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浙江大学学报(工学版)
Journal of ZheJiang University (Engineering Science)  2021, Vol. 55 Issue (8): 1548-1557    DOI: 10.3785/j.issn.1008-973X.2021.08.016
    
Design and modification of harmonic double circular arc tooth profile based on finite element method
Xing-yu ZENG(),Jun-yang LI*(),Jia-xu WANG,Ting TANG,Cong LI
State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, China
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Abstract  

There is a certain deviation between the deformation of the flexspline of the harmonic drive and the linear deformation according to the actual working conditions. The multi-body contact finite element analysis was carried out using ANSYS Workbench to explore the influence and sensitivity of the structural parameters of the flexspline on the actual radial deformation of the flexspline. Results show that the tooth width, the thickness, the length, and the distance between the wave generator and the bottom of the flexspline, etc., all have a certain degree of influence on the radial deformation of the flexspline. Among them, the distance between the wave generator and the bottom of the cup had the greatest influence on the radial deformation of the flexspline, and the sensitivity was the highest, followed by the tooth width and the length. And the thickness had the least influence on the radial deformation of the flexspline. The modified kinematics method and the precise rotation angle relationship of the flexspline were used to design the non-common tangent double-arc tooth profile based on the actual deformation of the flexspline, and combined with MATLAB parametric programming to perform three maintenance on the flexspline tooth profile, and perform finite element simulation on the solid model. Analysis results show that the maximum stress of the flexspline designed and modified by the finite element method (FEM) was reduced by 925 MPa than that without modification, and 144 MPa less than the modified and designed stress by the linear method. The circumferential stress of each section of the flexspline designed and modified by FEM was also lower than that of the linear method.

Key words： flexspline deformation      structural parameters      finite element analysis      tooth profile design and modification      flexspline stress     
Received: 12 August 2020      Published: 01 September 2021
CLC:  TH 132.43  
Fund:  国家重点研发计划资助项目（2018YFB1304800）；广东省重点领域研发计划资助项目（2020B090926002）
Corresponding Authors: Jun-yang LI     E-mail: 479376471@qq.com;lijunyang1982@sina.com
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Xing-yu ZENG
Jun-yang LI
Jia-xu WANG
Ting TANG
Cong LI
Cite this article:

Xing-yu ZENG,Jun-yang LI,Jia-xu WANG,Ting TANG,Cong LI. Design and modification of harmonic double circular arc tooth profile based on finite element method. Journal of ZheJiang University (Engineering Science), 2021, 55(8): 1548-1557.

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https://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2021.08.016     OR     https://www.zjujournals.com/eng/Y2021/V55/I8/1548


基于有限元法的谐波双圆弧齿廓设计和修形

针对实际工况中，谐波传动的柔轮的实际变形与线性假设变形存在一定偏差的问题，利用ANSYS Workbench进行多体接触有限元分析，探究柔轮的结构参数对柔轮有限元径向变形的影响规律和敏感度. 结果表明：齿宽、壁厚、筒长、波发生器与柔轮杯底距离等参数都对柔轮径向变形有一定程度的影响，其中波发生器与杯底的距离对柔轮径向变形影响程度最大，敏感度最高，其次是齿宽和筒长，壁厚对柔轮径向变形的影响程度最小. 基于柔轮的有限元变形，采用改进运动学法和柔轮精确转角关系设计无公切线双圆弧齿廓，并结合MATLAB参数化编程对柔轮齿廓进行三维修形，对仿真实体模型进行有限元分析，结果表明：有限元法设计修形的柔轮最大应力比未修形时减小925 MPa，比线性法设计修形减少144 MPa，有限元法设计修形的柔轮齿各截面的周向应力也低于线性法设计修形.


关键词： 柔轮变形,  结构参数,  有限元分析,  齿廓设计修形,  柔轮应力 
Fig.1 Coordinate system of double circular arc tooth profile
Fig.2 Harmonic drive coordinate system
Fig.3 Axial deflection diagram of flexspline tooth
Fig.4 Deformation state diagram of flexspline
Fig.5 Schematic diagram of flexspline tooth modification
Fig.6 Schematic diagram of flexspline structure
Fig.7 Radial deformation diagram of long and short axis of flexspline with different tooth widths
Fig.8 Influence of tooth width on difference between finite element and linear assumed radial deformation of flexspline
Fig.9 Radial deformation diagram of long and short axis of flexspline with different thicknesses
Fig.10 Influence of thickness on difference between finite element and linear assumed radial deformation of flexspline
Fig.11 Radial deformation of long and short axis of flexspline with different lengths
Fig.12 Influence of length on difference between finite element and linear assumed radial deformation of flexspline
Fig.13 Radial deformation of long and short axis of flexspline with different locations of wave generator
Fig.14 Influence of location of wave generator on difference between finite element and linear assumed radial deformation of flexspline
参数 符号 数值/mm 参数 符号 数值/mm
齿宽 b 10.8 柔轮内壁直径 d 61.32
筒长 L 30.7 筒体壁厚 t 0.38
齿根壁厚 tc 0.66 波发生器距离杯底的距离 η 21.2
Tab.1 Key structural parameters of flexspline
参数 柔轮齿廓参数 刚轮齿廓参数
ha*/mm hf*/mm xa/mm ya/mm ρa/mm t1/mm γ/(°) xf/mm yf/mm ρf/mm ha*/mm hf*/mm xa/mm ya/mm ρa/mm ρf/mm xf/mm yf/mm
数值/mm 0.7 0.9 0.396 3 0.608 4 0.63 0.653 6 6 0.876 7 0.742 2 0.65 0.65 0.90 0.871 9 1.154 4 0.644 9 0.631 9 0.397 9 1.021 7
Tab.2 Tooth profile parameters of theoretical method
参数 柔轮齿廓参数 刚轮齿廓参数
ha*/mm hf*/mm xa/mm ya/mm ρa/mm t1/mm γ/(°) xf/mm yf/mm ρf/mm ha*/mm hf*/mm xa/mm ya/mm ρa/mm ρf/mm xf/mm yf/mm
数值/mm 0.7 0.9 0.396 3 0.608 4 0.63 0.653 6 6 0.876 7 0.742 2 0.65 0.65 0.90 0.874 8 1.181 7 0.647 3 0.639 5 0.404 6 1.044
Tab.3 Tooth profile parameters of FEM
Fig.15 Track diagram of tooth movement of flexspline before modification
Fig.16 Track diagram of tooth movement of front section 1 after modification
Fig.17 3D view of tooth profile of flexspline after modification
Fig.18 Comparison chart of 3D axial modification amount of flexspline tooth
部件 材料 E/GPa μ ρ/（kg·m?3）
柔轮 30CrMnSiA 196 0.300 775 0
刚轮 45 210 0.269 785 0
波发生器 45 210 0.269 785 0
Tab.4 Material performance parameters of harmonic gear
Fig.19 Stress comparison of flexspline with different design and modification methods
Fig.20 Circumferential stress distribution diagram of flexspline
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