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机械可靠性英文文献及翻译
Mechanical Reliability
Simulation
Jianfeng Xu, Yalian Xie, Dan Xu
Shanghai Institute of Process Automation
Instrumentation Shanghai, PRC
Generally a mechanical system is composed
Abstract—This article provides a numerical method
to simulate the reliability of a complicated
mechanical system with multiple input random
parameters. The method bases on the theory of
mechanical reliability and uses a commercial FEM
software, ANSYS. The quantitative reliability of
the mechanical system is obtained by Monte Carlo
simulation. The method is verified by theoretical
analysis and is applied to a real structure. This
method has prevented the tedious mathematical
calculation and is more economical than
experimental methods. It can be used as a
complement of traditional experimental methods.
Keywords—MechanicalReliability;Finite
Element
Method; Monte Carlo Simulation
I. INTRODUCTION
Based on the national standard GB/T 3187-
1994 [1], reliability has been defined as
“the probability that a product will
satisfactorily perform its intended function
under given circumstances for a specified
period of time”. Reliability has been viewed
as one of the most critical parameters for the
products. For example, it is said in US that
reliability will be the focus of competition
for the future products. Most Japanese
companies take more reliable product as their
main objective for company development.
Therefore, to accurately predict the
reliability of the product, and to optimize
the design based on the reliability
requirements is very meaningful.
by many parts. A system’s reliability is
determined by many parameters, such as the
number of components, the quality of the
components and also how those components are
assembled. It is also related to the material
properties, manufacturing technologies,
dimension tolerances and so on. Most of the
references
currently available are focused on the
theoretical derivation for reliability
calculation, however, due the complexity, it
Normally the structural maximum stress and
the material strength are distributed with
certain probability density. Figure 1 shows an
example of the distribution, in
requires tedious mathematical calculation.
Another method to obtain the reliability of a
which
f （σ ）
and
g（ s）
represent the
system is by experiments, which requires
enough samples and testing time, thus the
experimental method can only be applicable to
those most critical systems.
This article uses mechanical strength
reliability as an example, illustrates how FEM
can be applied to the reliability simulation.
maximum stress
and material strength, respectively. In Figure
1（a）, since
is always larger than
reliability of the structure is 1. However, in
Figure 1（b）, it is not
The method employs commercial FEM software,
ANSYS, and takes advantage of the integrated
Monte Carlo simulation function. A simple
example is analyzed to verify the correctness
guaranteed that
） . There is an
g（ s）
be larger than
f （σ
of the method. The verified method is then
applied to a real structure to obtain the
quantitative reliability of a complex system
with multiple input random parameters.
II. STRENGTH-STRESS INTERFERENCE THEORY
interference zone in the distribution, as
shown in the shaded area and zoomed in Figure
2. Thus the reliability of the structure is
less than 1. Based on the mathematical
derivation, the reliability can be calculated
by integration with the distribution.
978-1-61284-666-8/11$26.002011 IEEE 1147
R = P（σ< s） =
σ +g（ s）ds dσ
−∞ + f （σ ）
III. METHOD DESCRIPTION
The commercial FEM package ANSYS has a PDS
f （σ ）g（s）
（a）
（Probabilistic Design Simulation） module [2].
The module defines dimension, material
properties and other parameters as random
variables with certain distributions. The output
stress and displacement can be defined as
results variables. Samples are collected based
on Monte
Carlo method to obtain the distribution of
f （σ
）
（b）
g（s
）
output variables.
With the PDS module, a suitable analysis
file should be firstly generated to define the
simulation process and to define the input and
output variables. If necessary, correlation
between input parameters can also be defined.
Monte Carlo method is then employed to collect
enough samples for simulation. Since each run is
independent, the
Figure 1 Stress and strength
distribution
g（s）f （σ ）
Figure 2 Stress-strength interference
zone
To create analysis file and
to define
parameters
To define the input random
parameters and
distributions
To define correlations
between input
To define output random
variables
To define method for probabilistic simulation
Response surface simulation
Results review
Figure 3. Flow chart for strength
method is suitable for parallel
run to save time. Generally
50~200 samples are collected in
Monte Carlo method to insure
the correctness of the results.
In some cases, the output
variable is a smooth function
of the random input variables.
with limited time. Basically the flowchart to
simulate the strength reliability with ANSYS
can be shown in Figure 3, in which the dashed
steps indicate they are optional.
IV. VERIFICATIONS AND APPLICATIONS
The correctness of the method is verified
by a tension rod which is an example in the
reference [3]. The
In such cases, the response can
be approximately evaluated by a
tension
P ~ N（29400,441）N
material
mathematical function. By
assessing the coefficients in
the function with the sampling
points and runs, response
strength
S ~ N（1054.48,41.36） N mm2 , radius of
the rod
surface can be determined and
numerous runs can be conducted
r ~ N（3.195,0.016）mm
theoretical
to obtain better distributions
calculation
shows the reliability of the rod is 0.999.
1148
reliability of a pressure vessel. The
shape and parameter distributions are
shown in Figure 6.
Bar element
P
Figure 4. FEM model
（a） Monte Carlo Simulation
the strength-stress difference, where KPa is
unit.
Themethodisthenappliedto
analyzethe
（b） Response Surface Simulation
Figure 5. Histogram of strength-stress difference
With ANSYS, the rod is modeled as a bar
element which is fixed in on end. The tension
is applied at the other end. The axial stress
is calculated in ANSYS. The tension force,
radius and material strength are defined as
random input variables in PDS. The output
variable is defined as the difference between
the strength and maximum stress. A value less
than 0 indicate a strength failure. One
hundred samples are collected with Monte Carlo
method. Results show that the difference
approximately follows a normal distribution
with normal 137.73N and variance 45.42N. Later
on, response surface are obtained based on the
previous samples and 10000 runs are conducted.
The normal and variance can be better obtained
as 137.65N and 44.52N, respectively. From
which it is concluded that the reliability of
the rod is 0.9988 and 0.9990 with the two
methods. This result matches well with the
theoretical calculation which demonstrates the
correctness of the method. Figure 4 shows the
FEM model, and Figure 5 is the histogram of
r
L
Fixed
r ~ U[1990,2010]mm
L ~ U[9990,10010]mm
Thickness~ U[7.5,8.5]mm
Pressure~ U[2800,3200]KPa
Young'sModule~ N（200,10）GPa
YieldStres ~ N（1000,50）MPa
Figure 6 Pressure Vessel
In PDS module, the random input
parameters are defined in accordance
with Figure 6. With 50 Monte Carlo
sampling points, the simulation shows
that the difference between strength
and stress can be approximately
described as a normal distribution
N （229.22,67.251）MPa . A further response
surface
simulation demonstrates that the
difference match the distribution as N
（228.86,65.823）MPa , as shown in
Figure 7. From these distributions,
the reliability of the
1149
pressure vessel can obtained as
0.99967 and 0.99974,
respectively.
（a） Monte Carlo Simulation
（b） Response Surface Simulation
Figure 7. Strength-stress deviation
histogram of pressure vessel
V. CONCLUSIONS
FEM and Monte Carlo simulation method can
be integrated to obtain the mechanical
reliability of a complex structure with
multiple input random parameters. Although
this article considers only the static cases,
the method can also be applied to dynamic
cases with suitable analysis file.
The method described in this article
prevents tedious mathematical calculation. The
structure reliability can be obtained with less
time and money. This method can be a complement
of the traditional experimental methods.
REFERENCES
[1] Reliability and Maintainability terms, GB/T
3187-94
[2] ANSYS Help, ANSYS Corp
[3] Liu P., Basic of Reliability Engineering,
Metrology Press, 2002
中文翻译：
机械可靠性仿真
Jianfeng Xu, Yalian Xie, Dan Xu
上海工业自动化仪表研究所
上海，中华人民共和国
摘要：
本文提供了一种多输入随机参数模拟复杂机械系统的可靠性计算方法。
该方法基于可靠性理论，采用商业有限元软件 ANSYS。
机械系统的可靠性定量
是通过蒙特卡罗模拟。
该方法是通过理论分析验证和实际应用的结构。
该方法
避免了繁琐的数学计算和比实验方法更经济。
它可以作为传统实验方法的一种
补充。
关键词：
机械可靠性；有限元；蒙特卡罗仿真方法。
一、引言
根据国家标准 GB／T 3187-1994 [ 1 ]，可靠性被定义为“在一定的条件
下在指定的一段时间内，一个产品很好地履行其预期的功能的概率”。
可靠性
一直被看作是一个产品的最重要的参数。
例如，据说在美国，可靠性将是今后
产品竞争的焦点。
大多数日本企业以更可靠的产品作为公司发展的主要目标。
因此，为了准确地预测产品的可靠性，并优化基于可靠性要求的设计是非常有
意义的。
一般的机械系统是由许多部分。
一个系统的可靠性取决于许多参数，如组
件的数量，质量的部件和这些部件组装。
这也与材料的性质，相关的制造技术，
尺寸公差等有关。
大多数现有的文献主要集中在可靠性理论推导计算，然而，
由于它的复杂性，需要繁琐的数学计算。
还有一种实验法来获得系统的可靠性
试验，需要足够的样本和测试时间，因此，实验方法只能适用于那些最重要的
系统。
本文采用机械强度可靠性为例，说明了有限元法可以应用于可靠性仿真。
该方法采用商业有限元软件，ANSYS，并利用 Monte Carlo 模拟功能集成。
一个
简单的实例分析验证了该方法的正确性。
这种验证的方法随后便应用到实际结
构，来获得多输入随机参数的一个复杂系统的可靠性定量。
二、拉压干涉理论。
通常情况下，结构的最大应力和材料强度的分布有一定的概率密度。
图 1
显示的分布的一个例子，在
f （σ ）和 g（ s）分别代表最大应力和材料强度的情
况下。
图 1（a）中因为
g（ s）总是大于 f （σ ），结构的可靠度是 1。
然而，在
图 1（b），它是不能保证
g（ s）大于 f （σ ）。
图中分布着一个干扰区，如
图所示，在阴影区域和放大的图 2。
因此，结构的可靠度小于 1。
基于数学推导
的可靠性，可以按照分布做一体化计算。
R = P（σ< s） =−∞+f （σ ）σ
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