A tooth thickness detection method for gear machining process

Abstract

The present application relates to gear machining detection technical field, specifically for a kind of tooth thickness detection method in gear machining process, comprising: after gear machining, its digitized three-dimensional model is obtained, actual processing surface is scanned by structured light scanning device, and tooth surface point cloud data is collected, according to theoretical design parameter, theoretical tooth surface model containing theoretical tooth profile curve and helix is constructed, and the measured point cloud is aligned with theoretical model to complete coordinate system unification.In unified coordinate system, the addendum point, dedendum point and two sides tooth surface measurement area of single gear tooth are identified, the measured point cloud of two sides tooth surface measurement of single tooth is extracted and matched with theoretical tooth profile curve, deviation vector set is calculated, the deviation vector set is processed by improving tooth thickness mapping algorithm, and the difference between actual tooth thickness and theoretical tooth thickness of gear is determined.The present application can accurately obtain gear tooth thickness machining deviation by accurately dividing tooth surface measurement area and processing deviation data relying on special algorithm.

Description

Technical Field

[0001] This invention relates to the field of gear processing and inspection technology, and in particular to a method for detecting tooth thickness during gear processing. Background Technology

[0002] Conventional gear tooth thickness measurement often employs contact measurements using mechanical measuring tools or collects overall gear point cloud data using ordinary 3D scanning equipment. After roughly matching the overall point cloud with the gear's theoretical model, tooth thickness-related deviations are obtained through conventional geometric calculations. These methods do not provide fine differentiation between different regions of individual teeth, and the use of point cloud data and matching with the theoretical model are both based on the entire gear. The extraction of tooth profile deviations does not focus on the core tooth surface region corresponding to the tooth thickness.

[0003] When matching point clouds with theoretical tooth surface models, existing detection technologies cannot eliminate point cloud interference from non-critical measurement areas such as tooth tip and tooth root. The calculated tooth profile deviation data is mixed with invalid information, making it difficult to accurately reflect the actual machining deviation of the tooth surfaces on both sides of a single tooth. Conventional tooth thickness calculation algorithms can only simplify the overall deviation data and cannot perform targeted calculations on the deviation vector sets of the tooth profiles on both sides of a single tooth. The final tooth thickness difference results have a low degree of consistency with the actual machining state of the gear and cannot accurately reflect the true machining deviation of the tooth thickness of a single tooth.

[0004] Therefore, it is urgent to solve the problems of unclear regional division and crude deviation handling in existing detection methods. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the existing technology and propose a tooth thickness detection method for gear machining.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: a method for detecting tooth thickness during gear machining, comprising: After the gear is machined, a digital 3D model of the gear is obtained, and the actual machined surface of the gear is scanned using a structured light scanning device to collect point cloud data of the tooth surface. Based on the theoretical design parameters of the gear, a theoretical tooth surface model of the gear is constructed in a virtual coordinate system. The theoretical tooth surface model includes the theoretical tooth profile curve and theoretical helix of each tooth. The collected tooth surface point cloud data is aligned with the theoretical tooth surface model using coordinate system unification to obtain a set of measured point clouds in a unified coordinate system. In the measured point cloud set, the tooth vertex, tooth root point, and the tooth surface measurement areas on both sides are identified for each tooth; Points belonging to the two sides of the tooth surface measurement area of ​​a single tooth are extracted from the measured point cloud set and matched with the theoretical tooth profile curve respectively to calculate the set of deviation vectors between the actual tooth profile position and the theoretical tooth profile position on both sides of the tooth. An improved tooth thickness mapping algorithm is used to process the calculated set of deviation vectors to determine the difference between the actual tooth thickness and the theoretical tooth thickness of the gear, including: For a single tooth, two sets of deviation components corresponding to the left tooth surface and the right tooth surface are separated from the set of deviation vectors; On the theoretical circumference of the gear pitch circle, a series of radial reference lines with equal angular intervals are selected; Each set of deviation components is decomposed by projection along the corresponding radial reference line to obtain the tooth profile radial deviation component along the gear radial direction and the tooth profile tangential deviation component along the gear circumferential tangential direction. On each of the radial reference lines, the radial deviation component of the tooth profile on the left tooth surface and the radial deviation component of the tooth profile on the right tooth surface are algebraically summed to obtain the comprehensive tooth thickness deviation value at the position of the radial reference line. Along the circumferential direction of the gear pitch circle, connect the combined tooth thickness deviation values ​​at all radial reference line positions to form the tooth thickness deviation distribution curve of a single tooth in the gear.

[0007] As a further aspect of the present invention, obtaining a digital three-dimensional model of the gear after the gear machining is completed includes: After the gear completes its final machining process, the final machining path data and CNC program instructions for the gear are extracted from the computer-aided manufacturing system. Based on the final machining path data and CNC program instructions, the machining process simulation of the gear is reconstructed in computer-aided design software to generate a solid model of the gear's manufacturing features. The manufacturing feature entity model is subjected to feature extraction processing to remove machining auxiliary features and process reference features, resulting in a simplified three-dimensional model that only contains tooth geometry features. The simplified 3D model is subjected to surface meshing to generate a discretized gear surface model composed of triangular facets; By performing a Boolean subtraction operation between the discretized model of the gear surface and the original 3D design model of the gear, the correctness of the machining process simulation is verified, and finally a digital 3D model of the gear is obtained.

[0008] As a further aspect of the present invention, the process of aligning the collected tooth surface point cloud data with the theoretical tooth surface model using coordinate system unification includes: Feature points of positioning holes or center holes on the gear end face are identified from the tooth surface point cloud data, and the measured center axis of the gear is fitted based on the feature points. The theoretical central axis of the gear is extracted from the digital 3D model of the gear; Calculate the rotation matrix and translation vector required to align the measured center axis with the theoretical center axis; All points in the tooth surface point cloud data are spatially transformed by applying the rotation matrix and translation vector; After aligning the measured center axis with the theoretical center axis, the point cloud data after spatial transformation is finely adjusted by rotating around the axis based on the normal direction of the gear end face, so that the tooth groove symmetry center of the point cloud data is aligned with the tooth groove symmetry center of the theoretical tooth surface model, thus completing the coordinate system unification process.

[0009] As a further aspect of the present invention, in the measured point cloud set, the tooth apex, tooth root point, and the tooth surface measurement areas on both sides are identified for each tooth, including: Calculate the radial distance of each point in the measured point cloud set along the circumferential direction of the gear; Select several points with the largest radial distance, calculate the geometric center of these points, and fit the spatial circumference where the geometric center is located to the tooth tip circle; Select several points with the smallest radial distance, calculate the geometric center of these points, and fit the spatial circumference where the geometric center is located to the tooth root circle; Based on the tooth tip circle and tooth root circle, the effective measurement range of the tooth profile is defined in the radial direction of the gear; Within the effective measurement range of the tooth profile, the measured point cloud set is divided into regions using the spatial grid method. The point cloud regions located on both sides of a single tooth and having an angle with the gear end face are respectively classified as the left tooth surface measurement region and the right tooth surface measurement region.

[0010] As a further aspect of the present invention, the step of extracting points belonging to the two tooth surface measurement areas of a single tooth from the measured point cloud set and matching them with the theoretical tooth profile curve includes: Under the unified coordinate system, the theoretical left and right tooth surface curves of a single tooth in the theoretical tooth surface model of the gear are invoked; From the measured point cloud set, obtain all the point clouds of the pre-identified left tooth surface measurement area belonging to the single tooth, and form the left measured point set; From the measured point cloud set, obtain all the point clouds that were pre-identified as belonging to the right tooth surface measurement area of ​​the single tooth, and form the right measured point set; For each measured point A in the set of measured points on the left, calculate the shortest spatial distance a from it to the theoretical left tooth surface curve, and the coordinate α of the nearest point A on the theoretical left tooth surface curve. The value of the shortest spatial distance a is taken as the normal deviation value of the measured point A, and the vector pointing from the measured point A to the nearest point coordinate α is taken as the deviation vector of the measured point A. For each measured point B in the set of measured points on the right, calculate the shortest spatial distance b from it to the theoretical right tooth surface curve, and the coordinate β of the nearest point B on the theoretical right tooth surface curve. The value of the shortest spatial distance b is taken as the normal deviation value of the measured point B, and the vector pointing from the measured point B to the nearest point coordinate β is taken as the deviation vector of the measured point B.

[0011] As a further aspect of the present invention, the step of selecting a series of radial reference lines with equal angular intervals on the theoretical circumference of the gear pitch circle includes: Calculate the pitch circle diameter and number of teeth of the gear based on its design parameters; Determine the axial section plane of the gear used to evaluate the tooth thickness, and establish a polar coordinate system with the gear center as the origin; In the polar coordinate system, with the origin as the center and the radius of the pitch circle as the radius, the circumference is divided into 360 equal parts; Starting from the zero-degree position of the gear, a series of straight lines passing through the center and extending to the outside of the tooth tip circle are generated according to a preset angular step size and used as the radial reference line; Record the angle value of each radial reference line in the polar coordinate system for correlation with the overall tooth thickness deviation value.

[0012] As a further aspect of the present invention, the projection decomposition of each set of deviation components along the corresponding radial reference line direction includes: For a measurement point on the left tooth surface of the gear, obtain its corresponding deviation vector, which contains three coordinate axis components in the Cartesian coordinate system; Obtain the radial reference line corresponding to the measurement point, and calculate the unit direction vector of the radial reference line in the polar coordinate system; The deviation vector is projected onto the unit direction vector to obtain the projection length, which is used as the radial deviation component of the tooth profile. Calculate the deviation vector and subtract the product of the projection length and the unit direction vector to obtain the vector C perpendicular to the radial reference line. Calculate the magnitude of the vector C to obtain the tooth profile tangential deviation component. Repeat the above projection decomposition process for all measurement points located on the left and right tooth surfaces of the gear teeth to generate the tooth profile radial deviation component and tooth profile tangential deviation component corresponding to each measurement point.

[0013] As a further aspect of the present invention, the method further includes a step of evaluating the overall gear error based on the tooth thickness deviation distribution curve: For all teeth of the gear, the improved tooth thickness mapping algorithm is executed sequentially to obtain the tooth thickness deviation distribution curve for each tooth; The tooth thickness deviation distribution curve of each tooth is plotted on a unified chart, with the horizontal axis representing the tooth number and circumferential angle position, and the vertical axis representing the comprehensive tooth thickness deviation value. From the tooth thickness deviation distribution curves of all gear teeth, extract the maximum and minimum comprehensive tooth thickness deviations for each gear tooth. Calculate the maximum value among the maximum values ​​of the comprehensive tooth thickness deviation of all gear teeth, and take it as the upper deviation of the gear tooth thickness. Calculate the minimum value among the minimum values ​​of the comprehensive tooth thickness deviation of all gear teeth, and take it as the lower deviation of the gear tooth thickness. The average value of the overall tooth thickness deviation of each tooth is compared with the theoretical nominal tooth thickness of the gear to calculate the tooth thickness variation of each tooth. The upper tooth thickness deviation, lower tooth thickness deviation, and tooth thickness variation of all gear teeth are integrated to form a gear tooth thickness accuracy assessment report.

[0014] As a further aspect of the present invention, the step of extracting the maximum and minimum values ​​of the comprehensive tooth thickness deviation for each tooth from the tooth thickness deviation distribution curves of all teeth includes: Read the tooth thickness deviation distribution curve of a single tooth. The tooth thickness deviation distribution curve consists of a series of discrete coordinate points, each of which contains the angular position and the corresponding comprehensive tooth thickness deviation value. Traverse all discrete coordinate points of the tooth thickness deviation distribution curve, compare the comprehensive tooth thickness deviation values ​​of each point, find the coordinate point with the largest value, record its angular position and the maximum deviation value, and take it as the maximum value of the comprehensive tooth thickness deviation of the tooth. Traverse all discrete coordinate points of the tooth thickness deviation distribution curve, compare the comprehensive tooth thickness deviation values ​​of each point, find the coordinate point with the smallest value, record its angular position and minimum deviation value, and take it as the minimum comprehensive tooth thickness deviation value of the tooth. Store the maximum and minimum combined tooth thickness deviations of the gear teeth in a data list; Repeat the steps for each tooth of the gear to extract the maximum and minimum combined tooth thickness deviations for all teeth.

[0015] Compared with the prior art, the advantages and positive effects of the present invention are as follows: The tooth vertex and root points of a single tooth are identified from the measured point cloud set under a unified coordinate system. Dedicated tooth surface measurement areas are delineated on both sides of the tooth. The point cloud data in these areas is extracted and matched with the theoretical tooth profile curve. The deviation vector set of the actual tooth profile position on both sides of the tooth relative to the theoretical tooth profile position is calculated. The point cloud data of non-measured areas such as the tooth tip and tooth root do not participate in the tooth profile deviation calculation. The generation of the deviation vector is only carried out around the effective tooth profile area corresponding to the tooth thickness detection. The source of the deviation data directly corresponds to the core machining area of ​​the tooth surface on both sides of a single tooth. The calculation result of the tooth profile position deviation can completely fit the actual machining shape of the tooth surface of a single tooth. The composition of the deviation vector set is not affected by irrelevant point cloud data, and the directionality and purity of the deviation data are optimized.

[0016] An improved tooth thickness mapping algorithm is used to process the set of deviation vectors corresponding to the tooth surfaces on both sides of a single tooth. The algorithm's operation logic fits the spatial geometric relationship between tooth profile deviation and tooth thickness. Various deviation data in the deviation vector set can be directly converted into characterization parameters of tooth thickness difference. The calculation of the difference between the actual and theoretical tooth thickness values ​​is completed based on the deviation data specific to a single tooth. The generation of the difference results perfectly matches the actual machining shape of a single tooth. The algorithm's processing method for deviation vectors abandons the simplified operation mode of conventional algorithms. The characterization form of tooth thickness difference fits the spatial distribution characteristics of gear tooth surfaces. The difference results can directly reflect the actual machining deviation state of the tooth thickness of a single tooth. Attached Figure Description

[0017] Figure 1 This is a flowchart of a tooth thickness detection method for gear machining according to the present invention; Figure 2 A flowchart for obtaining a digital 3D model of a gear; Figure 3 A flowchart for identifying the tooth apex, tooth root, and tooth surface measurement area. Detailed Implementation

[0018] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0019] Example See Figure 1 This invention provides a method for detecting tooth thickness during gear machining, the specific method including: After gear machining, a digital 3D model of the gear is acquired, and a structured light scanning device is used to scan the actual machined surface of the gear, collecting tooth surface point cloud data. Based on the theoretical design parameters of the gear, a theoretical tooth surface model of the gear is constructed in a virtual coordinate system. This theoretical tooth surface model includes the theoretical tooth profile curve and theoretical helix of each tooth. The collected tooth surface point cloud data and the theoretical tooth surface model are aligned using coordinate system unification processing to obtain a set of measured point clouds in a unified coordinate system. In this set of measured point clouds, the tooth vertex, tooth root point, and the tooth surface measurement areas on both sides are identified for each tooth. Points belonging to the tooth surface measurement areas on both sides of a single tooth are extracted from the set of measured point clouds and matched with the theoretical tooth profile curves to calculate the set of deviation vectors between the actual and theoretical tooth profile positions on both sides of the tooth. An improved tooth thickness mapping algorithm is used to process the calculated set of deviation vectors to determine the difference between the actual and theoretical tooth thickness of the gear.

[0020] In one embodiment of the present invention, see [reference] Figure 2 After the gear completes its final machining process, the final machining path data and CNC program instructions are extracted from the computer-aided manufacturing system. Based on these data, the gear machining process simulation is reconstructed in computer-aided design software, generating a manufacturing feature entity model of the gear. Feature extraction is performed on the manufacturing feature entity model to remove machining auxiliary features and process datum features, resulting in a simplified 3D model containing only tooth geometry features. This simplified 3D model is then meshed to generate a discretized gear surface model composed of triangular facets. A Boolean subtraction operation is performed between the discretized gear surface model and the original 3D design model of the gear to verify the correctness of the machining process simulation, ultimately yielding a digital 3D model of the gear.

[0021] The coordinate system unification and alignment process is as follows: Feature points of the positioning holes or center holes on the gear end face are identified from the tooth surface point cloud data. The measured central axis of the gear is fitted based on these feature points. The theoretical central axis of the gear is extracted from the digital 3D model of the gear. The rotation matrix and translation vector required to align the measured central axis with the theoretical central axis are calculated. All points in the tooth surface point cloud data are spatially transformed by applying the rotation matrix and translation vector. After aligning the measured and theoretical central axes, the spatially transformed point cloud data is fine-tuned by rotating around the axis based on the normal direction of the gear end face, aligning the tooth groove symmetry center of the point cloud data with the tooth groove symmetry center of the theoretical tooth surface model, thus completing the coordinate system unification process.

[0022] In practical implementation, taking a helical gear with a module of 3, 30 teeth, and a helix angle of 15 degrees as an example, the process of acquiring the digital 3D model and aligning it with the coordinate system is as follows: After the gear completes the final grinding process, the final machining path data and CNC program instructions are extracted from the computer-aided manufacturing system. The final machining path data includes the 3D coordinate sequence of the tool center trajectory point, and the CNC program instructions include G-code and M-code. Based on the final machining path data and CNC program instructions, the gear machining process simulation is reconstructed in the computer-aided design software. By parsing the CNC instructions one by one to drive the virtual tool to move along the path, Boolean operations for material removal are performed on the 3D model of the gear blank to generate the manufacturing feature entity model of the gear. The manufacturing feature entity model includes not only the tooth profile but also the transition fillets and tool relief grooves generated during machining. Feature extraction processing is performed on the manufacturing feature entity model. Geometric recognition algorithms are used to automatically identify and remove machining auxiliary features and process datum features, such as the chamfer of the process chuck boss and the center hole, to obtain a simplified 3D model containing only the tooth profile geometric features. The simplified 3D model was meshed, with the maximum triangle side length set to 0.05 mm, generating a discretized gear surface model composed of over 500,000 triangular facets. A Boolean subtraction operation was performed between the discretized gear surface model and the original 3D design model of the gear. If the Boolean subtraction result was an empty entity, the correctness of the machining process simulation was verified, ultimately yielding the digital 3D model of the gear.

[0023] In some embodiments, the coordinate system unification alignment process includes feature recognition and spatial transformation. Two hundred feature points (20 mm in diameter for a central hole) on the gear end face are identified from the tooth surface point cloud data acquired using a structured light scanning device. Based on these feature points, the measured central axis of the gear is fitted using a spatial circle fitting algorithm. The theoretical central axis of the gear is extracted from the digital 3D model by drawing the line connecting the centers of the two end faces. The rotation matrix and translation vector required to align the measured central axis with the theoretical central axis are calculated. The rotation matrix corrects angular deviations between the axes, and the translation vector corrects spatial positional deviations. All points in the tooth surface point cloud data are spatially transformed using the rotation matrix and translation vector. After axis alignment, the transformed point cloud data is fine-tuned by rotating around the axis based on the normal direction of the gear end face, aligning the tooth groove symmetry center of the point cloud data with the tooth groove symmetry center of the theoretical tooth surface model, thus completing the coordinate system unification process. The coordinate transformation relationship is described by the following formula:

[0024] in: This represents the coordinate vector of a point in the measured point cloud. This represents the calculated rotation matrix. This represents the calculated translation vector. This represents the coordinate vector of the aligned points in a unified coordinate system.

[0025] It is understandable that, in practical implementation, the process of calculating the rotation matrix and translation vector involves least-squares optimization, with the optimization objective being to minimize the average distance between the sampling points on the measured central axis and the theoretical central axis. In some embodiments, the feature types removed from the manufacturing feature entity model can be adjusted according to the gear process design. Optionally, the size of the triangular facets in the surface mesh generation process can be set to different values ​​according to the detection accuracy requirements. After completing the coordinate system unification process, the measured point cloud set under the unified coordinate system and the theoretical tooth surface model have the same origin and coordinate axis direction, and the orientation of the teeth in the measured point cloud set corresponds to the orientation of the teeth in the theoretical model. It is understandable that spatial transformation ensures the geometric consistency when matching the tooth surface points with the theoretical curve.

[0026] In one embodiment of the present invention, see [reference] Figure 3 In the measured point cloud set, the radial distance of each point in the set is calculated along the circumference of the gear. Several points with the largest radial distances are selected, their geometric centers are calculated, and the spatial circumference containing these centers is fitted to the addendum circle. Similarly, several points with the smallest radial distances are selected, their geometric centers are calculated, and the spatial circumference containing these centers is fitted to the dedendum circle. Based on the addendum and dedendum circles, the effective measurement range of the tooth profile is defined in the radial direction of the gear. Within the effective measurement range, the measured point cloud set is divided into regions using a spatial grid method. Point cloud regions located on both sides of a single tooth and having an angle with the gear end face are classified as the left tooth surface measurement region and the right tooth surface measurement region, respectively.

[0027] The process of extracting points belonging to the two tooth surface measurement areas of a single tooth from the measured point cloud set and matching them with the theoretical tooth profile curves is as follows: Under a unified coordinate system, the theoretical left and right tooth surface curves of a single tooth in the theoretical tooth surface model of the gear are called. From the measured point cloud set, all pre-identified point clouds belonging to the left tooth surface measurement area of ​​the single tooth are obtained, forming the left measured point set. From the measured point cloud set, all pre-identified point clouds belonging to the right tooth surface measurement area of ​​the single tooth are obtained, forming the right measured point set. For each measured point A in the set of measured points on the left, calculate its shortest spatial distance a to the theoretical left tooth surface curve, and the coordinate α of the nearest point of measured point A on the theoretical left tooth surface curve; use the value of the shortest spatial distance a as the normal deviation value of measured point A, and use the vector from measured point A to the coordinate α of the nearest point as the deviation vector of measured point A; for each measured point B in the set of measured points on the right, calculate its shortest spatial distance b to the theoretical right tooth surface curve, and the coordinate β of the nearest point of measured point B on the theoretical right tooth surface curve; The value of the shortest spatial distance b is taken as the normal deviation value of the measured point B, and the vector pointing from the measured point B to the nearest point coordinate β is taken as the deviation vector of the measured point B.

[0028] In practical implementation, taking a spur gear with a module of 4 and 25 teeth as an example, the process of identifying feature regions in the measured point cloud set and matching the point cloud with the theoretical tooth profile curve is as follows: The measured point cloud set in a unified coordinate system contains approximately 1.2 million data points. Along the circumference of the gear, the radial distance of each point in the measured point cloud set is calculated. The radial distance is defined as the perpendicular distance from the point to the theoretical central axis of the gear. The 500 points with the largest radial distances are selected, and the geometric centers of these points are calculated. Based on these geometric centers and the distribution of points, the spatial circumference where the geometric centers are located is fitted as the addendum circle, and the diameter of the fitted addendum circle is 108.12 mm. Then, the 500 points with the smallest radial distances are selected, and the geometric centers of these points are calculated. The spatial circumference where the geometric centers are located is fitted as the dedendum circle, and the diameter of the fitted dedendum circle is 99.80 mm. Based on the addendum circle and dedendum circle, the effective measurement range of the tooth profile is defined in the radial direction of the gear. The lower boundary of the effective measurement range is the dedendum circle diameter plus 0.5 mm, and the upper boundary is the addendum circle diameter minus 0.5 mm.

[0029] In some embodiments, within the effective measurement range of the tooth profile, the point cloud is divided into regions using a spatial grid method. A cylindrical coordinate grid is established with the gear axis as the Z-axis and the end face as the XY plane. The grid size is set to 0.5 mm in both the radial and circumferential directions. Point cloud regions located on both sides of a single tooth and having an angle with the gear end face are classified as the left tooth surface measurement region and the right tooth surface measurement region, respectively. The angle is determined by the angle between the projection direction of the point cloud normal vector on the end face and the radial direction. Points belonging to the two tooth surface measurement regions of a single tooth are extracted from the measured point cloud set and matched with the theoretical tooth profile curves. Under a unified coordinate system, the theoretical left and right tooth surface curves of the third tooth in the theoretical tooth surface model of the gear are called. From the measured point cloud set, all pre-identified point clouds belonging to the left tooth surface measurement region of the third tooth are obtained, forming a left measured point set containing 8,000 points. From the measured point cloud set, all pre-identified point clouds belonging to the right tooth surface measurement region of the third tooth are obtained, forming a right measured point set containing 7,500 points.

[0030] In practical implementation, for each measured point A in the left-side measured point set, the shortest spatial distance *a* to the theoretical left-side tooth surface curve is calculated. This calculation involves finding parameter points on the theoretical curve that minimize the distance. The coordinates *α* of the nearest point A on the theoretical left-side tooth surface curve are obtained through projection. The value of the shortest spatial distance *a* is used as the normal deviation value of measured point A, and the vector pointing from measured point A to the nearest point coordinate *α* is used as the deviation vector of measured point A. For each measured point B in the right-side measured point set, the shortest spatial distance *b* to the theoretical right-side tooth surface curve and the coordinates *β* of the nearest point B on the theoretical right-side tooth surface curve are calculated. The value of the shortest spatial distance *b* is used as the normal deviation value of measured point B, and the vector pointing from measured point B to the nearest point coordinate *β* is used as the deviation vector of measured point B. The distance from the point to the theoretical tooth profile curve is... Determined by the following relationship:

[0031] in: This represents the three-dimensional coordinate vector of a point in either the measured point set on the left or the measured point set on the right. The parametric equations representing the theoretical left or right tooth surface curves. These are curve parameters. This represents the Euclidean norm of a vector.

[0032] Optionally, the algorithm for calculating the shortest spatial distance can employ a variant of the iterative nearest-point algorithm. It is understood that the left and right measured point sets are processed independently. In some embodiments, the theoretical tooth profile curve is stored in the form of a non-uniform rational B-spline. The calculated set of deviation vectors contains the magnitude and direction of the normal deviation of each measured point relative to its theoretical position. It is understood that the matching process establishes an exact correspondence between each measured point and the theoretical model.

[0033] In one embodiment of the present invention, an improved tooth thickness mapping algorithm is used to process the calculated set of deviation vectors, including the following operations: For a single tooth, two sets of deviation components corresponding to the left and right tooth surfaces of the gear are separated from the set of deviation vectors. A series of equally spaced radial reference lines are selected on the theoretical circumference of the gear's pitch circle. Each set of deviation components is projected and decomposed along the corresponding radial reference line to obtain the radial deviation component of the tooth profile along the gear's radial direction and the tangential deviation component of the tooth profile along the gear's circumferential tangent direction. On each radial reference line, the radial deviation component of the tooth profile on the left tooth surface and the radial deviation component of the tooth profile on the right tooth surface are algebraically summed to obtain the comprehensive tooth thickness deviation value at the radial reference line position. The comprehensive tooth thickness deviation values ​​at all radial reference line positions are connected along the circumferential direction of the gear's pitch circle to form the tooth thickness deviation distribution curve of a single gear tooth.

[0034] In practical implementation, taking the seventh tooth of a helical gear with a module of 5, 40 teeth, and a helix angle of 10 degrees as an example, the improved tooth thickness mapping algorithm processes as follows: For a single tooth, two sets of deviation components corresponding to the left and right tooth surfaces of the gear are separated from the deviation vector set. The deviation components for the left tooth surface contain 3,200 vectors, and the deviation components for the right tooth surface contain 3,100 vectors. On the theoretical circumference of the gear's pitch circle, a series of radial reference lines with equal angular intervals are selected, with a pitch circle diameter of 200.00 mm. Each set of deviation components is projected and decomposed along the corresponding radial reference line direction to obtain the radial deviation component of the tooth profile along the gear's radial direction and the tangential deviation component of the tooth profile along the gear's circumferential tangential direction. The projection calculation is based on the radial reference line direction to which the measurement point belongs.

[0035] In some embodiments, when selecting radial reference lines, the angle step size is set to 1.8 degrees, generating a total of eleven radial reference lines within the profile of one tooth of the gear. On each radial reference line, the radial deviation components of the tooth profile on the left and right sides of the gear are algebraically summed to obtain the comprehensive tooth thickness deviation value at the radial reference line position. The summation calculation is performed on the left and right deviation component pairs projected onto the same reference line. Along the circumferential direction of the gear pitch circle, the comprehensive tooth thickness deviation values ​​at all radial reference line positions are connected to form the tooth thickness deviation distribution curve of a single tooth in the gear. The connection method is to connect each discrete data point sequentially with line segments. The calculation of the comprehensive tooth thickness deviation value is expressed by the following relationship:

[0036] in: Representing the The specific circumferential angle corresponding to each radial reference line. Represents the angle The radial deviation component of the tooth profile is obtained by projecting the deviation vector from the measurement point on the left tooth surface. Represents the angle The radial deviation component of the tooth profile is obtained by projecting the deviation vector of the measurement point on the right side of the tooth surface. Represents the angle The calculated comprehensive deviation value of tooth thickness.

[0037] It is understood that the numerical sign of the radial deviation component of the tooth profile reflects the excess or deficiency of material relative to the theoretical position. In some embodiments, the number of radial reference lines can be adjusted according to the tooth width and accuracy requirements. Optionally, the projection decomposition process is performed in a unified coordinate system, with the origin of the coordinate system located at the gear center. The calculated sequence of comprehensive tooth thickness deviation values ​​forms the data basis for the tooth thickness deviation distribution curve. For the seventh tooth in the example, the calculation results of some radial reference line positions are shown in Table 1: Table 1: Calculation Results of Radial Reference Line Position

[0038] It is understandable that the tooth thickness deviation distribution curve visually reflects the variation in tooth thickness at different circumferential angular positions of a single tooth. In practical implementation, a positive comprehensive tooth thickness deviation value indicates that the actual tooth thickness at that angular position is greater than the theoretical nominal tooth thickness. The tooth thickness deviation distribution curve formed by connecting the discrete data points can be used to further analyze the tooth thickness accuracy characteristics of the gear teeth.

[0039] In one embodiment of the present invention, a radial reference line is selected on the gear pitch circle, and the deviation component is projected and decomposed along the radial reference line direction. A spur gear with a module of 2 and 60 teeth is used as a specific example. Based on the gear's design parameters, the pitch circle diameter and number of teeth are calculated. The pitch circle diameter is 120.00 mm, and the number of teeth is 60. An axial section plane for evaluating tooth thickness is determined. This axial section plane is perpendicular to the gear axis, and a polar coordinate system is established with the gear center as the origin. In the polar coordinate system, the circumference is divided into 360 equal parts with the origin as the center and the pitch circle radius as the radius. Starting from the zero-degree position of the gear, a series of straight lines passing through the center and extending to the outside of the addendum circle are generated according to a preset angular step size of 0.5 degrees. These generated straight lines are the radial reference lines. The number of radial reference lines is determined by the angular step size. After dividing the circumference, a total of 720 radial reference lines are generated. Record the angle value of each radial reference line in the polar coordinate system so that it can be correlated with the tooth thickness comprehensive deviation value. For example, the angle value of the first radial reference line is 0 degrees and the second is 0.5 degrees.

[0040] In some embodiments, for a measurement point on the left tooth surface of the gear tooth, its corresponding deviation vector is obtained. The deviation vector contains three coordinate axis components in the Cartesian coordinate system, denoted as . Obtain the radial reference line corresponding to the measurement point. The measurement point is mapped to the nearest radial reference line through its circumferential angle. Calculate the unit direction vector corresponding to the radial reference line in the polar coordinate system. Project the deviation vector onto the unit direction vector to obtain the projection length, which is the tooth profile radial deviation component. Calculate the deviation vector minus the product of the projection length and the unit direction vector to obtain the vector perpendicular to the radial reference line. Calculate the magnitude of this vector to obtain the tooth profile tangential deviation component. Repeat the above projection decomposition process for all measurement points located on the left and right tooth surfaces to generate the tooth profile radial deviation component and tooth profile tangential deviation component corresponding to each measurement point. The projection calculation relationship is defined by the following formula:

[0041] in: This represents the deviation vector corresponding to a measurement point. This represents the unit direction vector of the radial reference line associated with the measurement point in the Cartesian coordinate system, and its direction is determined by the angle of the radial reference line in the polar coordinate system. The symbol "·" indicates the dot product operation of vectors. This refers to the calculated radial deviation component of the tooth profile.

[0042] In practical implementation, when calculating the unit direction vector, the angle of the radial reference line in the polar coordinate system... The direction vector is converted to Cartesian coordinates. It can be understood that the tangential deviation components of the tooth profile can be calculated by the vector difference. The modulus is obtained. Refer to Table 2, which shows the unit direction vectors calculated for a portion of the radial reference lines: Table 2: Radial Reference Line and Unit Direction Vector Table

[0043] Optionally, the angle step size can be set to 1.0 degree or 0.25 degrees depending on the density and accuracy assessment requirements of the measured point cloud. It can be understood that the projection decomposition process transforms the spatial three-dimensional deviation vector into a two-dimensional deviation description along a specific radial and tangential direction of the gear. In some embodiments, when a measurement point does not fall exactly on a radial reference line, its corresponding unit direction vector is determined by finding the radial reference line closest to the circumferential angle of that measurement point. After performing projection decomposition on each measurement point, each measurement point is assigned a tooth profile radial deviation component and a tooth profile tangential deviation component.

[0044] In one embodiment of the present invention, the step of evaluating the overall gear error based on the tooth thickness deviation distribution curve includes: sequentially executing an improved tooth thickness mapping algorithm on all teeth of the gear to obtain the tooth thickness deviation distribution curve for each tooth. The tooth thickness deviation distribution curve for each tooth is then plotted on a unified chart, with the horizontal axis representing the tooth number and circumferential angle position, and the vertical axis representing the overall tooth thickness deviation value.

[0045] Then, from the tooth thickness deviation distribution curves of all teeth, the maximum and minimum comprehensive tooth thickness deviations for each tooth are extracted. The maximum value among the maximum comprehensive tooth thickness deviations of all teeth is calculated as the upper tooth thickness deviation of the gear, and the minimum value among the minimum comprehensive tooth thickness deviations of all teeth is calculated as the lower tooth thickness deviation of the gear. The average comprehensive tooth thickness deviation of each tooth is compared with the theoretical nominal tooth thickness of the gear to calculate the tooth thickness variation of each tooth. The upper tooth thickness deviation, lower tooth thickness deviation, and tooth thickness variation of all teeth are integrated to form a gear tooth thickness accuracy assessment report.

[0046] The specific steps for extracting the maximum and minimum combined tooth thickness deviations for each tooth from the tooth thickness deviation distribution curves of all gear teeth are as follows: First, read the tooth thickness deviation distribution curve for a single tooth. This curve consists of a series of discrete coordinate points, each containing an angular position and a corresponding combined tooth thickness deviation value. Then, iterate through all the discrete coordinate points on the tooth thickness deviation distribution curve, comparing the combined tooth thickness deviation values ​​at each point. Find the coordinate point with the largest value and record its angular position and maximum deviation value as the maximum combined tooth thickness deviation. Next, iterate through all the discrete coordinate points on the tooth thickness deviation distribution curve, comparing the combined tooth thickness deviation values ​​at each point. Find the coordinate point with the smallest value and record its angular position and minimum deviation value as the minimum combined tooth thickness deviation. Store the maximum and minimum combined tooth thickness deviations in a data list. Repeat these steps for each tooth of the gear to extract the maximum and minimum combined tooth thickness deviations for all teeth.

[0047] In practical implementation, a gear with 50 teeth is used as a specific example. For all teeth of the gear, the improved tooth thickness mapping algorithm is executed sequentially to obtain the tooth thickness deviation distribution curve for each tooth, generating 50 tooth thickness deviation distribution curves for each of the 50 teeth. The tooth thickness deviation distribution curve for each tooth is plotted on a unified chart, with the horizontal axis representing the tooth number and circumferential angle position (tooth numbers from 1 to 50), and the vertical axis representing the comprehensive tooth thickness deviation value in millimeters. From the tooth thickness deviation distribution curves of all teeth, the maximum and minimum comprehensive tooth thickness deviation values ​​for each tooth are extracted. For example, the maximum comprehensive tooth thickness deviation for the 5th tooth is +0.015 mm, and the minimum comprehensive tooth thickness deviation for the 5th tooth is -0.003 mm.

[0048] In some embodiments, the operation of extracting the maximum and minimum comprehensive tooth thickness deviations for each tooth is as follows: Read the tooth thickness deviation distribution curve of a single tooth. The tooth thickness deviation distribution curve consists of a series of discrete coordinate points, each containing an angular position and a corresponding comprehensive tooth thickness deviation value. Traverse all discrete coordinate points on the tooth thickness deviation distribution curve, compare the comprehensive tooth thickness deviation values ​​at each point, find the coordinate point with the largest value, and record its angular position and maximum deviation value as the maximum comprehensive tooth thickness deviation. Traverse all discrete coordinate points on the tooth thickness deviation distribution curve, compare the comprehensive tooth thickness deviation values ​​at each point, find the coordinate point with the smallest value, and record its angular position and minimum deviation value as the minimum comprehensive tooth thickness deviation. Store the maximum and minimum comprehensive tooth thickness deviations in a data list. Repeat the steps for each tooth of the gear to complete the extraction of the maximum and minimum comprehensive tooth thickness deviations for all teeth. The final data list contains 50 sets of maximum and minimum value records.

[0049] In practical implementation, the maximum value among the maximum comprehensive tooth thickness deviations of all teeth is calculated as the upper tooth thickness deviation of the gear; for example, the maximum value selected from 50 maximum values ​​is +0.018 mm. The minimum value among the minimum comprehensive tooth thickness deviations of all teeth is calculated as the lower tooth thickness deviation of the gear; for example, the minimum value selected from 50 minimum values ​​is -0.006 mm. The average comprehensive tooth thickness deviation of each tooth is compared with the theoretical nominal tooth thickness of the gear to calculate the tooth thickness variation of each tooth. The tooth thickness variation reflects the change in the uniformity of the tooth thickness of a single tooth. The tooth thickness variation of the k-th tooth is... Calculated by the following formula:

[0050] in: This represents the arithmetic mean of the combined tooth thickness deviations of the k-th tooth at all radial reference line positions. The theoretical nominal tooth thickness design value represents the gear. This represents the calculated tooth thickness variation of the k-th tooth. Integrating the upper tooth thickness deviation, lower tooth thickness deviation, and the tooth thickness variation of all teeth, a gear tooth thickness accuracy assessment report is generated. The report lists the tooth thickness variation of all teeth and the overall deviation limits in tabular form.

[0051] It is understood that the average value of the overall tooth thickness deviation is obtained by calculating the arithmetic mean of the deviation values ​​at all discrete points on the tooth thickness deviation distribution curve of a single tooth. Optionally, the theoretical nominal tooth thickness design value is determined by design parameters such as gear module and pressure angle. The calculated upper and lower tooth thickness deviations define the overall allowable variation range of the actual tooth thickness of the gear. In some embodiments, the tooth thickness accuracy assessment report may also include a graphical representation of the tooth thickness deviation distribution curve. It is understood that performing the extraction operation sequentially for each tooth is the basis for obtaining the overall assessment data. The gear tooth thickness accuracy assessment report comprehensively reflects the consistency and compliance of the tooth thickness dimensions after gear machining.

[0052] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention in any other way. Any person skilled in the art may make changes or modifications to the above-disclosed technical content to create equivalent embodiments that can be applied to other fields. However, any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the protection scope of the present invention.

Claims

1. A method for detecting tooth thickness during gear machining, characterized in that, The method includes: After the gear is machined, a digital 3D model of the gear is obtained, and the actual machined surface of the gear is scanned using a structured light scanning device to collect point cloud data of the tooth surface. Based on the theoretical design parameters of the gear, a theoretical tooth surface model of the gear is constructed in a virtual coordinate system. The theoretical tooth surface model includes the theoretical tooth profile curve and theoretical helix of each tooth. The collected tooth surface point cloud data is aligned with the theoretical tooth surface model using coordinate system unification to obtain a set of measured point clouds in a unified coordinate system. In the measured point cloud set, the tooth vertex, tooth root point, and the tooth surface measurement areas on both sides are identified for each tooth; Points belonging to the two sides of the tooth surface measurement area of ​​a single tooth are extracted from the measured point cloud set and matched with the theoretical tooth profile curve respectively to calculate the set of deviation vectors between the actual tooth profile position and the theoretical tooth profile position on both sides of the tooth. An improved tooth thickness mapping algorithm is used to process the calculated set of deviation vectors to determine the difference between the actual tooth thickness and the theoretical tooth thickness of the gear, including: For a single tooth, two sets of deviation components corresponding to the left tooth surface and the right tooth surface are separated from the set of deviation vectors; On the theoretical circumference of the gear pitch circle, a series of radial reference lines with equal angular intervals are selected; Each set of deviation components is decomposed by projection along the corresponding radial reference line to obtain the tooth profile radial deviation component along the gear radial direction and the tooth profile tangential deviation component along the gear circumferential tangential direction. On each of the radial reference lines, the radial deviation component of the tooth profile on the left tooth surface and the radial deviation component of the tooth profile on the right tooth surface are algebraically summed to obtain the comprehensive tooth thickness deviation value at the position of the radial reference line. Along the circumferential direction of the gear pitch circle, connect the combined tooth thickness deviation values ​​at all radial reference line positions to form the tooth thickness deviation distribution curve of a single tooth in the gear.

2. The method for detecting tooth thickness during gear machining according to claim 1, characterized in that, The process of obtaining a digital 3D model of the gear after machining includes: After the gear completes its final machining process, the final machining path data and CNC program instructions for the gear are extracted from the computer-aided manufacturing system. Based on the final machining path data and CNC program instructions, the machining process simulation of the gear is reconstructed in computer-aided design software to generate a solid model of the gear's manufacturing features. The manufacturing feature entity model is subjected to feature extraction processing to remove machining auxiliary features and process reference features, resulting in a simplified three-dimensional model that only contains tooth geometry features. The simplified 3D model is subjected to surface meshing to generate a discretized gear surface model composed of triangular facets; By performing a Boolean subtraction operation between the discretized model of the gear surface and the original 3D design model of the gear, the correctness of the machining process simulation is verified, and finally a digital 3D model of the gear is obtained.

3. The method for detecting tooth thickness during gear machining according to claim 2, characterized in that, The process of aligning the collected tooth surface point cloud data with the theoretical tooth surface model using coordinate system unification includes: Feature points of positioning holes or center holes on the gear end face are identified from the tooth surface point cloud data, and the measured center axis of the gear is fitted based on the feature points. The theoretical central axis of the gear is extracted from the digital 3D model of the gear; Calculate the rotation matrix and translation vector required to align the measured center axis with the theoretical center axis; All points in the tooth surface point cloud data are spatially transformed by applying the rotation matrix and translation vector; After aligning the measured center axis with the theoretical center axis, the point cloud data after spatial transformation is finely adjusted by rotating around the axis based on the normal direction of the gear end face, so that the tooth groove symmetry center of the point cloud data is aligned with the tooth groove symmetry center of the theoretical tooth surface model, thus completing the coordinate system unification process.

4. The method for detecting tooth thickness during gear machining according to claim 3, characterized in that, In the measured point cloud set, the tooth vertex, tooth root point, and the measurement areas on both sides of the tooth surface are identified for each tooth, including: Calculate the radial distance of each point in the measured point cloud set along the circumferential direction of the gear; Select several points with the largest radial distance, calculate the geometric center of these points, and fit the spatial circumference where the geometric center is located to the tooth tip circle; Select several points with the smallest radial distance, calculate the geometric center of these points, and fit the spatial circumference where the geometric center is located to the tooth root circle; Based on the tooth tip circle and tooth root circle, the effective measurement range of the tooth profile is defined in the radial direction of the gear; Within the effective measurement range of the tooth profile, the measured point cloud set is divided into regions using the spatial grid method. The point cloud regions located on both sides of a single tooth and having an angle with the gear end face are respectively classified as the left tooth surface measurement region and the right tooth surface measurement region.

5. The method for detecting tooth thickness during gear machining according to claim 4, characterized in that, The step of extracting points belonging to the two tooth surface measurement areas of a single tooth from the measured point cloud set and matching them with the theoretical tooth profile curve includes: Under the unified coordinate system, the theoretical left and right tooth surface curves of a single tooth in the theoretical tooth surface model of the gear are invoked; From the measured point cloud set, obtain all the point clouds of the pre-identified left tooth surface measurement area belonging to the single tooth, and form the left measured point set; From the measured point cloud set, obtain all the point clouds that were pre-identified as belonging to the right tooth surface measurement area of ​​the single tooth, and form the right measured point set; For each measured point A in the set of measured points on the left, calculate the shortest spatial distance a from it to the theoretical left tooth surface curve, and the coordinate α of the nearest point A on the theoretical left tooth surface curve. The value of the shortest spatial distance a is taken as the normal deviation value of the measured point A, and the vector pointing from the measured point A to the nearest point coordinate α is taken as the deviation vector of the measured point A. For each measured point B in the set of measured points on the right, calculate the shortest spatial distance b from it to the theoretical right tooth surface curve, and the coordinate β of the nearest point B on the theoretical right tooth surface curve. The value of the shortest spatial distance b is taken as the normal deviation value of the measured point B, and the vector pointing from the measured point B to the nearest point coordinate β is taken as the deviation vector of the measured point B.

6. A method for detecting tooth thickness during gear machining according to claim 5, characterized in that, The selection of a series of radial reference lines with equal angular intervals on the theoretical circumference of the gear pitch circle includes: Calculate the pitch circle diameter and number of teeth of the gear based on its design parameters; Determine the axial section plane of the gear used to evaluate the tooth thickness, and establish a polar coordinate system with the gear center as the origin; In the polar coordinate system, with the origin as the center and the radius of the pitch circle as the radius, the circumference is divided into 360 equal parts; Starting from the zero-degree position of the gear, a series of straight lines passing through the center and extending to the outside of the tooth tip circle are generated according to a preset angular step size and used as the radial reference line; Record the angle value of each radial reference line in the polar coordinate system for correlation with the overall tooth thickness deviation value.

7. A method for detecting tooth thickness during gear machining according to claim 6, characterized in that, The step of projecting and decomposing each set of deviation components along the corresponding radial reference line includes: For a measurement point on the left tooth surface of the gear tooth, obtain its corresponding deviation vector, which contains three coordinate axis components in the Cartesian coordinate system; Obtain the radial reference line corresponding to the measurement point, and calculate the unit direction vector of the radial reference line in the polar coordinate system; The deviation vector is projected onto the unit direction vector to obtain the projection length, which is used as the radial deviation component of the tooth profile. Calculate the deviation vector and subtract the product of the projection length and the unit direction vector to obtain the vector C perpendicular to the radial reference line. Calculate the magnitude of the vector C to obtain the tooth profile tangential deviation component. Repeat the above projection decomposition process for all measurement points located on the left and right tooth surfaces of the gear teeth to generate the tooth profile radial deviation component and tooth profile tangential deviation component corresponding to each measurement point.

8. A method for detecting tooth thickness during gear machining according to claim 7, characterized in that, The method further includes a step of evaluating the overall gear error based on the tooth thickness deviation distribution curve: For all teeth of the gear, the improved tooth thickness mapping algorithm is executed sequentially to obtain the tooth thickness deviation distribution curve for each tooth; The tooth thickness deviation distribution curve of each tooth is plotted on a unified chart, with the horizontal axis representing the tooth number and circumferential angle position, and the vertical axis representing the comprehensive tooth thickness deviation value. From the tooth thickness deviation distribution curves of all gear teeth, extract the maximum and minimum comprehensive tooth thickness deviations for each gear tooth. Calculate the maximum value among the maximum values ​​of the comprehensive tooth thickness deviation of all gear teeth, and take it as the upper deviation of the gear tooth thickness. Calculate the minimum value among the minimum values ​​of the comprehensive tooth thickness deviation of all gear teeth, and take it as the lower deviation of the gear tooth thickness. The average value of the overall tooth thickness deviation of each tooth is compared with the theoretical nominal tooth thickness of the gear to calculate the tooth thickness variation of each tooth. The upper tooth thickness deviation, lower tooth thickness deviation, and tooth thickness variation of all gear teeth are integrated to form a gear tooth thickness accuracy assessment report.

9. A method for detecting tooth thickness during gear machining according to claim 8, characterized in that, The step of extracting the maximum and minimum comprehensive tooth thickness deviations for each tooth from the tooth thickness deviation distribution curves of all teeth includes: Read the tooth thickness deviation distribution curve of a single tooth. The tooth thickness deviation distribution curve consists of a series of discrete coordinate points, each of which contains the angular position and the corresponding comprehensive tooth thickness deviation value. Traverse all discrete coordinate points of the tooth thickness deviation distribution curve, compare the comprehensive tooth thickness deviation values ​​of each point, find the coordinate point with the largest value, record its angular position and the maximum deviation value, and take it as the maximum value of the comprehensive tooth thickness deviation of the tooth. Traverse all discrete coordinate points of the tooth thickness deviation distribution curve, compare the comprehensive tooth thickness deviation values ​​of each point, find the coordinate point with the smallest value, record its angular position and minimum deviation value, and take it as the minimum comprehensive tooth thickness deviation value of the tooth. Store the maximum and minimum combined tooth thickness deviations of the gear teeth in a data list; Repeat the steps for each tooth of the gear to extract the maximum and minimum combined tooth thickness deviations for all teeth.

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