Modeling measuring and positioning method for standard involute gear tooth profile parameters

Abstract

The invention discloses a standard involute gear tooth profile parameter modeling measurement positioning method, and belongs to the field of gear parameter measurement. The implementation method comprises the steps that a gear tooth profile involute is similar to a parabola in shape in a tooth profile application interval section, so that when a tooth profile is represented, a measurement waveform is divided into a left section and a right section, and an optimal demarcation point between the two sections is optimized and determined through respective fitting residual error effective value balancing modes; respectively carrying out local parabola approximation to characterize an involute by using two sections of curve waveforms on the left side and the right side of the optimal demarcation point, carrying out regression residual estimation in a local parabola fitting mode, carrying out approximate parameterization characterization on the involute by using double-parabola model parameters, and finding out a starting point parameter and an ending point parameter of a fitting curve interval section; and representing an involute interval boundary corresponding to the tooth profile of the gear according to a modeling corresponding linear mapping relation between a fitting curve interval endpoint and an involute interval endpoint, so as to realize modeling measurement and positioning of standard involute gear tooth profile parameters.

Description

Technical Field

[0001] This invention relates to a model-based 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 model-based measurement and positioning method for standard involute gear tooth profile parameters. Utilizing the involute shape of the gear tooth profile, which approximates a parabola within the applied range, the measured waveform is divided into left and right segments during profile characterization. The optimal boundary point between the two segments is determined by balancing the effective values ​​of their respective fitting residuals. Based on this, the two curve waveforms on either side of the optimal boundary point are used to perform local parabolic approximations to characterize the involute. Regression residual estimation is performed using a local parabolic fitting method, and the involute is approximated and parameterized using double parabolic model parameters. The starting and ending point parameters of the fitted curve range are then identified. The model-based linear mapping relationship between the endpoints of the fitted curve range and the endpoints of the involute range characterizes the boundary of the corresponding involute range, thus achieving the model-based measurement and positioning of standard involute gear tooth profile parameters.

[0009] The objective of this invention is achieved through the following technical solution:

[0010] This invention discloses a model-based measurement and positioning method for the tooth profile parameters of standard involute gears. Since the starting point of the involute intersects the base circle, it is the only minimum point in the initial segment of the involute, with a first derivative of 0. The fitting parabola used for involute waveform fitting also has a unique minimum point coinciding with the involute in this segment. This minimum point can be used as the optimal estimation point for the starting point of the involute intersecting the base circle, and as a reference point for the measurement and characterization of the involute measurement curve. Using this reference, the geometric position of the actual measured curve segment within the involute is uniquely determined by the length and slope of the straight line segment between the actual measured point and this reference point, thus obtaining complete measurement, positioning, and characterization results of the tooth profile parameters with model-based features. This invention provides a foundation for the evaluation and quality assessment of the tooth profile parameters of standard involute gears.

[0011] This invention discloses a modeling measurement and positioning method for standard involute gear tooth profile parameters, comprising the following steps:

[0012] Step 1: Divide the measured waveform into two segments, left and right, and fit them separately. By balancing the effective values ​​of the fitting residuals of each segment, find the optimal dividing point between the two segments.

[0013] For example Figure 1 The sequence of sampled measurement coordinate points (x0, y0),...,(x0, y0) of the gear tooth profile segment PQ shown. n-1 ,y n-1), select positive integer indices q∈[10,n-10], and divide the sampled measurement coordinate point sequence into α parts: (x0,y0),...,(x q-1 ,y q-1 ) and β part: (x q ,y q ),...,(x n-1 ,y n-1 The waveform of the involute segment α is fitted to obtain the fitting parameters a. α b α c α ρ α y gα x gα The waveform of the involute segment in the β region is fitted to obtain the fitting parameter a. β b β c β ρ β y gβ x gβ ;

[0014] Starting with q = 10, piecewise fitting is performed to obtain fitting parameters. Subsequently, the q value is sequentially increased, and fitting is repeated to obtain fitting parameters, continuing until q = n-10, thus completing the entire fitting process. The effective value of the fitting residual ρ is then calculated. α ρ β The variation of q is plotted onto the same curve, such as... Figure 3 As shown; according to this curve, ρ α The envelope of ρ increases monotonically with increasing q value. β The envelope of ρ decreases monotonically with increasing q value. α With ρ β When the two values ​​are closest, q = q0 is taken as the optimal dividing point for the involute segment.

[0015] Step 2: Using q0 as the optimal dividing point, divide the measured curve segment into two segments, α and β. Perform parabolic fitting on the involute segments on both sides of the optimal dividing point to obtain the best fitting result, denoted as α / β. As a parameter characterization result of fitting the involute using the double parabolic method; the degree of agreement between the fitted curve and the measured waveform is as follows: Figure 4 As shown.

[0016] Based on the fitting parameters, the fitted parabola is obtained, and the fitting residual sequence Δy between the involute tooth profile segments α and β and the fitted parabola is calculated respectively. i , (i=0,1,...,n-1), such as Figure 5 As shown;

[0017] The valley value fitted by the left α segment and its location The parameters serve as the position coordinates of the starting point (θ = 0°) of the involute. Determine the starting reference point D for fitting the involute.

[0018] Step 3: Based on the measured involute segment PQ at the initial endpoint P(x0,y0) and the reference point... The relative position of the coordinates enables the modeling measurement and positioning of the standard involute gear tooth profile parameters.

[0019] Furthermore, the initial endpoints of the α-segment waveform fitting curve are used to characterize the initial point position of the involute.

[0020] Furthermore, the endpoint of the fitted curve of the β segment waveform is used to characterize the termination point of the involute.

[0021] Furthermore, the method for implementing step two is as follows:

[0022] 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:

[0023]

[0024] 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 D be the coordinates of point D, represented as: coordinates of point D(x) D ,y D In equation (1), it is obvious that x D =r b y D =0;

[0025] 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:

[0026]

[0027] 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 kKD 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.

[0028] The involute of a circle is an open curve with only a starting point and no end point. The tooth profile used in gears is only a part of the initial stage in which the horizontal coordinate x and the vertical coordinate y change monotonically.

[0029] When the base circle radius is r b When the expansion angle θ∈[0,π / 2], the range of values ​​for the abscissa x and ordinate y in the rectangular coordinate system is, x∈[r b ,r b ·π / 2],y∈[0,r b ];

[0030] Therefore, the measurement and characterization of the involute tooth profile used in gears only needs to be performed on the interval θ∈[0,π / 2]. This invention limits the interval θ∈[0,π / 2]. In this case, x∈[r b ,r b ·π / 2],y∈[0,r b From the differential of equation (1), we can obtain

[0031]

[0032] 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;

[0033] From the shape of the involute within the finite interval θ∈[0,π / 2] and the equation described in (1), it can be seen that the ordinate of the involute is a concave function with a single peak relative to the abscissa, and the only "valley" value appears at the boundary θ=0. The coordinate value of this point is denoted as (x g ,y g );

[0034] Within the interval θ∈[0,π / 2], the shape of the involute is approximately the same as that of a parabola. The sampling and measurement coordinates of the involute in the rectangular coordinate system XOY are (x,y)=[x(r)]. b ,θ),y(r b Using the parabola y(x) to fit the involute, and assuming that both have a common "valley" point, the function expression of its least squares fitted curve is:

[0035]

[0036] Where a, b, and c are three fitting parameters;

[0037] Let the x-coordinates of the sampling and 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 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

[0038]

[0039] The effective value of the fitting residual is:

[0040]

[0041] The estimated value of the "valley" of the fitted waveform is then:

[0042]

[0043] The locations where the "valley" values ​​of the fitted waveform appear are:

[0044]

[0045] The goodness or badness of the fit is judged by the effective value ρ of the fitting residual, and the peak value obtained is judged by this.

[0046] The fitting process is as follows:

[0047] For the sampling measurement coordinate point [x i ,y i The sequence (i = 0, 1, ..., n-1) is given by equation (5):

[0048]

[0049] When ε reaches its minimum value, we have:

[0050]

[0051]

[0052] Solving this system of linear equations yields fitting parameters a, b, and c, which serve as approximate parameters representing the parabolic form of the fitted involute. The corresponding y is then calculated according to equations (8) and (9). g x g The effective value of its fitting residual ρ is calculated according to equation (7);

[0053] The regression residual sequence of the fitted curve is

[0054]

[0055] The valley point of the fitted parabola is used as the coordinate estimate of the intersection point D of the involute and the base circle, denoted as D(x). g ,y g It can be used as a reference coordinate point for involute measurement fitting and positioning;

[0056] Then, any point K(x,y) on the involute and the reference point D(x) g ,y g The length of the straight segment and slope They are respectively:

[0057]

[0058] Each point K(x,y) on the involute is related to the reference point D(x). g ,y g The length of the straight segment and slope The combinations are all unique at the reference point D(x). g ,y g When the slope is known, find the theoretically identical slope value 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.

[0059] Furthermore, the method for implementing step three is as follows:

[0060] Measuring the initial endpoint P(x0,y0) and reference point on the involute. Length of straight line segment and slope They are respectively:

[0061]

[0062] 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:

[0063]

[0064] Calculate according to formulas (16) to (19) Then, by searching for theoretically identical slope values ​​on the standard involute... The location of the point determines the geometric relative position of the measured involute segment PQ in the standard involute, thus realizing the model-based self-reference measurement and positioning of the tooth profile parameters of the standard involute gear.

[0065] Beneficial effects:

[0066] 1. This invention discloses a modeling measurement and positioning method for standard involute gear tooth profile parameters. It utilizes a sampling measurement sequence of the involute in rectangular coordinates, and uses a double parabola with optimal features to fit the involute curve of the gear tooth profile, obtaining an approximate representation of the involute using parabolic parameters. It also obtains the parabolic coordinate position values ​​corresponding to the starting and ending points of the involute. Furthermore, it provides the fitting residual sequence Δy between the gear tooth profile involute and the fitted curve. i , (i=0,1,...,n-1), and the effective value ρ of the fitting residual between the involute profile of the gear and the fitted curve. α With ρ β It is used to quantitatively evaluate the quality of the involute gear tooth profile.

[0067] 2. This invention discloses a model-based measurement and positioning method for the tooth profile parameters of a standard involute gear, utilizing 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 model-based measurement and positioning of the tooth profile parameters of the standard involute gear can be realized.

[0068] 3. The present invention discloses a model-based measurement and positioning method for standard involute gear tooth profile parameters, which does not require direct measurement of the base circle radius or the development angle of the involute development line. It is applicable to the parameter measurement and evaluation of any involute within the interval θ∈[0,π / 2], and has universality and versatility. Attached Figure Description

[0069] Figure 1 This is a schematic diagram showing the relationship between the measurement points of an involute curve.

[0070] Figure 2The waveform sequence for measuring involutes.

[0071] Figure 3 The effective values ​​of the fitting residuals for the α and β parts vary with the position of the boundary point q.

[0072] Figure 4 This is the fitted curve of an involute double parabola.

[0073] Figure 5 The regression residual curve is fitted to an involute double parabola.

[0074] Figure 6 This is the fitted curve for an involute single parabola.

[0075] Figure 7 The regression residual curve is fitted to an involute single parabola. Detailed Implementation

[0076] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0077] Example 1: Double Parabolic Fitting Method

[0078] This embodiment discloses a modeling measurement and positioning method for standard involute gear tooth profile parameters, the specific implementation steps of which are as follows:

[0079] A. Parabolic model fitting.

[0080] like Figure 1 As shown, with O as the center and radius r b The circle is the base circle of the involute, the straight line BK is the involute generating line, OB⊥BK, the angle θ between OB and the horizontal axis is the involute generating line development angle, as the angle θ increases from 0, the trajectory of point K is the involute, the intersection point D of the base circle and the horizontal axis X is the starting point of the involute, point P is the starting endpoint of the actual gear involute segment, point Q is the ending endpoint of the actual gear involute segment, and the involute sampling measurement coordinate point sequence {[x i ,y i Let} be the sequence of measured values ​​for line segment PQ (i = 0, 1, ..., n-1). Where the base circle radius 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;

[0081] 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:

[0082]

[0083] 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 D be the coordinates of point D, represented as: coordinates of point D(x) D ,y D In equation (1), it is obvious that x D =r b y D =0.

[0084] 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:

[0085]

[0086] 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 can be used to determine the position of point K(x,y) in the involute.

[0087] The involute of a circle is an open curve with only a starting point and no ending point. Usually, the tooth profile used for gears is only a small part of the initial stage in which the horizontal coordinate x and the vertical coordinate y change monotonically.

[0088] When the base circle radius is r b When the expansion angle θ∈[0,π / 2], the range of values ​​for the abscissa x and ordinate y in the rectangular coordinate system is, x∈[r b ,r b ·π / 2],y∈[0,r b ];

[0089] Therefore, the measurement and characterization of the involute tooth profile used in gears only needs to be performed on the interval θ∈[0,π / 2]. With the interval θ∈[0,π / 2] restricted, x∈[r] b ,rb ·π / 2],y∈[0,r b From the differential of equation (1), we can obtain

[0090]

[0091] 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.

[0092] From the shape of the involute within the finite interval θ∈[0,π / 2] and the equation described in (1), it can be seen that the ordinate of the involute is a concave function with a single peak relative to the abscissa, and the only "valley" value appears at the boundary θ=0. The coordinate value of this point is denoted as (x g ,y g ).

[0093] Within the interval θ∈[0,π / 2], the shape of the involute is approximately the same as that of a parabola. The sampling and measurement coordinates of the involute in the rectangular coordinate system XOY are (x,y)=[x(r)]. b ,θ),y(r b Let ,θ)] be the parabola y(x) fitted to the involute, and assume that both have a common "valley" point. Then the function expression of its least squares fitted curve is:

[0094]

[0095] Where a, b, and c are three fitting parameters.

[0096] Let the x-coordinates of the sampling and measurement points of the gear tooth profile segment 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 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

[0097]

[0098] The effective value of the fitting residual is:

[0099]

[0100] The estimated value of the "valley" of the fitted waveform can then be obtained as follows:

[0101]

[0102] The locations where the "valley" values ​​of the fitted waveform appear are:

[0103]

[0104] The goodness or badness of the fit is judged by the effective value ρ of the fitting residual, and the peak value obtained is judged by this.

[0105] The fitting process is as follows:

[0106] For the sampling measurement coordinate point [x i ,y i The sequence (i = 0, 1, ..., n-1) is given by equation (5):

[0107]

[0108] When ε reaches its minimum value, we have:

[0109]

[0110] Solving this system of linear equations yields fitting parameters a, b, and c, which serve as approximate parameters representing the parabolic form of the fitted involute. The corresponding y is then calculated according to equations (8) and (9). g x g The effective value of the fitting residual ρ is calculated according to equation (7).

[0111] The regression residual sequence of the fitted curve is

[0112]

[0113] The valley point of the fitted parabola is used as the coordinate estimate of the intersection point D of the involute and the base circle, denoted as D(x). g ,y g It can be used as a reference coordinate point for involute measurement fitting and positioning.

[0114] Then, any point K(x,y) on the involute and the reference point D(x) g ,y g The length of the straight segment and slope They are respectively:

[0115]

[0116] Each point K(x,y) on the involute is related to the reference point D(x). g ,y g The length of the straight segment and slope The combinations are all unique at the reference point D(x). g ,y g When the slope is known, find the theoretically identical slope value 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.

[0117] B. Double parabola fitting process

[0118] Because of the morphological differences between involute and parabolic waveforms, when fitting an involute waveform to a parabola, the effective value of the fitting residual tends to increase with the increase of the fitted curve segment. The fitted involute waveform segment is divided into two segments, α and β. The optimal boundary point is found that makes the effective values ​​of the fitting residuals of the two segments closest. Then, the fitting process described in A. above is performed on the waveform segments on both sides of the optimal boundary point to obtain the fitting parameters, thus completing the involute fitting using the double parabola method.

[0119] B.1 Boundary Point Optimization

[0120] For the above-mentioned gear tooth profile segment PQ, the sampled measurement coordinate point sequence (x0, y0),...,(x n-1 ,y n-1 ), select positive integer indices q∈[10,n-10], and divide the sampled measurement coordinate point sequence into α parts: (x0,y0),...,(x q-1 ,y q-1 ) and β part: (x q ,y q ),...,(x n-1 ,y n-1 The involute segment waveform of part α is fitted using the method described in A above to obtain the fitting parameters a. α b α c α ρ α y gα x gα The waveform of the involute segment in the β part is fitted according to the method described in A above to obtain the fitting parameter a. β b β c β ρ β y gβ x gβ .

[0121] Let's start piecewise fitting from q=10 to obtain the fitting parameters. Then, we sequentially increase the q value and perform fitting again, obtaining the fitting parameters each time, until q=n-10, completing the entire fitting process. The effective value ρ of the fitting residuals... α ρ β The variation of q is plotted onto the same curve, such as... Figure 3 As shown.

[0122] from Figure 3 As can be seen from ρ α The envelope of ρ increases monotonically with increasing q value. β The envelope of ρ decreases monotonically with increasing q value. α With ρ β When the two values ​​are closest, q = q0 is taken as the optimal dividing point for the involute segment.

[0123] B.2 Double Parabolic Fitting

[0124] Using q0 as the optimal dividing point, the measured curve segment is divided into two segments, α and β. Parabolic fitting is then performed on the involute segments on both sides of the optimal dividing point to obtain the optimal fitting result, denoted as . This serves as the parameter characterization result for fitting an involute using the double parabola method.

[0125] The fitted parabola is obtained based on the fitting parameters. The fitting residual sequence Δy between the involute tooth profiles of the two gear segments α and β and the fitted parabola is calculated according to Equation (13). i ,(i=0,1,...,n-1).

[0126] Valley values ​​fitted from the left α segment and its location The parameters serve as the position coordinates of the starting point (θ = 0°) of the involute. The starting reference point D for fitting the involute is determined and used to determine and characterize the positions of other points in the involute.

[0127] Based on actual measurements, the involute segment PQ at its initial endpoint P(x0,y0) and reference point... The relative position of the coordinates completes the modeling and self-reference measurement positioning of the standard involute gear tooth profile parameters.

[0128] Measuring the initial endpoint P(x0,y0) and reference point on the involute. Length of straight line segment and slope They are respectively:

[0129]

[0130] Measuring the end point Q(x) on an involuten-1 ,y n-1 ) and reference point Length of straight line segment and slope They are respectively:

[0131]

[0132] In particular, the results obtained by calculation according to formulas (16) to (19) Then, by searching along the standard involute line described in equation (1), we find the theoretically identical slope values. The location of the point determines the geometrical relative position of the measured involute segment PQ within the standard involute. The reference point is...

[0133] Figure 2 The sequence of sampled measurement coordinate points for the involute waveform curve obtained by calculating using the involute formula {[x i ,y i ]},(i=0,...,n-1), where the base circle radius 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;

[0134] Optimization of the dividing point:

[0135] Selecting positive integer indices q∈[10,n-10] as the splitting point indices, the involute sampling measurement coordinate point sequence is divided into two parts, α and β. Using the method described in section B above, the respective fitted residual effective value sequences ρ are obtained. α ρ β like Figure 3 As shown in the figure. The black curve represents the effective value ρ of the waveform fitting residual for the α portion. α As the boundary point q changes, the red curve represents the effective value ρ of the waveform fitting residual for the β portion. β How it changes with the boundary point q. (From...) Figure 3 It can be seen that the changing pattern of the red and black curve envelopes is consistent with the analysis in B. Furthermore, the two curves intersect with similar values ​​near q = 1571, which is determined to be the optimal dividing point.

[0136] Double parabola fitting:

[0137] Let the optimal boundary point q0 = 1571, and the α-part sampled measurement coordinate point sequence be: (x0, y0),...,(x 1570 ,y1570 Following the process described in section A above, a fitting curve waveform sequence y is obtained. i like Figure 4 The red part (α:y) i The blue curve y represents the original measured curve waveform. Its fitted residual waveform sequence Δy i like Figure 5 The red portion shown is (α:Δy) i The maximum residual was 0.0144 cm. The fitting parameters were:

[0138]

[0139] The sequence of sampled coordinate points for the β portion: (x 1571 ,y 1571 ),...,(x n-1 ,y n-1 Following the process described in section A above, a fitting curve waveform sequence y is obtained. i like Figure 4 The purple portion (β:y) i ), its fitted residual waveform sequence Δy i like Figure 5 The purple portion shown (β:Δy) i The maximum residual was 0.0146 cm. The fitting parameters were:

[0140]

[0141] Depend on Figure 5 Analysis shows that the two have a very high fitting accuracy.

[0142] Example 2: Single Parabolic Fitting Method

[0143] Figure 2 The sequence of sampled measurement coordinate points {[x] is used to calculate the involute waveform curve 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;

[0144] For this set of examples, if a single parabolic waveform is used for fitting according to the process described in A above, the fitted curve waveform sequence y is obtained. 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:

[0145] a=1.208445; b=-4.980468;

[0146] c=5.162574; ρ=2.537980×10 -2 cm;

[0147] y g =3.096793×10 -2 cm; x g = 2.060692cm;

[0148] Depend on Figure 7 Analysis shows that the two have high fitting accuracy.

[0149] 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 model-based measurement and positioning method for standard involute gear tooth profile parameters, characterized in that: Includes the following steps, Step 1: Divide the measured waveform into two segments, left and right, and fit them separately. By balancing the effective values ​​of the fitting residuals of each segment, find the optimal dividing point between the two segments. The sequence of sampled measurement coordinate points (x0, y0),...,(x0, y0) for gear tooth profile segment PQ n-1 ,y n-1 ), select positive integer indices q∈[10,n-10], and divide the sampled measurement coordinate point sequence into α parts: (x0,y0),...,(x q-1 ,y q-1 ) and β part: (x q ,y q ),...,(x n-1 ,y n-1 Parabolic fitting is performed on the involute segment waveform of part α to obtain the fitting parameters a. α b α c α ρ α y gα x gα Parabolic fitting is performed on the waveform of the involute segment in the β region to obtain the fitting parameters a. β b β c β ρ β y gβ x gβ ; Starting with q = 10, piecewise parabolic fitting is performed to obtain fitting parameters. Subsequently, the q value is sequentially increased, and fitting is repeated to obtain fitting parameters, continuing until q = n-10, thus completing the entire fitting process. The effective value of the fitting residual ρ is then calculated. α ρ β Plot the variation of ρ with q onto the same curve; based on this curve, ρ α The envelope of ρ increases monotonically with increasing q value. β The envelope of ρ decreases monotonically with increasing q value. α With ρ β When the two values ​​are closest, the value of q = q0 is taken as the optimal dividing point of the involute segment; Step 2: Using q0 as the optimal dividing point, divide the measured curve segment into two segments, α and β. Perform parabolic fitting on the involute segments on both sides of the optimal dividing point to obtain the best fitting result, denoted as α / β. As a parameter characterization result of fitting an involute using the double parabola method; Based on the fitting parameters, the fitted parabola is obtained, and the fitting residual sequence Δy between the involute tooth profile segments α and β and the fitted parabola is calculated respectively. i (i = 0, 1, ..., n-1); Valley values ​​fitted from the left α segment and its location The parameters serve as the position coordinates of the starting point (θ = 0°) of the involute. Determine the starting reference point D for fitting the involute; Step 3: Based on the measured involute segment PQ at the initial endpoint P(x0,y0) and the reference point... The relative position of the coordinates enables the modeling and self-reference measurement positioning of the standard involute gear tooth profile parameters.

2. The method as described in claim 1, characterized in that: The initial endpoints of the curve fitted by the α-segment waveform are used to characterize the initial point position of the involute.

3. The method as described in claim 2, characterized in that: The endpoint of the fitted curve of the β segment waveform is used to characterize the termination point of the involute.

4. The method as described in claim 3, characterized in that: The second step is implemented as follows: 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: 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 D be the coordinates of point D, represented as: coordinates of point D(x) D ,y D In equation (1), it is obvious that 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 in gears 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 the expansion angle θ∈[0,π / 2], the range of values ​​for the abscissa x and ordinate y in the rectangular coordinate system is, x∈[r b ,r b ·π / 2],y∈[0,r b ]; The measurement and characterization of the involute tooth profile used in gears, with the restriction interval θ∈[0,π / 2], where x∈[r b ,r b ·π / 2],y∈[0,r b From the differential of equation (1), we can obtain 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; Analysis of the involute shape within the finite interval θ∈[0,π / 2] and the equation described in (1) shows that the ordinate of the involute is a concave function with a single peak relative to the abscissa, and the only "valley" value appears at the boundary θ=0. The coordinate value of this point is denoted as (x g ,y g ); Within the interval θ∈[0,π / 2], the shape of the involute is approximately the same as that of a parabola. The sampling and measurement coordinates of the involute in the rectangular coordinate system XOY are (x,y)=[x(r)]. b ,θ),y(r b Given an involute curve fitted with a parabola y(x), and a common valley point, the least squares fitted curve's functional expression is: Where a, b, and c are three fitting parameters; Let the abscissas of the 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 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 effective value of the fitting residual is: The estimated value of the "valley" of the fitted waveform is then: The locations where the "valley" values ​​of the fitted waveform appear are: The goodness or badness of the fit is judged by the effective value ρ of the fitting residual, and the peak value obtained is judged by this. The fitting process is as follows: For the sampling measurement coordinate point [x i ,y i The sequence (i = 0, 1, ..., n-1) is given by equation (5): When ε reaches its minimum value, we have: Solving this system of linear equations yields fitting parameters a, b, and c, which serve as approximate parameters representing the parabolic form of the fitted involute. The corresponding y is then calculated according to equations (8) and (9). g x g The effective value of its fitting residual ρ is calculated according to equation (7); The regression residual sequence of the fitted curve is The valley point of the fitted parabola is used as the coordinate estimate of the intersection point D of the involute and the base circle, denoted as D(x). g ,y g It can be used as a reference coordinate point for involute measurement fitting and positioning; Then, any point K(x,y) on the involute and the reference point D(x) g ,y g The length of the straight segment and slope They are respectively: Each point K(x,y) on the involute is related to the reference point D(x). g ,y g The length of the straight segment and slope The combinations are all unique at the reference point D(x). g ,y g When the slope is known, find the theoretically identical slope value 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.

5. The method as described in claim 4, characterized in that: The method for implementing step three is as follows: Measuring the initial endpoint P(x0,y0) and reference point 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: Calculate according to formulas (16) to (19) Then, by finding the same slope value on the standard involute... The location of the point determines the geometric relative position of the measured involute segment PQ in the standard involute, thus realizing the model-based measurement and positioning of the tooth profile parameters of the standard involute gear.

Images