Design method of precise meshing harmonic tooth profile

Abstract

The invention relates to a design method for precisely meshing harmonic tooth profiles, and belongs to the technical field of mechanical transmission. Comprising the steps of establishing a harmonic drive meshing theory and a coordinate conversion system; the tooth profile curve of a basic tooth profile convex tooth profile is determined, and the basic tooth profile is a convex tooth profile curve of a flexible gear or a convex tooth profile curve of a rigid gear; on the basis of a two-time contact phenomenon of one-time enveloping, a continuous concave-convex complete tooth profile curve of a rigid gear or a flexible gear is formed after the basic tooth profile is enveloped for one time; and on the basis of a secondary envelope principle of harmonic transmission, a continuous concave-convex complete tooth profile curve with a two-point contact phenomenon of the flexible gear or the rigid gear is obtained. The tooth profile design is continuously completed in a concave-convex mode, accurate meshing can be achieved, and the problem that in the prior art, approximate meshing transmission precision is insufficient is solved; the problem that only a single basic tooth form fitting an S tooth form exists in the current market is solved, and a theoretical foundation is laid for the design of a harmonic drive accurate meshing new tooth form.

Description

Technical Field

[0001] The present invention relates to a design method for precisely meshing harmonic tooth profiles, belonging to the technical field of mechanical transmission. Background Art

[0002] Due to the advantages of high reduction ratio, small size, light weight, high precision, small backlash, high efficiency, and low noise, harmonic gear reducers are widely used in precision industries such as robotics, aerospace, etc. The harmonic gear meshing pair mainly relies on the continuous and controllable elastic deformation of the thin-walled flexspline under the action of the wave generator to mesh with the rigid gear tooth profile, thereby completing the transmission of rotational speed and torque. In order to achieve or improve the tooth meshing performance in harmonic drive, cycloid, double circular arc tooth profiles, and triple circular arc tooth profiles have been applied to harmonic gear tooth profiles in the prior art. For example, 1) Chinese Patent, publication number: CN110020509A, patent name: "A harmonic gear with a variable coefficient cycloid tooth profile", in which the rigid gear and the flexspline with a variable coefficient cycloid tooth profile have line contact on the tooth surface during the meshing process. 2) Chinese Patent, publication number: CN101135357B, patent name: "Harmonic gear drive with a double circular arc tooth profile", this patent analyzed the conjugate principle of the double circular arc harmonic tooth profile by improving the kinematic method, and improved the contact state and meshing interval of the teeth. In this solution, the tooth surfaces of the flexspline and the rigid gear are point contacts, and there is still a large room for improvement in the conjugate meshing interval and the number of meshing tooth pairs. 3) Chinese Patent, publication number: CN107191570B, patent name: "Design of triple circular arc tooth profiles for continuously conjugate cup-shaped or top-hat-shaped harmonic gears", this patent fits the intermediate transition curve by circular arcs, and has a higher envelope existence interval and conjugate meshing tooth surfaces. 4) US Patent, publication number: US5458023, patent name: "Flexingcontact type gear drive of non-profile-shifted two-circular-arc compositetooth profile", this patent starts from the perspective of easy processing in terms of technology, proposes a combined double circular arc tooth profile, considers the taper characteristics of the flexspline deformation, uses a 1 / 2 reduced scale linear mapping, and fits a double circular arc on the basis of the s tooth profile to obtain the flexspline tooth profile, which can achieve continuous meshing transmission.

[0003] The above patents: 1) "A harmonic gear with a variable coefficient cycloidal tooth profile", the entire tooth profile is composed of upper and lower parts of cycloids, resulting in a discontinuous tooth shape; patent 2) "Harmonic gear drive with a double circular arc tooth profile", the linear transition curve has a small tooth profile angle and is not easy to generate conjugation, and further improvement is needed to solve the problem of tooth root meshing interference; patent 3) "Three circular arc tooth profile design of a continuous conjugate cup-shaped or top hat-shaped harmonic gear", the transition curve obtained by circular arc fitting improves the conjugation angle problem but does not solve the interference problem and the problem of reduced meshing angle after fitting; patent 4) "Flexing contact type geardrive of non-profile-shifted two-circular-arc composite tooth profile" is based on the mapping principle and also considers the machining problem. In summary, these tooth profiles basically adopt the fitted S tooth shape, most of which cannot meet the meshing theory, belong to approximate meshing, have insufficient meshing accuracy and discontinuous tooth shape, affecting the bending strength of the tooth shape; at the same time, the fitted S tooth shape causes the basic tooth shape to be solidified, which leads to the lack of innovation in the harmonic tooth shape and cannot get rid of some inherent problems of the S tooth shape design. Therefore, it is urgent to solve the technical problem of how to design a more reasonable and effective meshing pair tooth profile in this technical field. Summary of the Invention

[0004] The object of the present invention is to solve the tooth profile design problem of precise meshing in harmonic drive. To achieve this object, the technical solution adopted by the present invention is to provide a design method for obtaining a harmonic concave-convex tooth profile meshing pair that realizes precise meshing by using a given convex curve as the basic tooth profile and through double enveloping.

[0005] The present invention provides a concave-convex harmonic tooth profile design method that can achieve precise meshing based on the phenomenon of secondary contact and double enveloping;

[0006] The terms applied in the generation process of the conjugate tooth profile of the harmonic gear in the design method provided by the present invention include the original curve, the neutral layer curve, the convex tooth profile curve of the basic tooth shape, the concave-convex tooth profile of the rigid gear, and the concave-convex tooth profile of the flexible gear.

[0007] The original curve refers to the contour curve of the wave generator that forces the flexible gear to deform, generally an elliptical curve or a cosine curve, or a combined curve, all of which are convex curves; the neutral layer curve is the equidistant curve of the original curve, and the equidistant distance is the sum of half of the difference between the inner and outer diameters of the flexible bearing and the wall thickness of the flexible gear; the convex tooth profile curve of the basic tooth shape is a convex curve that meets the given standard of the basic tooth shape, and this given standard is defined in the text; the concave-convex tooth profile of the rigid gear is a continuous concave-convex complete tooth profile curve formed by the first enveloping of the convex tooth profile curve of the basic tooth shape; the concave-convex tooth profile of the flexible gear is a continuous concave-convex complete tooth profile curve formed by the second enveloping of the convex tooth profile curve of the basic tooth shape.

[0008] The present invention provides a concave-convex harmonic tooth profile design method that can achieve precise meshing. The design method includes the following steps:

[0009] Step 1: Establish the meshing theory and coordinate transformation system of harmonic drive;

[0010] Step 2: Determine the tooth profile curve of the basic tooth profile convex tooth profile, which can be the convex tooth profile curve of the flexspline or the convex tooth profile curve of the rigid spline;

[0011] Step 3: Based on the two-contact phenomenon of the first envelope, form the continuous concave-convex complete tooth profile curve of the rigid (flexible) spline after the first envelope of the basic tooth profile;

[0012] Step 4: Based on the secondary envelope principle of harmonic drive, obtain the continuous concave-convex complete tooth profile curve of the flexspline (rigid spline) with a two-contact phenomenon;

[0013] Preferably, in the above Step 1, when establishing the meshing theory and coordinate transformation system of harmonic drive, the design method is as follows:

[0014] As shown in the accompanying drawings of the specification, according to the relationship between the coordinate systems in the figure, the transformation matrix from the flexspline tooth profile coordinate system {X t ,O t ,Y t} to the rigid spline fixed coordinate system {X 2 ,O 2 ,Y 2} is as shown in Equation (1). Similarly, the transformation matrix from the rigid spline fixed coordinate system {X 2 ,O 2 ,Y 2} to the flexspline tooth profile coordinate system {X t ,O t ,Y t} is as shown in Equation (2).

[0015]

[0016] Among them, β is the angle between the moving coordinate system of the rigid spline and the moving coordinate system of the flexspline; ρ is the vector radius of the moving coordinate system of the rigid spline relative to the moving coordinate system of the flexspline; μ is the normal deflection angle of the flexspline teeth; γ is the angle between the moving coordinate system of the rigid spline and the moving coordinate system fixedly connected to the center of the flexspline.

[0017] According to the gear meshing principle, by transforming the rigid spline tooth profile from the rigid spline coordinate system {X 2 ,O 2 ,Y 2} to the flexspline coordinate system {X t ,O t ,Y t}, the tooth profile equation (3) of the rigid gear tooth profile in the flexspline coordinate system {X t , O t , Y t} is obtained. Then, through the meshing equation (4), the envelope solution is carried out in the meshing area to obtain the flexspline tooth profile conjugate to the rigid gear tooth profile.

[0018] Γ (t) = M t2 Γ (2) Equation (3)

[0019] where β is the angle between the moving coordinate system of the rigid gear and the moving coordinate system of the flexspline; ρ is the vector radius of the moving coordinate system of the rigid gear relative to the moving coordinate system of the flexspline; μ is the normal deflection angle of the flexspline tooth; Γ (2) is the rigid gear tooth profile function; Γ (t) is the coordinate of the rigid gear tooth profile transformed to the flexspline coordinate system.

[0020] Lcosγ + Msinγ = O

[0021]

[0022] where ρ is the vector radius of the moving coordinate system of the rigid gear relative to the moving coordinate system of the flexspline; μ is the normal deflection angle of the flexspline tooth; β is the angle between the moving coordinate system of the rigid gear and the moving coordinate system of the flexspline; ρ is the vector radius of the moving coordinate system of the rigid gear relative to the moving coordinate system of the flexspline; γ is the angle between the moving coordinate system of the rigid gear and the moving coordinate system fixedly connected to the center of the flexspline; is the rotation angle of the undeformed section of the flexspline; are respectively obtained by differentiating β, ρ, γ with respect to with respect to; n x2 , n y2 are the normal vector projections; x 2 , y 2 are the given coordinates of the convex tooth profile of the rigid gear; L, M, O are intermediate parameters.

[0023] The flexspline tooth profile is transformed from the flexspline coordinate system {X t , O t , Y t} to the rigid gear coordinate system {X 2 , O 2 , Y 2} through the transformation matrix equation (2), and the tooth profile equation (5) of the flexspline tooth profile in the rigid gear coordinate system {X 2 , O 2 , Y 2} is obtained. Then, through the meshing equation (6), the envelope solution is carried out in the meshing area to obtain the rigid gear tooth profile conjugate to the flexspline tooth profile:

[0024] Γ (2) = M 2t Γ(t) Equation (5)

[0025] where β is the angle between the moving coordinate system of the rigid gear and the moving coordinate system of the flexible gear; ρ is the radius vector of the moving coordinate system of the rigid gear relative to the moving coordinate system of the flexible gear; γ is the angle between the moving coordinate system of the rigid gear and the moving coordinate system fixedly connected to the center of the flexible gear; Γ (2) is the coordinate of the flexible gear tooth profile transformed into the rigid gear coordinate system; Γ (t) is the flexible gear tooth profile function.

[0026] Icosμ + Jsinμ = K

[0027]

[0028] where ρ is the radius vector of the moving coordinate system of the rigid gear relative to the moving coordinate system of the flexible gear; β is the angle between the moving coordinate system of the rigid gear and the moving coordinate system of the flexible gear; γ is the angle between the moving coordinate system of the rigid gear and the moving coordinate system fixedly connected to the center of the flexible gear; is the rotation angle of the undeformed section of the flexible gear; are respectively obtained by differentiating β, ρ, γ with respect to with respect to; n xt , n yt is the normal vector projection; x t , y t are the flexible gear tooth profile functions obtained by the first enveloping; I, J, K are intermediate parameters.

[0029] Preferably, for the tooth profile curve of the basic tooth form convex tooth profile determined in step 2 above, its design method is as follows:

[0030] Determine the module m, the number of teeth z of the rigid gear g , the number of teeth z of the flexible gear r , the addendum height h a , the dedendum height h f , the wall thickness d at the root of the tooth, the wave generator radial deformation coefficient k and the tooth thickness ratio k t , and give the expression of the convex tooth profile equation of the rigid (flexible) gear, such as Equation (7):

[0031]

[0032] where θ is the parameter of the convex tooth profile equation expression; x, y are the convex tooth profile equation expressions.

[0033] Preferably, based on the two-contact phenomenon of the first enveloping in step 3 above, the continuous concave and convex complete tooth profile curves of the rigid (flexible) gear are formed after the first enveloping of the basic tooth form;

[0034] The design method is as follows: According to the phenomenon of primary envelope and secondary contact, from the coordinate transformation matrix formula (1) and the meshing function formula (4) of the flexible gear enveloping the rigid gear based on the gear meshing principle, the concave tooth profile convex tooth profile of the rigid gear can be obtained through the envelope of the convex tooth profile of the flexible gear, or according to the coordinate transformation matrix formula (2) and the meshing function formula (6) of the rigid gear enveloping the flexible gear based on the gear meshing principle, the concave tooth profile convex tooth profile of the flexible gear can be obtained through the envelope of the convex tooth profile of the rigid gear.

[0035] Preferably, in the above step 4, based on the secondary envelope principle of harmonic drive, a continuous concave-convex complete tooth profile curve with two-point contact phenomenon of the flexible (rigid) gear is obtained;

[0036] The design method is as follows: According to the phenomenon of secondary envelope and two-point contact, from the coordinate transformation matrix formula (2) and the meshing function formula (6) of the rigid gear enveloping the flexible gear based on the gear meshing principle, the concave tooth profile of the flexible gear can be obtained through the envelope of the convex tooth profile of the rigid gear, or according to the coordinate transformation matrix formula (1) and the meshing function formula (4) of the rigid gear enveloping the flexible gear based on the gear meshing principle, the concave tooth profile convex tooth profile of the rigid gear can be obtained through the envelope of the convex tooth profile of the flexible gear.

[0037] Compared with the prior art, the present invention has the following beneficial effects:

[0038] 1) The concave-convex continuous complete tooth profile design provided by the present invention can achieve precise meshing and solve the problem of insufficient transmission accuracy of approximate meshing;

[0039] 2) The present invention provides a method for specifying the basic tooth profile of harmonic drive, solves the problem of a single basic tooth profile of only S tooth profile fitting on the current market, and lays a theoretical foundation for the design of a new tooth profile with precise meshing of harmonic drive;

[0040] 3) The concave-convex tooth profile of the present invention is a continuous tooth profile, which can solve the problem of the blank meshing interval of the current fitted tooth profile and can increase the meshing interval;

[0041] 4) The tooth root curve of the concave-convex tooth profile of the flexible gear designed by the present invention is a continuous curve, and the bending strength is greatly increased. The load-bearing capacity of the harmonic gear drive is effectively increased. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 It is the front view of a concave-convex harmonic tooth profile design method based on the secondary contact phenomenon and secondary envelope of the present invention, which can achieve precise meshing.

[0043] Figure 2A It is a schematic diagram of the concave-convex tooth profile curve of the rigid gear.

[0044] Figure 2B It is a schematic diagram of the concave-convex tooth profile curve of the flexible gear.

[0045] Figure 3It is a schematic diagram of the first enveloping process of the continuous concave and convex tooth profiles of the flexspline when the convex tooth profile of the rigid spline is selected as an arc in the specific implementation manner.

[0046] Figure 4 It is a schematic diagram of the movement trajectory of the tooth profile of the rigid spline formed with an arc as the basic tooth profile in the specific implementation manner.

[0047] Figure 5 It is the movement trajectory of the flexspline in the specific implementation manner.

[0048] Figure 6 It is a harmonic drive coordinate transformation system.

[0049] Reference numerals: 1. original curve; 2. neutral layer curve; 3. basic tooth profile convex tooth profile curve; 4. rigid spline concave and convex tooth profile; 5. flexspline concave and convex tooth profile. Specific implementation manner

[0050] To make the present invention more obvious and understandable, a preferred embodiment is hereby described in detail in conjunction with the accompanying drawings as follows:

[0051] As Figures 1-6 shown, the present invention provides an implementation case of a harmonic meshing pair with precise meshing of continuous concave and convex tooth profiles formed by the method provided by the present invention when the basic tooth profile convex tooth profile is an arc curve and the contour line of the wave generator is a standard ellipse.

[0052] The terms applied in the generation process of the conjugate tooth profiles of the harmonic gear provided by the present invention include an original curve 1, a neutral layer curve 2, a basic tooth profile convex tooth profile curve 3, a rigid spline concave and convex tooth profile 4, and a flexspline concave and convex tooth profile 5.

[0053] The original curve 1 refers to the contour line of the wave generator that forces the flexspline to deform, generally an elliptical curve or a cosine curve, and may also be a composite curve, all of which are convex curves; the neutral layer curve 2 is an equidistant curve of the original curve 1, and the equidistant distance is the sum of half of the difference between the inner and outer diameters of the flexible bearing and half of the wall thickness of the flexspline; the basic tooth profile convex tooth profile curve 3 is a convex curve that meets the given standard of the basic tooth profile; the rigid spline concave and convex tooth profile 4 is a continuous concave and convex complete tooth profile curve formed by the first enveloping of the basic tooth profile convex tooth profile curve 3; the flexspline concave and convex tooth profile 5 is a continuous concave and convex complete tooth profile curve formed by the second enveloping of the basic tooth profile convex tooth profile curve 3.

[0054] The design steps of the harmonic gear meshing pair are as follows:

[0055] Step 1. Determine the convex tooth profile C of the arc rigid spline according to the geometric parameters of the harmonic drive 2 M 2 Basic tooth profile parameters;

[0056] Step 2: According to the envelope principle of harmonic transmission, based on the primary envelope of the harmonic meshing pair, the convex tooth profile C of the enveloping rigid wheel is 2 M 2 Get as attached Figure 2B Flexspline concave tooth profile M 1 D 1 Convex tooth profile C 1 M 1 ;

[0057] Step 3: According to the envelope principle of harmonic transmission, based on the secondary envelope of the harmonic meshing pair, the convex tooth profile C of the enveloping flexible wheel is 1 M 1 Get as attached Figure 2A Concave tooth profile of the steel wheel M 2 D 2 ;

[0058] The original curve 1 is a standard ellipse;

[0059] The basic tooth shape of the rigid wheel is a circular arc convex tooth profile and its design method is as follows:

[0060] Determine the module m and the number of teeth z of the designed harmonic gear g 、Number of flexspline teeth z r 、Tooth top height h a , tooth root height h f , tooth root wall thickness d, wave generator radial deformation coefficient k and tooth thickness ratio k t , according to the specific parameters, the equation of the basic tooth profile of the rigid wheel is given as follows:

[0061]

[0062] Among them, r t is the arc radius of the rigid wheel, and θ is the arc rotation angle function.

[0063] According to step 2, the concave and convex tooth profile C of the flex spline tooth profile is obtained 1 D 1 Expression, formula (9):

[0064]

[0065] According to step 3, the expression of the secondary enveloping rigid wheel tooth profile is obtained, formula (10):

[0066]

[0067] The flex spline tooth profile curve drawn according to the flex spline concave-convex tooth profile equation is shown in Figure 2, which is a continuous concave-convex curve, and the tooth root part is continuous, such as Figure 4 As shown in the figure, the bending strength of the tooth root is effectively improved. The conjugate curve of the concave and convex tooth profile is used to simulate the meshing process according to the motion principle of harmonics.Figure 5 As shown, the red curve in the figure is the profile of the rigid gear, and the black curve cluster is a series of trajectories of the flexible gear's movement. It can be seen that during the movement, the flexible gear is always in contact with the rigid gear. Thus, it can be seen that through double enveloping, the load-carrying capacity of the harmonic gear can be effectively increased to meet the precise meshing requirements of harmonic drive. When designing a harmonic gear drive, only by setting the parameters of the convex tooth profile of the rigid gear can the complete concave and convex tooth profiles of the rigid gear and the flexible gear be obtained.

[0068] The shape of the convex tooth profile mentioned in the present invention can be changed, and various deformed convex tooth profiles are within the protection scope of the present invention.

[0069] The above are only the preferred embodiments of the present invention, and do not impose any formal or substantial limitations on the present invention. It should be noted that for those of ordinary skill in the art in this technical field, without departing from the premise of the present invention, several improvements and supplements can still be made, and these improvements and supplements should also be regarded as within the protection scope of the present invention. For those skilled in the art, without departing from the spirit and scope of the present invention, any equivalent changes made by using the technical content disclosed above, such as minor modifications and evolutions, are equivalent embodiments of the present invention; at the same time, any equivalent changes, modifications and evolutions made to the above embodiments based on the essential technology of the present invention still fall within the scope of the technical solution of the present invention.

Claims

1. A design method for a precise meshing harmonic tooth profile, characterized in that: The design method comprises the following steps: Step 1: Establish the meshing theory and coordinate transformation system of harmonic drive; Step 2: Determine a tooth profile curve of a basic tooth profile convex tooth profile, where the basic tooth profile is a convex tooth profile curve of a flexible spline, or a convex tooth profile curve of a rigid spline; Step 3: Based on the double contact phenomenon of one enveloping, the basic tooth shape is enveloping once to form a continuous concave and convex complete tooth profile curve of the rigid wheel or the flexible wheel; Step 4: Based on the secondary envelope principle of harmonic drive, a continuous concave-convex complete tooth profile curve with two-point contact phenomenon of the flexible wheel or the rigid wheel is obtained.

2. A method for designing a precise meshing harmonic tooth profile according to claim 1, characterized in that: In step 1, the meshing theory and coordinate transformation system of harmonic drive are established, and the design method is as follows: As shown in the attached drawings of the specification, according to the relationship between the coordinate systems in the drawings, it can be obtained that from the flex spline tooth profile coordinate system {X t ,O t ,Y t } to the fixed coordinate system {X2, O2, Y2} of the rigid wheel as shown in equation (1). Similarly, the transformation matrix from the fixed coordinate system {X2, O2, Y2} of the rigid wheel to the flex spline tooth profile coordinate system {X t ,O t ,Y t The transformation matrix of} is as shown in formula (2); Among them, β is the angle between the rigid wheel moving coordinate system and the flexible wheel moving coordinate system; ρ is the radius vector of the rigid wheel moving coordinate system relative to the flexible wheel moving coordinate system; μ is the normal deflection angle of the flexible wheel tooth; γ is the angle between the rigid wheel moving coordinate system and the moving coordinate system fixed to the flexible wheel center; According to the gear meshing principle, the tooth profile of the rigid wheel is transformed from the rigid wheel coordinate system {X2, O2, Y2} to the flex wheel coordinate system {X t ,O t ,Y t }, and the tooth profile of the rigid wheel is obtained in the flexwheel coordinate system {X t ,O t ,Y t } tooth profile equation (3) under the meshing condition, and then solve the envelope in the meshing area through the meshing equation (4) to obtain the flexible wheel tooth profile conjugate with the rigid wheel tooth profile; C (t) =M t2 C (2) expression(3) Where, β is the angle between the rigid wheel moving coordinate system and the flexible wheel moving coordinate system; ρ is the radius vector of the rigid wheel moving coordinate system relative to the flexible wheel moving coordinate system; μ is the normal deflection angle of the flexible wheel tooth; Γ (2) is the gear tooth profile function; Γ (t) It is the coordinate of the rigid wheel tooth profile converted to the flexible wheel coordinate system; Among them, ρ is the radial vector of the rigid wheel moving coordinate system relative to the flexible wheel moving coordinate system; μ is the normal deflection angle of the flexible wheel tooth; β is the angle between the rigid wheel moving coordinate system and the flexible wheel moving coordinate system; ρ is the radial vector of the rigid wheel moving coordinate system relative to the flexible wheel moving coordinate system; γ is the angle between the rigid wheel moving coordinate system and the moving coordinate system fixed to the flexible wheel center; is the rotation angle of the undeformed section of the flexspline; They are β, ρ, and γ The derivative of n x2 、n y2 is the normal projection; x2 and y2 are the given convex tooth profile coordinates of the rigid wheel; L, M, O are the intermediate parameters; The flexspline tooth profile is transformed from the flexspline coordinate system {X t ,O t ,Y t } is transformed to the rigid wheel coordinate system {X2, O2, Y2}, and the tooth profile equation (5) of the flex spline tooth profile in the rigid wheel coordinate system {X2, O2, Y2} is obtained. Then, through the meshing equation (6), the envelope solution is performed in the meshing area to obtain the rigid wheel tooth profile conjugate with the flex spline tooth profile: C (2) =M 2t C (t) expression(5) Among them, β is the angle between the rigid wheel moving coordinate system and the flexible wheel moving coordinate system; ρ is the radius vector of the rigid wheel moving coordinate system relative to the flexible wheel moving coordinate system; γ is the angle between the rigid wheel moving coordinate system and the moving coordinate system fixed to the flexible wheel center; Γ (2) is the coordinate of the flex spline tooth profile transformed to the rigid spline coordinate system; Γ (t) is the flexspline tooth profile function; Among them, μ is the normal deflection angle of the flexspline tooth; ρ is the radius vector of the rigid wheel moving coordinate system relative to the flexspline moving coordinate system; β is the angle between the rigid wheel moving coordinate system and the flexspline moving coordinate system; γ is the angle between the rigid wheel moving coordinate system and the moving coordinate system fixed to the flexspline wheel center; is the rotation angle of the undeformed section of the flexspline; They are β, ρ, and γ The derivative of n xt 、n yt is the normal projection; x t ,y t is the flexspline tooth profile function obtained by primary enveloping; I, J, and K are intermediate parameters.

3. The design method of a precise meshing harmonic tooth profile according to claim 1, characterized in that: In step 2, the tooth profile curve of the basic tooth profile convex tooth profile is determined, and the design method thereof is as follows: Determine the module m and the number of teeth z of the designed harmonic gear g 、Number of flexspline teeth z r 、Tooth top height h a , tooth root height h f , tooth root wall thickness d, wave generator radial deformation coefficient k and tooth thickness ratio k t , the equation of the convex tooth profile of the rigid (flexible) wheel is given as formula (7): Among them, θ is the parameter of the convex tooth profile equation expression; x and y are the convex tooth profile equation expressions.

4. The method for designing a precise meshing harmonic tooth profile according to claim 1, characterized in that: In step 3, based on the double contact phenomenon of one enveloping, the basic tooth shape is enveloped once to form a continuous concave-convex complete tooth profile curve of a rigid wheel or a flexible wheel, and the design method thereof is as follows: According to the phenomenon of primary enveloping secondary contact, from the coordinate transformation matrix (1) and the gear meshing principle, the flexible wheel enveloping the rigid wheel meshing function (4), the convex tooth profile of the rigid wheel can be obtained by enveloping the convex tooth profile of the flexible wheel, or according to the coordinate transformation matrix (2) and the gear meshing principle, the rigid wheel enveloping the flexible wheel meshing function (6), the convex tooth profile of the flexible wheel concave tooth profile can be obtained by enveloping the convex tooth profile of the rigid wheel.

5. The method for designing a precise meshing harmonic tooth profile according to claim 1, characterized in that: In step 4: based on the secondary envelope principle of harmonic drive, a continuous concave-convex complete tooth profile curve of the flexible (rigid) wheel with two-point contact phenomenon is obtained, and the design method thereof is as follows: According to the phenomenon of secondary enveloping two-point contact, from the coordinate transformation matrix (2) and the gear meshing principle, the rigid wheel enveloping the flexible wheel meshing function (6), the concave tooth profile of the flexible wheel can be obtained by enveloping the convex tooth profile of the rigid wheel, or according to the coordinate transformation matrix (1) and the gear meshing principle, the rigid wheel enveloping the flexible wheel meshing function (4), the concave tooth profile of the rigid wheel can be obtained by enveloping the convex tooth profile of the flexible wheel.

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