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CN-122020951-A - Double-argument fitting method for standard involute template profile

CN122020951ACN 122020951 ACN122020951 ACN 122020951ACN-122020951-A

Abstract

The invention relates to a double-argument fitting method of a standard involute template profile, and belongs to the technical field of gear parameter measurement. The method utilizes the characteristic that the profile of a standard involute template is similar to the shape of a parabola, a waveform section is divided into an alpha section and a beta section after optimizing in a rectangular coordinate system xoy, the alpha section carries out waveform fitting in a parabola x=g (y) mode to obtain a fitting residual effective value rho α and a fitting parameter, the beta section carries out waveform fitting in a parabola y=f (x) mode to obtain a fitting residual effective value rho β and a fitting parameter, and double-argument fitting is completed. The method of the invention does not need to directly measure the radius of the base circle or measure the expansion angle of the involute expansion line, is applicable to the parameter measurement and evaluation of any involute in the section of theta epsilon 0, pi, and has universality.

Inventors

  • LIANG ZHIGUO
  • Geng Shuya
  • WU TENGFEI

Assignees

  • 中国航空工业集团公司北京长城计量测试技术研究所

Dates

Publication Date
20260512
Application Date
20251203

Claims (8)

  1. 1. A double-argument fitting method of a standard involute template profile is characterized by comprising the following steps: The method comprises the steps of firstly, using coordinate transformation on a classical involute equation curve to enable the curve to rotate 90 degrees clockwise and then to be characterized as a rectangular coordinate xoy form, wherein x is an abscissa and y is an ordinate, (x, y) is a measurement point coordinate on an involute, sampling and measuring coordinate point sequences of involute waveforms { [ x i ,y i ] } (i=0, the..the n-1), selecting positive integer sequence q as a segmentation point sequence, and dividing the sampling and measuring coordinate point sequences into an alpha part (x 0 ,y 0 ),...,(x q-1 ,y q-1 ) and a beta part (x q ,y q ),...,(x n-1 ,y n-1 ); Fitting the alpha part involute waveform by taking an independent variable element as an ordinate sequence value { y i }, taking the dependent variable element as an abscissa value x to obtain a fitting curve waveform sequence value { x i } and a fitting parameter a α 、b α 、c α 、ρ α 、y gα 、x gα , calculating an ordinate fitting residual sequence delta y i according to a fitting curve 'longitudinal' regression residual sequence, (i=0, 1,..q-1), fitting the beta part involute waveform by taking the independent variable element as an abscissa sequence value { x i }, taking the dependent variable element as an ordinate value y to obtain a fitting curve waveform sequence value { y i } and a fitting parameter a β 、b β 、c β 、ρ β 、y gβ 、x gβ , calculating an ordinate fitting residual sequence delta y i according to a fitting curve regression residual sequence, (i=q, q+1,., n-1); Drawing curves of the rho α and the rho β along with the change of Q values in the same graph, and finding out an optimal demarcation point from the curves, wherein when the values of the rho α and the rho β are closest, the q=q 0 value is used as the optimal demarcation point Q of an involute segment; dividing the measurement curve section into an alpha section and a beta section by taking Q as an optimal demarcation point; Fitting the alpha part sampling measurement coordinate point sequence by taking the argument as an ordinate sequence value { y i }, taking the argument as an abscissa value x to obtain a fitting curve waveform sequence value { x i }, and marking the best fitting result as Calculating a fitting residual sequence delta y i between the alpha-section gear tooth profile involute and the fitting parabola according to the fitting curve 'longitudinal' regression residual sequence, (i=0, 1., q-1); Fitting the beta part sampling measurement coordinate point sequence by taking the argument as an abscissa sequence value { x i }, taking the argument as an ordinate value y to obtain a fitting curve waveform sequence value { y i }, and marking the best fitting result as Calculating a fitted residual sequence delta y i between the beta-segment gear tooth profile involute and the fitted parabola according to the fitted curve regression residual sequence, (i=q, 1..once., n-1); Valley fitted with left alpha segment And its location Position coordinates with parameters as involute starting point (θ=0°) Finishing the determination of the initial reference point D of the fitting involute, and judging and characterizing the positions of other points in the involute; according to the relative position of the actual measurement involute PC at the initial endpoint P and the reference point D, modeling measurement positioning of standard involute gear tooth profile parameters is achieved.
  2. 2. The method of claim 1, wherein the step of dividing the sequence of sampled measured coordinate points into an alpha portion and a beta portion is performed by, The base circle radius of the involute is r b , the expansion angle of the involute generating line BK is theta, the abscissa is h, and the ordinate is z, and then the parameter equation of the involute is: Wherein r b is the base circle radius, the coordinate point (h 0 ,z 0 )＝(r b , 0) is the involute starting point, θ is the angle of the expansion of the generating line BK, and θ D =0 at the involute starting point; Coordinate transformation is performed according to x=z, y= -h, and then the involute equation of the above formula (1) is converted into an involute equation of the form described by the formula (2): In the involute curve after coordinate transformation, r b is the radius of a base circle, θ is the expansion angle of an occurrence line BK, the point corresponding to θ D = 0;D at the involute starting point D is the involute starting point positioned on the radius of the base circle, and (x D ,y D ) is the coordinate of the point D, which is characterized by a coordinate point D (x D ,y D ); With the point D (x D ,y D ) as the reference point, the straight line segment length r KD and the slope K KD of any point K (x, y) on the involute and the reference point D (x D ,y D ) are respectively: The combination of each point K (x, y) on the involute with the straight line segment length r KD and slope K KD of reference point D (x D ,y D ) is unique and is used to determine the location of point K (x, y) in the involute when reference point D (x D ,y D ) is known; The involute of a circle is an open curve with a starting point and no end point; When the base circle radius is r b and the expansion angle theta epsilon 0 pi, the value range of the abscissa x and the ordinate y in the rectangular coordinate system is x epsilon 0, r b ·π],y∈[-r b ,-r b pi/2; fitting and characterizing the involute sample plate, wherein the limit interval range is theta epsilon 0, pi, and x epsilon 0, r b ·π],y∈[-r b ,-r b pi/2, and differentiating by formula (2) When theta is 0, pi, x is 0, r b pi, X (r b , θ) increases monotonically over the interval; when theta is 0, pi/2, there is y E-r b ,-r b pi/2, Y (r b , θ) decreases monotonically over the interval; when theta is [ pi/2, pi ], y is [ pi ] -r b ,-r b . Pi/2 ], Y (r b , θ) increases monotonically over the interval; analysis of the involute shape in the finite interval theta epsilon 0, pi and the equation described in formula (2) shows that the ordinate of the involute is a concave function with unimodal characteristics relative to the abscissa; waveform analysis of the involute shows that: In the interval theta epsilon [0, pi/3 ], the involute shape approximates to a left-oriented transverse parabolic shape, and a parabolic equation x=g (y) with y as an independent variable and x as a dependent variable is suitable for fitting; in the interval theta epsilon [ pi/3, pi ], the involute shape approximates to a vertical parabola shape with the extreme point downwards, and a parabolic equation y=f (x) with x as an independent variable and y as a dependent variable is suitable for fitting; The abscissa of the sampling measurement coordinate point sequence of the involute template line segment PC is x 0 ,x 1 ,...,x n-1 , the ordinate is y 0 ,y 1 ,...,y n-1 , the corresponding involute generating line expansion angle is theta 0 ,θ 1 ,...,θ n-1 , and due to the selection and characterization of measurement reference points, constant coordinate offsets x d and y d exist between the sampling measurement coordinate point sequence [ x i ,y i ] and the involute model theoretical value sequence [ x (r b ,θ i ),y(r b ,θ i ) ], namely The middle of the measuring line segment PC is selected to be divided into a left half segment PQ and a right half segment QC, which are respectively called an alpha segment part and a beta segment part, wherein the left half segment PQ corresponds to a sampling measuring coordinate point sequence [ x 0 ,y 0 ],[x 2 ,y 2 ],...,[x q-1 ,y q-1 ], and the right half segment QC corresponds to a sampling measuring coordinate point sequence [ x q ,y q ],[x q+1 ,y q+1 ],...,[x n-1 ,y n-1 ]).
  3. 3. The method of claim 1, wherein the fitting of the alpha portion involute waveform is performed by, In the alpha segment part, the involute shape and the shape of a transverse parabola with the extreme point towards the left are approximate, the curve equation x=g (y) with y as an independent variable and x as a dependent variable is suitable for fitting, and the function expression of the least square fitting curve is as follows: wherein a α 、b α 、c α is 3 fitting parameters; The fit residual effective value is: then the fitted waveform "valley" estimate is obtained as: The position where the "valley" of the fitted waveform occurs is: the fitting process is as follows: The sequence of coordinate points is measured for samples [ x i ,y i ], (i=0, 1..q-1), as represented by equation (7): when ε α takes the minimum value, there are: Solving the linear equation set to obtain a fitting parameter a α 、b α 、c α which is used as an approximate characterization parameter of the parabolic form of the fitting involute, calculating according to formulas (9) and (10) to obtain a corresponding x gα 、y gα value, and calculating a fitting residual effective value rho α according to formula (8); fitting curve "transverse" regression residual sequence is Fitting curve "longitudinal" regression residual sequence as
  4. 4. The method of claim 3, wherein the fitting the waveform of the beta portion involute is performed by, In the section beta, the involute shape and the vertical parabola shape with downward extreme points are approximate, the curve is fit by using a parabolic equation y=f (x) with x as an independent variable and y as a dependent variable, and the function expression of the least square fitting curve is as follows: Wherein a β 、b β 、c β is 3 fitting parameters; The fit residual effective value is: Then the fitted waveform "valley" estimate is obtained as: The position where the "valley" of the fitted waveform occurs is: the fitting process is as follows: the sequence of coordinate points is measured for samples [ x i ,y i ], (i=q, q+1,..n-1), as represented by equation (16): When ε β takes the minimum value, there are: Solving the linear equation set to obtain a fitting parameter a β 、b β 、c β which is used as an approximate characterization parameter of the parabolic form of the fitting involute, calculating according to formulas (18) and (19) to obtain a corresponding y gβ 、x gβ value, and calculating a fitting residual effective value rho β according to formula (17); Fitting curve regression residual sequence is
  5. 5. The method of claim 4, wherein the initial point P (x 0 ,y 0 ) and the reference point are measured on the involute Length of straight line segment And slope of The method comprises the following steps of: Measuring end point C (x n-1 ,y n-1 ) and reference point on involute Length of straight line segment And slope of The method comprises the following steps of: Calculated according to the steps (24) - (27) Then, the standard involute of the formula (2) is searched for a standard involute with the same slope value in theory The position of a point is used for determining the geometric relative position of an actually measured involute PC in a standard involute, and the reference point is Namely, the double-argument parabolic involute fitting is realized, wherein y is taken as an independent variable of the left alpha segment part, x is taken as a dependent variable of the left alpha segment part, and x is taken as an independent variable of the right beta segment part.
  6. 6. The method of claim 5, wherein the application range is an involute generating line expansion angle theta E [0, pi ] interval.
  7. 7. The method of claim 5, wherein the initial end of the fitted curve of the alpha segment of curve segmented using the optimal demarcation point characterizes the initial point location of the involute.
  8. 8. The method of claim 7 wherein the end points of the fitted curve of the beta curve divided using the optimal demarcation point characterize the end point location of the involute.

Description

Double-argument fitting method for standard involute template profile Technical Field The invention relates to a double-argument fitting method of a standard involute template profile, in particular to waveform fitting of a cam, a spline, a gear tooth profile of a standard involute template profile curve and parameterization representation of the profile curve of the standard involute template profile curve, and belongs to the technical field of gear parameter measurement. Background The standard involute is a curve waveform with the most wide application in the gear industry due to excellent compact mechanical meshing characteristics, is widely applied to gear tooth profiles, spline tooth profiles and cam profiles, and is also manufactured into a standard involute template for characterizing the gear tooth profiles. In general, when used for gear tooth profiles, the involute length used is shorter, and when used for cam profiles, the involute length involved is longer, and at this time, curve waveform fitting is more difficult, which is mainly represented by the difficulty in finding a proper fitting curve to obtain higher fitting precision. The standard involute gear is the most widely applied gear in the practical engineering technology, the measurement and characterization of the gear profile parameters are the most important means for evaluating the gear quality, including the tooth profile deviation, the effective involute length, the available involute length and the like, and the gear profile parameters are generally measured by using a special gear measuring center, a universal gear measuring machine, a three-coordinate measuring machine and the like, and the gear profile parameters are characterized by the deviation and distribution between the actual measurement result and the standard involute. The same is true of the measurement and characterization of other involute profiles such as cams, splines, standard involute templates, etc. Under the measurement characterization condition of the three-coordinate measuring machine, the measurement result of the involute template is completely characterized in a coordinate point mode of each measurement point in a rectangular coordinate system. And then, performing curve fitting on a measurement sequence formed by all the measurement points, calculating regression deviation, and taking the fitted curve parameters as the characterization parameters in involute approximate characterization. The problems still existing at present mainly include: 1) The standard involute equation is an overdetermined equation with complex functional relation, the radius of a base circle and the starting point of an involute are required to be known firstly, and then the corresponding coordinate relation of other contour points on the involute can be determined according to the expansion angle theta of the involute generating line. However, the base circle radius and the involute starting point may not be in the tooth profile measuring points of the gear, and the expansion angle theta of the involute generating line is also unknown, in the involute measuring sequence represented by the normal rectangular coordinates, reference values such as the base circle radius, the involute starting point and the like are lacking, so that the determination of the involute representing reference point of the gear measuring result is problematic, and it is difficult for people to directly find and determine the involute reference point from the tooth profile measuring points and to measure the base circle radius; 2) In actual work, the base radius needs to be additionally measured and given, when only an involute exists, the base radius is unknown or not precisely known, the actual tooth profile involute template parameters are only a part of the involute, and the starting point of the involute is probably not included due to the processing technology and the like, so that the difference between the involute and the standard involute is difficult to determine, therefore, other curves such as an arc curve and the like are often used for carrying out local curve fitting instead of the involute, and the fitting regression residual is evaluated, so that the processing quality is evaluated. 3) The involute curve of tooth profile is a gradual change curve with different curvature radius at each point, and the fitting itself is carried out by using the circular arc with the same curvature radius at each point, thus the fitting error is unavoidable. 4) The curve represented by the standard involute equation has the advantages that the single-value interval is smaller, and the wider involute range faces the fitting difficulty of the non-single-value interval in a rectangular coordinate system due to the ordinate of the independent variable relative to the abscissa of the independent variable. Disclosure of Invention The invention aims to provide a double-argument fitting method of a standard involute