Based on the obtained feature points and their corresponding meshing angles, the direction and distribution of the tooth surface contact lines are analyzed to determine two contact boundary lines. These two contact boundary lines are the basis for dividing the tooth surface meshing region; they divide the face gear tooth surface into three regions with different meshing characteristics, such as... Figure 8 As shown. Within each meshing region, multiple discrete points are selected based on a certain discrete density. The coordinates of each discrete point are calculated using the tooth surface equation. Combined with the variation law of the meshing angle, the meshing angle corresponding to each discrete point is determined, thus completing the discretization process of the tooth surface.

[0097] S4: Performance Analysis: Based on the coordinates of the discrete points obtained in step S3 and the meshing angles corresponding to each discrete point, calculate the total length of the tooth surface contact line, the sliding ratio, and the principal curvature of the face gear pair, and generate the corresponding distribution map.

[0098] In step S4, for the total length of the tooth surface contact line, based on the discrete point coordinates obtained in S3, the spatial distance between two adjacent discrete points is calculated. The distances of all adjacent discrete points are then summed sequentially to obtain the total length of the tooth surface contact line. This ultimately allows for the visualization of the contact line. Figure 9 As shown. Based on the meshing angle and contact line length of the face gear teeth, the meshing state of the face gear teeth can be analyzed, such as... Figure 10 As shown, the longer the contact line, the larger the contact area of ​​the tooth surface and the stronger the load-bearing capacity.

[0099] When calculating the slip ratio, first determine the calculation methods for the moving arc lengths ΔS1 and ΔS2 and the arc length differentials dS1 and dS2 based on the kinematic relationship between the two conjugate tooth profiles. Then, substitute these values ​​into the slip ratio calculation formula to calculate the slip ratio of each tooth profile at the contact point. The resulting solution yields the variation law of the slip ratio of the combined tooth profile gear pair along the tooth profile direction, as shown below. Figure 11 As shown, by calculating the slip ratio at different discrete points, a slip ratio distribution map is generated, and the variation law of slip ratio along the tooth profile is analyzed.

[0100] The calculation of principal curvature is based on the differential geometry theory of surfaces. By solving for the second fundamental quantities (L, M, N) and the first fundamental quantities (E, F, G) of the tooth surface at discrete points, and substituting them into the principal curvature calculation formula, the principal curvature value at each discrete point is obtained. This generates a principal curvature distribution map, allowing for the study of the distribution of principal curvature along the tooth height and tooth width directions. The solution yields the variation law of the sliding rate of the combined tooth profile gear pair along the tooth height direction, as shown below. Figure 12 As shown, the variation law of the slip ratio along the tooth width direction is as follows: Figure 13 As shown.

[0101] S5: Result Evaluation: Based on the analysis results of the total contact line length, sliding rate distribution and principal curvature distribution obtained in step S4, the load-bearing capacity and meshing transmission performance of the face gear pair are quantitatively evaluated.

[0102] In step S5, the analysis results of the total contact line length, slip ratio distribution, and principal curvature distribution obtained in S4 are used to quantitatively evaluate the load-bearing capacity and meshing transmission performance of the face gear pair. A longer total contact line length indicates a larger tooth surface contact area and stronger load-bearing capacity; a uniform and low slip ratio distribution indicates low tooth surface wear and high transmission efficiency; and a smooth change in principal curvature within a reasonable range indicates high tooth surface contact strength and good lubrication conditions. Based on the comprehensive performance of these indicators, it is determined whether the face gear pair meets the design requirements and can operate stably and reliably under complex conditions such as high speed and heavy load.

[0103] This application breaks through the limitations of traditional single tooth profiles by establishing a tooth surface equation for a composite tooth profile, combining the advantages of involute and equiangular helical profiles, thus laying a theoretical foundation for further improving the performance of face gears. Compared with traditional involute tooth profile face gears, the composite tooth profile can improve load-bearing capacity while ensuring transmission smoothness; compared with single equiangular helical tooth profile face gears, it can effectively alleviate the root undercut problem and extend the service life of the gear.

[0104] Generating a visual 3D model allows for an intuitive display of the gear tooth profile, enabling designers to promptly identify issues such as tooth distortion and undercut, reducing risks in subsequent manufacturing and lowering R&D costs. Simultaneously, the 3D model provides a foundation for subsequent assembly and motion simulations, facilitating a comprehensive evaluation of the transmission performance of face gear pairs.

[0105] Tooth surface discretization transforms the complex problem of continuous tooth surfaces into an analysis problem of discrete points, simplifying the calculation of performance parameters and improving analysis efficiency. Precise analysis of discrete points allows for a more comprehensive and detailed understanding of the meshing characteristics of the tooth surface, providing accurate data support for performance optimization and avoiding the limitations of traditional holistic analysis methods.

[0106] In the performance analysis step, by calculating key parameters such as slip ratio and principal curvature and generating distribution maps, the performance indicators of the face gear pair can be quantitatively evaluated, providing a clear direction for design improvement. For example, based on the slip ratio distribution map, tooth profile parameters can be adjusted to reduce wear in high slip ratio areas; based on the principal curvature distribution, the tooth surface shape can be optimized to improve contact strength.

[0107] The results evaluation phase comprehensively and quantitatively assesses the load-bearing capacity and meshing transmission performance of the face gear pair by integrating various performance indicators. This ensures that the designed face gear pair can meet the usage requirements of high-end mechanical equipment such as helicopter main reducers under high-speed, heavy-load, and complex working conditions, improve the operational reliability, energy efficiency, and service life of the equipment, and promote technological progress in the machinery industry and industrial upgrading in related fields.

[0108] In some embodiments, the formula for calculating the equation of the cylindrical gear tooth surface with a combined involute and equiangular helix tooth profile, based on the basic gear parameters, is as follows:

[0109]

[0110] Where k is the constant of the equiangular helix, r0 is the initial radius of the equiangular helix, and r is the pitch circle radius of the cylindrical gear; bs μ is the base circle radius of the cylindrical gear. s For the tooth width parameter, θ s Let θ be the tooth profile expansion angle. os This is the angle parameter from the vertical axis of symmetry to the starting point of the tooth profile.

[0111] In this embodiment, the constant k of the equiangular helix is ​​a key parameter that determines the shape of the equiangular helix. It reflects the tightness of the helix. The larger the value of k, the steeper the helix; the smaller the value of k, the gentler the helix. This directly affects the load-bearing capacity and root strength of the tooth profile of the equiangular helix.

[0112] In a composite cylindrical gear, the arbitrary circle radius r0 refers to the radius of the circle extending from the gear center to any point on the tooth profile. It is used to distinguish different parts of the tooth profile. When r0∈(r,r...) a When r0∈(r), the tooth profile is an involute; when r0∈(r f When r), the tooth profile is an equiangular helix, which is an important parameter for determining the tooth profile type.

[0113] The pitch circle radius r refers to half the product of the module and the number of teeth on a cylindrical gear, i.e., r = m × z / 2 (where m is the module and z is the number of teeth). It is the reference circle radius for gear design and the boundary circle radius for dividing involute tooth profiles and equiangular helical tooth profiles. Its size determines the basic dimensions and transmission ratio of the gear.

[0114] Base circle radius r bs This refers to the radius of the base circle of a cylindrical gear. The base circle is the reference circle from which the involute curve is generated. The base circle radius r bs = r × cosα (α is the pressure angle). The size of the base circle radius affects the shape of the involute tooth profile and the meshing performance of the gear. The larger the base circle, the smoother the involute and the more stable the transmission.

[0115] Tooth width parameter μ s This parameter describes the position coordinates of a point along the tooth width of a cylindrical gear. Its value range corresponds to the tooth width length of the gear, and is expressed through μ. s It can determine the tooth surface shape at different positions in the tooth width direction, ensuring that the gear can mesh well throughout the entire tooth width range.

[0116] Tooth profile expansion angle θ sIn an involute tooth profile, the angle refers to the central angle of the base circle from the starting point of the involute to the current tooth profile point, reflecting the degree of development of the involute; in an equiangular spiral tooth profile, it is related to the polar angle of the spiral and is an important angular parameter for determining the position of the tooth profile point, directly affecting the shape and size of the tooth profile.

[0117] The angle parameter θ from the vertical axis of symmetry to the starting point of the tooth profile os The angle between the vertical axis of symmetry of the cylindrical gear (the axis passing through the center of the gear and perpendicular to the tooth width direction) and the starting point of the tooth profile is used to determine the circumferential position of the tooth profile, ensuring uniform tooth pitch and correct meshing. It is a key parameter in the tooth profile equation for determining the circumferential position of the tooth profile.

[0118] When r0∈(r,r) a According to the principle of involute formation, when a straight line rolls purely on a base circle, the trajectory of any point on that line is an involute. When establishing the coordinate system of a cylindrical gear, the center of the gear is taken as the origin, the tooth width direction as the z-axis, and the coordinates are expressed through the base circle radius r. bs Tooth profile expansion angle θ s The coordinate equations of the involute tooth profile are derived using isoparametric methods.

[0119] In the xy plane, the polar coordinates (ρ, θ) of any point on the involute satisfy ρ = r bs / cos(θ s ), θ=θ s +invα (invα is the involute function, α is the pressure angle). Convert polar coordinates to rectangular coordinates, and combine with the tooth width parameter μ. s The equation for the tooth surface of the involute tooth profile is obtained as follows:

[0120] The plus or minus sign is used to distinguish the left and right tooth profiles of the gear, θ os The starting position of the tooth profile in the circumferential direction was determined to ensure the symmetrical distribution of the left and right tooth profiles and to meet the tooth pitch requirements during gear meshing.

[0121] When r0∈(r f When r), the polar equation of the equiangular helix is ​​ρ=r×e^(k×θ) (θ is the polar angle). Its characteristic is that the angle between the helix and the radial line remains unchanged, which has good strength characteristics and is suitable as the tooth profile of the tooth root part. It can effectively avoid the problem of tooth root undercut.

[0122] Similarly, in the cylindrical gear coordinate system, the polar coordinate equation of the equiangular helix is ​​converted to rectangular coordinates, considering the tooth width parameter μ. s and angle parameter θ os The equation of the tooth surface of the equiangular helical tooth profile is obtained as follows:

[0123]

[0124] Where k is the constant of the equiangular helix, which determines the steepness of the helix. By adjusting the value of k, the tooth profile shape of the root part can be optimized and the tooth root strength can be improved; r is the pitch circle radius, which serves as the dividing point between the equiangular helix tooth profile and the involute tooth profile, ensuring a smooth transition between the two tooth profiles and avoiding impact during meshing.

[0125] By combining the tooth surface equations of cylindrical gears with different tooth profiles, the applicable ranges of involute and equiangular helical lines are clearly distinguished. Through reasonable parameter settings, a smooth transition between the two tooth profiles is achieved, ensuring the continuity and integrity of the gear tooth profile, avoiding meshing impact caused by abrupt changes in tooth profile, and improving the smoothness of transmission.

[0126] The tooth profile of the equiangular helix (r0∈(r f By utilizing the strength advantage of equiangular helixes, the load-bearing capacity of the tooth root is enhanced, effectively alleviating the problems of stress concentration and undercutting at the tooth root of traditional involute tooth profile gears, extending the service life of the gear, and enabling the gear to adapt to high-speed and heavy-load working conditions.

[0127] Involute tooth profile (r0∈(r,r) a It retains the advantages of smooth and accurate transmission ratio of involute gear transmission, ensuring the transmission performance of gears in the main meshing area, meeting the requirements of high-end equipment such as helicopter main reducers for transmission accuracy and stability, and achieving a balance between load-bearing capacity and transmission smoothness.

[0128] The equation includes the constant k of the equiangular helix and the tooth profile development angle θ. s With multiple adjustable parameters, designers can optimize the tooth profile shape by adjusting these parameters according to actual working conditions and performance indicators, so that the gear can achieve the best performance in different application scenarios, thus improving the flexibility and applicability of gear design.

[0129] This tooth surface equation provides an accurate mathematical basis for the subsequent generation of 3D models of cylindrical gears and the derivation of tooth surface equations for face gears. It ensures the accuracy and reliability of subsequent design steps, reduces design deviations caused by errors in the tooth surface equation, lowers R&D costs and time, and promotes the practical application and development of face gear transmission technology.

[0130] In some embodiments, in step S1, based on the meshing principle of face gears and coordinate transformation theory, solving for the face gear tooth surface equation conjugate with the cylindrical gear includes:

[0131] S11: Based on the principle of face gear meshing and coordinate transformation theory, the coordinate transformation matrix M is obtained. 2s and the relative velocity v at the meshing point s2 ;

[0132] The coordinate transformation matrix M 2s The calculation formula is as follows:

[0133]

[0134] The relative velocity v at the engagement point s2 The calculation formula is as follows:

[0135]

[0136] Where, φ s φ2 is the rotation angle of the cylindrical gear, ω is the rotation angle of the face gear. s The rotational speed of the cylindrical gear is m. 2s The gear ratio is the face gear pair transmission ratio.

[0137] S12: Based on the coordinate transformation matrix M 2s and the relative velocity v at the engagement point s2 The equation of the tooth surface of the face gear, which is conjugate to the cylindrical gear, is obtained by solving the following formula:

[0138]

[0139] Where, φ θ =θ s +θ os ±φ s m is the modulus, h a h is the tooth tip height. f The tooth root height is given by z, which is the z-coordinate of the face gear.

[0140] In some embodiments, the coordinate transformation matrix M 2s A matrix used to transform points in the coordinate system of a cylindrical gear to the coordinate system of a face gear. It takes into account the angular relationship between the cylindrical gear and the face gear, and realizes the coordinate transformation between the two different coordinate systems through matrix operations. It is a key tool for establishing the tooth surface equation of a face gear, ensuring the correspondence between the coordinates of the two gear tooth surfaces and the accuracy of meshing.

[0141] Relative velocity v at the engagement point s2 It refers to the velocity of the face gear relative to the cylindrical gear at the meshing point during the meshing process of a cylindrical gear and a face gear. It reflects the kinematic relationship between the two gears at the contact point. Its component in the direction of the common normal at the meshing point must be zero. This is the basic condition for correct gear meshing and an important basis for deriving the meshing equation and solving the face gear tooth surface equation.

[0142] Cylindrical gear rotation angle φ s The rotational speed ω of a cylindrical gear refers to the angle of rotation about its own axis. It is a parameter describing the motion state of the cylindrical gear, and its rate of change is the rotational speed ω of the cylindrical gear. sThe size of the rotation angle determines the position of the point on the tooth surface of the cylindrical gear, and as φ... s As the gear changes, different points on the cylindrical gear tooth surface successively come into contact with and mesh with the face gear tooth surface.

[0143] The face gear rotation angle φ2 refers to the angle of rotation of the face gear about its own axis, which is different from the cylindrical gear rotation angle φ. s There exists a fixed transmission ratio relationship (φ2=(z_s / z_2)×φs, where z_s is the number of teeth of the cylindrical gear and z_2 is the number of teeth of the face gear), which reflects the motion state of the face gear and is an indispensable parameter in coordinate transformation and solving the meshing equation.

[0144] Cylindrical gear rotation speed ω s ωs refers to the angle of rotation of a cylindrical gear per unit time. It is a parameter describing the speed of a cylindrical gear and determines the angular velocity and power transmitted by the gear transmission. In the calculation of relative velocity, ωs directly affects the magnitude of the relative velocity, and thus affects the solution of the meshing equation.

[0145] Face gear pair transmission ratio m 2s This refers to the ratio of the rotational speeds of a face gear and a cylindrical gear, which is equal to the inverse ratio of the number of teeth of the two gears, i.e., m. 2s =ω2 / ω s =z s / z2 reflects the motion transmission relationship between the two gears. It is an important parameter for determining the rotation angle relationship between the two gears and calculating the relative speed, thus ensuring the constant ratio transmission characteristics of the gear transmission.

[0146] The module *m* is one of the fundamental parameters of a gear and an important indicator determining its dimensions. A larger module results in a larger tooth pitch, greater tooth thickness and height, and a stronger load-bearing capacity. In the tooth surface equation of a face gear, the module *m* is used to determine the addendum *h*. a (h a =m×h a* h a* (for the tooth tip height coefficient) and tooth root height h f (h f =m×(h) a +c*), where c* is the clearance coefficient, directly affects the tooth profile size and meshing performance of the face gear.

[0147] Tooth tip height h a h refers to the distance from the pitch circle to the addendum circle of a gear, and its size is determined by the module and the addendum coefficient. a =m×h a* The size of the tooth tip height affects the meshing depth and tooth tip strength of the gear. A reasonable tooth tip height setting can ensure normal meshing of the gear and sufficient tooth tip load-bearing capacity.

[0148] Tooth root height h fh refers to the distance from the pitch circle to the root circle of a gear. f =m×(h) a* +c*), where c* is the clearance coefficient, which is used to ensure that there is a certain clearance between the tooth tip and tooth root of the two meshing gears, to avoid interference between the tooth tip and tooth root, and also to provide space for storing lubricating oil and reduce tooth surface wear.

[0149] The tooth surface equation of the face gear derived from the above formula corresponds to the equiangular helix and involute tooth profile of the cylindrical gear according to different tooth height ranges. It achieves precise conjugation with the combined tooth profile cylindrical gear, ensuring good meshing between the face gear tooth profile and the cylindrical gear tooth profile. This allows the face gear to inherit the advantages of the combined tooth profile, taking into account both load-bearing capacity and transmission smoothness, and adapting to high-speed and heavy-load working conditions.

[0150] In some embodiments, in step S2, calculating the coordinates of the tooth surface points based on the tooth surface equation to generate a three-dimensional digital model of the cylindrical gear and the face gear includes:

[0151] The left and right tooth profile curves are generated from the coordinate points of the left and right tooth profiles, then stretched into the left and right tooth surfaces, and then the tooth surfaces are arrayed to obtain the tooth surfaces of all the teeth of the cylindrical gear, thus generating a three-dimensional digital model of the cylindrical gear.

[0152] In a computer simulation environment, the three-dimensional digital model of the cylindrical gear is set as a machining tool. Boolean subtraction is used to simulate the cutting process of the machining tool on the face gear blank to obtain a face gear single tooth with machining marks. The face gear single tooth is then surface-finished to obtain a smooth single tooth. All the smooth single teeth are arranged in an array to obtain the three-dimensional digital model of the face gear.

[0153] The three-dimensional digital model of the cylindrical gear is assembled with the three-dimensional digital model of the face gear to obtain the face gear pair transmission model.

[0154] In some embodiments, the verification of tooth profile morphology in step S2 includes:

[0155] Compare the tooth surface morphology of the rendered face gear transmission model with the expected tooth surface morphology to check whether there is tooth profile distortion or undercutting phenomenon in the tooth surface morphology of the face gear transmission model.

[0156] By verifying the tooth profile morphology, issues such as tooth profile distortion and undercut can be identified during the 3D modeling stage. This avoids workpiece scrap due to design flaws during subsequent solid machining, reducing R&D costs and timelines. It is particularly suitable for the design of high-precision, high-cost parts such as aerospace face gears. The verified 3D model ensures that the tooth profile shape meets the design expectations, with a smooth transition between the involute and equiangular helix, and no obvious defects. This guarantees a continuous distribution of contact points during meshing of the face gear pair, avoiding localized stress concentration caused by tooth profile abnormalities, improving transmission smoothness, and reducing tooth surface wear.

[0157] In some embodiments, in step S3, a contact boundary is determined based on the plurality of feature points, the tooth surface of the face gear is divided into a plurality of meshing regions according to the contact boundary, and the coordinates of a plurality of discrete points are calculated in each of the meshing regions, specifically including:

[0158] Step S3, which involves determining the contact boundary based on four feature points and calculating discrete points within each engagement region, specifically includes:

[0159] Based on four characteristic points—the tooth tip engagement point, the tooth root engagement point, the apex of the internal tooth profile, and the tooth root point of the external tooth profile—and their corresponding meshing angles, two contact boundary lines are determined. These two contact boundary lines divide the tooth surface of the face gear into three meshing regions. The two contact boundary lines include a first contact boundary line and a second contact boundary line. The three meshing regions include:

[0160] The first region, located between the tooth tip engagement point and the first contact boundary line, is used to complete the discretization of the upper half of the external tooth profile and the tooth tip of the face gear.

[0161] The second region, located between the first contact boundary line and the second contact boundary line, is used to complete the discretization of the lower half of the outer tooth profile and the upper half of the inner tooth profile of the face gear.

[0162] The third region, located between the second contact boundary line and the transition curve, is used to complete the discretization of the face gear transition curve and the lower half of the internal tooth profile.

[0163] In this embodiment, the first contact boundary line refers to the point of engagement P at the tooth tip. in and the vertex P of the internal tooth profile ha Using the endpoints, fit a curve on the tooth surface. The fitting is based on the meshing angle between the two points, from φ. sin Gradient to φ sha The corresponding tooth profile expansion angle is from θ sin Gradient to θ sha Furthermore, the curve must be consistent with the direction of the tooth surface contact line (the tooth surface contact line of a face gear is a non-equidistant oblique line).

[0164] The second contact boundary line refers to the point P, the apex of the internal tooth profile. haand tooth root engagement point P out Using the endpoints as examples, similarly fit the curve, with the meshing angle starting from φ. sha Gradient to φ sout The tooth profile expansion angle is from θ sha Gradient to θ sout This ensures a smooth transition of the curve without obvious inflection points.

[0165] In some embodiments, in step S3, the engagement angle corresponding to the tooth tip engagement point, the engagement angle corresponding to the tooth vertex of the internal tooth profile, the engagement angle corresponding to the tooth root engagement point, and the engagement angle corresponding to the tooth root point of the external tooth profile are obtained by substituting the inner diameter, outer diameter, tooth height parameters of the face gear and the tooth tip development angle parameter of the cylindrical gear into the face gear tooth surface equation for solving.

[0166] The preferred tooth surface discretization method is as follows:

[0167] (1) When the meshing region is at the engagement point P in Between the first contact boundary line and the first contact boundary line, the range of θs is (θ sha ,θ sin By determining the outer diameter of the face gear, different meshing angles φ can be obtained. s The corresponding span angle θ s This completes the upper half of the external tooth profile, T. 1in Discretization; based on the obtained meshing angle φ sin and φ sha With known parameters, the tooth tip T can be completed. top Discretization.

[0168] (2) When the meshing area is between the first contact boundary line and the second contact boundary line, i.e. θ s The range of values ​​is (θ) shf ,θ sha By determining the inner and outer diameters of the face gear, different meshing angles φ can be obtained. s The corresponding span angle θ s This completes the lower half of the external tooth profile, T. 2in and the upper half of the internal tooth profile T 1out Discretization.

[0169] (3) When the meshing area is between the second contact boundary line and the transition curve, i.e. θ s The range of values ​​is (θ) sout ,θ shf Determine the development angle θ on the transition curve. s* This allows for the discretization of the transition curve; by determining the inner diameter of the face gear, different meshing angles φ can be obtained. s The corresponding span angle θ s This completes the discretization of the lower half of the internal tooth profile, T2out.

[0170] (4) Based on the meshing angle φ corresponding to the internal and external tooth profiles s Then, based on the inner and outer radii of the face gear, different meshing angles φ can be obtained. s The angle of development θ corresponding to the inner and outer radii s This completes the discretization of the contact line of the gear tooth surface.

[0171] The meshing angles corresponding to feature points on the tooth surface of a faceted gear can be solved using a discretization method. The method for calculating the meshing angles is as follows:

[0172] (1) Calculate the engagement angle φ corresponding to the tooth tip of the outer tooth profile. sin ;

[0173] The equation (x2) 2 +y2 2 ) 1 / 2 =R2 and z2 = -(rh a Substituting into the equation for the tooth surface of a faceted gear, the engagement angle φ can be calculated. sin Where R2 is the outer diameter of the face gear.

[0174] (2) Calculate the meshing angle φ corresponding to the tooth tip of the gear tooth profile. sha ;

[0175] The equation (x2) 2 +y2 2 ) 1 / 2 =R1 and z2 = -(rh a Substituting into the equation for the tooth surface of a face gear, the meshing angle φ can be calculated. sha Where R1 is the inner diameter of the face gear.

[0176] (3) Calculate the engagement angle φ corresponding to the starting point of the tooth profile transition curve. sout ;

[0177] The equation (x2) 2 +y2 2 ) 1 / 2 Substituting R1 into the equation for the tooth surface of the face gear, and then changing the development angle θ corresponding to the tooth tip of the cylindrical gear... s =θ sha =ln(r a Substituting / r0) / k into the equation, the disengagement angle φ can be calculated. sout .

[0178] (4) Calculate the meshing angle φ corresponding to the end point of the transition curve of the external tooth profile of the gear teeth. shf ;

[0179] The equation (x2) 2 +y2 2 ) 1 / 2 Substituting R2 into the equation for the tooth surface of the face gear, and then changing the development angle θ corresponding to the tooth tip of the cylindrical gear...s =θ sha =ln(r a Substituting / r0) / k into the equation, the meshing angle φ can be calculated. shf .

[0180] The above scheme decomposes the complex continuous tooth surface into three regions with clearly defined characteristics by locating feature points and dividing them by boundary lines. Discrete points are selected for each region, avoiding potential issues of insufficient local point density or redundancy during overall discretization. This improves the accuracy of discretization and provides an accurate data foundation for subsequent performance parameter calculations (such as slip ratio and principal curvature). Each region corresponds to different meshing stages of the gear teeth (engagement, intermediate engagement, and disengagement). After discretization, the meshing state of each stage can be analyzed separately. For example, the first region reflects the contact characteristics when the tooth tip engages, and the third region reflects the stress distribution when the tooth root disengages, which helps to accurately locate weak points in performance.

[0181] In some embodiments, in step S4, the total length of the tooth surface contact line of the face gear pair is obtained by iteratively calculating the spatial coordinates of adjacent discrete points on the contact line, and the calculation formula is as follows:

[0182]

[0183] Where L represents the total length of the tooth surface contact line, n represents the number of discrete points, and x i y i z i Let x and x represent the spatial coordinates of the i-th discrete point, respectively. i+1 y i+1 z i+1 Let l represent the spatial coordinates of the (i+1)th discrete point, respectively. i This represents the distance between the (i+1)th discrete point and the ith discrete point.

[0184] When calculating the total length L of the tooth surface contact line, discrete points on the contact line are first extracted from the three meshing regions and sorted in ascending order of meshing angle to obtain the discrete point sequence P1(x1,y1,z1), P2(x2,y2,z2), ..., P n+1 (x n+1 ,y n+1 ,z n+1 There are a total of n+1 points, corresponding to n line segments.

[0185] Then, the distance between adjacent points is calculated, specifically including: according to the distance formula between two points in space, the length li of each line segment (i=1,2,…,n) is calculated, and then the lengths of all line segments are added together to obtain the total length L of the contact line. The accumulated result is the total length of the contact line during the entire meshing process. If the density of discrete points is high enough (e.g., n≥20), the calculation result can approximately replace the actual length of the continuous contact line.

[0186] The formula for calculating the slip ratio is as follows:

[0187]

[0188] σ1 reflects the slip ratio of the face gear relative to the cylindrical gear, and σ2 reflects the slip ratio of the cylindrical gear relative to the face gear. If the absolute values ​​of σ1 and σ2 are small and change gradually, it indicates that the relative slip of the tooth surfaces is small and the wear is low.

[0189] Where ΔS1 and ΔS2 are the moving arc lengths, and dS1 and dS2 are the differential arc lengths of the two conjugate tooth profiles. The formulas for calculating dS1 and dS2 are as follows:

[0190]

[0191] The formula for calculating the principal curvature is as follows:

[0192]

[0193] Where L, M, and N represent the second fundamental quantities of the surface, and E, F, and G represent the first fundamental quantities of the surface. The formula is based on the principal curvature solution theory in surface differential geometry. Two principal curvatures are obtained by solving the characteristic equation, where "+" corresponds to the maximum principal curvature K1 and "-" corresponds to the minimum principal curvature K2. The numerical signs of the principal curvatures reflect the concavity and convexity direction of the tooth surface (positive for convex and negative for concave).

[0194] In this embodiment, the total length L of the tooth surface contact line refers to the total length of the contact line on the tooth surface during the meshing process of the face gear and the cylindrical gear. It is calculated by summing the spatial distances between all adjacent discrete points on the contact line and is a key indicator for evaluating the tooth surface contact area. The longer the contact line, the larger the contact area, the smaller the load per unit area of ​​the tooth surface, and the stronger the load-bearing capacity.

[0195] The moving arc lengths ΔS1 and ΔS2 refer to the arc lengths along the respective tooth profile curves of the cylindrical gear tooth profile (ΔS1) and the face gear tooth profile (ΔS2) at the contact point during an extremely short meshing time. They reflect the relative motion amplitude of the two tooth profiles at the instant of contact and are the basic parameters for calculating the slip ratio. Their difference directly affects the magnitude of the slip ratio.

[0196] The arc length differentials dS1 and dS2 refer to the limit values ​​of the moving arc lengths ΔS1 and ΔS2 as they approach zero. They are differential parameters describing the local length changes of the tooth profile curve. They are obtained by differentiating the tooth surface equation and can accurately reflect the curvature characteristics of the tooth profile curve at the contact point, ensuring the accuracy of the slip ratio calculation.

[0197] The second fundamental quantities (L, M, N) of the surface refer to the parameters that describe the local geometric characteristics of the tooth surface of the face gear. They are calculated based on the first-order partial derivatives of the tooth surface equation and reflect the rate of change of the length of the tooth surface in different directions. They are the basis for constructing the differential geometric model of the tooth surface and are used to solve for surface characteristic parameters such as principal curvature and Gaussian curvature.

[0198] The first fundamental quantities of the surface (E, F, G) are parameters that describe the bending characteristics of the tooth surface of a face gear. They are calculated based on the second-order partial derivatives of the tooth surface equation and the unit normal vector. They reflect the degree of concavity and convexity of the tooth surface at the contact point and are directly related to the distribution of contact stress on the tooth surface. They are the core parameters for calculating the principal curvature.

[0199] Principal curvatures K1 and K2 refer to the curvatures of the tooth surface in two mutually perpendicular principal directions at the contact point. They are the extreme values ​​of the surface curvature. K1 is the maximum principal curvature, and K2 is the minimum principal curvature. The value and sign of the principal curvatures determine the bending shape of the tooth surface and directly affect the tooth surface contact strength and the conditions for the formation of the lubricating film.

[0200] The above-mentioned scheme improves the efficiency of face gear performance evaluation and optimization from multiple dimensions by calculating the total length of the tooth surface contact line, the slip ratio, and the principal curvature. The total length of the contact line can intuitively reflect the tooth surface contact area, and combined with load parameters, it can accurately estimate the contact stress, providing a quantitative basis for load-bearing capacity assessment and verifying the effectiveness of the combined tooth profile in improving load-bearing capacity. The slip ratio calculation clearly presents the relative sliding law of the tooth surface. A uniform and small slip ratio can predict slow tooth surface wear, which helps in the design of the maintenance cycle of the transmission system. The principal curvature result reflects the bending characteristics of the tooth surface. A gently changing principal curvature means that the contact stress distribution is uniform and the contact strength is high, and it can also be used to optimize the tooth profile parameters in a targeted manner. At the same time, based on the same method, the key performance indicators of the combined tooth profile and the traditional involute tooth profile face gear can be compared, quantitatively highlighting the advantages of the combined tooth profile, providing data support for patent innovation, and promoting the application of face gear transmission technology in the field of high-end equipment.

[0201] In a second aspect, the present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the face gear tooth surface discretization method as described in the first aspect of the present invention.

[0202] The computer-readable storage medium may be volatile memory or non-volatile memory, or may include both volatile and non-volatile memory.

[0203] The non-volatile memory may be a read-only memory (ROM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), a magnetic random access memory (FRAM), a flash memory, a magnetic surface memory, an optical disc, or a compact disc read-only memory (CD ROM); the magnetic surface memory may be a disk storage device or a magnetic tape storage device.

[0204] The volatile memory may be random access memory (RAM), which serves as an external cache. By way of example, but not limitation, many forms of RAM are available, such as static random access memory (SRAM), synchronous static random access memory (SSRAM), dynamic random access memory (DRAM), synchronous dynamic random access memory (SDRAM), double data rate synchronous dynamic random access memory (DDRSDRAM), enhanced synchronous dynamic random access memory (ESDRAM), synclink dynamic random access memory (SLDRAM), and direct memory bus random access memory (DRRAM). The computer-readable storage media described in the embodiments of the present invention are intended to include these and any other suitable types of memory.

[0205] like Figure 14 As shown, in a third aspect, the present invention provides an electronic device 10, including a processor 101 and a storage medium 102, wherein a computer program is stored on the storage medium, and the computer program, when executed by the processor, implements the face gear tooth surface discretization method as described in the first aspect of the present invention.

[0206] In some embodiments, the processor may be implemented by software, hardware, firmware, or a combination thereof, and may use at least one of the following: circuit, single or multiple application-specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field-programmable gate arrays (FPGAs), central processing units (CPUs), controllers, microcontrollers, and microprocessors, thereby enabling the processor to execute some or all of the steps or any combination thereof in the face gear tooth surface discretization method described in the various embodiments of this application.

[0207] Finally, it should be noted that although the above embodiments have been described in the text and drawings of this application, this should not limit the scope of patent protection of this application. Any technical solutions that are based on the essential concept of this application and utilize the content described in the text and drawings of this application, resulting in equivalent structural or procedural substitutions or modifications, as well as the direct or indirect application of the technical solutions of the above embodiments to other related technical fields, are all included within the scope of patent protection of this application.

Claims

1. A method for discretizing the tooth surface of a face gear, characterized in that, The method includes the following steps: S1: Establish tooth surface equation: Based on the basic parameters of the gear, calculate the tooth surface equation of the cylindrical gear with the combined tooth profile of involute and equiangular helix. Based on the meshing principle of face gears and coordinate transformation theory, solve to obtain the tooth surface equation of the face gear conjugate with the cylindrical gear. The cylindrical gear and the face gear constitute a face gear pair. S2: Generate and visualize 3D models: Calculate the coordinates of the tooth surface points according to the tooth surface equation, generate 3D digital models of cylindrical gears and face gears, and render and display them for verification of tooth profile morphology. S3: Feature point solution and tooth surface discretization: Based on the tooth surface equation of the face gear, solve for the coordinates of multiple feature points on the tooth surface of the face gear and their corresponding meshing angles. The multiple feature points include the tooth tip engagement point, the tooth root engagement point, the tooth vertex of the internal tooth profile and the tooth root point of the external tooth profile. Based on the multiple feature points, determine the contact boundary. According to the contact boundary, divide the tooth surface of the face gear into multiple meshing regions. In each of the meshing regions, calculate the coordinates of multiple discrete points and the meshing angles corresponding to each discrete point. S4: Performance Analysis: Based on the coordinates of the discrete points obtained in step S3 and the meshing angles corresponding to each discrete point, calculate the total length of the tooth surface contact line, the sliding ratio, and the principal curvature of the face gear pair, and generate the corresponding distribution map. S5: Result Evaluation: Based on the analysis results of the total contact line length, sliding rate distribution and principal curvature distribution obtained in step S4, the load-bearing capacity and meshing transmission performance of the face gear pair are quantitatively evaluated.

2. The method for discretizing the tooth surface of a face gear as described in claim 1, characterized in that, In step S1, based on the basic parameters of the gear, the calculation formula for the equation of the cylindrical gear tooth surface with a combined involute and equiangular helix tooth profile is as follows: Where k is the constant of the equiangular helix, r0 is the initial radius of the equiangular helix, and r is the pitch circle radius of the cylindrical gear; bs μ is the base circle radius of the cylindrical gear. s For the tooth width parameter, θ s Let θ be the tooth profile expansion angle. os α is the angle parameter from the vertical axis of symmetry to the starting point of the tooth profile, ra is the radius of the addendum circle, and rf is the radius of the dedendum circle.

3. The method for discretizing the tooth surface of a face gear as described in claim 2, characterized in that, In step S1, based on the meshing principle of face gears and coordinate transformation theory, the equation of the face gear tooth surface conjugate with the cylindrical gear is obtained by solving the following: S11: Based on the principle of face gear meshing and coordinate transformation theory, the coordinate transformation matrix M is obtained. 2s and the relative velocity v at the meshing point s2 ; The coordinate transformation matrix M 2s The calculation formula is as follows: The relative velocity v at the engagement point s2 The calculation formula is as follows: Where, φ s φ2 is the rotation angle of the cylindrical gear, ω is the rotation angle of the face gear. s The rotational speed of the cylindrical gear is m. 2s The gear ratio is the face gear pair transmission ratio. S12: Based on the coordinate transformation matrix M 2s and the relative velocity v at the engagement point s2 Solving for r2, the equation of the tooth surface of the face gear conjugate with the cylindrical gear is obtained, and the calculation formula is as follows: Where, φ θ =θ s +θ os ±φ s m is the modulus, h a h is the tooth tip height. f Tooth root height, z is the z-coordinate of the face gear, θ s θ is the meshing angle. os Let φ be the tooth aspect ratio. s φ1 is the rotation angle of the cylindrical gear, and φ2 is the rotation angle of the face gear.

4. The method for discretizing the tooth surface of a face gear as described in claim 1, characterized in that, In step S2, the coordinates of the tooth surface points are calculated based on the tooth surface equation to generate a three-dimensional digital model of the cylindrical gear and the face gear, including: The left and right tooth profile curves are generated from the coordinate points of the left and right tooth profiles, then stretched into the left and right tooth surfaces, and then the tooth surfaces are arrayed to obtain the tooth surfaces of all the teeth of the cylindrical gear, thus generating a three-dimensional digital model of the cylindrical gear. In a computer simulation environment, the three-dimensional digital model of the cylindrical gear is set as a machining tool. Boolean subtraction is used to simulate the cutting process of the machining tool on the face gear blank to obtain a face gear single tooth with machining marks. The face gear single tooth is then surface-finished to obtain a smooth single tooth. All the smooth single teeth are arranged in an array to obtain the three-dimensional digital model of the face gear. The three-dimensional digital model of the cylindrical gear is assembled with the three-dimensional digital model of the face gear to obtain the face gear pair transmission model.

5. The method for discretizing the tooth surface of a face gear as described in claim 4, characterized in that, In step S2, the verification of the tooth profile includes: Compare the tooth surface morphology of the rendered face gear transmission model with the expected tooth surface morphology to check whether there is tooth profile distortion or undercutting phenomenon in the tooth surface morphology of the face gear transmission model.

6. The method for discretizing the tooth surface of a face gear as described in claim 1, characterized in that, In step S3, the contact boundary is determined based on the multiple feature points, and the tooth surface of the face gear is divided into multiple meshing regions according to the contact boundary. The coordinates of multiple discrete points are calculated within each meshing region. Specifically, this includes: Step S3, which involves determining the contact boundary based on four feature points and calculating discrete points within each engagement region, specifically includes: Based on four characteristic points—the tooth tip engagement point, the tooth root engagement point, the apex of the internal tooth profile, and the root point of the external tooth profile—and their corresponding meshing angles, two contact boundary lines are determined. These two contact boundary lines divide the tooth surface of the face gear into three meshing regions. The two contact boundary lines include a first contact boundary line and a second contact boundary line. The first contact boundary line is the contact line introduced by the apex of the internal tooth, and the second contact boundary line is the contact line introduced by the root point of the external tooth. The three meshing regions include: The first region, located between the tooth tip engagement point and the first contact boundary line, is used to complete the discretization of the upper half of the external tooth profile and the tooth tip of the face gear. The second region, located between the first contact boundary line and the second contact boundary line, is used to complete the discretization of the lower half of the outer tooth profile and the upper half of the inner tooth profile of the face gear. The third region, located between the second contact boundary line and the transition curve, is used to complete the discretization of the face gear transition curve and the lower half of the internal tooth profile.

7. The method for discretizing the tooth surface of a face gear as described in claim 6, characterized in that, In step S3, the engagement angle corresponding to the tooth tip engagement point, the engagement angle corresponding to the tooth vertex of the internal tooth profile, the engagement angle corresponding to the tooth root engagement point, and the engagement angle corresponding to the tooth root point of the external tooth profile are obtained by substituting the inner diameter, outer diameter, tooth height parameters of the face gear and the tooth tip development angle parameter of the cylindrical gear into the face gear tooth surface equation for solving.

8. The method for discretizing the tooth surface of a face gear as described in claim 1, characterized in that, In step S4, the total length of the tooth surface contact line of the face gear pair is obtained by iteratively calculating the spatial coordinates of adjacent discrete points on the contact line, as shown in the following formula: Where L represents the total length of the tooth surface contact line, n represents the number of discrete points, and x i y i z i Let x and x represent the spatial coordinates of the i-th discrete point, respectively. i+1 y i+1 z i+1 Let l represent the spatial coordinates of the (i+1)th discrete point, respectively. i This represents the distance between the (i+1)th discrete point and the ith discrete point. The formulas for calculating the slip ratio σ1 of the cylindrical gear and the slip ratio σ2 of the face gear are as follows: Where ΔS1 and ΔS2 are the moving arc lengths, and dS1 and dS2 are the differential arc lengths of the two conjugate tooth profiles. The formulas for calculating dS1 and dS2 are as follows: In the formula, (x1, y1, z1) and (x2, y2, z2) are the tooth surface coordinates of the cylindrical gear and the face gear, respectively. The formula for calculating the principal curvature is as follows: Where E, F, and G represent the first fundamental quantities of the surface, L, M, and N represent the second fundamental quantities of the surface, K1 represents the maximum principal curvature, and K2 represents the minimum principal curvature.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the face gear tooth surface discretization method as described in any one of claims 1 to 8.

10. An electronic device having a computer program stored thereon, characterized in that, It includes a processor and a storage medium, wherein a computer program is stored on the storage medium, and the computer program, when executed by the processor, implements the face gear tooth surface discretization method as described in any one of claims 1 to 8.

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