Involute gear tooth surface three-dimensional morphology parametric modeling method based on fractal features

Abstract

The invention discloses an involute gear tooth surface three-dimensional morphology parameterized modeling method based on fractal features, which scans a real involute gear tooth surface through a three-dimensional morphology instrument, is different from a previous two-dimensional fractal theory, introduces a three-dimensional fractal theory W-M function to characterize a tooth surface microstructure, and achieves the parameterized modeling of the involute gear tooth surface three-dimensional morphology. And key parameters of a fractal theory W-M function are calculated through a rough length method. An ideal involute gear tooth surface discrete model is programmed through MATLAB, an involute tooth surface is expanded and overlaid with a three-dimensional fractal theory W-M function to obtain a real involute tooth surface microstructure, and the obtained tooth surface microstructure is overlaid on the ideal involute tooth surface model to obtain a real involute tooth surface discrete model. And finally, superposing the microstructure on an ideal involute tooth surface model to obtain a real involute tooth surface point cloud model. Real involute tooth surface point cloud data with a microstructure is imported into three-dimensional modeling software to generate a three-dimensional model, and theoretical support is provided for simulating real tooth surface contact simulation.

Description

Technical Field

[0001] The invention relates to the technical field of gear involute tooth surface tooth profile design and simulation, and relates to a fractal feature-based parameterized modeling method for three-dimensional morphology of involute gear tooth surfaces. Background Art

[0002] Involute gear transmission is the most widely used type of gear transmission at present. The simulation research of the tooth surface contact of the involute gear is generally based on the smooth tooth surface contact theory. Although the smooth tooth surface can also obtain characteristic values ​​such as contact pressure close to the real tooth surface in the finite element simulation, it is still inaccurate compared with the real tooth surface contact. In patent CN113204890A, the involute gear tooth profile equation is corrected, but it is based on the two-dimensional WM function, and the real gear tooth surface cannot be represented by an involute. The real surface of the involute gear is extremely complex, so a single involute can only represent part of the surface features of the tooth surface, and cannot represent the complete gear tooth surface; based on this, the present invention proposes an improvement, which involves a three-dimensional morphology parameterized modeling method of the involute gear tooth surface based on fractal features.

[0003] The microscopic morphology of the surface of mechanical parts is one of the main reasons affecting the tooth surface wear behavior and gear transmission accuracy. Therefore, if it is possible to discretize the gear tooth surface data and improve the discretized data based on the microscopic morphology parameters, and model the actual rough surface in the finite element software, the meshing process of the real involute gear tooth surface can be simulated to obtain more realistic data. The three most critical steps to solve the above problems are: 1. Acquisition of the three-dimensional microscopic morphology parameters of the real tooth surface; 2. Establishment of the three-dimensional microscopic morphology model of the tooth surface; 3. Establishment of the involute tooth surface based on the microscopic morphology parameters. After the real involute surface is obtained after discretization, it is imported into the three-dimensional modeling software and the finite element simulation software to simulate and establish the real tooth surface contact, and obtain the characteristic values ​​such as the real tooth surface contact pressure. Summary of the invention

[0004] The purpose of the present invention is to propose a design method for improving the three-dimensional size of the involute tooth surface based on tooth surface morphology parameters, to realize the correction and precision of the involute tooth surface roughness, and to provide a theoretical basis for simulating the contact of a truly rough tooth surface.

[0005] To achieve the above purpose, the technical solution adopted by the present invention is:

[0006] S1: The real involute tooth profile parameters are obtained by using a three-dimensional topography scanner. According to the roughness length method, the scanned involute tooth profile parameters are fitted to obtain the key parameters D and G of the fractal theory WM function based on the real involute tooth profile.

[0007] S2: Establish an ideal involute tooth surface. Use MATLAB to establish an ideal involute tooth surface and discretize the tooth surface data.

[0008] S3: Based on the fractal theory WM function, the involute tooth surface is unfolded and the WM function micro-morphology is superimposed on the ideal involute unfolded tooth surface, and the involute unfolded tooth surface model with micro-morphology is established using MATLAB.

[0009] S4: The tooth surface data obtained by unfolding the involute tooth surface with micro-morphology in MATLAB is superimposed on the discretized data generated by S2 to obtain a real involute tooth surface model with micro-morphology and curvature radius.

[0010] The specific process is as follows Figure 1 shown.

[0011] The involute tooth surface point cloud data with micro-morphology established in MATLAB is imported into mature 3D modeling software to generate a 3D model, which provides reliable theoretical and technical support for contact simulation experiments based on real rough tooth surfaces and obtains real characteristic values ​​such as tooth surface contact pressure and tooth surface clearance.

[0012] The S1 specifically includes the following steps:

[0013] A three-dimensional topography scanner is used to scan the tooth profile of the involute gear to obtain the real tooth surface microscopic morphology parameters. According to the real tooth surface morphology characteristics obtained by scanning, the roughness length method is used for processing and analysis to obtain the key parameters D and G of the fractal theory WM function based on the real involute tooth profile.

[0014] The roughness length method is based on the standard deviation of the surface profile height distribution, i.e., R q .

[0015] According to the reference, R based on the measured surface profile data q1 It can be expressed as:

[0016]

[0017] Where: τ is the sampling point spacing, which is generally a constant; R q1 represents the standard deviation of the microscopic profile height; N(τ) is the number of sampling points separated by the sampling interval τ; z(x i ) is x i The corresponding contour height of the microstructure.

[0018] According to the reference, R based on fractal theory parameters q2 It can be expressed as:

[0019]

[0020] Where: τ is the sampling point spacing, which is generally a constant; D represents the fractal dimension of the surface profile; G represents the characteristic length scale of the fractal surface; γ represents the size parameter of the spectral density, which is a constant greater than 1.

[0021] According to the relationship between the two, R q1 =R q2 , (1) and (2) can be combined to obtain the fractal dimension D and the characteristic length scale G of the fractal surface.

[0022] The S2 specifically includes the following steps:

[0023] Establish as Figure 2 In the rectangular coordinate system shown, the tooth shape of the involute gear is determined by the involute of a base circle. When a straight line BK rolls purely along a circle, the trajectory AK of any point K on the straight line is the involute of the circle.

[0024] During the gear meshing process, only the segmental involute part of the tooth profile participates in the meshing, so only the segmental involute tooth profile equation is established.

[0025] The rectangular coordinate equation of the involute tooth profile of the tooth profile AK segment is:

[0026]

[0027] Where:

[0028] x k and k are the two rectangular coordinates of any point K on the involute in the coordinate system, mm; r k is the distance from point K to point O, mm; the radius of the base circle is: r b =rcosα, the involute function is:θ k =invα k =tanα k -α k .

[0029] According to the ideal involute equation, MATLAB is used to establish Figure 3 The ideal involute tooth profile model is shown in Figure 1. The ideal involute tooth profile is obtained by setting the tooth width b on the y-axis and using MATLAB to establish the following Figure 4 The discretization model of the involute tooth surface is shown, and the ideal involute tooth surface point cloud data is obtained.

[0030] The S3 specifically includes the following steps:

[0031] Since the machined involute tooth surface has fractal characteristics and the tooth surface with fractal characteristics has local and overall self-similarity, the present invention adopts the function simulation method used in the fractal contact model to characterize the contour curve of the rough surface, and the expression is:

[0032]

[0033] x=s k =r b ×tan(α k )

[0034] Where:

[0035] z(x,y) represents the height of the micro-morphology of the involute tooth surface; x represents the arc length sk of the involute surface after expansion, y represents the tooth surface width; D represents the fractal dimension (indicating the proportion of micro-convex bodies in the linear space), its value is [2,3], G represents the fractal roughness (size is independent of the size of the micro-convex body set), and the D and G values ​​are obtained by step S2; φ m,n It is represented by a uniform random phase, whose value satisfies [0,2π]; L is the sampling length; γ is the profile frequency density, which is generally 1.5; L a It represents the minimum cutoff wavelength; M represents the degree of microconvexity coincidence, which ranges from 1 to 100 and is generally 10; n represents the spatial frequency index, which is generally an integer, and its lower limit is 0 or n min = int[log(1 / L) / log(γ)], the upper limit is n max = int[log(L / L s ) / log(γ)] or n max =int[log(1 / Δx) / log(γ)].

[0036] For formula (5), γ = 1.5, which is a constant. The contour length coordinates x, y and frequency index n are all independent variables. The size of z(x, y) depends on the fractal dimension D and the fractal surface characteristic length scale G. The fractal dimension D and the fractal surface characteristic length scale G obtained in step S1 are substituted into formula (5), and then MATLAB is used to obtain the following: Figure 5 The tooth surface micro-morphology image z(x, y) is developed based on the involute of the real tooth surface, and the obtained involute developed tooth surface data is discretized to obtain the tooth surface point cloud model.

[0037] The S4 specifically includes the following steps:

[0038] The discrete ideal involute tooth surface point cloud model obtained in step S2 is corrected, and the point cloud model of the involute unfolded tooth surface micro-morphology z(x,y) is discretized and superimposed on the discrete ideal involute tooth surface point cloud model obtained in step S2. The x of the unfolded involute tooth surface is the arc length sk before unfolding, and y is the tooth width. The obtained micro-morphology z(x,y) is the morphology change in the direction of the radius of curvature rk of the involute surface before unfolding; for the radius of curvature rk corresponding to any point on the involute surface, the micro-morphology profile height z(x,y) is superimposed, that is, the radius of curvature rkz after superimposing the micro-morphology is obtained, and then the correction equation of the involute tooth profile obtained after the corresponding superimposed micro-morphology is obtained:

[0039]

[0040] r kz =r k +z (7)

[0041] Where:

[0042] x kz and kz are the two rectangular coordinates of any point K on the involute after correction in the coordinate system, mm; r k is the distance from point K to point O, mm; z is the profile height of the micro-morphology of the involute tooth surface; r kz is the arc radius corresponding to any point on the corrected involute, mm; the involute function is: θ k =invα k =tanα k -α k .

[0043] The point cloud data of the involute tooth surface with microscopic tooth surface morphology established in MATLAB is imported into mature 3D modeling software to generate a 3D model, providing reliable theoretical and technical support for contact simulation experiments based on real rough tooth surfaces.

[0044] The advantages and positive effects of the present invention are:

[0045] The present invention uses a three-dimensional profilometer to scan a real involute gear, introduces a fractal theory three-dimensional WM function to characterize the tooth surface micro-morphology, uses a roughness length method to calculate key parameters of the fractal theory WM function, and uses MATLAB theory WM function to characterize the tooth surface micro-morphology after the curved tooth surface is unfolded and superimposed on an ideal involute tooth surface point cloud model to obtain an involute tooth surface model with tooth surface micro-morphology. The involute tooth surface cloud data of the MATLAB micro-tooth surface morphology is imported into a mature three-dimensional modeling software to generate a three-dimensional model, thereby providing theoretical and technical support for simulating the contact simulation of a real rough tooth surface. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 It is a flow chart of the technical solution of the present invention;

[0047] Figure 2 It is the involute diagram and the establishment of coordinate system;

[0048] Figure 3 It is an ideal involute tooth profile model;

[0049] Figure 4 It is an ideal point cloud model of involute tooth surface;

[0050] Figure 5 It is a tooth surface micro-morphology image z(x,y) based on the involute expansion of the real tooth surface.

Claims

1. A three-dimensional parametric modeling method for involute gear tooth surface based on fractal features, characterized in that: The implementation steps of the method are as follows: S1: using a three-dimensional topography scanner to obtain the real involute tooth profile parameters, and according to the roughness length method, obtaining the key parameters D and G of the fractal theory function based on the real involute tooth profile; S2: Establish an ideal involute tooth surface, use MATLAB to establish the ideal involute tooth surface, and discretize the tooth surface data; S3: Based on the three-dimensional fractal theory WM function, the involute tooth surface is unfolded and the three-dimensional WM function micro-morphology is superimposed on the ideal involute unfolded tooth surface, and the involute unfolded tooth surface model with micro-morphology is established using MATLAB; S4: The tooth surface data of the involute unfolded tooth surface with micro-morphology obtained in MATLAB is superimposed on the discretized data generated by S2 to obtain a real involute tooth surface model with micro-morphology and curvature radius.

2. The method for parameterizing the three-dimensional morphology of involute gear tooth surface based on fractal features according to claim 1 is characterized in that: The specific steps of S1 are: The tooth profile of the involute gear is scanned by a three-dimensional profile scanner to obtain the microscopic parameters of the real tooth surface. According to the real tooth surface morphology characteristics obtained by scanning, the roughness length method is used for processing and analysis to obtain the key parameters D and G of the fractal theory WM function based on the real involute tooth profile. The roughness length method is based on the standard deviation of the surface profile height distribution, i.e., R q ; R based on measured surface profile data q1 It is expressed as: Where: τ is the sampling point spacing; R q1 represents the standard deviation of the microscopic profile height; N(τ) is the number of sampling points separated by the sampling interval τ; z(x i ) is x i The corresponding contour height of the microscopic topography; R based on fractal theory parameters q2 It is expressed as: Where: τ is the sampling point spacing; D represents the fractal dimension of the surface profile; G represents the characteristic length scale of the fractal surface; γ represents the size parameter of the spectral density; According to the relationship between the two, R q1 =R q2 , (1) and (2) can be combined to obtain the fractal dimension D and the characteristic length scale G of the fractal surface.

3. The method for parameterized modeling of three-dimensional morphology of involute gear tooth surface based on fractal features according to claim 1, characterized in that: The specific steps of S2 are: Establish a rectangular coordinate system. The tooth shape of the involute gear is determined by the involute of a base circle. When a straight line BK rolls purely along a circle, the trajectory AK of any point K on the straight line is the involute of the circle. In the process of gear meshing, only the segmental involute part of the tooth profile participates in the meshing, so only the segmental involute tooth profile equation is established; The rectangular coordinate equation of the involute tooth profile of the tooth profile AK segment is: Where: x k and k are the two rectangular coordinates of any point K on the involute in the coordinate system; r k is the distance from point K to point O; the radius of the base circle is: r b =rcosα, the involute function is:θ k =invα k =tanα k -α k ; According to the established ideal involute equation, the ideal involute tooth profile model is established using MATLAB; the value of the tooth width b is set on the y-axis to obtain the ideal involute tooth surface, and the involute tooth surface discretization model is established using MATLAB, and the ideal involute tooth surface point cloud data is obtained.

4. The method for parameterizing the three-dimensional morphology of involute gear tooth surface based on fractal features according to claim 1 is characterized in that: The specific steps of S3 are: The function simulation method used in the fractal contact model is used to characterize the contour curve of the rough surface, and its expression is: x=s k =r b ×tan(a k ) Where: z(x,y) represents the profile height of the involute tooth surface micro-morphology; x represents the arc length sk of the involute surface after expansion, y represents the tooth surface width; D represents the fractal dimension, and its value is [2,3]; G represents the fractal roughness, and the values ​​of D and G are obtained by step S2; φ m，n It is represented by a uniform random phase whose value satisfies [0, 2π]; L is the sampling length; γ is the profile frequency density; L a It represents the minimum cutoff wavelength; M represents the degree of microconvexity coincidence; n represents the spatial frequency index, whose lower limit is 0 or n min = int[log(1 / L) / log(γ)], the upper limit is n max = int[log(L / L s ) / log(γ)] or n max = int[log(1 / Δx) / log(γ)]; For formula (5), γ = 1.5, the contour length coordinates x, y and the frequency index n are all independent variables, and the size of z(x, y) depends on the fractal dimension D and the fractal surface characteristic length scale G. The fractal dimension D and the fractal surface characteristic length scale G obtained in step S1 are substituted into formula (5), and then MATLAB is used to obtain the involute expansion tooth surface micro-morphology image z(x, y) based on the real tooth surface, and the obtained involute expansion tooth surface data is discretized to obtain the tooth surface point cloud model.

5. The method for parameterized modeling of three-dimensional morphology of involute gear tooth surface based on fractal features according to claim 1, characterized in that: The specific steps of S4 are: The discrete ideal involute tooth surface point cloud model obtained in step S2 is corrected, and the point cloud model of the discrete involute unfolded tooth surface micro-morphology z(x, y) is superimposed on the discrete ideal involute tooth surface point cloud model obtained in step S2; x of the unfolded involute tooth surface is the arc length sk before unfolding, y is the tooth width, and the obtained micro-morphology z(x, y) is the morphology change in the direction of the curvature radius rk of the involute surface before unfolding; for the curvature radius rk corresponding to any point on the involute surface, the micro-morphology profile height z(x, y) is superimposed, that is, the curvature radius rkz after superimposing the micro-morphology is obtained, and then the correction equation of the involute tooth profile obtained after the corresponding superimposed micro-morphology is obtained: r kz =r k +z (7) where: x kz and kz are the two rectangular coordinates of any point K on the involute after correction in the coordinate system; r k is the distance from point K to point O; z is the profile height of the microscopic morphology of the involute tooth surface; r kz is the arc radius corresponding to any point on the corrected involute; the involute function is: θ k =invα k =tanα k -α k .

Images