CN111912373B - A method for measuring tooth profile deviation using a roughness profiler - Google Patents

A method for measuring tooth profile deviation using a roughness profiler 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
involute
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measurement data
fitting
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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)

1.一种利用粗糙度轮廓仪的齿廓偏差测量方法，其特征在于：根据被测齿轮的参数建立渐开线齿廓模型，通过构造原始测量数据与渐开线齿廓模型在法向上的最小二乘目标函数，利用最优化求解的思想求解拟合参数，得到拟合后的原始测量数据曲线与渐开线齿廓模型的正交距离拟合结果；1. a tooth profile deviation measuring method utilizing a roughness profiler, is characterized in that: according to the parameter of the gear to be tested, an involute tooth profile model is set up, by constructing the original measurement data and the involute tooth profile model in the normal direction. The least squares objective function uses the idea of optimal solution to solve the fitting parameters, and obtains the fitting result of the orthogonal distance between the original measurement data curve after fitting and the involute tooth profile model; 根据求解得到的拟合参数，计算得到渐开线法向上的齿廓任意点偏差，将偏差值进行评定计算，得到国家标准定义的GB/T10095.1-2008定义的齿廓偏差与精度等级；具体步骤如下：According to the obtained fitting parameters, the deviation of the tooth profile at any point in the normal direction of the involute is calculated, and the deviation value is evaluated and calculated to obtain the tooth profile deviation and accuracy grade defined by the national standard GB/T10095.1-2008; Specific steps are as follows: 根据被测渐开线齿轮的参数，基于渐开线展成法使用参数方程建模；渐开线是发生线绕基圆作纯滚动时，发生线上任意一点的轨迹；渐开线直角坐标，根据渐开线直角坐标定义得渐开线上任意一点K的滚动角uk为：According to the parameters of the measured involute gear, the parametric equation is used for modeling based on the involute expansion method; the involute is the trajectory of any point on the generating line when the generating line is purely rolling around the base circle; the involute rectangular coordinates , the rolling angle u k of any point K on the involute is defined according to the Cartesian coordinates of the involute as:
Figure FDA0003262575160000011
Figure FDA0003262575160000011
 式中：θk为K点的展角，αk为K点的压力角；KN为K点到基圆上一点N的距离，ON为基圆圆心O到基圆上一点N的距离；In the formula: θ k is the spread angle of point K, α k is the pressure angle of point K; KN is the distance from point K to a point N on the base circle, ON is the distance from the center O of the base circle to a point N on the base circle; 渐开线齿廓参数方程表示为：The involute profile parameter equation is expressed as:
Figure FDA0003262575160000012
Figure FDA0003262575160000012
 式中：rb为基圆半径；where: r b is the radius of the base circle; 设测量位置相对于式(2)中渐开线齿廓参数方程建模初始位置的旋转角为
Figure FDA0003262575160000017
通过对式(2)中渐开线齿廓参数方程进行旋转，最终得到最佳测量位置的渐开线齿廓理论模型：The rotation angle of the measurement position relative to the initial position of the involute profile parameter equation modeling in Eq. (2) is
Figure FDA0003262575160000017
By rotating the involute tooth profile parameter equation in formula (2), the theoretical model of the involute tooth profile for the optimal measurement position is finally obtained:
Figure FDA0003262575160000013
Figure FDA0003262575160000013
 式(3)中滚动角u的最小值umin与最大值umax分别取值于渐开线的起点与终点处；对于渐开线圆柱齿轮，渐开线的起点F点即为过渡曲线与渐开线齿廓的交点：In formula (3), the minimum value u min and the maximum value u max of the rolling angle u 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 transition curve and the Intersection of involute tooth profiles:
Figure FDA0003262575160000014
Figure FDA0003262575160000014
Figure FDA0003262575160000015
Figure FDA0003262575160000015
 式中：αF为F点的压力角，αn为端面压力角，
Figure FDA0003262575160000016
为齿顶高系数，x为变位系数，z为齿数，dF为F点的直径，db为基圆直径；where α F is the pressure angle at point F, α n is the end face pressure angle,
Figure FDA0003262575160000016
is the addendum height coefficient, x is the displacement coefficient, z is the number of teeth, d F is the diameter of point F, and db is the diameter of the base circle; 若忽略齿顶倒角，渐开线的终点在齿顶圆上；滚动角u的最小值umin与最大值umax的计算公式为：If the addendum chamfer is ignored, the end point of the involute is on the addendum circle; the calculation formulas for the minimum value u min and the maximum value u max of the rolling angle u are:
Figure FDA0003262575160000021
Figure FDA0003262575160000021
Figure FDA0003262575160000022
Figure FDA0003262575160000022
 式中：da为齿顶圆直径；where: d a is the diameter of the addendum circle; 测量数据预处理；预处理在x方向上，将原始测量数据与渐开线齿廓模型两者的中点位置平移对齐；在y方向上，通过计算渐开线齿廓模型与原始测量数据在y方向上两者的数据均值差，将原始测量数据与渐开线齿廓模型对齐，预处理后的原始测量数据为：Preprocessing of measurement data; preprocessing in the x direction, aligning the midpoint position of the original measurement data and the involute tooth profile model; in the y direction, by calculating the involute tooth profile model and the original measurement data. The mean difference between the two data in the y direction is to align the original measurement data with the involute tooth profile model. The preprocessed original measurement data is:
Figure FDA0003262575160000023
Figure FDA0003262575160000023
 通过预处理，使得渐开线齿廓模型与原始测量数据曲线的位置更加接近；Through preprocessing, the position of the involute tooth profile model and the original measurement data curve is closer; 最优化求解；正交距离拟合算法需要求解预处理后的测量数据曲线与渐开线齿廓模型在法向上的最小二乘解，通过构造目标函数利用最优化求解的思想，求解拟合参数，得到预处理后的测量数据曲线与渐开线齿廓模型的正交距离拟合结果；Optimal solution; the orthogonal distance fitting algorithm needs to solve the least squares solution of the preprocessed measurement data curve and the involute tooth profile model in the normal direction. By constructing the objective function, the idea of optimal solution is used to solve the fitting parameters. , the fitting result of the orthogonal distance between the preprocessed measurement data curve and the involute tooth profile model is obtained; 利用列文伯格-马夸尔特算法，求解预处理后的每个测量数据点对应渐开线齿廓模型上的最小距离点，得到所有测量点对应渐开线齿廓模型上所有最小距离点的位置参数u；Mi为预处理后测量数据中的任意点，i＝1,2,…,n，n为测量数据总个数，Ti为Mi在渐开线齿廓模型上的最小距离点，即正交距离对应点，x(u)为渐开线齿廓模型，di即为Mi与最小距离点Ti的法向距离；Using the Levenberg-Marquardt algorithm, solve the minimum distance points on the involute tooth profile model corresponding to each measurement data point after preprocessing, and obtain all the minimum distances on the involute tooth profile model corresponding to all measurement points The position parameter u of the point; Mi is any point in the measured data after preprocessing, i =1,2,...,n, n is the total number of measured data, T i is Mi on the involute tooth profile model The minimum distance point of , that is, the corresponding point of the orthogonal distance, x( u ) is the involute tooth profile model, d i is the normal distance between Mi and the minimum distance point Ti; 设法向距离为：The managed distance is:
Figure FDA0003262575160000024
Figure FDA0003262575160000024
 求解最小距离点Ti的位置参数u即可转换为对目标函数D(u)极值的求解：Solving the position parameter u of the minimum distance point T i can be converted into solving the extreme value of the objective function D(u):
Figure FDA0003262575160000025
Figure FDA0003262575160000025
 列文伯格-马夸尔特算法为高斯-牛顿迭代法的改进，引入阻尼系数λ使得迭代具有更大的收敛区间，由此得到迭代公式：The Levenberg-Marquardt algorithm is an improvement of the Gauss-Newton iteration method. The damping coefficient λ is introduced to make the iteration have a larger convergence interval, and the iteration formula is obtained:
Figure FDA0003262575160000031
Figure FDA0003262575160000031
 列文伯格-马夸尔特算法实现所有预处理后的测量数据点对渐开线齿廓模型上最小距离点位置参数u求解，所求解的所有位置参数u用作高斯-牛顿迭代法所有位置参数u′的迭代初值，同时加入旋转参数
Figure FDA0003262575160000032
和平移参数x0、y0，通过高斯-牛顿迭代法最优化求解，实现预处理后的测量数据曲线与渐开线齿廓模型的正交距离拟合，并求解出相关拟合参数，旋转参数
Figure FDA0003262575160000033
平移参数x0、y0，所有位置参数u′：The Levenberg-Marquardt algorithm realizes that all preprocessed measurement data points solve the position parameter u of the minimum distance point on the involute tooth profile model, and all the solved position parameters u are used as the Gauss-Newton iteration method. The initial iterative value of the position parameter u', and the rotation parameter is added at the same time
Figure FDA0003262575160000032
and translation parameters x 0 , y 0 , optimized and solved by Gauss-Newton iterative method, to realize the orthogonal distance fitting between the preprocessed measurement data curve and the involute tooth profile model, and to solve the relevant fitting parameters, rotate parameter
Figure FDA0003262575160000033
Translation parameters x 0 , y 0 , all position parameters u′:
Figure FDA0003262575160000034
Figure FDA0003262575160000034
 设旋转和平移处理后的最小距离对应点为：Let the minimum distance corresponding point after rotation and translation processing be: T′＝R-1M+X0 (13)T'=R -1 M+X 0 (13) 基于正交距离拟合的思想，需要求解旋转和平移处理后的渐开线齿廓模型与预处理后的测量数据之间最小距离的平方和最小化，且预处理后的测量数据给定点与渐开线齿廓模型之间的每个距离也应最小化；旋转、平移处理后的渐开线齿廓模型与预处理后的测量数据曲线的距离平方和为：Based on the idea of orthogonal distance fitting, it is necessary to minimize the sum of the squares of the minimum distances between the involute tooth profile model after rotation and translation processing and the preprocessed measurement data, and the given point of the preprocessed measurement data is equal to Each distance between the involute tooth profile model should also be minimized; the sum of squared distances between the involute tooth profile model after rotation and translation processing and the preprocessed measurement data curve is:
Figure FDA0003262575160000035
Figure FDA0003262575160000035
 一阶必要条件为：The first-order necessary conditions are:
Figure FDA0003262575160000036
Figure FDA0003262575160000036
 基于高斯-牛顿迭代法的思想，得迭代公式：Based on the idea of Gauss-Newton iteration method, the iterative formula is obtained: J|kΔb＝(M-T′)|k,bk+1＝bk+αΔb (16)J| k Δb=(MT′)| k ,b k+1 =b k +αΔb (16) 迭代公式(16)的展开形式为：The expanded form of iterative formula (16) is:
Figure FDA0003262575160000037
Figure FDA0003262575160000037
 式(17)是关于Δb的线性超定方程组，需求最小二乘解；Equation (17) is a system of linear overdetermined equations about Δb, which requires a least-squares solution; 设定停止条件为：Set the stop condition as: |bk+1-bk|<ε (18)|b k+1 -b k |<ε (18) 对于需要求解的拟合参数b，最终求解结果中的
Figure FDA0003262575160000038
x0,y0决定着拟合的精度，进而影响着齿廓偏差计算的结果，迭代的停止条件为：For the fitting parameter b that needs to be solved, in the final solution result
Figure FDA0003262575160000038
x 0 , y 0 determine the accuracy of the fitting, which in turn affects the result of the tooth profile deviation calculation. The stopping condition of the iteration is:
Figure FDA0003262575160000041
Figure FDA0003262575160000041
Figure FDA0003262575160000042
Figure FDA0003262575160000042
Figure FDA0003262575160000043
Figure FDA0003262575160000043
 求解出的拟合参数b包含正交距离拟合的最佳旋转参数
Figure FDA0003262575160000044
平移参数x0、y0和所有位置参数u′；为了保留测量齿廓在整个齿轮上的相对位置，最终的拟合结果对渐开线齿廓模型旋转
Figure FDA0003262575160000045
对原始测量数据平移x0、y0，最终得到渐开线齿廓模型与预处理后的测量数据的正交距离拟合结果；The solved fitting parameter b contains the optimal rotation parameter for the orthogonal distance fitting
Figure FDA0003262575160000044
Translation parameters x 0 , y 0 and all position parameters u′; in order to preserve the relative position of the measured tooth profile on the entire gear, the final fitting result is rotated on the involute tooth profile model
Figure FDA0003262575160000045
Translate the original measurement data by x 0 and y 0 , and finally obtain the fitting result of the orthogonal distance between the involute tooth profile model and the pre-processed measurement data; 根据正交距离拟合算法得到的所有位置参数u计算得到正交距离拟合算法处理后的齿廓法向偏差：According to all the position parameters u obtained by the orthogonal distance fitting algorithm, the normal deviation of the tooth profile processed by the orthogonal distance fitting algorithm is obtained:
Figure FDA0003262575160000046
Figure FDA0003262575160000046
 根据式(22)计算得到正交距离拟合算法处理后的齿廓法向偏差结果，基于圆柱齿轮精度国家标准GB/T10095.1-2008中的定义进行计算与评价，其中齿廓总偏差Fα表示在计值范围Lα内，包容实际齿廓迹线的两条设计齿廓迹线间的距离；齿廓形状偏差
Figure FDA0003262575160000047
表示在计值范围Lα内，包容实际齿廓迹线的，与平均齿廓迹线完全相同的两条迹线间的距离，且两条曲线与平均齿廓迹线的距离为常数；齿廓倾斜偏差
Figure FDA0003262575160000048
表示在计值范围Lα内，两端与平均齿廓迹线相交的两条设计齿廓迹线间的距离；According to formula (22), the normal deviation result of the tooth profile after the orthogonal distance fitting algorithm is obtained, and the calculation and evaluation are carried out based on the definition in the national standard GB/T10095.1-2008 for the accuracy of cylindrical gears. The total deviation of the tooth profile is F. α represents the distance between the two designed tooth profile traces that contain the actual tooth profile trace within the value range L α ; tooth profile shape deviation
Figure FDA0003262575160000047
Represents the distance between two traces that are exactly the same as the average tooth profile trace within the value range L α , including the actual tooth profile trace, and the distance between the two curves and the average tooth profile trace is constant; Profile tilt deviation
Figure FDA0003262575160000048
Represents the distance between two designed tooth profile traces whose ends intersect with the average tooth profile trace within the value range L α ; 根据计算结果，基于圆柱齿轮精度得到齿廓总偏差Fα、齿廓形状偏差
Figure FDA0003262575160000049
齿廓倾斜偏差
Figure FDA00032625751600000410
三个项目的单项精度，完成对被测渐开线齿轮的精度评价。According to the calculation results, the total tooth profile deviation F α and the tooth profile shape deviation are obtained based on the accuracy of the cylindrical gear.
Figure FDA0003262575160000049
Tooth profile inclination deviation
Figure FDA00032625751600000410
The single precision of the three items completes the precision evaluation of the measured involute gear.
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