CN113434817B - Analysis method of gear single topology error map - Google Patents
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Abstract
The invention discloses an analysis method of a gear single topological error map, wherein the map represents a three-dimensional gear error form by the imaging of the gear error map, and clearly expresses various types of characteristic errors on a tooth surface to form the tooth surface topological error after being combined. The map intuitively reflects various characteristic errors on the tooth surface, not only covers the characteristic of the traditional line feature, but also has the function of evaluating the appearance of the whole tooth surface. According to the invention, the error on the tooth surface is mapped to the meshing surface coordinate system, so that the two-dimensional measurement error on the tooth surface is realized. The tooth surface error under the meshing surface coordinate system is represented by adopting a two-dimensional normalized Legendre polynomial, so that mathematical representation of the tooth surface topology error is realized. The three-dimensional gear error form is characterized by the graphics of a gear error map, so that various types of characteristic errors on the tooth surface are clearly expressed, an error map is formed after the combination, and the characteristic errors on the tooth surface are quantitatively calculated.
Description
Technical Field
The invention relates to an analysis method of a gear single topological error map, and belongs to the field of gear mechanical manufacturing.
Background
In the traditional gear detection, due to the lack of tooth surface three-dimensional measurement means, in order to improve the measurement efficiency, several characteristic feature line measurements on the tooth surface are commonly used to replace tooth surface three-dimensional measurement, and the evaluation of the tooth surface is replaced by the evaluation of the several lines, which is based on two-dimensional measurement and evaluation. According to the current gear precision standard, the statistical sampling thought is utilized, namely, a small amount of samples are taken from a large sample, and the evaluation result of the small sample is taken as the evaluation result of the large sample, so that the method is widely adopted in a series of gear precision standards in China. The basic method is that 3-4 teeth are measured on the circumference of the gear, and each tooth surface only measures 2 lines of tooth profile and spiral line, so as to evaluate the tooth surface quality; 1 point was measured on each tooth surface, whereby the distribution position of the tooth surface on the circumference was evaluated.
Three-dimensional errors are a concept of relative and two-dimensional errors. With the development of optical technology, the information available on curved surfaces is more and more rich, complete and comprehensive. The three-dimensional error of the gear contains a large amount of information which cannot be effectively utilized, and how to mine the content of the research of which the intrinsic value is a value. It is therefore necessary to explore how to exploit the large amount of information on the tooth surface, and develop error characterization, measurement and assessment methods based on three-dimensional curved surfaces of gears. The three-dimensional error can more comprehensively reflect the information of the tooth surface, is favorable for evaluating the appearance of the whole tooth surface, is favorable for tracing the processing error, and is favorable for predicting the service performance of the gear. The characterization and evaluation of the development three-dimensional errors have outstanding practical values, and the gear evaluation system can be perfected.
In order to solve the limitation and the deficiency of the traditional cylindrical gear error characterization, the patent provides a brand new gear three-dimensional error characterization method, the diversified gear errors are condensed into typical gear characteristic errors with representative significance, and new concepts of gear characteristic error patterns such as tooth profile deviation patterns, spiral line deviation patterns, tooth pitch deviation patterns and the like and various gear single error topology patterns combined with each other among the characteristic errors are provided. The three-dimensional gear error form is represented by the graph of the gear error graph, and various characteristic errors on the tooth surface are clearly expressed to form the tooth surface topology error after being combined. The atlas intuitively reflects various characteristic errors on the tooth surface, not only covers the characteristic of the traditional line feature, but also has the function of evaluating the appearance of the whole tooth surface.
Disclosure of Invention
The invention provides an analysis method of a gear single topological error map. The method comprises the following specific steps:
1. and establishing a gear three-dimensional error model. According to the gear meshing principle, any point on the tooth surface can find a point corresponding to the point on the meshing surface, the characteristic line on the tooth surface also has a corresponding mapping relation on the meshing surface, and the mapping process involves two coordinate systems, namely a gear coordinate system and a tooth surface meshing coordinate system.
As shown in FIG. 1, the origin of the gear coordinate system is located at the intersection of the center of rotation of the gear and the bottom surface, the x-axis is located in the middle of the first tooth slot, the y-axis is perpendicular to the first tooth slot, and the z-axis is identical to the center of rotation of the gear shaft. The plane shown in figure 1 being where the gears meshOn which a coordinate system (Y n ,Z n )。Z n The axis being in the direction of tooth height, the extent beingY n The axis along the tooth profile development direction, in the range +.>In order to characterize the normal deviation of the tooth surface in the meshing coordinate system, the third coordinate is the coordinate system of Y n ,Z n Perpendicular to axis, d norm (Y n ,Z n ) Representation, coordinate system origin Y n ＝0,Z n =0 is centered along the tooth width direction range b and along the selected analysis area of the tooth profile development direction L. Coordinate system under the engagement surface (Y n ,Y n ,δ norm ) The relation with the gear coordinate system is shown in the formula (1)
Fig. 2 shows a three-dimensional gear coordinate system of the measurement points (x m ,y m ,z m ) The three-dimensional point cloud (left) of (3) obtains a calculated normal deviation graph (right) under the meshing coordinate system through coordinate transformation.
The tooth surface normal error comprising the modification information and the tooth surface error information is obtained by the measurement coordinates of the actual points on the tooth surface, namely, various modification or tooth surface errors on the tooth surface are finally reflected on the normal error. The normal error is further decomposed into a modification of the tooth surface along the spiral line direction, a modification along the tooth profile direction and various errors on the tooth surface through error decomposition.
Legendre polynomial and tooth surface three-dimensional error. The Legendre polynomial is a polynomial obtained by using a separation variable method under a Jie Lapu Las equation in a spherical coordinate, has orthogonality in a section [ -1,1], and is mutually independent among various coefficients. The normal deviation on the tooth surface of the gear can be accurately represented by superposition of two-dimensional normalized Legendre polynomials, thereby characterizing the three-dimensional error of the gear.
The l-th order Legendre polynomial can be expressed as
In the middle of
The invention adopts a two-dimensional Legendre polynomial to reconstruct the tooth surface, wherein the two-dimensional Legendre polynomial can be expressed as the product of two one-dimensional polynomials
Q n (x,y)＝P k (x)P l (y) (3)
k and l represent the order of the axes, respectively. The lowest two-dimensional Legendre term is a zero-order constant, which just represents the gear pitch error, the first-order linear term represents the tilt error common in helix measurement and profile measurement, the second-order term represents the mid-drum or mid-pit error common in helix and profile, the second-order cross term may represent the twist bias present on the tooth surface, and the higher-order term represents the waviness in the helix direction and profile direction on the tooth surface.
Any measurable tooth deviation can be accurately represented by a superposition of the form of the two-dimensional normalized Legendre polynomials. The normal deviation of the jth tooth surface can be expressed as
Wherein C is j,kl Is the expansion coefficient, ψ k (Z n ) And psi is equal to l (Y n ) Is a normalized legendre polynomial on the tooth surface along the spiral direction and the tooth profile direction, respectively. k and l represent the order of the axes, respectively, i.e. the deviation on the tooth surface can be characterized as the sum of a series of Legendre polynomials.
3. Typical error patterns for gears versus Legendre functions. Typical characteristic deviation differences on individual tooth surfaces are classified into 5 types, namely 0-order error tooth surface deviation, 1-order error comprises tooth profile inclination and spiral line inclination, 2-order error comprises tooth profile middle drum or middle concave and spiral line middle drum or middle concave and high-order error comprises tooth profile ripple and spiral line ripple, and 2-order crossed error tooth surface distortion is 10 types in total.
After the three-dimensional errors on the tooth surface are decomposed by adopting a Legendre polynomial, representative characteristic errors are extracted, and are graphically represented, namely, a three-dimensional topological error characteristic map is obtained, and all the characteristic errors are combined, so that a single three-dimensional topological error map can be obtained.
1) Tooth surface offset map
The deviation of the actual position of the tooth surface from the theoretical position is called tooth surface deviation, and is denoted by the symbol f d The characteristic map is shown in fig. 3, and is a plane which is offset from the zero point position. Coordinate Z n Representing Y along the tooth width direction n Representing the normal of the tooth profile of the engagement surface, for better characterization of the normal deviation of the tooth surface, a sum (Y) n ,Z n ) Vertical axis d norm [Y n ,Z n ]And (3) representing. Tooth surface offset and expansion coefficient C in polynomial j,00 Closely related, if the tooth surface offset is large, C j,00 If the absolute value of (C) is large, otherwise, the tooth surface offset is small, C j,00 Is small. C (C) j,00 The sign of (a) indicates the direction of the offset, "+" indicates away from the tooth surface and "-" indicates toward the inside of the tooth surface. The Legendre polynomial characterization function is as follows
L j (0,0)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n ) (5)
Expanding the characterization function to obtain
L j (0,0)＝C j,00 (6)
Expansion coefficient C j,00 In relation to the tooth surface offset
f d ＝C j,00 (7)
2) Tooth profile inclination deviation map
When only the tooth profile inclination deviation exists on the tooth surface, the tooth profile inclination map is a plane inclined along the tooth profile direction, and two ends in the metering range are intersected with the average tooth profile traceThe distance between the designed tooth profile traces is called the tooth profile tilt deviation, denoted by the symbol f Hα Representing the following. The characteristic map is shown in figure 4. Coordinate Z n And Y is equal to n Axis d norm [Y n ,Z n ]The meaning of (1) is the same as the tooth profile offset error map. Degree of tooth profile inclination and expansion coefficient C in polynomial j,01 Closely related, if the tooth profile is inclined by a large amount, C j,01 The absolute value of (C) is large, and conversely, the tooth profile inclination is small j,01 Is small. C (C) j,01 The sign of (c) indicates the direction of tilt, "+" indicates the presence of a negative pressure angle error or a positive base radius error, and "-" indicates the presence of a positive pressure angle error or a negative base radius error. The Legendre polynomial characterization function is as follows
L j (0,1)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n )+C j,01 ·ψ 0 (Z n )ψ 1 (Y n ) (8)
Expanding the characterization function to obtain
Expansion coefficient C j,01 The relation with the tooth profile inclination deviation is that
3) Helix inclination deviation profile
When the spiral inclination deviation exists on the tooth surface only, the spiral inclination map is a plane inclined along the spiral direction, the distance between two designed spiral traces which are intersected with the average spiral trace at the two ends in the metering range is called the spiral inclination deviation, and the sign f is used Hβ And (3) representing. The characteristic map is shown in figure 5. Coordinate Z n And Y is equal to n Axis d norm [Y n ,Z n ]The meaning of (1) is the same as that of the tooth profile offset error map. The degree of helix inclination and the expansion coefficient C in the polynomial j,10 Closely related, the inclination of the spiral is large, C j,10 And conversely, if the absolute value of the helix is large, C is small j,10 Is small. C (C) j,10 The legend of (2) represents the direction of tilt, and the Legendre polynomial characterization function is as follows
L j (1,0)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n )+C j,10 ·ψ 1 (Z n )ψ 0 (Y n ) (11)
Expanding the characterization function to obtain
Expansion coefficient C j,10 The relation with the single error is that
4) Drum deviation map in tooth profile
When only the middle drum profile modification or deviation exists on the tooth surface, the middle drum profile is a paraboloid presenting a middle drum along the tooth profile direction. The mid-drum deviation on the tooth profile belongs to the tooth profile shape deviation. The distance between two curves identical to the average tooth profile trace is contained in the calculated range. And the distance between the two curves and the average tooth profile trace is constant, which is called tooth profile shape deviation and is denoted by the symbol f fα And (3) representing. The characteristic map is shown in figure 6. Coordinate Z n And Y is equal to n D norm [Y n ,Z n ]The meaning of (1) is the same as the tooth profile offset error map. Degree of drum in profile and expansion coefficient C in polynomial j,02 Closely related, if the drum volume in the tooth profile is large, C j,02 If the absolute value of (a) is large, otherwise, if the drum amount in the tooth profile is small, C j,02 Is small. C (C) j,02 The sign of (a) indicates the direction of the parabolic opening, i.e., "+" indicates the presence of a second order tooth surface error on the tooth surface that is concave in the direction of the tooth profile, and "-" indicates the presence of a second order tooth surface error on the tooth surface that is concave in the direction of the tooth profile, as shown in fig. 6Second order tooth surface deviation error of the drum in the direction. The Legendre polynomial characterization function is as follows
L j (0,2)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n )+C j,01 ·ψ 0 (Z n )ψ 1 (Y n )+C j,02 ·ψ 0 (Z n )ψ 2 (Y n ) (14)
From the polynomial it can be seen that there is C in the Legendre expansion coefficient associated with mid-drum bias j,00 、C j,01 And C j,02 Three influence coefficients are added, and the characterization function is unfolded to obtain
Wherein the constant termRepresenting the shift of the entire mid-drum surface relative to the theoretical position, first order termRepresenting the inclination of the error face in the tooth profile direction due to the presence of the drum, the second order term +.>Indicating the presence of a drum, C j,01 The relation with the tooth profile inclination error is that
Expansion coefficient C j,02 The relationship with the drum deviation in the tooth profile is
5) Drum deviation map in spiral line
With only spirals on the tooth surfaceWhen the in-line drum is modified or deviated, the spiral line in-line drum map is a paraboloid of the in-line drum along the spiral line direction. The mid-drum deviation on the spiral belongs to the spiral shape deviation. The distance between two curves identical to the average spiral trace of the actual spiral trace is contained in the range of the value. And the distance between the two curves and the average spiral line trace is constant, called spiral shape deviation, and is denoted by the symbol f fβ And (3) representing. The characteristic map is shown in figure 7. Coordinate Z n And Y is equal to n D norm [Y n ,Z n ]The meaning of (1) is the same as the tooth profile offset error map. The degree of drum in the spiral and the expansion coefficient C of the 2 nd order term in the polynomial j,20 Closely related, if the drum amount in the spiral line is large, C j,20 If the absolute value of (C) is large, otherwise, the drum quantity in the spiral line is small, C j,20 Is small. C (C) j,20 The sign of (c) indicates the direction of the parabolic opening, i.e., "+" indicates the presence of a second order tooth surface error on the tooth surface that is concave in the helical direction, and "-" indicates the presence of a second order tooth surface deviation error on the tooth surface that is convex in the helical direction as shown in fig. 7. The Legendre polynomial characterization function is as follows
L j (2,0)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n )+C j,10 ·ψ 1 (Z n )ψ 0 (Y n )+C j,20 ·ψ 2 (Z n )ψ 0 (Y n ) (18)
From the polynomial it can be seen that there is C in the Legendre expansion coefficient associated with mid-drum bias j,00 、C j,10 And C j,20 Three influence coefficients are added, and the characterization function is unfolded to obtain
Wherein the constant termRepresenting the shift of the entire mid-drum surface relative to the theoretical position, first order termRepresenting the inclination of the error surface in the spiral direction due to the presence of the drum, the second order term +.>Indicating the presence of a drum, C j,10 The relation with the tooth profile inclination error is that
Expansion coefficient C j,20 The relationship with the drum deviation in the tooth profile is
6) Concave deviation map of tooth profile
Similar to the profile-in-drum deviation profile, the profile-in-drum deviation profile is a concave paraboloid along the profile direction when only profile-in-drum modification or deviation exists on the tooth surface. The dishing deviation on the profile belongs to the profile shape deviation. The distance between two identical curves of the actual tooth profile trace and the average tooth profile trace is contained in the calculated range. And the distance between the two curves and the average tooth profile trace is constant, which is called tooth profile shape deviation and is denoted by the symbol f fα And (3) representing. The characteristic map is shown in figure 8. Coordinate Z n And Y is equal to n D norm [Y n ,Z n ]The meaning of (1) is the same as the tooth profile offset error map. Degree of drum in profile and expansion coefficient C in polynomial j,02 Closely related, the concave amount of the tooth profile is large, C j,02 And conversely, if the tooth profile is concave, C j,02 Is small. C (C) j,02 The sign of (a) indicates the direction of parabolic opening, i.e., "+" indicates the presence of a concave second-order tooth surface error along the tooth profile direction as shown in fig. 8 on the tooth surface, and "-" indicates the presence of a second-order tooth surface deviation error along the drum in the tooth profile direction as shown in fig. 6 on the tooth surface. Legendre thereofThe polynomial characterization function is as follows
L j (0,2)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n )+C j,01 ·ψ 0 (Z n )ψ 1 (Y n )+C j,02 ·ψ 0 (Z n )ψ 2 (Y n ) (22)
From the polynomial it can be seen that there is C in the Legendre expansion coefficient associated with mid-drum bias j,00 、C j,01 And C j,02 Three influence coefficients are added, and the characterization function is unfolded to obtain
Wherein the constant termRepresenting the deviation of the entire concave surface from the theoretical position, first order termRepresenting the inclination of the error face in the tooth profile direction due to the presence of dishing, second order termRepresenting the presence of dishing, C j,01 The relation with the tooth profile inclination error is that
Expansion coefficient C j,02 The relation with the concave deviation of the tooth profile is that
7) Spiral line concave deviation map
When there is only concave modification or deviation of spiral line on tooth surface, the concave graph of spiral lineThe spectrum is a concave paraboloid along the direction of the spiral. The dished deviation on the spiral belongs to the spiral shape deviation. The distance between two curves identical to the average spiral trace of the actual spiral trace is contained in the range of the value. And the distance between the two curves and the average spiral line trace is constant, called spiral shape deviation, and is denoted by the symbol f fβ And (3) representing. The characteristic map is shown in figure 9. Coordinate Z n And Y is equal to n D norm [Y n ,Z n ]The meaning of (1) is the same as the tooth profile offset error map. Degree of dishing in spiral and expansion coefficient C in polynomial j,20 Closely related, the concave amount of the spiral line is large, C j,20 The absolute value of (C) is large, whereas the concave amount in the spiral line is small, C j,20 Is small. C (C) j,20 The sign of (c) indicates the direction of the parabolic opening, as shown in fig. 7, "+" indicates the presence of a second order tooth surface error along the drum in the helical direction on the tooth surface, and "-" indicates the presence of a second order tooth surface deviation error along the concave in the helical direction on the tooth surface, as shown in fig. 9. The Legendre polynomial characterization function is as follows
L j (2,0)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n )+C j,10 ·ψ 1 (Z n )ψ 0 (Y n )+C j,20 ·ψ 2 (Z n )ψ 0 (Y n ) (26)
From the polynomial it can be seen that there is C in the Legendre expansion coefficient associated with dishing bias j,00 、C j,10 And C j,20 Three influence coefficients are added, and the characterization function is unfolded to obtain
Wherein the constant termRepresenting the deviation of the entire concave surface from the theoretical position, first order termRepresenting the inclination of the error plane in the spiral direction due to the presence of dishing, second order termRepresenting the presence of dishing, C j,10 The relation with the tooth profile inclination error is that
Expansion coefficient C j,20 The relation with the concave deviation of the tooth profile is that
8) Tooth profile ripple deviation map
The fluctuations (waviness, cyclic deviations) are deviations of the shape of the spiral with a constant wavelength and an almost constant height. The roughness of the waviness profile is a component of the surface shape characteristics, typically with which roughness is superimposed, the waviness spacing being significantly greater than the roughness spacing. Wavelength of waviness in tooth profile direction lambda α The distance from the central line of the profile to the wave crest or the wave trough is the profile waviness, f is used wα And (3) representing. The wave length of the waviness along the spiral line direction is lambda β The distance from the central line of the profile to the wave crest or the wave trough is the profile waviness, f is used wβ And (3) representing. Because of its periodic nature, waviness is a cause of noise generated when gears are used, and it is also necessary to study the waviness of the tooth surface from its surface roughness when high standards of performance and reliability are required.
When there is waviness deviation on the tooth profile, the tooth profile waviness spectrum is shown in fig. 10, and more than one curved surface of wave crest or wave trough exists along the tooth profile direction. Coordinate Z n And Y is equal to n D norm [Y n ,Z n ]The meaning of (1) is the same as the tooth profile offset error map. The orthonormal polynomial selected herein is a 5 th order polynomial, and theoretically, the higher the order the orthonormal polynomial is selected, the higher the accuracy of characterizing the waviness deviation on the tooth surface. Polynomial expansion coefficient C j,04 There is a close relationship with higher order tooth surface deviations. Waviness amplitude f on tooth profile wα Large, then C j,04 The absolute value of (a) is large, whereas the waviness amplitude f on the tooth profile is large wα If it is small, C j,04 Is small in absolute value, C j,04 The sign of (c) indicates the direction of the ripple. "+" indicates that a peak occurs at a midpoint along the profile direction and "-" indicates that a trough occurs at a midpoint along the profile direction. The Legendre polynomial characterization function is as follows
L j (0,4)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n )+C j,02 ·ψ 2 (Z n )ψ 2 (Y n )+C j,04 ·ψ 0 (Z n )ψ 4 (Y n ) (30)
From the polynomial it can be seen that there is C in the Legendre expansion coefficient associated with the profile waviness deviation j,00 、C j,02 And C j,04 Three influence coefficients are added, and the characterization function is unfolded to obtain
Wherein the constant termRepresenting the deviation of the entire tooth surface from the theoretical position, second order termA mid-drum or a mid-recess representing the error face in the tooth profile direction due to the presence of waviness, higher order term +>Indicating the presence of a profile waviness deviation on the tooth surface, when the profile on the tooth surfaceWavelength lambda of the ripple α The larger the 2 nd order term is, the larger the ratio to the higher order term is, the wavelength lambda of the profile ripple on the tooth surface α The smaller the ratio of the 2 nd order term to the higher order term is, the smaller the expansion coefficient C j,00 、C j,02 And C j,04 The size of the tooth profile waviness is determined together, and the relationship between the tooth profile waviness deviation and the tooth profile waviness deviation is that
9) Spiral line ripple deviation map
When the waviness deviation exists on the spiral line, the spiral line waviness spectrum is shown in figure 11, and more than one curved surface of wave crest or wave trough exists along the spiral line direction. Coordinate Z n And Y is equal to n D norm [Y n ,Z n ]The meaning of (1) is the same as the spiral offset error map. The selected orthonormal polynomial is herein a 5 th order polynomial, similar to the tooth profile waviness deviation, and theoretically, the higher the order of the selected orthonormal polynomial, the higher the accuracy of characterizing the waviness deviation on the tooth surface. Polynomial expansion coefficient C j,40 There is a close relationship with higher order tooth surface deviations. Waviness amplitude f on spiral wβ Large, then C j,40 The absolute value of (a) is large, whereas the amplitude f of the waviness on the spiral is large wβ If it is small, C j,40 Is small in absolute value, C j,04 The sign of (c) indicates the direction of the ripple. "+" indicates that a peak occurs at a midpoint along the spiral direction and "-" indicates that a trough occurs at a midpoint along the spiral direction. The Legendre polynomial characterization function is as follows
L j (4,0)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n )+C j,20 ·ψ 2 (Z n )ψ 0 (Y n )+C j,40 ·ψ 4 (Z n )ψ 0 (Y n ) (33)
From the polynomial it can be seen that the Legendre expansion coefficient associated with the spiral waviness deviation has C j,00 、C j,20 And C j,40 Three influence coefficients are added, and the characterization function is unfolded to obtain
Higher order item->Indicating the presence of a variation in the pitch of the helix on the tooth surface, when the wavelength lambda of the helix on the tooth surface is β The larger the 2 nd order term is, the larger the ratio to the higher order term is, the wavelength lambda of the helical corrugation on the tooth surface β The smaller the ratio of 2 nd order terms to higher order terms is, the smaller the expansion coefficient C j,00 、C j,20 And C j,40 The size of the spiral waviness is determined together, and the relation between the spiral waviness and the deviation of the spiral waviness is that
10 Tooth surface distortion deviation map
The twist is the result of the rotation of the upper face profile of the tooth flank along the helix. When the inclination deviation exists along the tooth profile direction and the spiral line direction at the same time and the deviation of the tooth top and the deviation of the tooth bottom are opposite in sign, the distortion phenomenon is characterized on the tooth surface, and as shown in fig. 12, the end face tooth profile is distorted S α And spiral twist S β The differences are distinguished. Face profile twist S α Is |S α |＝|C HαI -C HαII I, and C HαI ＝-C HαII Helical line twist S β Is |S β |＝|C HβNa -C HβNf I, and C HβNa ＝-C HβNf . The tooth surface distortion map is shown in figure 12, and the coordinate Z n And Y is equal to n D norm [Y n ,Z n ]The meaning of (1) is the same as the tooth profile offset error map. Expansion coefficient C j,11 Closely related to the degree of twisting, the degree of twisting of the tooth surface is large, C j,11 The larger the absolute value of (C), the smaller the degree of tooth surface distortion, C j,11 The smaller the absolute value of (c). C (C) j,11 The sign of (a) indicates the direction of twist, "+" indicates positive tooth profile inclination deviation at the tip, negative tooth profile inclination deviation at the bottom, and "-" indicates negative tooth profile inclination deviation at the tip, positive tooth profile inclination deviation at the bottom. In order to distinguish the simultaneous presence of a first-order pitch on the tooth surface from a helix pitch, the twist is given by the symbol L' j (1, 1) shows that the Legendre polynomial characterization function is as follows
L' j (1,1)＝C j,10 ·ψ 1 (Z n )ψ 0 (Y n )+C j,01 ·ψ 0 (Z n )ψ 1 (Y n )+C j,11 ·ψ 1 (Z n )ψ 1 (Y n ) (36)
From the polynomial it can be seen that the Legendre expansion coefficient associated with the tooth flank twist deviation has C j,01 、C j,10 And C j,11 Three influence coefficients are added, and the characterization function is unfolded to obtain
Wherein the first order termRepresenting the inclination in the direction of the spiral due to the presence of twist, first order termRepresenting the inclination in the tooth profile direction due to the presence of a twist, the second order cross term +.>Representing twist on tooth surface, C j,11 In relation to the tooth surface twist
Table 1 shows the correspondence between Legendre polynomial expansion coefficients and the characteristic errors of each order on the tooth surface.
TABLE 1 Legendre polynomial expansion coefficient C j,kl Relationship with characteristic errors of each order
4. Nine typical errors constitute a single error map for gears
Except for 0-order error tooth surface offset, the rest nine typical errors are combined pairwise to obtain a tooth surface single error spectrum. As shown in FIG. 13, the characteristic errors on the tooth surface are combined according to the tooth profile direction and the spiral line direction to form a single error map, 43 kinds of single error maps are formed, and various types of errors on the tooth surface are basically contained.
The gear topology error map of the invention has the following remarkable characteristics:
1. a tooth surface three-dimensional error model is presented herein. The two-dimensional measurement error on the tooth surface is realized by mapping the error on the tooth surface to the meshing surface coordinate system.
2. A method of characterizing tooth surface errors by Legendre polynomials is presented. The tooth surface error under the meshing surface coordinate system is represented by adopting a two-dimensional normalized Legendre polynomial, so that mathematical representation of the tooth surface topology error is realized.
3. A tooth surface topology error map is presented. The three-dimensional gear error form is characterized by the graphics of a gear error map, so that various types of characteristic errors on the tooth surface are clearly expressed, an error map is formed after the combination, and the characteristic errors on the tooth surface are quantitatively calculated.
4. The relationship between the characteristic errors and the single errors of the gears is vividly shown by utilizing the gear error patterns, and the gear single topological error patterns provide theoretical basis for realizing the pattern relationship between each characteristic error and the comprehensive error of the gears, the integral error and the gear pair error.
Drawings
Fig. 1 gear coordinate system and mesh plane coordinate system.
FIG. 2 illustrates a gear three-dimensional model to normal deviation two-dimensional process.
FIG. 3 is a tooth surface shift map.
FIG. 4 is a profile inclination map.
Fig. 5 helix inclination map.
Fig. 6 is a mid-profile drum graph.
Fig. 7 drum-in-spiral map.
FIG. 8 is a concave profile.
Fig. 9 is a concave pattern of spiral lines.
Figure 10 tooth profile ripple plot.
Fig. 11 spiral ripple map.
Fig. 12 tooth surface twist map.
FIG. 13 is a three-dimensional error map of a gear single item.
Fig. 14 is a topology error diagram of tooth surface 1 and tooth surface 1.
The topology error map of the tooth surface 1 of fig. 15 is decomposed into a characteristic error map.
Detailed Description
The invention is illustrated below with reference to specific examples:
the basic parameters of the characteristic gears are shown in table 2, and the topology data of the tooth surface 1 with errors on the gears are shown in fig. 14. The data processing is performed by taking the tooth surface 1 as an example.
Table 2 measurement of test gear basic parameters
1. And establishing a gear three-dimensional error model. According to the gear meshing principle, any point on the tooth surface can find a point corresponding to the point on the meshing surface, the characteristic line on the tooth surface also has a corresponding mapping relation on the meshing surface, and the mapping process involves two coordinate systems, namely a gear coordinate system and a tooth surface meshing coordinate system.
As shown in FIG. 1, the origin of the gear coordinate system is located at the intersection of the center of rotation of the gear and the bottom surface, the x-axis is located in the middle of the first tooth slot, the y-axis is perpendicular to the first tooth slot, and the z-axis is identical to the center of rotation of the gear shaft. The plane shown in fig. 1 is the engagement surface at the time of gear engagement, on which a coordinate system (Y n ,Z n )。Z n The axis is along the tooth height direction and ranges from (-5, 5), Y n The axis is along the tooth profile development direction, ranging from (-1.8,1.8). In order to characterize the normal deviation of the tooth surface in the meshing coordinate system, the third coordinate is the coordinate system of Y n ,Z n Vertical axis, d norm (Y n ,Z n ) Representation, coordinate system origin Y n ＝0,Z n =0 is centered along the tooth width direction range b and along the selected analysis area of the tooth profile development direction L. Coordinate system under the engagement surface (Y n ,Y n ,δ norm ) Relation to gear coordinate system in formula (39)
Legendre polynomial and tooth surface three-dimensional error. The Legendre polynomial is a polynomial obtained by using a separation variable method under a Jie Lapu Las equation in a spherical coordinate, has orthogonality in a section [ -1,1], and is mutually independent among various coefficients. The normal deviation on the tooth surface of the gear can be accurately represented by superposition of two-dimensional normalized Legendre polynomials, thereby characterizing the three-dimensional error of the gear.
The tooth surface deviation can be accurately represented by a superposition of the form two-dimensional normalized Legendre polynomials. The normal deviation of the 1 st tooth surface can be expressed as
Wherein C is 1,kl Is the expansion coefficient, ψ k (Z n ) And psi is equal to l (Y n ) Is normalization on tooth surface along spiral line direction and tooth profile direction respectivelyLegend polynomials. k and l represent the order of the axes, respectively, i.e. the deviation on the tooth surface can be characterized as the sum of a series of Legendre polynomials.
3. Typical error patterns for gears versus Legendre functions. According to the method of the present invention, typical characteristic deviations on individual tooth flanks are classified into 5 types, namely 0-order error tooth flank deviations, 1-order errors include tooth profile inclination and helix inclination, 2-order errors include tooth profile medium drum or medium pit and helix medium drum or medium pit, higher-order errors include tooth profile ripple and helix ripple, and 2-order cross error tooth flanks are distorted for a total of 10 types. Specific examples include pitch deviation of tooth profile, drum deviation in tooth profile, pitch deviation of spiral line, drum deviation in spiral line and higher order deviation.
After the three-dimensional error on the tooth surface is decomposed by adopting a Legendre polynomial, a representative characteristic error is extracted, and is graphically represented, so that the three-dimensional topological error characteristic map is obtained.
1) Tooth profile inclination deviation map
When only the tooth profile inclination deviation exists on the tooth surface, the tooth profile inclination map is a plane inclined along the tooth profile direction, the distance between two designed tooth profile traces which are intersected with the average tooth profile trace at the two ends in the metering range is called the tooth profile inclination deviation, and the symbol f is used Hα Representing the following. The characteristic map is shown in figure 4. Coordinate Z n And Y is equal to n Axis d norm [Y n ,Z n ]The meaning of (1) is the same as the tooth profile offset error map. Degree of tooth profile inclination and expansion coefficient C in polynomial j,01 Closely related, if the tooth profile is inclined by a large amount, C j,01 The absolute value of (C) is large, and conversely, the tooth profile inclination is small j,01 Is small. C (C) j,01 The sign of (c) indicates the direction of tilt, "+" indicates the presence of a negative pressure angle error or a positive base radius error, and "-" indicates the presence of a positive pressure angle error or a negative base radius error. The Legendre polynomial characterization function is as follows
L j (0,1)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n )+C j,01 ·ψ 0 (Z n )ψ 1 (Y n ) (41)
Expanding the characterization function to obtain
Expansion coefficient C j,01 The relation with the tooth profile inclination deviation is that
2) Helix inclination deviation profile
When the spiral inclination deviation exists on the tooth surface only, the spiral inclination map is a plane inclined along the spiral direction, the distance between two designed spiral traces which are intersected with the average spiral trace at the two ends in the metering range is called the spiral inclination deviation, and the sign f is used Hβ And (3) representing. The characteristic map is shown in figure 5. Coordinate Z n And Y is equal to n Axis d norm [Y n ,Z n ]The meaning of (1) is the same as that of the tooth profile offset error map. The degree of helix inclination and the expansion coefficient C in the polynomial j,10 Closely related, the inclination of the spiral is large, C j,10 And conversely, if the absolute value of the helix is large, C is small j,10 Is small. C (C) j,10 The legend of (2) represents the direction of tilt, and the Legendre polynomial characterization function is as follows
L j (1,0)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n )+C j,10 ·ψ 1 (Z n )ψ 0 (Y n ) (44)
Expanding the characterization function to obtain
Expansion coefficient C j,10 The relation with the single error is that
3) Drum deviation map in tooth profile
When only the middle drum profile modification or deviation exists on the tooth surface, the middle drum profile is a paraboloid presenting a middle drum along the tooth profile direction. The mid-drum deviation on the tooth profile belongs to the tooth profile shape deviation. The distance between two curves identical to the average tooth profile trace is contained in the calculated range. And the distance between the two curves and the average tooth profile trace is constant, which is called tooth profile shape deviation and is denoted by the symbol f fα And (3) representing. The characteristic map is shown in figure 6. Coordinate Z n And Y is equal to n D norm [Y n ,Z n ]The meaning of (1) is the same as the tooth profile offset error map. Degree of drum in profile and expansion coefficient C in polynomial j,02 Closely related, if the drum volume in the tooth profile is large, C j,02 If the absolute value of (a) is large, otherwise, if the drum amount in the tooth profile is small, C j,02 Is small. C (C) j,02 The sign of (a) indicates the direction of the parabolic opening, i.e., "+" indicates the presence of a second order tooth surface error on the tooth surface that is concave in the tooth profile direction, and "-" indicates the presence of a second order tooth surface deviation error on the tooth surface that is convex in the tooth profile direction as shown in fig. 6. The Legendre polynomial characterization function is as follows
L j (0,2)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n )+C j,01 ·ψ 0 (Z n )ψ 1 (Y n )+C j,02 ·ψ 0 (Z n )ψ 2 (Y n ) (47)
From the polynomial it can be seen that there is C in the Legendre expansion coefficient associated with mid-drum bias j,00 、C j,01 And C j,02 Three influence coefficients are added, and the characterization function is unfolded to obtain
Wherein the constant termRepresenting the deviation of the entire mid-drum surface from the theoretical position, first order term +.>Representing the inclination of the error face in the tooth profile direction due to the presence of the drum, the second order term +.>Indicating the presence of a drum, C j,01 The relation with the tooth profile inclination error is that
Expansion coefficient C j,02 The relationship with the drum deviation in the tooth profile is
4) Drum deviation map in spiral line
When the tooth surface only has the shape correction or deviation of the middle drum of the spiral line, the middle drum map of the spiral line is a paraboloid of the middle drum along the direction of the spiral line. The mid-drum deviation on the spiral belongs to the spiral shape deviation. The distance between two curves identical to the average spiral trace of the actual spiral trace is contained in the range of the value. And the distance between the two curves and the average spiral line trace is constant, called spiral shape deviation, and is denoted by the symbol f fβ And (3) representing. The characteristic map is shown in figure 7. Coordinate Z n And Y is equal to n D norm [Y n ,Z n ]The meaning of (1) is the same as the tooth profile offset error map. The degree of drum in the spiral and the expansion coefficient C of the 2 nd order term in the polynomial j,20 Closely related, if the drum amount in the spiral line is large, C j,20 If the absolute value of (C) is large, otherwise, the drum quantity in the spiral line is small, C j,20 Is small. C (C) j,20 The sign of (a) indicates the direction of the parabolic opening, i.e., "+" indicates the presence of a tooth surface concave in the direction of the helixThe second order tooth surface error, "-" indicates that there is a second order tooth surface deviation error along the drum in the spiral direction as shown in fig. 7 on the stored tooth surface. The Legendre polynomial characterization function is as follows
L j (2,0)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n )+C j,10 ·ψ 1 (Z n )ψ 0 (Y n )+C j,20 ·ψ 2 (Z n )ψ 0 (Y n ) (51)
From the polynomial it can be seen that there is C in the Legendre expansion coefficient associated with mid-drum bias j,00 、C j,10 And C j,20 Three influence coefficients are added, and the characterization function is unfolded to obtain
Wherein the constant termRepresenting the shift of the entire mid-drum surface relative to the theoretical position, first order termRepresenting the inclination of the error surface in the spiral direction due to the presence of the drum, the second order term +.>Indicating the presence of a drum, C j,10 The relation with the tooth profile inclination error is that
Expansion coefficient C j,20 The relationship with the drum deviation in the tooth profile is
5) Tooth profile ripple deviation map
The fluctuations (waviness, cyclic deviations) are deviations of the shape of the spiral with a constant wavelength and an almost constant height. The roughness of the waviness profile is a component of the surface shape characteristics, typically with which roughness is superimposed, the waviness spacing being significantly greater than the roughness spacing. Wavelength of waviness in tooth profile direction lambda α The distance from the central line of the profile to the wave crest or the wave trough is the profile waviness, f is used wα And (3) representing. The wave length of the waviness along the spiral line direction is lambda β The distance from the central line of the profile to the wave crest or the wave trough is the profile waviness, f is used wβ And (3) representing. Because of its periodic nature, waviness is a cause of noise generated when gears are used, and it is also necessary to study the waviness of the tooth surface from its surface roughness when high standards of performance and reliability are required.
When there is waviness deviation on the tooth profile, the tooth profile waviness spectrum is shown in fig. 10, and more than one curved surface of wave crest or wave trough exists along the tooth profile direction. Coordinate Z n And Y is equal to n D norm [Y n ,Z n ]The meaning of (1) is the same as the tooth profile offset error map. The orthonormal polynomial selected herein is a 5 th order polynomial, and theoretically, the higher the order the orthonormal polynomial is selected, the higher the accuracy of characterizing the waviness deviation on the tooth surface. Polynomial expansion coefficient C j,04 There is a close relationship with higher order tooth surface deviations. Waviness amplitude f on tooth profile wα Large, then C j,04 The absolute value of (a) is large, whereas the waviness amplitude f on the tooth profile is large wα If it is small, C j,04 Is small in absolute value, C j,04 The sign of (c) indicates the direction of the ripple. "+" indicates that a peak occurs at a midpoint along the profile direction and "-" indicates that a trough occurs at a midpoint along the profile direction. The Legendre polynomial characterization function is as follows
L j (0,4)＝C j,00 ·ψ 0 (Z n )ψ 0 (Y n )+C j,02 ·ψ 2 (Z n )ψ 2 (Y n )+C j,04 ·ψ 0 (Z n )ψ 4 (Y n ) (55)
From the polynomial it can be seen that there is C in the Legendre expansion coefficient associated with the profile waviness deviation j,00 、C j,02 And C j,04 Three influence coefficients are added, and the characterization function is unfolded to obtain
Wherein the constant termRepresenting the deviation of the entire tooth surface from the theoretical position, second order termA mid-drum or a mid-recess representing the error face in the tooth profile direction due to the presence of waviness, higher order term +>Indicating the presence of deviations in the waviness of the tooth profile on the tooth surface, when the wavelength lambda of the waviness of the tooth profile on the tooth surface α The larger the 2 nd order term is, the larger the ratio to the higher order term is, the wavelength lambda of the profile ripple on the tooth surface α The smaller the ratio of the 2 nd order term to the higher order term is, the smaller the expansion coefficient C j,00 、C j,02 And C j,04 The size of the tooth profile waviness is determined together, and the relationship between the tooth profile waviness deviation and the tooth profile waviness deviation is that
6) Tooth surface distortion deviation map
The twist is the result of the rotation of the upper face profile of the tooth flank along the helix. When the inclination deviation exists along the tooth profile direction and the spiral line direction at the same time and the deviation of the tooth top and the deviation of the tooth bottom are opposite in sign, the characteristic is formed on the tooth surfaceThat is, the distortion phenomenon, as shown in FIG. 12, the face profile distortion S α And spiral twist S β The differences are distinguished. Face profile twist S α Is |S α |＝|C HαI -C HαII I, and C HαI ＝-C HαII Helical line twist S β Is |S β |＝|C HβNa -C HβNf I, and C HβNa ＝-C HβNf . The tooth surface distortion map is shown in figure 12, and the coordinate Z n And Y is equal to n D norm [Y n ,Z n ]The meaning of (1) is the same as the tooth profile offset error map. Expansion coefficient C j,11 Closely related to the degree of twisting, the degree of twisting of the tooth surface is large, C j,11 The larger the absolute value of (C), the smaller the degree of tooth surface distortion, C j,11 The smaller the absolute value of (c). C (C) j,11 The sign of (a) indicates the direction of twist, "+" indicates positive tooth profile inclination deviation at the tip, negative tooth profile inclination deviation at the bottom, and "-" indicates negative tooth profile inclination deviation at the tip, positive tooth profile inclination deviation at the bottom. In order to distinguish the simultaneous presence of a first-order pitch on the tooth surface from a helix pitch, the twist is given by the symbol L' j (1, 1) shows that the Legendre polynomial characterization function is as follows
L' j (1,1)＝C j,10 ·ψ 1 (Z n )ψ 0 (Y n )+C j,01 ·ψ 0 (Z n )ψ 1 (Y n )+C j,11 ·ψ 1 (Z n )ψ 1 (Y n ) (58)
From the polynomial it can be seen that the Legendre expansion coefficient associated with the tooth flank twist deviation has C j,01 、C j,10 And C j,11 Three influence coefficients are added, and the characterization function is unfolded to obtain
Wherein the first order termThe representation is composed ofInclination in the direction of the spiral caused by the presence of twist, first order term +.>Representing the inclination in the tooth profile direction due to the presence of a twist, the second order cross term +.>Representing twist on tooth surface, C j,11 In relation to the tooth surface twist
The tooth surface characteristic error map spectrum as shown in fig. 15 can be obtained from the tooth surface topology error map of the tooth surface 1 by adopting the method provided by the invention. The topology error map on the tooth surface 1 is expressed as a tooth surface deviation, a tooth profile inclination deviation, a drum in tooth profile deviation, a helix inclination deviation, a drum in helix deviation, and a higher order deviation. Table 3 gives the specific parameters in the examples and the results of the comparison with the gear measurement centre.
Table 3 comparative measurement of test gear tooth surface deviation parameters
The method takes gear error forms processed by various processes as a research basis, scientifically characterizes three-dimensional errors of gears obtained by modern measurement means, congeals typical gear characteristic errors which can reflect the use performance of gears and have special significance, and characterizes the three-dimensional errors of gears in a form of a gear three-dimensional topological error map. The new concept of gear characteristic error patterns, such as tooth profile deviation patterns, spiral line deviation patterns, tooth pitch deviation patterns and the like, and various gear single error topology patterns combined with each other among characteristic errors is provided. The three-dimensional gear error form is characterized by the graph of the gear error graph, so that various types of characteristic errors on the tooth surface are clearly expressed to form an error graph after being combined, the characteristic errors on the tooth surface are quantitatively calculated, and a theoretical basis is provided for the subsequent realization of the graph relationship among various characteristic errors, comprehensive errors, integral errors and gear pair errors of the gear.
Claims (3)
1. A method for analyzing a single topological error map of a gear is characterized by comprising the following steps of: the analysis method comprises the following steps:
s1, establishing a gear three-dimensional error model; according to the gear meshing principle, any point on the tooth surface can find a point corresponding to the point on the meshing surface, a corresponding mapping relation exists on the tooth surface by the characteristic line on the meshing surface, and the mapping process involves two coordinate systems, namely a gear coordinate system and a tooth surface meshing coordinate system;
the position of the origin of the gear coordinate system is positioned at the intersection point of the rotation center of the gear and the bottom surface, the x-axis is positioned in the middle of the first tooth groove, the y-axis is vertical to the first tooth groove, and the z-axis is identical with the rotation center of the gear shaft; the engagement surface of the gears when engaged, on which a coordinate system (Y n ,Z n )；Z n The axis being in the direction of tooth height, the extent beingY n The axis along the tooth profile development direction, in the range +.>To characterize the normal deviation of the tooth surface in the meshing coordinate system, the third coordinate is the coordinate value of Y n ,Z n Perpendicular to axis, d norm (Y n ,Z n ) Representation, coordinate system origin Y n ＝0,Z n =0 is centered along the tooth width direction range b and along the selected analysis area of the tooth profile development direction L; decomposing the normal error into a modification of the tooth surface along the spiral line direction, a modification of the tooth surface along the tooth profile direction and various errors on the tooth surface through error decomposition;
s2Legendre polynomial and tooth surface three-dimensional error; the Legendre polynomial is a polynomial obtained by solving the Laplace equation under the spherical coordinates by using a separation variable method, has orthogonality in a section [ -1,1], and is mutually independent among various coefficients; accurately representing the normal deviation on the tooth surface of the gear by superposition of a two-dimensional normalized Legendre polynomial so as to represent the three-dimensional error of the gear;
s3, a typical error map of the gear and a Legendre function; typical characteristic deviation differences on single tooth surfaces are 5 types, namely 0-order error tooth surfaces are offset respectively, 1-order errors comprise tooth profile inclination and spiral line inclination, 2-order errors comprise tooth profile medium drum or medium concave and spiral line medium drum or medium concave, high-order errors comprise tooth profile ripple and spiral line ripple, and 2-order crossed error tooth surfaces are distorted, and the total number of the errors is 10;
s4, nine typical errors form a single error map of the gear
Except 0-order error tooth surface offset, the rest nine typical errors are combined pairwise to obtain a tooth surface single error map; according to the tooth profile direction and the spiral line direction, characteristic errors on the tooth surface are combined to form single error patterns, 43 kinds of single error patterns are formed, and various errors on the tooth surface are basically contained;
after the three-dimensional errors on the tooth surface are decomposed by adopting a Legendre polynomial, representative characteristic errors are extracted, graphical representation is carried out, namely, a three-dimensional topological error characteristic map is obtained, and all the characteristic errors are combined, so that a single three-dimensional topological error map can be obtained.
2. The method for analyzing a single topology error map of a gear according to claim 1, wherein: coordinate system under the engagement surface (Y n ,Y n ,δ norm ) The relation with the gear coordinate system is shown in the formula (1)
Measuring point (x) under three-dimensional gear coordinate system m ,y m ,z m ) Obtaining a calculation normal deviation graph under an engaged coordinate system through coordinate transformation of the three-dimensional point cloud of the (2);
the tooth surface normal error comprising the modification information and the tooth surface error information is obtained by the measurement coordinates of the actual points on the tooth surface, namely, various modification or tooth surface errors on the tooth surface are finally reflected on the normal error.
3. The method for analyzing a single topology error map of a gear according to claim 1, wherein: the l-th order Legendre polynomial is expressed as
In the middle of
Reconstructing the tooth surface using a two-dimensional Legendre polynomial expressed as the product of two one-dimensional polynomials
Q n (x,y)＝P k (x)P l (y) (3)
k and l represent the order of each axis, respectively; the lowest two-dimensional Legendre term is a zero-order constant and represents gear pitch error, the first-order linear term represents common inclination error in spiral line measurement and tooth profile measurement, the second-order term represents middle drum or middle concave error in spiral line and tooth profile, the second-order cross term can represent distortion deviation on a tooth surface, and the higher-order term represents waviness in spiral line direction and tooth profile direction on the tooth surface;
any measurable tooth deviation is accurately represented by a superposition of the form of the two-dimensional normalized Legendre polynomials; the normal deviation of the jth tooth surface is expressed as
Wherein C is j,kl Is the expansion coefficient, ψ k (Z n ) And psi is equal to l (Y n ) Normalized Legend polynomials along the spiral line direction and the tooth profile direction on the tooth surface; k and l represent the order of the axes, respectively, i.e. the deviation on the tooth surface is characterized by the sum of a series of Legendre polynomials.
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| CN107860313A (en) * | 2017-10-25 | 2018-03-30 | 北京工业大学 | It is a kind of based on the beveled gear teeth of line-structured light to bias measurement method |
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| CN107588742A (en) * | 2017-10-25 | 2018-01-16 | 北京工业大学 | A kind of cylindrical gear profile bias measurement method based on line-structured light |
| CN107860313A (en) * | 2017-10-25 | 2018-03-30 | 北京工业大学 | It is a kind of based on the beveled gear teeth of line-structured light to bias measurement method |
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