Double-variable-element fitting method for standard involute template contour

Abstract

The invention relates to a dual-variable fitting method for a standard involute template contour, and belongs to the technical field of gear parameter measurement. According to the method, the characteristic that the contour of a standard involute sample plate is similar to the shape of a parabola is utilized, a waveform segment is divided into an optimized segment alpha and an optimized segment beta in a rectangular coordinate system xoy, waveform fitting is carried out on the segment alpha in a parabola x = g (y) mode, and a fitting residual error effective value rho alpha and fitting parameters are obtained; performing waveform fitting on the beta section in a parabola y = f (x) mode to obtain a fitting residual error effective value rho beta and a fitting parameter; and double-variable method fitting is completed. The method does not need to directly measure the radius of the base circle, does not need to measure the expansion angle of the expansion line of the involute, can be suitable for parameter measurement and evaluation of any involute in the interval section of theta belonging to [0, pi], and has universality.

Description

Technical Field

[0001] This invention relates to a two-variable fitting method for the profile of a standard involute template, and particularly to waveform fitting of the profile curves of cams, splines, gear tooth profiles, and standard involute templates with standard involute morphology, as well as the parametric characterization of their profile curves, belonging to the field of gear parameter measurement technology. Background Technology

[0002] Due to its excellent and tight mechanical meshing characteristics, the standard involute curve has become the most widely used curve waveform in the gear industry. It is widely used in gear tooth profiles, spline tooth profiles, and cam profiles, and is also used to create standard involute templates for representing gear tooth profiles. Typically, when used for gear tooth profiles, the involute length is relatively short, while when used for cam profiles, the involute length involved is much longer. In these cases, curve fitting becomes more difficult, mainly because it is hard to find a suitable fitting curve to achieve high fitting accuracy.

[0003] Standard involute gears are the most widely used gears in practical engineering technology. Measuring and characterizing their tooth profile parameters is the most important means of evaluating gear quality. These parameters include tooth profile deviation, effective involute length, and usable involute length. They are typically measured using dedicated gear measuring centers, universal gear measuring machines, and coordinate measuring machines. The quality of the gear tooth profile parameters is characterized by the deviation and distribution between the measured results and the standard involute. The same applies to the measurement and characterization of other involute shapes such as cams, splines, and standard involute templates.

[0004] In the case of measurement and characterization using a coordinate measuring machine, the measurement results of the involute template are represented entirely by the coordinates of each measurement point in a rectangular coordinate system. Subsequently, curve fitting is performed on the measurement sequence composed of each measurement point, and the regression deviation is calculated. The parameters of the fitted curve are then used as the characterization parameters for the approximate representation of the involute.

[0005] The main problems that still exist are:

[0006] 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.

[0007] 2) In practice, the base circle radius needs to be measured and given separately. When only the involute curve is available, the base circle radius is often unknown or not precisely known. The actual tooth profile involute template parameters are only a part of the involute curve, and due to processing technology and other reasons, the starting point of the involute curve may not be included, making it difficult to determine its difference from the standard involute curve. Therefore, other curves such as circular arc curves are often used to replace the involute curve for local curve fitting, and the fitting regression residual is evaluated to assess the processing quality.

[0008] 3) Using circular arcs to fit the involute template curve of the tooth profile faces problems such as complex selection of circular arc parameters, possible non-unique results, and difficulty in optimization; moreover, the involute is a gradual curve with different radii of curvature at each point, and fitting it with a circular arc with the same radii of curvature at each point itself will inevitably result in fitting error.

[0009] 4) The curve represented by the standard involute equation has a small single-value interval relative to the x-coordinate of the independent variable in the rectangular coordinate system. A wider range of involute curves faces the challenge of fitting non-single-value intervals. Summary of the Invention

[0010] The purpose of this invention is to address the challenge of high-precision curve fitting for a wide range of involute template contours by providing a dual-variable fitting method for standard involute template contours.

[0011] A two-variable fitting method for a standard involute template profile includes the following steps:

[0012] Step 1: Apply coordinate transformation to the classic involute curve, rotating it 90° clockwise to represent it in rectangular coordinates xoy, where x is the abscissa and y is the ordinate; (x, y) represents the coordinates of the measurement point on the involute; the involute waveform sampling measurement coordinate point sequence {[x i ,y i ]},(i=0,...,n-1); Select positive integer index q as the split point index to 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 );

[0013] Step 2: For the involute segment waveform of part α, use the independent variable as the ordinate sequence value {y}. i The dependent variable is the horizontal coordinate value x. A fitting curve waveform sequence value {x} is obtained. i} and fitting parameter a α bα c α ρ α y gα x gα ; Calculate the ordinate of the fitted residual sequence Δy based on the "vertical" regression residual sequence of the fitted curve. i (i = 0, 1, ..., q-1); At this time, for the involute segment waveform of the β part, the independent variable is the sequence value {x} of the abscissa. i The dependent variable is the ordinate value y. A fitting curve waveform sequence value {y} is obtained by fitting the curve with the ordinate value y as the dependent variable. i} and fitting parameter a β b β c β ρ β y gβ x gβ ; Calculate the ordinate of the fitted residual sequence Δy based on the regression residual sequence of the fitted curve. i (i = q, q+1, ..., n-1);

[0014] Step 3, ρ α With ρ β Plot the curves of both as a function of q on the same graph, and find the optimal dividing point, i.e., ρ. α With ρ β When the two values ​​are closest, the value of q = q0 is taken as the optimal dividing point Q of the involute segment;

[0015] Step 4: Using Q as the optimal dividing point, divide the measured curve segment into two segments, α and β.

[0016] The α-part sampled measurement coordinate point sequence is used, with the independent variable as the ordinate sequence value {y}. i The dependent variable is the horizontal coordinate value x. A fitting curve waveform sequence value {x} is obtained. i} and the best fit result is denoted as The fitting residual sequence Δy between the involute gear tooth profile segment α and the fitted parabola is calculated based on the longitudinal regression residual sequence of the fitted curve. i (i = 0, 1, ..., q-1);

[0017] The β-part sampled measurement coordinate point sequence is used, with the independent variable being the x-coordinate sequence value {x}. i The dependent variable is the ordinate value y. A fitting curve waveform sequence value {y} is obtained by fitting the curve with the ordinate value y as the dependent variable. i} and the best fit result is denoted as Calculate the fitting residual sequence Δy between the involute gear tooth profile segment β and the fitted parabola based on the regression residual sequence of the fitted curve. i(i = q, 1, ..., n-1);

[0018] 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 of the fitted involute is determined and used to determine and characterize the positions of other points in the involute.

[0019] Based on the measured relative position of the involute segment PC at the initial endpoint P and the reference point D, the model-based measurement and positioning of the standard involute gear tooth profile parameters are realized.

[0020] The method described in step one for dividing the sampled measurement coordinate point sequence into α and β parts is as follows:

[0021] The base circle radius of the involute is r b If the development angle of the involute generator BK is θ, the abscissa is h, and the ordinate is z, then the parametric equation of the involute is:

[0022]

[0023] Where, r b Let r be the radius of the base circle, and let (h0, z0) = (r b ,0) is the starting point of the involute, θ is the development angle of the generating line BK, and θ is the starting point of the involute. D =0;

[0024] By performing coordinate transformations according to x = z and y = -h, the involute equation described in equation (1) above is transformed into the involute equation of the form described in equation (2):

[0025]

[0026] In the involute curve after coordinate transformation, 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 (2), it is obvious that x D =0, y D =-r b ;

[0027] 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:

[0028]

[0029] 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.

[0030] The involute of a circle is an open curve with only a starting point and no ending point;

[0031] When the base circle radius is r b When the expansion angle θ∈[0,π], the range of values ​​for the abscissa x and ordinate y in the rectangular coordinate system is, x∈[0,π]. b ·π],y∈[-r b ,-r b ·π / 2];

[0032] Measurement, fitting, and characterization of the involute template, with a restricted interval of θ∈[0,π], where x∈[0,r] b ·π],y∈[-r b ,-r b ·π / 2];Differentiating from equation (2) yields

[0033]

[0034] When θ∈[0,π], we have x∈[0,r] b ·π], x(r b ,θ) increases monotonically within the interval;

[0035] When θ∈[0,π / 2], we have y∈[-r b ,-r b ·π / 2], y(r b ,θ) decreases monotonically within the interval;

[0036] When θ∈[π / 2,π], we have y∈[-r] b ,-r b ·π / 2], y(r b ,θ) increases monotonically within the interval;

[0037] Analysis of the shape of the involute within the finite interval θ∈[0,π] and the equation described in (2) shows that the ordinate of the involute is a concave function with a single-peak characteristic relative to the abscissa.

[0038] Waveform analysis of the involute curve shows that:

[0039] Within the interval θ∈[0,π / 3], the shape of the involute is approximately the same as the shape of the "horizontal parabola" with the extreme point pointing to the left, and it is suitable to fit the parabola equation x=g(y) with y as the independent variable and x as the dependent variable.

[0040] Within the interval θ∈[π / 3,π], the shape of the involute is approximately the same as the shape of the "vertical parabola" with the extreme point pointing downwards, making it suitable for fitting with the parabolic equation y=f(x) with x as the independent variable and y as the dependent variable;

[0041] Let the abscissas of the sampling measurement coordinate point sequence of the involute template segment PC 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 sequence of sampling measurement coordinate points [x] will be affected. i ,y i ] and the theoretical value sequence 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

[0042]

[0043] Select point Q in the middle of the measurement line segment PC, and divide it into two parts: the left half PQ and the right half QC, referred to as the α segment and the β segment, respectively; the left half PQ corresponds to the sampling measurement coordinate point sequence: [x0,y0],[x2,y2],...,[x q-1 ,y q-1 ]; The right half of the QC corresponding sampling measurement coordinate point sequence: [x q ,y q ],[x q+1 ,y q+1 ],...,[x n-1 ,y n-1 ]).

[0044] The method for fitting the waveform of the involute segment α is as follows:

[0045] Within segment α, the involute shape approximates the shape of the "transverse parabola" with its extreme point pointing to the left, making it suitable for fitting with the parabolic equation x = g(y) where y is the independent variable and x is the dependent variable; the functional expression of its least-squares fitted curve is:

[0046]

[0047] Among them, a α b α c α There are 3 fitting parameters;

[0048] The effective value of the fitting residual is:

[0049]

[0050] The estimated value of the "valley" of the fitted waveform is then:

[0051]

[0052] The locations where the "valley" values ​​of the fitted waveform appear are:

[0053]

[0054] The fitting process is as follows:

[0055] For the sequence of sampled measurement coordinate points [x i ,y i ],(i=0,1,...,q-1), from equation (7) we have:

[0056]

[0057] In ε α When the minimum value is obtained, we have:

[0058]

[0059] Solving this system of linear equations yields the fitting parameters a. α b α c α As an approximate characteristic parameter of the parabolic form of the fitted involute, the corresponding x is calculated according to equations (9) and (10). gα y gα The value, its fitting residual effective value ρ α Calculate according to formula (8);

[0060] The fitted curve's "lateral" regression residual sequence is

[0061]

[0062] The longitudinal regression residual sequence of the fitted curve is

[0063]

[0064] The method for fitting the waveform of the involute segment in the β part is as follows:

[0065] Within the β segment, the involute shape approximates the shape of a "vertical parabola" with its extreme points pointing downwards, making it suitable for fitting with the parabolic equation y = f(x) where x is the independent variable and y is the dependent variable; the function expression of its least-squares fitted curve is:

[0066]

[0067] Among them, a β b β c β There are 3 fitting parameters;

[0068] The effective value of the fitting residual is:

[0069]

[0070] The estimated value of the "valley" of the fitted waveform can then be obtained as follows:

[0071]

[0072] The locations where the "valley" values ​​of the fitted waveform appear are:

[0073]

[0074] The fitting process is as follows:

[0075] For the sequence of sampled measurement coordinate points [x i ,y i ],(i=q,q+1,...,n-1), from equation (16) we have:

[0076]

[0077] In ε β When the minimum value is obtained, we have:

[0078]

[0079] Solving this system of linear equations yields the fitting parameters a. β b β c β As an approximate representation parameter of the parabolic form of the fitted involute, the corresponding y is calculated according to equations (18) and (19). gβ x gβ The value, its fitting residual effective value ρ β Calculate according to formula (17);

[0080] The regression residual sequence of the fitted curve is

[0081]

[0082] Measuring the starting endpoint P(x0,y0) and reference point on the involute curve Length of straight line segment and slope They are respectively:

[0083]

[0084] Measuring the end point C(x) on the involute n-1 ,y n-1 ) and reference point Length of straight line segment and slope They are respectively:

[0085]

[0086] Calculate according to formulas (24) to (27) Then, by searching along the standard involute described in equation (2), we find those with theoretically identical slope values. The location of the point determines the geometrical relative position of the measured involute segment PC within the standard involute. The reference point is...

[0087] This means achieving involute fitting using a two-variable parabolic method with y as the independent variable and x as the dependent variable for the left α segment and x as the independent variable and y as the dependent variable for the right β segment.

[0088] The applicable range is the involute generator line development angle θ∈[0,π].

[0089] The initial endpoints of the fitted curve of the α-segment curve, divided using the optimal dividing point, characterize the initial point position of the involute.

[0090] The endpoint of the fitted curve of the β segment, divided using the optimal dividing point, represents the termination point of the involute.

[0091] Beneficial effects

[0092] 1. The present invention provides a two-variable fitting method for a standard involute template profile, characterized in that it first uses an involute coordinate representation method after coordinate transformation to expand the involute fitting interval to the interval θ∈[0,π].

[0093] 2. Using the involute sampling measurement sequence in rectangular coordinates, different variable parabolas with optimal features are used to fit the involute template curves of different curve segments, in order to fit the effective value of the residual ρ. α With ρβ The optimal segmentation point Q is obtained by using an approximately equal equilibrium method, with the index q of Q. The two sides of Q are defined as the α segment and the β segment, respectively.

[0094] 3. An approximate representation of the involute template profile characterized by two-variable parabolic parameters is obtained. The parabolic fitting results for the α and β segments are obtained, along with the parabolic coordinate positions corresponding to the starting and ending points of the involute. The residual sequence Δy between the involute template profile and the fitted curve is also presented. i , (i = 0, 1, ..., n-1), and the effective value of the fitting residual between the involute template and the fitted curve. and Used to quantitatively evaluate the quality of involute templates.

[0095] 4. Using 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 positional relationship between the measurement endpoints P and C on the standard involute, the modeling measurement and positioning of the tooth profile parameters of the standard involute gear can be completed. The reference point is...

[0096] 5. The method described in this invention does not require direct measurement of the base circle radius, nor does it require measurement of the development angle of the involute. It is applicable to the parameter measurement and evaluation of any involute within the interval θ∈[0,π], and has universality and versatility. Attached Figure Description

[0097] Figure 1 This is a schematic diagram showing the relationship between the measurement points of the involute corresponding to the standard involute equation, with the horizontal axis being h and the vertical axis being z.

[0098] 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 C 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 ​​of line segment PC, (i = 0, 1, ..., n-1).

[0099] Figure 2 This is a schematic diagram showing the relationship between the involute measurement points used in this invention. It is based on a standard involute. Figure 1 A schematic diagram of the involute with a new functional relationship formed after rotating it 90° clockwise. Its horizontal axis is x, and its vertical axis is y.

[0100] Figure 3 The involute curve of the gear tooth profile;

[0101] Figure 4 The variation of the effective values ​​of the fitting residuals for the α and β parts;

[0102] Figure 5 The curve is a two-variable fitting curve for an involute.

[0103] Figure 6 The regression residual curve is fitted to an involute two-variable system.

[0104] Figure 7 The curve is a fitted curve of an involute simple parabola;

[0105] Figure 8 The regression residual curve is fitted to an involute single parabola. Detailed Implementation

[0106] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0107] Example 1: Two-variable fitting

[0108] A two-variable fitting method for a standard involute template contour is described below:

[0109] A. Principles and methods of parabolic model fitting

[0110] like Figure 1 As shown, the radius of the base circle is r. b If the development angle of the involute generator BK is θ, the abscissa is h, and the ordinate is z, then the parametric equation of the involute is:

[0111]

[0112] Where, r b Let r be the radius of the base circle, and let (h0, z0) = (r b ,0) is the starting point of the involute, θ is the development angle of the generating line BK, and θ is the starting point of the involute. D =0;

[0113] By performing coordinate transformations according to x = z and y = -h, the involute equation described in equation (1) above is transformed into the involute equation of the form described in equation (2):

[0114]

[0115] The involute curve after coordinate transformation is as follows: Figure 2 As shown. 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 (2), it is obvious that x D =0, y D =-r b .

[0116] 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:

[0117]

[0118] 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.

[0119] like Figure 2 As shown, the involute of a circle is an open curve with only a starting point and no ending point.

[0120] When the base circle radius is r b When the expansion angle θ∈[0,π], the range of values ​​for the abscissa x and ordinate y in the rectangular coordinate system is, x∈[0,π]. b ·π],y∈[-r b ,-r b ·π / 2];

[0121] Therefore, the measurement, fitting, and characterization of the involute template are only applicable to the interval θ∈[0,π]. The method described in this invention restricts the interval to θ∈[0,π]. In this case, x∈[0,π]. b ·π],y∈[-r b ,-rb ·π / 2]. By differentiating equation (2), we can obtain

[0122]

[0123] When θ∈[0,π], we have x∈[0,r] b ·π], x(r b ,θ) increases monotonically within the interval.

[0124] When θ∈[0,π / 2], we have y∈[-r b ,-r b ·π / 2], y(r b ,θ) decreases monotonically within the interval.

[0125] When θ∈[π / 2,π], we have y∈[-r] b ,-r b ·π / 2], y(r b ,θ) increases monotonically within the interval.

[0126] from Figure 2 The shape of the involute within the finite interval θ∈[0,π] shown, and the equation described in (2), show that the ordinate of the involute is a concave function with a single-peak characteristic relative to the abscissa.

[0127] After the Figure 2 Waveform analysis of the involute shown reveals the following:

[0128] Within the interval θ∈[0,π / 3], the shape of the involute is approximately the same as the shape of the "horizontal parabola" with the extreme point pointing to the left, and it is suitable to fit the parabola equation x=g(y) with y as the independent variable and x as the dependent variable.

[0129] Within the interval θ∈[π / 3,π], the shape of the involute is approximately the same as the shape of the "vertical parabola" with the extreme point pointing downwards, making it suitable for fitting with the parabolic equation y=f(x) with x as the independent variable and y as the dependent variable;

[0130] like Figure 2 As shown, let x0,x1,...,x n-1 Let y0, y1, ..., y1 be the abscissas of the sampling measurement coordinate point sequence of the involute template segment PC. n-1 The ordinates of the sampled measurement coordinates are given, and the corresponding involute generator unfolding angles are θ0, θ1, ..., θ. n-1 Due to the selection and characterization of the measurement reference point location, the sequence of sampling measurement coordinate points [x] will be affected. i ,y i] and the theoretical value sequence 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

[0131]

[0132] Select point Q in the middle of the measurement line segment PC, and divide it into the left half segment PQ (corresponding to the sampling measurement coordinate point sequence: [x0,y0],[x2,y2],...,[x...). q-1 ,y q-1 ]) and the right half of QC (corresponding to the sampling measurement coordinate point sequence: [x q ,y q ],[x q+1 ,y q+1 ],...,[x n-1 ,y n-1 It consists of two parts, referred to as the α segment and the β segment, respectively.

[0133] A. Fitting the partial parabolic model of segment α.

[0134] Within segment α, the involute shape approximates the shape of the "transverse parabola" with its extreme point pointing to the left, making it suitable for fitting with the parabolic equation x = g(y) where y is the independent variable and x is the dependent variable; the functional expression of its least-squares fitted curve is:

[0135]

[0136] Among them, a α b α c α There are 3 fitting parameters.

[0137] The effective value of the fitting residual is:

[0138]

[0139] The estimated value of the "valley" of the fitted waveform can then be obtained as follows:

[0140]

[0141] The locations where the "valley" values ​​of the fitted waveform appear are:

[0142]

[0143] The specific fitting process is as follows:

[0144] For the sequence of sampled measurement coordinate points [x i ,yi ],(i=0,1,...,q-1), from equation (7) we have:

[0145]

[0146] In ε α When the minimum value is obtained, we have:

[0147]

[0148] Solving this system of linear equations yields the fitting parameters a. α b α c α As an approximate characteristic parameter of the parabolic form of the fitted involute, the corresponding x is calculated according to equations (9) and (10). gα y gα The value, its fitting residual effective value ρ α Calculate according to formula (8).

[0149] The fitted curve's "lateral" regression residual sequence is

[0150]

[0151] The longitudinal regression residual sequence of the fitted curve is

[0152]

[0153] A. Fitting the partial parabolic model of segment 2β

[0154] Within the β segment, the involute shape approximates the shape of a "vertical parabola" with its extreme points pointing downwards, making it suitable for fitting with the parabolic equation y = f(x) where x is the independent variable and y is the dependent variable; the function expression of its least-squares fitted curve is:

[0155]

[0156] Among them, a β b β c β There are 3 fitting parameters.

[0157] The effective value of the fitting residual is:

[0158]

[0159] The estimated value of the "valley" of the fitted waveform can then be obtained as follows:

[0160]

[0161] The locations where the "valley" values ​​of the fitted waveform appear are:

[0162]

[0163] The specific fitting process is as follows:

[0164] For the sequence of sampled measurement coordinate points [x i ,y i ],(i=q,q+1,...,n-1), from equation (16) we have:

[0165]

[0166] In ε β When the minimum value is obtained, we have:

[0167]

[0168] Solving this system of linear equations yields the fitting parameters a. β b β c β As an approximate representation parameter of the parabolic form of the fitted involute, the corresponding y is calculated according to equations (18) and (19). gβ x gβ The value, its fitting residual effective value ρ β Calculate according to formula (17).

[0169] The regression residual sequence of the fitted curve is

[0170]

[0171] B. Double parabola fitting process

[0172] 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 will increase as the fitted curve segment increases.

[0173] The sampling measurement coordinate point sequence (x0, y0),...,(x0, y0) of the involute template line segment PC n-1 ,y n-1 ), x0,x1,...,x n-1 Let y0, y1, ..., y1 be the abscissas of the sampling measurement coordinate point sequence of the involute template segment PC. n-1 The ordinate of the sequence of sampled measurement points.

[0174] Point Q is selected to divide the involute waveform segment to be fitted into two segments, α and β. The sampling measurement coordinate point sequence of the left α segment is (x0, y0),...,(x q-1 ,y q-1 The right-side β segment sampling measurement coordinate point sequence is (x q ,y q ),...,(xn-1 ,y n-1 ).

[0175] Find the optimal boundary point Q that makes the effective values ​​of the two fitted residuals closest. Then, perform the fitting process described in A above on the waveform segments on both sides of the optimal boundary point to obtain the fitting parameters, thus completing the involute fitting using the two-variable parabolic method. The process is as follows:

[0176] Regarding the above, Figure 2 As shown, the sampling measurement coordinate point sequence (x0, y0),...,(x0, y0), of the involute template line segment PC is... n-1 ,y n-1 ), select positive integer indices q∈[10,n / 3], 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 ).

[0177] The involute segment waveform of part α is fitted according to the method described in A.1 above to obtain the fitting parameter a. α b α c α ρ α y gα x gα .

[0178] The waveform of the involute segment in the β part is fitted according to the method described in A.2 above to obtain the fitting parameter a. β b β c β ρ β y gβ x gβ .

[0179] 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 / 3, thus completing the entire fitting process.

[0180] ρ α With ρ β The curves of both as a function of q are plotted as follows: Figure 4 In the same curve graph shown, from Figure 4 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 βWhen the two values ​​are closest, q = q0 is taken as the optimal dividing point for the involute segment.

[0181] By fitting the involute segments on both sides of the optimal dividing point, the optimal fitting result of the α-segment involute is obtained, denoted as […]. The fitting residual sequence Δy between the involute profile of the gear segment α and the fitted parabola is calculated according to equation (15). i (i = 0, 1, ..., q-1); obtain the best fitting result for the β segment of the involute, denoted as The fitting residual sequence Δy between the involute profile of the β-segment gear and the fitted parabola is calculated according to equation (23). i , (i = q, 1, ..., n-1). This represents the parameter characterization result of fitting the involute using the biparabolic method.

[0182] 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.

[0183] Based on the measured relative positions of the involute segment PC at the initial endpoint P and the reference point D, the modeling measurement and positioning of the standard involute gear tooth profile parameters are completed. The reference point is...

[0184] Measuring the starting endpoint P(x0,y0) and reference point on the involute curve Length of straight line segment and slope They are respectively:

[0185]

[0186] Measuring the end point C(x) on the involute n-1 ,y n-1 ) and reference point Length of straight line segment and slope They are respectively:

[0187]

[0188] In particular, the results obtained by calculating according to equations (24) to (27) Then, by searching along the standard involute described in equation (2), we find those with theoretically identical slope values. The location of the point determines the geometrical relative position of the measured involute segment PC within the standard involute. The reference point is...

[0189] Specific calculation example:

[0190] Figure 3 The sequence of sampled measurement coordinate points for the involute waveform 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 = 4.712389 rad., the horizontal axis sampling interval is: Δx i =2.3561945×10 -3 cm; Number of sampling points n = 2000;

[0191] Selecting positive integer indices q∈[10,666] as the segmentation point indices, the sequence of sampled measurement coordinate points is divided into α parts: (x0,y0),...,(x q-1 ,y q-1 ) and β part: (x q ,y q ),...,(x n-1 ,y n-1 ).

[0192] The involute segment waveform of part α is fitted according to the method described in A.1 above to obtain the fitting parameter a. α b α c α ρ α y gα x gα .

[0193] The waveform of the involute segment in the β part is fitted according to the method described in A.2 above to obtain the fitting parameter a. β b β c β ρ β y gβ x gβ .

[0194] 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=666, thus completing the entire fitting process.

[0195] ρ α With ρ β The curves of both as a function of q are plotted as follows: Figure 4 In the same curve graph shown, from Figure 4 As can be seen from ρ α The envelope of ρ increases monotonically with increasing q value. βThe envelope of ρ decreases monotonically with increasing q value. α With ρ β When the two values ​​are closest, q = q0 = 350.

[0196] Let q0 = 350 be the optimal dividing point of the involute segment. The α-part sampled measurement coordinate point sequence is: (x0, y0), ..., (x 350 ,y 350 Following the process described in A.1 above, a fitting curve waveform sequence y is obtained. i like Figure 5 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 6 The red portion shown is (α:Δy) i The maximum fitting residual was 0.09 cm. Furthermore, in all regions other than θ = 0°, the fitting residual was below 0.02 cm.

[0197] The fitting parameters are:

[0198]

[0199] The β portion is sampled and measured from the coordinate point sequence: (x 351 ,y 351 ),...,(x 1999 ,y 1999 Following the process described in A.2 above, a fitting curve waveform sequence y is obtained. i like Figure 5 The black portion (β:y) i The blue curve y represents the original measured curve waveform. Its fitted residual waveform sequence Δy i like Figure 6 The black portion shown (β:Δy) i The maximum residual was 0.023 cm. The fitting parameters were:

[0200]

[0201]

[0202] As a parameter characterization result of fitting the involute using the double parabola method. Figure 6 It is evident that the fitting accuracy between the two is quite high.

[0203] Example 2: Single Parabolic Fitting Method

[0204] Figure 3 The sequence of sampled measurement coordinate points for the involute waveform obtained by calculating using the involute formula {[xi ,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 = 4.712389 rad., the horizontal axis sampling interval is: Δx i =2.3561945×10 -3 cm; Number of sampling points n = 2000;

[0205] For this set of examples, if a single parabolic waveform is used for fitting according to the process described in A.2 above, the fitted curve waveform sequence y is obtained. i like Figure 7 The red portion refers to the original measured curve waveform, where the purple curve y represents the original measured curve waveform. Its fitted residual waveform sequence Δy... i like Figure 8 As shown, the maximum value of the fitting residual is 0.31 cm. The fitting parameters are:

[0206] a=0.1847593; b=-0.7913342;

[0207] c=-2.305503; ρ=4.0581×10 -2 cm;

[0208] y g = -3.152835cm; x g = 2.141527cm;

[0209] Depend on Figure 8 It is evident that the fit between the two is far less than that of Example 1.

[0210] The above description is merely a preferred embodiment of the present invention, and the present invention should not be limited to the content disclosed in this embodiment and the accompanying drawings. Any equivalent or modified versions made without departing from the spirit of the present invention fall within the scope of protection of the present invention.

Claims

1. A two-variable fitting method for the profile of a standard involute template, characterized in that: Includes the following steps: Step 1: Apply coordinate transformation to the classic involute curve, rotating it 90° clockwise to represent it in rectangular coordinates xoy, where x is the abscissa and y is the ordinate; (x, y) represents the coordinates of the measurement point on the involute; the involute waveform sampling measurement coordinate point sequence {[x i ,y i ]},(i=0,...,n-1); Select positive integer index q as the split point index to 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 ); Step 2: For the involute segment waveform of part α, use the independent variable as the ordinate sequence value {y}. i The dependent variable is the horizontal coordinate value x. A fitting curve waveform sequence value {x} is obtained. i } and fitting parameter a α b α c α ρ α y gα x gα ; Calculate the ordinate of the fitted residual sequence Δy based on the "vertical" regression residual sequence of the fitted curve. i (i = 0, 1, ..., q-1); At this time, for the involute segment waveform of the β part, the independent variable is the sequence value {x} of the abscissa. i The dependent variable is the ordinate value y. A fitting curve waveform sequence value {y} is obtained by fitting the curve with the ordinate value y as the dependent variable. i } and fitting parameter a β b β c β ρ β y gβ x gβ ; Calculate the ordinate of the fitted residual sequence Δy based on the regression residual sequence of the fitted curve. i (i = q, q+1, ..., n-1); Step 3, ρ α With ρ β Plot the curves of both as a function of q on the same graph, and find the optimal dividing point, i.e., ρ. α With ρ β When the two values ​​are closest, the value of q = q0 is taken as the optimal dividing point Q of the involute segment; Step 4: Using Q as the optimal dividing point, divide the measured curve segment into two segments, α and β. The α-part sampled measurement coordinate point sequence is used, with the independent variable as the ordinate sequence value {y}. i The dependent variable is the horizontal coordinate value x. A fitting curve waveform sequence value {x} is obtained. i } and the best fit result is denoted as The fitting residual sequence Δy between the involute gear tooth profile segment α and the fitted parabola is calculated based on the longitudinal regression residual sequence of the fitted curve. i (i = 0, 1, ..., q-1); The β-part sampled measurement coordinate point sequence is used, with the independent variable being the x-coordinate sequence value {x}. i The dependent variable is the ordinate value y. A fitting curve waveform sequence value {y} is obtained by fitting the curve with the ordinate value y as the dependent variable. i } and the best fit result is denoted as Calculate the fitting residual sequence Δy between the involute gear tooth profile segment β and the fitted parabola based on the regression residual sequence of the fitted curve. i (i = q, 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. The starting reference point D of the fitted involute is determined and used to determine and characterize the positions of other points in the involute. Based on the measured relative position of the involute segment PC at the initial endpoint P and the reference point D, the model-based measurement and positioning of the standard involute gear tooth profile parameters are realized.

2. The method as described in claim 1, characterized in that: The method described in step one for dividing the sampled measurement coordinate point sequence into α and β parts is as follows: The base circle radius of the involute is r b If the development angle of the involute generator BK is θ, the abscissa is h, and the ordinate is z, then the parametric equation of the involute is: Where, r b Let r be the radius of the base circle, and let (h0, z0) = (r b ,0) is the starting point of the involute, θ is the development angle of the generating line BK, and θ is the starting point of the involute. D =0; By performing coordinate transformations according to x = z and y = -h, the involute equation described in equation (1) above is transformed into an involute equation of the form described in equation (2): In the involute curve after coordinate transformation, 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 (2), it is obvious that x D =0, y D =-r b ; 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 ending point; When the base circle radius is r b When the expansion angle θ∈[0,π], the range of values ​​for the abscissa x and ordinate y in the rectangular coordinate system is, x∈[0,π]. b ·π],y∈[-r b ,-r b ·π / 2]; Measurement, fitting, and characterization of the involute template, with a restricted interval of θ∈[0,π], where x∈[0,r] b ·π],y∈[-r b ,-r b ·π / 2];Differentiating from equation (2) yields When θ∈[0,π], we have x∈[0,r] b ·π], x(r b ,θ) increases monotonically within the interval; When θ∈[0,π / 2], we have y∈[-r b ,-r b ·π / 2], y(r b ,θ) decreases monotonically within the interval; When θ∈[π / 2,π], we have y∈[-r] b ,-r b ·π / 2], y(r b ,θ) increases monotonically within the interval; Analysis of the shape of the involute within the finite interval θ∈[0,π] and the equation described in (2) shows that the ordinate of the involute is a concave function with a single-peak characteristic relative to the abscissa. Waveform analysis of the involute curve shows that: Within the interval θ∈[0,π / 3], the shape of the involute is approximately the same as the shape of the "horizontal parabola" with the extreme point pointing to the left, and it is suitable to fit the parabola equation x=g(y) with y as the independent variable and x as the dependent variable; Within the interval θ∈[π / 3,π], the shape of the involute is approximately the same as the shape of the "vertical parabola" with the extreme point pointing downwards, making it suitable for fitting with the parabolic equation y=f(x) with x as the independent variable and y as the dependent variable; Let the abscissas of the sampling measurement coordinate point sequence of the involute template segment PC 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 sequence of sampling measurement coordinate points [x] will be affected. i ,y i ] and the theoretical value sequence 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 Select point Q in the middle of the measurement line segment PC, and divide it into two parts: the left half PQ and the right half QC, referred to as the α segment and the β segment, respectively; the left half PQ corresponds to the sampling measurement coordinate point sequence: [x0,y0],[x2,y2],...,[x q-1 ,y q-1 ]; The right half of the QC corresponding sampling measurement coordinate point sequence: [x q ,y q ],[x q+1 ,y q+1 ],...,[x n-1 ,y n-1 ]).

3. The method as described in claim 1, characterized in that: The method for fitting the waveform of the involute segment α is as follows: Within segment α, the shape of the involute approximates the shape of the "horizontal parabola" with its extreme point pointing to the left, making it suitable for fitting with the parabolic equation x = g(y) with y as the independent variable and x as the dependent variable; the functional expression of its least squares fitted curve is: Among them, a α b α c α There are 3 fitting parameters; 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 fitting process is as follows: For the sequence of sampled measurement coordinate points [x i ,y i ],(i=0,1,...,q-1), from equation (7) we have: In ε α When the minimum value is obtained, we have: Solving this system of linear equations yields the fitting parameters a. α b α c α As an approximate characteristic parameter of the parabolic form of the fitted involute, the corresponding x is calculated according to equations (9) and (10). gα y gα The value, its fitting residual effective value ρ α Calculate according to formula (8); The fitted curve's "lateral" regression residual sequence is The longitudinal regression residual sequence of the fitted curve is 4. The method as described in claim 3, characterized in that: The method for fitting the waveform of the involute segment in the β part is as follows: Within the β segment, the involute shape approximates the shape of a "vertical parabola" with its extreme points pointing downwards, making it suitable for fitting with the parabolic equation y = f(x) where x is the independent variable and y is the dependent variable; the function expression of its least squares fitted curve is: Among them, a β b β c β There are 3 fitting parameters; The effective value of the fitting residual is: The estimated value of the "valley" of the fitted waveform can then be obtained as follows: The locations where the "valley" values ​​of the fitted waveform appear are: The fitting process is as follows: For the sequence of sampled measurement coordinate points [x i ,y i ],(i=q,q+1,...,n-1), from equation (16) we have: In ε β When the minimum value is obtained, we have: Solving this system of linear equations yields the fitting parameters a. β b β c β As an approximate representation parameter of the parabolic form of the fitted involute, the corresponding y is calculated according to equations (18) and (19). gβ x gβ The value, its fitting residual effective value ρ β Calculate according to formula (17); The regression residual sequence of the fitted curve is 5. The method as described in claim 4, characterized in that: Measuring the starting endpoint P(x0,y0) and reference point on the involute curve Length of straight line segment and slope They are respectively: Measuring the end point C(x) on the involute n-1 ,y n-1 ) and reference point Length of straight line segment and slope They are respectively: Calculate according to formulas (24) to (27) to obtain Then, by searching along the standard involute described in equation (2), we find those with theoretically identical slope values. The location of the point determines the geometrical relative position of the measured involute segment PC within the standard involute. The reference point is... This means achieving involute fitting using a two-variable parabolic method with y as the independent variable and x as the dependent variable for the left α segment and x as the independent variable and y as the dependent variable for the right β segment.

6. The method as described in claim 5, characterized in that: The applicable range is the involute generator line development angle θ∈[0,π].

7. The method as described in claim 5, characterized in that: The initial endpoints of the fitted curve of the α-segment curve, divided using the optimal dividing point, characterize the initial point position of the involute.

8. The method as described in claim 7, characterized in that: The endpoint of the fitted curve of the β segment, divided using the optimal dividing point, represents the termination point of the involute.

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