Gear transmission conjugate tooth profile design method of harmonic reducer

Abstract

The invention discloses a gear transmission conjugate tooth profile design method of a harmonic reducer, which comprises the following steps of: obtaining a motion track curve cluster of a tooth profile to be solved through coordinate conversion based on a geometric relationship of harmonic gear transmission, dispersing the curve cluster into a point set, extracting a two-dimensional space boundary line of the point set by utilizing a boundary function, and calculating a two-dimensional space boundary line of the point set; according to the method, the S-shaped tooth profile and the difference and advantages of the tooth profile meshing performance based on the meshing theory are integrated, the meshing theory and numerical calculation are combined, the meshing performance of the S-shaped tooth profile and the meshing performance of the tooth profile based on the meshing theory are improved, and the meshing performance of the S-shaped tooth profile and the meshing performance of the tooth profile based on the meshing theory is improved. Based on the forming characteristics of an S-shaped tooth profile, a tooth profile design method of effective edge conjugate points is provided, the meshing interval is increased, and two-point meshing and secondary meshing can be achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of harmonic reducers, and in particular to a method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer. Background Art

[0002] Harmonic gear transmissions offer advantages such as high reduction ratios, low noise, lightweight, compact dimensions, and high transmission accuracy. They are currently used in numerous high-tech fields, including robotics, aerospace, optical and medical equipment, precision machining centers, and military equipment. The tooth profiles of the flexspline and rigid-spindle are key factors influencing the transmission performance of harmonic reducers. Therefore, gear tooth profile design that achieves high meshing transmission performance has been a research hotspot for scholars both domestically and internationally. In-depth research has been conducted both domestically and internationally on the meshing theory and tooth profile design of harmonic gear transmissions.

[0003] Existing technical solutions are divided into two categories:

[0004] ①S-shaped tooth profile design

[0005] Step 1: Construct the gear tooth surface coordinate system and establish the kinematic model based on the gear-rack mapping relationship;

[0006] Step 2: Derive the parametric equation of the S-shaped tooth profile based on the geometric constraints of the rigid wheel tooth tip circle and the flex spline tooth root circle;

[0007] Step 3: Use cubic spline curve to fit the continuous tooth profile to ensure the curvature continuity of the tooth top-tooth root transition zone;

[0008] Step 4: Verify the single tooth meshing contact stress distribution through finite element simulation.

[0009] The advantage of this solution is: achieving continuous contact of single tooth profile

[0010] The limitation of this solution is that the failure rate of conjugate solution caused by complex curvature equation is greater than 25%.

[0011] ②Conjugate meshing theory design

[0012] Step 1: Construct the elastic deformation field of the flexible wheel

[0013] Step 2: Introduce the instantaneous center line method to solve the conjugate contact conditions

[0014] Step 3: Establish tooth contact equation

[0015] The advantage of this solution is that it is easy to achieve two-point meshing

[0016] The limitations of this solution are: the complex tooth profile equation leads to divergence of the numerical solution; the tooth profile accuracy loss rate in actual processing is greater than 15%

[0017] In summary, existing solutions present a conflict between improved meshing performance and engineering feasibility: while the S-shaped tooth profile offers good process adaptability, it is limited by meshing characteristics; while conjugate theory can optimize meshing performance, it faces challenges in mathematical modeling. This technical conflict provides a technical entry point for the innovation of new tooth profile design methods. Summary of the Invention

[0018] The purpose of this section is to summarize some aspects of the embodiments of the present invention and briefly introduce some preferred embodiments. Some simplifications or omissions may be made in this section and the abstract and title of this application to avoid blurring the purpose of this section, the abstract and the title of the invention, and such simplifications or omissions should not be used to limit the scope of the present invention.

[0019] Therefore, the purpose of the present invention is to provide a method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer, which increases the meshing range and can achieve two-point meshing and secondary meshing.

[0020] To solve the above technical problems, according to one aspect of the present invention, the present invention provides the following technical solutions:

[0021] A method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer, comprising the following steps:

[0022] S1. Based on the geometric relationship of the harmonic gear transmission, a motion trajectory curve cluster of the tooth profile to be solved is obtained through coordinate transformation, and the curve cluster is discretized into a point set;

[0023] S2. using a boundary function to extract the two-dimensional spatial boundary line of the point set, screening out points that satisfy the meshing equation from the boundary line, and obtaining a valid edge conjugate point set;

[0024] S3. Fitting the effective edge conjugate point set to obtain a conjugate tooth profile.

[0025] As a preferred solution of the method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer according to the present invention, the coordinate system is established based on the geometric relationship of the harmonic gear transmission, including:

[0026] Establish a dynamic coordinate system fixedly connected to the wave generator, a dynamic coordinate system fixedly connected to the flexspline, and a fixed coordinate system fixedly connected to the rigid wheel;

[0027] The origin of the wave generator coordinate system coincides with the origin of the rigid wheel coordinate system, and the origin of the flexspline coordinate system is located on the flexspline neutral layer curve in the wave generator coordinate system.

[0028] As a preferred solution of the method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer according to the present invention, the coordinate transformation includes:

[0029] The coordinate transformation matrix from the flexible pulley coordinate system to the rigid pulley coordinate system is expressed as:

[0030]

[0031] The coordinate expression of the flexspline tooth profile in the flexspline coordinate system in the rigid wheel coordinate system is:

[0032]

[0033] Where x1 and y1 are the coordinates of the flexspline tooth profile in coordinate system S1, and x1′ and y1′ are the coordinates of the flexspline tooth profile in coordinate system S2;

[0034] The coordinate transformation matrix from the rigid wheel coordinate system to the rigid wheel coordinate system is expressed as:

[0035]

[0036] The coordinate expression of the rigid wheel tooth profile coordinate point in the flexible wheel coordinate system is:

[0037]

[0038] Where x2 and y2 are the coordinates of the wheel tooth profile in the coordinate system S2, and x′2 and y′2 are the coordinates of the wheel tooth profile in the coordinate system S1.

[0039] As a preferred solution of the method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer according to the present invention, the method for obtaining the motion trajectory curve cluster includes:

[0040] When the flexspline is fixed and the rigid spline is moving, the motion trajectory curve cluster of the rigid spline tooth profile in the fixed coordinate system is obtained;

[0041] When the rigid wheel is fixed and the flexspline is moving, the motion trajectory curve cluster of the flexspline tooth profile in the fixed coordinate system is obtained.

[0042] As a preferred solution of the gear transmission conjugate tooth profile design method of the harmonic reducer described in the present invention, the meshing equation is: i ·v i =0, where n i is the common normal vector of the two conjugate surfaces at the contact point, v i is the relative velocity vector.

[0043] As a preferred solution of the method for designing the conjugate tooth profile of the gear transmission of a harmonic reducer described in the present invention, when the flexspline is fixed and the rigid wheel is moving, the meshing equation is expressed in the rigid wheel coordinate system as follows:

[0044] Among them, W 12is the base vector transformation matrix from the rigid wheel coordinate system to the flexspline coordinate system, n2 is the normal vector of the rigid wheel tooth profile in the rigid wheel coordinate system, and v1 is the velocity vector of the flexspline relative to the rigid wheel.

[0045] As a preferred solution of the conjugate tooth profile design method for the gear transmission of a harmonic reducer described in the present invention, when the rigid wheel is fixed and the flexspline is moving, the meshing equation is expressed in the flexspline coordinate system as follows:

[0046] Among them, W 21 is the base vector transformation matrix from the flexspline coordinate system to the rigid wheel coordinate system, n1 is the normal vector of the flexspline tooth profile in the flexspline coordinate system, and v2 is the velocity vector of the steel wheel relative to the flexspline.

[0047] As a preferred solution of the conjugate tooth profile design method for a gear transmission of a harmonic reducer described in the present invention, the process of extracting the boundary line using the boundary function includes: calculating the boundary of a point set in two-dimensional space using the boundary function in Matlab software, the expression is k = boundary(x, y), where (x, y) is the coordinate of the point set and k is the boundary point index vector.

[0048] As a preferred solution of the gear transmission conjugate tooth profile design method of a harmonic reducer described in the present invention, the screening conditions of the effective edge conjugate point set include: satisfying the condition that the arc length of the neutral layer of the flexible wheel before and after deformation is equal, the expression is where r m is the radius of the neutral layer of the undeformed flexspline, and r(θ) is the curve expression of the neutral layer of the flexspline after deformation.

[0049] As a preferred solution of the gear transmission conjugate tooth profile design method of a harmonic reducer described in the present invention, the process of fitting to obtain the conjugate tooth profile includes: using a cubic spline curve to fit the effective edge conjugate point set to ensure the continuity of the curvature of the tooth top-tooth root transition zone.

[0050] Compared with the existing technology, the present invention has the following beneficial effects: S-shaped tooth profiles are easy to realize but have limited performance, while conjugate theory has excellent performance but is difficult to implement. The present invention finds a breakthrough point in this contradiction, which is to expand the meshing area and reduce stress while ensuring that the design method is solvable and machinable. By integrating the differences and advantages of S-shaped tooth profiles and those based on meshing theory in terms of tooth profile meshing performance, the present invention combines meshing theory with numerical calculations. Based on the formation characteristics of S-shaped tooth profiles, a tooth profile design method with effective edge conjugate points is proposed, which increases the meshing range and can realize two-point meshing and secondary meshing. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the present invention will be described in detail below with reference to the accompanying drawings and detailed embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be derived from these drawings without inventive effort. Among them:

[0052] Figure 1 Schematic diagram of the harmonic gear transmission coordinate system provided by the present invention. DETAILED DESCRIPTION

[0053] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the specific embodiments of the present invention are described in detail below with reference to the accompanying drawings.

[0054] The present invention provides a method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer, which increases the meshing range and can realize two-point meshing and secondary meshing.

[0055] A method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer, comprising the following steps:

[0056] S1. Based on the geometric relationship of the harmonic gear transmission, a motion trajectory curve cluster of the tooth profile to be solved is obtained through coordinate transformation, and the curve cluster is discretized into a point set;

[0057] S2. using a boundary function to extract the two-dimensional spatial boundary line of the point set, screening out points that satisfy the meshing equation from the boundary line, and obtaining a valid edge conjugate point set;

[0058] S3. Fitting the effective edge conjugate point set to obtain a conjugate tooth profile.

[0059] In step S1, the geometric relationship of the harmonic gear transmission is used to establish a coordinate system, which includes the following steps: establishing a dynamic coordinate system fixed to the wave generator, a dynamic coordinate system fixed to the flexspline, and a fixed coordinate system fixed to the rigid wheel, and the origin of the wave generator coordinate system coincides with the origin of the rigid wheel coordinate system, and the origin of the flexspline coordinate system is located on the neutral layer curve of the flexspline in the wave generator coordinate system, wherein the coordinate system is as follows: Figure 1 As shown in the figure, S0{OXY} and S1{O1 X1 Y1} are the moving coordinate systems of the wave generator and flexspline, respectively; S2{O2 X2 Y2} is the fixed coordinate system fixed to the rigid pulley. The origin O of the wave generator coordinate system S0 coincides with the origin O2 of the rigid pulley coordinate system S2, and the origin O2 of the flexspline coordinate system S1 coincides with the origin O2 of the flexspline coordinate system S1. The origin O1 of the flexspline coordinate system S1 lies on the neutral layer curve of the flexspline in the wave generator coordinate system S0. Figure 1 middle, is the rotation angle of the flexspline deformation end relative to the center axis of the wave generator coordinate system; is the rotation angle of the undeformed end of the flexspline relative to the Y-axis in the wave generator coordinate system; is the rotation angle of the wave generator relative to the rigid pulley; μ is the normal rotation angle of the flexspline; γ is the rotation angle of the flexspline deformation end relative to the Y2 axis; β is the normal rotation angle of the flexspline relative to the rigid pulley. The angle direction is positive in the counterclockwise direction and negative in the clockwise direction.

[0060] Based on the basic assumption of harmonic gear transmission, the neutral layer arc length of the flexspline is equal before and after deformation, so we can get

[0061]

[0062] Where, is the radius of the neutral layer of the undeformed flexspline; r is the radius of the neutral layer of the undeformed flexspline; The curve expression of the neutral layer of the flexible pulley after deformation with the independent variable being the deformation variable.

[0063] According to the kinematic theory of harmonic gear transmission friction model, and Has the following relationship:

[0064]

[0065] Where zr and zg are the number of teeth of the flexspline and rigid spline respectively.

[0066] according to Figure 1 The geometric relationship shown in Figure 2 is as follows: μ, γ, and β can be expressed as

[0067]

[0068] Will As the independent variable of all motion parameters, the subsequent conjugate tooth profile of harmonic gear transmission can be accurately solved.

[0069] In order to obtain the motion state of the flexspline tooth profile relative to the rigid wheel tooth profile to intuitively reflect the meshing condition of the harmonic gear in the assembly deformation state, the motion trajectory curve cluster of the rigid wheel tooth profile in the fixed coordinate system when the flexspline is fixed and the rigid wheel is in motion, or the motion trajectory curve cluster of the flexspline teeth in the fixed coordinate system when the rigid wheel is fixed and the flexspline is in motion, can be obtained. Figure 1 The geometric relationship between the main components of the harmonic gear transmission shown in the figure, the coordinate transformation matrix M from the flexible wheel coordinate system S1 to the rigid wheel coordinate system S2 21 It can be expressed as:

[0070]

[0071] Coordinate transformation matrix M from the rigid wheel coordinate system S1 to the rigid wheel coordinate system S2 21 It can be expressed as:

[0072]

[0073] Combined with the above expressions, the coordinate expression of the flexspline tooth profile in the flexspline coordinate system S1 in the rigid wheel coordinate system S2 is:

[0074]

[0075] Where x1 and y1 are the coordinates of the flexspline tooth profile in coordinate system S1; x1′ and y1′ are the coordinates of the flexspline tooth profile in coordinate system S2.

[0076] Similarly, the coordinate expression of the rigid wheel tooth profile coordinate point in the flexible wheel coordinate system S1 is:

[0077]

[0078] Where x2 and y2 are the coordinates of the wheel tooth profile in the coordinate system S2; x′2 and y′2 are the coordinates of the wheel tooth profile in the coordinate system S1.

[0079] Next is the effective edge conjugate point tooth profile design:

[0080] In Matlab software, the boundary function can be used to calculate the boundary of a point in two-dimensional space, k = boundary(x, y). Among them, point (x, y) is the point index vector of a single compatible two-dimensional boundary, and the points [x(k), y(k)] constitute the boundary, and the boundary can be contracted inward to surround these points. Based on the boundary function, the boundary line of the cluster of points in two-dimensional space is obtained, and not all the points obtained can be used as conjugate points of the tooth profile. According to the kinematic method, at the contact point of the mutually enveloping tooth profile, the velocity vector of the relative motion should be perpendicular to the normal vector of the tooth profile, that is, the two surfaces must satisfy the following meshing equation at the contact point:

[0081] n i ·v i =0(9)

[0082] Where n i and V i The coordinate system S i The common normal vector and relative velocity vector of the two conjugate surfaces at the contact point; i = 0, 1, 2.

[0083] When the rigid wheel is fixed and the flexible wheel is moving, in the coordinate system S2, according to formula (9), substitute n2=W 21 n1 and We can get:

[0084]

[0085] Where W21 is the base vector transformation matrix from the flexible pulley coordinate system S1 to the rigid pulley coordinate system S2, and its expression is:

[0086]

[0087] Similarly, when the flexspline is fixed and the rigid wheel is moving, in the coordinate system S1, according to formula (9), substitute n1=W 12 n2 and We can get:

[0088]

[0089] Where W12 is the base vector transformation matrix from the rigid wheel coordinate system S2 to the flexible wheel coordinate system S1, and its expression is:

[0090]

[0091] According to the collection point criteria determined by equations (10) and (12), the point set of effective edge conjugate points can be obtained by using the boundary function.

[0092] In summary, the effective edge conjugate point tooth profile design method is as follows: based on the geometric relationship of the harmonic gear transmission, the motion trajectory curve cluster of the tooth profile to be solved is obtained through coordinate transformation, and it is discretized into a point set; then the boundary line of the two-dimensional space point cluster is extracted according to the boundary function, and the points that meet equations (10) and (12) are screened out to obtain the effective edge conjugate point set, and the smooth curve obtained by fitting is the conjugate tooth profile. Among them, the screening conditions of the effective edge conjugate point set include: satisfying the condition that the arc length of the neutral layer of the flexspline before and after deformation is equal, and the expression is: where r m is the radius of the neutral layer of the undeformed flexspline, r(θ) is the curve expression of the neutral layer of the deformed flexspline, and the process of fitting the conjugate tooth profile includes: fitting the effective edge conjugate point set using a cubic spline curve to ensure the continuity of the curvature of the tooth top-tooth root transition zone.

[0093] Although the present invention has been described above with reference to embodiments, various modifications may be made thereto and equivalent components may be substituted without departing from the scope of the present invention. In particular, as long as there are no structural conflicts, the various features of the embodiments disclosed herein may be combined with each other in any manner, and the omission of an exhaustive description of such combinations in this specification is solely for the sake of space and resource conservation. Therefore, the present invention is not limited to the specific embodiments disclosed herein, but includes all technical solutions falling within the scope of the claims.

Claims

1. A method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer, characterized in that: Here are the steps: S1. Based on the geometric relationship of the harmonic gear transmission, a motion trajectory curve cluster of the tooth profile to be solved is obtained through coordinate transformation, and the curve cluster is discretized into a point set; S2. using a boundary function to extract the two-dimensional spatial boundary line of the point set, screening out points that satisfy the meshing equation from the boundary line, and obtaining a valid edge conjugate point set; S3. Fitting the effective edge conjugate point set to obtain a conjugate tooth profile.

2. The method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer according to claim 1, characterized in that: The coordinate system is established based on the geometric relationship of the harmonic gear transmission, including: Establish a dynamic coordinate system fixedly connected to the wave generator, a dynamic coordinate system fixedly connected to the flexspline, and a fixed coordinate system fixedly connected to the rigid wheel; The origin of the wave generator coordinate system coincides with the origin of the rigid wheel coordinate system, and the origin of the flexspline coordinate system is located on the flexspline neutral layer curve in the wave generator coordinate system.

3. The method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer according to claim 1, characterized in that: The coordinate transformation includes: The coordinate transformation matrix from the flexible pulley coordinate system to the rigid pulley coordinate system is expressed as: The coordinate expression of the flexspline tooth profile in the flexspline coordinate system in the rigid wheel coordinate system is: Where x1 and y1 are the coordinates of the flexspline tooth profile in coordinate system S1, and x1′ and y1′ are the coordinates of the flexspline tooth profile in coordinate system S2; The coordinate transformation matrix from the rigid wheel coordinate system to the rigid wheel coordinate system is expressed as: The coordinate expression of the rigid wheel tooth profile coordinate point in the flexible wheel coordinate system is: Where x2 and y2 are the coordinates of the wheel tooth profile in the coordinate system S2, and x′2 and y′2 are the coordinates of the wheel tooth profile in the coordinate system S1.

4. The method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer according to claim 1, characterized in that: The method for obtaining the motion trajectory curve cluster includes: When the flexspline is fixed and the rigid spline is moving, the motion trajectory curve cluster of the rigid spline tooth profile in the fixed coordinate system is obtained; When the rigid wheel is fixed and the flexspline is moving, the motion trajectory curve cluster of the flexspline tooth profile in the fixed coordinate system is obtained.

5. The method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer according to claim 1, characterized in that: The meshing equation is: i ·v i =0, where n i is the common normal vector of the two conjugate surfaces at the contact point, v i is the relative velocity vector.

6. The method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer according to claim 5, characterized in that: When the flexspline is fixed and the rigid wheel is moving, the meshing equation in the rigid wheel coordinate system is expressed as: Among them, W 12 is the base vector transformation matrix from the rigid wheel coordinate system to the flexspline coordinate system, n2 is the normal vector of the rigid wheel tooth profile in the rigid wheel coordinate system, and v1 is the velocity vector of the flexspline relative to the rigid wheel.

7. The method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer according to claim 5, characterized in that: When the rigid wheel is fixed and the flexspline is moving, the meshing equation in the flexspline coordinate system is expressed as: Among them, W 21 is the base vector transformation matrix from the flexspline coordinate system to the rigid wheel coordinate system, n1 is the normal vector of the flexspline tooth profile in the flexspline coordinate system, and v2 is the velocity vector of the steel wheel relative to the flexspline.

8. The method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer according to claim 1, characterized in that: The process of extracting boundary lines using the boundary function includes: calculating the boundary of a point set in two-dimensional space using the boundary function in Matlab software, the expression is k=boundary(x,y), where (x,y) is the coordinates of the point set and k is the boundary point index vector.

9. The method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer according to claim 1, characterized in that: The screening conditions of the effective edge conjugate point set include: satisfying the condition that the arc length of the neutral layer of the flexible wheel before and after deformation is equal, and the expression is: where r m is the radius of the neutral layer of the undeformed flexspline, and r(θ) is the curve expression of the neutral layer of the flexspline after deformation.

10. The method for designing a conjugate tooth profile of a gear transmission of a harmonic reducer according to claim 1, characterized in that: The process of fitting to obtain the conjugate tooth profile includes: fitting the effective edge conjugate point set using a cubic spline curve to ensure the curvature continuity of the tooth top-tooth root transition zone.

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