CN111912373B - Tooth profile deviation measuring method using roughness profilometer - Google Patents

Tooth profile deviation measuring method using roughness profilometer Download PDF

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CN111912373B
CN111912373B CN202010670501.6A CN202010670501A CN111912373B CN 111912373 B CN111912373 B CN 111912373B CN 202010670501 A CN202010670501 A CN 202010670501A CN 111912373 B CN111912373 B CN 111912373B
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tooth profile
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deviation
distance
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CN111912373A (en
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林家春
滕辰
石照耀
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Beijing University of Technology
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B21/00—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant
    • G01B21/20—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant for measuring contours or curvatures, e.g. determining profile
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B11/00—Measuring arrangements characterised by the use of optical techniques
    • G01B11/24—Measuring arrangements characterised by the use of optical techniques for measuring contours or curvatures
    • G01B11/2416—Measuring arrangements characterised by the use of optical techniques for measuring contours or curvatures of gears

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Abstract

The invention discloses a tooth profile deviation measuring method by using a roughness profiler, which comprises the steps of obtaining tooth profile data of an involute cylindrical gear on the roughness profiler, establishing an involute tooth profile model according to parameters of a measured gear, solving fitting parameters by using an optimization solution idea through constructing a least square objective function of original measured data and the involute tooth profile model in a normal direction, obtaining an original measured data curve and the involute tooth profile model after orthogonal distance fitting, further calculating to obtain the tooth profile deviation of any point in the involute normal direction, and finally evaluating and calculating the deviation value to obtain the tooth profile deviation and the precision grade defined by national standards.

Description

Tooth profile deviation measuring method using roughness profilometer
Technical Field
The invention belongs to the field of precision measurement, and particularly relates to measurement and evaluation of involute cylindrical gear tooth profile deviation, in particular to a processing method of involute tooth profile measurement data and an evaluation calculation method of involute tooth profile deviation data.
Background
The gear is a transmission part, and the quality of the tooth surface of the gear has direct influence on the performance of the gear transmission, such as transmission error, bearing capacity, vibration noise and the like. The tooth profile deviation is an important parameter of the gear precision and needs to be obtained by evaluating the gear tooth profile information. Therefore, it is very important to acquire and process the gear tooth profile information.
In order to measure the tooth profile deviation, it is common practice to form a theoretical involute by a mechanical or electronic generating method according to an involute forming principle, and then record the deviation of the actual tooth profile from the theoretical involute by a measuring head, which is typically a gear measuring center. In addition, three-coordinate measuring machines and optical measuring instruments, which are different from the generating measuring principle, are also used for gear measurement.
The roughness contourgraph is a contact comprehensive measuring instrument, can be used for detecting two-dimensional form and position errors of workpieces, and is widely applied to detection of microscopic profile parameters such as surface roughness, waviness, original profile and the like.
Disclosure of Invention
The invention provides a processing method of involute tooth profile measurement data, which is characterized in that an involute tooth profile model is established according to parameters of a measured gear, and a fitted parameter is solved by constructing a least square objective function of original measurement data and the involute tooth profile model in the normal direction and utilizing the idea of optimal solution to obtain a fitted original measurement data curve and the involute tooth profile model.
The invention provides an evaluation calculation method of involute tooth profile deviation data, which is characterized in that the tooth profile deviation of any point in the normal direction of an involute is calculated according to a fitting parameter obtained by solving, and the deviation value is evaluated and calculated to obtain the tooth profile deviation and the precision grade defined by national standards.
The method comprises the steps of obtaining tooth profile data of an involute cylindrical gear on a roughness contourgraph, establishing an involute tooth profile model according to parameters of a measured gear, constructing a least square objective function of original measured data and the involute tooth profile model in a normal direction, solving fitting parameters by using an optimization solving idea to obtain a fitted original measured data curve and the involute tooth profile model, further calculating to obtain a tooth profile random point deviation value in the involute normal direction, and finally evaluating and calculating the deviation value to obtain tooth profile deviation and precision grade defined by national standards.
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FIG. 1 is an isometric illustration of a three-dimensional structure of the present invention.
Fig. 2 is a side view of the structure of the present invention.
Fig. 3 is a structural front view of the present invention.
FIG. 4 is a schematic view of the gear tooth of the present invention in a preferred measurement position.
Figure 5 is a schematic diagram of the involute rectangular coordinates of the present invention.
FIG. 6 is a schematic diagram of the pretreatment of the present invention.
FIG. 7 is a schematic diagram of a minimum distance point according to the present invention.
FIG. 8 is a diagram illustrating the fitting result of the orthogonal distance according to the present invention.
FIG. 9 is a schematic diagram of involute profile deviation calculation and evaluation in accordance with the present invention.
Detailed Description
The invention provides a method for acquiring involute cylindrical gear tooth profile data on a roughness profile instrument. As shown in fig. 1, the roughness profiler comprises a column 1, a driving box 2, a measuring head system 3, an axial moving table 4 and a base 5. As shown in fig. 2, the involute gear clamp 6 includes a motor 7, a coupling 8, a spindle shafting 9, a circular grating system 10, and an involute gear to be measured 11. The measuring head system 3 consists of a measuring contact pin and a sensor measuring rod and is connected to the driving box 2 to enable the measuring head system 3 and the driving box 2 to move along a measured tooth surface in an X-axis plane (a tangential axis), the stand column 1 enables the measuring head system 3 and the driving box 2 to move up and down in a Z-axis (a vertical axis), the involute gear clamp 6 clamps the measured involute gear and rotates in the X-axis plane, the axial moving platform 4 enables the involute gear clamp 6 to move in a Y-axis (a radial axis), so that the measured involute gear 11 is adjusted in a measuring position in the tooth width direction, and the circular grating system 10 is installed on the involute gear clamp 6 and used for detecting the rotating angle of the measured involute gear 11 after each measurement. The left end of a main shaft system 9 is connected to the motor 7 through a coupler 8, the right end of the main shaft system is connected with a measured involute gear 11, one end of the measured involute gear 11 is limited through a shaft shoulder, the other end of the measured involute gear 11 is clamped and fixed axially through a screw, a nut and a shaft sleeve, and the main shaft system 9 is supported and fixed on the axial moving platform 4 through a bearing. Wherein, the motor shaft, the circular grating and the tested involute gear rotate synchronously.
In order to accurately and comprehensively obtain the involute tooth profile of the measured involute gear, the measured involute gear needs to be accurately positioned at a measuring position after being installed. After the clamping of the measured involute gear is finished, the measured gear tooth is driven to rotate to the optimal measuring position state through the motor, so that a tooth root forming point F on the tooth root position of the tooth surface of the measured gear tooth is enabled to be formedfThe tooth tip forming point F on the tooth tip positionaOn a horizontal line, as shown in figure 4, the measurement variation range of the stylus probe system is minimised, reducing the non-linear error in the measurement of the involute profile.
And (5) measuring the tooth profile data after the clamping of the involute gear to be measured is finished and the adjustment of the tooth surface measuring position is finished. The measurements were carried out as follows: firstly, controlling an axial moving platform, and adjusting a measuring head on the axial center position of a gear tooth profile; secondly, moving the measuring head to a tooth root forming point F of the measured gear toothfAnd continuing to move 0.05mm in the direction of the tooth root, and operating the profilometer to enable the probe to be properFront position measurement beyond the tooth tip forming point FaAnd (5) completing the measurement of the tooth profile of the involute gear to be measured at a position of 0.05mm nearby, and storing the measured data in a computer.
The invention provides a method for processing involute tooth profile measurement data. Firstly, according to the parameters of the measured involute gear, modeling is carried out by using a parameter equation based on an involute generating method. The involute is a track of any point on a line when the line rolls around a base circle in a pure rolling mode. The involute rectangular coordinate is shown in FIG. 5, and the roll angle u of any point K on the involute can be obtained according to the definitionkComprises the following steps:
Figure BDA0002582112380000031
in the formula: thetakIs the spread angle of K points, alphakThe pressure angle at point K.
The involute profile parameter equation is expressed as:
Figure BDA0002582112380000032
in the formula: r isbIs the base circle radius.
The rotation angle of the optimal measurement position relative to the initial position modeled by the involute tooth profile parameter equation in the formula (2) is set as
Figure BDA0002582112380000039
Finally, the involute tooth profile theoretical model of the optimal measurement position is obtained by rotating the involute tooth profile parameter equation in the formula (2):
Figure BDA0002582112380000033
minimum value u of roll angle u in equation (3)minAnd maximum value umaxThe values are respectively taken at the starting point and the ending point of the involute. For the involute cylindrical gear, the starting point F of the involute is the intersection point of the transition curve and the involute tooth profile:
Figure BDA0002582112380000034
Figure BDA0002582112380000035
in the formula: alpha is alphaFPressure angle of point F, αtIs an end face pressure angle, and is,
Figure BDA0002582112380000036
is the crest coefficient, x is the displacement coefficient, z is the number of teeth, dFIs the diameter of point F, dbIs the base circle diameter.
If the tooth top chamfer is neglected, the end point of the involute is on the tooth top circle. Therefore, the minimum value u of the roll angle uminAnd maximum value umaxThe calculation formula of (2) is as follows:
Figure BDA0002582112380000037
Figure BDA0002582112380000038
in the formula: daThe diameter of the addendum circle.
And secondly, preprocessing the measurement data. In order to ensure the stability of the algorithm and improve the convergence speed of the algorithm, the preprocessing enables the involute tooth profile model and an original measured data curve to be as close as possible before orthogonal distance fitting.
The raw measurement data obtained by measuring the gear tooth profile by the profilometer has no position information but only length and shape information. Preprocessing shifts and aligns the original measurement data with the midpoint position of both involute profile models in the x-direction. In the y direction, aligning the original measurement data with the involute tooth profile model by calculating the data average value difference of the involute tooth profile model and the original measurement data in the y direction, wherein the preprocessed original measurement data are as follows:
Figure BDA0002582112380000041
as shown in fig. 6, the positions of the involute profile model and the raw measurement data curve are brought closer to each other as much as possible by preprocessing.
And thirdly, optimizing and solving. An Orthogonal Distance Fitting Algorithm (Orthogonal Distance Fitting Algorithm) needs to solve a least square solution of the preprocessed measured data curve and the involute tooth profile model in the normal direction, and solves Fitting parameters by constructing an objective function and utilizing an optimization solving idea to obtain an Orthogonal Distance Fitting result of the preprocessed measured data curve and the involute tooth profile model.
Firstly, solving the minimum distance point on the involute tooth profile model corresponding to each preprocessed measurement data point by utilizing Levenberg-Marquardt Algorithm (Levenberg-Marquardt Algorithm), and obtaining the position parameters u of all the minimum distance points on the involute tooth profile model corresponding to all the measurement data points. As shown in FIG. 4, MiT is any point in the pre-processed measurement data (i ═ 1, 2.. times.n, n is the total number of measurement data), TiIs MiThe minimum distance point on the involute profile model, i.e., the orthogonal distance corresponding point, x (u) is the involute profile model, diIs MiFrom the point of minimum distance TiIs measured.
The direction distance is managed as follows:
Figure BDA0002582112380000042
solving for the minimum distance point TiThe position parameter u of (a) can be converted into a solution to the extremum of the objective function d (u):
Figure BDA0002582112380000043
the L-M algorithm is an improvement of a Gauss-Newton iteration method, and a damping coefficient lambda is introduced to enable iteration to have a larger convergence interval, so that an iteration formula is obtained:
Figure BDA0002582112380000044
the L-M algorithm realizes that all preprocessed measurement data points solve the position parameter u of the minimum distance point on the involute tooth profile model, all the solved position parameters u are used as iteration initial values of all the position parameters u' of the Gaussian-Newton iteration method, and simultaneously, the rotation parameter is added
Figure BDA0002582112380000045
And a translation parameter x0、y0The method comprises the steps of optimally solving by a Gauss-Newton iterative method, realizing orthogonal distance fitting of a preprocessed measured data curve and an involute tooth profile model, and solving related fitting parameters (rotation parameters)
Figure BDA0002582112380000046
Translation parameter x0、y0All position parameters u'):
Figure BDA0002582112380000047
and setting the minimum distance corresponding points after rotation and translation processing as follows:
T′＝R-1M+X0 (13)
based on the idea of orthogonal distance fitting, the square sum minimization of the minimum distance between the involute tooth profile model after rotation and translation processing and the preprocessed measurement data needs to be solved, and each distance between a given point of the preprocessed measurement data and the involute tooth profile model should be minimized. The sum of squares of distances between the involute tooth profile model after rotation and translation processing and the measurement data curve after preprocessing is as follows:
Figure BDA0002582112380000051
the first requirement is:
Figure BDA0002582112380000052
based on the idea of the gauss-newton iterative method, an iterative formula can be obtained:
J|kΔb＝(M-T′)|k，bk+1＝bk+αΔb (16)
the expanded form of the iterative equation (16) is:
Figure BDA0002582112380000053
equation (17) is a linear overdetermined system of equations for Δ b, requiring a least squares solution.
The stop conditions are set as follows:
|bk+1-bk|＜ε (18)
for the fitting parameter b needing to be solved, in the final solving result
Figure BDA0002582112380000054
The accuracy of the fit is determined, which in turn affects the result of the tooth profile deviation calculation. Thus, the stop condition for the iteration is:
Figure BDA0002582112380000055
Figure BDA0002582112380000056
Figure BDA0002582112380000057
solve outThe fitting parameters b comprise optimal rotation parameters of orthogonal distance fitting
Figure BDA0002582112380000058
Translation parameter x0、y0And all position parameters u'. To preserve the relative position of the measured tooth profile throughout the gear, the final fit results are rotated on the involute profile model
Figure BDA0002582112380000059
Translating x to raw measurement data0、y0And finally obtaining the orthogonal distance fitting result of the involute tooth profile model and the preprocessed measurement data, as shown in fig. 8 (deviation amplification).
The invention provides an evaluation and calculation method of involute tooth profile deviation data. Firstly, calculating to obtain the normal deviation of the tooth profile after ODF treatment according to all position parameters u obtained by the ODF algorithm:
Figure BDA00025821123800000510
and secondly, calculating according to the formula (22) to obtain the normal deviation result of the tooth profile after ODF treatment, and calculating and evaluating based on the definition in the current national standard GB/T10095.1-2008 of cylindrical gear precision manufacturing. Wherein, the tooth profile deviation calculation is based on the regulation of tooth profile deviation in the current national standard GB/T10095.1-2008 for cylindrical gear precision manufacturing: total deviation of tooth profile (F)α) Is shown in the evaluation range LαThe distance between two designed tooth profile traces containing the actual tooth profile trace; deviation of tooth profile shape
Figure BDA0002582112380000061
Is shown in the evaluation range LαThe distance between two traces which contain the actual tooth profile trace and are completely the same as the average tooth profile trace is constant; deviation of tooth profile inclination
Figure BDA0002582112380000062
Is shown in the evaluation range LαAnd the distance between the two designed-profile traces whose two ends intersect the mean profile trace, as shown in fig. 9.
Thirdly, according to the calculation result, table look-up is carried out based on the current national standard GB/T10095.1-2008 of the cylindrical gear precision to obtain the total tooth profile deviation (F)α) Deviation of tooth profile shape
Figure BDA0002582112380000063
Deviation of tooth profile inclination
Figure BDA0002582112380000064
And the accuracy evaluation of the measured involute gear is finished by the single accuracy of the three items.

Claims (1)

1. A tooth profile deviation measuring method using a roughness profiler is characterized in that: establishing an involute tooth profile model according to parameters of a measured gear, solving fitting parameters by using an optimization solution idea through constructing a least square objective function of original measurement data and the involute tooth profile model in a normal direction, and obtaining an orthogonal distance fitting result of a fitted original measurement data curve and the involute tooth profile model;
calculating to obtain the deviation of any point of the tooth profile in the normal direction of the involute according to the fitting parameters obtained by solving, and evaluating and calculating the deviation value to obtain the tooth profile deviation and the precision grade defined by GB/T10095.1-2008 defined by the national standard; the method comprises the following specific steps:
according to the parameters of the measured involute gear, modeling by using a parameter equation based on an involute generating method; the involute is a track of any point on a line when the line rolls around a base circle for pure rolling; the rectangular coordinate of the involute is defined according to the rectangular coordinate of the involute to obtain the roll angle u of any point K on the involutekComprises the following steps:
Figure FDA0003262575160000011
in the formula: thetakIs the spread angle of K points, alphakThe pressure angle at point K; KN is the distance from the point K to a point N ON the base circle, and ON is the distance from the center O of the base circle to the point N ON the base circle;
the involute profile parameter equation is expressed as:
Figure FDA0003262575160000012
in the formula: r isbIs the base circle radius;
the rotation angle of the measurement position relative to the initial position modeled by the involute tooth profile parameter equation in the formula (2) is set as
Figure FDA0003262575160000017
Finally, the involute tooth profile theoretical model of the optimal measurement position is obtained by rotating the involute tooth profile parameter equation in the formula (2):
Figure FDA0003262575160000013
minimum value u of roll angle u in equation (3)minAnd maximum value umaxRespectively taking values at the starting point and the end point of the involute; for the involute cylindrical gear, the starting point F of the involute is the intersection point of the transition curve and the involute tooth profile:
Figure FDA0003262575160000014
Figure FDA0003262575160000015
in the formula: alpha is alphaFPressure angle of point F, αnIs an end face pressure angle, and is,
Figure FDA0003262575160000016
is the crest coefficient, x is the displacement coefficient, z is the number of teeth, dFIs the diameter of point F, dbIs the base circle diameter;
if the tooth crest chamfer is neglected, the end point of the involute is on the tooth crest circle; minimum value u of roll angle uminAnd maximum value umaxThe calculation formula of (2) is as follows:
Figure FDA0003262575160000021
Figure FDA0003262575160000022
in the formula: daThe diameter of the addendum circle;
preprocessing measurement data; preprocessing, in the x direction, carrying out translational alignment on the midpoint positions of the original measurement data and the involute tooth profile model; in the y direction, aligning the original measurement data with the involute tooth profile model by calculating the data average value difference of the involute tooth profile model and the original measurement data in the y direction, wherein the preprocessed original measurement data are as follows:
Figure FDA0003262575160000023
the positions of the involute tooth profile model and the original measured data curve are closer through preprocessing;
optimizing and solving; the orthogonal distance fitting algorithm needs to solve the least square solution of the preprocessed measured data curve and the involute tooth profile model in the normal direction, and solves fitting parameters by constructing a target function and utilizing the idea of optimized solution to obtain the orthogonal distance fitting result of the preprocessed measured data curve and the involute tooth profile model;
solving the minimum distance point on the involute tooth profile model corresponding to each preprocessed measurement data point by utilizing a Levenberg-Marquardt algorithm to obtain the position parameters u of all the minimum distance points on the involute tooth profile model corresponding to all the measurement points; miI is 1,2, …, n is the total number of measured data, T is any point in the pre-processed measured dataiIs MiThe minimum distance point on the involute profile model, i.e., the orthogonal distance corresponding point, x (u) is the involute profile model, diIs MiFrom the point of minimum distance TiThe normal distance of (d);
the direction distance is managed as follows:
Figure FDA0003262575160000024
solving for the minimum distance point TiThe position parameter u of (a) can be converted into a solution to the extremum of the objective function d (u):
Figure FDA0003262575160000025
the Levenberg-Marquardt algorithm is an improvement of a Gaussian-Newton iteration method, and a damping coefficient lambda is introduced to enable iteration to have a larger convergence interval, so that an iteration formula is obtained:
Figure FDA0003262575160000031
solving all the preprocessed measurement data points by using Levenberg-Marquardt algorithm on the position parameter u of the minimum distance point on the involute tooth profile model, using all the solved position parameters u as the iteration initial values of all the position parameters u' of the Gaussian-Newton iteration method, and simultaneously adding the rotation parameter
Figure FDA0003262575160000032
And a translation parameter x0、y0The method comprises the steps of optimizing and solving through a Gauss-Newton iterative method, realizing orthogonal distance fitting of a preprocessed measured data curve and an involute tooth profile model, solving related fitting parameters and rotation parameters
Figure FDA0003262575160000033
Translation parameter x0、y0All position parameters u':
Figure FDA0003262575160000034
and setting the minimum distance corresponding points after rotation and translation processing as follows:
T′＝R-1M+X0 (13)
based on the idea of orthogonal distance fitting, the square sum minimization of the minimum distance between the involute tooth profile model after rotation and translation processing and the preprocessed measurement data needs to be solved, and each distance between a given point of the preprocessed measurement data and the involute tooth profile model is also minimized; the sum of squares of distances between the involute tooth profile model after rotation and translation processing and the measurement data curve after preprocessing is as follows:
Figure FDA0003262575160000035
the first requirement is:
Figure FDA0003262575160000036
based on the thought of the Gauss-Newton iteration method, an iteration formula is obtained:
J|kΔb＝(M-T′)|k,bk+1＝bk+αΔb (16)
the expanded form of the iterative equation (16) is:
Figure FDA0003262575160000037
equation (17) is a linear overdetermined system of equations for Δ b, requiring a least squares solution;
the stop conditions are set as follows:
|bk+1-bk|<ε (18)
for the fitting parameter b needing to be solved, in the final solving result
Figure FDA0003262575160000038
x0,y0Determining the fitting precision and further influencing the tooth profile deviation calculation result, wherein the iteration stop conditions are as follows:
Figure FDA0003262575160000041
Figure FDA0003262575160000042
Figure FDA0003262575160000043
the solved fitting parameter b comprises the optimal rotation parameter of orthogonal distance fitting
Figure FDA0003262575160000044
Translation parameter x0、y0And all position parameters u'; to preserve the relative position of the measured tooth profile throughout the gear, the final fit results are rotated on the involute profile model
Figure FDA0003262575160000045
Translating x to raw measurement data0、y0Finally obtaining the orthogonal distance fitting result of the involute tooth profile model and the preprocessed measurement data;
calculating according to all position parameters u obtained by the orthogonal distance fitting algorithm to obtain the normal deviation of the tooth profile processed by the orthogonal distance fitting algorithm:
Figure FDA0003262575160000046
calculating according to the formula (22) to obtain the tooth profile normal deviation result processed by the orthogonal distance fitting algorithm, and calculating and evaluating based on the definition in the national standard GB/T10095.1-2008 of the accuracy of the cylindrical gear, wherein the total tooth profile deviation FαIs shown in the evaluation range LαThe distance between two designed tooth profile traces containing the actual tooth profile trace; deviation of tooth profile shape
Figure FDA0003262575160000047
Is shown in the evaluation range LαThe distance between two traces which contain the actual tooth profile trace and are completely the same as the average tooth profile trace is constant; deviation of tooth profile inclination
Figure FDA0003262575160000048
Is shown in the evaluation range LαThe distance between two designed tooth profile traces with two ends intersected with the average tooth profile trace;
obtaining the total deviation F of the tooth profile based on the precision of the cylindrical gear according to the calculation resultαDeviation of tooth profile shape
Figure FDA0003262575160000049
Deviation of tooth profile inclination
Figure FDA00032625751600000410
And the accuracy evaluation of the measured involute gear is finished by the single accuracy of the three items.
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EP0510431A2 (en) * 1991-04-22 1992-10-28 Toyota Jidosha Kabushiki Kaisha Method and apparatus for evaluating gear motion characteristics, based on tooth profile deflection differentiated by rotation angle of the gear
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