CN112903288A - Unified characterization method for characteristic lines of three-dimensional errors of gear - Google Patents
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Abstract
The invention discloses a unified characterization method for a characteristic line of a gear three-dimensional error, which establishes a relation between a coordinate system on a gear and a meshing surface coordinate system and provides an equation of the gear characteristic line under the gear coordinate system under the meshing surface coordinate system: the involute is a sectional line of the tooth surface cut by the end plane, the contact line is an intersecting line of the base circle tangent plane and the tooth surface, the normal meshing tooth profile is a track of an instantaneous contact point generated on the tooth surface when two tooth surfaces are meshed in the transmission of the helical gear, and the helical line is an intersecting line of a coaxial cylindrical surface of the gear axis and the tooth surface. And (3) performing two-dimensional characterization on the three-dimensional measurement error on the tooth surface, analyzing to obtain that four characteristic lines on the tooth surface are straight lines in a meshing surface coordinate system, and realizing uniform mathematical characterization of each characteristic line in the meshing surface coordinate system. A method for extracting tooth profile deviation curves, spiral deviation curves, contact line deviation curves and normal meshing tooth deviation curves on a tooth surface along the normal direction of meshing under a meshing plane coordinate system is provided, and the deviation is evaluated. By utilizing the method, the four characteristic curves at any position on the tooth surface are extracted based on a large amount of tooth surface measurement data and are comprehensively evaluated, so that the measurement and evaluation results are more comprehensive.
Description
Technical Field
The invention relates to a method for unifying characteristic lines of three-dimensional errors of gears, and belongs to the technical field of gear machinery manufacturing.
Background
The parameter curve of particular significance on the tooth surface is called a characteristic line. Although there are many kinds of parametric curves of the tooth surface, the selection of the characteristic line is limited by many factors such as the working mechanism, design principle, processing method, use state, curve testability, etc. of the workpiece. The errors affecting various performances of the gear are finally attributed to the errors on four characteristic lines, namely a spiral line, an involute, a normal meshing tooth form and a contact line, on the tooth surface of the gear. The spiral line and the involute are common tooth surface characteristic lines, but normal meshing tooth shapes and contact lines which have special significance and can better reflect the use performance of the gear are less in use, and the measurement and evaluation are difficult to realize mainly due to the traditional measurement method of the latter two characteristic lines.
The mainstream gear measurement and evaluation method at present is to measure two characteristic lines of a spiral line and an involute on a gear measurement center, so as to realize the evaluation of two single errors of the spiral line deviation and the tooth profile deviation of the gear. The current special gear measuring instrument has no measuring function specially used for the characteristic line of the normal meshing tooth shape. The contact line measurement and evaluation are often performed based on a combination of the measurement results in the spiral direction and the tooth profile direction.
When the main precision index is evaluated, the deviation in the ISO1328 series precision standard document only uses a few error curves as evaluation bases. For tooth profile deviations and helix deviations, only a few teeth are measured, each tooth measuring only one curve in the middle of the tooth flank. The geometric error based on the local (characteristic point or characteristic line) on the tooth surface of the gear can only reflect partial information of the complex appearance of the tooth surface, and the measurement data acquired as the evaluation basis is only a very small sample of the information of the whole geometric shape of the gear to be measured. With the development of optical non-contact measurement, the measurement mode of the gear is changed, the three-dimensional measurement of the tooth surface becomes a reality, and complete three-dimensional information on the tooth surface can be acquired simultaneously. For the large amount of data obtained, the use of a small sample to evaluate the whole necessarily results in a significant difference between the evaluation result and the actual use performance of the gear.
With the development of optical non-contact measurement, the measurement mode of the gear is changed, the tooth surface three-dimensional measurement becomes a reality, and complete three-dimensional information on the tooth surface can be acquired simultaneously. In order to evaluate the gear under three-dimensional measurement, four characteristic lines under traditional evaluation, namely tooth profile, spiral line, normal meshing tooth profile and contact line on the tooth surface are extracted from a large amount of gear measurement data, and become the key for evaluating the three-dimensional error of the tooth surface. Therefore, how to combine the tooth surface measurement data acquired by the new measurement means with the traditional evaluation requires establishing a unified model of four characteristic lines, and the tooth surface measurement data can conveniently and quickly extract characteristic errors such as tooth profile deviation, spiral line deviation and the like, and normal meshing tooth profile deviation and contact line deviation. For the spiral straight gear, the extracted normal meshing tooth shape and the extracted contact line can reflect the use performance of the gear better. The gear error unified model is provided based on the requirements, the three-dimensional errors of the gears are two-dimensionally realized, the representation of the errors is unified, the tooth surface characteristic line errors are rapidly extracted, and the evaluation of the three-dimensional errors of the gears is concise and convenient.
Disclosure of Invention
In order to solve the limitation and the deficiency of the error representation of the traditional cylindrical gear, the invention provides a unified model of a characteristic line of the three-dimensional error of the gear. By utilizing the model, various characteristic lines at any position on the three-dimensional tooth surface can be rapidly extracted, the evaluation of the three-dimensional error of the tooth surface is simplified, the model has important significance on a gear error theory, and a theoretical basis is provided for rapid evaluation under the three-dimensional measurement of the gear.
The invention provides a unified characterization method for a characteristic line of a three-dimensional error of a gear. The method comprises the following specific steps:
s1, establishing a relation between a coordinate system on the gear and a meshing surface coordinate system. According to the gear meshing principle, any point on the tooth surface can find a point corresponding to the tooth surface 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, one is a gear coordinate system, and the other is a tooth surface meshing coordinate system. During the design phase, a gear coordinate system is preferred. In the analysis of the measurement data, a flank engagement coordinate system is preferred. The gear coordinate system is defined as shown in fig. 2. The origin of the coordinate system is located at the intersection point of the gear revolution axis and the bottom surface, the x axis is located in the middle of the first tooth groove, and the y axis is perpendicular to the first tooth grooveAnd the x-Oy plane is vertical to the axis, and the z-axis is coincident with the rotation axis of the gear shaft. When the gears are meshed, the base circle tangent plane passing through the meshing line of the two gears is used as a meshing surface on which a coordinate system (Y) is establishedn,Zn)。ZnThe shaft is along the gear axis in the range of
b is the tooth width, YnThe axis is along the tooth profile development direction and ranges
L is the engagement range. The coordinate system origin is located at the axial and radial centers of the selected analysis zone along the axial direction extent b and along the tooth profile development direction L.
Coordinate system under the meshing plane (Y)n,Zn) The relation with the gear coordinate system is shown in formula (1)
Wherein
In the formula, rbThe radius of the base circle is shown,
the angle of spread of the involute is shown,
showing the spread angle at the tooth tip,
denotes the flare angle at the tooth root and b denotes the gear tooth width.
The measurement data on the tooth flank is converted into normal deviations in the meshing plane coordinate system.
S2, based on the relation between the coordinate system on the gear and the meshing surface coordinate system in the S1, the characteristic line of the gear in the gear coordinate system is given in the meshing surface coordinate systemThe equation of (c). For an involute helicoid, there are four characteristic lines: the involute is a sectional line of the tooth surface cut by the end plane. Corresponding to the engagement coordinate system along YnA straight line of coordinates. The contact line is the intersection line of the base circle tangent plane and the tooth surface. Corresponding to Z and Z in the meshing coordinate systemnThe included angle of the direction is a base circle helix angle betabIs measured. And the normal meshing tooth form refers to a track of an instantaneous contact point generated on a tooth surface when two tooth surfaces are meshed in the transmission of the helical gear. Corresponding to Y and Y in the meshing coordinate systemnThe included angle in the direction is a base circle helix angle betabIs measured. The spiral line is the intersection line of the coaxial cylindrical surface of the gear axis and the gear surface. Corresponding to the engagement coordinate system along ZnA straight line of coordinates. In theory, any combination of two of these four characteristic lines can completely describe the helical surface.
The equation of the characteristic curve on the ith tooth surface in the gear coordinate system is set as follows:
wherein
The coordinate relation of the ith tooth in the meshing area is obtained by a characteristic line equation (2) in a gear coordinate system and a mapping relation (1) between the gear coordinate system and a meshing surface coordinate system
Wherein
The involute is represented by the formula (4) in the meshing surface coordinate system
Wherein
z*Is the z value at any position on the tooth surface.
The representation of the helix obtained in the meshing plane coordinate system is represented by the formula (5)
Wherein
At any position on the tooth surface
The value is obtained.
The representation of the normal meshing tooth profile under the meshing surface coordinate system is represented by the formula (6)
Wherein
The contact line in the mating surface coordinate system is represented by the formula (7)
Wherein
Tooth profile, spiral line, contact line and normal mesh tooth on tooth surfaceThe characteristic lines of the gear in the meshing coordinate system are all straight lines, and the spiral line, the tooth profile line and the coordinate axis ZnAnd YnThe included angle between the normal meshing tooth form and the tooth profile direction is betabThe angle between the contact line and the helix is betab。
And S3, unified model and error extraction of the gear characteristic line. Because the four key characteristic lines on the gear are all straight lines in the meshing coordinate system, the straight lines can be represented uniformly in a two-dimensional plane, and therefore a uniform error model corresponding to the characteristic lines in the gear coordinate system and the characteristic lines in the meshing plane is established, and the formula is shown in (8).
Wherein
A. And different values of B are different, so that different types of tooth surface deviations can be obtained. When k is 1, B is 0, and the intersecting line represents the deviation of the tooth profile; when k is 2, A is 0, and the intersecting line represents the helical deviation; when k is 3, the number of the groups is 3,
the intersecting line represents the tooth surface normal meshing tooth form deviation; when k is equal to 4, the number of the first symbols is 4,
the intersecting lines represent the deviations of the contact lines, so the model can extract the characteristic lines at any position on the tooth surface.
In the meshing plane coordinate system (Y)n,Zn) Then, the tooth surface error delta is measured with the corresponding characteristic planenorm(Yn,Zn) The normal error curve at any position in any direction on the tooth surface can be obtained by intercepting different positions, a two-dimensional curve is obtained at the moment, namely information obtained under three-dimensional measurement and error representation is converted to a two-dimensional plane for evaluation, and the evaluation method in ISO1328 is still applicable. But is different from the conventional methodThe model provided by the method realizes the two-dimension of the three-dimensional tooth surface error, simplifies the evaluation and realizes the unified representation of the error on the tooth surface. Four characteristic deviations at any position on any tooth surface are obtained, and the obtained information is more comprehensive and complete.
The unified model of the gear three-dimensional error characteristic line has the following remarkable characteristics:
1. a unified tooth surface characteristic line error model is provided by the method, three-dimensional measurement errors on a tooth surface are represented in a two-dimensional mode, and unified mathematical representation of tooth profile deviation, spiral line deviation, normal meshing tooth shape deviation and contact line deviation on the tooth surface is achieved.
2. A method for extracting tooth profile deviation curves, helical line deviation curves, contact line deviation curves and normal meshing tooth deviation curves on a tooth surface along the normal direction of meshing under a meshing plane coordinate system is provided, and the deviation is evaluated.
3. By the method, the four characteristic curves at any position on the tooth surface can be extracted based on a large amount of tooth surface measurement data and comprehensively evaluated, and the measurement and evaluation results are more comprehensive.
4. The unified model of the tooth surface error provided by the method provides a theoretical basis for analysis of big data obtained in optical measurement and gear precision evaluation, and provides theoretical support for measurement of a new generation of gears.
Drawings
FIG. 1 Gear coordinate System
FIG. 2 meshing plane coordinate system
FIG. 3 is a view showing a tooth surface characteristic line and a characteristic line in a meshing coordinate system
FIG. 4 tooth surface feature deviation extraction diagram
FIG. 5 Gear diagram with tooth surface errors
FIG. 6 Normal deviation 3 characteristic lines on graph
FIG. 7 deviation of tooth profile 1 under unified model and deviation of tooth profile evaluation on same position on tooth surface
FIG. 8 is a graph for extracting 4 characteristic deviations of a test gear
Detailed Description
The invention is illustrated below with reference to specific processing examples:
1. the basic parameters of the characteristic gear are shown in a table 1, and the tooth surface point cloud data with errors on the gear is shown in fig. 5. The data processing is performed using the tooth surface 1 as an example.
The position of the origin of the gear coordinate system is located at the intersection point of the gear rotation axis and the bottom surface, the x axis is located in the middle of the first tooth groove, the y axis is perpendicular to the y axis, the xOy plane is perpendicular to the axis, and the z axis is superposed with the gear shaft rotation axis. When the gears are meshed, the base circle tangent plane of the meshing line of the two gears is used as a meshing surface on which a coordinate system (Y) is establishedn,Zn)。ZnThe shaft is along the gear axis in the range of
YnThe axis is along the tooth profile development direction and ranges
The origin of the coordinate system is located at the axial and radial centers of the selected analysis zone along the axial direction range b and along the tooth profile development direction L.
Coordinate system under the meshing plane (Y)n,Zn) Relation with gear coordinate system see formula (9)
Wherein
In the formula, rbThe radius of the base circle is shown,
the angle of spread of the involute is shown,
showing the spread angle at the tooth tip,
denotes the flare angle at the tooth root and b denotes the gear tooth width.
The measurement data on the tooth surface can be converted into normal deviations in the meshing plane coordinate system.
TABLE 1 measurement of basic parameters of test gears
2. And analyzing an equation of four characteristic lines under the gear coordinate system under the meshing surface coordinate system.
The equation of the characteristic curve on the ith tooth surface in the gear coordinate system is set as follows:
wherein
The coordinate relation of the ith tooth in the meshing area can be obtained by a characteristic line equation (10) in a gear coordinate system and a mapping relation (9) between the gear coordinate system and a meshing surface coordinate system
Wherein
The representation of the involute under the meshing surface coordinate system can be represented by the formula (4)
Wherein
z*Is the z value at any position on the tooth surface.
The representation of the helix in the coordinate system of the meshing surface can be obtained by the formula (13)
Wherein
At any position on the tooth surface
The value is obtained.
The representation of the normal meshing tooth shape under the meshing surface coordinate system can be represented by the formula (14)
Wherein
The characterization of the contact line in the mating surface coordinate system can be represented by the formula (15)
Wherein
The tooth profile, the spiral line, the contact line and the characteristic line of the normal meshing tooth form on the tooth surface are all straight lines under a meshing coordinate system, and the spiral line, the tooth profile and the coordinate axis ZnAnd YnThe included angle between the normal meshing tooth form and the tooth profile direction is betabContact line and snailIncluded angle of rotation line direction is betab。
3. And (5) unified model and error extraction of the gear characteristic line. Because the four key characteristic lines on the gear are all straight lines under the meshing coordinate system, the straight lines can be represented uniformly in a two-dimensional plane, so that a corresponding uniform error model of the characteristic lines under the gear coordinate system and the characteristic lines under the meshing plane can be established, and a formula is shown in (16).
Wherein
A. And different values of B are different, so that different types of tooth surface deviations can be obtained. When k is 1, B is 0, and the intersecting line represents the deviation of the tooth profile; when k is 2, A is 0, and the intersecting line represents the helical deviation; when k is 3, the number of the groups is 3,
the intersecting line represents the tooth surface normal meshing tooth form deviation; when k is equal to 4, the number of the first symbols is 4,
the intersecting lines represent the deviations of the contact lines, so the model can extract the characteristic lines at any position on the tooth surface.
In the meshing plane coordinate system (Y)n,Zn) Then, the tooth surface error delta is measured with the corresponding characteristic planenorm(Yn,Zn) The normal error curve at any position in any direction on the tooth surface can be obtained by intercepting different positions, a two-dimensional curve is obtained at the moment, namely information obtained under three-dimensional measurement and error representation is converted to a two-dimensional plane for evaluation, and the evaluation method in ISO1328 is still applicable. But different from the traditional method, the model provided by the method realizes the two-dimension of the three-dimensional tooth surface error, simplifies the evaluation, and realizes the unified representation of the error on the tooth surface. Can acquire four characteristics at any position on any tooth surfaceAnd the obtained information is more comprehensive and complete due to deviation.
Fig. 6 shows the extraction of the tooth profile lines at 5 profile positions at 1mm, 3mm, 5mm, 7mm, 9mm on the tooth flank 1.
4. And after the gear three-dimensional error is subjected to two-dimension, evaluating all three-dimensional errors of the extracted tooth surface according to the current gear standard ISO 1328. Fig. 7 shows the error of 5 profile characteristic lines of the tooth flank 1 in the meshing plane coordinate system. Table 2 shows the evaluation results of 5 tooth profiles, including the tooth profile inclination deviation, the tooth profile shape deviation, and the tooth profile total deviation.
TABLE 25 evaluation values of tooth profiles
5. Fig. 8 shows the extracted tooth profile deviations, helix deviations, normal tooth profile deviations, contact line deviations on the tooth surface 1. The evaluation results of these 4 deviations are given in table 3.
TABLE 3 evaluation of four characteristic lines
According to the method, the relationship between a gear coordinate system and a meshing surface coordinate system in the gear meshing process is utilized, the three-dimensional gear error is converted into a two-dimensional plane, four characteristic lines on a tooth surface are analyzed and summarized to be straight lines in the meshing surface coordinate system, then the unified representation of each characteristic line in the meshing surface coordinate system is provided, the extraction method of the characteristic lines on the tooth surface under three-dimensional measurement is provided, and the feasibility of the unified model and the extraction method of the characteristic lines are proved through experiments. The model realizes the rapid extraction of various characteristic lines at any position on the three-dimensional tooth surface, simplifies the evaluation of the three-dimensional error of the tooth surface, has important significance on the gear error theory, and provides theoretical basis for the rapid evaluation under the three-dimensional measurement of the gear.
Claims (4)
1. A unified characterization method for a characteristic line of a gear three-dimensional error is characterized by comprising the following steps: the method comprises the following specific steps:
s1, establishing a relation between a coordinate system on a gear and a coordinate system of an engagement surface; according to the gear meshing principle, any point on the tooth surface can find a point corresponding to the tooth surface 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 relates to two coordinate systems, wherein one coordinate system is a gear coordinate system, and the other coordinate system is a tooth surface meshing coordinate system; the position of the origin of the coordinate system is positioned at the intersection point of the gear rotation axis 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, the xOy plane is vertical to the axis, and the z axis is superposed with the gear shaft rotation axis; when the gears are meshed, a coordinate system (Y) is established by taking a base circle tangent plane of a meshing line of the two gears as a meshing surfacen,Zn)；ZnThe shaft is along the gear axis in the range of
b is the tooth width, YnThe axis is along the tooth profile development direction and ranges
L is the engagement range; the coordinate system origin is located at the axial and radial centers of the selected analysis zone along the axial direction extent b and along the tooth profile development direction L;
s2, based on the relation between the coordinate system on the gear and the meshing surface coordinate system in the S1, giving an equation of the gear characteristic line under the gear coordinate system under the meshing surface coordinate system;
s3, unified model and error extraction of the gear characteristic line; because the four key characteristic lines on the gear are all straight lines under the meshing coordinate system, the straight lines are uniformly characterized in a two-dimensional plane, and a corresponding uniform error model of the characteristic lines under the gear coordinate system and the characteristic lines under the meshing plane is established;
in the meshing plane coordinate system (Y)n,Zn) Then, the tooth surface error delta is measured with the corresponding characteristic planenorm(Yn,Zn) Intercepting at different positions to obtain any position in any direction on tooth surfaceAnd (4) obtaining a two-dimensional curve by using the normal error curve, namely converting the information obtained under the three-dimensional measurement and error characterization to a two-dimensional plane for evaluation, and uniformly characterizing the error on the tooth surface.
2. The method for uniformly characterizing the characteristic line of the three-dimensional error of the gear according to claim 1, wherein the method comprises the following steps: coordinate system (Y) under mesh surface in S1n,Zn) The relationship with the gear coordinate system is shown in the following formula:
wherein
In the formula, rbThe radius of the base circle is shown,
the angle of spread of the involute is shown,
showing the spread angle at the tooth tip,
represents the flare angle at the tooth root, b represents the gear tooth width; the measured data on the tooth surface are converted into normal deviations in the coordinate system of the meshing surface.
3. The method for uniformly characterizing the characteristic line of the three-dimensional error of the gear according to claim 1, wherein the method comprises the following steps: in S2, there are four characteristic lines for the involute helicoid: firstly, the involute is a sectional line of the tooth surface cut by an end plane; corresponding to the engagement coordinate system along YnA straight line of coordinates; the contact line is the intersection line of the base circle tangent plane and the tooth surface; corresponding to Z and Z in the meshing coordinate systemnThe included angle of the direction is a base circle helix angle betabA straight line of (a); ③ Normal directionThe meshing tooth shape refers to a track of instantaneous contact points generated on tooth surfaces when two tooth surfaces are meshed in the transmission of the helical gear; corresponding to Y and Y in the meshing coordinate systemnThe included angle of the direction is a base circle helix angle betabA straight line of (a); the spiral line is the intersection line of the coaxial cylindrical surface of the gear axis and the tooth surface; corresponding to the engagement coordinate system along ZnA straight line of coordinates; any two combinations of these four characteristic lines can completely describe the helical curved surface;
the equation of the characteristic curve on the ith tooth surface in the gear coordinate system is set as follows:
wherein
The coordinate relation of the ith tooth in the meshing area is obtained by a characteristic line equation (2) in a gear coordinate system and a mapping relation (1) between the gear coordinate system and a meshing surface coordinate system
Wherein
The involute is represented by the formula (4) in the meshing surface coordinate system
Wherein
z*Is the z value of any position on the tooth surface;
the representation of the helix obtained in the meshing plane coordinate system is represented by the formula (5)
Wherein
At any position on the tooth surface
A value;
the representation of the normal meshing tooth profile under the meshing surface coordinate system is represented by the formula (6)
Wherein
The contact line in the mating surface coordinate system is represented by the formula (7)
Wherein
The tooth profile, the spiral line, the contact line and the characteristic line of the normal meshing tooth form on the tooth surface are all straight lines under a meshing coordinate system, and the spiral line, the tooth profile and the coordinate axis ZnAnd YnThe included angle between the normal meshing tooth form and the tooth profile direction is betabThe angle between the contact line and the helix is betab。
4. The method for uniformly characterizing the characteristic line of the three-dimensional error of the gear according to claim 1, wherein the method comprises the following steps: a unified error model in S3, wherein the formula is shown in (8);
wherein
A. The values of B are different, so that different types of tooth surface deviations can be obtained; when k is 1, B is 0, and the intersecting line represents the tooth profile deviation; when k is 2, A is 0, and the intersecting line represents the helical deviation; when k is 3, the number of the groups is 3,
the intersecting line represents the tooth surface normal meshing tooth form deviation; when k is equal to 4, the number of the first symbols is 4,
the intersecting line represents the deviation of the contact line, and a characteristic line at any position on the tooth surface is extracted.
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| CN113496356A (en) * | 2021-07-05 | 2021-10-12 | 河南理工大学 | Gear digital measurement and evaluation cloud system and measurement and evaluation method |
| CN113496356B (en) * | 2021-07-05 | 2023-08-08 | 河南理工大学 | Gear digital measurement and evaluation cloud system and measurement and evaluation method |
| CN113591238A (en) * | 2021-07-19 | 2021-11-02 | 北京工业大学 | Gear tangential comprehensive deviation calculation method based on gear point cloud data |
| CN113591238B (en) * | 2021-07-19 | 2024-03-22 | 北京工业大学 | Gear tangential integrated deviation calculation method based on gear point cloud data |
| CN114993230A (en) * | 2022-06-19 | 2022-09-02 | 北京工业大学 | Normal meshing tooth profile measuring and evaluating method based on gear characteristic line unified model |
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