CN-122019936-A - Modeling measurement positioning method for standard involute gear tooth profile parameters

Abstract

A modeling measurement positioning method for standard involute gear tooth profile parameters belongs to the field of gear parameter measurement. The realization method of the invention comprises the steps of utilizing the involute of the gear tooth profile to approximate to the parabolic shape in a section applied by the tooth profile, dividing a measurement waveform into a left section and a right section when the tooth profile is characterized, optimizing and determining the optimal demarcation point between the two sections by respectively fitting a residual error effective value balancing mode, respectively carrying out partial parabolic approximation to characterize the involute by using two curve waveforms at the left side and the right side of the optimal demarcation point, carrying out regression residual error estimation by using a partial parabolic fitting mode, carrying out approximate parameterization characterization on the involute by using double parabolic model parameters, searching out a starting point parameter and a terminal point parameter of a fitted curve section, and representing the boundary of the involute section corresponding to the gear tooth profile by modeling corresponding linear mapping relation between the end points of the fitted curve section and the end points of the involute section, thereby realizing the modeling measurement positioning of the standard involute tooth profile parameters.

Inventors

• LIANG ZHIGUO
• HE XUAN
• WU TENGFEI
• QIAN FENG
• Geng Shuya

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 (5)

1. 1. A modeling measurement positioning method for standard involute gear tooth profile parameters is characterized by comprising the following steps, Step one, dividing a measured waveform into a left section and a right section for fitting respectively, and optimizing and determining an optimal demarcation point between the two sections in a way of balancing the effective values of the fitting residual errors; Sampling and measuring coordinate point sequences (x 0 ,y 0 ),...,(x n-1 ,y n-1 ) of a gear tooth profile line segment PQ, selecting positive integer sequence numbers q epsilon [10, n-10], dividing the sampling and measuring coordinate point sequences into alpha parts (x 0 ,y 0 ),...,(x q-1 ,y q-1 ) and beta parts (x q ,y q ),...,(x n-1 ,y n-1 ), performing parabolic fitting on the alpha part involute waveform to obtain a fitting parameter a α 、b α 、c α 、ρ α 、y gα 、x gα , performing parabolic fitting on the beta part involute waveform to obtain a fitting parameter a β 、b β 、c β 、ρ β 、y gβ 、x gβ ; The method comprises the steps of starting from q=10, performing piecewise parabolic fitting to obtain fitting parameters, sequentially increasing q values, performing fitting again and obtaining fitting parameters until q=n-10 is reached, and drawing the effective value rho α 、ρ β of a fitting residual error into the same curve along with the change rule of q, wherein according to the curve, the envelope line of rho α is monotonically increased and changed along with the increase of q values, the envelope line of rho β is monotonically decreased and changed along with the increase of q values, and when the values of rho α and rho β are closest, the q=q 0 is taken as the optimal demarcation point of an involute line segment; Step two, taking q 0 as an optimal demarcation point, dividing a measured curve section into an alpha section and a beta section, respectively performing parabolic fitting on involute sections on two sides of the optimal demarcation point to obtain an optimal fitting result, and marking as Fitting the parameter characterization result of the involute by a double-parabola method; Obtaining a fitting parabola according to the fitting parameters, and respectively calculating fitting residual sequences delta y i between the involute of the tooth profile of the two sections of the alpha and beta gears and the fitting parabola, (i=0, 1,., 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 a fitting involute initial reference point D; Step three, according to the measured involute PQ, the initial end point P (x 0 ,y 0 ) and the reference point The relative positions of the coordinates realize the modeling self-reference measurement positioning of the standard involute gear tooth profile parameters.
2. 2. The method of claim 1 wherein the initial end of the curve is fitted using an alpha segment waveform to characterize the initial point location of the involute.
3. 3. The method of claim 2, wherein the end points of the fitted curve of the beta waveform are used to characterize the end point positions of the involute.
4. 4. The method of claim 3, wherein the second implementation method is, The base circle radius of the involute is r b , the expansion angle of the involute generating line BK is theta, and then the parameter equation of the involute is: Wherein r b is a base circle radius, θ is an angle of expansion of the generating line BK, and the involute starting point D corresponds to a point θ D = 0;D, which is an involute starting point located on the base circle radius, (x D ,y D ) is a coordinate of the point D, and 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 the circle is an open curve with a starting point and no end point, and the tooth profile contour line used by the gear is only a part of the starting phase in which the abscissa x and the ordinate y change monotonically; When the radius of the base circle is r b and the expansion angle theta epsilon [0, pi/2 ], the value range of the abscissa x and the ordinate y in the rectangular coordinate system is x epsilon [ r b ,r b ·π/2],y∈[0,r b ]; Measuring and characterizing involute profile for gear, limiting interval theta epsilon 0, pi/2, where x epsilon r b ,r b ·π/2],y∈[0,r b , obtained by differentiating formula (1) When theta is E [0, pi/2 ], x is E [ r b ,r b ·π/2],y∈[0,r b ], X (r b , θ) and y (r b , θ) both belong to a monotonically increasing function curve within the interval; Analysis of the involute shape and equation described in equation (1) within the finite interval θ e 0, pi/2 reveals that the ordinate of the involute is a concave function with unimodal features relative to the abscissa, and that a unique "valley" occurs at the boundary θ=0, and this point coordinate value is noted as (x g ,y g ); In the interval theta epsilon [0, pi/2 ], the involute shape and the parabolic shape are approximate, the involute sampling measurement coordinate point in the rectangular coordinate system XOY is (x, y) = [ x (r b ,θ),y(r b , theta) ], the involute is fitted by using the parabolic y (x), and the involute and the parabolic y (x) are judged to have a common 'valley' value point, and then the function expression of the least square fitting curve is as follows: Wherein a, b and c are 3 fitting parameters; Let the abscissa of the sampling measurement coordinate point of the gear tooth profile line segment PQ be x 0 ,x 1 ,...,x n-1 , the ordinate be y 0 ,y 1 ,...,y n-1 , the corresponding involute generating line expansion angle be θ 0 ,θ 1 ,...,θ n-1 , because of the selection and characterization of the measurement reference point position, constant coordinate offsets x d and y d exist between the sampling measurement coordinate point [ x i ,y i ] and the involute model theoretical value point [ x (r b ,θ i ),y(r b ,θ i ) ], namely 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: judging and comparing the fitting quality by using the fitting residual effective value rho, and judging the obtained peak value result by using the fitting residual effective value rho; the fitting procedure was as follows: The sequence of coordinate points [ x i ,y i ] is measured for samples, (i=0, 1,..n-1), as represented by equation (5): when ε takes the minimum value, there are: Solving the linear equation set to obtain fitting parameters a, b and c, wherein the fitting parameters a, b and c are used as approximate characterization parameters of the parabolic form of a fitting involute, corresponding y g 、x g values are obtained through calculation according to formulas (8) and (9), and the fitting residual effective value rho is calculated according to formula (7); Fitting curve regression residual sequence is Fitting a parabolic valley point, taking the parabolic valley point as a coordinate estimated value of an intersection point D of the involute and a base circle, and marking the parabolic valley point as D (x g ,y g ), wherein the parabolic valley point can be used as a reference coordinate point for involute measurement fitting and positioning; Then, the length of the straight line segment between any point K (x, y) on the involute and the reference point D (x g ,y g ) And slope of The method comprises the following steps of: length of straight line segment of each point K (x, y) on involute and reference point D (x g ,y g ) And slope of Is unique, by finding the standard involute of formula (1) that has theoretically the same slope value when the reference point D (x g ,y g ) is known The geometric position of the point K (x, y) in the involute can be determined.
5. 5. The method of claim 4, wherein the third implementation method is, Measuring initial endpoint P (x 0 ,y 0 ) and reference point on involute Length of straight line segment And slope of The method comprises the following steps of: measuring end point Q (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 (16) - (19) Then, by searching for values with the same slope on the standard involute And determining the geometric relative position of the actually measured involute PQ in the standard involute, namely realizing the modeling measurement and positioning of the standard involute gear tooth profile parameters.

Description

Modeling measurement positioning method for standard involute gear tooth profile parameters Technical Field The invention relates to a modeling measurement positioning method of standard involute gear tooth profile parameters, in particular to a gear tooth profile parameter and a position positioning and quantitative characterization method thereof on a standard involute, belonging to the technical field of gear parameter measurement. Background 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. Under the condition of polar coordinate measurement and characterization, the current national standard GB/T13924 'involute cylindrical gear precision inspection rule' prescribes that the measurement result takes each tooth form period of a gear as a reference period, the gear axis as a reference shaft, and the rotation angle and the corresponding radius are used for representing tooth profile measurement data points, and each tooth form period corresponds to 0-360 degrees. Under the measurement characterization condition of the three-coordinate measuring machine, the measurement result is completely characterized by a coordinate point mode of each measurement point in a rectangular coordinate system. And then, curve fitting is carried out on the tooth profile measuring points of the gear so as to calculate and acquire regression deviation. The problems still existing at present are: 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 radius of the base circle is unknown or not known precisely, the actual tooth profile parameter is only a part of the involute, and the initial point of the involute is probably not included due to the reasons of processing technology and the like, so that the difference between the initial point and the standard involute is difficult to determine, therefore, other curves such as arc curves and the like are often used for replacing the involute to perform local tooth profile fitting, and the fitting regression residual error is evaluated, so that the processing quality of the gear is evaluated. 3) The tooth profile curve fitted by using the arc curve faces the problems of complex selection of arc parameters, possibly non-unique result, difficult optimization and the like. Disclosure of Invention The invention provides a modeling measurement positioning method for involute gear tooth profile parameters, aiming at the problems existing in measurement and characterization of involute gear tooth profile, which utilizes a gear tooth profile involute to approximate to parabolic shapes in a section applied by the tooth profile, so that during tooth profile characterization, a measurement waveform is divided into a left section and a right section, an optimal demarcation point between the two sections is determined by optimizing in a balanced mode of effective values of fitting residual errors, on the basis, two sections of curve waveforms at the left side and the right side of the optimal demarcation point are used for respectively carrying out local parabolic approximation to characterize an involute, regression residual error estimation is carried out in a local parabolic fitting mode, parametric characterization is carried out on the involute by approximate parameters of a double-parabolic model, and starting point parameters and ending point parameters of the fitted cu