Gear error evaluation method based on full coordinate detection

Abstract

The application discloses a kind of gear error evaluation methods based on full coordinate detection, steps are as follows: S1, tooth profile data point collection;S2, tooth profile data point pre-processing;S3, curve fitting and the drawing of involute;S4, involute optimization;S5, error calculation.The evaluation method of the application carries out full coordinate detection to all data points of gear tooth profile using three-coordinate measuring instrument;Curve fitting is carried out to the detected data points, and the involute and involute cluster near the data points are solved according to the fitting result, the involute in the involute cluster closest to the detected gear data points is found, and the involute is used as the basis for solving the cumulative deviation of gear pitch and the radial runout tolerance of gear ring, to realize gear error evaluation.The error is small, which verifies the accuracy of the gear error evaluation method, realizes high-precision gear precision calculation and error evaluation, provides accurate data basis for the design of gear shaping machine error compensation system, can optimize processing technology, and improve the gear shaping accuracy of gear shaping machine.

Description

Technical Field

[0001] This invention belongs to the field of gear deviation measurement technology, and in particular relates to a gear error evaluation method based on full coordinate detection. Background Technology

[0002] The development of industrial technology places increasingly higher demands on the precision of mechanical products. Gears are crucial components for motion transmission in mechanical equipment, and their precision directly affects the stability and accuracy of gear transmission. Improving gear precision has become an important means of ensuring the working accuracy and service life of mechanical products. Therefore, researching how to improve the inspection and manufacturing precision of gear products is of great significance. Among gear error data, the cumulative pitch deviation and the radial runout tolerance of the gear ring directly determine the smoothness of gear transmission and the accuracy of motion transmission. Therefore, it is essential to optimize the motion position and speed parameters of the tool radial feed axis and the table rotation axis in the gear shaper machining process path, and to compensate for errors in the cumulative pitch deviation and the radial runout tolerance of the gear ring, thereby improving the shaping accuracy of the gear shaper and the precision of the machined gears.

[0003] Traditional coordinate measuring machines (CMMs) detect single points near the pitch circle of gears and then evaluate errors through calculations. However, the measurement process is susceptible to environmental factors and noise, leading to abnormal data points and significant errors in the detection results. To address this issue and improve gear error detection accuracy, Lin Jiachun from Beijing University of Technology proposed a method for measuring the tooth profile shape deviation of cylindrical gears based on a roughness profile analyzer. The measurement results were consistent with the measurement report from the measurement center, with a maximum error not exceeding 5 μm. Miao Jianwei from Jilin University, in his research on visual measurement technology for involute cylindrical gear tooth profiles and radial runout, improved the measurement accuracy of gear tooth profiles using the least squares geometric fitting method. Regarding improving the accuracy of gear error evaluation algorithms, Sun Yonghou from Guilin University of Electronic Technology used a CMM to detect tooth profile data points and applied the moving least squares method to fit the tooth profile curve of the involute gear, obtaining an error evaluation report for the gear. This provided a reference for tooth profile curve fitting and pitch deviation calculation. The above adjustments to the accuracy detection method and error calculation method resulted in a more accurate gear accuracy detection report. However, the measured error results are only used to evaluate gear errors and cannot serve as an accurate basis for optimizing the machining process of the gear shaper to achieve high-precision machining.

[0004] To optimize the machining process of a gear shaper, it is necessary to determine the specific location and accurate parameters of the involute curve corresponding to the cumulative deviation of gear pitch and the radial runout tolerance of the gear ring. Then, based on the specific location and accurate parameters of the involute curve, the position and speed process parameters of the tool radial feed axis and the table rotation axis of the gear shaper need to be adjusted to improve the shaping accuracy of the gear shaper. Therefore, it is necessary to design a gear error evaluation method based on full-coordinate detection. Summary of the Invention

[0005] To address the problems and shortcomings of existing technologies, this invention provides a gear error evaluation method based on full-coordinate measurement. This method utilizes a coordinate measuring machine to perform full-coordinate measurement on all data points of the gear tooth profile. Curve fitting is performed on the detected data points, and the involute and involute cluster near the data points are calculated based on the fitting results. The involute closest to the detected gear data point within the involute cluster is then identified. This involute is used as the basis for calculating the cumulative pitch deviation and radial runout tolerance of the gear ring, thus achieving gear error evaluation. Comparison with error detection reports generated by traditional methods shows a smaller error, verifying the accuracy of this gear error evaluation method. This achieves high-precision gear accuracy calculation and error evaluation, providing accurate data for the design of gear shaper error compensation systems, optimizing machining processes, and improving the shaping accuracy of gear shapers.

[0006] This invention is implemented as follows: a gear error evaluation method based on full coordinate detection, with the specific steps as follows:

[0007] S1. Acquisition of tooth profile data points

[0008] A coordinate measuring machine was used to collect the full coordinate data points of the gear tooth profile in the middle part of the cylindrical gear;

[0009] S2, Preprocessing of Tooth Profile Data Points

[0010] S21. Filtering of Tooth Profile Data Points

[0011] The involute tooth profile of the gear is analyzed, and the detected full-coordinate data points are screened. The involute data points of the left and right tooth profiles of the gear are retained, while the data points belonging to the transition curve of the addendum circle and the root circle are filtered out. The filtered data points are then grouped and stored in the database according to the tooth profile of each tooth in the gear.

[0012] S22. Removal of outlier data points

[0013] Each group of data in the filtered and grouped data is inspected separately, and outlier data points are removed from each group, retaining only valid data points;

[0014] S3. Curve Fitting and Involute Drawing

[0015] S31. Extract a set of data from the valid data of the above outlier removal and analyze it. Use a curve fitting algorithm to fit the data points in the set of data and solve the component form of the fitted curve equation.

[0016] S32. Using the component form of the fitted curve equation obtained in S31, and combining it with the gear pitch circle equation, solve for the coordinates of the intersection point P.

[0017] S33. Drawing Involutes

[0018] Using point P as the reference point for solving the involute in S32, solve the involute passing through point P, and solve the coordinates of the starting and ending points of the involute so that the involute starts at the base circle and ends at the addendum circle.

[0019] S34, Involute Traversal Operation

[0020] Determine the machining accuracy of the gear shaper. Within the machining accuracy range of the gear shaper, define the error range of the involute base circle radius and the error range of the involute starting angle. The error range of the involute base circle radius is Δd, and the error range of the involute starting angle is Δθ. Using the involute passing through point P in S33 as the decision boundary, solve for the involute family within the error range of the involute base circle radius and the error range of the involute starting angle. Plot all involutes within the error interval to obtain the involute family.

[0021] S4, Involute Optimization

[0022] S41. Calculate the total distances Sd1, Sd2, Sd3, ... from different involutes in the involute family to all data points in the corresponding data group analyzed in S31 using the equidistant curve formula;

[0023] S42. Select the total distance Sd that satisfies the following constraints. min Locating the optimal involute within the involute family:

[0024] Sd min =min(Sd1, Sd2, Sd3, ...)

[0025] In the formula, Sd min This is the total distance between the involute closest to all data points in the involute family and all data points.

[0026] According to Sd min The result determines the position of the involute, and obtains the base circle radius corresponding to the involute and the coordinates of the intersection point of the involute and the pitch circle of the gear; wherein, the base circle radius is determined according to the traversal interval of the base circle radius in S34, and the coordinates of the intersection point of the involute and the pitch circle of the gear are solved by simultaneously solving the rectangular coordinate equation of the involute and the pitch circle equation.

[0027] S43. Repeat steps S3, S41, and S42 multiple times to determine the position of the involute corresponding to each group of data in the valid data, and obtain the base circle radius rd of the involute corresponding to each group of data and the coordinates (ρ, θ) of the intersection point of the involute and the pitch circle of the gear, where θ∈[0,2π]. Then terminate the algorithm and output the result.

[0028] S5, Error Calculation

[0029] Cumulative pitch deviation ΔF p The radial runout tolerance ΔF of the gear ring is the maximum absolute value of the difference between the actual arc length and the nominal arc length between any two tooth surfaces on the same side. r It is the specific value of the base circle radius. The error is calculated using the base circle radius rd corresponding to the involute and the coordinates (ρ, θ) of the intersection point of the involute and the pitch circle. The error calculation result is output and a gear error evaluation report is generated.

[0030] Preferably, in step S21, the method for filtering out data points belonging to the transition curve of the tooth tip circle and tooth root circle is to convert the data points (x, y) in the rectangular coordinate system into polar coordinates (ρ, θ) by using a coordinate transformation formula, where θ∈[0, 2π].

[0031] The coordinate transformation formula is as follows:

[0032]

[0033] In the formula, x is the abscissa of the detected data point, y is the ordinate of the detected data point, ρ is the polar radius in polar coordinates, and θ is the angle between the line connecting the data point and the origin and the polar axis in polar coordinates, where θ∈[0, 2π].

[0034] The extreme diameter ρ is compared with the sizes of the addendum circle, base circle, and dedendum circle. When the base circle radius of the gear is greater than the dedendum circle radius, data points where ρ > base circle radius and ρ < addendum circle radius are retained; when the base circle radius of the gear is less than the dedendum circle radius, data points where ρ > dedendum circle radius and ρ < addendum circle radius are retained.

[0035] Preferably, in step S22, the method for detecting outlier data points is to determine the remaining data points (x, y, y) in the corresponding group of data, excluding the first and last data points. i y i ) to its two adjacent data points (x i-1 y i-1 ) and (x i+1 y i+1 Does the distance between the connecting lines satisfy the following constraints?

[0036] |Δd i |>3σ

[0037] In the formula, Δdi For data points (x) i y i ) to its two adjacent data points (x i-1 y i-1 ) and (x i+1 y i+1 The absolute error of the distance between the lines connecting them is:

[0038]

[0039] In the formula, d i For data points (x) i y i ) to its two adjacent data points (x i-1 y i-1 ) and (x i+1 y i+1 The distance between the lines connecting the two points is:

[0040]

[0041] In the formula, Ax + By + C represents the data point (x... i y i The two adjacent data points (x) i-1 y i-1 ) and (x i+1 y i+1 The general form of the straight line equation obtained by solving the system of equations (x, y) is given by A and B, where A and B are the data points (x, y). i-1 y i-1 ) and (x i+1 y i+1 The coefficients of the unknowns x and y in the simultaneous equations of the lines are given. The unknowns x and y are the data points to be solved (x, y). i y i The x and y coordinates of )

[0042] For data points (x) i y i ) to its two adjacent data points (x i-1 y i-1 ) and (x i+1 y i+1 The distance d between the lines connecting them i The arithmetic mean, that is:

[0043]

[0044] In the formula, σ represents the data point (x i y i ) to its two adjacent data points (x i-1 y i-1 ) and (x i+1 yi+1 Distance d of the line i The standard deviation, i.e.:

[0045]

[0046] If a certain data point (x) i y i ) to its two adjacent data points (x i-1 y i-1 ) and (x i+1 y i+1 The absolute error Δd of the distance between the lines i Satisfy the constraint condition |Δd i If |>3σ, then the data point (x) is considered to be... i y i Data points that are outliers are removed.

[0047] Preferably, in step S31, when fitting the data points in the extracted corresponding group of data, four data points are selected from the group of data, the two closest to the scale circle above and below, for a total of four data points, for curve fitting.

[0048] The polar radius ρ in the polar coordinate form (ρ, θ) of the above four data points must satisfy the following constraints:

[0049] ρ i <ρ0<ρ1<r<ρ2<ρ3<ρ j

[0050] Where ρ0, ρ1, ρ2, and ρ3 are the polar radii in polar coordinates corresponding to the above four data points, respectively. i For all polar radii less than ρ0, ρ j For all pole diameters greater than ρ3, r is the pitch circle radius of the gear;

[0051] (ρ0, θ0), (ρ1, θ1), (ρ2, θ2), and (ρ3, θ3) that satisfy the above constraints are the four data points closest to the scale circle. Curve fitting is performed using their rectangular coordinates.

[0052] According to the B-spline curve fitting algorithm, the equation of the fitted curve is as follows:

[0053] Component form of the fitted curve equation:

[0054]

[0055] In the formula:

[0056]

[0057] In the formula, (x0,y0), (x1,y1), (x2,y2), and (x3,y3) are the four data points in the set of data that are closest to the pitch circle of the gear. Among them, (x0,y0) and (x1,y1) are located below the pitch circle, and (x2,y2) and (x3,y3) are located above the pitch circle.

[0058] Preferably, in step S33, when solving for the involute passing through point P, the polar coordinate equation and rectangular coordinate equation of the involute are used to calculate the involute originating from the x-axis; then, the involute originating from the x-axis is combined with the coordinate rotation equation to obtain the involute passing through point P.

[0059] Polar equation of involute:

[0060]

[0061] θ k =tana k -a k

[0062] The rectangular coordinate equation of the involute:

[0063] x = r b *cosθ+r b *sinθ

[0064] y = r b *sinθ-r b *cosθ

[0065] In the formula, r k Let r be the radius vector at point K on the involute. b Let θ be the base circle radius of the involute. k For the unfolding angle of point K on the involute line, a k The pressure angle at point K of the involute;

[0066] Coordinate rotation equation formula:

[0067]

[0068] That is:

[0069]

[0070] or

[0071]

[0072] In the formula, (X,Y) are the data points where the involute curve occurs on the x-axis, and θ is the polar coordinate angle corresponding to the intersection point P of the fitted curve and the pitch circle, θ∈[0,2π].

[0073] Preferably, in step S41, when calculating the total distance from all data points in the corresponding group of data analyzed in S31 to different involutes in the involute cluster, the total distance from all data points in the group of data to the same involute in the involute cluster is calculated first, and then the process is repeated to calculate the total distance from all data points in the group of data to different involutes in the involute cluster, so as to obtain the total distances Sd1, Sd2, Sd3, ... from all data points in the group of data to different involutes in the involute cluster.

[0074] The method for calculating the total distance from all data points in this dataset to the same involute within the involute family is as follows:

[0075] Formula for isometric curves:

[0076]

[0077]

[0078] In the formula, Δd is the distance from the detected data point to the involute, and (x1, y1) are the coordinates of the detected data point. To find the partial derivative of the involute equation with respect to x, substitute y1 into the derivative equation and solve. To find the partial derivative of the involute equation with respect to x, substitute x1 into the derivative equation and solve for the solution. (X1, Y1) are the coordinates of the point on the involute that (x1, y1) is mapped to.

[0079] The distances Δd1, Δd2, Δd3, ... from each data point in the dataset to the same involute within the involute family are calculated using the equidistant curve formula. Then, the calculated distances are summed to obtain the total distance from all data points in the dataset to the same involute within the involute family.

[0080]

[0081] In the formula, n is the total number of data points in the data set, and Sd is the total distance from all data points in the data set to an involute.

[0082] The advantages and positive effects of this invention are:

[0083] This invention, under existing processing and detection technology conditions, utilizes a coordinate measuring machine (CMM) to measure gear tooth profile data points, and employs interval restriction and outlier removal algorithms to filter data points and eliminate outliers. Furthermore, a curve fitting algorithm is used to fit the data points, and based on the fitting results, the involute of the gear tooth profile is drawn, traversed, and optimized. Based on the specific parameters of the optimal involute, the cumulative pitch deviation and radial runout tolerance of the gear ring are calculated. Compared with the gear error report detected by a traditional CMM, the error is less than 5μm. Moreover, the error data obtained using this method can be directly used to optimize the machining process by adjusting the process parameters of the gear shaper, thereby improving the shaping accuracy of the gear shaper. Attached Figure Description

[0084] Figure 1 This is a schematic diagram of the gear shaper coordinate system and the workpiece coordinate system of the present invention;

[0085] Figure 2 This is a schematic diagram of the gear shaping machine's machining process.

[0086] Figure 3 This is a schematic diagram of the full coordinate data points of the gear tooth profile of the present invention;

[0087] Figure 4 This is a schematic diagram of the filtered gear tooth profile data points of the present invention;

[0088] Figure 5 This is a schematic diagram of outlier data point removal according to the present invention;

[0089] Figure 6 This is a schematic diagram of curve fitting data points of the present invention;

[0090] Figure 7 This is a schematic diagram of the involute principle of the present invention;

[0091] Figure 8 This is a schematic diagram of the involute passing through point P of the present invention;

[0092] Figure 9 This is a schematic diagram illustrating the establishment of the error range of the present invention;

[0093] Figure 10 This is the involute traversal diagram of the present invention;

[0094] Figure 11 This is a schematic diagram showing the total distance from all data points in a set of data analyzed in S31 of the present invention to different involutes in the involute cluster; wherein, the vertical axis represents the total distance and the horizontal axis represents the involute number;

[0095] Figure 12 This is the traversal analysis diagram of the present invention;

[0096] Figure 13The cumulative deviation ΔF of the left tooth surface pitch in this invention p picture;

[0097] Figure 14 The cumulative deviation ΔF of the right tooth surface pitch in this invention p picture;

[0098] Figure 15 The radial runout tolerance ΔF of the gear ring of this invention r picture. Detailed Implementation

[0099] To make the objectives, technical solutions, and advantages of this invention clearer, and to further understand the invention's content, features, and effects, the following specific embodiments are provided to further illustrate the invention in detail. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of the invention.

[0100] Example

[0101] First, the processing principle of the gear shaper and the error evaluation index of the gear will be explained:

[0102] like Figure 1 As shown, a coordinate system (x1, y1) is established with O1 as the origin. This coordinate system is fixed to the gear shaper and rotates with the tool, thus becoming the gear shaper coordinate system. A coordinate system (x2, y2) is established with O2 as the origin. This coordinate system is fixed to the workpiece and rotates with the workpiece, thus becoming the workpiece coordinate system. In the initial position, the coordinate axes y1 and y2 coincide with the y-axis, and x1 and x2 are parallel to the x-axis. In the roughing stage of gear shaping, the cutting path of the gear shaping machine adopts a helical feed method. That is, while the gear shaping cutter and workpiece perform generating motion, the gear shaping cutter also performs radial feed motion towards the center of the workpiece. This ensures that the maximum cutting force for each tooth is relatively stable and does not exceed the rated cutting force provided by the machine tool. The center distance a = O1O2. During the gear shaping cutting process, the two coordinate centers gradually approach each other, and the center distance gradually decreases. The center distance at the depth of cut is the final center distance. When the gear shaping cutter reaches the final radial cutting position relative to the workpiece, the finishing stage begins. The radial feed stops, and the gear shaping cutter and workpiece perform generating motion. After the workpiece completes one generating motion, the cutting stops, completing the cutting process. The relative machining path between the gear shaping cutter and the workpiece is as follows: Figure 2 As shown. Transmission ratio i 12 It remains unchanged during the cutting process.

[0103] In gear errors, cumulative pitch deviation refers to the maximum absolute value of the difference between the actual arc length and the nominal arc length between any two tooth surfaces on the same side of the gear's pitch circle. Radial runout tolerance of the gear ring refers to the maximum variation of the probe relative to the gear axis within one revolution of the gear, when the probe is in double contact with the tooth height at the midpoint of the tooth groove. Cumulative pitch deviation and radial runout tolerance directly determine the smoothness of gear transmission and the accuracy of motion transmission. Based on the accurate parameters of the involute curve corresponding to the cumulative pitch deviation and radial runout tolerance, the parameters of the tool radial feed axis and the table rotation axis in the gear shaping machine's machining path can be optimized to compensate for the cumulative pitch deviation and radial runout tolerance errors, thereby improving the accuracy of the machined gears.

[0104] The gear error evaluation method based on full coordinate detection has the following specific steps:

[0105] S1. Acquisition of tooth profile data points

[0106] A coordinate measuring machine (CMM) was used to collect full coordinate data points of the gear tooth profile in the middle section of the cylindrical gear. Specifically, the collected gear tooth profile data points were plotted in a Cartesian coordinate system, such as... Figure 3 As shown.

[0107] S2, Preprocessing of Tooth Profile Data Points

[0108] S21. Filtering of Tooth Profile Data Points

[0109] In order to obtain the cumulative pitch deviation and radial runout tolerance of the gear ring, it is necessary to analyze the involute tooth profile of the gear, screen the detected full coordinate data points, filter out the data points belonging to the transition curve of the addendum circle and the root circle, and retain the involute data points of the left and right tooth profiles of the gear.

[0110] The method for filtering out data points belonging to the transition curve of the tooth tip circle and tooth root circle is to convert the data points (x, y) in the rectangular coordinate system into polar coordinates (ρ, θ) using the coordinate transformation formula, where θ∈[0, 2π].

[0111] The coordinate transformation formula is as follows:

[0112]

[0113] In the formula, x is the abscissa of the detected data point, y is the ordinate of the detected data point, ρ is the polar radius in polar coordinates, and θ is the angle between the line connecting the data point and the origin and the polar axis in polar coordinates, where θ∈[0, 2π].

[0114] The extreme diameter ρ is compared with the sizes of the addendum circle, base circle, and dedendum circle. Specifically, when the number of teeth of the gear is less than or equal to 41, the base circle radius of the gear is greater than the dedendum circle radius. Therefore, it is necessary to compare the sizes of the extreme diameter ρ with the base circle radius and the extreme diameter ρ with the addendum circle radius to perform curve constraints, filter the data points, and retain data points where ρ > base circle radius and ρ < addendum circle radius. When the number of teeth of the gear is greater than or equal to 42, the base circle radius of the gear is less than the dedendum circle radius. Therefore, it is necessary to compare the sizes of the extreme diameter ρ with the dedendum circle radius and the extreme diameter ρ with the addendum circle radius to perform interval constraints, filter the data points, and retain data points where ρ > dedendum circle radius and ρ < addendum circle radius.

[0115] The filtered data points are grouped and stored, with each tooth in the Z-tooth matrix divided into left and right profiles, resulting in 2Z groups of data. After filtering, the retained data points are plotted in a Cartesian coordinate system, such as... Figure 4 As shown.

[0116] S22. Removal of outlier data points

[0117] Because the test results are affected by environmental factors such as noise during the testing process, outliers may appear in the measured data points, and it is necessary to remove the outlier data points.

[0118] The method for detecting outlier data points is as follows: extract one set from the 2Z sets of data, and determine the remaining data points (x, y, z) excluding the first and last data points. i y i ) to its two adjacent data points (x i-1 y i-1 ) and (x i+1 y i+1 Does the distance between the connecting lines satisfy the following constraints?

[0119] |Δd i |>3σ

[0120] In the formula, Δd i For data points (x) i y i ) to its two adjacent data points (x i-1 y i-1 ) and (x i+1 y i+1 The absolute error of the distance between the lines connecting them is:

[0121]

[0122] In the formula, d i For data points (x) i y i ) to its two adjacent data points (x i-1 yi-1 ) and (x i+1 y i+1 The distance between the lines connecting the two points is:

[0123]

[0124] In the formula, Ax + By + C represents the data point (x... i y i The two adjacent data points (x) i-1 y i-1 ) and (x i+1 y i+1 The general form of the straight line equation obtained by solving the system of equations (x, y) is given by A and B, where A and B are the data points (x, y). i-1 y i-1 ) and (x i+1 y i+1 The coefficients of the unknowns x and y in the simultaneous equations of the lines are given. The unknowns x and y are the data points to be solved (x, y). i y i The horizontal and vertical coordinates of ).

[0125] For data points (x) i y i ) to its two adjacent data points (x i-1 y i-1 ) and (x i+1 y i+1 The distance d between the lines connecting them i The arithmetic mean, that is:

[0126]

[0127] In the formula, σ represents the data point (x i y i ) to its two adjacent data points (x i-1 y i-1 ) and (x i+1 y i+1 Distance d of the line i The standard deviation, i.e.:

[0128]

[0129] If a certain data point (x) i y i ) to its two adjacent data points (x i-1 y i-1 ) and (x i+1 y i+1 The absolute error Δd of the distance between the lines i Satisfy the constraint condition |Δd i If |>3σ, then the data point (x) is considered to be...i y i ) represents outlier data points, such as Figure 5 As shown, perform the removal.

[0130] Step S22 is repeated. After 2Z calculations, all data in the 2Z groups are calculated, and outlier data points in each group are removed, leaving only valid data points.

[0131] S3. Curve Fitting and Involute Drawing

[0132] S31. Extract one set of data from the 2Z sets of valid data after outlier removal and analyze it. Use a curve fitting algorithm to fit the extracted data points and solve for the component form of the fitted curve equation. To avoid large errors in the fitting results due to too many fitting points, select four data points from this set for B-spline curve fitting. To reduce computational complexity and optimize the algorithm, select the two closest points above and below the pitch circle for curve fitting. The polar radius ρ in the polar coordinate form (ρ, θ) of these four data points must satisfy the following constraints:

[0133] ρ i <ρ0<ρ1<r<ρ2<ρ3<ρ j

[0134] Where ρ0, ρ1, ρ2, and ρ3 are the polar radii in polar coordinates corresponding to the above four data points, respectively. i For all polar radii less than ρ0, ρ j For all extreme diameters greater than ρ3, r is the pitch circle radius of the gear.

[0135] (ρ0, θ0), (ρ1, θ1), (ρ2, θ2), and (ρ3, θ3) that satisfy the above constraints are the four data points closest to the scale circle. Curve fitting is performed using their rectangular coordinates.

[0136] According to the B-spline curve fitting algorithm, the equation of the fitted curve is as follows:

[0137] Component form of the fitted curve equation:

[0138]

[0139] In the formula:

[0140]

[0141] In the formula, (x0, y0), (x1, y1), (x2, y2), and (x3, y3) are the four data points closest to the pitch circle of the gear in this set of data. Among them, (x0, y0) and (x1, y1) are located below the pitch circle, and (x2, y2) and (x3, y3) are located above the pitch circle. The fitting result is as follows. Figure 6 As shown.

[0142] S32. Using the component form of the fitted curve equation obtained in S31, and combining it with the gear pitch circle equation, solve for the coordinates of the intersection point P.

[0143] Gear pitch circle equation:

[0144] r = m * z / 2

[0145] In the formula, r is the pitch circle radius of the gear, m is the gear module, and z is the number of gear teeth.

[0146] S33. Drawing Involutes

[0147] Using point P as the reference point for solving the involute in S32, solve for the involute passing through the intersection point P, analyze the formation principle of the involute, and solve for the coordinates of the starting and ending points of the involute so that the involute starts at the base circle and ends at the addendum circle.

[0148] Polar equation of involute:

[0149]

[0150] θ k =tan a k -a k

[0151] The rectangular coordinate equation of the involute:

[0152] x = r b *cosθ+r b *sinθ

[0153] y = r b *sinθ-r b *cosθ

[0154] In the formula, r k Let r be the radius vector at point K on the involute. b Let θ be the base circle radius of the involute. k For the unfolding angle of point K on the involute line, a k The pressure angle at point K of the involute is shown in the schematic diagram of the involute principle. Figure 7 As shown.

[0155] Using the polar and rectangular equations of the involute, calculate the involute originating from the x-axis; then, solve the system of equations combining the involute originating from the x-axis with the coordinate rotation equations to obtain the involute passing through point P. The principle is as follows:

[0156] Coordinate rotation equation formula:

[0157]

[0158] That is:

[0159]

[0160] or

[0161]

[0162] In the formula, (X, Y) represent the data points where the involute curve occurs along the x-axis, and θ is the polar coordinate angle corresponding to the intersection point P of the fitted curve and the pitch circle, θ∈[0, 2π]. Figure 8 As shown.

[0163] S34, Involute Traversal Operation

[0164] Determine the machining accuracy of the gear shaper. Within the machining accuracy range of the gear shaper, define the error range of the involute base circle radius and the error range of the involute starting angle. The error range of the involute base circle radius is Δr, and the error range of the involute starting angle is Δθ. Figure 9 As shown. Taking the involute passing through point P in S33 as the decision boundary, the involute family is solved within the error range of the involute base circle radius and the error range of the involute starting angle. All involutes within the error interval are plotted, resulting in the involute family, as shown. Figure 10 As shown.

[0165] S4, Involute Optimization

[0166] S41. Calculate the total distance from different involutes in the involute family to all data points in the corresponding data group analyzed in S31 using the equidistant curve formula.

[0167] Specifically, the method for calculating the total distance from all data points in the corresponding set of data analyzed in S31 to different involutes in the involute family is as follows:

[0168] Formula for isometric curves:

[0169]

[0170]

[0171] In the formula, Δd is the distance from the detected data point to the involute, and (x1, y1) are the coordinates of the detected data point. To find the partial derivative of the involute equation with respect to x, substitute y1 into the derivative equation and solve. To find the partial derivative of the involute equation with respect to x, substitute x1 into the derivative equation to obtain the solution. (X1, Y1) are the coordinates of the point on the involute that (x1, y1) is mapped to.

[0172] The distances Δd1, Δd2, Δd3, ... from each data point in the dataset to the same involute within the involute family are calculated using the equidistant curve formula. Then, the calculated distances are summed to obtain the total distance from all data points in the dataset to the same involute within the involute family.

[0173] In the formula, n is the total number of data points in the data set, and Sd is the total distance from all data points in the data set to an involute.

[0174] Repeat the above process to calculate the total distance from all data points in the data set to different involutes in the involute family, and obtain the total distances Sd1, Sd2, Sd3, ... from all data points in the data set to different involutes in the involute family.

[0175] Specifically, the total distance from all data points in this dataset to different involutes within the involute family is described, and a schematic diagram illustrating the total distance from all data points in this dataset to different involutes within the involute family is provided, as shown below. Figure 11 As shown.

[0176] S42. Select the total distance Sd that satisfies the following constraints. min Locating the optimal involute within the involute family:

[0177] Sd min =min(Sd1, Sd2, Sd3, ...)

[0178] In the formula, Sd min This represents the total distance between the involute closest to all data points in the involute family and all data points.

[0179] According to Sd min The result determines the position of the involute, and obtains the base circle radius corresponding to the involute and the coordinates of the intersection point of the involute and the pitch circle of the gear.

[0180] Specifically, the base circle radius is determined based on the traversal interval of the base circle radius in S34, and the coordinates of the intersection point of the involute and the pitch circle of the gear are solved by simultaneously solving the involute rectangular coordinate equation and the pitch circle equation:

[0181] Involute rectangular coordinate equation:

[0182] x = r b *cosθ+r b *sinθ

[0183] y = r b *sinθ-r b *cosθ

[0184] In the formula, r k Let r be the radius vector at point K on the involute. b Let θ be the base circle radius of the involute. k For the unfolding angle of point K on the involute line, a k The pressure angle at point K on the involute.

[0185] Gear pitch circle equation:

[0186] r = m * z / 2

[0187] In the formula, r is the pitch circle radius of the gear, m is the module of the gear being tested, and z is the number of teeth of the gear being tested.

[0188] S43. Repeat steps S3, S41, and S42 multiple times, that is, determine the position of the involute corresponding to each group of data in the 2Z groups of valid data. After 2Z calculations, obtain the base circle radius rd of the involute corresponding to each group of data and the coordinates (ρ, θ) of the intersection point of the involute and the pitch circle of the gear, where θ∈[0, 2π]. Then terminate the algorithm and output the result.

[0189] S5, Error Calculation

[0190] Cumulative pitch deviation ΔF p The radial runout tolerance ΔF of the gear ring is the maximum absolute value of the difference between the actual arc length and the nominal arc length between any two tooth surfaces on the same side. r It is the specific value of the base circle radius. The error is calculated using the base circle radius rd corresponding to the involute and the coordinates (ρ, θ) of the intersection point of the involute and the pitch circle. The error calculation result is output and a gear error evaluation report is generated.

[0191] In summary, this invention performs full coordinate detection on gear tooth profile data points, uses coordinate transformation formulas to filter and retain involute tooth profile data points that need analysis, and uses an outlier removal algorithm to remove outlier data points. It then uses a curve fitting algorithm to fit data points near the pitch circle and draws a standard involute based on the fitting results. With the goal of identifying the involute closest to the data point, it iterates through all involutes within the error range to find the optimal involute, calculating the total distance between the data point and all involutes in the involute cluster, and using the minimum total distance to locate the optimal involute. Finally, it calculates the cumulative pitch deviation and radial runout tolerance of the gear ring based on the specific parameters of the optimal involute, obtaining accurate values ​​for these two parameters.

[0192] The following example uses a gear with a module of 3, 36 teeth, and a pressure angle of 20° to demonstrate the feasibility of the error evaluation algorithm by calculating the cumulative pitch deviation and radial runout tolerance of the gear ring.

[0193] A coordinate measuring machine (CMM) was used to perform full coordinate measurement of the gear tooth profile data points. A Hexagon CMM was used, and the measurement room temperature was maintained at 25-27℃. Before measurement, the gear tooth surface needed to be cleaned to avoid significant errors in the measurement results due to contamination. The test results are shown in Table 1 below:

[0194] Table 1. Coordinates of gear tooth profile data points

[0195]

[0196] The data point measurement gap is approximately 0.2mm. All data points from this test are stored in the database.

[0197] In MATLAB, coordinate transformation equations are used to convert all data points in the rectangular coordinate system to polar coordinates, retaining the polar radius ρ value, as shown in Table 2:

[0198] Table 2. Polar radius of data points in polar coordinates.

[0199]

[0200] The gear used in this study has 36 teeth. Therefore, the relationship between the extreme diameter ρ and the base circle radius and the addendum circle radius was compared. Data points with ρ > base circle radius and ρ < addendum circle radius were retained. The filtered data points were grouped and stored. Each of the 36 teeth was divided into left and right tooth profiles, for a total of 72 groups of data.

[0201] An outlier removal algorithm was used to remove outlier data points detected during the detection process. A total of 25 outlier data points were removed in this case.

[0202] After analyzing and removing the tooth profile data points, the first set of data from the 72 valid sets is extracted for analysis. Step S3 is executed to obtain the involute cluster corresponding to this set of data. The machining accuracy of the gear shaper used meets the national standard grade 7 accuracy. Under the national standard grade 7 accuracy, the error range of the involute base circle radius and the error range of the involute starting angle are determined. Specifically, the error range of the base circle radius for the involute traversal calculation under the national standard grade 7 accuracy is Δd = ±0.05mm, and the error range of the involute starting angle is Δθ = ±0.05°. The results are plotted as follows. Figure 10 As shown.

[0203] Execute step S4 to calculate the total distances Sd1, Sd2, Sd3... from all data points in this set of data to different involutes, and select the total distance Sd that satisfies the following constraints. min Determine whether the following constraints are met to locate the optimal involute in the involute family:

[0204] Sd min =min(Sd1, Sd2, Sd3, ...)

[0205] Determine Sd min and return Sd min The specific parameters of the corresponding involute include the base circle radius of the involute and the coordinates of the intersection point of the involute and the pitch circle.

[0206] Repeat steps S3-S4 multiple times, such as Figure 12 As shown, the accurate position of the involute corresponding to all data in the 72 valid data sets is calculated. After 72 calculations, the base circle radius rd of the involute corresponding to each data set and the coordinates (ρ, θ) of the intersection point of the involute and the pitch circle of the gear are obtained. The base circle radius of the optimal involute corresponding to each data set and the angle θ corresponding to the intersection point (ρ, θ) of the involute and the pitch circle are output, where θ∈[0,2π], as shown in Table 3.

[0207] Table 3. Involute Parameter Table

[0208]

[0209] Based on the specific parameters of the involute curve in Table 3, error calculations were performed to obtain the cumulative pitch deviation and radial runout tolerance data of the gear with 36 teeth and a module of 3. A gear error evaluation report was then generated. Figures 13-15 As shown.

[0210] In summary, this invention performs full coordinate detection on gear tooth profile data points, uses coordinate transformation formulas to filter and retain involute tooth profile data points that need analysis, and uses an outlier removal algorithm to remove outlier data points. It then uses a curve fitting algorithm to fit data points near the pitch circle and draws a standard involute based on the fitting results. With the goal of identifying the involute closest to the data point, it iterates through all involutes within the error range to find the optimal involute, calculating the total distance between the data point and all involutes in the involute family. The optimal involute is determined based on the minimum total distance. Finally, based on the specific parameters of the optimal involute, it calculates the cumulative pitch deviation and radial runout tolerance of the gear, obtaining accurate values ​​for these tolerances.

[0211] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A gear error evaluation method based on full coordinate detection, characterized in that, The specific steps are as follows: S1. Acquisition of tooth profile data points A coordinate measuring machine was used to collect the full coordinate data points of the gear tooth profile in the middle part of the cylindrical gear; S2, Preprocessing of Tooth Profile Data Points S21. Filtering of Tooth Profile Data Points The involute tooth profile of the gear is analyzed, and the detected full-coordinate data points are screened. The involute data points of the left and right tooth profiles of the gear are retained, while the data points belonging to the transition curve of the addendum circle and the root circle are filtered out. The filtered data points are then grouped and stored in the database according to the tooth profile of each tooth in the gear. S22. Removal of outlier data points Each group of data in the filtered and grouped data is inspected separately, and outlier data points are removed from each group, retaining the valid data points; S3. Curve Fitting and Involute Drawing S31. Extract a set of data from the valid data of the above outlier removal and analyze it. Use a curve fitting algorithm to fit the data points in the set of data and solve the component form of the fitted curve equation. When fitting the data points in the extracted corresponding set of data, select four data points in the set of data that are closest to the scale circle above and below, for curve fitting; polar coordinates of the above four data points polar diameter The following constraints must be met: in, These are the polar radii in polar coordinates for the four data points mentioned above. Less than All polar radii, greater than All polar radii, The pitch circle radius of the gear; Satisfying the above constraints For the four data points closest to the pitch circle, curve fitting is performed using their rectangular coordinates. According to the B-spline curve fitting algorithm, the equation of the fitted curve is as follows: Component form of the fitted curve equation: In the formula: In the formula, These are the four data points in the data set closest to the gear pitch circle. Located below the pitch circle, Located above the pitch circle; S32. Using the component form of the fitted curve equation obtained in S31, and combining it with the gear pitch circle equation, solve for the coordinates of the intersection point P. S33. Drawing Involutes Using point P as the reference point for solving the involute in S32, solve for the involute passing through point P, and solve for the coordinates of the starting and ending points of the involute, so that the involute starts at the base circle and ends at the addendum circle. S34, Involute Traversal Operation Determine the machining accuracy of the gear shaper, and within the machining accuracy range of the gear shaper, define the error range of the involute base circle radius and the error range of the involute starting angle. The error range of the involute base circle radius is... The error range of the involute starting angle is Using the involute passing through point P in S33 as the decision boundary, the involute family is solved within the error range of the base circle radius of the involute and the error range of the starting angle of the involute. All involutes within the error interval are plotted to obtain the involute family. S4, Involute Optimization S41. Calculate the total distance from different involutes in the involute family to all data points in the corresponding data set analyzed in S31 using the equidistant curve formula. ; S42. Select the total distance that satisfies the following constraints. Locating the optimal involute within the involute family: In the formula, This is the total distance between the involute closest to all data points in the involute family and all data points. according to The result determines the position of the involute, and obtains the base circle radius corresponding to the involute and the coordinates of the intersection point of the involute and the pitch circle of the gear; wherein, the base circle radius is determined according to the traversal interval of the base circle radius in S34, and the coordinates of the intersection point of the involute and the pitch circle of the gear are solved by simultaneously solving the rectangular coordinate equation of the involute and the pitch circle equation. S43. Repeat steps S3, S41, and S42 multiple times to determine the position of the involute corresponding to each set of valid data, and obtain the base circle radius of the involute corresponding to each set of data. Coordinates of the intersection point of the involute and the pitch circle of the gear When θ∈[0,2π], the algorithm terminates and the result is output. S5, Error Calculation Cumulative pitch deviation The radial runout tolerance of the gear ring is the maximum absolute value of the difference between the actual arc length and the nominal arc length between any two tooth surfaces on the same side. It is the specific value of the base circle radius, using the base circle radius corresponding to the involute. Coordinates of the intersection of the involute and the pitch circle Perform error calculation, output the error calculation results, and generate a gear error evaluation report.

2. The gear error evaluation method based on full coordinate detection according to claim 1, characterized in that, In step S21, the method for filtering out data points belonging to the transition curve between the addendum circle and the root circle is to use a coordinate transformation formula to convert the data points in the rectangular coordinate system. Convert to polar coordinates ), θ∈[0, 2π]; The coordinate transformation formula is as follows: In the formula, x is the abscissa of the detected data point, y is the ordinate of the detected data point, ρ is the polar radius in polar coordinates, and θ is the angle between the line connecting the data point and the origin and the polar axis in polar coordinates, where θ∈[0, 2π]. for polar diameter Compare the dimensions with the addendum circle, base circle, and dedendum circle. When the base circle radius of the gear is larger than the dedendum circle radius, retain... Base circle radius and <Data points for the addendum circle radius; when the base circle radius of the gear is smaller than the root circle radius, retain... >Root radius and <Data points for the addendum circle radius.

3. The gear error evaluation method based on full coordinate detection according to claim 1, characterized in that, In step S22, the method for detecting outlier data points is to determine the remaining data points in the corresponding group of data, excluding the first and last two data points. , ) to its two adjacent data points ( , )and( , Does the distance between the connecting lines satisfy the following constraints? In the formula, For data points ( , ) to its two adjacent data points ( , )and( , The absolute error of the distance between the lines connecting them is: In the formula, For data points ( , ) to its two adjacent data points ( , )and( , The distance between the lines connecting the two points is: In the formula, For data points ( , The two adjacent data points () , )and( , The general form of the straight line equation obtained by solving the system of equations (A and B are data points). , )and( , The coefficients of the unknowns x and y in the simultaneous equations of the lines are given, where the unknowns x and y are the data points to be solved. , The x and y coordinates of ) For data points ( , ) to its two adjacent data points ( , )and( , Distance between the lines The arithmetic mean, that is: In the formula, n is the total number of data points in the set of data; In the formula, For data points ( , ) to its two adjacent data points ( , )and( , Distance between lines The standard deviation, i.e.: If a certain data point ( , ) to its two adjacent data points ( , )and( , The absolute error of the distance between the lines Satisfy constraints Then it is considered that the data point ( , Data points that are outliers are removed.

4. The gear error evaluation method based on full coordinate detection according to claim 1, characterized in that, In step S33, when solving for the involute passing through point P, the polar and rectangular equations of the involute are used to calculate the value from point P. The involute originates from the axis; then from... The involute originating from the axis, combined with the coordinate rotation equation, yields the involute passing through point P; Polar equation of involute: The rectangular coordinate equation of the involute: In the formula, Let K be the radius vector of the involute. Let be the base circle radius of the involute. To gradually open the angle of K point on the line, The pressure angle at point K of the involute; Coordinate rotation equation formula: That is: or In the formula, For the involute data points on the x-axis, Let θ be the polar coordinate angle corresponding to the intersection point P of the fitted curve and the scale circle, where θ∈[0,2π].

5. The gear error evaluation method based on full coordinate detection according to claim 1, characterized in that, In step S41, when calculating the total distance from all data points in the corresponding data set analyzed in S31 to different involutes in the involute family, the total distance from all data points in the data set to the same involute in the involute family is first calculated. Then, the process is repeated to calculate the total distance from all data points in the data set to different involutes in the involute family, thus obtaining the total distance from all data points in the data set to different involutes in the involute family. ; The method for calculating the total distance from all data points in this dataset to the same involute within the involute family is as follows: Formula for isometric curves: In the formula, To detect the distance from the data point to the involute, Detect the coordinates of the data points. For the involute equation pair Find the partial derivative Substitute the solution into the derivative equation. For the involute equation pair Find the partial derivative Substitute the solution into the derivative equation. for The coordinates of the point mapped on the involute; The distance from each data point in the dataset to the same involute within the involute family can be calculated using the equidistant curve formula.

1.

2.

3. ..., then add the obtained data together to calculate the total distance from all data points in the data set to the same involute in the involute family: In the formula, n is the total number of data points in the data set, and Sd is the total distance from all data points in the data set to an involute.

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