Tooth direction deviation correction method based on cubic spline method

Abstract

The invention provides a tooth direction deviation correction method based on a cubic spline method, and the method comprises the steps: measuring tooth surface data based on an online measurement technology, processing the tooth surface data to obtain a tooth surface deviation correction value, and processing the tooth surface deviation correction value through a cubic spline interpolation method to obtain an interpolation function. And finally, correcting the tooth direction deviation by adopting an additional radial and rotation angle composite motion double-side modification method. The method comprises the following steps: firstly, carrying out grid division on a tooth surface according to a tooth profile curvature, and measuring by utilizing an online measurement technology to obtain tooth surface data; fitting the tooth surface by using a Gaussian curved surface fitting method, and compensating the radius of the measuring head to obtain an actual tooth surface equation; then, calculating a straight line parameter equation of the grid points along the end face normal vector, carrying out simultaneous operation on the straight line parameter equation and an actual tooth surface equation to obtain a tooth direction deviation value, and carrying out mirroring on the tooth direction deviation value to obtain a tooth direction deviation correction value; processing the tooth direction correction by adopting a cubic spline interpolation method to obtain an interpolation function; and finally, correcting the tooth direction deviation by adopting an additional radial and rotation angle composite motion double-side modification method. According to the tooth direction deviation correction method provided by the invention, the tooth direction deviation value is efficiently obtained; on the basis that the tooth direction deviation is effectively improved, the machining efficiency is improved; and the method can correct taper errors, helical angle errors and nonlinear tooth direction errors at the same time.

Description

Technical Field

[0001] The present invention relates to the field of CNC mechanical technology, and specifically to a tooth deviation correction method based on the cubic spline method, so as to solve the tooth deviation problem generated during the tooth shaping process, and to efficiently obtain the tooth deviation amount by using online measurement technology; and the method improves the processing efficiency on the basis of effectively improving the tooth deviation. Background Art

[0002] CNC profile gear grinding machines are widely used in high-precision gear processing, including straight / helical cylindrical gears, complex tooth profile modified gears, complex tooth profile modified gears and other gear grinding processes. For profile gear grinding, long-term grinding will cause the column posture to change, thereby generating nonlinear tooth profile deviation, which cannot be corrected by conventional taper and helix angle compensation. In view of this, the present invention proposes a tooth profile deviation correction method based on the cubic spline method, which can simultaneously correct the taper error, helix angle error and nonlinear tooth profile error. Summary of the invention

[0003] The object of the present invention is to provide a tooth profile deviation correction method based on the cubic spline method, so as to correct the tooth profile deviation generated during the tooth profile modification process.

[0004] In order to achieve the above technical objectives, the technical solution adopted by the present invention is as follows:

[0005] A tooth guide deviation correction method based on cubic spline method, characterized by comprising the following steps:

[0006] Step 1: Measure the tooth surface data and plan the measurement path according to the curvature of the tooth profile;

[0007] Step 2: Use Gaussian surface fitting method to calculate the actual tooth surface point data;

[0008] Step 3: Calculate the tooth deviation and tooth deviation correction;

[0009] Step 4: Perform cubic spline interpolation processing on the tooth deviation correction amount;

[0010] Step 5: Correct the tooth deviation according to the obtained interpolation function.

[0011] The step 1 specifically includes the following steps:

[0012] The equation of the involute helical tooth surface is:

[0013]

[0014] Among them, u and θ are parameter variables; r b is the base circle radius; σ0 is the tooth groove half angle on the base circle; p is the helical parameter;

[0015] According to the curvature of the tooth profile, grid points are selected in the rotational projection plane of the tooth surface. The tooth profile needs to be divided into intervals. The greater the curvature, the more equal divisions are appropriately increased. Therefore, the method of dividing the curve by a constant is used to divide it. First, m points are taken on the curve, and the length of the equally divided curve is calculated; then the equally divided points are connected with straight lines to calculate the distance of each straight line; finally, the difference between the length of each curve and the straight line is calculated, and the difference is used as the basis for adding equal divisions. Where the difference is large, more equally divided points are inserted;

[0016] Divide the curve into equal parts:

[0017] (1) When the difference is greater than 1.5 times the average value, two equal points are inserted to divide the interval into three segments;

[0018] (2) When the difference is greater than the mean value divided by 2N, where N is the number of equal segments, an equal division point is inserted to divide the interval into two segments;

[0019] Finally, (m+M)×n can be obtained in the tooth surface rotation projection plane, where m=1, 2, …i; n=1, 2…j; M is the number of equally spaced grid points inserted, mapped to the tooth surface, and the tooth surface grid points are determined. The involute equidistance method is used to plan the measurement trajectory, and then the measurement is performed. The coordinates of the tooth surface points are recorded as (ρ 11 ,θ 11 ,Z 11 ),(ρ 12 ,θ 12 ,Z 12 ),……,(ρ ij ,θ ij ,Z ij ), the measured tooth surface coordinate data is converted to the workpiece coordinate system and then converted to rectangular coordinates, recorded as (x ij ,y ij ,z ij ).

[0020] The step 2 specifically includes the following steps:

[0021] According to the measured data, the Gaussian surface fitting method is used to perform surface fitting on the measured tooth surface point data. The formula of the Gaussian function is as follows:

[0022]

[0023] In the above formula, a represents the peak value of the curve; b represents the horizontal coordinate corresponding to the peak value of the curve; c represents the standard deviation; based on this, the function of the three-dimensional Gaussian surface can be obtained as:

[0024]

[0025] In the above formula, a represents the peak value of the surface; bx Indicates the x-coordinate corresponding to the peak value of the surface; c x represents the standard deviation of the x-axis; b y Indicates the y coordinate corresponding to the peak value of the surface; c y represents the standard deviation on the y-axis;

[0026] According to the data points obtained by measurement (x ij ,y ij ,z ij ), perform Gaussian surface fitting based on x and y to obtain the peak value of the Gaussian surface;

[0027] Based on Gauss's three-dimensional formula, let the peak be z0, its coordinates be x0, y0, and the standard deviation of the x-axis be c x , the standard deviation on the y-axis is c y , then the formula for any point can be written as:

[0028]

[0029] Solve the above formula by taking the logarithm, and we can get

[0030]

[0031] The above formula can be transformed into:

[0032]

[0033] make:

[0034]

[0035] Then the surface fitting equation can be obtained as follows:

[0036]

[0037] Based on the least squares method, the parameters of the above equation can be solved to obtain the surface fitting equation:

[0038]

[0039] Then the normal vector of a point on the surface can be written as:

[0040]

[0041] Then the probe compensation is performed on the measured tooth surface coordinate point data, and the actual tooth surface point can be obtained as:

[0042]

[0043] In the formula, R c is the radius of the measuring sphere.

[0044] The step three specifically includes the following steps:

[0045] The expression of the end normal vector of the involute mesh point is:

[0046]

[0047] In the formula, u and θ are parameter variables, σ0 is the half angle of the tooth groove on the base circle;

[0048] The parametric equation of the straight line passing through the theoretical tooth surface grid point along the normal direction of the gear end face is expressed as:

[0049]

[0050] In the formula, (x s ,y s ,z s ) is the actual measured tooth surface coordinate, (x p ,y p ,z p ) is the theoretical tooth surface point coordinate, l is the deviation, (n x ,n y ) is the end surface normal vector at the theoretical point;

[0051] The tooth surface deviation ε of each grid point can be obtained by combining and solving the linear parameter equation determined by the normal vector of each grid point and the end face point with the equation of the actual tooth surface.

[0052] The tooth deviation data at the pitch circle is obtained by interpolation. The tooth deviation correction value can be obtained by taking the inverse of the data and recording it as y i , i=1,2,…n.

[0053] The step 4 specifically includes the following steps:

[0054] In order to improve the smoothness and avoid the Runge-Kutta phenomenon, the tooth shape deviation is processed by interpolation using the piecewise cubic spline method, and the measured tooth width interval is divided into n intervals [z i ,z i +1], i=1,2,…n, each interval is expressed by a spline curve:

[0055] ε k (z) = a k +b k (zz i )+c k (zz i ) 2 +d k (zz i ) 3 ,(k=0,1,...n-1)

[0056] The entire cubic spline curve needs to determine a k ,b k ,c k ,d k , (k=0,1,...n-1), a total of 4n undetermined coefficients, let h i =z i+1 -z i , according to the boundary of each spline curve:

[0057]

[0058] Let ε i (z i )=ε i , ε i (z i+1 )=ε i+1 , we can get:

[0059]

[0060] Let m i =ε i ”(z i )=2c i ,but:

[0061] d i =(m i+1 -m i ) / 6 / h i

[0062] A i , c i , d i Substituting into the above formula, we can get

[0063]

[0064] Under natural boundary conditions, m0=0, m n = 0, we can get,

[0065]

[0066] According to the above matrix equation, the unique solution of each interval spline coefficient can be obtained to realize cubic spline interpolation.

[0067] The step five specifically includes the following steps:

[0068] Using the bilateral modification method of additional radial plus angular compound motion, the relationship between the end face tooth modification amounts ε1, ε2 at the left and right tooth pitch circles and the additional radial motion amount Δx and the additional angular motion amount Δc is:

[0069]

[0070] Right now,

[0071]

[0072] According to the fitting curve obtained by cubic spline interpolation, the tooth deviation is corrected by using the additional radial plus angular compound motion bilateral modification method.

[0073] The technical effects of the present invention have the following advantages:

[0074] The present invention provides a tooth deviation correction method based on the cubic spline method to solve the tooth deviation problem generated during the tooth shaping process. The method uses online measurement technology to obtain tooth surface data, which can improve efficiency and processing accuracy; the correction method can simultaneously correct taper error, helix angle error and nonlinear tooth error. BRIEF DESCRIPTION OF THE DRAWINGS

[0075] Figure 1 A schematic diagram of the mesh point division of the tooth surface of the present invention;

[0076] Figure 2 This is a schematic diagram of online measurement path planning of the present invention;

[0077] Figure 3 It is a schematic diagram of tooth deviation evaluation of the present invention. DETAILED DESCRIPTION

[0078] like Figures 1 to 3 As shown, the present invention provides a method for correcting tooth deviation based on the cubic spline method, which is described by taking the SKMC-1250 / 10 gear grinding machine of Nanjing University of Technology CNC Technology Co., Ltd. as an example, and its preferred specific implementation is as follows:

[0079] The step 1 specifically includes the following steps:

[0080] The equation of the involute helical tooth surface is:

[0081]

[0082] Among them, u and θ are parameter variables; r b is the base circle radius; σ0 is the tooth groove half angle on the base circle; p is the helical parameter;

[0083] According to the curvature of the tooth profile, grid points are selected in the rotational projection plane of the tooth surface. The tooth profile needs to be divided into intervals. The greater the curvature, the more equal divisions are appropriately increased. Therefore, the method of dividing the curve by a constant is used to divide it. First, m points are taken on the curve, and the length of the equally divided curve is calculated; then the equally divided points are connected with straight lines to calculate the distance of each straight line; finally, the difference between the length of each curve and the straight line is calculated, and the difference is used as the basis for adding equal divisions. Where the difference is large, more equally divided points are inserted;

[0084] Divide the curve into equal parts:

[0085] (1) When the difference is greater than 1.5 times the average value, two equal points are inserted to divide the interval into three segments;

[0086] (2) When the difference is greater than the mean value divided by 2N, where N is the number of equal segments, an equal division point is inserted to divide the interval into two segments;

[0087] Finally, (m+M)×n can be obtained in the tooth surface rotation projection plane, where m=1, 2, …i; n=1, 2…j; M is the number of equally spaced grid points inserted, mapped to the tooth surface, and the tooth surface grid points are determined. The involute equidistance method is used to plan the measurement trajectory, and then the measurement is performed. The coordinates of the tooth surface points are recorded as (ρ 11 ,θ 11 ,Z 11 ),(ρ 12 ,θ 12 ,Z 12 ),……,(ρ ij ,θ ij ,Z ij ), the measured tooth surface coordinate data is converted to the workpiece coordinate system and then converted to rectangular coordinates, recorded as (x ij ,y ij ,z ij );

[0088] The step 2 specifically includes the following steps:

[0089] According to the measured data, the Gaussian surface fitting method is used to perform surface fitting on the measured tooth surface point data. The formula of the Gaussian function is as follows:

[0090]

[0091] In the above formula, a represents the peak value of the curve; b represents the horizontal coordinate corresponding to the peak value of the curve; c represents the standard deviation; based on this, the function of the three-dimensional Gaussian surface can be obtained as:

[0092]

[0093] In the above formula, a represents the peak value of the surface; b x Indicates the x-coordinate corresponding to the peak value of the surface; c x represents the standard deviation of the x-axis; b y Indicates the y coordinate corresponding to the peak value of the surface; c y represents the standard deviation on the y-axis;

[0094] According to the data points obtained by measurement (x ij ,y ij ,z ij), perform Gaussian surface fitting based on x and y to obtain the peak value of the Gaussian surface;

[0095] Based on Gauss's three-dimensional formula, let the peak be z0, its coordinates be x0, y0, and the standard deviation of the x-axis be c x , the standard deviation on the y-axis is c y , then the formula for any point can be written as:

[0096]

[0097] Solve the above formula by taking the logarithm, and we can get

[0098]

[0099] The above formula can be transformed into:

[0100]

[0101] make:

[0102]

[0103] Then the surface fitting equation can be obtained as follows:

[0104]

[0105] Based on the least squares method, the parameters of the above equation can be solved to obtain the surface fitting equation:

[0106]

[0107] Then the normal vector of a point on the surface can be written as:

[0108]

[0109] Then the probe compensation is performed on the measured tooth surface coordinate point data, and the actual tooth surface point can be obtained as:

[0110]

[0111] In the formula, R c is the radius of the measuring sphere;

[0112] The step three specifically includes the following steps:

[0113] The expression of the end normal vector of the involute mesh point is:

[0114]

[0115] In the formula, u and θ are parameter variables, σ0 is the half angle of the tooth groove on the base circle;

[0116] The parametric equation of the straight line passing through the theoretical tooth surface grid point along the normal direction of the gear end face is expressed as:

[0117]

[0118] In the formula, (x s ,y s ,z s ) is the actual measured tooth surface coordinate, (x p ,y p ,z p ) is the theoretical tooth surface point coordinate, l is the deviation, (n x ,n y ) is the end surface normal vector at the theoretical point;

[0119] The tooth surface deviation ε of each grid point can be obtained by combining and solving the linear parameter equation determined by the normal vector of each grid point and the end face point with the equation of the actual tooth surface.

[0120] The tooth deviation data at the pitch circle is obtained by interpolation. The tooth deviation correction value can be obtained by taking the inverse of the data and recording it as y i , i=1,2,…n;

[0121] The step 4 specifically includes the following steps:

[0122] In order to improve the smoothness and avoid the Runge-Kutta phenomenon, the tooth shape deviation is processed by interpolation using the piecewise cubic spline method, and the measured tooth width interval is divided into n intervals [z i ,z i +1], i=1,2,…n, each interval is expressed by a spline curve:

[0123] ε k (z) = a k +b k (zz i )+c k (zz i ) 2 +d k (zz i ) 3 ,(k=0,1,...n-1)

[0124] The entire cubic spline curve needs to determine a k ,b k ,c k ,d k , (k=0,1,...n-1), a total of 4n undetermined coefficients, let h i =z i+1 -z i , according to the boundary of each spline curve:

[0125]

[0126] Let ε i (z i )=ε i , ε i (z i+1 )=ε i+1 , we can get:

[0127]

[0128] Let m i =ε i ”(z i )=2c i ,but:

[0129] d i =(m i+1 -m i ) / 6 / h i

[0130] A i , c i , d i Substituting into the above formula, we can get

[0131]

[0132] Under natural boundary conditions, m0=0, m n = 0, we can get,

[0133]

[0134] According to the above matrix equation, the unique solution of each interval spline coefficient can be obtained to realize cubic spline interpolation;

[0135] The step five specifically includes the following steps:

[0136] Using the bilateral modification method of additional radial plus angular compound motion, the relationship between the end face tooth modification amounts ε1, ε2 at the left and right tooth pitch circles and the additional radial motion amount Δx and the additional angular motion amount Δc is:

[0137]

[0138] Right now,

[0139]

[0140] According to the fitting curve obtained by cubic spline interpolation, the tooth deviation is corrected by using the additional radial plus angular compound motion bilateral modification method.

Claims

1. A tooth deviation correction method based on cubic spline method, characterized in that The steps include: Step 1: Measure the tooth surface data and plan the measurement path according to the curvature of the tooth profile; Step 2: Use Gaussian surface fitting method to calculate the actual tooth surface point data; Step 3: Calculate the tooth deviation and tooth deviation correction; Step 4: Perform cubic spline interpolation processing on the tooth deviation correction amount; Step 5: Correct the tooth deviation according to the obtained interpolation function.

2. The method for correcting tooth deviation based on cubic spline method according to claim 1 is characterized in that The step 1 specifically includes the following steps: The equation of the involute helical tooth surface is: Among them, u and θ are parameter variables; r b is the base circle radius; σ0 is the tooth groove half angle on the base circle; p is the helical parameter; According to the curvature of the tooth profile, grid points are selected in the rotational projection plane of the tooth surface. The tooth profile needs to be divided into intervals. The greater the curvature, the more equal divisions are needed. Therefore, the method of dividing the curve by a constant is used to divide it. First, m points are taken on the curve, and the length of the equally divided curve is calculated. Then, the equally divided points are connected with straight lines to calculate the distance of each straight line. Finally, the difference in length between each curve and the straight line is calculated, and the difference is used as the basis for adding equal divisions. Where the difference is large, more equally divided points are inserted. Divide the curve into equal parts: (1) When the difference is greater than 1.5 times the average value, two equal points are inserted to divide the interval into three segments; (2) When the difference is greater than the mean value divided by 2N, where N is the number of equal segments, an equal division point is inserted to divide the interval into two segments; Finally, (m+M)×n can be obtained in the tooth surface rotation projection plane, where m=1, 2, …i; n=1, 2…j; M is the number of equally spaced grid points inserted, mapped to the tooth surface, and the tooth surface grid points are determined. The involute equidistance method is used to plan the measurement trajectory, and then the measurement is performed. The coordinates of the tooth surface points are recorded as (ρ 11 ,θ 11 ,Z 11 ),(ρ 12 ,θ 12 ,Z 12 ),……,(ρ ij ,θ ij ,Z ij ), the measured tooth surface coordinate data is converted to the workpiece coordinate system and then converted to rectangular coordinates, recorded as (x ij ,y ij ,z ij ).

3. The method for correcting tooth deviation based on cubic spline method according to claim 1 is characterized in that The step 2 specifically includes the following steps: According to the tooth surface data obtained in step 1, the Gaussian surface fitting method is used to perform surface fitting on the measured tooth surface point data. The formula of the Gaussian function is as follows: In the above formula, a represents the peak value of the curve; b represents the horizontal coordinate corresponding to the peak value of the curve; c represents the standard deviation; based on this, the function of the three-dimensional Gaussian surface can be obtained as: In the above formula, a represents the peak value of the surface; b x Indicates the x-coordinate corresponding to the peak value of the surface; c x represents the standard deviation of the x-axis; b y Indicates the y coordinate corresponding to the peak value of the surface; c y represents the standard deviation on the y-axis; According to the data points obtained by measurement (x ij ,y ij ,z ij ), perform Gaussian surface fitting based on x and y to obtain the peak value of the Gaussian surface; Based on Gauss's three-dimensional formula, let the peak be z0, its coordinates be x0, y0, and the standard deviation of the x-axis be c x , the standard deviation on the y-axis is c y , then the formula for any point can be written as: Solve the above formula by taking the logarithm, and we can get The above formula can be transformed into: make: Then the surface fitting equation can be obtained as follows: Based on the least squares method, the parameters of the above equation can be solved to obtain the surface fitting equation: Then the normal vector of a point on the surface can be written as: Then the probe compensation is performed on the measured tooth surface coordinate point data, and the actual tooth surface point can be obtained as: In the formula, R c is the radius of the measuring sphere.

4. The method for correcting tooth deviation based on cubic spline method according to claim 1, characterized in that The step three specifically includes the following steps: The expression of the end normal vector of the involute mesh point is: In the formula, u and θ are parameter variables, σ0 is the half angle of the tooth groove on the base circle; The parametric equation of the straight line passing through the theoretical tooth surface grid point along the normal direction of the gear end face is expressed as: In the formula, (x s ,y s ,z s ) is the actual measured tooth surface coordinate, (x p ,y p ,z p ) is the theoretical tooth surface point coordinate, l is the deviation, (n x ,n y ) is the end surface normal vector at the theoretical point; The tooth surface deviation ε of each grid point can be obtained by combining and solving the linear parameter equation determined by the normal vector of each grid point and the end face point with the equation of the actual tooth surface. The tooth deviation data at the pitch circle is obtained by interpolation. The tooth deviation correction value can be obtained by taking the inverse of the data and recording it as y i , i=1,2,…n.

5. The method for correcting tooth deviation based on cubic spline method according to claim 1, characterized in that The step 4 specifically includes the following steps: In order to improve the smoothness and avoid the Runge-Kutta phenomenon, the tooth shape deviation is processed by interpolation using the piecewise cubic spline method, and the measured tooth width interval is divided into n intervals [z i ,z i +1], i=1,2,…n, each interval is expressed by a spline curve: ε k (from)=a k +b k (z i )+c k (z i ) 2 +d k (z i ) 3 ,(k=0,1,...n-1) The entire cubic spline curve needs to determine a k ,b k ,c k ,d k , (k=0,1,...n-1), a total of 4n undetermined coefficients, let h i =z i+1 -z i , according to the boundary of each spline curve: Order i (z i )=e i ,he i (z i+1 )=e i+1 , available: Let m i =ε i ”(z i )=2c i ,but: d i =(m i+1 -m i ) / 6 / h i A i , c i , d i Substituting into the above formula, we can get Under natural boundary conditions, m0=0, m n =0, we can get, According to the above matrix equation, the unique solution of each interval spline coefficient can be obtained to realize cubic spline interpolation.

6. The method for correcting tooth deviation based on cubic spline method according to claim 1, characterized in that The step five specifically includes the following steps: Using the bilateral modification method of additional radial plus angular compound motion, the relationship between the end face tooth modification amounts ε1, ε2 at the left and right tooth pitch circles and the additional radial motion amount Δx and the additional angular motion amount Δc is: Right now, According to the piecewise linear interpolation polynomial obtained by the piecewise low-order interpolation method in step 4, the tooth deviation is corrected by using the additional radial plus angular compound motion bilateral modification method.

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