CN108533715B - Bidirectional conjugate tooth profile design method for harmonic gear transmission - Google Patents
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Abstract
A bidirectional conjugate tooth profile design method for harmonic gear transmission comprises the steps of firstly representing coordinates of a convex tooth profile of a top section of a flexible gear through a parameter equation, converting the coordinates into a rigid gear coordinate system, solving to obtain a theoretical conjugate concave tooth profile of the convex tooth profile of the top section of the flexible gear and discrete point coordinates of the convex tooth profile, determining the working tooth height of a tooth profile of a rigid gear through least square fitting, determining a fitting curve section as a concave tooth profile of a bottom section of the rigid gear and a convex tooth profile of a top section of the rigid gear, and calculating to complete tooth profile design of the rigid gear; then, expressing the convex tooth profile coordinates of the rigid wheel top section by using a parameter equation, converting the convex tooth profile coordinates into a flexible wheel coordinate system, solving to obtain discrete point coordinates of a theoretical concave conjugate tooth profile in the flexible wheel bottom section range, determining the tooth height of the flexible wheel through least square fitting, determining a fitting curve as a flexible wheel bottom section concave tooth profile, and calculating to obtain a flexible wheel working tooth profile; the invention directly obtains the tooth profiles of the flexible gear and the rigid gear with double conjugation and secondary conjugation, improves the conjugation contact area and the meshing area, reduces the contact stress of the tooth surface and increases the transmission precision of the harmonic reducer.
Description
Technical Field
The invention relates to the technical field of harmonic reducers, in particular to a bidirectional conjugate tooth profile design method for harmonic gear transmission.
Background
The harmonic transmission technology mainly utilizes the elastic deformation wave of a flexible working member to realize motion or power transmission, and because the number of teeth between a flexible gear and a rigid gear is different, the flexible gear and the rigid gear perform staggered tooth motion in the rotation process of a wave generator, thereby realizing the power transmission. The tooth shapes of a flexible gear and a rigid gear in the harmonic reducer have great influence on the performance. The involute tooth profile is good in manufacturability, so that the involute tooth profile is widely applied to harmonic gear transmission, but the involute tooth profile is not a conjugate tooth profile, the contact surface of a flexible gear tooth profile and a rigid gear tooth profile is small during transmission, most flexible gear teeth are in a sharp point meshing state, and the optimal performance cannot be achieved. The service life and the transmission performance of the harmonic reducer can be improved by selecting the conjugate tooth profile. The working tooth profile of the double-circular-arc tooth profile can be divided into a convex circular arc at the top section and a concave circular arc at the bottom section by a pitch circle, and the two tooth profiles are connected by a common tangent line segment, so that the double-circular-arc tooth profile has a larger conjugate area and can realize continuous conjugate contact.
The existing bidirectional conjugate tooth profile design method comprises the steps of firstly determining tooth profile parameters of a flexible gear, then respectively calculating conjugate tooth profiles of two sections of circular arc tooth profiles of the flexible gear, and then fitting the conjugate tooth profiles to obtain a rigid gear tooth profile. The theoretical conjugate concave tooth profile CT1 and the conjugate convex tooth profile CT2 of the flexible gear top-section convex tooth profile ST1 and the theoretical conjugate concave tooth profile CA1 of the bottom-section concave tooth profile SA1 can be obtained through an envelope method. The tooth profiles ST2 and SA2 of the rigid wheel should be fitted from the theoretical conjugate tooth profile of the flexible wheel. Since the two circular arc tooth profiles of the flexible gear are meshed with the convex circular arc segment ST2 of the rigid gear, conjugate tooth profiles CT2 and CA1 of the top segment of the rigid gear are obtained in calculation, but only the rightmost one of the conjugate tooth profiles can be selected as the convex tooth profile of the rigid gear in order to ensure that the gear teeth do not interfere. When CA1 is on the right side, ST2 is determined by CA1, a section of meshing area exists in the deepest part of the meshing, the convex arc section and the concave arc section of the flexible gear and the two sections of arc tooth profiles of the rigid gear form conjugate tooth profiles (namely, double conjugate phenomenon), so that a larger conjugate contact area can be achieved, but in a very long part of meshing and nibbled areas, the convex tooth profile of the flexible gear and the convex tooth profile of the rigid gear do not have conjugate relation and do not mesh, so that the meshing area is very small, and the number of teeth participating in meshing is relatively small; when the CT2 is on the right side, the ST2 is determined by the CT2, and in the harmonic transmission motion, the tooth profiles of the two sections of the rigid gear are conjugate tooth profiles of the convex arc section of the flexible gear (namely, a secondary conjugate phenomenon), so that the range of a meshing area is maximum, but the concave arc of the flexible gear does not participate in meshing, and the meshing contact surface is smaller; when CA1 intersects CT2, ST2 is obtained by fitting the rightmost part of the two lines CA1 and CT2, which has both advantages and disadvantages.
To achieve the maximum conjugate contact area and the maximum engagement zone, CA1 is aligned with CT2, which cannot be achieved with the prior art design; although CA1 can be made to coincide as much as possible with CT2 by optimizing the flexspline profile parameters. However, because the parameters of the double-circular-arc tooth shape of the flexible gear are more, a large amount of calculation is usually required to find out the proper parameters, so that the design process of the double-circular-arc tooth shape is very complicated, the consumed time is long, and only an approximate result can be obtained.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention aims to provide a bidirectional conjugate tooth profile design method for harmonic gear transmission, which directly obtains tooth profiles of a flexible gear and a rigid gear with double conjugate and secondary conjugate phenomena, improves conjugate contact area and meshing area, reduces tooth surface contact stress and increases transmission precision of a harmonic reducer.
In order to achieve the purpose, the invention adopts the technical scheme that:
a bidirectional conjugate tooth profile design method for harmonic gear transmission comprises the following steps:
1) firstly, selecting preset parameters according to the design requirements of a harmonic reducer, wherein the preset parameters comprise the inner diameter of a flexible gear, the wall thickness of the flexible gear, the pitch-circle tooth thickness ratio, the tooth height coefficient and the like, the basic design parameters are used for determining the convex tooth profile ST1 of the top section of the flexible gear, and the coordinates (x) of the convex tooth profile ST1 of the top section of the flexible gear are expressed through a parameter equation1(s),y1(s))；
2) Coordinates (x) of the flexible gear top segment convex tooth profile ST1 are transformed through coordinates1(s),y1(s)) converting the conjugate convex tooth profile into a rigid wheel coordinate system, substituting the coordinate system into an envelope equation, and solving through numerical calculation to obtain discrete point coordinates of a theoretical conjugate concave tooth profile CT1 and a theoretical conjugate convex tooth profile CT2 of a flexible wheel top section convex tooth profile ST 1;
3) by utilizing the discrete point coordinates of the theoretical conjugate concave tooth profile CT1, the concave tooth profile of the rigid wheel bottom section is fitted through least squares, and fitting parameters are adjusted to enable the discrete points of the theoretical conjugate concave tooth profile CT1 to be positioned on the inner side of a fitting curve, so that tooth profile interference is avoided; similarly, the fitting rigid gear top section convex tooth profile is obtained through least square fitting by using the discrete point coordinates of the theoretical conjugate convex tooth profile CT2, and fitting parameters are adjusted to enable the theoretical conjugate convex tooth profile CT2 discrete points to be on the outer side of the fitting curve;
determining the working tooth height range of the rigid gear tooth profile according to the tooth height coefficient of the rigid gear, and respectively determining fitting curve segments in the tooth height range of the rigid gear as a concave tooth profile SA2 at the bottom section of the rigid gear and a convex tooth profile ST2 at the top section of the rigid gear;
4) calculating a common tangent line segment according to the rigid gear bottom section concave tooth profile SA2 and the rigid gear top section convex tooth profile ST2, and combining the rigid gear bottom section concave tooth profile SA2, the rigid gear top section convex tooth profile ST2 and the common tangent line segment to obtain a complete rigid gear working tooth profile; calculating a transition curve of the working tooth profile of the rigid wheel tangent to the root circle of the rigid wheel to complete the tooth profile design of the rigid wheel;
5) the coordinate (x) of the convex tooth profile ST2 of the top section of the rigid wheel is expressed by using a parameter equation2(s),y2(s)), coordinates (x) of the convex tooth profile ST2 of the crown section of the rigid wheel are transformed by coordinate transformation2(s),y2(s)) converting the conjugate into a flexible gear coordinate system, substituting into an envelope equation, and performing numerical solution to obtain discrete point coordinates of the theoretical concave conjugate tooth profile CT3 in the range of the bottom section of the flexible gear;
6) by utilizing the discrete point coordinates of the theoretical conjugate concave tooth profile CT3, fitting the concave tooth profile of the bottom section of the flexible gear through least square, and adjusting fitting parameters to enable the discrete points of the theoretical conjugate concave tooth profile CT3 to be positioned on the inner side of a fitting curve so as to ensure that no tooth profile interference occurs; determining a flexible gear tooth height range according to the tooth height coefficient of the flexible gear, and determining a fitting curve in the flexible gear tooth height range as a flexible gear bottom section concave tooth profile SA 1;
7) calculating a common tangent line according to the flexible gear top section convex tooth profile ST1 and the flexible gear bottom section concave tooth profile SA1, and combining the flexible gear top section convex tooth profile ST1, the flexible gear bottom section concave tooth profile SA1 and the common tangent line segment thereof to obtain a complete flexible gear working tooth profile; and calculating a tangent transition curve of the working tooth profile of the flexible gear and the root circle of the flexible gear to complete the tooth profile design of the flexible gear.
The bidirectional conjugate tooth profile design method for harmonic gear transmission is suitable for a double-circular-arc tooth profile formed by a tooth root partial arc section and a tooth top partial arc section, and is also suitable for a non-circular-curve tooth profile meeting the following condition that tooth profiles of a flexible gear and a rigid gear are divided into a top profile and a bottom profile by a pitch circle, wherein the top profile is a convex tooth profile, and the bottom profile is a concave tooth profile.
The invention has the beneficial effects that:
in the method, the flexible gear tooth profile to the rigid gear tooth profile and the flexible gear tooth profile to the flexible gear tooth profile are subjected to bidirectional envelope conjugate calculation, and the designed flexible gear top section convex tooth profile ST1 is in conjugate contact with the rigid gear bottom section concave tooth profile SA2 and the top section convex tooth profile ST2 in the meshing motion, so that secondary conjugate is realized, the range of a meshing area is enlarged, and the number of pairs of teeth simultaneously participating in meshing is increased. The designed concave tooth profile SA1 at the bottom section of the flexible gear is in conjugate contact with the convex tooth profile ST2 of the rigid gear, and meanwhile, the convex tooth profile ST1 at the top section of the flexible gear is in conjugate contact with the concave tooth profile SA2 at the bottom section of the rigid gear, so that double conjugation is realized, the conjugate contact area is increased, the contact stress of the tooth surface is reduced, and the abrasion of the tooth surface is reduced. The torsional rigidity and the transmission precision of the harmonic reducer are improved.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a relative position and coordinate system of harmonic gear driven flexspline and rigid spline teeth.
FIG. 3 is a flexspline addendum convex tooth profile segment in a flexspline coordinate system.
Detailed Description
The invention is described in detail below with reference to the figures and examples.
Taking the design of a double-circular-arc tooth profile of harmonic gear transmission as an example, referring to fig. 1, a bidirectional conjugate tooth profile design method for harmonic gear transmission comprises the following steps:
1) firstly, completing the design of structural parameters according to the requirements of load and the like, and then determining the meshing parameters of the tooth profile required to be designed; as shown in fig. 2, firstly establishing each coordinate system, establishing a wave generator coordinate system { OXY }, wherein the origin is located at the rotation center of the wave generator, the Y axis is coincident with the long axis of the wave generator, and the X axis is coincident with the short axis of the wave generator; establishing a flexible gear tooth coordinate system (o)1x1y1Of y of1The axis coincides with the central line of the flexible gear teeth and has an origin o1Then lies on the flexspline neutral layer curve (i.e., the original curve), x1Axis and y1The axes are mutually vertical and tangent to the original curve; establishing a rigid wheel coordinate system o1x1y1At the origin o2Coincident with point O, y2The central lines of the shaft and the tooth socket of the rigid wheel are coincident, x2Axis and y2The axes are mutually vertical; the distance from a point on the original curve to the origin O is represented as
In the formula (I), the compound is shown in the specification,
is a non-deformed end corner r of the flexible gearmThe radius of a neutral layer of the flexible gear is shown, and omega is the radial deformation of the flexible gear; the tangential deformation is
According to the geometric relationship in the figure, y1And o1o2Is at an included angle of
According to the assumption that the length of the neutral layer of the flexible gear is not changed, the arc lengths before and after the deformation of the flexible gear are equal, and an equation is solved
Neglecting the high-order quantity, obtaining the rotation angle of the flexible gear deformation end as
Calculating to obtain the rotation angle of the rigid wheel according to the transmission ratio
In the formula, z1Number of teeth of flexible gear, z2The number of teeth of the rigid gear; the difference of the rotation angles of the flexible wheel and the rigid wheel is
Included angle between flexible wheel coordinate system and rigid wheel coordinate system
φ＝μ+γ
According to the geometric relation, a coordinate conversion matrix for converting the flexible wheel coordinate system to the rigid wheel coordinate system is obtained
Determining preset parameters according to design parameters of harmonic drive, as shown in FIG. 3, wherein the AB-segment arc is a top-segment convex tooth profile ST1 of the flexible gear, determining the thickness ratio K of the segment circle of the flexible gear and the tooth crest height h of the flexible geara1Height h of convex arc working tooth of flexible gearlAnd the arc dip angle beta to obtain the pitch circle tooth thickness of
In the formula (d)1The radius of the convex tooth profile ST1 of the top section of the flexible gear is the pitch circle diameter of the flexible gear
Circle center position of flexible gear top section convex tooth profile ST1
Determining the coordinate point (x) of the convex tooth profile ST1 of the top section of the flexible gear1(s),y1(s)) a parametric equation of
Wherein s is the arc length from point A and ranges from 0 to rhoa(α1-β)；
2) Coordinates (x) of the flexible gear top segment convex tooth profile ST1 are transformed through coordinates1(s),y1(s)) to the rigid wheel coordinate system,
substitution of envelope equations
Solving to obtain the coordinates of the conjugate tooth profile satisfying the theory
The theoretical conjugate tooth profile obtained here is solved into discrete coordinate points, as shown in fig. 2; can be divided into two solution intervals, namely a theoretical conjugate concave tooth profile CT1 and a theoretical conjugate convex tooth profile CT 2;
3) obtaining a circle center and a radius by fitting a least square circular arc on the theoretical conjugate concave tooth profile CT1, and changing the radius to ensure that all points of the theoretical conjugate concave tooth profile CT1 fall in the circle so as to avoid tooth profile interference; according to the range of the tooth bottom section of the rigid gear, taking the fitting circular arc in the range as a concave tooth profile section SA2 of the rigid gear; in a similar way, the circle center and the radius of the theoretical conjugate convex tooth profile CT2 are obtained through least square circular arc fitting, and the radius is changed to enable all points of the theoretical conjugate convex tooth profile CT2 to fall outside the circle; then according to the tooth crest height of the rigid wheel, taking the fitting circular arc in the top section range as a convex tooth profile section ST2 of the rigid wheel;
4) calculating a common tangent line segment of a concave tooth profile SA2 of a bottom section of the rigid gear and a convex tooth profile ST2 of a top section of the rigid gear to obtain a complete working tooth profile of the rigid gear;
5) the tooth profile coordinate (x) of the convex circular arc tooth profile ST2 of the rigid wheel is in the rigid wheel coordinate system2(s),y2(s)) converting it into a flexspline coordinate system by coordinate transformation
Substitution of envelope equations
Solving to obtain the coordinates of the conjugate tooth profile satisfying the theory
Obtaining a theoretical conjugate concave tooth profile CT3 in the range of the flexible gear tooth bottom;
6) using the discrete point coordinates of the theoretical conjugate concave profile CT3, the conjugate profile point (x) is first aligned1,y1) Performing least square circular arc fitting to obtain a fitting circle center and a radius, and then adjusting the radius to ensure that the CT3 point of the theoretical conjugate concave tooth profile is positioned on the inner side of the fitting circular arc so as to ensure that no tooth profile interference occurs; determining the tooth root height range of the flexible gear according to the tooth height coefficient of the flexible gear, and determining a fitting curve in the tooth height range of the flexible gear as a concave tooth profile SA1 of the bottom section of the flexible gear;
7) calculating a common tangent line according to the flexible gear top section convex tooth profile ST1 and the flexible gear bottom section concave tooth profile SA1, and combining the flexible gear top section convex tooth profile ST1, the flexible gear bottom section concave tooth profile SA1 and the common tangent line segment thereof to obtain a complete flexible gear working tooth profile; and calculating a tangent transition curve of the working tooth profile of the flexible gear and the root circle of the flexible gear to complete the tooth profile design of the flexible gear.
The invention obtains a flexible gear tooth profile and a rigid gear tooth profile: at the deepest meshing position, the whole tooth profile can participate in meshing, a double-conjugate phenomenon exists, the conjugate contact area is increased, the tooth surface contact stress is reduced, and the tooth surface abrasion is reduced; the meshing positions of the tooth profiles of the flexible gear and the rigid gear at two moments of meshing and deepest meshing are known, the convex tooth profile of the flexible gear has a secondary conjugate phenomenon, the range of a meshing area is enlarged, the number of pairs of teeth participating in meshing simultaneously is increased, and the torsional rigidity and the transmission precision of the harmonic reducer are improved.
Claims (2)
1. A bidirectional conjugate tooth profile design method for harmonic gear transmission is characterized by comprising the following steps:
1) firstly, selecting preset parameters according to the design requirements of a harmonic reducer, wherein the preset parameters comprise the inner diameter of a flexible gear, the wall thickness of the flexible gear, the pitch-circle tooth thickness ratio and the tooth height coefficient, the basic design parameters are used for determining the convex tooth profile ST1 of the top section of the flexible gear, and the coordinates (x) of the convex tooth profile ST1 of the top section of the flexible gear are expressed through a parameter equation1(s),y1(s))；
2) Coordinates (x) of the flexible gear top segment convex tooth profile ST1 are transformed through coordinates1(s),y1(s)) converting the conjugate convex tooth profile into a rigid wheel coordinate system, substituting the coordinate system into an envelope equation, and solving through numerical calculation to obtain discrete point coordinates of a theoretical conjugate concave tooth profile CT1 and a theoretical conjugate convex tooth profile CT2 of a flexible wheel top section convex tooth profile ST 1;
3) by utilizing the discrete point coordinates of the theoretical conjugate concave tooth profile CT1, the concave tooth profile of the rigid wheel bottom section is fitted through least squares, and fitting parameters are adjusted to enable the discrete points of the theoretical conjugate concave tooth profile CT1 to be positioned on the inner side of a fitting curve, so that tooth profile interference is avoided; similarly, the fitting rigid gear top section convex tooth profile is obtained through least square fitting by using the discrete point coordinates of the theoretical conjugate convex tooth profile CT2, and fitting parameters are adjusted to enable the theoretical conjugate convex tooth profile CT2 discrete points to be on the outer side of the fitting curve;
determining the working tooth height range of the rigid gear tooth profile according to the tooth height coefficient of the rigid gear, and respectively determining fitting curve segments in the tooth height range of the rigid gear as a concave tooth profile SA2 at the bottom section of the rigid gear and a convex tooth profile ST2 at the top section of the rigid gear;
4) calculating a common tangent line segment according to the rigid gear bottom section concave tooth profile SA2 and the rigid gear top section convex tooth profile ST2, and combining the rigid gear bottom section concave tooth profile SA2, the rigid gear top section convex tooth profile ST2 and the common tangent line segment to obtain a complete rigid gear working tooth profile; calculating a transition curve of the working tooth profile of the rigid wheel tangent to the root circle of the rigid wheel to complete the tooth profile design of the rigid wheel;
5) the coordinate (x) of the convex tooth profile ST2 of the top section of the rigid wheel is expressed by using a parameter equation2(s),y2(s)), coordinates (x) of the convex tooth profile ST2 of the crown section of the rigid wheel are transformed by coordinate transformation2(s),y2(s)) converting the conjugate into a flexible gear coordinate system, substituting into an envelope equation, and performing numerical solution to obtain discrete point coordinates of the theoretical concave conjugate tooth profile CT3 in the range of the bottom section of the flexible gear;
6) by utilizing the discrete point coordinates of the theoretical conjugate concave tooth profile CT3, fitting the concave tooth profile of the bottom section of the flexible gear through least square, and adjusting fitting parameters to enable the discrete points of the theoretical conjugate concave tooth profile CT3 to be positioned on the inner side of a fitting curve so as to ensure that no tooth profile interference occurs; determining a flexible gear tooth height range according to the tooth height coefficient of the flexible gear, and determining a fitting curve in the flexible gear tooth height range as a flexible gear bottom section concave tooth profile SA 1;
7) calculating a common tangent line according to the flexible gear top section convex tooth profile ST1 and the flexible gear bottom section concave tooth profile SA1, and combining the flexible gear top section convex tooth profile ST1, the flexible gear bottom section concave tooth profile SA1 and the common tangent line segment thereof to obtain a complete flexible gear working tooth profile; and calculating a tangent transition curve of the working tooth profile of the flexible gear and the root circle of the flexible gear to complete the tooth profile design of the flexible gear.
2. The method of claim 1, wherein the tooth profile is a tooth profile of a harmonic gear drive, the method comprising: the bidirectional conjugate tooth profile design method for harmonic gear transmission is suitable for a double-circular-arc tooth profile formed by a tooth root partial arc section and a tooth top partial arc section, and is also suitable for a non-circular-curve tooth profile meeting the following condition that tooth profiles of a flexible gear and a rigid gear are divided into a top profile and a bottom profile by a pitch circle, wherein the top profile is a convex tooth profile, and the bottom profile is a concave tooth profile.
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| CN110263492B (en) * | 2019-07-15 | 2023-05-02 | 北京工业大学 | Method for calculating torsional rigidity of double-arc tooth profile of harmonic reducer |
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| JP2015075149A (en) * | 2013-10-08 | 2015-04-20 | キヤノン株式会社 | Process of manufacture of wave gear device |
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| CN204729582U (en) * | 2015-05-13 | 2015-10-28 | 东莞市鑫拓智能机械科技有限公司 | A kind of firm wheel construction of S shape tooth being applied to harmonic speed reducer |
| EP3306132B1 (en) * | 2015-06-02 | 2020-04-01 | Harmonic Drive Systems Inc. | Strain wave gearing device with compound meshing that involves congruity of tooth surfaces |
| JP6587064B2 (en) * | 2016-03-18 | 2019-10-09 | 日立オートモティブシステムズ株式会社 | Actuator and wave gear reducer for link mechanism for internal combustion engine |
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