Anisotropic surface fractal dimension calculation method

Abstract

The present application relates to the technical field of model construction, and specifically discloses a method for calculating the fractal dimension of anisotropic surfaces, comprising the following steps: Step 1: In order to make the calculation results reasonable and universal, fractal surfaces are generated in three ways, namely, WM function, Fourier transformation and mechanical processing; Step 2: The relationship between the surface dimension of the anisotropic surface and the vertical and horizontal profiles is measured, and the relationship between the two profile dimensions with an included angle of 90° and the rough surface dimension is established; Step 3: The relationship between the profile dimension at any angle and the vertical and horizontal profile dimensions is measured, and then the relationship between any two profile dimensions and the surface dimension is obtained; Step 4: The fractal parameters are changed, and the steps of Step 2 and Step 3 are repeated to improve the relationship between the anisotropic surface dimension and the profile dimension. The uniqueness of the calculation results of the anisotropic surface is ensured, the calculation accuracy of the fractal contact model is improved, and a theoretical basis for the establishment of the subsequent fractal contact model is provided.

Description

Technical Field

[0001] This invention relates to the field of function model algorithm technology, specifically to a method for calculating the fractal dimension of anisotropic surfaces. Background Technology

[0002] In anisotropic surfaces, although the surface fractal dimension is a constant, the profile fractal dimension differs in each direction. Different profile fractal dimensions, despite a fixed surface fractal dimension, result in different surface morphologies. The fractal dimension is an essential parameter in fractal contact models. Classical fractal contact models use the surface fractal dimension, leading to fractal surfaces with the same surface dimension but different morphologies possessing the same mechanical properties. This is clearly unreasonable, especially considering the more pronounced anisotropy in actual machined surfaces such as milled and ground rough surfaces. Summary of the Invention

[0003] To address the problems raised in the background technology, this invention proposes a segmentation model based on a graph neural network structure, which ensures the uniqueness of the calculation results for anisotropic surfaces, improves the calculation accuracy of fractal contact models, and provides a preliminary theoretical foundation for the establishment of subsequent fractal contact models.

[0004] The present invention provides a method for calculating the fractal dimension of anisotropic surfaces, comprising the following steps:

[0005] Step 1: To ensure the rationality and universality of the calculation results, fractal surfaces are generated using three methods: WM function, Fourier transform, and machining.

[0006] Step 2: Measure the relationship between the surface dimension of the irregular surface and the vertical and horizontal profiles, and establish the relationship between the dimensions of the two profiles with an included angle of 90° and the dimension of the rough surface;

[0007] Step 3: Measure the relationship between the dimension of the profile at any angle and the dimensions of the profile in the horizontal and vertical directions, and then obtain the relationship between the dimensions of any two profiles and the surface dimension;

[0008] Step 4: Change the fractal parameters and repeat steps 2 and 3 to refine the relationship between the dimension of the anisotropic surface and the dimension of the contour.

[0009] Preferably, step 1 is performed as follows:

[0010] Step 1.1: Construct an anisotropic rough surface using the WM function;

[0011] The rough surface Z(x, y) is generated by superimposing WM functions in both the vertical and horizontal directions using formula (1):

[0012]

[0013] In formula (1): x and y are the coordinates of two directions, G x G is the scale parameter in the horizontal direction X. y D is the scale parameter in the vertical direction Y. x Let D be the fractal dimension of the profile in the X direction. y Let be the fractal dimension of the profile in the Y direction, γ be the frequency coefficient (taken as γ = 1.5), and n be the frequency exponent, where n min The minimum frequency index;

[0014] Step 1.2: Fourier transform to generate anisotropic rough surfaces;

[0015] The rough surface Z(x, y) is simulated by superimposing Fourier transforms in both the vertical and horizontal directions, and is generated by formula (2):

[0016]

[0017] In formula (2): p and q are the coordinates of two directions, and G x Let G be the scale parameter of P in the horizontal direction. y Let φ be the scale parameter in the vertical direction q, M and N be the number of sampling points in the two directions, k and l be the frequency indices in the two directions, and Δx and Δy be the sampling intervals in the two directions; k φ l The phases are random and usually uniformly distributed;

[0018] Step 1.3: The machined surface yields anisotropic roughness.

[0019] By selecting any one of the milling, grinding, or turning methods to process the plane and observing the results, a true anisotropic rough surface of the machined material can be obtained.

[0020] Preferably, step 2 is performed as follows:

[0021] Step 2.1: Use the WM function to generate the surface;

[0022] Choose different values ​​of the fractal dimension D in the X and Y directions. x D y , (1 < D x <2)(1<D y <2), the two-dimensional scale parameter G, frequency exponent n, number of sampling points N, and sampling length L are all taken to the same value within the range. Various anisotropic rough surfaces are simulated and generated. Multiple data measurements are performed, and the data are fitted. The following data are based on the scale parameter G. x =Scale parameter G y=1e-15, frequency index n=1:50, number of sampling points 1000×1000, sampling length 0.0667 meters, the dimension of the rough surface is measured by fractal Brownian method, the fractal Brownian method formula is (3):

[0023]

[0024] In formula (3): N Δr This represents the number of pixel pairs within a distance Δr in the region, where Δr represents a series of integers k = 1, 2…M, and M is an integer; ΔI Δr Let be the absolute value of the height difference between two adjacent points with height Δr; H is a random exponent ranging from (0,1); the dimension D is usually calculated. s =3-H; Solve for H in logarithmic coordinates to obtain the surface dimension; then obtain the relationship between the vertical and horizontal profile dimensions and the surface dimension;

[0025] By performing nonlinear surface fitting on this data, we obtain the empirical formula (4):

[0026] D s = -0.075D y +0.877D x +1.288 D x <D y (4);

[0027] D s D is the surface fractal dimension, derived from an empirical formula. y The initial coefficient is very small, D y When varying between 1 and 2, -0.075D y The maximum value is 0.15, which has a very small impact on the calculation. The maximum error calculated is about 5%. Therefore, the formula is approximated as formula (5):

[0028] D s =min(D x D y )+1 (5);

[0029] min(D x D y ) is to take D x D y The smaller fractal dimension value is obtained, so the surface dimension can be calculated by formula (5) under the same parameters;

[0030] Step 2.2: Generate the surface using Fourier transform;

[0031] The parameter selection is exactly the same as for the WM function; repeat the above method.

[0032] After nonlinear fitting, formula (6) can be obtained;

[0033] D s = -0.005D y +0.91D x +1.12 D x <D y (6);

[0034] The conclusions obtained from the WM function are basically consistent, so formula (6) is approximated as formula (5).

[0035] Preferably, step (3) is performed as follows:

[0036] The contour dimension in different directions is measured using the structure function method, and the measurement methods are formulas (7) and (8).

[0037] S(τ) = <[z(x+τ)-z(x)] 2 >=cτ 4-2D (7);

[0038]

[0039] In formulas (7) and (8): τ is the interval between two points; z(x+τ)-z(x) is the height difference between two points with an interval of τ; c is a constant, obtained from formula (8); G is the scale parameter; D is the fractal dimension; Γ is the gamma function; S(τ) is the structure function, and the mean square error of the profile height is calculated.

[0040] D x The empirical formulas (9) and (10) are derived by fitting the values ​​of the contour dimension and the angle α between the two contours;

[0041] D r =1.2×min(D) x D y )·cosα 0°<α<45° (9);

[0042]

[0043] D x The empirical formulas (11) and (12) are derived by fitting the values ​​of the contour dimension and the angle between the two contours.

[0044] D r =1.15×min(D) x D y )·cosα0°<α<45° (11);

[0045]

[0046] In the formula: Dr α is the numerical value of the contour in any direction; α is the angle between the two contours.

[0047] Therefore, the specific relationships of the fit are given by formulas (13) and (14);

[0048] D r =1.175×min(D) x D y )·cosα 0°<α<45° (13);

[0049]

[0050] Preferably, step (4) is performed as follows:

[0051] 4.1 The effect of changing the scale parameter G on the surface dimension;

[0052] The influence of changing the scale parameter G in two directions on the surface dimension was investigated, and the initial parameters were obtained through multiple data measurements.

[0053] The fitting formula is in the form of (16);

[0054]

[0055] In formula (16): D x Let D be the fractal dimension of the profile in the X direction. y Let D be the fractal dimension of the profile in the Y direction, where D y +1 represents the maximum dimension controlling the roughness of the surface, D x -D y Controlling the range of dimension variation for rough surfaces, where x0 is the G value at which the intermediate dimension is reached. x / G y The numerical magnitude under logarithmic conditions, the p-value controls the steepness of the dimension change; D is obtained from multiple data measurements. x D y It is not related to the magnitude of x0 and p at the same time;

[0056] D x D y The value of x is fitted with the values ​​of x0 and p using a nonlinear surface to obtain the final formula for the surface fractal dimension:

[0057]

[0058] 4.2 Surface generation using Fourier transform method;

[0059] Repeat step 4.1 to obtain the Fourier surface scale parameter G and the surface dimension D. s Empirical formula (18);

[0060]

[0061] Therefore, by combining the WM function with the empirical formula for Fourier surface fitting, the total scale parameter G and the surface dimension D are obtained. s The relation (19);

[0062]

[0063] Compared with the prior art, the beneficial effects of the present invention are as follows: it provides the relationship between the fractal dimension of anisotropic surfaces and the fractal dimension of contours, making the fractal contact model more consistent with the actual situation when in contact in different directions, improving the calculation accuracy, distinguishing the fractal contact situation of anisotropic surfaces from that of isotropic surfaces, and providing a theoretical basis for the fractal contact model of anisotropic surfaces. Attached Figure Description

[0064] Figure 1 Anisotropic rough surface generated by the WM function of this invention;

[0065] Figure 2 The anisotropic rough surface generated by the Fourier transform of this invention;

[0066] Figure 3 The following are images showing different machined surfaces according to the present invention: (a) Reconstruction of surface data from a milled plane; (b) Reconstruction of surface data from a ground plane; (c) Reconstruction of surface data from a turned plane.

[0067] Figure 4 D of the present invention x D y The value size and D s Relationship;

[0068] Figure 5 D of the present invention x D y Value size and D s Relationship;

[0069] Figure 6 The scale parameter ratio of the present invention and D s Relationship diagram;

[0070] Figure 7 This is a graph showing the relationship between the scale parameter of the present invention and Ds in logarithmic coordinates. Detailed Implementation

[0071] The following illustrations disclose several embodiments of the present invention. For clarity, many physical details will be described in the following description. However, it should be understood that these physical details are not intended to limit the invention. That is, in some embodiments of the invention, these physical details are not essential. Furthermore, for the sake of simplicity, some conventional structures and components will be shown in the illustrations in a simple schematic manner.

[0072] Furthermore, the technical solutions of the various embodiments can be combined with each other, but only if they are feasible for those skilled in the art. If the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such combination of technical solutions does not exist and is not within the scope of protection claimed by this invention.

[0073] like Figure 1-7 As shown, the present invention provides a method for calculating the fractal dimension of anisotropic surfaces, comprising the following steps:

[0074] Step 1: To ensure the rationality and universality of the calculation results, fractal surfaces are generated using three methods: WM function, Fourier transform, and machining.

[0075] Step 2: Measure the relationship between the surface dimension of the irregular surface and the vertical and horizontal profiles, and establish the relationship between the dimensions of the two profiles with an included angle of 90° and the dimension of the rough surface;

[0076] Step 3: Measure the relationship between the dimension of the profile at any angle and the dimensions of the profile in the horizontal and vertical directions, and then obtain the relationship between the dimensions of any two profiles and the surface dimension;

[0077] Step 4: Change the fractal parameters and repeat steps 2 and 3 to refine the relationship between the dimension of the anisotropic surface and the dimension of the contour.

[0078] Further, step 1 is performed as follows:

[0079] Step 1.1: Construct an anisotropic rough surface using the WM function;

[0080] The rough surface Z(x, y) is generated by superimposing WM functions in both the vertical and horizontal directions using formula (1):

[0081]

[0082] In formula (1): G x G y These are the scale parameters in the horizontal X direction and the vertical Y direction, respectively, D. x D y Let be the fractal dimension of the profile in the X and Y directions, γ be the frequency coefficient (usually taken as γ = 1.5), and n be the frequency exponent, where nmin It is the minimum frequency index.

[0083] The parameter variation range is shown in Table 1.

[0084] Table 1. Range of WM function parameters

[0085] 1～2 1e-16~1e-8 1～20 50～80 1.5

[0086] Simulated anisotropic rough surfaces, such as Figure 1 As shown;

[0087] Step 1.2: Fourier transform to generate anisotropic rough surfaces;

[0088] The rough surface Z(x, y) is simulated by superimposing Fourier transforms in both the vertical and horizontal directions, and is generated by formula (2):

[0089]

[0090] In formula (2): p and q are the coordinates of two directions, and G x G y These are the scale parameters P in the horizontal direction and Q in the vertical direction, respectively, and D. x D y Let M and N be the fractal dimensions of the contours in the two directions, respectively; M and N be the number of sampling points in the two directions, respectively; k and l be the frequency indices in the two directions, respectively; Δx and Δy be the sampling intervals in the two directions, respectively; and φ be the frequency index in the two directions, respectively. k φ l The phases are random and usually uniformly distributed.

[0091] The parameter variation range is shown in Table 2.

[0092] Table 2. Range of Fourier Transform Parameters

[0093] <![CDATA[2 10～15 ]]> 1e-16~1e-8 [0,M-1] [0,2] L / (M-1) L / (N-1)

[0094] Simulated anisotropic rough surfaces, such as Figure 2 As shown

[0095] Step 1.3: The machined surface yields anisotropic rough surfaces, such as... Figure 3 As shown;

[0096] By selecting any one of the milling, grinding, or turning methods to process the plane and observing the results, a true anisotropic rough surface of the machined material can be obtained.

[0097] Further, step 2 is performed as follows:

[0098] Step 2.1: Use the WM function to generate the surface;

[0099] Choose different values ​​of the fractal dimension D in the X and Y directions. x D y (1<D x <2)(1<D y <2), the two-dimensional scale parameter G, frequency exponent n, number of sampling points N, and sampling length L are all taken to the same value within the range. Various anisotropic rough surfaces are simulated and generated. Multiple data measurements are performed, and the data are fitted. The following data are based on the scale parameter G. x =Scale parameter G y =1e-15, frequency index n=1:50, number of sampling points 1000×1000, sampling length 0.0667 meters, the dimension of the rough surface is measured by fractal Brownian method, the fractal Brownian method formula is (3):

[0100]

[0101] In formula (3): N Δr This represents the number of pixel pairs within a distance Δr in the region, where Δr typically represents a series of integers k = 1, 2…M, where M is an integer; ΔI Δr Let be the absolute value of the height difference between two adjacent points with height Δr; H is a random exponent ranging from (0,1); the dimension D is usually calculated. s =3-H; Solve for H in logarithmic coordinates to obtain the surface dimension; then obtain the relationship between the vertical and horizontal profile dimensions and the surface dimension, such as Figure 4 As shown;

[0102] By performing nonlinear surface fitting on this data, we obtain the empirical formula (4):

[0103] D s = -0.075D y +0.877D x +1.288 D x <D y (4);

[0104] D is derived from empirical formulas. y The initial coefficient is very small, D y When varying between 1 and 2, -0.075D y The maximum value is 0.15, which has a very small impact on the calculation. The maximum error calculated is about 5%. Therefore, the formula is approximated as formula (5):

[0105] D s =min(D x D y )+1 (5);

[0106] min(D x D y) is to take D x D y The smaller fractal dimension value in the middle, therefore, under the same parameters, the surface dimension can be calculated by formula (5);

[0107] Step 2.2: Generate the surface using Fourier transform;

[0108] The parameter selection is exactly the same as for the WM function. Repeat the above method to obtain the following result: Figure 5 As shown;

[0109] After nonlinear fitting, we can obtain formula (6):

[0110] D s = -0.005D y +0.91D x +1.12 D x <D y (6);

[0111] The conclusions obtained from the WM function are basically consistent, so formula (6) can be approximated as formula (5).

[0112] The machining and measurement surfaces are as follows:

[0113] The dimensions of the vertical and horizontal profiles were measured, and the surface dimension was also measured, yielding the following data:

[0114] Table 3-1 Measurement data of plane milling surfaces

[0115]

[0116]

[0117] Table 3-2 Measurement data of surface grinding

[0118] <![CDATA[D x ]]> 1.49 1.46 1.45 <![CDATA[D y ]]> 1.58 1.56 1.52

[0119] Table 3-3 Measurement data of machined surfaces

[0120] <![CDATA[D x ]]> 1.31 1.28 1.30 <![CDATA[D y ]]> 1.48 1.44 1.51

[0121] Since the surfaces are machined, the amplitude and shape of the contours can be considered to be similar, so they can be considered to have equal fractal parameters. As can be seen from the above data, the relationship between the dimension of the machined surface and the dimension of the contour also conforms to formula (5).

[0122] Further, step (3) is performed as follows:

[0123] The contour dimension in different directions is measured using the structural function method, and the measurement methods are formulas (7) and (8):

[0124] S(τ) = <[z(x+τ)-z(x)] 2 >=cτ 4-2D (7);

[0125]

[0126] In formulas (7) and (8): τ is the interval between two points; z(x+τ)-z(x) is the height difference between two points with an interval of τ; c is a constant, obtained from formula (8); G is the scale parameter; D is the fractal dimension; Γ is the gamma function; S(τ) is the structure function, used to calculate the mean square error of the profile height.

[0127] The WM function generates surfaces in the following way:

[0128] Take D x =1.3, D y =1.7, frequency exponents are all 1 to 50, number of sampling points is 1000×1000, sampling length L = 0.0667 meters, the transformation of the angle with the horizontal direction and the dimension of the profile is shown in Table 4:

[0129] Table 4. Profile dimension values ​​at different angles to the horizontal profile.

[0130] fractal dimension 1.31 1.58 1.557 1.556 1.555 1.554 1.552 1.548 Measuring angles 9.46° 11.3° 14.03° 18.43° 26.56° 45° 63.44° 71.56° fractal dimension 1.538 1.53 1.51 1.48 1.395 1.42 1.418 1.415 Measuring angles 75.96° 78.69° 80.54° 82.88° 84.29° 85.24° 85.91° 86.42° fractal dimension 1.413 1.41 1.407 1.405 1.402 1.4 1.395 1.386 Measuring angles 86.82° 87.14° 90° fractal dimension 1.368 1.383 1.690

[0131] By fitting the Dx values ​​from Table 3 with the contour dimension values ​​and the angle α between the two contours, empirical formulas (9) and (10) are derived:

[0132] D r =1.2×min(D) x D y )·cosα 0°<α<45° (9);

[0133]

[0134] It is found that the fractal dimension of the profile is related to the smaller dimension values ​​in the horizontal and vertical directions, and the calculation relationship is shown in formula (9) and formula (10);

[0135] The Fourier transform method for generating surfaces is as follows:

[0136] The fractal parameters are chosen in the same way as the WM function, and the transformation of the angle between the measured and horizontal directions and the profile dimension is as follows:

[0137] Table 5. Profile dimension values ​​at different angles to the horizontal profile.

[0138]

[0139]

[0140] By fitting the Dx values ​​in Table 4 with the contour dimension values ​​and the angle between the two contours, empirical formulas (11) and (12) are obtained.

[0141] D r =1.15×min(D x D y )·cosα 0°<α<45° (11);

[0142]

[0143] In formulas (11) and (12): D r denoted as the numerical value of the contour dimension in any direction; α is the angle between the two contours.

[0144] It can be seen that the fitting formula is not much different from the WM function fitting formula, so the specific relationship of the fitting is formula (13) and formula (14);

[0145] D r =1.175×min(D) x D y )·cosα 0°<α<45° (13);

[0146]

[0147] The surface to be measured using machining is:

[0148] Select a rough surface milled on a plane and measure its horizontal and vertical profile dimensions. Measure the profile dimension at any angle and compare it with the data calculated using formulas (13) and (14) as follows:

[0149] Table 6 D x =1.22D y The profile dimension when the angle with the horizontal direction is different at 1.52.

[0150]

[0151]

[0152] As can be seen from the table, the maximum error between the data calculated by the empirical formula and the measured data of the machined surface is 5.6%, which is within the permissible range, verifying the correctness of formulas (13) and (14).

[0153] Further, step (4) is performed as follows:

[0154] 4.1 The effect of changing the scale parameter G on the surface dimension:

[0155] The effect of changing the scale parameter G in two directions on the surface dimension was investigated, and initial parameters were obtained through multiple data measurements; n min =1, n max =50, sampling points 1000×1000, sampling length L=0.0667m, the ratio of G to the roughness of surface D is obtained. s Relationship such as Figure 5 As shown

[0156] Fitting in logarithmic coordinates yields the following: Figure 6 :

[0157] We obtain the empirical formula (15):

[0158]

[0159] That is, the fitting formula is in the form of (16):

[0160]

[0161] In formula (16): D x D y Let D be the fractal dimension of the profile in the X and Y directions, where D is the fractal dimension of the profile in the X and Y directions. y +1 represents the maximum dimension controlling the roughness of the surface, D x -D y Controlling the range of dimension variation for rough surfaces, where x0 is the G value at which the intermediate dimension is reached. x / G y The numerical magnitude under logarithmic conditions, the p-value controls the steepness of the dimension change; D is obtained from multiple data measurements. x D y It does not depend on the magnitude of x0 and p simultaneously; for example:

[0162] Table 7 shows D. x D y Changes with x0

[0163] <![CDATA[D y ]]> 1.4 1.5 1.6 1.7 1.8 1.9 1.5 1.6 1.7 <![CDATA[x0]]> 10.41 10.69 11.3 11.49 11.72 11.77 7.81 8.76 9.49 <![CDATA[D x ]]> 1.2 1.2 1.3 1.3 1.3 1.3 1.3 1.4 1.4 <![CDATA[D y ]]> 1.8 1.9 1.5 1.6 1.7 1.8 1.9 1.5 1.6 <![CDATA[x0]]> 9.99 10.3 5.15 6.6 7.76 8.28 8.72 2.19 4.23 <![CDATA[D x ]]> 1.4 1.4 1.4 1.5 1.5 1.5 1.6 <![CDATA[D y ]]> 1.7 1.8 1.9 1.7 1.8 1.9 1.9 <![CDATA[x0]]> 5.65 6.52 7.2 3.53 4.77 5.67 4.16

[0164] Table 8 is for D. x D y Changes with p

[0165] <![CDATA[D y ]]> 1.4 1.5 1.6 1.7 1.8 1.9 1.5 1.6 1.7 p 14.03 23.66 22.66 26.71 30.35 35.53 17.11 15.62 21.84 <![CDATA[D x ]]> 1.2 1.2 1.3 1.3 1.3 1.3 1.3 1.4 1.4 <![CDATA[D y ]]> 1.8 1.9 1.5 1.6 1.7 1.8 1.9 1.5 1.6 p 21.4 32.34 10.19 14.67 15.64 23.17 30.42 4.67 8.57 <![CDATA[D x ]]> 1.4 1.4 1.4 1.5 1.5 1.5 1.6 <![CDATA[D y ]]> 1.7 1.8 1.9 1.7 1.8 1.9 1.9 p 18.67 21.85 28.55 12.73 12.5 20.49 16.1

[0166] By performing nonlinear surface fitting on the values ​​of Dx and Dy with the values ​​of x0 and p, the final surface fractal dimension formula is obtained:

[0167]

[0168] 4.2 Surface generation using Fourier transform method:

[0169] Repeat step 4.1 to obtain the Fourier surface scale parameter G and the surface dimension D. s Empirical formula (18):

[0170]

[0171] Therefore, by combining the WM function and the empirical formula for Fourier surface fitting, the relationship between the total scale parameter G and the surface dimension Ds can be obtained (19).

[0172]

[0173] 5. Conclusion

[0174] Based on steps (1) to (4) above, an intrinsic relationship between the fractal dimension of anisotropic surfaces and the fractal dimension of contours can be obtained, which includes the following cases:

[0175] ①When other surface parameters are the same, the relationship between the surface dimension and the vertical and horizontal profile dimensions is as follows:

[0176] D s =min(D x D y )+1 (20);

[0177] ② When the amplitudes of the surface profiles in two directions differ, i.e., when the scale parameter G is different, the relationship between the surface dimension and the dimensions of the two profile directions and their corresponding amplitudes is as follows:

[0178]

[0179] When other surface parameters are the same, the relationship between the angular profile in any direction and the vertical and horizontal profiles is as follows:

[0180] D r =1.175×min(D) x D y )·cosα 0°<α<45° (22);

[0181]

[0182] Based on the above calculations, the intrinsic relationship between the fractal dimension of anisotropic surfaces and the fractal dimension of contours can be obtained. It can be summarized that when the amplitudes of the two directions are not much different, the contour dimension of any direction is related to the smaller dimension of the vertical and horizontal contours, which can be calculated according to formulas (22) and (23). The surface dimension is the smaller dimension of the vertical and horizontal contours plus 1. When the amplitudes of the two directions are different, the surface dimension can be calculated according to formula (21). This model well represents the relationship between the fractal dimension of anisotropic surfaces and the fractal dimension of contours, and is more suitable for rough surfaces with obvious directionality produced by actual processing such as turning, milling, grinding, etc., providing a more accurate theoretical basis for subsequent fractal contact models of anisotropic rough surfaces.

[0183] The above description is merely an embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principle of the present invention should be included within the scope of the claims of the present invention.

Claims

1. A method for calculating the fractal dimension of anisotropic surfaces, characterized in that, Includes the following steps: Step 1: To ensure the calculation results are reasonable and universal, fractal surfaces are generated using three methods: WM function, Fourier transform, and machining. Step 2: Measure the relationship between the surface dimension of the irregular surface and the vertical and horizontal profiles, and establish the relationship between the dimensions of the two profiles with an included angle of 90° and the dimension of the rough surface; Step 3: Measure the relationship between the dimension of the profile at any angle and the dimensions of the profile in the horizontal and vertical directions; Step 4: Change the fractal parameters and repeat steps 2 and 3 to refine the relationship between the anisotropic surface dimension and the profile dimension; Step 4 is performed as follows: 4.1 The effect of changing the scale parameter G on the surface dimension; The influence of the scale parameter G on the surface dimension in two directions was changed, and the initial parameters were obtained through multiple data measurements; the empirical formula (15) was obtained by fitting the two directions: (15)； In formula (15): x and y are coordinates in two directions; The fitting formula is in the form of (16); (16)； In formula (16): D x Let D be the fractal dimension of the profile in the X direction. y Let D be the fractal dimension of the profile in the Y direction, where D y +1 represents the maximum dimension controlling the roughness of the surface, D x -D y Controlling the range of dimension variation for rough surfaces, where x0 is the G value at which the intermediate dimension is reached. x / G y The numerical magnitude under logarithmic conditions, the p-value controls the steepness of the dimension change; D is obtained from multiple data measurements. x D y It is not related to the magnitude of x0 and p at the same time; D x D y The value of is fitted with a nonlinear surface to obtain the final surface fractal dimension formula: (17)； 4.2 Surface generation using Fourier transform method; Repeat step 4.1 to obtain the Fourier surface scale parameter G and the surface dimension D. s Empirical formula (18); (18)； Therefore, by combining the WM function with the Fourier surface fitting empirical formula, the total scale parameter G and surface dimension D are obtained. s The relation (19); (19)。 2. The method for calculating the fractal dimension of anisotropic surfaces according to claim 1, characterized in that, Step 1 is performed as follows: Step 1.1: Construct anisotropic surfaces using the WM function; Rough surfaces are simulated by superimposing WM functions in both vertical and horizontal directions. Generated from formula (1): (1)； In formula (1): x and y are the coordinates of two directions, G x G is the scale parameter in the horizontal direction X. y D is the scale parameter in the vertical direction Y. x Let D be the fractal dimension of the profile in the X direction. y Let be the fractal dimension of the profile in the Y direction, γ be the frequency coefficient (taken as γ=1.5), and n be the frequency exponent, where n min The minimum frequency index; Step 1.2: Fourier transform to generate anisotropic surfaces; Rough surfaces are simulated by superimposing Fourier transforms in both the vertical and horizontal directions. It is generated by formula (2): (2)； In formula (2): p and q are the coordinates of two directions, and G x Let G be the scale parameter of P in the horizontal direction. y Let be the scale parameter for the vertical direction q, M and N be the number of sampling points in the two directions, and k and l be the frequency indices in the two directions, respectively. , These represent the sampling intervals in both directions; , The phases are random and usually uniformly distributed; Step 1.3: Obtain an anisotropic surface by machining the surface; By selecting any one of the milling, grinding, or turning methods to process the plane and observing the results, a true anisotropic surface of the machined material can be obtained.

3. The method for calculating the fractal dimension of anisotropic surfaces according to claim 1, characterized in that: Step 2 is performed as follows: Step 2.1: Use the WM function to generate the surface; Choose different values ​​of the fractal dimension D in the X and Y directions. x D y ,1<D x <2, 1 <D y <2, with the two-dimensional scale parameter G, frequency exponent n, number of sampling points N, and sampling length L all taking the same value within the range, various anisotropic surfaces were simulated and generated. Multiple data measurements were performed, and the data were fitted. The following data is based on the scale parameter G. x =Scale parameter G y =1e-15, frequency index n=1:50, number of sampling points 1000×1000, sampling length 0.0667 meters, the dimension of the rough surface is measured by fractal Brownian method, the fractal Brownian method formula is (3): (3)； In formula (3): This indicates that all distances within the sampling point area are... The number of pixel pairs, Let k represent a series of integers k = 1, 2, ..., M, where M is an integer. For adjacent The absolute value of the height difference between the two points; H is a random exponent in the range (0, 1); the dimension D is usually calculated. s =3-H; Solve for H in logarithmic coordinates to obtain the surface dimension; then obtain the relationship between the vertical and horizontal profile dimensions and the surface dimension; By performing nonlinear surface fitting on this data, we obtain the empirical formula (4): (4)； in, D s D is the surface fractal dimension, derived from an empirical formula. y The initial coefficient is very small, D y When varying between 1 and 2, -0.075D y The maximum value is 0.15, which has a very small impact on the calculation. The maximum error calculated is about 5%. Therefore, the formula is approximated as formula (5): (5)； min(D) x D y ) is to take D x D y The smaller fractal dimension value is obtained, so the surface dimension can be calculated by formula (5) under the same parameters; Step 2.2: Generate the surface using Fourier transform; The parameter selection is exactly the same as for the WM function; repeat the above method. After nonlinear fitting, formula (6) can be obtained; (6)； The conclusions obtained from the WM function are basically consistent, so formula (6) is approximated as formula (5).

4. The method for calculating the fractal dimension of anisotropic surfaces according to claim 1, characterized in that: Step 3 is performed as follows: The contour dimension in different directions is measured using the structure function method, and the measurement methods are formulas (7) and (8). （7）； （8）； In formulas (7) and (8): τ is the interval between two points; Let τ be the height difference between two points with an interval of τ; c is a constant, obtained from formula (8); G is the scale parameter; D is the fractal dimension; For gamma function, Given the structure function, calculate the mean square error of the profile height; Based on WM functions, D x Values ​​related to the contour dimension and the angle between the two contours Empirical formulas (9) and (10) were derived through fitting. (9)； (10)； Based on Fourier transform, D x Values ​​related to the contour dimension and the angle between the two contours The empirical formulas (11) and (12) were obtained by fitting the data. (11)； (12)； In the formula: D r denoted as the numerical value of the contour dimension in any direction; α is the angle between the two contours. Therefore, the specific relationships of the fit are given by formulas (13) and (14); (13)； (14)。

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