CN107977503B - Multi-tool approximation method for machining small-size tooth top fillet by worm grinding wheel - Google Patents

Multi-tool approximation method for machining small-size tooth top fillet by worm grinding wheel Download PDF

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CN107977503B
CN107977503B CN201711208764.XA CN201711208764A CN107977503B CN 107977503 B CN107977503 B CN 107977503B CN 201711208764 A CN201711208764 A CN 201711208764A CN 107977503 B CN107977503 B CN 107977503B
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rack
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李国龙
任唯贤
何坤
董鑫
徐凯
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Chongqing University
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Abstract

The invention belongs to the field of gear manufacturing, relates to a multi-tool approximation method for machining small-size tooth top fillets by using a worm grinding wheel, and solves the problem that the worm grinding wheel cannot be repaired due to too small curvature radius when the small-size tooth top fillets are machined. A multi-section curve is processed by translating an X axis and a Y axis of a worm grinding wheel, a fillet curve is enveloped by the multi-section curve, a point vector pairing method is provided, the offset of the X axis and the Y axis of the grinding wheel is solved, the principle error of an approximation method is analyzed, and the optimal approximation circular arc size and the approximation tool number are established according to the upper error limit required in design.

Description

Multi-tool approximation method for machining small-size tooth top fillet by worm grinding wheel
Technical Field
The invention belongs to the field of gear manufacturing, relates to an approximation method for enveloping a small circular arc by using a plurality of sections of large circular arcs, and solves the problem that a worm grinding wheel cannot be trimmed because the minimum curvature radius is smaller than the radius of a roller when a small-size tooth top fillet is machined (hereinafter referred to as a small circular arc).
Background
A large number of theories and practices show that the edge edges and corners of the gear can easily cause tooth surface protrusion due to small collision to generate noise and damage a meshing tooth surface, and the edge edges and corners can easily cause stress concentration in the heat treatment process, influence the mechanical property of the gear to cause the gear to be easily subjected to fatigue wear, and reduce the service life of the gear. Therefore, chamfering and rounding of gears have become an important process in the field of gear manufacturing.
The gear radius is divided into tooth profile radius and tooth top radius. Tooth profile rounding is usually performed on a chamfering machine through a chamfering tool or a forming milling cutter, and the tooth top rounding effect is more important than tooth profile rounding because in gear transmission, the tooth top point of a driving gear is directly contacted and meshed with the tooth surface of a driven gear, so that impact is easy to generate, and vibration and noise are caused.
The tooth top rounding still adopts the mode of manual grinding at present, and its main reason is that the worm grinding wheel of processing fillet is difficult to maintain, and especially when the fillet size is less, the minimum radius of curvature of worm grinding wheel will be less than the radius of curvature of diamond roller, leads to the emery wheel unable to maintain.
Disclosure of Invention
Aiming at the problem that the worm grinding wheel for processing the small tooth top fillet cannot be repaired, the invention provides a multi-tool approximation method for processing the small tooth top fillet by using the worm grinding wheel, and the purpose of processing the small tooth top fillet within a given error allowable range by using the large-fillet worm grinding wheel is realized.
In order to solve the technical problems, the invention adopts the following technical scheme:
a multi-tool approach method for machining a small tooth top fillet by using a worm grinding wheel is characterized in that the worm grinding wheel for machining the large tooth top fillet is subjected to translation in an X axis and a Y axis to machine a plurality of sections of different tooth top curves, and finally the small tooth top fillet is enveloped by the plurality of sections of curves. The approximation principle is as shown in fig. 1, a worm grinding wheel with a processing radius of 0.5 round corner is used, and a 5-knife method is adopted to approximate the round corner with the radius of 0.05, so that the approximation result is very accurate.
The method is realized by a point vector pairing mode, and concretely comprises the steps of comparing the profile of the shape generating rack with a large tooth top fillet and a small tooth top fillet, pairing point vectors with the same normal vector, and matching the coordinate difference of each pair of paired point vectors, namely the offset of a group of X axes and Y axes of the grinding wheel. Aiming at solving the translation quantity of an X axis and a Y axis of each transformation of the grinding wheel, a point vector pairing method is provided, taking 0.5 circular arc approaching 0.05 circular arc as an example, comparing a product shape rack point vector with a machining radius of 0.5 circular arc with a product shape rack point vector with a machining radius of 0.05, pairing point vectors with equal slopes, and using the coordinate difference of matched point vectors as the translation quantity (delta X, delta Y) of the grinding wheel:
Figure GDA0003004042820000021
wherein ((X)2，Y2) Is to process a halfProduct shape rack discrete point coordinate with 0.5 round corner diameter, (x)2，y2) Is the discrete point coordinate of the shaping rack with the processing radius of 0.05.
As another preferred scheme of the invention, the approach method of enveloping the small round angle by the multi-segment curve has a sharp point error, the sharp point is the intersection point of two adjacent curves, the distance between the intersection point and the circle center of the small round angle is the height of the sharp point, and the approximation error value is obtained by subtracting the radius of the small round angle from the height of the sharp point. The approximation method of the tangent envelope has a cusp error, the cusp is formed by the intersection point of the tooth profiles processed by two adjacent cutters, as shown in fig. 2, the error magnitude E can be represented by the difference between the distance from the intersection point of the tooth profiles to the center of the circular bead and the radius:
Figure GDA0003004042820000022
wherein (X)j，Yj) Is the coordinates of the tooth profile intersection point, (X)0，Y0) Circular center coordinates, r is the small circular corner radius.
As an improvement scheme of the invention, according to the maximum rounding error given by design, the minimum approximate cutter number and the maximum fillet radius under the error are calculated. A series of data points can be obtained by calculating error values generated by different large fillet radii and the number of cutters, an error curve is obtained by fitting the data points by adopting cubic splines, and the most appropriate cutter number and large fillet radius can be selected by a multi-cutter approximation method according to the error curve.
The invention has the technical effects that: the problem that when a small-size tooth top fillet is machined by using a worm grinding wheel, the small-size tooth top fillet cannot be trimmed due to the fact that the minimum curvature radius of the grinding wheel is too small is solved, the approach error is reduced to the greatest extent by adopting the proper number of cutters and the radius of the large fillet, and the rounding precision is improved.
Drawings
FIG. 1 is a schematic diagram of a 0.5 fillet penta-tool approach to a 0.05 fillet;
FIG. 2 is a schematic illustration of a tip error for a first tool machining and a second tool machining;
FIG. 3 is a graph showing the relationship between the radius of a large fillet and the approximation error for different numbers of cutters.
Detailed Description
The invention is described in further detail below with reference to the figures and the detailed description.
A multi-tool approach method for machining small tooth top fillet by using worm grinding wheel is disclosed, the approach principle is shown in figure 1, firstly, a shape generating rack point vector for machining large tooth top fillet is determined, the radius of the large fillet is R, and a large fillet equation (X) is obtained1，Y1) Comprises the following steps:
Figure GDA0003004042820000031
normal vector of large fillet tooth profile (N)x，Ny) Comprises the following steps:
Figure GDA0003004042820000032
wherein: raThe radius of the addendum circle is represented, alpha is a solving parameter of a circle center coordinate, theta 1 represents an arc angle on the addendum circle, and theta 2 represents an arc angle on the involute of the tooth profile. Substituting the meshing equation:
Figure GDA0003004042820000033
wherein
Figure GDA0003004042820000034
The tooth profile normal vector is represented as,
Figure GDA0003004042820000035
indicating relative velocity, subscripts 1 and 2 indicate that under different coordinate systems S1 and S2, respectively, the a matrix is:
Figure GDA0003004042820000036
substituting the formula (4) into the formula (3), solving the meshing equation to obtain the tooth profile point entering meshThe required corner phi is matched, and then the profile (X) of the large-circular-angle shaping rack is obtained through coordinate transformation2，Y2) Comprises the following steps:
Figure GDA0003004042820000041
normal vector of large fillet shape-producing rack (N2)x，N2y) Comprises the following steps:
Figure GDA0003004042820000042
(X2，Y2) And (N2)x，N2y) And forming a point vector of the large-fillet shape-producing rack.
And then solving the point vector of the shape generating rack of which the small fillet approaches to the tangent point, wherein the solving method is the same as the method for solving the point vector of the shape generating rack of which the large fillet approaches to the tangent point. When a multi-cutter method with the cutter number of D is adopted for processing, approximate tangent points are selected at equal division points of D of the small round angle, the radius of the small round angle is r, after the D is equally divided,
Figure GDA0003004042820000043
equation of small fillet (x)1，y1) Comprises the following steps:
Figure GDA0003004042820000044
small fillet normal vector (n)x，ny) Comprises the following steps:
Figure GDA0003004042820000045
substituting the formula (3) into the formula meshing equation, and performing coordinate transformation to obtain the profile (x) of the small round angle product-shaped rack2，y2) Comprises the following steps:
Figure GDA0003004042820000046
normal vector of small fillet shape-generating rack (n 2)x，n2y) Comprises the following steps:
Figure GDA0003004042820000047
(x2，y2) And (n 2)x，n2y) And forming point vectors of the shape-generating racks at the small round corner tangent points.
And finally, matching the point vectors of the large fillet shape generating rack at the point of the tangent point of the small fillet with the point vectors of the shape generating rack at the point of the tangent point of the small fillet, namely, searching the point in the large fillet shape generating rack in the same direction as the normal vector of the small fillet shape generating rack, and considering that the slopes of the normals of the two are in the same interval, wherein the interval is (0, arctana), and a is an involute pressure angle, so that the point vector of the shape generating rack at the tangent point of each small fillet can find the matched point vector in the large fillet shape generating rack.
And comparing matched point vectors with the same normal slope, and obtaining the X-axis and Y-axis offset (delta X and delta Y) of each tool transformation of the grinding wheel by the coordinate difference.
Figure GDA0003004042820000051
Assuming that the first knife offset amount is (delta X1, delta Y1), the profile discrete point (X 'after grinding wheel offset can be obtained according to the offset amount of the grinding wheel'2，Y′2):
Figure GDA0003004042820000052
Substituting the discrete points of the profile into the meshing equation:
Figure GDA0003004042820000053
wherein the B matrix is:
Figure GDA0003004042820000054
solving an engagement equation to obtain a rotation angle phi required by the tooth profile point entering into engagement, and obtaining the gear profile (X ') processed after the first cutter transformation through coordinate transformation'1，Y′1) Comprises the following steps:
Figure GDA0003004042820000055
drawing a first machined gear profile (X ') in CAD'1，Y′1) The fitting curve can be obtained by drawing results, the curve processed after the translation transformation of the grinding wheel is seen to be no longer a circular arc, the gear profile during the next processing can be calculated by the same method, the fitting curve can be drawn in the CAD, the intersection point of the two curves is the sharp point with the maximum error, the difference value between the distance from the sharp point to the circle center of the fillet and the radius is the error size E, and the formula is expressed as follows:
Figure GDA0003004042820000056
the intersection point exists when every two adjacent cutters are processed, and the maximum value of the error of the cusp is selected as the final approximation error EmError value E generated by calculating different large fillet radii and the number of cuttersmObtaining a series of data points, obtaining an error curve by fitting the data points by a cubic spline, and guiding a multi-tool approximation method to select the most appropriate tool number and the most appropriate radius of the large fillet according to the error curve as shown in FIG. 2; FIG. 3 is a graph showing the relationship between the radius of a large fillet and the approximation error for different numbers of cutters.
Finally, the above embodiments are only for illustrating the technical solutions of the present invention and not for limiting, although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications or equivalent substitutions may be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all of them should be covered in the claims of the present invention.

Claims (1)

1. A multi-tool approach method for machining small-size tooth top fillet by worm grinding wheel is characterized by firstly determining the resultant rack point vector for machining the large tooth top fillet, wherein the radius of the large fillet is R, and the equation of the large fillet is X1，Y1) Comprises the following steps:
Figure FDA0003004042810000011
normal vector of large fillet tooth profile (N)x，Ny) Comprises the following steps:
Figure FDA0003004042810000012
wherein: raRepresenting the radius of the addendum circle, wherein alpha is a solving parameter of a circle center coordinate, theta 1 represents an arc angle on the addendum circle, and theta 2 represents an arc angle on the involute of the tooth profile; substituting the meshing equation:
Figure FDA0003004042810000013
wherein
Figure FDA0003004042810000014
The tooth profile normal vector is represented as,
Figure FDA0003004042810000015
indicating relative velocity, subscripts 1 and 2 indicate that under different coordinate systems S1 and S2, respectively, the a matrix is:
Figure FDA0003004042810000016
substituting the formula (4) into the formula (3), solving an engagement equation to obtain a rotation angle phi required by the tooth profile point to enter into engagement, and obtaining the profile (X) of the large-circular-angle generating rack through coordinate transformation2，Y2) Comprises the following steps:
Figure FDA0003004042810000017
normal vector of large fillet shape-producing rack (N2)x，N2y) Comprises the following steps:
Figure FDA0003004042810000018
(X2，Y2) And (N2)x，N2y) Forming a point vector of the large-fillet shape-producing rack;
then solving the point vector of the shape generating rack of which the small fillet approaches to the tangent point, wherein the solving method is the same as the method for solving the point vector of the shape generating rack of which the large fillet approaches to the tangent point; when a multi-cutter method with the cutter number of D is adopted for processing, approximate tangent points are selected at equal division points of D of the small round angle, the radius of the small round angle is r, after the D is equally divided,
Figure FDA0003004042810000021
equation of small fillet (x)1，y1) Comprises the following steps:
Figure FDA0003004042810000022
small fillet normal vector (n)x，ny) Comprises the following steps:
Figure FDA0003004042810000023
put into the equation (3) and go toLine coordinate transformation is carried out to obtain the profile (x) of the small round angle shaping rack2，y2) Comprises the following steps:
Figure FDA0003004042810000024
normal vector of small fillet shape-generating rack (n 2)x，n2y) Comprises the following steps:
Figure FDA0003004042810000025
(x2，y2) And (n 2)x，n2y) Forming point vectors of the shape-producing racks at the small round corner tangent points;
finally, matching the point vectors of the large fillet shape generating rack with the point vectors of the shape generating rack at the tangent point of the small fillet, namely, searching the point in the large fillet shape generating rack with the same direction as the normal vector of the small fillet shape generating rack, and considering that the slopes of the normals of the two are in the same interval, wherein the interval is (0, arctana), and a is an involute pressure angle, so that the point vector of the shape generating rack at the tangent point of each small fillet can find the matched point vector in the large fillet shape generating rack;
comparing the matched point vectors with the same normal slope, and obtaining the X-axis and Y-axis offset (delta X and delta Y) of each tool transformation of the grinding wheel by the coordinate difference,
Figure FDA0003004042810000026
let the first offset be (Δ X1, Δ Y1)
From the amount of deflection of the grinding wheel, the profile discrete point (X 'after the grinding wheel deflection) can be obtained'2，Y′2):
Figure FDA0003004042810000027
Substituting the discrete points of the profile into the meshing equation:
Figure FDA0003004042810000031
wherein the B matrix is:
Figure FDA0003004042810000032
solving an engagement equation to obtain a rotation angle phi required by the tooth profile point entering into engagement, and obtaining the gear profile (X ') processed after the first cutter transformation through coordinate transformation'1，Y′1) Comprises the following steps:
Figure FDA0003004042810000033
drawing a first machined gear profile (X ') in CAD'1，Y′1) The fitting curve of (2) can be seen from the drawing result, the curve processed after the translation transformation of the grinding wheel is no longer a circular arc, the same method can calculate the gear profile when the next cut is processed and draw the fitting curve in CAD, the intersection point of the two curves is the sharp point with the maximum error, the difference value between the distance from the sharp point to the circle center of the fillet and the radius is the error size E, and the formula is expressed as follows:
Figure FDA0003004042810000034
wherein (X)j，Yj) Is the coordinates of the tooth profile intersection point, (X)0，Y0) Circle center coordinates of the round corners; the intersection point exists when every two adjacent cutters are processed, and the maximum value of the error of the cusp is selected as the final approximation error EmError value E generated by calculating different large fillet radii and the number of cuttersmA series of data points can be obtained, and three samples are taken for the data pointsAnd fitting the strips to obtain an error curve, and selecting the most appropriate number of cutters and the radius of the large fillet according to the error curve by guiding a multi-cutter approximation method.
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CN108953550B (en) * 2018-08-01 2019-12-31 中南大学 Design method for point tooth surface of straight gear
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