﻿ 模糊-随机混合参数的机构运动可靠度计算方法<sup>*</sup>
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1. 北京航空航天大学 可靠性与系统工程学院, 北京 100083;
2. 北京航空航天大学 可靠性与环境工程技术国防科技重点实验室, 北京 100083

Computation method on motional reliability of mechanism under mixed parameters with fuzziness and randomness
YOU Lingfei1,2, ZHANG Jianguo1,2, ZHAI Hao1,2, LI Qiao1,2
1. School of Reliability and System Engineering, Beihang University, Beijing 100083, China;
2. Science and Technology on Reliability and Environmental Engineering Laboratory, Beihang University, Beijing 100083, China
Received: 2018-07-17; Accepted: 2018-10-15; Published online: 2018-10-29 14:59
Foundation item: National Natural Science Foundation of China (51675026)
Corresponding author. ZHANG Jianguo, E-mail: zjg@buaa.edu.cn
Abstract: Mixed uncertainties of random variables and fuzzy variables are ubiquitous in the parameters of the current mechanism products, but the existing fuzzy reliability model mainly aims at static problems, which cannot describe the time-dependent problem with mixed uncertainty. This paper proposes a reliability modeling and computation method of the fuzzy time-dependent mechanism based on the advanced envelope function through the kinematic error analysis of mechanism and considering the fuzziness of failure criterion and the variables. First, fuzzy criterion can be transferred into random variables in the limit state function. Then, the cut set of fuzzy theory can be used to deal with the fuzzy and random variables, and thus the fuzzy time-dependent reliability model is built. After that, the advanced envolope function is used to calculate the time-dependent reliability of the mechanism. Finally, the feasibility of the method is verified by the motion error issue of four-bar linkage. The results show that the method has high computational accuracy.
Keywords: envelope function     fuzzy     time-dependent reliability     motion error     cut set

1 模糊失效判据的等效

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μG(z)为递减函数，即使得机构运动的失效程度随z值的减小而增大，根据文献[15]，结合随机变量的概率分布函数的定义，可以把1－μG(z)看作一个新的随机变量(记为Z′)概率分布函数，模糊-随机失效域可以描述为{X|G(X)≤Z′}，等效的功能函数为Ge=G(X)－Z′。

μG(z)为递增函数，则可以把μG(z)看作一个新的随机变量(记为Z″)概率分布函数，同理，这种情况下机构运动的失效域描述为{X|G(X)≥Z″}，对应的等效功能函数为Ge=Z″－G(X)。

2 运动可靠度的一般包络方法

 图 1 机构运动误差的包络函数 Fig. 1 Envelope functions of mechanism motion error

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3 模糊-随机时变可靠性建模

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 图 2 运动误差的失效隶属度函数 Fig. 2 Membership function of motion error failure
 图 3 阈值模糊的时变可靠性失效事件描述 Fig. 3 Failure event description of time-dependent reliability based on fuzzy threshold

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Gα+(X)=0表达式为

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Gα(X)=0表达式为

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4 基于改进包络函数的时变可靠度计算 4.1 模糊-随机时变可靠度的近似求解

Gα+(U)=0表达式为

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Gα(U)=0表达式为

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α在[0, 1]上离散n等份，则阈值为ε下的可靠度可表示为

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4.2 算法流程

 图 4 基于改进包络函数的可靠度计算方法流程 Fig. 4 Flowchart of reliability computation method based on advanced envelope function
5 四连杆机构运动可靠度计算 5.1 四连杆机构运动学建模与分析

 图 5 四连杆机构 Fig. 5 Four-bar linkage mechanism

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α水平下的概率密度函数为

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α水平下R2的均值和标准差分别为

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 变量 均值 标准差 分布类型 R1 10 0.1 正态分布 (R2)α 50 3(α－1)2/20000 正态分布 R3 40 0.1 正态分布 R4 40 0.1 正态分布

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5.2 四连杆机构模糊-随机时变可靠度求解

 图 6 变量取均值时的运动误差变化 Fig. 6 Motion error change at means of variables

 图 7 ε=0.5时可靠度随截集水平α的变化 Fig. 7 Change of reliability with cut set level α when ε=0.5

ε在[0.2, 1.4]上变化时计算结果如图 8所示，由失效概率曲线图表明本文提出的模糊-随机时变可靠度计算方法与MCS法在失效概率的计算上差别较小，计算结果准确度可以认为满足机构运动误差的分析要求，对工程实际应用有一定的参考价值。部分误差阈值对应的结果见表 2

 图 8 0°~90°的失效概率 Fig. 8 Probability of failure at 0°-90°

 ε/(°) 本文方法 MCS法 0.70 0.714 25 0.713 83 0.80 0.836 45 0.836 11 0.90 0.914 07 0.913 67 1.00 0.958 25 0.957 94 1.10 0.981 31 0.981 03 1.20 0.992 32 0.992 18

6 结论

1) 本文提出的模糊-随机时变可靠度求解方法相较于传统时变可靠度求解方法，不但考虑了参数的模糊性，同时还考虑了判据的模糊性，解决了模糊-随机混合参数下的机构时变可靠性问题，更符合实际工程应用，对类似的机构运动可靠性分析具有一定的指导意义。

2) 本文方法求解方便，相比于MCS法计算误差较小，本案例中2种方法的计算误差最大不超过0.000 8，贴合度较高；同时计算效率大大提高，每α水平下MCS法计算次数为107次，而本文方法为50次左右。

3) 本文提出的机构运动模糊-随机时变可靠性分析方法适用于随机样本不完善，或样本数量不够的情况，可计算出较精确的结果。

 [1] 拓耀飞, 陈建军, 陈永琴. 区间参数弹性连杆机构的非概率可靠性分析[J]. 中国机械工程, 2007, 18(5): 528-563. TUO Y F, CHEN J J, CHEN Y Q. Non-probabilistic reliability analysis of elastic linkage mechanism with interval parameters[J]. China Mechanical Engineering, 2007, 18(5): 528-563. DOI:10.3321/j.issn:1004-132X.2007.05.007 (in Chinese) [2] TUO Y F, CHEN J J, ZHANG C J, et al. Reliability analysis of kinematic acuuracy for the elastic slider-crank mechanism[J]. Frontiers of Mechanical Engineering in China, 2007, 2(2): 214-217. DOI:10.1007/s11465-007-0037-3 [3] 董玉革, 陈心昭, 赵显德, 等. 模糊可靠性理论在机构运动可靠性分析中的应用[J]. 应用科学学报, 2002, 20(3): 316-320. DONG Y G, CHEN X Z, ZHAO X D, et al. An application of fuzzy reliability theory in the reliability analysis of mechanism movement[J]. Journal of Applied Sciences, 2002, 20(3): 214-217. (in Chinese) [4] 张义民, 黄贤振, 贺向东, 等. 不完全概率信息牛头刨床机构运动精度可靠性稳健设计[J]. 机械工程学报, 2009, 45(4): 105-109. ZHANG Y M, HUANG X Z, HE X D, et al. Reliability-based robust design for kinematic accuracy of the shaper mechanism under incomplete probability information[J]. Journal of Mechanical Engineering, 2009, 45(4): 105-109. (in Chinese) [5] 张义民, 黄贤振, 贺向东. 任意分布参数平面连杆机构运动精度可靠性稳健设计[J]. 农业机械学报, 2008, 39(7): 139-143. ZHANG Y M, HUANG X Z, HE X D. Reliability-based robust design for kinematic accuracy of the planar linkage mechanism with arbitrary distribution parameters[J]. Transactions of the Chinese Society for Agricultural Machinery, 2008, 39(7): 139-143. (in Chinese) [6] ZHANG J F, DU X P. Time-dependent reliability analysis for function generator mechanisms[J]. Journal of Mechanical Design, 2011, 133(3): 031005. DOI:10.1115/1.4003539 [7] ANDERIEU-RENAUD C, SUDERT B, LEMAIRE M, et al. The PHI2 method:A way to compute time-variant reliability[J]. Reliability Engineering and System Safety, 2004, 84(1): 74-86. [8] MEJRI M, CAZUGEL M, COGNARD J Y, et al. A time-variant reliability approach for aging marine structures with nonlinear behavior[J]. Computers & Structures, 2011, 89(19-20): 1742-1753. [9] LI J, CHEN J B, FAN W L. The equivalent extreme-value event and evaluation of the structural system reliability[J]. Structure Safety, 2007, 29(2): 112-131. DOI:10.1016/j.strusafe.2006.03.002 [10] 马小兵, 任宏道, 蔡义坤. 高温结构可靠性分析的时变响应面法[J]. 北京航空航天大学学报, 2015, 41(2): 198-202. MA X B, REN H D, CAI Y K. Time-varying response surface method for high-temperature structural reliability analysis[J]. Journal of Beijing University of Aeronautics and Astronautics, 2015, 41(2): 198-202. (in Chinese) [11] DU X P. Time-dependent mechanism reliability analysis with envelope function and first-order approximation[J]. Journal of Mechanical Design, 2014, 136(8): 081010. DOI:10.1115/1.4027636 [12] ZHANG J F, DU X P. Time-dependent reliability analysis for function generation mechanisms with random joint clearances[J]. Mechanism and Machine Theory, 2015, 92: 184-199. DOI:10.1016/j.mechmachtheory.2015.04.020 [13] WEI P F, SONG J W, LU Z Z, et al. Time-dependent reliability sensitivity analysis of motion mechanisms[J]. Reliability Engineering and System Safety, 2016, 149: 107-120. DOI:10.1016/j.ress.2015.12.019 [14] WEI P F, WANG Y Y, TANG C H. Time-variant global reliability sensitivity analysis of structures with both input random variables and stochastic processes[J]. Structural and Multidisciplinary Optimization, 2017, 55(5): 1883-1898. DOI:10.1007/s00158-016-1598-8 [15] 张明. 结构可靠度分析-方法与程序[M]. 北京: 科学出版社, 2009: 204-206. ZHANG M. Structural reliability analysis:Method and prosedures[M]. Beijing: Science Press, 2009: 204-206. (in Chinese) [16] WANG Z L, HUANG H Z, LI Y F, et al. An approach to system reliability analysis wiyh fuzzy random variables[J]. Mechanism and Machine Theory, 2012, 52: 35-46. DOI:10.1016/j.mechmachtheory.2012.01.007 [17] 张萌, 陆山. 模糊可靠性模型的收敛性及改进的截集分布[J]. 北京航空航天大学学报, 2014, 40(8): 1109-1115. ZHANG M, LU S. Convergence of fuzzy reliability models and an improved cut-set distribution[J]. Journal of Beijing University of Aeronautics and Astronautics, 2014, 40(8): 1109-1115. (in Chinese)

#### 文章信息

YOU Lingfei, ZHANG Jianguo, ZHAI Hao, LI Qiao

Computation method on motional reliability of mechanism under mixed parameters with fuzziness and randomness

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(4): 714-721
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0433