An anisotropic three-dimensional rough surface fitting method

Abstract

The application discloses a kind of anisotropic three-dimensional rough surface fitting method comprising the following steps: step one, the height matrix of the original rough surface topography discrete point needing fitting is obtained;Step two, the standard deviation of original rough surface discrete point height is calculated, step three, the maximum height value of Zy is calculated;Step four, conversion operator A and coefficient vector B are calculated;Step five, the anisotropic three-dimensional rough surface of adjustable horizontal and vertical texture is obtained.The application considers the fractal dimension in X, Y direction, fractal roughness, establishes two-dimensional fractal profile curve using fractal theory, generates three-dimensional rough surface model in X, Y direction by combining conversion operator A and coefficient vector B, and establishes anisotropic three-dimensional rough surface after superimposition, with smaller fitting error, to obtain anisotropic three-dimensional rough surface that is more consistent with real surface, and the size of fractal dimension and fractal roughness in X, Y direction can be adjusted to effectively control surface texture morphology.

Description

Technical Field

[0001] This invention relates to the field of materials, and in particular to a method for fitting anisotropic three-dimensional rough surfaces. Background Technology

[0002] Due to their geometry and manufacturing processes, mechanical components such as gears and guide rails exhibit anisotropic textures on their surfaces, composed of groups of micro-protrusions. The presence of these micro-protrusions of varying sizes means that, during calculations, the actual contact area is much smaller than the nominal contact area, while the localized actual contact pressure is much greater than the nominal contact pressure. The contact area and contact pressure have a significant impact on the calculation and prediction of the lifespan and wear of mechanical components.

[0003] However, three-dimensional rough surfaces are composed of irregular shapes, making them difficult to describe using the general Euler formula. Current techniques typically employ statistical and fractal models to fit the actual rough surface. Statistical models reconstruct the three-dimensional rough surface using parameters such as the average radius of curvature and average height of a group of micro-protrusions measured on the surface. While they have strong universality, they have certain drawbacks for surfaces that do not conform to probability density distributions. Generally, two-dimensional parabolic micro-protrusions combined with average height are used to fit the three-dimensional rough surface, or three-dimensional ellipsoidal micro-protrusions combined with the watershed algorithm are used.

[0004] Fractal models utilize parameters such as fractal dimension and fractal roughness measured on the surface to reconstruct three-dimensional rough surfaces based on the Weierstrass-Mandelbrot function. However, this model exhibits strong randomness, resulting in anisotropic surfaces, which is unsuitable for real-world rough surfaces with significant anisotropy. The polar coordinate WM function, proposed based on this model, can simulate anisotropic surfaces, but it is difficult to control their texture characteristics. Summary of the Invention

[0005] To address the aforementioned technical problems, this invention proposes a method for fitting anisotropic three-dimensional rough surfaces.

[0006] The objective of this invention is achieved through the following technical solution:

[0007] A method for fitting an anisotropic three-dimensional rough surface includes the following steps:

[0008] Step 1: Obtain the height matrix Zs(x) of the discrete points of the original rough surface topography to be fitted. i ,y j ), where x i Represents the X-axis coordinates and y-axis coordinates of discrete points. j Zs(x) represents the Y-axis coordinate of a discrete point. i ,y j ) represents the discrete points (x) of the original rough surface. i ,y jThe Z-axis coordinate of ).

[0009] Step 2: Calculate the standard deviation of the height of the discrete points on the original rough surface, denoted as σ; calculate the fractal dimension D of each row of discrete points along the X direction on the original rough surface. x Fractal roughness G x The fractal dimension D of each row of discrete points along the Y direction y Fractal roughness G y The average height and maximum height are denoted as the original rough surface; the curve Sms, composed of the average height values ​​of each row of discrete points along the Y direction, and the curve Szs, composed of the maximum height values ​​of the curves composed of each row of discrete points along the Y direction, are obtained; the fractal dimension D of curve Sms is calculated. Sms and fractal roughness G Sms ;

[0010] Step 3: Based on the Weierstrass-Mandelbrot function, utilize D x G x D y G y D Sms G Sms Establish a two-dimensional fractal profile curve, denoted as Z. x Z y S x ; Calculate Z y The maximum height value is denoted as Szy; Z x Indicated by D x G x The discrete points and Z of the calculated two-dimensional fractal profile curve y Indicated by D y G y The discrete points of the calculated two-dimensional fractal profile curve, S x Indicated by D Sms G Sms Discrete points of the calculated two-dimensional fractal profile curve;

[0011] Step 4: Compare curve Sms and curve Sx, denoted as transformation operator A; compare Szs and Szy, denoted as coefficient vector B;

[0012] Step 5: Based on the two-dimensional fractal contour curves Zx and Zy, combine them with the transformation operator A and the coefficient vector B respectively to obtain the three-dimensional rough surfaces in the X and Y directions; superimpose the three-dimensional rough surfaces in the X and Y directions to obtain an anisotropic three-dimensional rough surface with adjustable transverse and longitudinal textures.

[0013] In a further improvement, the formula for calculating the standard deviation of the height of discrete points on the original rough surface in step two is as follows:

[0014]

[0015] In the formula, This represents the average height of discrete points on the original rough surface, where N represents the total number of discrete points.

[0016] Calculate the fractal dimension D of the original rough surface. x D y D Sms The calculation formula is as follows:

[0017]

[0018] In the formula: D represents the fractal dimension, k = 1, 2, 3 represent the slopes of the double logarithmic curves of the power spectral density function, structure function, and root mean square function, respectively; D = D x D y Or D Sms ;

[0019] Calculate the fractal roughness G of the original rough surface x G y G Sms At that time, D x D y D Sms Substitute into the following formula:

[0020]

[0021] In the formula, G represents fractal roughness, ω L The lowest frequency representing the spatial frequency ω is determined by the sampling length L. U The highest frequency representing spatial frequency ω is determined by the sampling interval δ and is related to instrument resolution and filtering; ω = γ n , is the reciprocal of the random profile spatial frequency and the profile wavelength, γ represents the ratio parameter of self-affineness and spectral density and the relative phase difference between spectral density and spectral mode. In reality, the phase is random, n represents the micro-convexity level, which is a non-negative integer. Given any γ, the rough profile curve is represented by the fractal dimension D, fractal roughness G, and sampling length L. To accommodate the high spectral density and phase randomness, γ = 1.5 is taken, and integer values ​​less than 5 are discarded for n.

[0022] In a further improvement, in step three, a two-dimensional fractal profile curve Z(x) is established based on the Weierstrass-Mandelbrot function. Z(x) is obtained by substituting the fractal dimension D and fractal roughness G of the corresponding curve into the following formula:

[0023]

[0024] Maximum height S zyG represents the sum of the maximum peak and maximum valley depth of the two-dimensional fractal profile curve Zy; D-1 It means G raised to the power of (D-1), L D-2 Let L be the power of (D-2), and n0 represent the minimum value of n. max Z(x) represents the maximum value of n, and x represents the x-th discrete point; Z(x) = Z x Z y or S x Z x From fractal dimension D x Fractal roughness G x Substituting into the above formula, we get: Z y From fractal dimension D y and fractal roughness G y Substituting into the above formula, S x From fractal dimension D Sms and fractal roughness G Sms The result is obtained by substituting into the above formula.

[0025] 4. The anisotropic three-dimensional rough surface fitting method as described in claim 1, characterized in that, in step four, the calculation formula for transformation operator A is as follows:

[0026]

[0027] In the formula, Ns is the number of discrete points, t represents the t-th discrete point, T represents the matrix transpose, Zst represents the discrete point in the t-th row along the Y direction, and Smst represents the average height value of the discrete point in the t-th row along the Y direction.

[0028] The computational count of coefficient vector B is as follows:

[0029] B=Szy●Szs -1

[0030] Where Szy represents the maximum height value of the Zy curve, and Szs represents the curve formed by the maximum height values ​​of the discrete points along the Y direction.

[0031] In a further improvement, the calculation formula for superimposing the discrete points of the anisotropic three-dimensional rough surface in step five is as follows:

[0032]

[0033] In the formula, A i B represents the maximum absolute value of the i-th row in transformation operator A. j This represents the j-th value in the coefficient vector B.

[0034] The beneficial effects of this invention are as follows:

[0035] (1) It exhibits significant anisotropy and better conforms to the original rough surface.

[0036] For the original rough surface with obvious stripes, the resulting anisotropic three-dimensional rough surface fits the original rough surface better than the isotropic fractal model.

[0037] (2) Adjustable horizontal and vertical textures

[0038] The morphology of anisotropic three-dimensional rough surfaces can be altered by changing the amplitude and smoothness of the two-dimensional fractal profile curve according to the fractal dimensions Dx and Dy and the fractal roughness Gx and Gy. Attached Figure Description

[0039] The invention will be further illustrated with reference to the accompanying drawings, but the contents of the drawings do not constitute any limitation on the invention.

[0040] Figure 1 The original rough surface of the example;

[0041] Figure 2 These are the discrete points on the original rough surface of the example;

[0042] Figure 3 An anisotropic three-dimensional rough surface fitted for the example;

[0043] Figure 4 Discrete points of anisotropic three-dimensional rough surface fitted for the example;

[0044] Figure 5 To establish a schematic diagram of an anisotropic three-dimensional rough surface;

[0045] Figure 6 Anisotropic three-dimensional rough surfaces with different fractal dimensions and fractal roughness. Detailed Implementation

[0046] To make the purpose, technical solution, and advantages of the invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and examples.

[0047] Using the method of the present invention, for example Figure 1 The original rough surface with stripes shown is fitted.

[0048] The formula for calculating the standard deviation of the original rough surface is as follows:

[0049]

[0050] The calculated fractal dimensions Dx are 1.6521, Dy is 1.7161, and the fractal roughness Gx is 3.09 × 10⁻⁶. -5 mm, Gy is 6.17×10 -5 mm,

[0051] Original rough surface ( Figure 1 and discrete points of the original rough surface ( Figure 2 The surface exhibits greater undulation along the X-direction and less undulation along the Y-direction, consistent with the model assumptions. The anisotropic three-dimensional rough surface in this invention (… Figure 3 As can be seen in the graph, the undulations are larger along the X-direction and smaller along the Y-direction, satisfying the assumptions in the model. The fractal dimension describes the irregularity and complexity of the rough surface profile curve at all scales and is independent of the profile curve height Z(x). A smaller fractal dimension results in a larger profile amplitude; a larger fractal dimension results in a smaller profile amplitude. The characteristic scale coefficient reflects the magnitude of the rough surface profile and is size-dependent, reflecting the smoothness within a unit scale, i.e., the increase or decrease in surface roughness. A larger characteristic scale coefficient results in a larger profile amplitude; a smaller characteristic scale coefficient results in a smaller profile amplitude and better profile smoothness. The profile curve does not change significantly, meaning the characteristic scale coefficient is directly proportional to the profile curve height.

[0052] The main innovation of this invention lies in the following: Based on the fractal dimension and fractal roughness in the X and Y directions, a two-dimensional fractal contour curve is established using fractal theory. Combined with transformation operator A and coefficient vector B, a three-dimensional rough surface model in the X and Y directions is generated. These models are then superimposed to establish an anisotropic three-dimensional rough surface. This conforms to the assumption of larger fluctuations in the X direction and smaller fluctuations in the Y direction; the fitting error is small, resulting in an anisotropic three-dimensional rough surface that more closely resembles the real surface. Adjusting the fractal dimension and fractal roughness in the X and Y directions can effectively control the surface texture morphology.

[0053] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit the scope of protection of the present invention. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the essence and scope of the technical solutions of the present invention.

Claims

1. A method for fitting anisotropic three-dimensional rough surfaces, characterized in that, Includes the following steps: Step 1: Obtain the height matrix Zs(x) of the discrete points of the original rough surface topography to be fitted. i ,y j ), where x i Represents the X-axis coordinates and y-axis coordinates of discrete points. j Zs(x) represents the Y-axis coordinate of a discrete point. i ,y j ) represents the discrete points (x) of the original rough surface. i ,y j The Z-axis coordinate of ). Step 2: Calculate the standard deviation of the height of the discrete points on the original rough surface, denoted as σ; calculate the fractal dimension D of each row of discrete points along the X direction on the original rough surface. x Fractal roughness G x The fractal dimension D of each row of discrete points along the Y direction y Fractal roughness G y The average height and maximum height are denoted as the original rough surface; the curve Sms, composed of the average height values ​​of each row of discrete points along the Y direction, and the curve Szs, composed of the maximum height values ​​of the curves composed of each row of discrete points along the Y direction, are obtained; the fractal dimension D of curve Sms is calculated. Sms and fractal roughness G Sms ; Step 3: Based on the Weierstrass-Mandelbrot function, utilize D x G x D y G y D Sms G Sms Establish a two-dimensional fractal profile curve, denoted as Z. x Z y S x ; Calculate Z y The maximum height value is denoted as Szy; Z x Indicated by D x G x The discrete points and Z of the calculated two-dimensional fractal profile curve y Indicated by D y G y The discrete points of the calculated two-dimensional fractal profile curve, S x Indicated by D Sms G Sms Discrete points of the calculated two-dimensional fractal profile curve; Step 4: Compare curve Sms and curve Sx, denoted as transformation operator A; compare Szs and Szy, denoted as coefficient vector B; Step 5: Based on the two-dimensional fractal contour curves Zx and Zy, combine them with the transformation operator A and the coefficient vector B respectively to obtain the three-dimensional rough surfaces in the X and Y directions; superimpose the three-dimensional rough surfaces in the X and Y directions to obtain an anisotropic three-dimensional rough surface with adjustable transverse and longitudinal textures.

2. The anisotropic three-dimensional rough surface fitting method as described in claim 1, characterized in that, In step two, the formula for calculating the standard deviation of the height of discrete points on the original rough surface is as follows: In the formula, This represents the average height of discrete points on the original rough surface, where N represents the total number of discrete points. Calculate the fractal dimension D of the original rough surface. x D y D Sms The calculation formula is as follows: In the formula: D represents the fractal dimension, k = 1, 2, 3 represent the slopes of the double logarithmic curves of the power spectral density function, structure function, and root mean square function, respectively; D = D x D y Or D Sms ; Calculate the fractal roughness G of the original rough surface x G y G Sms At that time, D x D y D Sms Substitute into the following formula: In the formula, G represents fractal roughness, ω L The lowest frequency representing the spatial frequency ω is determined by the sampling length L. U The highest frequency representing spatial frequency ω is determined by the sampling interval δ and is related to instrument resolution and filtering; ω = γ n , is the reciprocal of the random profile spatial frequency and the profile wavelength, γ represents the ratio parameter of self-affineness and spectral density and the relative phase difference between spectral density and spectral mode. In reality, the phase is random, n represents the micro-convexity level, which is a non-negative integer. Given any γ, the rough profile curve is represented by the fractal dimension D, fractal roughness G, and sampling length L. To accommodate the high spectral density and phase randomness, γ = 1.5 is taken, and integer values ​​less than 5 are discarded for n.

3. The anisotropic three-dimensional rough surface fitting method as described in claim 1, characterized in that, In step three, a two-dimensional fractal profile curve Z(x) is established based on the Weierstrass-Mandelbrot function. Z(x) is obtained by substituting the fractal dimension D and fractal roughness G of the corresponding curve into the following formula: Maximum height S zy G represents the sum of the maximum peak and maximum valley depth of the two-dimensional fractal profile curve Zy; D-1 It means G raised to the power of (D-1), L D-2 Let L be the power of (D-2), and n0 represent the minimum value of n. max Z(x) represents the maximum value of n, and x represents the x-th discrete point; Z(x) = Z x Z y or S x Z x From fractal dimension D x Fractal roughness G x Substituting into the above formula, we get: Z y From fractal dimension D y and fractal roughness G y Substituting into the above formula, S x From fractal dimension D Sms and fractal roughness G Sms The result is obtained by substituting into the above formula.

4. The anisotropic three-dimensional rough surface fitting method as described in claim 1, characterized in that, In step four, the calculation formula for transformation operator A is as follows: In the formula, Ns is the number of discrete points, t represents the t-th discrete point, T represents the matrix transpose, Zst represents the discrete point in the t-th row along the Y direction, and Smst represents the average height value of the discrete point in the t-th row along the Y direction. The computational count of coefficient vector B is as follows: B=Szy●Szs -1 Where Szy represents the maximum height value of the Zy curve, and Szs represents the curve formed by the maximum height values ​​of the discrete points along the Y direction.

5. The anisotropic three-dimensional rough surface fitting method as described in claim 1, characterized in that, In step five, the calculation formula for superimposing discrete points of the anisotropic three-dimensional rough surface is as follows: In the formula, A i B represents the maximum absolute value of the i-th row in transformation operator A. j This represents the j-th value in the coefficient vector B.

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