Arc two-dimensional surface contour fractal characterization method based on coordinate transformation

Abstract

The invention relates to an arc two-dimensional surface contour fractal characterization method based on coordinate transformation. According to the technical scheme, a 3D confocal laser microscope is adopted to observe the appearance of the arc forming grinding surface, and arc contour data are obtained; converting the arc contour data into plane contour data based on a coordinate conversion method; calculating a fractal dimension D and a scale coefficient G by adopting a structure function method and combining plane contour data, and substituting the fractal dimension D and the scale coefficient G into the optimized W-M function model so as to carry out plane contour fractal simulation; and converting the simulated plane contour into an arc surface contour by using a coordinate transformation method so as to finish the characterization of the arc forming grinding surface contour. The fractal theory and the arc forming grinding experiment are combined, the W-M function model is optimized, the fractal simulation model of the arc forming grinding machining surface appearance is given, and therefore the fractal characterization method of the arc two-dimensional surface appearance is determined, and the real condition of the surface appearance can be well reflected.

Description

Technical Field

[0001] The invention relates to a surface fitting method, in particular to a fractal characterization method of circular arc two-dimensional surface contour based on coordinate transformation. Background Art

[0002] Arc wheel grinding technology is a method of precision machining using a grinding wheel with an arc shape. This grinding wheel can be used to process various complex geometries, especially those workpieces that require inner or outer arc surfaces. Arc wheel grinding technology occupies an important position in modern manufacturing, and its importance will continue to grow with the continuous advancement of technology and the expansion of application areas.

[0003] Comprehensive evaluation and optimization of the surface topography of arc grinding is crucial to improving the machining quality and functional characteristics of workpieces. At present, research on the characterization of part surface topography is mostly focused on the surface ground by flat grinding wheels, but due to the influence of the curvature of the arc surface profile, the existing research methods cannot well reflect the true condition of its surface topography. Summary of the invention

[0004] The present invention provides a fractal characterization method for the two-dimensional surface contour of an arc based on coordinate transformation. By combining fractal theory and arc forming grinding experiments, the WM function model is optimized, and a fractal simulation model of the surface morphology of the arc forming grinding process is given. Thus, a fractal characterization method for the two-dimensional surface morphology of an arc is established, which can well reflect the real condition of its surface morphology.

[0005] The technical solution of the present invention is as follows:

[0006] A fractal characterization method for arc two-dimensional surface contour based on coordinate transformation comprises the following steps:

[0007] Step 1: Use a 3D confocal laser microscope to observe the arc forming grinding surface morphology and obtain arc profile data;

[0008] Step 2: Based on the coordinate transformation method, the arc contour data is converted into plane contour data;

[0009] Step 3: Using the structure function method and combining the plane contour data, the fractal dimension D and the scale coefficient G are calculated, and then brought into the optimized WM function model to perform plane contour fractal simulation;

[0010] Step 4: Use the coordinate transformation method to transform the simulated plane profile into an arc surface profile, thereby completing the characterization of the arc forming grinding surface profile.

[0011] Furthermore, in the fractal characterization method of the arc two-dimensional surface contour based on coordinate transformation, in the step 2, the coordinates of the arc contour are transformed into the coordinates of the plane contour by using translation transformation and rotation transformation; the translation transformation refers to moving the coordinate point along a certain direction in the coordinate system by a fixed distance, and changing the position of the coordinate point by translation to achieve the effect of moving the figure in the coordinate system; the rotation transformation is to rotate the coordinate around a point or axis by a specific angle, and change the direction or angle of the figure by rotation transformation to achieve the effect of rotation;

[0012] The translation transformation matrix T and the rotation transformation matrix R are:

[0013]

[0014] Among them, T x is the distance that the data point moves along the x-axis, T z is the distance the data point moves along the z-axis, and θ is the rotation angle of the data point around the origin;

[0015] In the coordinate system xoy, let point A(x0,z0) rotate counterclockwise around the origin by an angle of θ degrees, and after transformation, point A1(x1,z1) is obtained; the matrix expression of the homogeneous coordinate transformation process is:

[0016]

[0017] After transformation, the coordinates of the point under xoy are:

[0018]

[0019] Furthermore, the fractal characterization method of arc two-dimensional surface contour based on coordinate transformation transforms the measured arc contour data into the rotation coordinates of the plane contour data, and the steps are as follows:

[0020] (a) Fit the arc contour baseline using the least squares method, move the lowest point of the fitted arc contour baseline to the origin of the coordinate system through translation transformation, and convert all the original arc contour data points p ai (x ai ,z ai ) moves accordingly, let the arc contour data point after the movement be p i (x i ,z i );

[0021] (b) Combine the equations of the tangent line at any point on the arc contour baseline, l, and the line passing through p i (x i ,z i ) to solve the intersection point e of the two equations on the arc contour baseline i (a i ,bi ), the angle required to transform the arc contour data into the plane contour data is θ i =arctan(b i | / a i );

[0022] (c) Use the transformation matrix to transform p i (x i ,z i ) is rotated and transformed to the plane contour data, with p i '(x i ',z i ')express;

[0023] When the coordinates of all data points of the arc contour are expressed in homogeneous form, the coordinate transformation relationship is as follows:

[0024] X'=XTR (4)

[0025] Among them, X is the original data point coordinate matrix, and X' is the transformed data point coordinate matrix;

[0026] The matrix expression of the transformation process from arc profile data to plane profile data is as follows:

[0027]

[0028] Among them, T xi is the distance that the i-th original data point moves along the x-axis; T zi is the distance that the i-th original data point moves along the z axis; θ i = -arctan(b i / a i ), i = 1, 2, 3, ... n, where n is the number of data point coordinates.

[0029] Further, in the fractal characterization method of the arc two-dimensional surface profile based on coordinate transformation, in the step three, according to the measured arc profile data analysis, the fractal dimension D is inversely proportional to the amplitude of the arc profile. The larger D is, the smaller the simulated arc profile amplitude is, that is, it is negatively correlated with the surface Ra; the scale coefficient G is directly proportional to the arc profile amplitude. The larger G is, the larger the simulated arc profile amplitude is, that is, it is positively correlated with Ra; the surface roughness Ra of each position of the arc surface after grinding (from one end of the arc to the other end) shows a large-small-large law, and the WM function model is optimized, and the expression is as follows:

[0030]

[0031] Where P(x) is the optimization coefficient, Where B = 1 + k, k is the proportionality coefficient, which is expressed as the ratio of the bottom roughness to the edge roughness in the groove direction of the arc profile; ω is the frequency, ω = L1 / L, L is the length of the simulation sample, and L1 is the distance from the intersection of the two equations y1 = 1-B and y2 = cosx; is the phase, which is related to L; φ is the random phase;

[0032] The fractal dimension D and scale coefficient G of plane contour data are calculated using the structure function method, and the fractal dimension D and scale coefficient G are brought into the optimized WM function model to perform fractal simulation of plane contour.

[0033] Furthermore, in the fractal characterization method of the circular arc two-dimensional surface contour based on coordinate transformation, in the step 4, the simulated plane contour data is subjected to an inverse transformation process of the rotational coordinate transformation to obtain the circular arc surface contour characterized by the fractal characterization;

[0034] The data transformation relationship is as follows:

[0035] X”=X'R' (7)

[0036] In the formula, R is the inverse transformation matrix of the rotation transformation, and X" is the final required circular arc surface contour coordinate point matrix;

[0037] The matrix expression of the coordinate rotation inverse transformation process is as follows:

[0038]

[0039] Among them, x i ”,z i " is the final required arc surface contour coordinate point, θ' i is the rotation angle of the inverse transformation, θ' i =-θ i =arctan(b i / a i );

[0040] The mathematical expression of the arc two-dimensional surface profile data based on coordinate transformation is:

[0041]

[0042] The beneficial effects of the present invention are as follows: the present invention combines fractal theory and arc forming grinding experiments to optimize the WM function model, and provides a fractal simulation model of the surface morphology of the arc forming grinding process, thereby establishing a fractal characterization method for the two-dimensional surface morphology of the arc, which can well reflect the true condition of its surface morphology. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 This is a flow chart of the fractal characterization method of arc two-dimensional surface contour based on coordinate transformation;

[0044] Figure 2 Schematic diagram of the principle of coordinate transformation;

[0045] Figure 3 This is a schematic diagram of the straightening transformation of arc contour data;

[0046] Figure 4 Schematic diagram of the two-dimensional fractal contour simulation of the optimized WM function;

[0047] Figure 5 Schematic diagram of the characterization profile and the actual measured profile of a circular arc two-dimensional surface; (a) is a schematic diagram of the characterization profile, and (b) is a schematic diagram of the actual measured profile. DETAILED DESCRIPTION

[0048] like Figure 1 As shown, a fractal characterization method for arc two-dimensional surface contour based on coordinate transformation includes the following steps:

[0049] Step 1: Use a 3D confocal laser microscope to observe the arc forming grinding surface morphology and obtain arc profile data;

[0050] Step 2: Based on the coordinate transformation method, the arc contour data is converted into plane contour data;

[0051] like Figure 2 As shown in the figure, the coordinates of the arc contour are transformed into the coordinates of the plane contour by translation and rotation transformation; translation transformation refers to moving the coordinate point along a certain direction in the coordinate system by a fixed distance, and changing the position of the coordinate point by translation to achieve the effect of moving the figure in the coordinate system; rotation transformation is to rotate the coordinate around a point or axis by a specific angle, and change the direction or angle of the figure by rotation transformation to achieve the effect of rotation;

[0052] The translation transformation matrix T and the rotation transformation matrix R are:

[0053]

[0054] Among them, T x is the distance that the data point moves along the x-axis, T z is the distance the data point moves along the z-axis, and θ is the rotation angle of the data point around the origin;

[0055] In the coordinate system xoy, let point A(x0,z0) rotate counterclockwise around the origin by an angle of θ degrees, and after transformation, point A1(x1,z1) is obtained; the matrix expression of the homogeneous coordinate transformation process is:

[0056]

[0057] After transformation, the coordinates of the point under xoy are:

[0058]

[0059] like Figure 3 As shown, the measured arc profile data is transformed into the rotation coordinates of the plane profile data. The steps are as follows:

[0060] (a) Fit the arc contour baseline using the least squares method, move the lowest point of the fitted arc contour baseline to the origin of the coordinate system through translation transformation, and convert all the original arc contour data points p ai (x ai ,z ai ) moves accordingly, let the arc contour data point after the movement be p i (x i ,z i );

[0061] (b) Combine the equations of the tangent line at any point on the arc contour baseline, l, and the line passing through p i (x i ,z i ) to solve the intersection point e of the two equations on the arc contour baseline i (a i ,b i ), the angle required to transform the arc contour data into the plane contour data is θ i =arctan(b i | / a i );

[0062] (c) Use the transformation matrix to transform p i (x i ,z i ) is transformed into the plane contour data by p i '(x i ',z i ')express;

[0063] When the coordinates of all data points of the arc contour are expressed in homogeneous form, the coordinate transformation relationship is as follows:

[0064] X'=XTR (4)

[0065] Among them, X is the original data point coordinate matrix, and X' is the transformed data point coordinate matrix;

[0066] The matrix expression of the transformation process from arc profile data to plane profile data is as follows:

[0067]

[0068] Among them, T xiis the distance that the i-th original data point moves along the x-axis; T zi is the distance that the i-th original data point moves along the z axis; θ i = -arctan(b i / a i ); i = 1, 2, 3...n, n is the number of data point coordinates;

[0069] Step 3: Using the structure function method and combining the plane contour data, the fractal dimension D and the scale coefficient G are calculated, and then brought into the optimized WM function model to perform plane contour fractal simulation;

[0070] like Figure 4 As shown in the figure, according to the analysis of the measured arc profile data, the fractal dimension D is inversely proportional to the amplitude of the arc profile. The larger the D, the smaller the simulated arc profile amplitude, that is, it is negatively correlated with the surface Ra; the scale coefficient G is directly proportional to the arc profile amplitude. The larger the G, the larger the simulated arc profile amplitude, that is, it is positively correlated with Ra; the surface roughness Ra of each position of the arc surface after grinding (from one end of the arc to the other end) shows a large-small-large law. The WM function model is optimized and the expression is as follows:

[0071]

[0072] Where P(x) is the optimization coefficient, Where B = 1 + k, k is the proportionality coefficient, which is expressed as the ratio of the bottom roughness to the edge roughness in the groove direction of the arc profile; ω is the frequency, ω = L1 / L, L is the length of the simulation sample, and L1 is the distance from the intersection of the two equations y1 = 1-B and y2 = cosx; is the phase, which is related to L; φ is the random phase;

[0073] The fractal dimension D and scale factor G of the plane contour data are calculated using the structure function method, and the fractal dimension D and scale factor G are brought into the optimized WM function model to perform fractal simulation of the plane contour;

[0074] Step 4: The simulated plane profile is transformed into an arc surface profile by using a coordinate transformation method, thereby completing the characterization of the arc forming grinding surface profile;

[0075] The simulated plane profile data is subjected to inverse transformation of the rotation coordinate transformation to obtain the circular arc surface profile represented by the fractal;

[0076] The data transformation relationship is as follows:

[0077] X”=X'R' (7)

[0078] In the formula, R is the inverse transformation matrix of the rotation transformation, and X" is the final required circular arc surface contour coordinate point matrix;

[0079] The matrix expression of the coordinate rotation inverse transformation process is as follows:

[0080]

[0081] Among them, x i ”,z i " is the final required arc surface contour coordinate point, θ' i is the rotation angle of the inverse transformation, θ' i =-θ i =arctan(b i / a i );

[0082] The mathematical expression of the arc two-dimensional surface profile data based on coordinate transformation is:

[0083]

[0084] Now given the fractal dimension D = 1.8, the scale factor G = 2×10 -7 , the simulation data length L = 1.25mm, that is, the value range of x is 0mm~1.25mm, and the simulated fractal surface contour curve is as follows Figure 4 As shown; Figure 4 The simulation area L of the overall contour is divided into 5 parts on average, and the surface roughness corresponding to the l1, l2, l3, l4, and l5 areas are calculated respectively, and it can be obtained that Ra1=1.589μm, Ra2=1.477μm, Ra3=1.456μm, Ra4=1.605μm, Ra5=1.659μm; It can be seen from the above Ra values ​​that based on the optimized WM function simulation, the overall surface contour curve shows a high-low-high change trend, which is the same as the Ra experimental measurement results; This proves that after optimizing the WM function, the changing trend of the arc forming grinding surface contour can be simulated more accurately.

[0085] The optimized WM function simulates the two-dimensional contour and performs a rotation coordinate transformation. The data generated by the simulation of formula (6) is (x i ',z i '), then after the transformation of formula (9), the coordinates of the data are (x i ”,z i ”), the above transformation can complete the simulation of the groove profile of the arc forming grinding surface; the combined model at this time is the fractal representation model of the arc surface morphology.

[0086] The arc two-dimensional surface representation profile and the actual measured profile are as follows Figure 5 shown.

Claims

1. A fractal characterization method for circular arc two-dimensional surface contour based on coordinate transformation, characterized in that: The following steps are involved: Step 1: Use a 3D confocal laser microscope to observe the arc forming grinding surface morphology and obtain arc profile data; Step 2: Based on the coordinate transformation method, the arc contour data is converted into plane contour data; Step 3: Using the structure function method and combining the plane contour data, the fractal dimension D and the scale coefficient G are calculated, and then brought into the optimized WM function model to perform plane contour fractal simulation; Step 4: Use the coordinate transformation method to transform the simulated plane profile into an arc surface profile, thereby completing the characterization of the arc forming grinding surface profile.

2. The fractal characterization method of circular arc two-dimensional surface contour based on coordinate transformation according to claim 1 is characterized in that: In the step 2, the coordinates of the arc contour are transformed into the coordinates of the plane contour by using translation transformation and rotation transformation; the translation transformation refers to moving the coordinate point along a certain direction in the coordinate system by a fixed distance, and changing the position of the coordinate point by translation to achieve the effect of moving the figure in the coordinate system; the rotation transformation is to rotate the coordinate around a point or axis by a specific angle, and change the direction or angle of the figure by rotation transformation to achieve the effect of rotation; The translation transformation matrix T and the rotation transformation matrix R are: Among them, T x is the distance that the data point moves along the x-axis, T z is the distance the data point moves along the z-axis, and θ is the rotation angle of the data point around the origin; In the coordinate system xoy, let point A(x0,z0) rotate counterclockwise around the origin by an angle of θ degrees, and after transformation, point A1(x1,z1) is obtained; the matrix expression of the homogeneous coordinate transformation process is: After transformation, the coordinates of the point under xoy are:

3. The fractal characterization method of circular arc two-dimensional surface contour based on coordinate transformation according to claim 2, characterized in that: The steps to transform the measured arc profile data into the rotation coordinates of the plane profile data are as follows: (a) Fit the arc contour baseline using the least squares method, move the lowest point of the fitted arc contour baseline to the origin of the coordinate system through translation transformation, and convert all the original arc contour data points p ai (x ai ,z ai ) moves accordingly, let the arc contour data point after the movement be p i (x i ,z i ); (b) Combine the equations of the tangent line at any point on the arc contour baseline, l, and the line passing through p i (x i ,z i ) to solve the intersection point e of the two equations on the arc contour baseline i (a i ,b i ), the angle required to transform the arc contour data into the plane contour data is θ i =arctan(b i | / a i ); (c) Use the transformation matrix to transform p i (x i ,z i ) is transformed into the plane contour data by p i '(x i ',z i ')express; When the coordinates of all data points of the arc contour are expressed in homogeneous form, the coordinate transformation relationship is as follows: X'=XTR (4) Among them, X is the original data point coordinate matrix, and X' is the transformed data point coordinate matrix; The matrix expression of the transformation process from arc profile data to plane profile data is as follows: Among them, T xi is the distance that the i-th original data point moves along the x-axis; T zi is the distance that the i-th original data point moves along the z axis; θ i = -arctan(b i / a i ), i = 1, 2, 3, ... n, where n is the number of data point coordinates.

4. The method for fractal characterization of arc two-dimensional surface contour based on coordinate transformation according to claim 3 is characterized in that: In the step 3, according to the measured arc profile data analysis, the fractal dimension D is inversely proportional to the amplitude of the arc profile. The larger D is, the smaller the simulated arc profile amplitude is, that is, it is negatively correlated with the surface Ra; the scale coefficient G is directly proportional to the arc profile amplitude. The larger G is, the larger the simulated arc profile amplitude is, that is, it is positively correlated with Ra; the surface roughness Ra of each position of the arc surface after grinding (from one end of the arc to the other end) shows a large-small-large rule, and the WM function model is optimized, and the expression is as follows: Where P(x) is the optimization coefficient, Where B = 1 + k, k is the proportionality coefficient, which is expressed as the ratio of the bottom roughness to the edge roughness in the groove direction of the arc profile; ω is the frequency, ω = L1 / L, L is the length of the simulation sample, and L1 is the distance from the intersection of the two equations y1 = 1-B and y2 = cosx; is the phase, which is related to L; φ is the random phase; The fractal dimension D and scale coefficient G of plane contour data are calculated using the structure function method, and the fractal dimension D and scale coefficient G are brought into the optimized WM function model to perform fractal simulation of plane contour.

5. The method for fractal characterization of arc two-dimensional surface contour based on coordinate transformation according to claim 3, characterized in that: In the step 4, the simulated plane contour data is subjected to an inverse transformation of the rotational coordinate transformation to obtain a circular arc surface contour represented by a fractal; The data transformation relationship is as follows: X”=X'R' (7) In the formula, R is the inverse transformation matrix of the rotation transformation, and X" is the final required circular arc surface contour coordinate point matrix; The matrix expression of the coordinate rotation inverse transformation process is as follows: Among them, x i ”,z i " is the final required arc surface contour coordinate point, θ' i is the rotation angle of the inverse transformation, θ' i =-θ i =arctan(b i / a i ); The mathematical expression of the arc two-dimensional surface profile data based on coordinate transformation is:

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