High-precision and high-reliability harmonic reducer tooth profile simulation machining design method

Abstract

The invention discloses a high-precision and high-reliability harmonic reducer tooth profile simulation machining design method, and belongs to the field of precision transmission. A composite cycloid flexible gear tooth profile is designed by establishing a harmonic reducer geometrical relationship model, deducing a flexible gear deformation equation, an accurate rotation angle relationship and a meshing theory based on an envelope method, and a conjugate region and a conjugate tooth profile are calculated by utilizing MATLAB. Actual deformation and theoretical deformation errors of the flexible gear are analyzed through finite element simulation, flexible gear profile parameters are optimized, an actual neutral layer curve is generated, and then a rigid gear profile is enveloped and calculated. The tooth profile design adopts a non-zero inclination angle composite cycloid, so that the meshing area is expanded by more than 30%, and the maximum stress of the flexible gear is reduced by 22.75% in combination with a linear modification method and a finite element modification method. In addition, through a multi-physical field simulation verification system (including static deformation, dynamic contact and fatigue life prediction) and a neural network optimization tooth shape parameter, the transmission performance is remarkably improved. According to the method, the torsional rigidity can be improved by 66%, and the transmission error is controlled within a high-precision range.

Description

Technical Field

[0001] The present invention relates to the technical field of harmonic reducers, and particularly to a method for simulating and machining the tooth profile of a high-precision and high-reliability harmonic reducer, which is applicable to scenarios with requirements for precision transmission such as industrial automation, robotics, aerospace, etc. Background Art

[0002] In the traditional cycloidal tooth profile, due to limitations in the meshing area and problems such as tooth tip interference, its meshing performance is restricted. The traditional cycloidal tooth profile adopts a fixed common tangent inclination angle design, and the conjugate area only accounts for about 50% of the tooth surface, resulting in insufficient load-bearing capacity and limited meshing area. It is necessary to adjust the tooth profile parameters to expand the conjugate area. The traditional linear deformation assumption model cannot accurately predict the actual meshing position, resulting in a transmission error of ±72. These problems lead to an increase in transmission error (the transmission error of traditional designs is usually greater than 50 arcsec), making it difficult to meet the requirements of high-precision applications. For example, in the application of industrial robot joint drives, due to poor meshing of the traditional tooth profile, the decrease in motion accuracy will directly affect the accuracy of trajectory control.

[0003] In terms of the deviation between the flexible gear deformation theory and the actual situation, there are significant differences between the traditional theoretical model based on the linear deformation assumption and the actual elastic deformation of the flexible gear (the error between the theory and finite element simulation can reach more than 1μm). This makes the tooth profile design out of touch with the actual working conditions, leading to assembly stress concentration (the maximum stress of the unmodified flexible gear can reach 729.5MPa) and uneven contact pressure distribution (the peak contact pressure exceeds 700MPa), thus shortening the service life.

[0004] In terms of processing technology and detection means, it is difficult for traditional processing technology to achieve high-precision modification of the tooth profile (the wall thickness tolerance is usually greater than ±0.05mm, and the surface roughness Ra is greater than 1.6μm), and there is a lack of a detection system for multi-physical field collaborative optimization, which cannot verify the transmission performance under complex working conditions (such as the torsional stiffness fluctuation under multiple working conditions exceeds 30%).

[0005] In addition, most current studies only focus on the geometric design of the tooth profile and do not combine finite element simulation with machine learning optimization. For example, the traditional linear modification method relies on empirical parameters. After modification, the maximum stress of the flexible gear only decreases by 16.96%, and at the same time, the number of engaged teeth decreases to 45%, failing to achieve the collaborative optimization of stress and the number of engaged teeth.

[0006] Moreover, the existing technology lacks a closed-loop system of "design - simulation - machining - detection". For example, traditional machining does not consider the deformation compensation of simulation feedback (such as the origin offset of the flexspline coordinate system ΔX can reach 2.5×10⁻³mm), resulting in a significant difference between the actual tooth profile and the theoretical model (the maximum gap is 1.12μm), which affects the transmission accuracy. In terms of testing technology, there is a lack of high-precision testing platforms in China, making it difficult to verify the transmission performance under complex working conditions. The traditional test bench can only test the transmission error under a single load, while the comprehensive performance test bench built in the present invention can achieve multi-parameter coupling testing (the accuracy of the torque sensor is ±0.5%, and the resolution of the angle encoder is ≤0.1°), filling the domestic technical gap. Summary of the Invention

[0007] The technical problem to be solved by the present invention is to provide a method for simulating the machining design of the tooth profile of a harmonic reducer with high precision and high reliability.

[0008] It is intended to establish a geometric relationship model, design a composite cycloid flexspline tooth profile, use finite element simulation to analyze the deformation error and optimize the tooth profile parameters. Solve the problems of insufficient meshing performance of the traditional tooth profile, deviation between the flexspline deformation theory and the actual situation, lag in machining technology, and lack of collaborative optimization of the whole process, and improve the accuracy and reliability of the harmonic reducer.

[0009] To solve the above technical problems, the present application provides a method for simulating the machining design of the tooth profile of a harmonic reducer with high precision and high reliability, which includes the following steps: A. Establish a geometric relationship model of the harmonic reducer, deduce the flexspline deformation equation, the precise rotation angle relationship, and the meshing theory based on the envelope method; B. Design a composite cycloid flexspline tooth profile, and calculate the conjugate region and conjugate tooth profile through MATLAB; C. Use finite element simulation to analyze the actual deformation and theoretical deformation error of the flexspline, and determine the origin offset of the flexspline coordinate system and the normal rotation angle deviation; D. Optimize the flexspline tooth profile parameters according to the simulation results, generate the actual neutral layer curve, and envelope and calculate the rigid gear tooth profile.

[0010] Among them, based on the material mechanics characteristics of the flexspline, a neutral layer equation considering the coupling deformation of the radial, tangential, and normal directions is deduced:

[0011] (In the formula: is the radial deformation coefficient, is the radius of the flexspline neutral layer, is the radius of the outer rolling circle, is the rotation angle of the wave generator) The method for simulating the machining design of the tooth profile of a harmonic reducer with high precision and high reliability according to the present invention, the design of the composite cycloid tooth profile includes: a. Generate an epicycloid tooth profile outside the pitch circle and a hypocycloid tooth profile inside; b. Achieve a non-zero inclination design of the tooth profile by adjusting the inclination angle α0 of the common tangent; c. Establish a coordinate transformation matrix to realize the translation and fitting of the tooth profile; It can adopt the design formula: Equation of the outer tooth profile: (1) Equation of the inner tooth profile: (2) (3) Among them, is the radius of the outer rolling circle, is the tooth height. By adjusting the inclination angle of the common tangent, a non-zero inclination design of the tooth profile is achieved, and the meshing area is expanded by more than 30%.

[0012] Preferably, the finite element simulation analysis includes: a. Establish a three-dimensional contact model of the flexspline and the wave generator; b. Set the mesh division parameters (C3D8R element, the number of meshes ≥ 580,000); c. Define the material properties (flexspline 30CrMnSiA, rigid ring 45# steel) and the contact friction coefficient (0.15); The flexspline deformation model adopts the following non-linear equations: (4) Among them, is the radial deformation coefficient, and its value range is 0.8 - 1.2. By introducing the normal rotation angle correction term, the error between the theoretical deformation and the finite element simulation is reduced to less than 0.2 μm.

[0013] Preferably, the tooth profile modification includes: a. Linear modification method: Design the wall thickness modification amount based on the radial displacement gradient of the axial section; b. Finite element modification method: Fit the modification amount polynomial equation through the simulation data; Among them, the tooth profile modification adopts a piecewise linear model: (5) (6) Among them, is the wall thickness modification gradient coefficient. Through fitting with the finite element simulation data formula (6), the maximum stress of the flexspline after modification is reduced by 22.75%.

[0014] And the meshing characteristic analysis can include: a. Verify non-interfering meshing through the motion trajectory simulation; b. The assembly state analysis shows that 66.25% of the tooth surfaces are meshed; c. The backlash distribution is controlled within the range of 0 - 1.6 μm; Based on the involute meshing principle and combined with the geometric relationship after the deformation of the flexspline, the meshing backlash formula (7) is derived: (7) where, are the addendum circle radii of the flexspline and the rigid spline, is the pressure angle. By constraining , the meshing smoothness is improved by 40%. As Figure 10 shown, the meshing backlash distribution curve shows the dynamic change range of the backlash j.

[0015] In the experimental verification, it may include: a. Design and build a comprehensive performance test bench for the harmonic reducer. The test bench includes: A servo motor drive system with a rated torque ≥ 30 N·m; A loading system with a torque sensor accuracy of ±0.5%; An angle encoder with a resolution ≤ 0.1°; A data acquisition system with a sampling rate ≥ 1 kHz; b. Transmission error test (the error range after modification is ±31″ vs ±72″ without modification); c. The torsional stiffness is increased by 66% (32840 Nm / rad after modification vs 19140 Nm / rad without modification).

[0016] Furthermore, the tooth profile parameters can be optimized by using a neural network: (8) The input parameters include tooth surface coordinates, contact stress and strain. The modification amount is optimized through training data, and the modification efficiency is increased by 3 times.

[0017] In the manufacturing process requirements, the processing of the flexspline needs to meet the following process parameters: wall thickness tolerance: ±0.02 mm; tooth profile surface roughness: Ra ≤ 0.8 μm; heat treatment hardness: ≥ 45 HRC.

[0018] In addition, a multi - physical - field simulation verification system can be established, which includes: Static deformation analysis, solved implicitly by ABAQUS; dynamic contact analysis, solved explicitly by ABAQUS; Fatigue life prediction, using FE - SAF. The present invention also includes an integrated design system, including: A parametric design module, using the MATLAB platform; Simulation analysis module with ABAQUS / ANSYS interfaces; Performance prediction module based on Python neural network model.

[0019] The high-precision and high-reliability tooth profile simulation machining design method of harmonic reducer in the present invention systematically solves the core bottlenecks of domestic harmonic reducers in terms of accuracy, reliability, and processability through the three-dimensional coordination of theoretical model innovation, method system innovation, and process technology innovation. Compared with international similar products, the present invention is close to the international advanced level in terms of transmission error (±31″ vs international ±15″), torsional stiffness (32840 Nm / rad vs international level), etc., providing an independent and controllable technical solution for the precision transmission of high-end equipment such as robots and aerospace, and having significant engineering application value and promoting effect on industrial upgrading.

[0020] Compared with the prior art, the beneficial effects of the present invention are as follows: Expansion of meshing area: Through the design of variable common tangent inclination angle, the meshing area is increased from 50% to 66.25% (simulation data); Improvement of transmission accuracy: After modification, the transmission error is reduced from ±72″ to ±31″ (experimental data); Extension of service life: The peak value of contact stress is reduced by 22.75%, and the fatigue life is extended to more than 10,000 hours; Improvement of design efficiency: The integrated design system shortens the R & D cycle by 40%.

[0021] The following further describes the high-precision and high-reliability tooth profile simulation machining design method of harmonic reducer in the present invention with reference to the accompanying drawings. Description of the Drawings

[0022] Figure 1 ( Figure 1 a, Figure 1 b) is the schematic diagram of the generation principle of the composite cycloid tooth profile of a high-precision and high-reliability tooth profile simulation machining design method of harmonic reducer in the present invention; Figure 2 is the schematic diagram of the deformation of the neutral layer of the flexspline in a high-precision and high-reliability tooth profile simulation machining design method of harmonic reducer in the present invention; Figure 3 is the influence curve of the common tangent inclination angle on the conjugate region in a high-precision and high-reliability tooth profile simulation machining design method of harmonic reducer in the present invention ( Figure 3 c is the conjugate region diagram, Figure 3 a, Figure 3 b, Figure 3 d is the result diagram); Figure 4 is the influence curve of the flexspline wall thickness on the coordinate system offset; Figure 5 It is the neutral layer curve fitting error diagram of a high-precision and high-reliability harmonic reducer tooth profile simulation machining design method of the present invention, where Figure 5 a (simulation data points and fitting curve), Figure 5 b (error distribution); Figure 6 It is the rigid gear tooth profile solution result diagram of a high-precision and high-reliability harmonic reducer tooth profile simulation machining design method of the present invention.

[0023] Figure 7 It is a schematic diagram of the deformation of the flexspline after being installed in the wave generator under the condition of linear deformation assumption; Figure 8 It is the schematic diagram of the interference of the axial section motion trajectory of the present invention; Figure 9 It is the motion trajectory diagram after linear method modification of the present invention; Figure 10 It is the meshing backlash distribution curve diagram of a high-precision and high-reliability harmonic reducer tooth profile simulation machining design method of the present invention; Figure 11 ( Figure 11 a, Figure 11 b ) is the material parameter table of a high-precision and high-reliability harmonic reducer tooth profile simulation machining design of the present invention. Specific implementation mode

[0024] The present invention takes the composite cycloid tooth profile as the research object, systematically explores the meshing mechanism and mechanical properties of the harmonic drive system. Based on the conjugate design theory of harmonic drive, with the help of the MATLAB parametric calculation platform, it reveals the influence mechanism of the flexspline parameters on the conjugate meshing characteristics, and analyzes the reasons for the differences between the actual and theoretical deformations of the flexspline. On this basis, the harmonic gear planar tooth profile is designed, a simplified three-dimensional model of the harmonic reducer is established, the tooth profile of the flexspline is modified by the linear method and the finite element method respectively, and the stress deformation and transmission performance of the harmonic tooth profile before and after modification are analyzed through finite element simulation and experiment comparison.

[0025] According to the basic principle, composition and transmission ratio calculation of the harmonic reducer, the basic assumptions of harmonic drive are proposed, the geometric relationship models of each component of the harmonic gear drive are established, the flexspline deformation equation, the accurate rotation angle relationship and the harmonic drive meshing theory based on the envelope method are deduced, and the calculation process of the composite cycloid flexspline tooth profile is obtained. Taking the composite cycloid flexspline tooth profile as the object, the tooth profile parameters are initially selected, the conjugate region and conjugate tooth profile are deduced based on the accurate rotation angle relationship of the flexspline and the envelope method, and the influence of the flexspline parameters on it is studied by MATLAB. The reasons for the differences between the actual and theoretical deformation errors of the flexspline are analyzed through finite element simulation, as well as the influence of the flexspline wall thickness and tooth thickness on the offset of the flexspline coordinate system origin and the deviation of the normal rotation angle.

[0026] Finally, according to the research, the profile of the flexspline is designed. The origin of the actual flexspline coordinate system is obtained through simulation. The actual neutral layer curve is obtained by fitting the origins of each tooth. The profile of the rigid spline is calculated by the envelope method. The meshing characteristics of the designed profile are analyzed. The motion trajectory, assembly state, and meshing backlash of the profile are analyzed using MATLAB.

[0027] Meanwhile, aiming at the axial inclination of the flexspline profile after the assembly of the wave generator, the profile modification of the flexspline is carried out respectively based on the linear assumption radial deformation and the finite element radial deformation of the flexspline. The motion trajectory of the modified profile is checked by combining MATLAB. Finite element analysis is carried out on the harmonic reducer models with linear method modification, finite element method modification, and without modification, and the deformation and stress conditions are compared.

[0028] In this invention, a comprehensive performance test bench for the harmonic reducer is built to carry out performance test experiments. The harmonic reducer prototypes before and after the flexspline modification by finite element are compared. The performance indexes such as transmission efficiency, transmission error, backlash, and torsional stiffness are analyzed to verify the improvement effect of the flexspline profile modification on the overall performance of the harmonic reducer. As Figure 1 shown is the generation principle diagram of the composite cycloid profile of a high-precision and high-reliability harmonic reducer tooth profile simulation machining design method of this invention ( Figure 1 a is the generation diagram of the composite cycloid profile, Figure 1 b is the schematic diagram of the inclination angle of the common tangent of the cycloid profile). It shows that the outer rolling circle rolls along the outside of the pitch circle to generate the outer cycloid profile, and the inner rolling circle rolls along the inside of the pitch circle to generate the inner cycloid profile. This is the generation principle of the composite cycloid profile. The outer cycloid outside the pitch circle (radius R) is generated by the outer rolling circle (radius rw) rolling along its outside, and the inner cycloid inside is generated by the inner rolling circle (radius rn) rolling along its inside. There are also the common tangent inclination angle α0 and the rolling direction of the rolling circle (shown by the arrow) in the figure, which shows the construction process of the composite cycloid profile parametric model and the geometric generation mechanism of the basic profile. The basic parameters of the precision roller gear transmission mechanism of this invention are shown in Table 1.

[0029] Table 1 Basic parameter table of the precision roller gear transmission mechanism

[0030] I. Calculation of the transmission ratio of harmonic drive Let ω1, ω2, and ω3 be the absolute angular velocities of the wave generator, flexspline, and rigid spline respectively. An angular velocity with the same magnitude and opposite direction as the angular velocity of the wave generator is applied to the entire transmission system. In this way, in the converted gear train, the angular velocity of the wave generator can be regarded as zero, that is, the wave generator is in a stationary state. The angular velocities of the flexspline and the rigid spline relative to the wave generator and are respectively: (2-1) The transmission ratio of the conversion gear train relative to the wave generator is as follows: (2-2) In the formula, Z2 is the number of teeth of the rigid gear; Z1 is the number of teeth of the flexible gear.

[0031] The following are the calculation methods for the transmission ratios of harmonic gear drives in different working forms: ① When the rigid gear is fixed, the wave generator inputs, and the flexible gear outputs, the transmission ratio is as follows: (2-3) ② When the rigid gear is fixed, the flexible gear inputs, and the wave generator outputs, the transmission ratio is as follows: (2-4) ③ When the flexible gear is fixed, the wave generator inputs, and the rigid gear outputs, the transmission ratio is as follows: (2-5) ④ When the flexible gear is fixed, the rigid gear inputs, and the wave generator outputs, the transmission ratio is as follows: (2-6) ⑤ When the wave generator is fixed, the flexible gear inputs, and the rigid gear outputs, the transmission ratio is as follows: (2-7) ⑥ When the wave generator is fixed, the rigid gear inputs, and the flexible gear outputs, the transmission ratio is as follows: (2-8) Regarding the geometric relationship model and conjugate theory of harmonic gear drive During the research process, in order to improve the analytical calculation feasibility of the model, the secondary influencing factors were systematically eliminated, a theoretical model with engineering generality was established, and the following basic assumptions were proposed: (1) Under the action of the wave generator, the thin-walled structure of the flexible gear forms a neutral surface with a constant perimeter, and this neutral surface maintains an equidistant relationship with the outer contour of the wave generator.

[0032] (2) During the operation of the harmonic reducer, the tooth grooves of the flexible gear teeth will change, while the tooth shape of the flexible gear remains unchanged.

[0033] (3) After the wave generator is assembled, the characteristic curves such as the pitch circle, root circle, and addendum circle of the flexible gear teeth form an equidistant mapping relationship with the neutral surface.

[0034] (4) When the normal direction of the flexible gear deforms on the neutral layer surface, it will not be distorted, and this normal direction is still perpendicular to the neutral layer after deformation.

[0035] Neutral layer equation and deformation equation of harmonic drive As Figure 2As shown in the figure, it is a schematic diagram of the deformation of the neutral layer of the flexspline in a high-precision and high-reliability harmonic reducer tooth profile simulation processing design method of the present invention. The deformation characteristics of the neutral layer of the flexspline are presented in the figure. It shows the geometric relationship between the undeformed neutral layer (dashed line 206) and the neutral layer after standard ellipse deformation (solid line 207), and marks the radial deformation amount (208). The dashed line in the middle represents the neutral layer curve of the flexspline in the undeformed state; the solid ellipse represents the curve of the neutral layer after deformation under the action of the cam wave generator. The coordinate system takes the rotation center of the flexspline as the origin, the X1 axis coincides with the minor axis of the ellipse line, and the Y1 axis coincides with the major axis of the ellipse line.

[0036] Taking the standard ellipse curve as the neutral layer curve, according to the parametric equation of the standard ellipse, we can obtain Figure 2 The polar coordinate equation of the neutral layer curve of the flexspline after deformation is: (2-9) In the formula, a is the major axis length of the neutral layer curve of the flexspline after deformation, b is the minor axis length of the neutral layer curve of the flexspline after deformation, and φ1 is the angle between the radius vector ρ and the major axis of the neutral layer curve.

[0037] Figure 2 Among them, ω0 is the maximum radial deformation amount after the flexspline is deformed, and its expression is: (2-10) In the formula is the radial deformation coefficient of the flexspline, and its value range is 0.8~1.2. Generally, it can be taken as 1, and m is the module.

[0038] Figure 2 Among them, rm is the radius of the neutral layer curve of the flexspline before deformation. Therefore, the major axis length of the neutral layer curve of the flexspline after deformation is: (2-11) II. Design of the composite cycloid flexspline tooth profile 1. Generation of the composite cycloid tooth profile The composite cycloid tooth profile and its generation process are as Figure 1 shown in Figure a, which is a schematic diagram of the generation principle of the composite cycloid tooth profile. It shows that the outer rolling circle (201) rolls along the outside of the pitch circle (202) to generate the outer cycloid tooth profile (203), and the inner rolling circle (204) rolls along the inside of the pitch circle to generate the inner cycloid tooth profile (205); the tooth profile outside the pitch circle is the outer cycloid, which is formed by the movement trajectory of the moving point K on it when the outer rolling circle rolls along the outside of the pitch circle; the tooth profile inside the pitch circle is the inner cycloid, which is generated by the movement trajectory of the moving point P on it when the inner rolling circle rolls along the inside of the pitch circle.

[0039] In the figure, the epicycloid outside the pitch circle (radius R) is generated by the outer rolling circle (radius rw) rolling along the outside of the pitch circle, and the hypocycloid inside is generated by the inner rolling circle (radius rn) rolling along the inside of the pitch circle. t1 and t2 are the angles through which the rolling circle rotates. Points D and E are the points on the tooth tip and tooth root profile curves with the same slope. θ1 is half of the central angle of the tooth thickness of the flexspline pitch circle. The inclination angle of the common tangent is α0, and the rolling direction of the rolling circle (shown by the arrow) demonstrates the geometric generation mechanism of the basic tooth profile.

[0040] The curve AB is the tooth profile outside the pitch circle formed by the outer rolling circle. The tooth profile equation within is: (2-31) The curve BC is the tooth profile inside the pitch circle formed by the inner rolling circle. The tooth profile equation within {O,X,Y} is: (2-32) 2. Composite cycloid flexspline tooth profile with variable common tangent inclination angle As Figure 1 shown in b, it is a schematic diagram of the common tangent inclination angle of the cycloid tooth profile. To obtain a tooth profile with a non-zero common tangent inclination angle, take points D and E where the tangent slopes of the tooth profile outside the pitch circle and the tooth profile inside the pitch circle are equal, remove the BD and BE segments, and make points D and E coincide at point B to form a composite cycloid tooth profile with a non-zero common tangent inclination angle.

[0041] Assume the angle of the common tangent inclination angle is α0, which can be obtained through calculation: (2-33) Substitute Equation (2-33) into Equation (2-31) to obtain the coordinates of point D : (2-34) Translate the tooth profile curve outside the pitch circle so that point D coincides with point B, and remove the BD segment. Thus, a tooth profile curve outside the pitch circle with a common tangent inclination angle of α0 is obtained, and then perform a coordinate transformation. The coordinate transformation matrix is: (2-35) (2-36) After the coordinate transformation, the tooth profile equation in {Of,Xf,Yf} is: (2-37) In the formula, rm represents the radius of the neutral layer of the flexspline; ; K represents the tooth thickness ratio.

[0042] The tooth profile outside the pitch circle is obtained through calculation: (2-38) In the formula,

[0043]

[0044]

[0045]

[0046] Continuing the formula (2-32) through the above process can obtain the inner profile of the pitch circle in (2-39) In the formula,

[0047]

[0048]

[0049]

[0050] III. Tooth Profile Design and Meshing Characteristic Analysis of Harmonic Gear Drive 1. Analysis of Conjugate Characteristics and Parameter Influence of Harmonic Gear Drive 1.1 Conjugate Region and Conjugate Tooth Profile First, preliminarily determine the basic parameters of the flexspline. The specific parameters are shown in Table 3-1. According to the meshing theory of the harmonic reducer, calculate through MATLAB to obtain the conjugate region.

[0051] Table 3-1 Flexspline Tooth Profile Parameters

[0052] 1.2 Influence of Flexspline Parameters on Conjugate Region and Conjugate Tooth Profile The main parameters that affect the meshing characteristics of harmonic drive include: common tangent inclination angle α 0, outer rolling circle radius r w , inner rolling circle radius r n and flexspline radial deformation coefficient .

[0053] (1) Influence of common tangent inclination angle α 0 on conjugate region and conjugate tooth profile ( Figure 3 ) Figure 3 a is conjugate region I (epicycloid), Figure 3 b: conjugate region II (hypocycloid), 3c: total conjugate region.

[0054] Figure 3 As can be observed in Fig. c, the cycloidal gear profile harmonic drive system has two significantly different conjugate meshing intervals. Conjugate region 1 (interval parameter [0.0663°, 9.9109°]) is marked with a solid black line and has a limited spatial span; conjugate region 2 (interval parameter [9.9886°, 54.721°]) is identified with a dashed red line and has a wider meshing coverage. There is a non-conjugate transition zone with a parameter span of [9.9109°, 9.9886°] between the two regions, that is, blank region 3, where there is no conjugate tooth profile.

[0055] From the perspective of spatial distribution, the two conjugate regions are separated in the system. The left meshing region corresponds to the external cycloidal tooth profile of the flexspline, and the right corresponds to the internal cycloidal tooth profile. Curves 1 and 2 respectively represent the variation law of the meshing angle of the external and internal cycloidal tooth profiles in conjugate region 1, and curves 3 and 4 reflect the variation law in conjugate region 2.

[0056] Based on kinematic analysis, a normal line is constructed along the horizontal axis to intersect the conjugate meshing region. The intersection points A and B reveal the phenomenon of secondary conjugation with double meshing angles at the same coordinate position of the flexspline tooth profile; a horizontal reference line is constructed through the vertical axis to intersect the conjugate region at points B and C, indicating the two-point conjugate characteristic of the flexspline tooth profile having double conjugate positions at a specific meshing angle. The figure shows the difference in the width of the conjugate intervals of the curves and the conjugate region distribution calculated based on the envelope method. Three also superimposes the influence curves of different common tangent inclination angles α0 (10°, 14°, 18°) on the conjugate interval, reflecting the law that the conjugate interval decreases as α0 increases, intuitively demonstrating the rationality of taking the conjugate blank area S as the optimization target.

[0057] Using the variable control method, while keeping the other parameters of the flexspline tooth profile constant, the common tangent inclination angle between the external and internal cycloids is taken as an independent variable for analysis. Five gradient levels are set, and the common tangent inclination angle is adjusted to 10°, 12°, 14°, 16° and 18° for comparative study. Figure 3 d, Figure 3 a and Figure 3 b are the graphs of the influence of α0 on the conjugate region. Among them, Figure 3 a is the tooth profile of conjugate region 1, Figure 3 b is the tooth profile of conjugate region 2.

[0058] Generally speaking, a smaller inclination angle can expand the conjugate range of the rigid gear and the flexspline tooth profiles.

[0059] According to Figs. Figure 3 a and 3b, it can be seen that the change in the inclination angle has a significant impact on the tooth profile characteristics of the rigid gear. Specifically, as the inclination angle increases, the tooth tip thickness of the rigid gear gradually decreases, while the tooth root thickness increases accordingly. At the same time, the conjugate arc length of the tooth profile in conjugate region 1 of the flexspline also increases, and shows a rotational trend in the counterclockwise direction.

[0060] 2. Comparative Analysis of the Actual Deformation and Theoretical Deformation of the Flexible Gear Plane Tooth Ring When the wave generator is installed into the flexible gear plane tooth ring, there is often a certain error between the actual deformation and the theoretical deformation of the flexible gear profile. This error will have a certain impact on the design of the harmonic reducer. In this section, the reasons for this error will be studied.

[0061] 2.1 Establishing the Finite Element Analysis Model The wave generator is simplified into an elliptical rigid body with a thickness of 2 mm, and one end is chamfered so that it can be smoothly inserted into the inner cylinder of the flexible gear. The flexible gear is simplified into an outer cylindrical tooth ring with an axial length of only 2 mm. Subsequently, a three-dimensional model is established and imported into the ABAQUS software for finite element analysis. The specific steps are as follows: (1)Material selection: The flexible gear is made of 30CrMnSiA (density is 7.75×10 3 kg / m³, elastic modulus is 196 GPa, Poisson's ratio is 0.3); the wave generator is made of 45# steel (density is 7.81×10 3 kg / m³, elastic modulus is 210 GPa, Poisson's ratio is 0.269).

[0062] (2)Mesh generation: Use HYPERMESH for mesh generation.

[0063] (3)Contact setting: The contact mode between the wave generator and the flexible gear is selected as "surface - surface" contact. The outer surface of the wave generator is used as the target surface, and the inner surface of the flexible gear is used as the contact surface. The friction coefficient is set to 0.15.

[0064] (4)Constraint and load setting: This section mainly conducts finite element analysis on the deformation of the flexible gear after the wave generator is assembled. Therefore, only one analysis step is set. The flexible gear is fixed in the Z-axis direction, and the wave generator is installed into the flexible gear.

[0065] Finally, through finite element calculation, the actual deformation of the flexible gear after the wave generator is assembled can be obtained, and the tooth profile data of the deformed flexible gear is imported into MATLAB for comparative analysis with the tooth profile of the theoretically deformed flexible gear.

[0066] 2.2 Comparison of the Actual Deformation and Theoretical Deformation of the Flexible Gear The theoretical deformation of the flexible gear is to calculate the deformation of the flexible gear after the wave generator is installed into the flexible gear through MATLAB. Before deformation, the tooth profile of the flexible gear is a circle with an equal - angle distribution; after deformation, the tooth profile turns into a quasi - elliptical shape with an equal - arc - length distribution. According to the basic assumptions, the rotation angle j of the φ 1j th flexible gear tooth profile can be calculated as: (3-1) To construct the circumferential distribution model of the flexspline tooth profile, it is necessary to introduce a coordinate transformation matrix. Let the original position of the flexspline point set be , and use to represent the initial tooth profile point set. For the j th tooth profile, its coordinate expression is: (3-2) From the above formula, the full tooth profile plane state diagram of the deformed flexspline can be obtained. Since the deformed tooth profile distribution is symmetric about the coordinate system, the deformation state of 1 / 4 tooth profile in the first quadrant is selected for analysis and compared with the actual results.

[0067] Through MATLAB calculation, the maximum gap between the actual deformation and the theoretical deformation of each tooth profile of the flexspline is obtained. From the major axis to the minor axis, the maximum gap of each tooth of the flexspline first increases and then decreases, and the gap is the smallest at the positions of the major axis and the minor axis, and the largest in the middle of the major axis and the minor axis. The maximum gap is 3.05 microns. This is because there is a certain deviation between the origin position of the flexspline coordinate system after actual deformation and the theoretical value. Secondly, during the deformation process of the flexspline, the flexspline teeth will generate a normal rotation angle, and there will also be a certain gap between the actual result of the normal rotation angle and the theoretical value. Therefore, it is necessary to calculate the offset of the origin position of the flexspline coordinate system after actual deformation relative to the origin position of the flexspline coordinate system after theoretical deformation and the angle deviation of the normal rotation angle through finite element simulation. After finite element simulation, the offset of the abscissa X of the actual origin of the flexspline coordinate system relative to the theoretical value ΔX , and the ordinate Y of the offset relative to the theoretical value ΔY , and the deviation of the normal rotation angle relative to the theoretical value Δμ .

[0068] Assume that the origin of the flexspline coordinate system of the j th flexspline after actual deformation is represented as . By calculation, the form of the rectangular coordinate system is converted into ρ j and φ 1 j to obtain: (3-3) (3-4) Substitute the ρ j and φ 1 j values of the origin of the flexspline coordinate system of each tooth after actual deformation into the theoretical deformation formula to obtain the coordinate point expression of the deformed flexspline as: (3 - 5) From the above formula, it is possible to obtain the comparison between the theoretical and actual deformation curves of the flexspline after considering the offset of the origin position of the flexspline coordinate system relative to the origin position of the flexspline coordinate system after theoretical deformation and the deviation of the normal rotation angle.

[0069] After considering the offset of the origin position of the flexspline coordinate system relative to the origin position of the flexspline coordinate system after theoretical deformation and the deviation of the normal rotation angle, the difference between the theoretical and actual deformation curves of the flexspline obtained through theory becomes significantly smaller, and the maximum difference for each tooth is less than 0.2 micrometers, within a reasonable error range.

[0070] 2.3 Influence of Different Parameters of the Flexspline on the Offset of the Origin Position of the Flexspline Coordinate System Relative to the Origin Position of the Flexspline Coordinate System after Theoretical Deformation and the Deviation of the Normal Rotation Angle Since the origin of the coordinate system of each tooth of the flexspline is on the neutral layer curve, and the actual neutral layer curve after the deformation of the flexspline is affected by the wall thickness and tooth thickness of the flexspline, the wall thickness and tooth thickness of the flexspline will have an impact on the offset of the origin position of the flexspline coordinate system relative to the origin position of the flexspline coordinate system after theoretical deformation and the deviation of the normal rotation angle. Therefore, in this section, the influence laws of the wall thickness and tooth thickness of the flexspline are analyzed respectively. Also, since the origin of the flexspline coordinate system of each deformed flexspline tooth is symmetric about the rigid spline coordinate system, the deformation state of 1 / 4 tooth profile in the first quadrant of the rigid spline coordinate system is selected for analysis.

[0071] Analysis of the influence law of the flexspline wall thickness: The wall thicknesses of the flexspline are taken as 0.4 mm, 0.6 mm, 0.8 mm, and 1.0 mm respectively, and the other parameters are kept unchanged. Then, the influence law of the wall thickness of the flexspline on the offset of the origin position of the flexspline coordinate system relative to the origin position of the flexspline coordinate system after theoretical deformation and the deviation of the normal rotation angle is as Figure 4 shown, presenting the distribution of the ΔX offset for wall thicknesses of 0.4 mm (curve 401), 0.6 mm (curve 402), 0.8 mm (curve 403), and 1.0 mm (curve 404). In the figure, the abscissa is the number of teeth of the flexspline from the major axis to the minor axis, the first tooth is the tooth at the major axis of the flexspline, and the 41st tooth is the tooth at the minor axis of the flexspline. The ordinates are the theoretical and actual offsets and the deviations of the normal rotation angle of the origin of each flexspline coordinate system respectively.

[0072] As can be seen from the figure, overall, from the major axis to the minor axis, the abscissa X and ordinate Y of the origin position of the flexspline coordinate system after the actual deformation of the flexspline relative to the origin position of the flexspline coordinate system after the theoretical deformation both increase first and then decrease, and the offsets at the major axis and minor axis positions are the smallest. The offset of the abscissa X is the largest at about the 17th tooth away from the major axis, and the ordinate YThe offset is the largest at about the 25th tooth from the major axis. It can also be seen from the figure that different wall thicknesses have a greater impact on the horizontal coordinate X and the vertical coordinate Y of the origin position of the flexspline coordinate system relative to the theoretical deformation after actual deformation. For the parameters of this flexspline tooth profile, when the wall thickness ranges from 0.6 mm to 1.2 mm, the horizontal coordinate X and the vertical coordinate Y both show a trend of first decreasing and then increasing, while the deviation of the normal rotation angle is generally small, close to 0°, and the change law is not obvious, with a small change.

[0073] 3. Example of Tooth Profile Design of Harmonic Gear Drive In this study, a typical double-wave harmonic drive system is selected to build a mathematical model: the module is set to 0.3958, the number of teeth of the flexspline is set to 160, the number of teeth of the rigid gear is set to 162, forming a standard meshing pair with a tooth difference of 2. The kinematic configuration of the system is set as the rigid gear is fixed and constrained, the flexspline outputs power, and the wave generator is used as the driving input end. The specific parameters are shown in Table 3-2: Table 3-2 Flexspline Design Parameter Values

[0074] 3.1 Solution of Rigid Gear Tooth Profile (see Figure 6 ) Since the solution of the rigid gear tooth profile is related to the flexspline tooth profile equation and the neutral layer curve after deformation, and through the comparative analysis of the finite element deformation and theoretical deformation of the flexspline plane tooth ring in the previous section, it can be known that there is a certain deviation between the theoretical position and the actual position of the origin of the flexspline coordinate system of each tooth of the flexspline after assembly, and the origin of the flexspline coordinate system is located on the neutral layer curve of the flexspline. Therefore, there are two methods to obtain a reasonable rigid gear tooth profile. One is to find a reasonable flexspline structure so that the theoretical deformation of the flexspline is consistent with the actual deformation; the second is to obtain the actual deformation of the neutral layer curve through simulation and conjugate the rigid gear tooth profile through the neutral layer curve after actual deformation. Since the first method requires finite element analysis of flexsplines with various different structural parameters to find a reasonable flexspline structure, which takes a long time, the second method is selected in this paper.

[0075] First, finite element analysis is carried out on the assembly of the flexspline plane tooth ring and the cam to obtain the coordinates of each node of the neutral layer curve of the flexspline. Substitute them into Equations 3-3 and 3-4 to obtain the rotation angle and radius vector of each point of the actual neutral layer of the flexspline after deformation. At the same time, fit the rotation angle and radius vector through MATLAB to obtain the fitting equation, that is, the actual neutral layer expression is: (3-6) Figure 5 ( Figure 5 a Simulation data points and fitting curve, Figure 5The (b) error distribution shows the simulation results of the neutral layer curve, its fitting curve, and the fitting error. It includes simulation data points, the fitting curve, and the error band, with a maximum error of 0.1 μm. Among them, Figure 5 (a) The simulation data points are represented by circular markers, and the solid line represents the fitting curve; Figure 5 (b) shows the error distribution during the fitting process. The analysis results indicate that the fitting error has obvious periodic fluctuation characteristics, and the maximum deviation value does not exceed 0.1 micrometer, meeting the requirements of engineering applications.

[0076] From the comparison of the profile of the flexspline's actual deformation and the theoretical deformation of the fitted neutral line curve and the maximum gap of each tooth profile, it can be seen from the figure that the gap between the profile calculated by fitting the neutral layer curve and the actual profile becomes significantly smaller, with a maximum gap of only less than 0.2 micrometers and a small error. Therefore, the profile of the rigid spline can be calculated using the fitted neutral layer curve.

[0077] Substitute the obtained fitting equation into the above rigid spline conjugate equation to obtain the rigid spline tooth profile curve conjugated by the actual neutral layer curve as Figure 6 shown, which shows the topological structure of the conjugate region.

[0078] 3.2 Tooth Profile Interference Inspection Tooth profile interference refers to the occurrence of unexpected contact or collision between the tooth profiles of the flexspline and the rigid spline during the gear transmission process, resulting in a decrease in transmission accuracy, increased tooth surface wear, and even possible transmission failure. Therefore, tooth profile interference inspection is a key link in the design and analysis of harmonic gear transmissions, aiming to ensure smooth and efficient transmission of the tooth profiles during meshing. In harmonic gear transmissions, tooth profile interference mainly includes tooth profile overlap interference, transition curve interference, and tooth tip interference.

[0079] (1) Tooth Tip Interference Tooth tip interference occurs when the tooth tip of the flexspline contacts the tooth tip or tooth root of the rigid spline at the moment of meshing in or out. Due to the symmetry of the harmonic drive, if tooth tip interference occurs at one end of the flexspline and rigid spline tooth profiles, tooth tip interference will inevitably occur at the symmetric other end. Therefore, the requirements for no interference at the moment of meshing in and meshing of the flexspline are the same.

[0080] The condition for no tooth tip interference between the flexspline and the rigid spline is that the tooth tip rotation angle ψ L2 of the rigid spline is greater than the tooth tip rotation angle ψ L1 of the flexspline, that is (3 - 7) (3 - 8) In the formula, φ ca is the central angle corresponding to the coincidence of the flexspline tooth tip coordinates and the rigid spline tooth tip coordinates at the moment of just disengaging from meshing; is the pressure angle at the addendum circle of the flexspline; s a1 is the addendum thickness of the flexspline; e a2 is the width of the tooth space at the addendum of the rigid spline; r a1 is the radius of the addendum circle of the flexspline; r a2 is the radius of the addendum circle of the rigid spline.

[0081] In the rigid spline coordinate system { O 2, X 2, Y 2}, at the moment when the flexspline meshes or disengages, the coordinate point of the addendum of the rigid spline ( X a2 , Y a2 ) and the coordinate point of the addendum of the flexspline ( X a1 , Y a1 ) have the same numerical values, that is (3-9) In the rigid spline coordinate system { O 2, X 2, Y 2}, the coordinates of the addendum of the flexspline are: (3-10) Where x a1 , y a1 are the coordinates of the addendum of the flexspline in { O 1, X 1, Y 1}.

[0082] After substituting the parameters, it can be obtained:

[0083] Therefore, holds, that is, there will be no addendum interference during operation.

[0084] (2) Profile overlap interference Profile overlap interference means that during the operation of the flexspline and the rigid spline, the contour curves in the meshing area should not cross. In the coordinate system { O 2, X 2, Y 2}, if a specific point on the tooth surface of the flexspline is set as P 1( X P1 , Y P1 ), then the coordinates of this point can be expressed as (3 - 11) With the point as the center and the radius being draw a circle, and the intersection point of its arc and the profile of the rigid gear is , then the condition for no profile interference is (3 - 12) Therefore, each tooth profile curve of the flexspline is discretized into a point set. Substituting it into the above formula, the interference situation at different meshing positions can be obtained. Through calculation, it is obtained that:

[0085] It can be known through calculation that the designed tooth profile meets the condition of no profile interference.

[0086] (3) Transition curve interference Transition curve interference means that during the meshing process of the harmonic gear drive, it is necessary to ensure that in the condition of the maximum tooth penetration depth, the tooth tip area of the flexspline and the rigid gear does not invade the transition curve area of the mating gear. The condition to avoid transition curve interference is: (3 - 13) In the formula, r a1 is the radius of the flexspline tooth tip circle when not deformed; r g1 is the radius of the flexspline tooth root circle when not deformed; r a2 is the radius of the rigid gear tooth tip circle; r g2 is the radius of the rigid gear tooth root circle; w 0 is the radial deformation amount when the flexspline is deformed.

[0087] Through calculation, the values of each parameter can be obtained as: , , , , . By substituting the above parameters into the constraint conditions of transition curve interference for verification, the calculation results show that this condition is satisfied. That is to say, the proposed harmonic gear tooth profile design scheme can avoid the interference problem between the tooth tip and the transition curve area during the transmission process.

[0088] IV. Meshing characteristic analysis 4.1 Tooth profile motion trajectory analysis According to the coordinate transformation matrix in the meshing principle, the coordinate transformation matrix S from the flexspline coordinate system S 1 to the rigid gear coordinate system M 21 is: (3 - 14) Assume that the coordinates of a certain point on the flexspline in the coordinate system are , then the coordinates of the flexspline tooth profile in the rigid spline coordinate system can be expressed as: (3 - 15) Due to the symmetry of the flexspline motion trajectory, it is only necessary to analyze the 1 / 4 motion trajectory. Discretize the flexspline tooth profile in this paper into a point set and substitute it into the above formula. Then, since M 21 the variables β and Δ φ in φ can both be represented by the independent variable φ 1, therefore, discretize the independent variable

[0089] 1 in the range of 0° to 90° and substitute it into the corresponding calculation formula to finally solve the motion trajectory of the flexspline relative to the rigid spline. According to the obtained tooth surface motion trajectory distribution, it can be seen that no interference phenomenon occurs during the meshing process. To evaluate its performance and reliability, the assembly state analysis mainly focuses on the relative positional relationship between the flexspline and the rigid spline. The tooth profile of the rigid spline will remain evenly distributed after assembly and will not change significantly due to the assembly process. In contrast, after the flexspline is assembled with the wave generator, it is no longer evenly distributed at equal angles along the circumferential direction. That is, the tooth profile of the flexspline will expand outward in the long axis direction and contract inward in the short axis direction. This uneven distribution will affect the meshing characteristics of the flexspline and the rigid spline. Therefore, it is necessary to use MATLAB to write a program to simulate the motion state of the flexspline after assembly and visualize the motion trajectory, so as to intuitively evaluate the motion characteristics of the flexspline tooth profile after assembly.

[0090] Since the tooth profile of the rigid spline is evenly distributed at equal angles around the circumference, for the i th rigid spline tooth, its included angle φ 2i can be expressed as: (3 - 16) Similar to the method of plotting the motion trajectory of the flexspline tooth profile, first discretize the rigid spline tooth profile into a point set. The processing process of the rigid spline tooth profile is relatively simplified. It only needs to be rotated around the origin of its own coordinate system through the base vector transformation matrix W 2i to complete the trajectory calculation. Its base vector transformation matrix W 2i is: (3 - 17) Assume that the initial position of the above rigid spline is , use Denote the set of points on the profile of the rigid gear. For the i th profile of the rigid gear, its coordinate points in the circumferential direction are expressed as follows: (3-18) The expression of the deformed profile of the flexible gear has been calculated above. From this, the assembly drawing of the deformed flexible gear and the rigid gear in the initial state can be obtained.

[0091] Since the assembled profile is symmetric about the coordinate axis, the 1 / 4 profile assembly state in the first quadrant is selected for enlarged observation. After observation, no interference phenomenon occurs in the designed profile in the assembly state, and 27 flexible gear teeth and the rigid gear profile are in the meshing state in the first quadrant, that is, 66.25% of the teeth of the flexible gear ring are in the meshing state.

[0092] 3.4.3 Meshing backlash analysis The meshing backlash refers to the small gap between the profiles of the flexible gear teeth and the rigid gear teeth in the meshing state in harmonic gear drive. In harmonic gear drive, the backlash is one of the key indicators to measure the meshing performance of gears. A reasonable backlash can effectively reduce the friction and wear between the tooth surfaces, thereby improving the transmission efficiency and extending the service life of the gears. At the same time, an appropriate backlash can avoid interference between the tooth surfaces and ensure the smoothness and reliability of the transmission process. In addition, the size and distribution of the backlash directly affect the distribution of the tooth load, and further affect the stability of the entire transmission system. Therefore, the calculation of the meshing backlash is crucial for evaluating the transmission performance and reliability. The basic principle of calculating the backlash is to determine the gap between the profiles through geometric relationships. Specifically, taking the distance from the meshing position point to the center of the global coordinate system as the rotation radius, after rotating around the origin of the coordinates and intersecting with the conjugate profile, the distance between the obtained intersection point and the original meshing point is taken as the backlash amount.

[0093] Establish a meshing backlash model for the flexible gear and the rigid gear. In the rigid gear coordinate system S 2, take the coordinate values of the points A ([[]] x jA , y jA ) of the profile curve of the flexible gear, and calculate the A polar radius r j corresponding to the point as: (3-19) Draw an arc with radius r j to intersect with the rigid gear profile and determine the coordinate position of the intersection point B ([[]] x jB , yjB )。By calculating the distance between point A and point B , the meshing backlash value of the flexspline at the l AB point position can be obtained: A (3 - 20) Discretize each tooth profile of the modified flexspline into several discrete points, and apply the aforementioned calculation formula to solve the backlash, then the backlash distribution characteristics at each position of the tooth surface can be obtained. During this process, the minimum value among all the calculated l AB values represents the meshing backlash parameter of this tooth profile. Figure 10 is the meshing backlash distribution curve of the flexspline and the rigid spline. It can be observed from the figure that within the conjugate region, the maximum meshing backlash of the tooth profile is 1.6 microns, the minimum backlash is close to 0 microns, and the backlash values in most regions are kept within 0.65 microns. Generally speaking, the backlash distribution is relatively uniform and the values are small. Under the loaded condition, these teeth can achieve good contact, make the load evenly distributed, and improve the transmission efficiency and reliability.

[0094] The present invention derives the conjugate region and conjugate tooth profile based on the precise rotation angle relationship of the flexspline and the envelope method, and uses MATLAB to calculate and study the influence of different parameters of the flexspline on the conjugate characteristics in the composite cycloid tooth profile. Through finite element simulation, the reasons for the error between the actual deformation and the theoretical deformation of the flexspline tooth profile are analyzed, as well as the influence of the wall thickness and tooth thickness of the flexspline on the offset of the origin position of the flexspline coordinate system and the deviation of the normal rotation angle after the actual deformation relative to the theoretical deformation. Then, the planar tooth profile of the harmonic reducer flexspline is designed. The actual neutral layer curve after the deformation of the flexspline planar tooth ring is obtained through finite element simulation. The conjugate rigid spline tooth profile is generated by the envelope method, and the tooth profile interference inspection is carried out. Finally, MATLAB is used to analyze the tooth profile motion trajectory, assembly state and meshing backlash.

[0095] V. Tooth Profile Modification of Harmonic Reducer The present invention uses the linear method and the finite element method to modify the tooth profile of the flexspline respectively, and compares the displacement, stress and contact pressure of the flexspline before and after modification in the assembly state and load operation state through finite element analysis, providing a theoretical basis and practical reference for the tooth profile design and modification of the harmonic reducer.

[0096] 1.1 Tooth Profile Modification of Flexspline Based on Linear Deformation, Analysis of Tooth Profile Motion Trajectory of Axial Section Based on Linear Deformation Before Modification: Figure 7 ​Shows the deformation of the flexspline after being installed in the wave generator under the condition of linear deformation assumption. In this paper, section 2 is selected as the main section (i.e., the design section of this paper), axial section 3 as the front section of the tooth profile, and section 1 as the rear section of the tooth profile. In the figure, S 1 represents the distance from the rear section 1 to the bottom of the flexspline cylinder, S 2 represents the distance from the main section 2 to the bottom of the flexspline cylinder, S 3 represents the distance from the front section 3 to the bottom of the flexspline cylinder.

[0097] Assume that the radial deformation of the bottom of the flexspline cylinder is zero. According to the linear assumption method, at a distance from the bottom of the cup in the tooth width direction S the deformation of the original curve of the flexspline w 0S can be expressed as: (4-1) Due to the large number of sections, in this paper, sections 1, 2, and 3 are selected as examples, and the tooth profile motion meshing conditions of these sections are calculated and analyzed by MATLAB using the above formula. Figure 8 Shows: Figure 8 (a) Section 1, Figure 8 (b) Section 2, Figure 8 (c) The conjugate locus after eliminating interference in section 3. Through the analysis of the deformation and meshing state of each observation section, it is found that the radial deformation of the front section 3 is significantly higher than that of the main section, resulting in a lot of contacts between the flexspline and the rigid gear at the initial stage of meshing, and a serious geometric interference is formed between the tooth tip of the flexspline and the tooth root of the rigid gear in the fully meshed state. And in the rear section 1, due to the relatively small radial deformation of the flexspline, dynamic impact is generated between the tooth tip of the flexspline and the tooth root of the rigid gear on the motion trajectory. While no interference phenomenon occurs in the main section 2.

[0098] 1.2. Flexspline modification based on linear deformation As Figure 9 shown, shows the running trajectories of each section after modification by the linear method: Figure 9 (a) Section 1, figure (b) Section 3. According to the analysis, during the movement of the harmonic reducer, interference may occur between the tooth profile of the flexspline and the rigid gear. To avoid this interference, the position of the tooth profile of the flexspline can be adjusted so that it is located below the tooth profile of the rigid gear. Specifically, it is to adjust the wall thickness of each section of the flexspline.

[0099] According to the principle of theoretical linear deformation, the maximum radial displacement of cross-sections at different axial positions along the flexspline tooth ring shows a linearly decreasing distribution from the front-end cross-section 3 to the rear-end cross-section 1. This indicates that when the distribution pattern of the wall thickness modification parameters in the tooth root region matches the radial displacement gradient of the neutral layer and ensures no interference between the front-end cross-section and the rear-end cross-section, non-interfering meshing states can be achieved for all cross-sections of the entire tooth ring structure. Based on the above rules, by determining the modification amounts of the front-end and rear-end cross-sections and establishing a linear calculation model according to the geometric relationship between the axial positions of each cross-section and the main cross-section, the modification scheme for the tooth root wall thickness of the entire tooth ring can be derived.

[0100] Calculations were performed using MATLAB, and the tooth root wall thickness modification amount of the front-end cross-section 3 was obtained as 0.0668 mm, and the tooth root wall thickness modification amount of the rear-end cross-section 1 was 0.0454 mm.

[0101] In the unmodified state, during the assembly process of the flexspline and the wave generator, a spatial skew effect will be caused, resulting in meshing interference between the flexspline and the rigid gear when the transmission system is working. Figure 9 The cross-section motion trajectories of the modified flexspline are shown. It can be observed that after the tooth root wall thickness modification, no tooth profile interference phenomenon appears in each cross-section of the flexspline. This modification method effectively balances the asymmetric deformation under the action of the wave generator by adjusting the tooth root wall thickness of the flexspline, enabling the motion trajectories of the front-end and rear-end cross-sections to both move out of the interference region and form a stable conjugate meshing state.

[0102] 2. Finite element model 2.1 Three-dimensional solid modeling Aiming at the highly non-linear characteristics of the multi-tooth meshing mechanism of the harmonic drive system, this study uses a parametric modeling method to establish a three-dimensional solid model that includes the full tooth profile geometric characteristics of the flexspline and the rigid gear. On the premise of ensuring the calculation accuracy, partial structural simplification is carried out on the harmonic drive model.

[0103] (1) During the modeling process of the wave generator, to optimize the calculation efficiency and maintain the model accuracy, an integrated processing method can be used to reconstruct the structure of the flexible bearing and the cam assembly. Specifically, the outer ring of the flexible bearing is retained as the core load-bearing component, the discrete distributed rolling elements are equivalent to a homogeneous annular structure, and through geometric fusion with the cam profile, a simplified model of the wave generator with equivalent mechanical characteristics is constructed. This method effectively solves the mesh generation problem caused by the multi-body contact characteristics of the flexible bearing in traditional modeling and significantly reduces the consumption of computing resources. And the result difference between the simplified model and the complete solid model is extremely small, fully meeting the accuracy requirements for subsequent analysis.

[0104] (2) When modeling the flexspline, remove the chamfers at the bottom of the flexspline cylinder and the cup mouth. Removing the bottom of the flexspline cylinder helps reduce the number of meshes, thereby reducing the operation time. In addition, the chamfer transition structure in the tooth profile area of the cup mouth will significantly increase the complexity of the mesh generation stage, resulting in a systematic decrease in the element quality evaluation index and ultimately affecting the solution accuracy.

[0105] (3) When modeling the rigid spline, only retain the toothed part and simplify it to an internal cylindrical gear ring. The other structures of the rigid spline except the gear ring do not contact the flexspline, and the influence on the simulation results of the contact mechanical characteristics can be ignored. Simplifying the rigid spline can significantly reduce the complexity of constructing the finite element model and the consumption of computing resources. The UG software can be used to construct a simplified three-dimensional model of the harmonic reducer.

[0106] 2.2 Finite element related settings (1) Element type and mesh generation Mesh generation is one of the most critical steps in finite element analysis, directly affecting the accuracy, reliability and computational efficiency of the simulation results. The density of the mesh is closely related to the accuracy. Generally, a finer mesh can improve the computational accuracy, but it will also increase the computational cost. Therefore, a balance needs to be found between accuracy and computing resources. In addition, the mesh quality will also affect the accuracy and reliability of the finite element simulation results. Improper mesh generation may lead to an increase in numerical errors and even affect the stability and convergence of the simulation results. Therefore, reasonable mesh generation is crucial for the analysis results. Since the tooth profiles of the flexspline and the rigid spline are relatively complex, the fineness of their meshes is extremely critical for the meshing analysis during the transmission process. In this paper, the HYPERMESH software is used to generate the mesh for the harmonic reducer model. All mesh generations use hexahedral elements C3D8R. Finally, the number of mesh elements of the flexspline, the rigid spline and the wave generator are 589,760, 347,616 and 26,040 respectively, and the total number of mesh elements is 963,416.

[0107] (2) Material properties Since the main failure form in harmonic gear transmission is the fatigue fracture of the flexspline, the material selection is particularly important. Materials with high toughness and good anti-fatigue characteristics should be selected. As a commonly used alloy steel, 30CrMnSiA has high strength, excellent anti-fatigue property, toughness and hardenability after heat treatment, and its mechanical properties and workability are both good. Therefore, 30CrMnSiA is selected as the flexspline material in this paper. The working conditions of the rigid spline are similar to those of ordinary gears, and its material requirements are also the same as those of ordinary gears. Since the outer ring of the equivalent wave generator simulates the outer ring of the bearing, bearing steel GCr15 is selected. The working conditions of the inner ring of the equivalent wave generator are relatively good and there are no special requirements for the material, so commonly used carbon steel can be selected. The specific material parameters are shown in Table 4-1.

[0108] Table 4-1 Material Parameter of Each Component of Harmonic Reducer

[0109] (3)Contact Setting After defining the material properties, the contact mode needs to be set next. When setting up the contact pair in ABAQUS, the contact type involved should be clarified first. Due to the simplification of the wave generator, the original five pairs of contacts are reduced to three pairs. The first pair of contacts is the contact between the outer surface of the ring equivalent to the inner ring of the wave generator and the outer raceway surface of the outer ring; the second pair of contacts is the contact between the outer surface of the outer ring of the equivalent wave generator and the inner surface of the flexspline; the third pair of contacts is the contact between the tooth surfaces of the rigid gear and the flexspline. All contacts adopt the "surface-to-surface" contact mode. In the setting of the first pair of contacts, the outer surface of the ring is designated as the contact surface, while the outer raceway surface of the outer ring of the equivalent wave generator is used as the target surface, and the friction coefficient is set to 0.02. In the setting of the second and third pairs of contacts, the flexspline is used as the contact surface, and the tooth surfaces of the rigid gear and the outer surface of the outer ring of the equivalent wave generator are used as the target surfaces, and the friction coefficient is taken as 0.15.

[0110] (4)Load and Analysis Step Setting After completing the contact setting, the analysis steps and loads need to be set. In order to better simulate the actual operating state of the harmonic reducer, this study realizes the progressive simulation of assembly and operation through four analysis steps: The first step is to simulate the assembly stage of the wave generator component, apply full degree-of-freedom constraints to the bottom of the rigid gear and the flexspline, and apply a displacement load of 10 mm in the positive Z-axis direction through the wave generator to cause pre-deformation of the flexspline; the second step is to simulate the assembly stage of the rigid-flexspline system, fix the displacement degrees of freedom of the bottom of the flexspline and the wave generator, and drive the rigid gear to move 13 mm in the positive Z-axis direction to realize system assembly; the third step is the static load-bearing characteristic verification stage, release the rotational degree-of-freedom constraint of the flexspline output end around the Z-axis, and apply a clockwise torque of 30 N·m to the flexspline flange; the fourth step is the dynamic transmission performance simulation stage, keep the boundary conditions of the previous stage, release the rotational degree-of-freedom of the inner ring of the wave generator around the Z-axis, and apply an angular displacement load to drive the system into the dynamic meshing state.

[0111] 3. Tooth Profile Modification Based on Finite Element Simulation 3.1 Analysis of Tooth Profile Motion Trajectory Based on Simulation Deformation in Axial Section Before Modification After the flexspline is properly modified, when the wave generator is assembled inside the flexspline, there will be no tooth profile interference between the flexspline and the rigid gear. Based on this characteristic, in this study, a finite element model was constructed to simulate and analyze the assembly process of the wave generator, with a focus on observing the radial displacement in the long and short axis directions of the flexspline during this process. After importing the displacement simulation data into MATLAB, a systematic evaluation of the motion trajectory of the flexspline tooth surface can be carried out, and then the modification amount of the flexspline tooth profile can be accurately calculated. Therefore, in this section, only the first analysis step of the finite element model established in the previous section needs to be run to complete the relevant calculations.

[0112] Finite element analysis was carried out using ABAQUS software to obtain the radial displacement of the unmodified flexspline after the wave generator was installed.

[0113] In ABAQUS software, the radial displacement data of the long and short axes of the neutral layer of the flexspline were extracted respectively and compared and analyzed with the theoretical values.

[0114] When the flexspline was not subjected to tooth profile modification, starting from the bottom of the cup, the deviation of the finite element radial displacement of the long axis from the theoretical value was relatively small at the beginning. However, when it reached a position about 20 mm from the bottom of the cup (near the rear cross-section of the flexspline), the deviation began to increase significantly, which was mainly caused by the elastic deformation of the tooth ring. The variation law of the deviation between the finite element radial displacement of the short axis of the flexspline and the theoretical value was similar, that is, it was small at the beginning and gradually increased when it reached a position about 20 mm from the bottom of the cup, and the deviation between the finite element radial displacement of the short axis and the theoretical value was larger than that of the long axis.

[0115] Extract respectively Figure 7 the radial deformation amounts of the long and short axes of the neutral layer of cross-sections 1, 2, and 3 in the middle section, and then calculate the theoretical radial deformation amounts of each cross-section of the flexspline according to formula (4-1). The specific values are shown in Table 4-2.

[0116] 4-2 Radial deformation amounts of the long and short axes of the neutral layer of each cross-section of the flexspline

[0117] After importing the finite element simulation results into MATLAB, the motion trajectory of a single flexspline tooth relative to the rigid gear tooth within a 90° range can be obtained.

[0118] According to the simulation analysis results, the meshing system shows different tooth surface interference characteristics on different cross-sections. Specifically, interference phenomena are observed on the entire cross-section on the meshing contact surface of the front cross-section, while on the main cross-section and the rear cross-section, there is also tooth profile interference between the flexspline and the rigid gear in some parts.

[0119] 3.2 Flexspline modification based on finite element deformation Profile modification based on finite element simulation obtains the radial displacement distribution data of each cross-section of the flexspline teeth along the tooth width direction through finite element analysis, and imports the data into MATLAB to accurately simulate the contact trajectory and relative motion relationship between a single flexspline tooth and the rigid spline tooth during the transmission process. Finally, according to the meshing interference analysis results, profile modification is implemented on the tooth root wall thickness at different axial positions of the flexspline, so as to eliminate the meshing interference risk on the premise of ensuring the structural strength.

[0120] By performing polynomial fitting on the modification amount of the finite element method, the modification amount equation for each cross-section can be obtained as follows: (4-4) In the formula, t is the wall thickness modification amount, d is the distance from each cross-section to the front-end cross-section 3.

[0121] After the flexspline is modified by the finite element method, there is no interference between the flexspline teeth and the rigid spline teeth at each cross-section during the meshing process.

[0122] 4. Assembly simulation analysis of the harmonic reducer before and after flexspline modification Adopt the finite element simulation method to compare and analyze the mechanical properties of the flexspline in the harmonic reducer before and after modification. Use the ABAQUS software to simulate the assembly of the unmodified, linearly modified, and finite element method modified flexsplines. The analysis shows that there are significant stress concentrations in the bottom regions of the cylinder corresponding to the major axis and minor axis of the wave generator. The peak values of the equivalent stress in the three groups of simulations are 729.5 MPa, 712.7 MPa, and 721.5 MPa respectively. The tooth profile modification has little effect on the maximum stress value of the flexspline, and the stress level remains basically unchanged.

[0123] To clearly analyze the stress distribution of the tooth profile of the flexspline before and after modification, extract the stress nephogram of the flexspline tooth profile for detailed analysis. The stress analysis shows that the maximum stress of the tooth profile of the unmodified flexspline is 520.4 MPa, that of the linearly modified one is 484.1 MPa, and that of the finite element method modified one is 402.0 MPa. The finite element method modification reduces by 22.75% and 16.96% respectively compared with the unmodified and linearly modified ones, indicating that the tooth profile modification based on finite element simulation can more effectively improve the stress distribution of the flexspline.

[0124] Observed from the top view, the maximum stress of the unmodified flexspline occurs at the front cross-section near the major axis of the wave generator and the rear cross-section in the middle of the major and minor axes. The maximum stress of the linearly modified flexspline is only at the rear cross-section in the middle of the major and minor axes. The stress distribution of the tooth profile of the finite element method modified flexspline is better, the uniformity of the stress field is improved, the high stress area is in the major axis direction and its adjacent areas, and there are stress extreme values at the tooth root. There is local interference in the tooth surface meshing of the unmodified and linearly modified flexsplines, and the stress concentration effect is strong, and the peak stress is in the tooth surface contact area. For the flexspline modified by the finite element method, 50 teeth on one side (100 teeth in total on both sides) participate in meshing, accounting for 62.5% of the total number of teeth. The number of meshing teeth is increased compared with the unmodified and linearly modified ones. The contact pressure distribution of the tooth profile is uniform, most of the tooth profiles are in contact at more than half of the positions, the contact area is increased, the peak contact pressure is decreased, and the meshing performance is improved.

[0125] The tooth profiles of the flexspline are modified by the linear method and the finite element method, and three finite element models are established and assembled and run for simulation. The assembly simulation results show that the maximum stress of the tooth profile of the flexspline modified by the finite element method is significantly lower than that of the unmodified and linearly modified ones. The unmodified and linearly modified flexsplines will have secondary deformation, while the flexspline modified by the finite element method hardly occurs. The load operation results show that when the wave generator rotates from 0° to 30°, the tooth profile stress and contact pressure of the linearly modified flexspline keep increasing, and the contact area decreases. The tooth profile stress and contact pressure of the flexspline modified by the finite element method fluctuate within a certain range, and the maximum value is lower. Through comprehensive analysis, the flexspline modified by the finite element method has lower stress values and contact pressure during assembly and load operation, and should be used as the preferred solution.

[0126] V. Experimental Verification of Harmonic Reducer In order to better confirm the improvement effect of the tooth profile modification of the flexspline on the transmission performance of the harmonic reducer, the transmission performance of the harmonic reducer prototypes before and after modification was tested by using the comprehensive performance test bench of the harmonic reducer to evaluate the actual influence of the tooth profile modification of the flexspline.

[0127] 5.1 Construction of the Comprehensive Performance Test Bench of the Harmonic Reducer The comprehensive performance test bench of the harmonic reducer consists of a drive system, a loading system, a measurement system, a control and monitoring system. The drive system uses a servo motor to provide power; the loading system applies a load through a loading motor; the measurement system includes torque and speed sensors and an angle encoder to measure parameters such as torque, speed, and angle; the control and monitoring system consists of an industrial computer and a motion control card to control the experiment and monitor the parameters in real time. This test bench has a wide range of uses, can accurately detect the transmission performance, such as transmission efficiency, etc., and can also evaluate the reliability and simulate the working conditions to monitor the long-term operation performance.

[0128] Its working process is as follows: Start the motor through the console to run at a preset speed, and the speed can be adjusted in real time to meet different test requirements; Apply a preset torque with the loading motor to simulate the actual load, and the magnitude of the loading torque can be controlled by the console. During the test, the torque sensor and the angle encoder measure the torque and rotation angle of the input shaft and the output shaft respectively. The data is transmitted to the computer through the multi-channel acquisition system, and the operator can monitor in real time through the interface to ensure the stability and accuracy of the test.

[0129] 2. Transmission error test For a harmonic reducer, the transmission error is defined as the difference between the actual rotation angle and the theoretical rotation angle of the output shaft, and its calculation formula is as follows: (5-1) In the formula, θ TE is the transmission accuracy of the harmonic reducer, θ out is the actual output angle of the harmonic reducer, θ in is the actual input angle of the harmonic reducer, i is the reduction ratio of the harmonic reducer.

[0130] When conducting the transmission error test of the harmonic reducer, first ensure that all components of the test bench are in good condition, and correctly install and connect the reducer to be tested. Set the test parameters through the console, start the motor at the input end, gradually increase the speed from low speed to the rated speed, and at the same time apply the rated torque to the harmonic reducer using the loading device. Then, the angle encoder starts to collect the rotation angle data of the input shaft and the output shaft in real time, and transmits it to the computer for recording and analysis.

[0131] Test results The test results of the transmission error are shown in Figure 11 (a, b), Figure 11 (a) Harmonic reducer with unmodified flexspline, Figure 11 (b) Harmonic reducer with modified flexspline. The horizontal axis represents the output angle, and the vertical axis represents the transmission error. It can be seen from the figure that for the harmonic reducer with modified flexspline, its maximum transmission error is between -31″ and 20″, and the transmission error curve fluctuates up and down near zero. For the harmonic reducer with unmodified flexspline, the range of transmission error is -72″ to 63″, and compared with the harmonic reducer with modified flexspline, its fluctuation amplitude is significantly larger.

[0132] As shown in the figure, an increase in load will cause the transmission efficiency of the harmonic reducer to rise, but when the load further increases, the growth rate of efficiency gradually decreases. At the rated speed, the transmission efficiency of the harmonic reducer with an unmodified flexspline is 57.39%, and after modification, it is increased to 65.35%. Under the load condition of 1.5 - 4 times the rated torque, the transmission efficiency of the modified harmonic reducer is higher than that of the unmodified one, indicating that the modification of the flexspline tooth profile is beneficial to improving the transmission efficiency.

[0133] Harmonic reducers are widely used in the field of precision technology due to their high precision, and the core of their transmission performance lies in the design of the harmonic gear tooth profile. In this invention, taking the composite cycloid tooth profile as the object, the meshing mechanism of harmonic gear transmission is analyzed, the influence laws of flexspline parameters on the conjugate interval and tooth profile are discussed, and the reasons for the differences between the actual and theoretical deformations of the flexspline after the assembly of the wave generator are explored. This invention designs the plane tooth profile of the harmonic gear, analyzes the meshing characteristics with MATLAB, modifies the flexspline tooth profile using the finite element method and the linear method, and compares the performance differences before and after modification through simulation and experiments.

[0134] It mainly includes: The basic structure and operating mechanism of the harmonic reducer are elaborated, and the motion relationships of various components in harmonic gear transmission are discussed. The mathematical expressions of the wave generator contour, the flexspline deformation equation, the rotation angle calculation formula, and the conjugate theoretical model based on the envelope method are derived. The design method of the composite cycloid flexspline tooth profile is introduced, and the cycloid tooth profile equation with the change of the common tangent angle is derived.

[0135] 1. Based on the composite cycloid flexspline tooth profile equation, the conjugate region and conjugate tooth profile are determined through envelope method analysis. The influence laws of flexspline parameters on the conjugate region and conjugate tooth profile are deeply analyzed. The error reasons between the actual and theoretical deformations of the flexspline are simulated and analyzed, the plane tooth profile of the flexspline is designed, the origin of the actual flexspline coordinate system is obtained through simulation, the rigid gear tooth profile is calculated, and the meshing characteristics of the tooth profile are analyzed.

[0136] 2. The flexspline tooth profile is modified by the linear method and the finite element method, and the stress distribution and tooth profile contact characteristics before and after modification are studied through finite element analysis. The results show that the flexspline modified by the finite element method has a lower stress level during assembly and load operation, the maximum value of the tooth profile stress is lower than that of the flexspline modified by the linear method, and it performs better in terms of stability and contact pressure.

[0137] 3. A performance test experimental platform for the harmonic reducer is designed and built, and the key performance indicators of the harmonic reducer before and after modification are tested. The results show that the harmonic reducer modified by the finite element method has improvements in transmission error, transmission efficiency, torsional stiffness, and backlash, indicating that the modification of the flexspline tooth profile can significantly improve the overall transmission performance of the harmonic reducer.

[0138] Harmonic reducers play an important role in the fields of industrial automation and precision equipment, and their design has a significant impact on transmission performance. Through theoretical analysis and experimental verification, this invention provides a new research direction and improvement method for the tooth profile design of harmonic reducers.

Claims

1. A method for simulating the machining design of the tooth profile of a high-precision and high-reliability harmonic reducer, characterized in that It includes the following steps: A. Establish a geometric relationship model of the harmonic reducer, derive the flexspline deformation equation, the precise rotation angle relationship, and the meshing theory based on the envelope method; B. Design the composite cycloid flexspline tooth profile, and calculate the conjugate region and conjugate tooth profile through MATLAB; C. Use finite element simulation to analyze the error between the actual deformation and the theoretical deformation of the flexspline, and determine the offset of the origin of the flexspline coordinate system and the deviation of the normal rotation angle; D. Optimize the flexspline tooth profile parameters according to the simulation results, generate the actual neutral layer curve, and envelope and calculate the rigid gear tooth profile.

2. The high-precision and high-reliability harmonic reducer tooth profile simulation machining design method according to claim 1, wherein The design of the composite cycloid tooth profile includes: a. Generate an epicycloid tooth profile outside the pitch circle and a hypocycloid tooth profile inside; b. Realize the design of non-zero inclination angle of the tooth profile by adjusting the inclination angle α0 of the common tangent; c. Establish a coordinate transformation matrix to realize the translation and fitting of the tooth profile; Design formula: Equation of the outer tooth profile: （1） Equation of the inner tooth profile: （2） （3） Among them, is the outer rolling circle radius, is the tooth height, the inner rolling circle radius, by adjusting the common tangent inclination angle realize the non-zero inclination angle design of the tooth profile, so that the meshing area is expanded by more than 30%.

3. The high-precision and high-reliability harmonic reducer tooth profile simulation machining design method according to claim 1, characterized in that, The flexspline deformation model adopts the following non-linear equations: （4） Among them, is the radial deformation coefficient, with a value range of 0.8 to 1.

2. By introducing the normal rotation angle correction term, the error between the theoretical deformation and the finite element simulation is reduced to less than 0.2 μm.

4. The high-precision and high-reliability harmonic reducer tooth profile simulation machining design method according to claim 1, characterized in that The tooth profile modification includes: a. Linear modification method: Design the wall thickness modification amount based on the radial displacement gradient of the axial section; b. Finite element modification method: Fit the modification amount polynomial equation through simulation data; Among them, the tooth profile modification adopts a piecewise linear model: （5） （6） Among them, is the wall thickness modification gradient coefficient, is the wall thickness modification increment in the adjacent tooth profile area, β is the stress distribution attenuation coefficient, and the value range is 0.1~0.

5. and are the boundaries of the segmented modification area, which are fitted through the finite element simulation data formula (6), so that the maximum stress of the modified flexspline is reduced by 22.75%.

5. The high-precision and high-reliability harmonic reducer tooth profile simulation machining design method according to claim 1, characterized in that The meshing characteristic analysis includes: a. The motion trajectory simulation verifies non-interference meshing; b. The assembly state analysis shows that 66.25% of the tooth surface is meshing; c. The side clearance distribution is controlled within the range of 0 - 1.6 μm; The dynamic formula is used for calculating the meshing side clearance: （7） Among them, is the addendum circle radius of the flexspline and the circular spline, is the pressure angle, is the dedendum circle radius of the flexspline, is the root circle radius of the rigid gear, and by constraining , the meshing smoothness is improved by 40%.

6. The high-precision and high-reliability harmonic reducer tooth profile simulation machining design method according to claim 1, characterized in that The experimental verification includes: a. Design and build a comprehensive performance test bench for the harmonic reducer, and the test bench includes: Servo motor drive system, where the rated torque ≥ 30 N·m; Loading system, with the torque sensor accuracy of ±0.5%; Angle encoder, with the resolution ≤ 0.1°; Data acquisition system, with the sampling rate ≥ 1 kHz; b. Transmission error test; c. The torsional stiffness is increased by 66%.

7. The high-precision and high-reliability harmonic reducer tooth profile simulation machining design method according to claim 1, wherein Optimize the tooth profile parameters by using neural network: （8） Among them, is the meshing contact stress, ε strain is the meshing strain, The input parameters include tooth surface coordinates, contact stress and strain, and optimize the modification amount through training data to increase the modification efficiency by 3 times.

8. The high-precision and high-reliability harmonic reducer tooth profile simulation machining design method according to claim 1, wherein, The processing of the flexspline needs to meet the following process parameters: · Wall thickness tolerance: ±0.02 mm; Tooth profile surface roughness: Ra ≤ 0.8 μm; Heat treatment hardness: ≥ 45 HRC.

9. The high-precision and high-reliability harmonic reducer tooth profile simulation machining design method according to claim 1, wherein, Establish a multi-physical field simulation verification system, including: Static deformation analysis, using ABAQUS implicit solution; Dynamic contact analysis, using ABAQUS explicit solution; Fatigue life prediction, using FE-SAF.

10. The high-precision and high-reliability harmonic reducer tooth profile simulation machining design method according to claim 1, wherein, Design an integrated design system, including: Parametric design module, using the MATLAB platform; Simulation analysis module, with an ABAQUS / ANSYS interface; Performance prediction module, based on the Python neural network model.

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