fatigue life analysis based on six sigma robust optimization for pantograph collector head support

811 1–9 Ó The Authors 2016 DOI: 10.1177/1687814016679314 aime.sagepub.com Fatigue life analysis based on six sigma robust optimization for pantograph collector head support Yonghua Li1,

Advances in Mechanical Engineering

2016, Vol 8(11) 1–9

Ó The Author(s) 2016 DOI: 10.1177/1687814016679314 aime.sagepub.com

Fatigue life analysis based on six sigma

robust optimization for pantograph

collector head support

Yonghua Li1, Mingguang Hu1and Feng Wang2

Abstract

In this article, a new fatigue life analysis method based on six sigma robust optimization is proposed, which considers the random effects of material properties, external loads, and dimensions on the fatigue life of a pantograph collector head support Some main random factors are identified through fatigue reliability sensitivity analysis, which are used as input variables during fatigue life analysis The six sigma optimization model is derived using the second-order response surface method The response surface is fitted by the Monte Carlo method, the samples are obtained by the Latin hypercube sampling technique, and the proposed model is optimized using the interior point algorithm Through the optimization, the collector head support weight is reduced, the mean and the standard deviation of fatigue life have been decreased, and the effect of design parameter variation on the fatigue life is reduced greatly The robustness of fatigue life prediction

of collector head support is improved The proposed method may be extended to fatigue life analysis of other compo-nents of electric multiple units

Keywords

Fatigue, P-S-N curve, six sigma robust optimization, collector head support, Monte Carlo

Date received: 27 June 2016; accepted: 14 October 2016

Academic Editor: Yongming Liu

Introduction

The pantograph is one of the most important electric

equipments in the electric multiple units (EMU), which

collects the electric power from catenary for the EMU

The pantograph is mainly composed of collector head

support (CHS) and carbon slipper, which is subjected

to shock loads between carbon slipper and contact

wire.1 The working conditions of pantograph are

becoming worse and worse with the increasing speed of

the EMU

The pantograph has been subjected to variable

load-ings due to the uneven track, the impact force of

panto-graph and catenary, and the air flow force, which may

cause fatigue damage of the pantograph, shorten its

ser-vice life, and affect the safety and reliability during

EMU operation.2,3When the pantograph worked for a

period, some cracks were found in the pantograph CHS

and the vane where fatigue failure may occur To ensure the normal usage of pantograph and reduce accident occurrences, it is necessary to investigate the fatigue life and reliability of the pantograph CHS

Fatigue reliability analysis, which combines the fati-gue life analysis and reliability-based design, is an effec-tive method to improve the reliability of engineering components.4In this method, the dispersion problem of

1 School of Traffic and Transportation Engineering, Dalian Jiaotong University, Dalian, China

2 College of Bullet Train Application and Maintenance Engineering, Dalian Jiaotong University, Dalian, China

Corresponding author:

Yonghua Li, School of Traffic and Transportation Engineering, Dalian Jiaotong University, Dalian 116028, China.

Email: yonghuali@163.com

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License

(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).

of fatigue reliability analysis is to establish an effective

fatigue reliability model, which should be consistent with

fatigue failure mechanism and also reflect the dispersion

of various factors during fatigue failure.9–18 At present,

the established models for fatigue reliability analysis

mainly include residual strength model,9cumulative

dam-age model,10strain energy model,11,12ductility exhaustion

model,13 damage-strengthening model,14 and

prob-abilistic life prediction models.15,16 However, the

robust optimization is rarely considered during

fati-gue reliability analysis of pantograph CHS In recent

years, the six sigma robust optimization has been

widely applied to engineering practice.18–22

In this article, fatigue life analysis based on six sigma

robust optimization for pantograph CHS is proposed,

where the fatigue reliability analysis is conducted to

find the uncertain factors which affected the fatigue life

of structure, and six sigma robust optimization analysis

is conducted to decrease the effects of uncertain factors

on fatigue life

From the engineering application prospect, the

com-bination of six sigma and robust optimization in

rail-way industry can improve the robustness of mechanical

components and reduce the impact of random factors

on the component performance Meanwhile, this

method lightens the component weight and contributes

to the light weight design

Through the fatigue reliability analysis of CHS, its

fatigue reliability can be predicted considering many

factors, such as surface quality, stress concentration,

external loads, and plate thickness, and the main factors

and secondary factors that cause its fatigue failure can

be obtained The dispersion problems of fatigue life are

analyzed using six sigma robust optimization design

Accordingly, the random factors affecting the

sensitiv-ity of fatigue life are reduced greatly

This article is organized as follows: section ‘‘Six

sigma robust optimization design method’’ provides six

sigma robust optimization design method The static

strength analysis of CHS based on finite element

analy-sis is given in section ‘‘Static strength analyanaly-sis of CHS

based on finite element method.’’ Section ‘‘Fatigue

reliability analysis of CHS’’ conducts fatigue reliability

achieve the goal of lower sensitivity value for target response under the random plenty of uncertainty fac-tors Six sigma robust optimization19,20is an advanced design method with combination of six sigma quality management theory and robust optimization It mini-mizes the objective response value to meet the reliabil-ity design requirement

Considering the complex nonlinear relationship between target response values and design parameters, the Monte Carlo method is used for numerical calcula-tion in this article The used sampling method is Latin hypercube in the ANSYS probabilistic design system (PDS) module The six sigma robust optimization model is built through the response surface method Finally, the six sigma robust optimization design of CHS is accomplished using the optimization toolbox of MATLAB 2010b

Static strength analysis of CHS based on finite element method

Structural analysis of CHS

In order to obtain an accurate simulation result, the integrated model of the pantograph was established including CHS and the contact strip The CHS stress was calculated by analyzing the strength of the inte-grated model The geometry of the CHS is shown in Figure 1

Finite element model of CHS

To improve the model calculation accuracy, a finite ele-ment model of CHS is meshed by hexahedral eleele-ment The loads and constraints of CHS are defined based on the force and boundary conditions of contact strip and CHS Figure 2 shows the integrated finite element model of contact strip and CHS In Figure 2, the grid size of contact strip and CHS finite element model is

5 mm, and the total grid numbers are 92,646

According to the actual loading conditions of CHS, the force in the pantograph can be simplified as follows First, the friction between pantograph and contact wire

can be simplified as the longitudinal force FX The load

caused by the car body vibration and the impact of

the pantograph-catenary is simplified as the vertical

force FZ Then, the air pressure can be simplified as aerodynamic load PRES The values of the above-mentioned simplified forces are 450 N, 350 N, and

6000 Pa, respectively The material of model for the CHS and carbon slipper is the aluminum alloy, and its yield strength is 435 MPa

Static strength check of CHS The load condition of static strength calculation includes longitudinal force FX, vertical force FZ, and aerodynamic load PRES.22 This analysis is conducted

in the ANSYS 14.0 When the calculation is completed, the maximum Von Mises stress results are obtained Figure 3 shows the maximum Von Mises stress results

of the CHS and the carbon slipper The Von Mises stress distribution of the CHS can be obtained as shown in Figure 4

From Figure 4, it can be seen that the maximum value is 93 MPa, which is smaller than the material yield strength 435 MPa Therefore, the static strength of the CHS meets the design requirements

The static strength calculation for the CHS and the carbon slipper indicates the position of the maximum Von Mises stress located in the installing hole of the CHS The simulation result is consistent with that of actual test conditions Thus, the simulation results pro-vide a certain reference value for the primary design work It can also shorten the product development cycle and reduce the cost to a certain extent

Fatigue reliability analysis of CHS Fatigue life assessment of CHS

Fatigue life of CHS under variable loading is evaluated according to the P-S-N curve of material and the load spectrum of CHS.23–26 However, the test data of

Figure 1 Geometry of the CHS.

Figure 2 The integrated finite element model of contact strip

and CHS.

Figure 3 Von Mises stress results of the CHS and the carbon

slipper.

Figure 4 Von Mises stress of the CHS.

and APand BPare material constants.

Mean stress correction During the operation of the

EMU, the pantograph will generate vibrations, which

makes the stress of the CHS fluctuant around a certain

average stress Often such cyclic loadings with mean

stress considerably influence the component damage

accumulation process The stress spectrum can be

expressed by average stress and stress amplitude The

average stress is zero according to the material P-S-N

curve Therefore, the simplified stress spectrum

equa-tion is modified based on Goodman diagram26 as

follows

s1= sa

where s1is the equivalent stress, sais the stress

ampli-tude, smis the mean stress, and sb is the material

ulti-mate tensile strength

The fatigue life of CHS can be calculated by

equa-tion (1), where the equivalent stress s1 is obtained by

equation (2) to consider mean stress corrections

Fatigue reliability calculation of CHS

The calculation of fatigue reliability life is completed

using the ANSYS PDS module and HyperMesh 11.0

the limit state G \ 0 represents the structural failure

Reliability calculation of CHS The Monte Carlo method is used to calculate the structural fatigue reliability.21,28In this article, the fatigue probability analysis for CHS is completed using ANSYS PDS module The sample points of 500, which are obtained using Latin hyper-cube sampling technique, are introduced to calculate the structural state function of G Figure 5 represents the variation trend of the sample mean values The vertical coordinate is the difference between the calcu-lated life and the designed life, and the horizontal coordinate is the number of sample points The mid-dle line represents the mean value, and the other two lines are the upper and lower bounds of the structural state function From Figure 5, it can be seen that the variations of mean value of structural state function

G tends to be stable, which indicates the reliability agrees well with the design requirement and the relia-bility is 99.60%

Sensitivity analysis of fatigue reliability Through the sensi-tivity analysis, the main factors that affect CHS fatigue reliability can be obtained Figure 6 shows the sensitiv-ity of reliabilsensitiv-ity results, in which the size of areas repre-sents the important degree of different influencing

Table 1 Distribution characteristics and numerical values of each random variable parameter.

factors The positive input parameters mean that the

parameters are positively correlated with the output,

while the negative input parameters mean that the

parameters are negatively correlated with the output

From Figure 6, note that the main factors are the

longi-tudinal force FX and the structure size T1 and T2 of

CHS The other factors have little impact on the fatigue

reliability of the CHS

Fatigue life analysis based on six sigma robust optimization for CHS

Establishment of approximate response surface model

The longitudinal force accuracy is difficult to control since such force is determined by many factors, such as the CHS structure, air flow impact, and vibrations

Figure 5 The variation trend of the mean value of structural state function samples.

Figure 6 The sensitivity of fatigue reliability results.

This article focuses on the influence of the size of CHS

structure on its fatigue reliability

However, the relationship between fatigue life and

structural dimensions is the implicit nonlinear, it is

dif-ficult to be formulated in the analytic expression The

response surface method may describe accurately

impli-cit nonlinear relationship of the fatigue life and

struc-tural dimensions The robust optimization model of six

sigma is built using the response surface method to

improve the analysis accuracy

In this research, a response surface–based model is

built considering the parameters of the plate thickness

and fatigue life values In engineering application, the

second-order response surface model is used widely,

and the basic formula is

y = Xn

i = 1

ciix2i +Xn

i.j

cijxixj+ Xn

i = 1

cixi+ c0 ð4Þ

where n is the number of design variables, c0is the

con-stant, and ci, cii, and cijare the polynomial coefficients

The Monte Carlo method is used to simulate and fit

an accurate response surface model of CHS The response value of the CHS fatigue life is obtained by

500 times Latin hypercube sampling technique The distribution types and the design variables are shown in Table 2

Figure 7 shows the variation trend of the mean value

of fatigue life samples The vertical coordinate is the fatigue life of the CHS, and the horizontal coordinate

is the number of sample points From Figure 7, the mean values of fatigue life samples have stabilized by

500 times simulation to the CHS Thus, the predicted fatigue life is reasonable

Figure 8 shows the sensitivity analysis for the CHS fatigue life, in which the area size represents the impor-tant degree of influencing factor From Figure 8, it can

be seen that the effect degrees of the analyzed design variables on the fatigue life are ranked as T1 T2 T3 Based on the sensitivity analysis, the design variables

on the fatigue life, T1, T2, and T3, are used to fit the response surface

Figure 7 The variation trend of the mean value of fatigue life samples.

The response surface of the mean value of CHS

fati-gue life is as follows

uN=  0:1902T12 0:05965T22

 0:1444T2

3+ 0:06946T1T3 + 0:1139T2T3+ 1:0441T1

+ 0:1652T2+ 0:6551T3+ 3:5981

ð5Þ

The response surface for standard deviation value of

fatigue life for CHS is as follows

sN= 0:03T12+ 0:0127T22

+ 0:0106T32+ 0:0235T1T3

+ 0:0175T2T3 0:2159T1

 0:1523T2 0:1989T3+ 0:9032

ð6Þ

where uN is the mean value of fatigue life and sN is the

standard deviation value of fatigue life T1, T2, and T3

are dimensions of the CHS

Evaluation of response surface model error

The fitting precision of response surface model is

evalu-ated by the determination coefficient The calculation

method of the determination coefficient,29R2, is as

follows

R2=XP

i = 1

(^yi yi)2 (yi yi)2 ð7Þ

where yi is the response value for the sample point of i,

^yi is predictive value, yi is the response mean value for the sample point of i, and P are the number of sample points When the determination coefficient R2is close

to 1, the prediction accuracy of the response surface is higher The value of R2is 98.6%, which indicates that the accuracy of the fitted response surface is higher

Six sigma robust optimization design of fatigue life The fatigue life of CHS is calculated using its para-metric model considering the first principal stress Then, the response surface models of the fatigue life mean and standard deviation are obtained using the Monte Carlo method Finally, the optimal solution is obtained using the interior point algorithm method The specific flow is shown in Figure 9

Fatigue life analysis based on six sigma robust opti-mization model is as follows

min F½uN(Ti), sN(Ti)

= w1½uN(Ti) N02+ w2s2N(Ti) s:t: uN(Ti) 6sN(Ti) N0

TiL+ 6sTi uT i TU

i  6sT i

ð8Þ

where T is dimension parameter, i is the number of con-straints, and N0= 1 3 107:0 is the design life.21uN(Ti) is

a function for fatigue life mean value response surface,

sN(Ti) is a function for the standard deviation values response surface of fatigue life, TiLand TUare the lower

Figure 8 The sensitivity of fatigue life.

and upper bounds of plate thickness, uTi is mean values

of the dimension parameter, sTi is the standard

devia-tion values of the dimension parameter, and w1 and w2

are weight coefficients

Result analysis

The interior point algorithm is performed by MATLAB

2010b software The results of pre-optimization and

post-optimization are shown in Table 3

life value N0 The CHS weight is reduced with the decrease in the sizes of design variables Considering the engineering practice, the optimized thickness values

of T1, T2, and T3 are round to 2.0, 3.0, and 4.0 mm, respectively Meantime, the mean and standard devia-tion of fatigue life have been decreased by 16%, which indicates that the effects of parameter variation on fati-gue life are reduced greatly Therefore, the robustness

of CHS design is improved

Conclusion This article analyzed the fatigue life based on six sigma robust optimization method for pantograph CHS A new six sigma optimization model is built by the second-order response surface The response surface is fitted by Monte Carlo method and the samples are obtained by the Latin hypercube sampling technique The design parameters are optimized and the robustness of fatigue life of CHS is improved Some conclusions are as follows:

1 Through the sensitivity analysis of the CHS fati-gue life, the effects of its main design variables affecting on the fatigue life are identified, which are thickness of carbon slipper mount T1, thick-ness of U shape mount T2, and thickness of spring mount T3

2 The sensitive design parameters of T1, T2, and

T3are optimized by the six sigma robust optimi-zation method Through the optimioptimi-zation, the CHS weight is reduced with the decrease in the sizes of design variables, the mean and standard deviation of fatigue life have been decreased by 16%, and the effect of parameter variation on fatigue life is greatly reduced Six sigma robust optimization for pantograph CHS is realized The proposed method may be extended for application on the other components of EMU Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Figure 9 Process of fatigue life analysis based on six sigma

robust optimization design.

The author(s) disclosed receipt of the following financial

sup-port for the research, authorship, and/or publication of this

article: The authors would like to acknowledge the partial

supports provided by the program of Educational

Commission of Liaoning Province under contract number

JDL2016001, the program of National Natural Science

Foundation of Liaoning Province under contract number

2014028020, and the program of the Dalian Science and

Technology Project under contract number 2015A11GX026.

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