﻿ 宇航单机元器件随机振动疲劳分析 Fatigue Analysis of Aerospace Electronic Equipments under Random Vibration

International Journal of Mechanics Research
Vol.06 No.03(2017), Article ID:22128,10 pages
10.12677/IJM.2017.63015

Fatigue Analysis of Aerospace Electronic Equipments under Random Vibration

Qiuju Zhu, Zimin Pan, Shenghao Li, Sheng Wu

Shanghai Aerospace Electronic Technology Institute, Shanghai

Received: Aug. 31st, 2017; accepted: Sep. 14th, 2017; published: Sep. 21st, 2017

ABSTRACT

The aerospace electronic equipment undergoes a variety of vibration environments throughout its life cycle. In the process of random vibration, the sensitive chips in the electronic equipment are prone to fatigue failure, and the fracture of the pins or the welding joints are broken out. In this paper, the fracture of the pin of the aerospace electronic equipment is discussed, and a detailed finite element analysis of the geometric model of the equipment structure and chips is carried out to obtain RMS stress field under random vibration conditions, and Miner’s cumulative fatigue damage theory and three bandwidth technique are used to analyze the fatigue of chips and its pins. After the structure optimization of the equipment, a detailed finite element analysis of the geometric model of the equipment structure and chips showed that the optimized structure safety margin has been greatly improved and verified by test. The results show that this method is an effective method for random vibration fatigue analysis of the aerospace electronic equipment.

Keywords:Random Vibration, Miner’s Cumulative Fatigue Damage, Three Bandwidth Technique, Fatigue Damage Ratio

1. 引言

2. Miner疲劳损伤累积理论与随机振动

2.1. Miner疲劳损伤累积理论

Miner线性损伤积累理论 [2] 认为，每经历一次应力循环，结构将消耗掉一部分的寿命，这里的应力循环是指随机振动引起的交变应力，该理论认为，结构在各种交变应力下的疲劳损伤是相互独立的，并且总损伤可以线性累积起来，当总疲劳损伤超过产品的疲劳极限后即造成疲劳破坏。用疲劳破坏率R表示已消耗结构寿命的百分比，如R = 0.5，表示实际应力循环已消耗掉结构疲劳寿命的一半，疲劳破坏率R [1] 定义如下：

$R=\sum \frac{{n}_{i}}{{N}_{i}}$ (1)

Miner线性损伤积累理论的成功之处在于它在工程上简便易用，且大量试验结果显示，结构发生疲劳破坏时的 $\sum {n}_{i}/{N}_{i}$ 均值确实接近于1 [3] 。而其他累积损伤理论和计算方法，要么计算过于复杂，要么并未给出所用参数的确定方法，要么必须进行大量的疲劳试验才能获取所需的计算参数，均难以应用于实际工程中，且计算精度也并不比Miner理论高，所以即使Miner理论自身并不完美，却一直在工程界广泛应用，尤其在随机载荷下，应力循环的大小次序完全是随机的，相邻两个应力循环由低到高与由高到低是等概率事件，削弱了载荷次序的影响 [3] 。本文采用Miner准则作为元器件管脚疲劳破坏的判据。

2.2. 随机振动及三带宽技术

2.2.1. 随机振动

$M\left\{\stackrel{¨}{u}\left(t\right)\right\}+C\left\{\stackrel{˙}{u}\left(t\right)\right\}+K\left\{u\left(t\right)\right\}=\left\{F\left(t\right)\right\}$ (2)

$H\left(i\omega \right)=\sum _{r=1}^{n}\frac{\left\{{\phi }_{r}\right\}{\left\{{\phi }_{r}\right\}}^{T}}{-{\omega }^{2}{m}_{r}+{k}_{r}+i\omega {c}_{r}}$ (3)

${G}_{out}\left(\omega \right)={|H\left(i\omega \right)|}^{2}{G}_{in}\left(\omega \right)$ (4)

2.2.2. 基于高斯(正态)分布的三带宽技术

$p\left(x\right)=\frac{1}{\sigma \sqrt{2\text{π}}}{\text{e}}^{-\frac{{x}^{2}}{2{\sigma }^{2}}}$ (5)

Figure 1. Gaussian distribution (Normal distribution) (σ = 1)

$\begin{array}{l}1 \sigma 数值发的生时间是68.3%\\ 2 \sigma 数值发的生时间是27.1%\\ 3 \sigma 数值发的生时间是27.1%\end{array}\right\}$ (6)

3. 元器件管脚随机振动疲劳分析实例

3.1. 元器件管脚随机振动疲劳失效分析实例

3.2. 有限元模型建模及仿真分析

Table 1. Qualification conditions for random vibration test

Figure 2. The fracture of the pin in the chip (the 1st pin)

3.3. 元器件管脚的疲劳寿命计算

3.3.1. 应用三带宽技术计算管脚发生疲劳所需的应力循环次数

Figure 3. The first order mode shapes of fault module

Figure 4. RMS stress field under random vibration conditions of the electronic equipment

Figure 5. The S-N curve of kovar

${N}_{1}={N}_{2}{\left(\frac{{S}_{2}}{{S}_{1}}\right)}^{b}$ (7)

N2 = 1000 (柯伐合金管脚在极限抗拉强度下故障前经受1000个循环)；

S1 = 157.64 MPa (管脚所受的最大均方根应力)

b = 6.4 (柯伐合金疲劳曲线的斜率)；

${N}_{1\sigma }=1000×{\left(\frac{579}{1×157.64}\right)}^{6.4}=4131213$

${N}_{2\sigma }=1000×{\left(\frac{579}{2×157.64}\right)}^{6.4}=48920$

${N}_{3\sigma }=1000×{\left(\frac{579}{3×157.64}\right)}^{6.4}=3652$

3.3.2. 元器件管脚实际应力循环次数

${n}_{1\sigma }=211.5\text{Hz}×60×6\mathrm{min}×0.683=52003.62$

${n}_{2\sigma }=211.5\text{Hz}×60×6\mathrm{min}×0.271=20633.94$

${n}_{3\sigma }=211.5\text{Hz}×60×6\mathrm{min}×0.0433=3296.862$

$R=\frac{{n}_{1\sigma }}{{N}_{1\sigma }}+\frac{{n}_{2\sigma }}{{N}_{2\sigma }}+\frac{{n}_{3\sigma }}{{N}_{3\sigma }}=1.337>1$

3.3.3. 结构改进及验证

${N}_{1\sigma }=1000×{\left(\frac{579}{1×96.57}\right)}^{6.4}=95093871$

${N}_{2\sigma }=1000×{\left(\frac{579}{2×96.57}\right)}^{6.4}=1126057$

${N}_{3\sigma }=1000×{\left(\frac{579}{3×96.57}\right)}^{6.4}=84058$

Figure 6. The first order mode shapes of the improved module

${n}_{1\sigma }=\text{249}\text{.4Hz}×60×6\mathrm{min}×0.683=61322.5$

${n}_{2\sigma }=\text{249}\text{.4Hz}×60×6\mathrm{min}×0.271=24331.5$

${n}_{3\sigma }=\text{249}\text{.4Hz}×60×6\mathrm{min}×0.0433=3887.6$

$R=\frac{{n}_{1\sigma }}{{N}_{1\sigma }}+\frac{{n}_{2\sigma }}{{N}_{2\sigma }}+\frac{{n}_{3\sigma }}{{N}_{3\sigma }}=0.0685,$

4. 结论

Figure 7. RMS stress field of the improved module

Fatigue Analysis of Aerospace Electronic Equipments under Random Vibration[J]. 力学研究, 2017, 06(03): 141-150. http://dx.doi.org/10.12677/IJM.2017.63015

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