留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

基于PCE的谐波减速器动态精度不确定性分析

张金洋 张建国 彭文胜 刘育强 汪龙

张金洋, 张建国, 彭文胜, 等 . 基于PCE的谐波减速器动态精度不确定性分析[J]. 北京航空航天大学学报, 2018, 44(5): 1056-1065. doi: 10.13700/j.bh.1001-5965.2017.0305
引用本文: 张金洋, 张建国, 彭文胜, 等 . 基于PCE的谐波减速器动态精度不确定性分析[J]. 北京航空航天大学学报, 2018, 44(5): 1056-1065. doi: 10.13700/j.bh.1001-5965.2017.0305
ZHANG Jinyang, ZHANG Jianguo, PENG Wensheng, et al. Dynamic accuracy uncertainty analysis of harmonic reducer based on PCE[J]. Journal of Beijing University of Aeronautics and Astronautics, 2018, 44(5): 1056-1065. doi: 10.13700/j.bh.1001-5965.2017.0305(in Chinese)
Citation: ZHANG Jinyang, ZHANG Jianguo, PENG Wensheng, et al. Dynamic accuracy uncertainty analysis of harmonic reducer based on PCE[J]. Journal of Beijing University of Aeronautics and Astronautics, 2018, 44(5): 1056-1065. doi: 10.13700/j.bh.1001-5965.2017.0305(in Chinese)

基于PCE的谐波减速器动态精度不确定性分析

doi: 10.13700/j.bh.1001-5965.2017.0305
基金项目: 

国家“973”计划 2013CB733000

详细信息
    作者简介:

    张金洋  男, 硕士研究生。主要研究方向:机械产品可靠性

    张建国  男, 博士, 教授, 博士生导师。主要研究方向:机械产品可靠性、机电产品可靠性

    彭文胜  男, 博士研究生。主要研究方向:机械设计、机电产品可靠性设计

    通讯作者:

    张建国, E-mail: zjg@buaa.edu.cn

  • 中图分类号: V442

Dynamic accuracy uncertainty analysis of harmonic reducer based on PCE

Funds: 

National Basic Research Program of China 2013CB733000

More Information
  • 摘要:

    谐波减速器的动态精度不仅与其各个部件的制造公差和装配间隙有关,还必须考虑谐波减速器柔性和摩擦的影响。目前谐波减速器精度问题研究大多只考虑单一因素,在进行精度分析时没有考虑到模型参数的不确定性对精度的影响。本文研究了谐波减速器在静态因素(加工、装配)和动力学因素(柔性、摩擦)综合作用下的动态精度问题;建立了考虑静态误差、柔性的非线性动力学模型;利用多项式混沌展开(PCE)方法进行参数灵敏度分析和不确定性分析,并和Monte Carlo方法作了比较,结果表明PCE方法效率更高。并基于动态精度PCE进行可靠性分析,得到动态精度可靠度。

     

  • 图 1  谐波减速器结构简图

    Figure 1.  Structure diagram of harmonic reducer

    图 2  谐波减速器往复运动动态误差曲线

    Figure 2.  Dynamic error curve of harmonic reducer reciprocation

    图 3  谐波减速器物理简化模型

    Figure 3.  A physical simplified model of harmonic reducer

    图 4  谐波减速器Dymola仿真模型

    Figure 4.  Dymola simulation model of harmonic reducer

    图 5  谐波减速器精度测试平台

    Figure 5.  Precision test platform of harmonic reducer

    图 6  谐波减速器输入转速

    Figure 6.  Speed of harmonic reducer input

    图 7  谐波减速器输出转速

    Figure 7.  Speed of harmonic reducer output

    图 8  谐波减速器动态误差

    Figure 8.  Dynamic error of harmonic reducer

    图 9  静态误差和动态误差曲线

    Figure 9.  Static error and dynamic error curves

    图 10  不同扭转刚度k1时动态误差变化曲线

    Figure 10.  Dynamic error curves at different torsional stiffness k1

    图 11  不同输出轴转动惯量Jl时动态误差变化曲线

    Figure 11.  Dynamic error curves at different output shaft moment of inertia Jl

    图 12  不同柔轮切向相邻齿综合误差Δff1时动态误差变化曲线

    Figure 12.  Dynamic error curves at different flexible wheel tangential adjacent gear comprehensive error Δff1

    图 13  2种方法动态误差均值比较

    Figure 13.  Dynamic error mean comparison between two methods

    图 14  2种方法动态误差均方差比较

    Figure 14.  Dynamic error's mean square error comparison between two methods

    图 15  动态误差的Monte Carlo仿真曲线

    Figure 15.  Monte Carlo simulation curves of dynamic error

    表  1  谐波减速器动力学模型参数

    Table  1.   Parameters of harmonic reducer dynamic model

    参数 数值
    Jm/(kg·m2) 3.2×10-4
    Jl/(kg·m2) 8.5×10-4
    Bm/(N·m·s·rad-1) 1.7×10-4
    Bl/(N·m·s·rad-1) 5.0×10-4
    Bsp/(N·m·s·rad-1) 2.8×10-4
    k1/(N·m·rad-1) 7 160
    k2/(N·m·rad-3) 21 576
    N 90
    下载: 导出CSV

    表  2  静态误差模型参数

    Table  2.   Parameters of static error model

    参数 数值
    Z2 182
    Z1 180
    αn/(°) 20
    Ef/m 3.1×10-5
    Ec/m 1.78×10-4
    Eb/m 2.05×10-4
    ΔFp2/m 3.6×10-5
    Δff2/m 1.14×10-4
    ΔFp1/m 3.6×10-5
    Δff1/m 1.14×10-4
    下载: 导出CSV

    表  3  不同电机输入角下动态误差实验值与仿真值

    Table  3.   Experimental values and simulation values of dynamic errors under different motor input corners

    电机
    输入角/(°)
    动态误差
    实验值/(°)
    动态误差
    仿真值/(°)
    真实误差/
    (°)
    相对
    误差/%
    50 0.023 473 0.021 908 0.001 565 4.7
    100 0.018 379 0.017 472 0.000 907 4.9
    200 0.016 727 0.015 486 0.001 241 7.4
    300 0.018 503 0.017 256 0.001 247 6.7
    400 0.016 205 0.017 369 0.001 164 7.1
    500 0.019 181 0.018 688 0.000 493 2.5
    700 0.017 800 0.019 320 0.002 820 8.5
    800 0.019 794 0.017 913 0.001 881 9.5
    1 000 0.020 950 0.022 243 0.001 293 4.1
    下载: 导出CSV

    表  4  谐波减速器不确定性参数及分布

    Table  4.   Uncertainty parameters and distribution of harmonic reducer

    参数 分布 均值 标准差
    Bm 正态分布 1.7×10-4 N·m·s·rad-1 0.000 03
    Jl 正态分布 8.5×10-4kg·m2 0.000 05
    k1 正态分布 7.16×103 N·m·rad-1 200
    k2 正态分布 2.157 6×104 N·m·rad-3 500
    Bsp 正态分布 2.8 ×10-4 N·m·s·rad-1 0.000 04
    Ef 正态分布 3.1 ×10-5 m 0.000 005
    Ec 正态分布 1.78 ×10-4 m 0.000 04
    Eb 正态分布 2.05 ×10-4 m 0.000 05
    Δff1 正态分布 1.14×10-4 m 0.000 04
    下载: 导出CSV

    表  5  动态精度多项式混沌展开式配点

    Table  5.   Collocation of dynamic accuracy polynomial chaos expansion

    参数 多项式混沌展开配点
    Bm (0.000 187,0.000 17,0.000 152)
    Jl (0.000 879,0.000 85,0.000 821)
    k1 (0.000 721 8,0.000 716 0,0.000 710 2)
    k2 (0.020 999,0.021 576,0.022 153)
    Bsp (0.000 303,0.000 28,0.000 257)
    Ef (0.000 033 8,0.000 031,0.000 028 1)
    Ec (0.000 201,0.000 178,0.000 154)
    Eb (0.000 176 3,0.000 205,0.000 233 7)
    Δff1 (0.000 116 9,0.000 14,0.000 163 1)
    下载: 导出CSV

    表  6  动态精度多项式混沌展开式系数

    Table  6.   Polynomial chaotic expansion coefficient of dynamic accuracy

    系数
    1 0.012 5
    ξ1 0.032 8
    ξ2 -0.154 0
    ξ3 0.357 6
    ξ4 -0.009 3
    ξ5 0.000 9
    ξ6 0.014 0
    ξ7 0.074 9
    ξ8 0.007 2
    ξ9 0.102 3
    下载: 导出CSV

    表  7  动态误差多项式混沌展开式系数

    Table  7.   Polynomial chaos expansion coefficients of dynamic error

    系数
    1 0.024 151
    ζ1 -0.236 1
    ζ2 0.036 6
    ζ3 0.720 9
    ζ4 -0.023 6
    ζ5 -0.021 8
    ζ1-1 0.0365
    ζ2-1 -0.000 8
    ζ3-1 0.049 0
    ζ4-1 -0.000 2
    ζ5-1 -0.001 5
    ζ1ζ2 0.003 7
    ζ1ζ3 -0.036 5
    ζ1ζ4 0.000 6
    ζ1ζ5 -0.005 3
    ζ2ζ3 -0.005 4
    ζ2ζ4 0.000 1
    ζ2ζ5 0.001 5
    ζ3ζ4 0.003 9
    ζ3ζ5 0.004 2
    ζ4ζ5 -0.000 5
    下载: 导出CSV

    表  8  FORM法可靠度计算结果

    Table  8.   Reliability calculation results based on FORM method

    迭代次数 β 可靠度
    1 2.783 0.997 3
    2 1.782 0.962 6
    3 1.737 0.958 8
    4 1.732 0.958 4
    5 1.732 0.958 4
    下载: 导出CSV

    表  9  Monte Carlo仿真实验估算失效概率

    Table  9.   Failure probability estimation under Monte Carlo simulation experiment

    nt nf Pf
    500 14 0.028 0
    2 000 61 0.030 5
    4 000 126 0.031 5
    6 000 205 0.034 2
    7 000 254 0.036 3
    8 000 305 0.038 1
    10 000 381 0.038 1
    下载: 导出CSV
  • [1] NYE T, KRAML R. Harmonic drive gear error: Characterization and compensation for precision pointing and tracking[C]//Processing of the 25th Aerospace Mechanics, Symposium. Washington, D. C. : NASA, 1991: 237-252.
    [2] EMELYANOV A F.Calculation of the kinematic error of a harmonic gear transmission taking into account the compliance of elements[J].Soviet Engineering Research, 1983, 3(7):7-10.
    [3] GRAVAGNO F, MUCINO V H, PENNESSTRI E.Influence of wave generator profile on pure kinematic error and centrodes of harmonic drive[J].Mechanism and Machine Theory, 2016, 104:100-107. doi: 10.1016/j.mechmachtheory.2016.05.005
    [4] 沙晓晨, 范元勋.谐波减速器传动误差的研究[J].机械制造, 2015, 44(5):50-54. http://www.cnki.com.cn/Article/CJFDTotal-ZZHD201505013.htm

    SHA X C, FAN Y X.Study of transmission error of harmonic drive reducer[J].Machine Building Automation, 2015, 44(5):50-54(in Chinese). http://www.cnki.com.cn/Article/CJFDTotal-ZZHD201505013.htm
    [5] HSIA L M. The analysis and design of harmonic gear drives[C]//Processing of the 1988 IEEE International Conference on Systems, Man, and Cybernetics. Piscataway, NJ: IEEE Press, 1988: 616-620.
    [6] TUTTLE T, SEERING W. Kinematic error, compliance, and friction in a harmonic drive gear transmission[C]//ASME Design Technical Conferences 19th Design Automation. New York: ASME, 1993: 319-324.
    [7] 游斌弟, 赵阳.考虑非线性因素的谐波齿轮传动动态误差研究[J].宇航学报, 2010, 31(5):1297-1282. http://d.wanfangdata.com.cn/Periodical_yhxb201005004.aspx

    YOU B D, ZHAO Y.Study on dynamic error of harmonic drive with nonlinear factors[J].Journal of Astronautics, 2010, 31(5):1297-1282(in Chinese). http://d.wanfangdata.com.cn/Periodical_yhxb201005004.aspx
    [8] PREISSNER C, ROYSTON T J, SHU D.A high-fidelity harmonic drive model[J].Journal of Dynamic Systems, Measurement, and Control, 2012, 134(1):011002. doi: 10.1115/1.4005041
    [9] KIRCANSKI N, GOLDENBERG A, ANGELS J. Nonlinear modeling and parameter identification of harmonic drive gear transmissions[C]//Processing of the 32nd IEEE Conference on Robotics and Automatic. Piscataway, NJ: IEEE Press, 1995: 3027-3032.
    [10] 王爱东. 机器人用谐波齿轮传动装置的运动精度分析[D]. 北京: 中国科学院, 2001: 11-15.

    WANG A D. Kinematic accuracy analysis of gear transmission for harmonic reducer of robot[D]. Beijing: Chinese Academy of Sciences, 2001: 11-15(in Chinese).
    [11] FATHI H, PRASANNA S, FRIEDHELM A.On the kinematic error in harmonic drive gears[J].Journal of Mechanical Design, 2001, 123(1):90-97. doi: 10.1115/1.1334379
    [12] WIENER N.The homogeneous chaos[J].American Journal of Mathematics, 1938, 60(4):897-936. doi: 10.2307/2371268
    [13] 赵珂, 高正红, 黄江涛, 等.基于PCE方法的翼型不确定性分析及稳健设计[J].力学学报, 2014, 46(1):11-19. http://www.cqvip.com/QK/91029X/201401/48361837.html

    ZHAO K, GAO Z H, HUANG J T, et al.Uncertainty quantification and robust design of airfoil based on polynomial chaos technique[J].Chinese Journal of Theoretical and Applied Mechanics, 2014, 46(1):11-19(in Chinese). http://www.cqvip.com/QK/91029X/201401/48361837.html
    [14] LOEVETT T, PONCI F, MONTI A.A polynomial chaos approach to measurement uncertainty[J].IEEE Transations on Instrumentation and Measurement, 2006, 55(3):729-736. doi: 10.1109/TIM.2006.873807
    [15] BENJAMIN L, HOSAM K, JEFFREY L.Recursive maximum likelihood parameter estimation for state space systems using polynomial chaos theory[J].Automatica, 2011, 47(11):2420-2424. doi: 10.1016/j.automatica.2011.08.014
    [16] 陶海川, 来新民.基于Dymola的无刷直流电机仿真模型[J].计算机仿真, 2005, 22(5):63-65. doi: 10.3969/j.issn.1006-9348.2005.05.018

    TAO H C, LAI X M. Computer simulation of brushless DC motor system based on Dymola[J].Computer Simulation, 2005, 22 (5):63-65(in Chinese). doi: 10.3969/j.issn.1006-9348.2005.05.018
    [17] SUDRET B.Global sensitivity analysis using polynomial chaos expansion[J].Reliability Engineering and System Safety, 2008, 93(7):964-979. doi: 10.1016/j.ress.2007.04.002
    [18] PENG W S, ZHANG J G, ZHU D T.ABCLS methods for high-reliability aerospace mechanism with truncated random uncertainties[J].Chinese Journal of Aeronautics, 2015, 28(4):1066-1075. doi: 10.1016/j.cja.2015.06.012
  • 加载中
图(15) / 表(9)
计量
  • 文章访问数:  729
  • HTML全文浏览量:  71
  • PDF下载量:  539
  • 被引次数: 0
出版历程
  • 收稿日期:  2017-05-12
  • 录用日期:  2017-06-16
  • 网络出版日期:  2018-05-20

目录

    /

    返回文章
    返回
    常见问答