A Modeling Method for the Single-Tooth Contact Stiffness and Meshing Clearance of a Harmonic Gear

Abstract

The present invention discloses a method for modeling the single - tooth contact stiffness and meshing clearance of a harmonic gear. A non - contact optical profiler is used to complete the non - contact scanning of the tooth profiles of the flexible gear and the rigid gear of the harmonic reducer and the extraction of tooth profile data points. Based on the W - M function, the height distribution characteristics of the micro - protrusions on the rough tooth surfaces of the rigid gear and the flexible gear of the harmonic reducer are described. Combining with the roughness length method, the roughness parameters G and the fractal dimension D of the tooth surfaces of the rigid gear and the flexible gear are calculated respectively. Finally, a single - tooth contact stiffness and meshing clearance model of the rigid gear and the flexible gear based on the roughness parameter G and the fractal dimension D is established. The present invention establishes a mathematical relationship between the dynamic characteristics of the gear and surface micro - parameters such as surface roughness, and can improve the gear dynamic performance by adjusting the gear surface topography, laying a foundation for the overall non - linear dynamic analysis of transmission gears.

Description

Technical Field

[0001] The present invention relates to the field of research on the dynamic characteristics of harmonic reducers, and particularly to a method for modeling the single-tooth contact stiffness and meshing clearance of harmonic gears. Background Art

[0002] The harmonic gear transmission device was proposed by the American scholar Musser in 1955 based on the thin-shell elastic deformation theory. It mainly consists of three major components: a flexible gear, a rigid gear, and a wave generator. At present, the research on harmonic reducers in China started relatively late, with a slow development process. The lack of mature mastery of key technologies and the backward manufacturing process have led to problems such as transmission errors, backlash, and relatively high noise during the use of domestic harmonic reducers, making them unsuitable for occasions with high requirements for transmission accuracy. The transmission process of a harmonic reducer belongs to a multi-tooth meshing transmission with small module and large deformation. When the rigid gear is fixed and the wave generator rotates counterclockwise, the flexible gear rotates clockwise. The rigid and flexible gears at both ends of the long axis of the wave generator participate in meshing, while the rigid and flexible gears at the short axis are disengaged. The contact stiffness and backlash of the harmonic reducer are directly related to the positioning accuracy and vibration characteristics of the robot. However, the tooth height of the tooth pair of the harmonic reducer is usually less than 1 mm, and the measurement and modeling of the backlash and contact stiffness are relatively complex. The contact stiffness of the joint surface is the weakest link in the gear meshing stiffness. For harmonic gear transmission, its multi-tooth meshing characteristics further increase the number of joint surfaces. Therefore, it is extremely urgent to propose a contact stiffness model and a backlash model suitable for harmonic gears. Summary of the Invention

[0003] The purpose of the present invention: In order to improve the positioning accuracy of harmonic reducers for robots and reduce vibration, to solve the problem that the current research on the stiffness of harmonic reducers mainly focuses on torsional stiffness, the existing dynamic models do not contain tooth surface topography parameters, lack a single-tooth pair meshing characteristic model that truly reflects the multi-tooth meshing transmission with small module and large deformation, and to verify the feasibility of improving the dynamic performance of gears by adjusting the gear surface topography, the present invention proposes a method for modeling the single-tooth contact stiffness and meshing clearance of harmonic gears.

[0004] To achieve the above purpose, the present invention adopts the following technical solutions:

[0005] Use a non-contact optical profiler to complete the non-contact scanning of the tooth profiles of the flexible gear and the rigid gear of the harmonic reducer and the extraction of tooth profile data points. Based on the W-M function, describe the height distribution characteristics of the micro-protrusions on the rough tooth surfaces of the rigid gear and the flexible gear of the harmonic reducer. Combine the roughness length method to calculate the roughness parameter G and the fractal dimension D of the tooth surfaces of the rigid gear and the flexible gear respectively. Finally, establish a single-tooth contact stiffness and meshing clearance model for the rigid gear and the flexible gear based on the roughness parameter G and the fractal dimension D.

[0006] The non-contact measurement scheme for the tooth profiles of the flexspline and the rigid spline of the harmonic reducer mainly includes: cutting the rigid spline ring by wire cutting, placing the rigid and flexspline to be measured under the probe and clamping them with a fixture, rotating the fine-tuning button to focus the white light spot emitted by the measuring probe on the gear surface, and starting the measurement after setting the scanning and moving parameters;

[0007] The method for extracting the tooth profile data points of the rigid spline and the flexspline mainly includes removing the multi-noise data of the circular arcs of the tooth profiles of the rigid spline and the flexspline, and extracting a rectangular area of 0.2mm×0.1mm on the tooth root surfaces of the rigid spline and the flexspline to replace the tooth profile data;

[0008] The description of the height distribution characteristics of the micro-protrusions on the rough tooth surfaces of the rigid spline and the flexspline of the harmonic reducer based on the W-M function:

[0009]

[0010] In formula (1), z(x) represents the height distribution of the micro-protrusions on the tooth surfaces of the rigid spline and the flexspline, which is obtained by selecting multiple straight line segments in the above-mentioned rectangular area of 0.2mm×0.1mm, γ represents the density of the micro-protrusions, usually γ = 1.5, x represents the abscissa of the sampling point, φ represents the random phase 0≤φ≤2π, r' represents the cross-sectional area of a single micro-protrusion, and n0 represents the frequency index. represents a uniformly distributed random phase, and its value satisfies [0, 2π].

[0011] The calculation of the roughness parameters G and the fractal dimension D of the tooth surfaces of the rigid spline and the flexspline respectively by combining the roughness length method includes formulas (2) to (5):

[0012] S(ξ) = <(z(x + ξ) - z(x)) 2 > (2)

[0013]

[0014]

[0015] D = <D s > + 1 (5)

[0016] In the formula, S(ξ) represents the variance of the height of the micro-protrusion profile, ξ represents the sampling interval of the data points, generally a constant, Γ represents the gamma equation, D s represents the two-dimensional fractal dimension, G s represents the fractal roughness, and the symbol <> represents taking the average value.

[0017] The establishment of the single-tooth contact stiffness K and the meshing clearance b models of the rigid spline and the flexspline based on the roughness parameter G and the fractal dimension D includes formulas 6, 7, and 8:

[0018]

[0019] b = G (D-2) (lnγ) 1 / 2 (2r') 3-D (7)

[0020]

[0021] In the formula, ψ is related to the fractal dimension, a l ' represents the cross-sectional area of the micro-convex bodies on the tooth surfaces of the rigid gear and the flexible gear, E is the equivalent normal elastic modulus, a l ' represents the maximum contact area of the micro-convex bodies on the tooth surfaces of the rigid gear and the flexible gear, a c ' represents the critical contact area of the micro-convex bodies on the tooth surfaces of the rigid gear and the flexible gear.

[0022] The calculation method of the equivalent normal elastic modulus E is as follows, where υ1 and υ2 are the Poisson's ratios of the two contacting materials, and E1 and E2 are the elastic moduli of the two contacting materials.

[0023] Compared with the prior art, the beneficial effects of the present invention are:

[0024] The present invention proposes a modeling method for the single-tooth contact stiffness and meshing clearance of harmonic gears. This method truly reflects the meshing characteristics of a single tooth pair in the multi-tooth meshing transmission with small module and large deformation. The model is not only applicable to the accurate modeling of the contact stiffness and backlash of harmonic gear transmission, but also applicable to the identification of other meshing parameters of small-module gears. The model establishes a mathematical relationship between the dynamic characteristics of gears and surface microscopic parameters such as surface roughness, and can improve the dynamic performance of gears by adjusting the gear surface topography, laying a foundation for the overall non-linear dynamic analysis of transmission gears. Description of the Drawings

[0025] Figure 1 Torsional vibration model of gear pair

[0026] Figure 2 Backlash function model of gear pair

[0027] Figure 3 Contact stiffness model

[0028] Figure 4 Meshing clearance evaluation model Detailed Embodiments

[0029] To further understand the content, features and effects of the present invention, the following embodiments are exemplified and described in detail with reference to the accompanying drawings as follows:

[0030] A modeling method for the single-tooth contact stiffness and meshing clearance of a harmonic gear, comprising the following steps: using a non-contact optical profiler to complete the non-contact scanning of the tooth profiles of the flexible gear and the rigid gear of the harmonic reducer and the extraction of tooth profile data points, describing the height distribution characteristics of the asperities on the rough tooth surfaces of the rigid gear and the flexible gear of the harmonic reducer based on the W-M function, calculating the roughness parameters G and the fractal dimension D of the tooth surfaces of the rigid gear and the flexible gear respectively by combining the roughness length method, and finally establishing a single-tooth contact stiffness and meshing clearance model for the rigid gear and the flexible gear based on the roughness parameters G and the fractal dimension D.

[0031] The non-contact measurement scheme for the tooth profiles of the flexible gear and the rigid gear of the harmonic reducer mainly includes: cutting the rigid gear ring by wire cutting, placing the measured flexible and rigid gears under the probe and clamping them with a fixture, rotating the fine-tuning button to focus the white light spot emitted by the measuring probe on the gear surface, and starting the measurement after setting the scanning and moving parameters, as Figure 1 shown;

[0032] The method for extracting the tooth profile data points of the rigid gear and the flexible gear mainly includes removing the multi-noise data of the circular arcs of the tooth profiles of the rigid gear and the flexible gear, and extracting a rectangular area of 0.2 mm × 0.1 mm on the tooth root surfaces of the rigid gear and the flexible gear as Figure 2 shown, so as to replace the tooth profile data;

[0033] Describing the height distribution characteristics of the asperities on the rough tooth surfaces of the rigid gear and the flexible gear of the harmonic reducer based on the W-M function:

[0034]

[0035] In formula (1), z(x) represents the height distribution of the asperities on the tooth surfaces of the rigid gear and the flexible gear, obtained by selecting multiple straight line segments in the above-mentioned rectangular area of 0.2 mm × 0.1 mm, γ represents the density of the asperities, usually γ = 1.5, x represents the abscissa of the sampling point, φ represents the random phase 0 ≤ φ ≤ 2π, r' represents the cross-sectional area of a single asperity, n0 represents the frequency exponent, represents a uniformly distributed random phase, and its value satisfies [0, 2π];

[0036] Calculating the roughness parameters G and the fractal dimension D of the tooth surfaces of the rigid gear and the flexible gear respectively by combining the roughness length method, including formulas (2) to (5):

[0037] S(ξ) = <(z(x + ξ) - z(x)) 2 > (2)

[0038]

[0039]

[0040] D = <D s>+1 (5)

[0041] In the formula, S(ξ) represents the variance of the micro-convex body contour height, ξ represents the sampling interval of data points, which is generally a constant, Γ represents the gamma equation, D s represents the two-dimensional fractal dimension, G s represents the fractal roughness, and the symbol < > represents taking the average value.

[0042] The establishment of the single-tooth contact stiffness K of the rigid gear and the flexible gear based on the roughness parameter G and the fractal dimension D is as Figure 3 shown and the meshing clearance b model is as Figure 4 shown, including Formula 6, Formula 7 and Formula 8:

[0043]

[0044] b = G (D-2) (lnγ) 1 / 2 (2r') 3-D (7)

[0045]

[0046] In the formula, ψ is related to the fractal dimension, a l ' represents the cross-sectional area of the micro-convex bodies on the tooth surfaces of the rigid gear and the flexible gear, E is the equivalent normal elastic modulus ( where υ1 and υ2 are the Poisson's ratios of the two contacting materials, and E1 and E2 are the elastic moduli of the two contacting materials), a l ' represents the maximum contact area of the micro-convex bodies on the tooth surfaces of the rigid gear and the flexible gear, a c ' represents the critical contact area of the micro-convex bodies on the tooth surfaces of the rigid gear and the flexible gear.

[0047] Compared with the prior art, the beneficial effects of the present invention are:

[0048] The method for modeling the single-tooth contact stiffness and meshing clearance of a harmonic gear proposed by the present invention truly reflects the meshing characteristics of a single tooth pair in a small-module large-deformation multi-tooth meshing transmission. The model is not only applicable to the accurate modeling of the contact stiffness and backlash of harmonic gear transmission, but also applicable to the identification of other small-module gear meshing parameters. The model establishes a mathematical relationship between the dynamic characteristics of the gear and surface micro-parameters such as surface roughness, and can improve the dynamic performance of the gear by adjusting the gear surface topography, laying a foundation for the overall non-linear dynamic analysis of transmission gears.

Claims

1. A modeling method for the single-tooth contact stiffness and meshing clearance of a harmonic gear, characterized in that: The method includes the following steps: using a non-contact optical profiler to complete the non-contact scanning of the tooth profiles of the flexspline and the circular spline of the harmonic reducer and the extraction of tooth profile data points; based on the W-M function to describe the height distribution characteristics of the micro-protrusions on the rough tooth surfaces of the circular spline and the flexspline of the harmonic reducer, and combining the roughness length method to calculate the roughness parameter G and the fractal dimension D of the circular spline teeth and the flexspline tooth surface respectively; finally, establishing the single-tooth contact stiffness and meshing clearance models of the circular spline and the flexspline based on the roughness parameter G and the fractal dimension D; The non-contact measurement of the tooth profiles of the flexspline and the circular spline of the harmonic reducer includes: cutting the circular spline tooth ring by wire cutting, placing the measured flexspline and circular spline under the probe and clamping them with a fixture, rotating the fine-tuning button to make the white light spot emitted by the measuring probe focus on the gear surface, and starting the measurement after setting the scanning movement parameters; The method for extracting the tooth profile data points of the circular spline and the flexspline includes eliminating the multi-noise data of the circular arcs of the tooth profiles of the circular spline and the flexspline, and extracting a rectangular area of 0.2 mm×0.1 mm on the tooth root surfaces of the circular spline and the flexspline to replace the tooth profile data; The establishment of the single-tooth contact stiffness K and meshing clearance b models of the circular spline and the flexspline based on the roughness parameter G and the fractal dimension D includes Formula 6, Formula 7 and Formula 8: b = G (D-2) (lnγ) 1 / 2 (2r') 3-D (7) In the formula, ψ is related to the fractal dimension, E is the equivalent normal elastic modulus, a l ' represents the maximum contact area of the micro-protrusions on the surfaces of the rigid gear and the flexible gear, a c ' represents the critical contact area of the micro-protrusions on the surfaces of the rigid gear and the flexible gear; γ represents the density of the micro-protrusions, and r' represents the cross-sectional area of a single micro-protrusion.

2. The modeling method for the single-tooth contact stiffness and meshing clearance of a harmonic gear according to claim 1, wherein: The description of the height distribution characteristics of the micro-protrusions on the rough tooth surfaces of the circular spline and the flexspline of the harmonic reducer based on the W-M function: In formula (1), z(x) represents the height distribution of the micro-convex bodies on the surfaces of the rigid gear and the flexible gear, which is obtained by selecting multiple straight line segments in a rectangular area of 0.2 mm × 0.1 mm. γ represents the density of the micro-convex bodies, x represents the abscissa of the sampling point, φ represents a random phase where 0 ≤ φ ≤ 2π, r' represents the cross-sectional area of a single micro-convex body, and n0 represents the frequency exponent. represents a uniformly distributed random phase.

3. A method for modeling the single-tooth contact stiffness and meshing clearance of a harmonic gear according to claim 2, characterized in that: The calculation of the roughness parameter G and the fractal dimension D of the circular spline teeth and the flexspline tooth surface respectively by combining the roughness length method includes Formula (2) to Formula (5): S(ξ) = <(z(x + ξ) - z(x)) 2 > (2) D = <D s > + 1 (5) In the formula, S(ξ) represents the variance of the micro-convex body profile height, ξ represents the sampling interval of data points, Γ represents the gamma equation, and D s represents the two-dimensional fractal dimension, and G s represents the fractal roughness, and the symbol < > represents taking the average value.

4. A method for modeling the single-tooth contact stiffness and meshing clearance of a harmonic gear according to claim 1, characterized in that: The calculation method of the equivalent normal elastic modulus E is as follows, where υ1 and υ2 are the Poisson's ratios of the two contacting materials, and E1 and E2 are the elastic moduli of the two contacting materials.

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