CN110245417B - Method for calculating normal slope of meshing point of double-arc tooth profile of harmonic reducer - Google Patents

Method for calculating normal slope of meshing point of double-arc tooth profile of harmonic reducer Download PDF

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CN110245417B
CN110245417B CN201910507971.8A CN201910507971A CN110245417B CN 110245417 B CN110245417 B CN 110245417B CN 201910507971 A CN201910507971 A CN 201910507971A CN 110245417 B CN110245417 B CN 110245417B
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equation
tooth profile
double
arc
normal slope
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CN110245417A (en
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刘志峰
张涛
杨聪彬
张彩霞
胡秋实
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Beijing University of Technology
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02—Toothed members; Worms
    • F16H55/17—Toothed wheels
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02P—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN THE PRODUCTION OR PROCESSING OF GOODS
    • Y02P70/00—Climate change mitigation technologies in the production process for final industrial or consumer products
    • Y02P70/10—Greenhouse gas [GHG] capture, material saving, heat recovery or other energy efficient measures, e.g. motor control, characterised by manufacturing processes, e.g. for rolling metal or metal working

Abstract

The invention discloses a method for calculating the normal slope of a meshing point of a double-arc tooth profile of a harmonic reducer, which comprises the following steps: and deducing a harmonic reducer rigid gear tooth profile equation according to the double-arc flexible gear tooth profile equation and the double-arc conjugate envelope theory. Step two: angle of different wave generators of rigid gear tooth form in single tooth envelope calculation process
Figure DDA0002092472030000011
Normal slope at; according to the invention, through analyzing the rigid-flexible gear conjugate envelope theory of the harmonic speed reducer, through solving the equation of rigid gear equation and analyzing the equation of normal slope, the mathematical characterization of the normal slope of the meshing point of the harmonic speed reducer under the condition of different wave generator rotation angles is obtained, and a theoretical basis is laid for the characterization of the pressure angle of the harmonic speed reducer and the improvement of transmission efficiency.

Description

Method for calculating normal slope of meshing point of double-arc tooth profile of harmonic reducer
Technical Field
The invention relates to the technical field of design and manufacture of harmonic reducers, in particular to a method for calculating normal slope of a meshing point of a double-arc tooth profile of a harmonic reducer.
Background
The harmonic reducer is a core element of the robot joint, and the engagement of the flexible gear and the rigid gear of the harmonic reducer belongs to small-modulus multi-tooth engagement under the condition of large deformation. The transmission efficiency and torsional rigidity of the harmonic reducer are related to the number of teeth engaged during the meshing process of the rigid-flexible gear, the contact stress of the meshing point and the speed of the meshing point. At present, the double-arc tooth form is one of tooth forms which are widely applied and have better performance parameters in the transmission of the harmonic speed reducer, but most of researches adopt a finite element simulation technology to acquire the distribution of meshing tooth pairs of the harmonic speed reducer and the rule of meshing point normals, but the simulation method assumes too many conditions and cannot effectively support a force distribution model in the meshing process, so that a calculation method for the slopes of the meshing point normals of the double-arc tooth form of the harmonic speed reducer is provided.
Disclosure of Invention
The invention aims at: in order to improve the meshing quality of a rigid-flexible gear of a harmonic speed reducer, a calculation method for the normal slope of a double-arc tooth-shaped meshing point of the harmonic speed reducer is provided by analyzing a mathematical expression of a rigid-gear equation in the enveloping process of the flexible gear of the harmonic speed reducer.
The technical scheme adopted by the invention is as follows:
a method for calculating the normal slope of a meshing point of a double-arc tooth profile of a harmonic reducer comprises the following steps: and deducing a harmonic reducer rigid gear tooth profile equation according to the double-arc flexible gear tooth profile equation and the double-arc conjugate envelope theory.
Figure BDA0002092472010000011
Wherein x is g And y g The abscissa and ordinate of the rigid wheel are respectively indicated. X is x r And y r The abscissa and ordinate of the flexspline are indicated. Beta represents an included angle between the central axis of the flexible gear meshing gear teeth and the vertical direction of the rigid gear fixed coordinate system,
Figure BDA0002092472010000012
the included angle between the sagittal diameter of the intersection point of the central shaft of the flexible gear meshing gear tooth and the neutral layer and the origin of the fixed coordinate system and the vertical direction is shown in +.>
Figure BDA0002092472010000021
The angle of the wave generator is represented, s represents the arc length parameter of the flexible gear profile, ρ represents the sagittal diameter of the flexible gear after the curve of the neutral layer of the flexible gear is deformed.
Step two: angle of different wave generators of rigid gear tooth form in single tooth envelope calculation process
Figure BDA0002092472010000022
Normal slope at:
Figure BDA0002092472010000023
/>
k g is expressed by
Figure BDA0002092472010000024
The solution steps of the composed hidden function equation are as follows.
Step 2.1:
Figure BDA0002092472010000025
also by the flexible gear tooth profile arc length parameter s and the wave generator rotation angle +>
Figure BDA0002092472010000026
The composed hidden function is first generated by the method of +.>
Figure BDA0002092472010000027
Equal interval values in the definition domain are calculated by dichotomy>
Figure BDA0002092472010000028
Corresponding tooth profile parameters s and form a matrix +.>
Figure BDA0002092472010000029
Step 2.2: equation(s)
Figure BDA00020924720100000210
Two sides are simultaneously opposite to (or->
Figure BDA00020924720100000211
Deriving and obtaining->
Figure BDA00020924720100000212
Is a matrix +.>
Figure BDA00020924720100000213
The carry-in equation solves the different +.>
Figure BDA00020924720100000214
Corresponding matrix
Figure BDA00020924720100000215
Step 2.3: matrix is formed
Figure BDA00020924720100000216
Carry in->
Figure BDA00020924720100000217
Solving to obtain the rotation angles of different wave generators>
Figure BDA00020924720100000218
Corresponding meshing point normal slope k g 。
Step three: by rotation angle based on single tooth conjugation
Figure BDA00020924720100000219
Obtaining normal slopes corresponding to different wave generator corners in the multi-tooth meshing process:
Figure BDA00020924720100000220
the invention has the advantages and positive effects that:
according to the invention, through analyzing the rigid-flexible gear conjugate envelope theory of the harmonic speed reducer, through solving the equation of rigid gear equation and analyzing the equation of normal slope, the mathematical characterization of the normal slope of the meshing point of the harmonic speed reducer under the condition of different wave generator rotation angles is obtained, and a theoretical basis is laid for the characterization of the pressure angle of the harmonic speed reducer and the improvement of transmission efficiency.
Drawings
FIG. 1 illustrates the direction of force applied to the engagement point of a rigid-flexible gear of a harmonic reducer;
fig. 2 is a flow of calculation of the single tooth normal slope of the harmonic reducer.
Detailed Description
For a further understanding of the invention, its features and advantages, reference is now made to the following examples, which are illustrated in the accompanying drawings in which:
a method for calculating the normal slope of a meshing point of a double-arc tooth profile of a harmonic reducer comprises the following steps:
step one: as shown in fig. 1, the harmonic reducer rigid gear tooth profile equation is deduced through a double-arc flexible gear tooth profile equation and a double-arc conjugate envelope theory.
Figure BDA0002092472010000031
Wherein x is g And y g The abscissa and ordinate of the rigid wheel are respectively indicated. X is x r And y r The abscissa and ordinate of the flexspline are indicated. Beta represents an included angle between the central axis of the flexible gear meshing gear teeth and the vertical direction of the rigid gear fixed coordinate system,
Figure BDA0002092472010000032
the included angle between the sagittal diameter of the intersection point of the central shaft of the flexible gear meshing gear tooth and the neutral layer and the origin of the fixed coordinate system and the vertical direction is shown in +.>
Figure BDA0002092472010000033
The angle of the wave generator is represented, s represents the arc length parameter of the flexible gear profile, ρ represents the sagittal diameter of the flexible gear after the curve of the neutral layer of the flexible gear is deformed.
Step two: angle of different wave generators of rigid gear tooth form in single tooth envelope calculation process
Figure BDA0002092472010000034
Normal slope at (a) as shown in fig. 2:
Figure BDA0002092472010000035
k g is expressed by
Figure BDA0002092472010000041
The present invention will therefore describe the solving step of this hidden function.
Step 2.1:
Figure BDA0002092472010000042
also by the parameters s and +.>
Figure BDA0002092472010000043
The composed hidden function is first generated by the method of +.>
Figure BDA0002092472010000044
Equal interval values in the definition domain are calculated by dichotomy>
Figure BDA0002092472010000045
Corresponding tooth profile parameters s and form a matrix +.>
Figure BDA0002092472010000046
Step 2.2: equation(s)
Figure BDA0002092472010000047
Two sides are simultaneously opposite to (or->
Figure BDA0002092472010000048
Deriving and obtaining->
Figure BDA0002092472010000049
Is a matrix +.>
Figure BDA00020924720100000410
The carry-in equation solves the different +.>
Figure BDA00020924720100000411
Corresponding matrix
Figure BDA00020924720100000412
Step 2.3: matrix is formed
Figure BDA00020924720100000413
Carry in->
Figure BDA00020924720100000414
Solving to obtain the rotation angles of different wave generators>
Figure BDA00020924720100000415
Corresponding meshing point normal slope k g 。
Step three: by rotation angle based on single tooth conjugation
Figure BDA00020924720100000416
Obtaining normal slopes corresponding to different wave generator corners in the multi-tooth meshing process: />
Figure BDA00020924720100000417
The invention has the advantages and positive effects that:
according to the invention, through analyzing the rigid-flexible gear conjugate envelope theory of the harmonic speed reducer, through solving the equation of rigid gear equation and analyzing the equation of normal slope, the mathematical characterization of the normal slope of the meshing point of the harmonic speed reducer under the condition of different wave generator rotation angles is obtained, and a theoretical basis is laid for the characterization of the pressure angle of the harmonic speed reducer and the improvement of transmission efficiency.

Claims (1)

1. A method for calculating the normal slope of a meshing point of a double-arc tooth profile of a harmonic reducer is characterized by comprising the following steps of: the method comprises the following steps of: deducing a harmonic reducer rigid gear tooth profile equation according to the double-arc flexible gear tooth profile equation and the double-arc conjugate envelope theory;
Figure FDA0004155206710000011
wherein x is g And y g Respectively representThe abscissa and ordinate of the rigid wheel; x is x r And y r The abscissa and the ordinate of the flexspline are represented; beta represents an included angle between the central axis of the flexible gear meshing gear teeth and the vertical direction of the rigid gear fixed coordinate system,
Figure FDA0004155206710000012
the included angle between the vector diameter and the vertical direction of the two points, namely the intersection point of the central axis of the flexible gear meshing gear tooth and the neutral layer and the origin of the fixed coordinate system, is +.>
Figure FDA0004155206710000013
The angle of the wave generator is represented, s represents a flexible gear tooth profile arc length parameter, ρ represents a sagittal diameter of a flexible gear neutral layer curve after deformation;
step two: angle of different wave generators of rigid gear tooth form in single tooth envelope calculation process
Figure FDA0004155206710000014
Normal slope at meshing point:
Figure FDA0004155206710000015
k g is expressed by
Figure FDA0004155206710000016
The solution steps of the composed hidden function equation are as follows;
step 2.1:
Figure FDA0004155206710000017
also by the flexible gear tooth profile arc length parameter s and the wave generator rotation angle +>
Figure FDA0004155206710000018
The composed hidden function is first generated by the method of +.>
Figure FDA0004155206710000019
Equal interval values in the definition domain are calculated by dichotomy>
Figure FDA00041552067100000110
Corresponding flexible tooth profile arc length parameters s and form a matrix +.>
Figure FDA00041552067100000111
Step 2.2: equation(s)
Figure FDA00041552067100000112
Two sides are simultaneously opposite to (or->
Figure FDA00041552067100000113
Deriving and obtaining->
Figure FDA00041552067100000114
Is a matrix +.>
Figure FDA0004155206710000021
The carry-in equation solves the different +.>
Figure FDA0004155206710000022
Corresponding matrix->
Figure FDA0004155206710000023
Step 2.3: matrix is formed
Figure FDA0004155206710000024
Carry in->
Figure FDA0004155206710000025
Solving to obtain the rotation angles of different wave generators>
Figure FDA0004155206710000026
Corresponding meshing point normal slope k g ；
Step three: by rotation angle based on single tooth conjugation
Figure FDA0004155206710000027
Obtaining normal slopes corresponding to different wave generator corners in the multi-tooth meshing process: />
Figure FDA0004155206710000028
/>
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CN110688614B (en) * 2019-09-18 2022-10-28 北京工业大学 Multi-tooth meshing composite stress solving method for cup-shaped flexible wheel of harmonic reducer
CN110688716B (en) * 2019-09-23 2021-03-26 大连理工大学 Method for obtaining harmonic gear transmission conjugate profile based on rotation transformation
CN113486476B (en) * 2021-08-11 2023-04-18 重庆大学 Grinding wheel tooth profile design method for grinding double-arc harmonic reducer rigid wheel slotting tool
CN114110136B (en) * 2021-11-30 2024-01-26 重庆大学 Method for designing internal tooth profile of complex wave type movable tooth speed reducer and two-stage speed reducer

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Publication number Priority date Publication date Assignee Title
US5662008A (en) * 1993-08-30 1997-09-02 Teijin Seiki Boston, Inc. Extended contact harmonic drive devices
CN101135357A (en) * 2006-08-31 2008-03-05 辛洪兵 Harmonic gear power transmission with double circular arc tooth outline
CN104074948A (en) * 2014-07-02 2014-10-01 天津工业大学 Cup-shaped harmonic gear with common tangent type double-circular arc tooth profile and tooth profile design method of gear
CN106641183A (en) * 2016-12-28 2017-05-10 重庆大学 Design method of harmonic drive rack approximation tooth profile
WO2017215621A1 (en) * 2016-06-16 2017-12-21 南通慧幸智能科技有限公司 Tooth profile design method for three-dimensional high-rigidity harmonic speed reducer
CN109630652A (en) * 2019-01-08 2019-04-16 四川大学 A kind of three-arc harmonic wave wheel gear shaped cutter and its tooth Profile Design method

Patent Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US5662008A (en) * 1993-08-30 1997-09-02 Teijin Seiki Boston, Inc. Extended contact harmonic drive devices
CN101135357A (en) * 2006-08-31 2008-03-05 辛洪兵 Harmonic gear power transmission with double circular arc tooth outline
CN104074948A (en) * 2014-07-02 2014-10-01 天津工业大学 Cup-shaped harmonic gear with common tangent type double-circular arc tooth profile and tooth profile design method of gear
WO2017215621A1 (en) * 2016-06-16 2017-12-21 南通慧幸智能科技有限公司 Tooth profile design method for three-dimensional high-rigidity harmonic speed reducer
CN106641183A (en) * 2016-12-28 2017-05-10 重庆大学 Design method of harmonic drive rack approximation tooth profile
CN109630652A (en) * 2019-01-08 2019-04-16 四川大学 A kind of three-arc harmonic wave wheel gear shaped cutter and its tooth Profile Design method

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