Self-reference measuring and positioning method for tooth profile parameters of standard involute gear

Abstract

The invention discloses a self-reference measuring and positioning method for tooth profile parameters of a standard involute gear, and belongs to the field of gear parameter measurement. The implementation method comprises the following steps of: measuring the tooth profile of the involute gear by using a three-coordinate measuring machine to obtain involute coordinate measurement sequence points; a measurement curve involute of a gear tooth profile is approximate to a sine curve in shape in a tooth profile application interval section, so that when the tooth profile is represented, an optimal demarcation point is optimized and determined in a residual effective value balancing mode; a left curve waveform and a right curve waveform are used, local sine curve approximate characterization of an involute is carried out, regression residual estimation is carried out in a local sine curve fitting mode, approximate parameterization characterization is carried out on the involute through double-sine model parameters, and starting point parameters and ending point parameters of a fitting sine curve interval section are found out; and an involute interval boundary corresponding to the tooth profile of the gear is obtained, a complete tooth profile parameter measurement and characterization result with a self-reference characteristic is obtained, and self-reference measurement positioning is realized.

Description

Technical Field

[0001] This invention relates to a self-reference measurement and positioning method for standard involute gear tooth profile parameters, and particularly to a method for locating and quantitatively characterizing gear tooth profile parameters and their positions on a standard involute, belonging to the field of gear parameter measurement technology. Background Technology

[0002] Standard involute gears are the most widely used gears in practical engineering technology. The measurement and characterization of their gear tooth profile parameters are the most important means of evaluating gear quality, including tooth profile deviation, effective involute length, and usable involute length. They are usually measured using dedicated gear measuring centers, universal gear measuring machines, coordinate measuring machines, etc., and the quality of gear tooth profile parameters is characterized by the deviation and distribution between the measured results and the standard involute.

[0003] Under polar coordinate measurement and characterization, the current national standard GB / T 13924 "Detailed Rules for Accuracy Inspection of Involute Cylindrical Gears" stipulates that the measurement results use the various tooth profile cycles of the gear as the reference cycle and the gear axis as the reference axis. The tooth profile measurement data points are characterized by rotation angles and corresponding radii, with each tooth profile cycle corresponding to 0–360°. Under coordinate measuring machine (CMM) measurement and characterization, the measurement results are characterized entirely by the coordinates of each measurement point in a rectangular coordinate system. Subsequently, curve fitting is performed on the gear tooth profile measurement points to calculate and obtain the regression deviation.

[0004] The problems that still exist are:

[0005] 1) The standard involute equation is an overdetermined equation with complex functional relationships. It requires that the base circle radius and the starting point of the involute be known first, and then the coordinate relationship of other profile points on the involute can be determined based on the development angle θ of the involute generator. However, the base circle radius and the starting point of the involute may not be in the gear tooth profile measurement points, and the development angle θ of the involute generator is also an unknown quantity. In the usual involute measurement sequence represented by rectangular coordinates, there is a lack of reference values ​​such as the base circle radius and the starting point of the involute. This leads to problems in determining the involute representation reference point of the gear measurement results. Furthermore, it is difficult to directly find and determine the involute reference point and perform base circle radius measurement from the tooth profile measurement points.

[0006] 2) In practice, the base circle radius is often unknown or not precisely known, while the actual tooth profile parameters are only a part of the involute. Due to the machining process and other reasons, the starting point of the involute may not be included, making it difficult to determine the difference between it and the standard involute. Therefore, other curves such as circular arcs are often used to replace the involute for local tooth profile fitting, and the fitting regression residual is evaluated to assess the machining quality of the gear.

[0007] 3) Using circular arc curve fitting for tooth profile curves faces problems such as complex selection of circular arc parameters, possible non-unique results, and difficulty in finding the optimal solution. Summary of the Invention

[0008] To address the problems existing in the measurement and characterization of involute gear tooth profiles, this invention proposes a self-reference measurement and positioning method for standard involute gear tooth profile parameters. Utilizing the fact that the involute of the gear tooth profile approximates a sine curve shape within the applied tooth profile interval, the method divides the measured curve segment into two parts and performs sine fitting on each. The optimal boundary point is determined by balancing the effective residual values ​​of the two parts. Based on this, the left and right curve waveforms are used to approximate the involute with local sine curves. Regression residual estimation is performed using local sine curve fitting, and the involute is approximated and parameterized using dual sine model parameters. The starting and ending point parameters of the fitted sine curve interval are found, and the boundary of the involute interval corresponding to the gear tooth profile is obtained accordingly, thus achieving self-reference measurement and positioning of standard involute gear tooth profile parameters.

[0009] This invention is achieved through the following technical solution.

[0010] This invention discloses a self-reference measurement and positioning method for standard involute gear tooth profile parameters, comprising the following steps:

[0011] Step 1: Use a coordinate measuring machine to measure the tooth profile of the involute gear and obtain the coordinate measurement sequence points {[x]}. i ,y i ]}, (i=0,1,...,n-1), such as Figure 1 As shown, where x0,x1,...,x n-1 Let y0, y1, ..., y be the abscissas of the equally spaced sampling measurement points on the gear tooth profile. n-1 The ordinate of the sequence of coordinate points for sampling and measuring the gear tooth profile;

[0012] Step 2: Since the starting point of the involute intersects the base circle is the only minimum point in the initial segment of the involute, its first derivative is 0. The fitting sine curve used for involute waveform fitting also has a unique minimum point coinciding with the involute in this segment. This point serves as the optimal estimate of the starting point of the involute intersecting the base circle and is used as the measurement reference point for the involute measurement curve. Using this measurement reference point as a reference, the geometric position of the actual measured curve segment in the involute is uniquely determined by the length and slope of the straight line segment between the actual measured point and this reference point. This achieves sine model fitting and obtains the geometric position of the actual measured curve segment in the involute.

[0013] Step 3: Due to the morphological differences between the involute waveform and the sine curve waveform, when fitting the involute waveform with a sine curve, the effective value of the fitting residual will increase with the increase of the fitted sine curve segment. The involute waveform segment to be fitted is divided into two segments, a and b. The boundary point that makes the effective values ​​of the fitting residuals of segments a and b closest is found. The sine model fitting process described in Step 2 is performed on the waveform segments on both sides of the boundary point to obtain the corresponding fitting parameters, thereby realizing the double sine involute fitting. Based on the double sine involute fitting results, combined with Step 1, the geometric relative position of the measured involute segment PQ in the standard involute is determined, thereby realizing the self-reference measurement and positioning of the standard involute gear tooth profile parameters.

[0014] Furthermore, the initial phase of the sinusoidal fitted curve of segment a, divided by the optimal dividing point, characterizes the initial point position of the involute.

[0015] Furthermore, the terminal phase of the sinusoidal fitted curve of the b-segment curve divided by the optimal dividing point characterizes the termination point position of the involute.

[0016] Furthermore, the implementation method for step two is as follows:

[0017] like Figure 1 As shown, the base circle radius of the involute is r. b If the development angle of the involute generator BK is θ, then the parametric equation of the involute is:

[0018]

[0019] Where, r b Let θ be the radius of the base circle, and θ be the development angle of the generating line BK. The starting point D of the involute corresponds to θ. D =0; Point D is the starting point of the involute located on the radius of the base circle, (x D ,y D Let be the coordinates of point D, represented as coordinate point D(x). D ,y D In equation (1), x D =r b y D =0.

[0020] With a bit of D(x) D ,y D Let K(x,y) be the reference point, and let D(x,y) be the reference point. D ,y D The length r of the straight line segment KD and slope k KD They are respectively

[0021]

[0022] Each point K(x,y) on the involute is related to the reference point D(x). D ,y D The length r of the straight line segment KD and slope k KD The combinations are all unique at the reference point D(x). D ,y D When the x, y is known, it is used to determine the position of point K(x,y) in the involute.

[0023] The involute of a circle is an open curve with only a starting point and no ending point. The tooth profile used on a gear is only a small part of the initial stage in which the horizontal coordinate x and the vertical coordinate y change monotonically.

[0024] When the base circle radius is r b When θ∈[0,π / 2], the range of values ​​for the x-coordinate and y-coordinate is, x∈[r b ,r b ·π / 2],y∈[0,r b ];

[0025] Therefore, the measurement and characterization of the involute gear tooth profile only needs to be performed within the aforementioned interval θ∈[0,π / 2]. The involute portion outside this interval is generally not involved or used in gear manufacturing and applications. Thus, this invention is limited to the interval θ∈[0,π / 2], where the range of values ​​for the abscissa x and ordinate y is x∈[r... b ,r b ·π / 2],y∈[0,r b ].

[0026] Differentiating from equation (1) yields

[0027]

[0028] When θ∈[0,π / 2], we have x∈[r b ,r b ·π / 2],y∈[0,r b ], x(r b ,θ) and y(r b ,θ) are all function curves that are monotonically increasing within the interval.

[0029] The shape of the involute within the finite interval θ∈[0,π / 2] approximates the shape of the sine curve within the interval [-π / 2,0], thus incorporating the abscissa information of the geometric position of the measurement point. The phase value of the measurement point representing the fitted sine curve approximated by the involute is represented by the rectangular coordinates of the points on the involute as [x(r b ,θ),y(rb ,θ)).

[0030] Let x0, x1, ..., x n-1 Let y0, y1, ..., y be the abscissas of the equally spaced sampling measurement points of the gear tooth profile segment PQ. n-1 Let θ be the ordinate of the sampling measurement point on the gear tooth profile, and let θ be the corresponding involute generator unfolding angle. n-1 The corresponding phase angle of the fitted sine curve is Therefore, due to the selection and characterization of the measurement reference point location, the sampling measurement coordinate point [x] will be affected. i ,y i ] and the theoretical value point of the involute model [x(r b ,θ i ),y(r b ,θ i There exists a constant coordinate offset x between them. d and y d ,Right now

[0031]

[0032] The amplitude A, digital angular frequency ω, frequency f, and initial phase are obtained through a change search. Several fitting parameters for the DC component d are used to minimize the effective value ρ of the fitting residual as described in the following equation.

[0033]

[0034] Includes the x-coordinate information of the geometric position of the measurement point. for

[0035]

[0036] Δx i =x i -x i-1 (i = 0, 1, ..., n-1) (9)

[0037] Obtain the sampling measurement coordinates [x] i ,y i The fitted sine curve of the sequence is:

[0038]

[0039] At each measurement coordinate point, the fitted value of the ordinate is...

[0040]

[0041] Amplitude A, digital angular frequency ω, frequency f, initial phase The DC component d is used as an approximate fitting parameter for the involute. The phase value corresponding to the measurement point is the fitted sine curve representing the involute waveform containing the abscissa information of the geometric position of the measurement point.

[0042] Involute measurement starting point P, ordinate fitting regression value

[0043]

[0044] Involute measurement end point Q ordinate fitting regression value

[0045]

[0046] Regression residuals

[0047]

[0048] RMS value of fitted residual

[0049]

[0050] Phase value corresponding to the valley of the fitted sine curve The involute angle corresponding to θ = 0, i.e., the coordinates of the intersection point D of the involute and the base circle. have

[0051]

[0052] Then, any point K(x,y) on the involute and the reference point length of straight line segment and slope They are respectively

[0053]

[0054] Each point K(x,y) on the involute and the reference point length of straight line segment and slope The combinations are all unique, at the reference point. When known, we can find theoretically identical slope values ​​on the standard involute described in equation (1). The position of point K(x,y) can determine the geometric position of point K(x,y) in the involute.

[0055] When the measured involute curve starts at point P and is compared with the reference point length of straight line segment and slope and the involute measurement end point Q and reference point length of straight line segment and slope After all calculations are determined, theoretically identical slope values ​​are found on the standard involute described in equation (1). The location of the point can determine the geometrical relative position of the measured involute segment PQ in the standard involute.

[0056] Furthermore, the implementation method for step three is as follows:

[0057] Step 3.1: Optimize the boundary point q.

[0058] For the gear tooth profile segment PQ, the coordinate points (x0, y0),...,(x n-1 ,y n-1 Let q ∈ [10, n-10] be a positive integer index, and let a be the sequence of sampled measurement coordinate points: (x0, y0), ..., (x q-1 ,y q-1 Parts ) and b: (x q ,y q ),...,(x n-1 ,y n-1 Then, fit the waveform of the involute segment a in step two to obtain the fitting parameters A. aq f aq , d aq ρ aq The involute segment waveform in part b is fitted according to step two to obtain the fitting parameters A. bq f bq , d bq ρ bq .

[0059] Let the piecewise fitting begin with q = 10 to obtain the fitting parameters. Then, the q value is increased sequentially to perform fitting and obtain the fitting parameters again, until q = n - 10, thus completing the entire fitting process.

[0060] ρ aq The envelope of ρ increases monotonically with increasing q value. bq The envelope of ρ decreases monotonically with increasing q value. aq With ρ bq When the two values ​​are closest, q = q0 is taken as the optimal dividing point for the involute segment.

[0061] Step 3.2: Perform double sine fitting and achieve self-reference measurement and positioning of standard involute gear tooth profile parameters.

[0062] Using the optimal dividing point q0 as the boundary, the sampling measurement points are divided into two parts, a and b. Sine fitting is then performed on the involute segments on both sides of the dividing point to obtain the optimal fitting result, denoted as q0. Reference point The parameters are used to characterize the involute curve fitted using the double sine method. The fitted sine curve is obtained based on the fitted parameters.

[0063]

[0064] Then, the starting endpoint P(x0,y0) and the reference point are measured on the involute. Length of straight line segment and slope They are respectively

[0065]

[0066] Measuring the end point Q(x) on an involute n-1 ,y n-1 ) and reference point Length of straight line segment and slope They are respectively

[0067]

[0068]

[0069] exist After the calculation is determined, theoretically identical slope values ​​are found on the standard involute described in equation (1). The location of the reference point determines the geometrical relative position of the measured involute segment PQ within the standard involute, thereby achieving self-reference measurement and positioning of the standard involute gear tooth profile parameters. The reference point is...

[0070] Beneficial effects:

[0071] 1. This invention discloses a self-reference measurement and positioning method for standard involute gear tooth profile parameters. It utilizes an equally spaced sampling measurement sequence of the involute in rectangular coordinates, and uses a double sine curve with optimal features to fit the involute curve of the gear tooth profile. This yields an approximate representation of the involute expressed in sine parameters, as well as the initial phase angle of the fitted sine curve corresponding to the measurement starting point P. The value of the involute measurement at the endpoint Q corresponds to the fitted sinusoidal instantaneous phase angle. The value is given, along with the fitting residual sequence Δy between the involute gear tooth profile and the fitted sine curve. i (i = 0, 1, ..., n-1), and the effective value of the fitting residual between the involute profile of the gear tooth and the fitted sine curve. Used to quantitatively evaluate the quality of the involute gear tooth profile.

[0072] 2. This invention discloses a self-reference measurement and positioning method for standard involute gear tooth profile parameters, which utilizes the coordinates of the intersection point D of the involute and the base circle. The determination of the reference point and the arbitrary measurement point K(x,y) and the reference point length of straight line segment and slope The uniqueness of the combination can be used to determine the geometric position of any point K(x,y) in the standard involute. By determining the corresponding geometric position relationship between the measurement endpoints P and Q of the involute on the standard involute, the self-reference measurement and positioning of the tooth profile parameters of the standard involute gear can be realized.

[0073] 3. The self-reference measurement and positioning method for standard involute gear tooth profile parameters disclosed in this invention requires no prior knowledge and eliminates the need for complex and tedious selection, optimization, and judgment of fitting parameters, exhibiting excellent adaptive characteristics through automatic optimization. It possesses universality and versatility. Attached Figure Description

[0074] Figure 1 This is a schematic diagram illustrating the relationship between the involute measurement points described in this invention.

[0075] Figure 2 The involute curve of the gear tooth profile;

[0076] Figure 3 The changes in the effective values ​​of the fitting residuals for parts a and b;

[0077] Figure 4 The curve is a double sine fit curve for an involute.

[0078] Figure 5 The regression residual curve is an involute double sine fit.

[0079] Figure 6 The curve is a fitted curve of an involute single sine wave.

[0080] Figure 7 The regression residual curve is a single sinusoidal fit for an involute curve. Detailed Implementation

[0081] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0082] Example 1: Fitting an involute to a double sine wave

[0083] This embodiment discloses a self-reference measurement and positioning method for standard involute gear tooth profile parameters, and the specific implementation steps are as follows:

[0084] Step 1: Use a coordinate measuring machine to measure the tooth profile of the involute gear and obtain the coordinate measurement sequence points {[x]}. i ,y i]}, (i=0,1,...,n-1), such as Figure 2 As shown, where x0,x1,...,x n-1 Let y0, y1, ..., y be the abscissas of the equally spaced sampling measurement points on the gear tooth profile. n-1 Here are the ordinates of the sequence of coordinate points for sampling and measuring the gear tooth profile; x0 = 2.0 cm, x n-1 =3.1cm, y0=0.0cm, y n-1 = 3.1cm, given the base circle radius r b =2.0cm, x D =r b =2.0cm, y D =0cm. At the starting point of the involute, θ0 = 0 rad. At the ending point of the involute, θ n-1 = 2.984513 rad., the horizontal axis sampling interval is: Δx i =1.493003×10 -3 cm; Number of sampling points n = 2000.

[0085] Step 2: Since the starting point of the involute intersects the base circle is the only minimum point in the initial segment of the involute, its first derivative is 0. The fitted sine curve used for involute waveform fitting also has a unique minimum point coinciding with the involute in this segment. This minimum point serves as the optimal estimate of the starting point of the involute intersecting the base circle and is used as the measurement reference point for the involute measurement curve. Using this measurement reference point as a reference, the geometric position of the actual measured curve segment in the involute is uniquely determined by the length and slope of the straight line segment between the actual measured point and this reference point. This achieves sine model fitting and obtains the geometric position of the actual measured curve segment in the involute.

[0086] If the radius of the base circle of the involute is r b If the development angle of the involute generator BK is θ, then the parametric equation of the involute is:

[0087]

[0088] Where, r b Let θ be the radius of the base circle, and θ be the development angle of the generating line BK. The starting point D of the involute corresponds to θ. D =0; Point D is the starting point of the involute located on the radius of the base circle, (x D ,y D Let be the coordinates of point D, represented as coordinate point D(x). D ,y D In equation (1), x D =r b y D =0.

[0089] With a bit of D(x) D ,y D Let K(x,y) be the reference point, and let D(x,y) be the reference point. D ,y D The length r of the straight line segment KD and slope k KD They are respectively

[0090]

[0091] Each point K(x,y) on the involute is related to the reference point D(x). D ,y D The length r of the straight line segment KD and slope k KD The combinations are all unique at the reference point D(x). D ,y D When the x, y is known, it is used to determine the position of point K(x,y) in the involute.

[0092] The involute of a circle is an open curve with only a starting point and no ending point. The tooth profile used on a gear is only a small part of the initial stage in which the horizontal coordinate x and the vertical coordinate y change monotonically.

[0093] When the base circle radius is r b When θ∈[0,π / 2], the range of values ​​for the x-coordinate and y-coordinate is, x∈[r b ,r b ·π / 2],y∈[0,r b The method involved in this invention is limited to the interval θ∈[0,π / 2]. In this case, the range of values ​​for the horizontal coordinate x and the vertical coordinate y is x∈[r...]. b ,r b ·π / 2],y∈[0,r b From the differential of equation (1), we can obtain

[0094]

[0095] When θ∈[0,π / 2], we have x∈[r b ,r b ·π / 2],y∈[0,r b ], x(r b ,θ) and y(r b ,θ) are all function curves that are monotonically increasing within the interval.

[0096] The shape of the involute within the finite interval θ∈[0,π / 2] approximates the shape of the sine curve within the interval [-π / 2,0], thus incorporating the abscissa information of the geometric position of the measurement point. The phase value of the measurement point representing the fitted sine curve approximated by the involute is represented by the rectangular coordinates of the points on the involute as [x(r b ,θ),y(r b ,θ)).

[0097] Let the x-coordinates of the equally spaced sampling measurement points of the gear tooth profile segment PQ be x0, x1, ..., x n-1 The ordinates are y0, y1, ..., y n-1 The corresponding involute generators unfold at angles θ0, θ1, ..., θ n-1 The corresponding phase angle of the fitted sine curve is Therefore, due to the selection and characterization of the measurement reference point location, the sampling measurement coordinate point [x] will be affected. i ,y i ] and the theoretical value point of the involute model [x(r b ,θ i ),y(r b ,θ i There exists a constant coordinate offset x between them. d and y d ,Right now

[0098]

[0099] The amplitude A, digital angular frequency ω, frequency f, and initial phase are obtained through a change search. Several fitting parameters for the DC component d are used to minimize the effective value ρ of the fitting residual as described in the following equation.

[0100]

[0101] Includes the x-coordinate information of the geometric position of the measurement point. for

[0102]

[0103] Δx i =x i -x i-1 (i = 0, 1, ..., n-1) (9)

[0104] Obtain the sampling measurement coordinates [x] i ,y i The fitted sine curve of the sequence is:

[0105]

[0106] At each measurement coordinate point, the fitted value of the ordinate is...

[0107]

[0108] Amplitude A, digital angular frequency ω, frequency f, initial phase The DC component d is used as an approximate fitting parameter for the involute. The phase value corresponding to the measurement point is the fitted sine curve representing the involute waveform containing the abscissa information of the geometric position of the measurement point.

[0109] Involute measurement starting point P, ordinate fitting regression value

[0110]

[0111] Involute measurement end point Q ordinate fitting regression value

[0112]

[0113] Regression residuals

[0114]

[0115] RMS value of fitted residual

[0116]

[0117] Phase value corresponding to the valley of the fitted sine curve The involute angle corresponding to θ = 0, i.e., the coordinates of the intersection point D of the involute and the base circle. have

[0118]

[0119] Then, any point K(x,y) on the involute and the reference point length of straight line segment and slope They are respectively:

[0120]

[0121] Each point K(x,y) on the involute and the reference point length of straight line segment and slope The combinations are all unique, at the reference point. When known, we can find theoretically identical slope values ​​on the standard involute described in equation (1). The position of point K(x,y) can determine the geometric position of point K(x,y) in the involute.

[0122] When the measured involute curve starts at point P and is compared with the reference point length of straight line segment and slope and the involute measurement end point Q and reference point length of straight line segment and slope After all calculations are determined, theoretically identical slope values ​​are found on the standard involute described in equation (1). The location of the point can determine the geometrical relative position of the measured involute segment PQ in the standard involute.

[0123] Step 3: Due to the morphological differences between the involute waveform and the sine curve waveform, when fitting the involute waveform with a sine curve, the effective value of the fitting residual will increase with the increase of the fitted sine curve segment. The involute waveform segment to be fitted is divided into two segments, a and b. The boundary point that makes the effective values ​​of the fitting residuals of segments a and b closest is found. The sine model fitting process described in Step 2 is performed on the waveform segments on both sides of the boundary point to obtain the corresponding fitting parameters, thereby realizing the double sine involute fitting. Based on the double sine involute fitting results, combined with Step 1, the geometric relative position of the measured involute segment PQ in the standard involute is determined, thereby realizing the self-reference measurement and positioning of the standard involute gear tooth profile parameters.

[0124] Step 3.1: Optimize the boundary point q.

[0125] For the gear tooth profile segment PQ, the coordinate points (x0, y0),...,(x n-1 ,y n-1 Let q ∈ [10, n-10] be a positive integer index, and let a be the sequence of sampled measurement coordinate points: (x0, y0), ..., (x q-1 ,y q-1 Parts ) and b: (x q ,y q ),...,(x n-1 ,y n-1 The waveform of the involute segment a is fitted according to the method described in A above, and the fitting parameters A are obtained. aq f aq , d aq ρ aq Fit the involute segment waveform of part b as described in A above to obtain the fitting parameters A. bq f bq , d bq ρ bq .

[0126] Let the piecewise fitting begin with q = 10 to obtain the fitting parameters. Then, the q value is increased sequentially to perform fitting and obtain the fitting parameters again, until q = n - 10, thus completing the entire fitting process.

[0127] like Figure 3 As shown, ρ aq The envelope of ρ increases monotonically with increasing q value. bq The envelope of ρ decreases monotonically with increasing q value. aq With ρ bq When the two values ​​are closest, q = q0 is taken as the optimal dividing point for the involute segment.

[0128] Among them, the black curve represents the effective value ρ of the waveform fitting residual for part a. aq As the boundary point q changes, the red curve represents the effective value ρ of the waveform fitting residual for part b. bq The changes with the dividing point q. The red and black curves show a pattern of waxing and waning, and their values ​​converge near q = 1571, which is considered the optimal dividing point.

[0129] Step 3.2: Perform double sine fitting and achieve self-reference measurement and positioning of standard involute gear tooth profile parameters.

[0130] The optimal dividing point is selected as q0 = 1571. The measurement points for part a are: (x0, y0), ..., (x...). 1570 ,y 1570 Following the above process, a fitted sine curve waveform sequence y is obtained. i like Figure 4 The red part (a:y) i The blue curve y represents the original measured curve waveform. Its fitted residual waveform sequence Δy i like Figure 5 The red part shown (a:Δy) i The maximum residual was 0.0143 cm. The fitting parameters were:

[0131]

[0132] The measurement points in part b: (x 1571 ,y 1571 ),...,(x n-1 ,y n-1 Following the above process, a fitted sine curve waveform sequence y is obtained. i like Figure 4 The purple portion (b:y) i ), its fitted residual waveform sequence Δy i like Figure 5 The purple part shown (b:Δy) i The maximum residual was 0.0146 cm. The fitting parameters were:

[0133]

[0134] Depend on Figure 5 Analysis shows that the two have a fairly high fitting accuracy.

[0135] Reference point This serves as the parameter characterization result for fitting the involute using the double sine method.

[0136] The fitted sine curve is obtained based on the fitting parameters, such as... Figure 4 As shown, the fitted residual waveform sequence Δy i like Figure 5 As shown. Among them

[0137]

[0138] Then, the starting endpoint P(x0,y0) and the reference point are measured on the involute. Length of straight line segment and slope They are respectively

[0139]

[0140] Measuring the end point Q(x) on an involute n-1 ,y n-1 ) and reference point Length of straight line segment and slope They are respectively

[0141]

[0142] exist After the calculation is determined, theoretically identical slope values ​​are found on the standard involute described in equation (1). The location of the reference point can determine the geometrical relative position of the measured involute segment PQ within the standard involute, thereby achieving self-reference measurement and positioning of the standard involute gear tooth profile parameters. The reference point is...

[0143] Example 2: Fitting an involute to a single sine wave

[0144] The involute gear tooth profile is measured using a coordinate measuring machine or other measuring system to obtain the sequence of involute sampling measurement coordinate points {[x i ,y i ]},(i=0,...,n-1), where x0,x1,...,x n-1 Let y0, y1, ..., y be the abscissas of the equally spaced sampling measurement points on the gear tooth profile. n-1 The vertical coordinate of the sampling measurement point of the gear tooth profile is given.

[0145] Figure 2 The sequence of sampled measurement coordinate points {[x] is used to calculate the involute waveform of the gear tooth profile obtained using the involute formula. i ,y i ]}, (i=0,...,n-1), where the radius of the base circle is r b =2cm, θ is the development angle of the generating line BK, θ0 = 0 rad at the starting point of the involute, and θ at the ending point of the involute. n-1 = 2.984513 rad., the horizontal axis sampling interval is: Δx i =1.493003×10 -3 cm; Number of sampling points n = 2000;

[0146] For the involute sampling measurement coordinate point sequence {[x i ,y i ]}, (i = 0, ..., n-1), using a single sine curve waveform, fit it according to the process described in A above to obtain the fitted sine curve waveform sequence y. i like Figure 6 The red portion refers to the original measured curve waveform, where the black curve y represents the original measured curve waveform. Its fitted residual waveform sequence Δy... i like Figure 7 As shown, the maximum value of the fitting residual is 0.118 cm. The fitting parameters are:

[0147] A=26082.33cm, f=2.438691×10 -4 cm -1 ;

[0148] d=26082.36cm, ρ=2.53146×10 -2 cm;

[0149]

[0150] Depend on Figure 7 Analysis shows that the two are well-fitted.

[0151] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A self-reference measurement and positioning method for standard involute gear tooth profile parameters, characterized in that: Includes the following steps: Step 1: Use a coordinate measuring machine to measure the tooth profile of the involute gear and obtain the coordinate measurement sequence points {[x]}. i ,y i ]},(i=0,1,...,n-1), where x0,x1,...,x n-1 Let y0, y1, ..., y be the abscissas of the equally spaced sampling measurement points on the gear tooth profile. n-1 The ordinate of the sequence of coordinate points for sampling and measuring the gear tooth profile; Step 2: Since the starting point of the involute intersecting the base circle is the only minimum point in the initial segment of the involute, with a first derivative of 0, the fitting sine curve used for involute waveform fitting also has a unique minimum point coinciding with the involute in this segment. This point serves as the optimal estimation point for the starting point of the involute intersecting the base circle and is used as the measurement reference point for the involute measurement curve. Using the measurement reference point as a reference, the geometric position of the actual measured curve segment in the involute is uniquely determined by the length and slope of the straight line segment between the actual measured point and the reference point, thus achieving sine model fitting and obtaining the geometric position of the actual measured curve segment in the involute. Step 3: Due to the morphological differences between the involute waveform and the sine curve waveform, when fitting the involute waveform with a sine curve, the effective value of the fitting residual will increase with the increase of the fitted sine curve segment. Divide the fitted involute waveform segment into two segments, a and b, and find the boundary point that makes the effective values ​​of the fitting residuals of segments a and b closest. Perform the sine model fitting process described in Step 2 on the waveform segments on both sides of the boundary point to obtain the corresponding fitting parameters, thereby realizing the double sine involute fitting. Based on the double sine involute fitting results, combined with the geometric relative position of the measured involute segment PQ in the standard involute determined in Step 1, realize the self-reference measurement and positioning of the standard involute gear tooth profile parameters.

2. The method as described in claim 1, characterized in that: The initial phase of the sine-fitted curve of segment a, divided by the optimal dividing point, characterizes the initial point position of the involute.

3. The method as described in claim 2, characterized in that: Based on claim 1, the terminal phase of the sinusoidal fitted curve of the b-segment curve divided by the optimal dividing point characterizes the termination point position of the involute.

4. The method as described in claim 3, characterized in that: The implementation method for step two is as follows: The radius of the base circle of the involute is r. b If the development angle of the involute generator BK is θ, then the parametric equation of the involute is: Where, r b Let θ be the radius of the base circle, and θ be the development angle of the generating line BK. The starting point D of the involute corresponds to θ. D =0; Point D is the starting point of the involute located on the radius of the base circle, (x D ,y D Let be the coordinates of point D, represented as coordinate point D(x) D ,y D In equation (1), x D =r b y D =0; With a bit of D(x) D ,y D Let K(x,y) be the reference point, and let D(x,y) be the reference point. D ,y D The length r of the straight line segment KD and slope k KD They are respectively Each point K(x,y) on the involute is related to the reference point D(x). D ,y D The length r of the straight line segment KD and slope k KD The combinations are all unique at the reference point D(x). D ,y D When the x, y is known, it is used to determine the position of point K(x, y) in the involute. The involute of a circle is an open curve with only a starting point and no end point. The tooth profile used on a gear is only a part of the initial stage in which the horizontal coordinate x and the vertical coordinate y change monotonically. When the base circle radius is r b When θ∈[0,π / 2], the range of values ​​for the x-coordinate and y-coordinate is, x∈[r b ,r b ·π / 2],y∈[0,r b ]; The measurement and characterization of the involute gear tooth profile are restricted to the interval θ∈[0,π / 2]. In this case, the range of values ​​for the abscissa x and ordinate y is [r...]. b ,r b ·π / 2],y∈[0,r b ]; Differentiating from equation (1) yields When θ∈[0,π / 2], we have x∈[r b ,r b ·π / 2],y∈[0,r b ], x(r b ,θ) and y(r b ,θ) are all function curves that increase monotonically within the interval; The shape of the involute within the finite interval θ∈[0,π / 2] approximates the shape of the sine curve within the interval [-π / 2,0], thus incorporating the abscissa information of the geometric position of the measurement point. The phase value of the measurement point representing the fitted sine curve approximated by the involute is represented by the rectangular coordinates of the points on the involute as [x(r b ,θ),y(r b ,θ)]; Let x0, x1, ..., x n-1 Let y0, y1, ..., y be the abscissas of the equally spaced sampling measurement points of the gear tooth profile segment PQ. n-1 Let θ be the ordinate of the sampling measurement point on the gear tooth profile, and let θ be the corresponding involute generator development angle. n-1 The corresponding phase angle of the fitted sine curve is Therefore, due to the selection and characterization of the measurement reference point location, the sampling measurement coordinate point [x] will be affected. i ,y i ] and the theoretical value point of the involute model [x(r b ,θ i ),y(r b ,θ i There exists a constant coordinate offset x between them. d and y d ,Right now The amplitude A, digital angular frequency ω, frequency f, and initial phase are obtained through a change search. Several fitting parameters for the DC component d are used to minimize the effective value ρ of the fitting residual as described in the following equation. Includes the x-coordinate information of the geometric position of the measurement point. for Δx i =x i -x i-1 (i=0,1,...,n-1) (9) Obtain the sampling measurement coordinates [x] i ,y i The fitted sine curve of the sequence is: At each measurement coordinate point, the fitted value of the ordinate is... Amplitude A, digital angular frequency ω, frequency f, initial phase The DC component d is used as an approximate fitting parameter for the involute. The phase value corresponding to the measurement point of the fitted sine curve representing the involute waveform containing the abscissa information of the geometric position of the measurement point; Involute measurement starting point P, ordinate fitting regression value Involute measurement end point Q ordinate fitting regression value Regression residuals RMS value of fitted residual Phase value corresponding to the valley of the fitted sine curve The involute angle corresponding to θ = 0, i.e., the coordinates of the intersection point D of the involute and the base circle. have Then, any point K(x,y) on the involute and the reference point length of straight line segment and slope They are respectively Each point K(x,y) on the involute and the reference point length of straight line segment and slope The combinations are all unique, at the reference point. When known, by finding the standard involute with the same slope value as described in equation (1). The position of point K(x,y) is used to determine the geometric position of point K(x,y) in the involute. When the measured involute curve starts at point P and is compared with the reference point length of straight line segment and slope and the involute measurement end point Q and reference point length of straight line segment and slope After all calculations are determined, theoretically identical slope values ​​are found on the standard involute described in equation (1). The location of the point can determine the geometrical relative position of the measured involute segment PQ in the standard involute.

5. The method as described in claim 4, characterized in that: The implementation method for step three is as follows: Step 3.1: Optimize the boundary point q; For the gear tooth profile segment PQ, the coordinate points (x0, y0),...,(x n-1 ,y n-1 Let q ∈ [10, n-10] be a positive integer index, and let a be the sequence of sampled measurement coordinate points: (x0, y0), ..., (x q-1 ,y q-1 Parts ) and b: (x q ,y q ),...,(x n-1 ,y n-1 Then, fit the waveform of the involute segment a in step two to obtain the fitting parameters A. aq f aq , d aq ρ aq The waveform of the involute segment in part b is fitted according to step two to obtain the fitting parameters A. bq f bq , d bq ρ bq ; Let the piecewise fitting begin with q = 10 to obtain the fitting parameters. Then, the q value is increased sequentially and the fitting is performed again to obtain the fitting parameters. This process continues until q = n-10, thus completing the entire fitting process. ρ aq The envelope of ρ increases monotonically with increasing q value. bq The envelope of ρ decreases monotonically with increasing q value. aq With ρ bq When the two values ​​are closest, the value of q = q0 is taken as the optimal dividing point of the involute segment; Step 3.2: Perform double sine fitting and achieve self-reference measurement and positioning of standard involute gear tooth profile parameters; Using the optimal dividing point q0 as the boundary, the sampling measurement points are divided into two parts, a and b. Sine fitting is then performed on the involute segments on both sides of the dividing point to obtain the optimal fitting result, denoted as q0. Reference point The parameters are used to characterize the involute curve fitted by the double sine method; the fitted sine curve is obtained based on the fitting parameters; where... Then, the starting endpoint P(x0,y0) and the reference point are measured on the involute. Length of straight line segment and slope They are respectively Measuring the end point Q(x) on an involute n-1 ,y n-1 ) and reference point Length of straight line segment and slope They are respectively exist After the calculation is determined, the same slope value is found on the standard involute line described in equation (1). The location of the reference point determines the geometrical relative position of the measured involute segment PQ within the standard involute, thereby achieving self-reference measurement and positioning of the standard involute gear tooth profile parameters. The reference point is...

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