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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

Fuzzy time-variant reliability analysis of mechanical structure based on generalized degradation
SUN Xuan1,2 , ZHANG Jianguo1,2 , WANG Pidong1,2 , PENG Wensheng1,2
1. School of Reliability and System Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China ;
2. Science and Technology on Reliability and Engineering Laboratory, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2015-08-17; Accepted: 2015-09-18; Published online: 2015-11-16 15:00
Foundation item: National Basic Research Program of China (2013CB733000)
Corresponding author. ZHANG Jianguo Tel.:010-82338356 E-mail:zjg@buaa.edu.cn
Abstract: The character that the specimen number of on-orbit mechanical products in space is small leads to the fuzziness of relative parameters and the dynamic degraded failure criterion. The existing fuzzy reliability model mainly aims at static problems, which cannot describe the time-variant and fuzzy problem. This paper proposes a fuzzy time-dependent reliability modeling and analysis method which is based on the interference model of generalized stress-strength and takes account of the fuzziness of both variables and failure criterion at the same time. First, fuzzy criterion can be transferred into random variables equivalently. Then the theory of cut set in fuzzy math can be used to deal with the fuzzy random variables, and thus the fuzzy time-variant reliability model is built. After that, a new method (FPHI2) is presented, based on the method of PHI2 which is a tool for time-variant reliability computation based on the outcrossing approach, to compute the fuzzy time-variant reliability. In the end, a numerical case and an engineering case are provided to verify the feasibility of the proposed method.
Key words: fuzziness     degradation     time-variant reliability     cut set     PHI2

1 模糊时变可靠性模型 1.1 模糊随机应力强度干涉模型

 (1)

τ时刻，假设分别为模糊随机强度和模糊随机应力的概率密度函数，代入式(1)可得τ时刻的失效概率为

 (2)

 图 1 在时间τ内的模糊随机应力-强度干涉模型 Fig. 1 Fuzzy random stress-resistance interference model in time τ

τ时刻的模糊极限状态面为，则它将模糊随机变量空间划分为模糊安全域和模糊失效域，模糊联合概率密度函数，因此，在τ时刻的模糊失效概率为

 (3)
1.2 模糊判据的等效

1) Z<0不表示产品完全失效，只是其适用性有所降低，Z值越小，降低的程度越多。

2) Z>0不表示产品一定处于安全区域。

3) Z=0不是产品可靠和失效状态的界限。

 图 2 模糊失效判据 Fig. 2 Fuzzy failure criterion

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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＂)的概率分布函数，同理[14]

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1.3 广义模糊应力的处理

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1.4 模糊时变可靠性模型的建立

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μg(g)为增函数，根据1.2节的方法，引入随机变量Z＂来表示失效判据的模糊性，它的概率分布函数和概率密度函数形如式(8)，得到此时的模糊失效域和等效的功能函数

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2 模糊时变可靠度求解 2.1 时变可靠度求解方法

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τ时刻的瞬时失效概率定义为

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2.2 模糊时变可靠度计算方法

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α水平下产品在时间区间[0,t]内的累积失效概率表示为

 (22)

τ时刻的瞬时失效概率定义为

 (23)

α水平下的等效极限状态函数中包含随机过程时，由于随机过程的时间相关性，不同时间点的可靠度具有相关性，因此基于上穿方法处理。设Nα+(0,t)表示α水平下的等效极限状态曲面在[0,t]内从安全域向失效域的上穿次数，则[0,t]内的累积失效概率为

 (24)

 (25)

 (26)

 (27)

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 图 3 根据FORM计算两时刻的可靠度指标 Fig. 3 Computation of reliability index of two instances according to FORM
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3 案例分析 3.1 数值算例

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 (36)

 变量 分布类型 均值 标准差 Y1 正态分布 0 0.01 Y2 正态分布 480 48 Z′ 均匀分布 0 1/12

 图 4 累积失效概率 Fig. 4 Cumulative failure probability

3.2 工程案例

 图 5 谐波减速器 Fig. 5 Harmonic reducer

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α水平下的概率密度函数为

 (40)

 (41)

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 变量 分布类型 均值 标准差 [W] 正态分布 5 0.5 σ 极值I型分布 106.111 5 15 L 正态分布 12.5 0.125 S 正态分布 0.467 7 0.004 677 Z＂ 均匀分布 0 0.028 868 ν1 正态分布 0.295 0.014 75 E1 正态分布 209 10.45 ν2 正态分布 0.3 0.015 E2 正态分布 206 10.3

 图 6 谐波减速器的累积失效概率 Fig. 6 Cumulative failure probability of harmonic reducer
 图 7 谐波减速器的可靠度 Fig. 7 Reliability of harmonic reducer

1) 根据相关项目要求，谐波减速器服役期为12 a，要求在12 a末产品的可靠度仍需达到0.999，计算结果表明考虑失效判据以及相关变量的模糊性时，产品时变失效概率较低，14 a末可靠度大于0.999，超过了设计要求，即产品在服役期内是可靠的。这样的分析结果也是符合实际的，因此对相关单位具有一定的指导意义。

2) 失效概率曲线图表明模糊时变可靠度计算方法与MCS方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工程分析要求,对工程实际应用有一定的参考价值。

4 结 论

1) 本文提出的模糊时变可靠度求解方法求解方便，计算误差较小，对工程应用具有一定的指导意义，例如数值案例中两种方法的计算误差最大不超过10%，工程案例中可靠度和累计失效概率曲线贴合度较高。

2) 该方法建立的航空在轨机械产品的模糊时变可靠性模型形式简单，便于后续的模糊时变可靠度求解。

3) 本文提出的机械产品模糊时变可靠性分析方法也适用于航空、通用机械等领域的类似可靠性分析问题。

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#### 文章信息

SUN Xuan, ZHANG Jianguo, WANG Pidong, PENG Wensheng

Fuzzy time-variant reliability analysis of mechanical structure based on generalized degradation

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(8): 1731-1738
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0528