Special Issue Article Fatigue life analysis based on six sigma robust optimization for pantograph collector head support Advances in Mechanical Engineering 2016, Vol 8(11) 1–9 Ó The Author(s) 2016 DOI: 10.1177/1687814016679314 aime.sagepub.com Yonghua Li1, Mingguang Hu1 and 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 components 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 loadings due to the uneven track, the impact force of pantograph and catenary, and the air flow force, which may cause fatigue damage of the pantograph, shorten its service life, and affect the safety and reliability during EMU operation.2,3 When 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 fatigue life analysis and reliability-based design, is an effective method to improve the reliability of engineering components.4 In this method, the dispersion problem of School of Traffic and Transportation Engineering, Dalian Jiaotong University, Dalian, China 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) 2 affecting the component fatigue life is fully considered Comparing the conventional fatigue analysis, however, a large number of random factors make the fatigue reliability analysis much more complex Moreover, fatigue reliability analysis is also related to probability distribution of fatigue life.5 Generally, the probability distribution needs to be statistically analyzed based on a large number of experimental data, and these experiments are often time consuming and costly Thus, the development of the investigation on structural fatigue reliability is relatively slow In recent years, many researchers have made some achievements on structural fatigue reliability analysis.6–8 The key to solve the issue 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,9 cumulative damage model,10 strain energy model,11,12 ductility exhaustion model,13 damage-strengthening model,14 and probabilistic life prediction models.15,16 However, the robust optimization is rarely considered during fatigue 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 combination of six sigma and robust optimization in railway 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 sensitivity 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 analysis is given in section ‘‘Static strength analysis of CHS based on finite element method.’’ Section ‘‘Fatigue reliability analysis of CHS’’ conducts fatigue reliability Advances in Mechanical Engineering analysis of CHS Section ‘‘Fatigue life analysis based on six sigma robust optimization for CHS’’ carries out fatigue life analysis based on six sigma robust optimization for CHS A brief discussion and conclusion closes the article Six sigma robust optimization design method Robust optimization design18 is an effective method to reduce the effects of various uncertain factors on the target response value Namely, the mean square value of the target response can be reduced It can also achieve the goal of lower sensitivity value for target response under the random plenty of uncertainty factors Six sigma robust optimization19,20 is an advanced design method with combination of six sigma quality management theory and robust optimization It minimizes the objective response value to meet the reliability design requirement Considering the complex nonlinear relationship between target response values and design parameters, the Monte Carlo method is used for numerical calculation 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 integrated model The geometry of the CHS is shown in Figure Finite element model of CHS To improve the model calculation accuracy, a finite element model of CHS is meshed by hexahedral element The loads and constraints of CHS are defined based on the force and boundary conditions of contact strip and CHS Figure 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 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 Li et al Figure Geometry of the CHS Figure Von Mises stress of the CHS force FZ Then, the air pressure can be simplified as aerodynamic load PRES The values of the abovementioned 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 Figure The integrated finite element model of contact strip and 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 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 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 provide 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 Figure Von Mises stress results of the CHS and the carbon slipper Fatigue reliability analysis of CHS 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 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 Fatigue life assessment of CHS Advances in Mechanical Engineering variable load spectrum are difficult to be obtained due to the limitation of experimental conditions In this study, a simplified fatigue load spectrum according to the actual situation is applied to calculate the CHS fatigue life together with the material P-S-N curve The parameter of P-S-N curve For the CHS, the P-S-N curve of the material is often used to calculate its fatigue life, and the formula can be expressed as follows ð1Þ ( lg N )P = AP + BP lg sÀ1 where N is the fatigue life, sÀ1 is the equivalent stress, and AP and BP are 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 equation is modified based on Goodman diagram26 as follows sÀ1 = sa À sm =sb ð2Þ where sÀ1 is the equivalent stress, sa is the stress amplitude, sm is the mean stress, and sb is the material ultimate tensile strength The fatigue life of CHS can be calculated by equation (1), where the equivalent stress sÀ1 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 software A parametric model of CHS is built using the APDL language.27 The parameters of variable loads, structural dimensions, and materials are defined as random inputs, and the distribution characteristics and values of each parameter are listed in Table A limit state equation can be expressed as the difference between the calculated life and the designed life28 G = N À N0 ð3Þ where G is the limit state function, N is the fatigue life calculated by equation (1), and N0 is designed life G represents the structural safety G = represents the limit state G \ represents the structural failure Reliability calculation of CHS The Monte Carlo method is used to calculate the structural fatigue reliability.21,28 In this article, the fatigue probability analysis for CHS is completed using ANSYS PDS module The sample points of 500, which are obtained using Latin hypercube sampling technique, are introduced to calculate the structural state function of G Figure represents the variation trend of the sample mean values The vertical coordinate is the difference between the calculated life and the designed life, and the horizontal coordinate is the number of sample points The middle 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 reliability is 99.60% Sensitivity analysis of fatigue reliability Through the sensitivity analysis, the main factors that affect CHS fatigue reliability can be obtained Figure shows the sensitivity of reliability results, in which the size of areas represents the important degree of different influencing Table Distribution characteristics and numerical values of each random variable parameter Random variables Sign Distribution Mean Coefficient of variation Elastic modulus (MPa) Poisson ratio Density Longitudinal load (N) Vertical load (N) Aerodynamic load (MPa) Thickness of carbon slipper mount (mm) Thickness of U shape mount (mm) Thickness of spring mount (mm) E U DENS FX FZ PRES T1 T2 T3 Gaussian Gaussian Uniform Log normal Log normal Log normal Gaussian Gaussian Gaussian 7.1 104 0.33 2.7 1029 450 350 6.0 1023 2.5 3.5 4.0 0.05 0.05 0.1 0.1 0.1 0.05 0.05 0.05 Li et al Figure The variation trend of the mean value of structural state function samples Figure The sensitivity of fatigue reliability results 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 longitudinal 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 6 Advances in Mechanical Engineering Table Upper and lower bounds of variables and the distribution types Design variables Lower (Ti L ) Upper (Ti U ) Type of distribution Standard deviation T1 (mm) T2 (mm) T3 (mm) 1.5 2.5 3.0 3.5 4.5 5.0 Gaussian Gaussian Gaussian 0.05 0.05 0.05 Figure The variation trend of the mean value of fatigue life samples 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 difficult to be formulated in the analytic expression The response surface method may describe accurately implicit nonlinear relationship of the fatigue life and structural 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= n X i=1 cii x2i + n X i.j cij xi xj + n X c i xi + c0 ð4Þ i=1 where n is the number of design variables, c0 is the constant, and ci , cii , and cij are 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 Figure 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 shows the sensitivity analysis for the CHS fatigue life, in which the area size represents the important 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 Li et al Figure The sensitivity of fatigue life The response surface of the mean value of CHS fatigue life is as follows uN = À 0:1902T12 À 0:05965T22 À 0:1444T32 + 0:06946T1 T3 + 0:1139T2 T3 + 1:0441T1 ð5Þ 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 R2 is close to 1, the prediction accuracy of the response surface is higher The value of R2 is 98.6%, which indicates that the accuracy of the fitted response surface is higher + 0:1652T2 + 0:6551T3 + 3:5981 The response surface for standard deviation value of fatigue life for CHS is as follows sN = 0:03T12 + 0:0127T22 + 0:0106T32 + 0:0235T1 T3 + 0:0175T2 T3 À 0:2159T1 À 0:1523T2 À 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 Six sigma robust optimization design of fatigue life The fatigue life of CHS is calculated using its parametric 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 Fatigue life analysis based on six sigma robust optimization model is as follows F½uN (Ti ), sN (Ti ) Evaluation of response surface model error The fitting precision of response surface model is evaluated by the determination coefficient The calculation method of the determination coefficient,29R2, is as follows R2 = P X (^yi À yi )2 yi ) i = (yi À 7ị = w1 ẵuN (Ti ) N0 + w2 s2N (Ti ) s:t: uN (Ti ) À 6sN (Ti ) ! N0 TiL + 6sTi uTi ð8Þ TiU À 6sTi where T is dimension parameter, i is the number of constraints, and N0 = 107:0 is the design life.21 uN (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, Ti L and TiU are the lower Advances in Mechanical Engineering Table Six sigma robust optimization results of fatigue life Variables Pre-optimization Post-optimization Change rate (%) Standard deviation, sN Mean value, uN Thickness, T1 (mm) Thickness, T2 (mm) Thickness, T3 (mm) Weight, m (kg) 2.75 1022 1.0 107.47 2.5 3.5 4.0 8.59 2.31 1022 1.0 107.06 1.9 3.0 3.9 8.04 16.0 61.1 24.0 14.3 2.5 6.4 From Table 3, it can be concluded that the fatigue life value of post-optimization is more close to designed 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 deviation of fatigue life have been decreased by 16%, which indicates that the effects of parameter variation on fatigue 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: Figure Process of fatigue life analysis based on six sigma robust optimization design and upper bounds of plate thickness, uTi is mean values of the dimension parameter, sTi is the standard deviation values of the dimension parameter, and w1 and w2 are weight coefficients Through the sensitivity analysis of the CHS fatigue life, the effects of its main design variables affecting on the fatigue life are identified, which are thickness of carbon slipper mount T1, thickness of U shape mount T2, and thickness of spring mount T3 The sensitive design parameters of T1, T2, and T3 are optimized by the six sigma robust optimization method Through the optimization, 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 Result analysis Declaration of conflicting interests The interior point algorithm is performed by MATLAB 2010b software The results of pre-optimization and post-optimization are shown in Table The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article Li et al Funding The author(s) disclosed receipt of the following financial support 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 14 15 16 References Li MG, Yang YQ and Su C Research on anti-fatigue performance of upper-arm cross beam of pantograph structure of electric multiple units J Dalian Jiaotong Univ 2013; 34: 23–26 (in Chinese) Ibrahim MH, Kharmanda G and Charki A Reliabilitybased design optimization for fatigue damage analysis Int J Adv Manuf Tech 2015; 76: 1021–1025 Zhang HF, Tong LS, Chen ZB, et al Improvement of pantograph framed structures for metro vehicles based on fatigue strength analysis Electr Locomot Mass Transit Veh 2010; 33: 30–33 (in Chinese) Zhu SP, Huang HZ, Li YF, et al Probabilistic modeling of damage accumulation for time-dependent fatigue reliability analysis of railway axle steels Proc IMechE, Part F: J Rail Rapid Transit 2015; 229: 23–33 Schmidt D, Manuel L, Nguyen HH, et al Fatigue reliability analysis for brace–column connection details in a semisubmersible hull J Offshore Mech Arct 2015; 137: 061301-1–061301-7 Zhu SP, Huang HZ, Peng W, et al Probabilistic physics of failure-based framework for fatigue life prediction of aircraft gas turbine discs under uncertainty Reliab Eng Syst Safe 2016; 146: 1–12 Jablonski M, Lucchini R, Bossuyt F, et al Impact of geometry on stretchable meandered interconnect uniaxial tensile extension fatigue reliability Microelectron Reliab 2014; 55: 143–154 Rafsanjani HM, Sørensen JD and Sciubba E Reliability analysis of fatigue failure of cast components for wind turbines Energies 2015; 8: 2908–2923 Jun W and Zhi PQ Fatigue reliability based on residual strength model with hybrid uncertain parameters Acta Mech Sinica 2012; 28: 112–117 10 Park TG, Choi CH, Won JH, et al An efficient method for fatigue reliability analysis accounting for scatter of fatigue test data Int J Precis Eng Man 2010; 11: 429–437 11 Zhu SP and Huang HZ A generalized frequency separation-strain energy damage function model for low cycle fatigue-creep life prediction Fatigue Fract Eng M 2010; 33: 227–237 12 Fan YN, Shi HJ and Tokuda K A generalized hysteresis energy method for fatigue and creep-fatigue life prediction of 316L (N) Mater Sci Eng A: Struct 2015; 625: 205–212 13 Zhu SP, Huang HZ, Liu Y, et al An efficient life prediction methodology for low cycle fatigue-creep based on 17 18 19 20 21 22 23 24 25 26 27 28 29 ductility exhaustion theory Int J Damage Mech 2013; 22: 556–571 Zhao LH, Zheng SL and Feng JZ Fatigue life prediction under service load considering strengthening effect of loads below fatigue limit Chin J Mech Eng 2014; 27: 1178–1184 Zhu SP, Huang HZ, Ontiveros V, et al Probabilistic low cycle fatigue life prediction using an energy-based damage parameter and accounting for model uncertainty Int J Damage Mech 2012; 21: 1128–1153 Hu D, Ma Q, Shang L, et al Creep-fatigue behavior of turbine disc of superalloy GH720Li at 650°C and probabilistic creep-fatigue modeling Mater Sci Eng A: Struct 2016; 670: 17–25 Karadeniz H Uncertainty modeling in the fatigue reliability calculation of offshore structures Reliab Eng Syst Safe 2001; 74: 323–335 Ke L, Zhang YJ, Qin S, et al A new robust design for imperfection sensitive stiffened cylinders used in aerospace engineering Sci China Technol Sci 2015; 58: 796–802 Sun GY, Li GY, Chen T, et al Sheet metal forming based six sigma robust optimization design J Mech Eng 2008; 44: 248–254 (in Chinese) Asafuddoula M, Singh HK and Ray T Six-sigma robust design optimization using a many-objective decomposition-based evolutionary algorithm IEEE T Evolut Comput 2014; 19: 490–507 Li YH, Hu MG and Wang J Six sigma robust optimization of fatigue life for the passenger car battery hanging device J Donghua Univ 2016; 33: 108–111 Kim JW and Yu SN Design variable optimization for pantograph system of high-speed train using robust design technique Int J Precis Eng Man 2013; 14: 37–43 Shimizu S, Tosha K and Tsuchiya K New data analysis of probabilistic stress-life (P–S–N) curve and its application for structural materials Int J Fatigue 2010; 32: 565–575 Gagnon M, Tahan A, Bocher P, et al Influence of load spectrum assumptions on the expected reliability of hydroelectric turbines: a case study Struct Saf 2014; 50: 1–8 Torgeir M and Efren AU Reliability-based assessment of deteriorating ship structures operating in multiple sea loading climates Reliab Eng Syst Safe 2008; 93: 433–446 Zhao YX, Yang B and Peng JC Drawing and application of Goodman-Smith diagram for the design of railway vehicle fatigue reliability China Railw Sci 2005; 26: 6–12 (in Chinese) Rui FS, Jesus AMPD, Correia JA FO, et al A probabilistic fatigue approach for riveted joints using Monte Carlo simulation J Constr Steel Res 2015; 110: 149–162 Lu H, He YH and Zhang YM Reliability-based robust design of mechanical components with correlated failure modes based on moment method Adv Mech Eng Epub ahead of print June 2014 DOI: 10.1155/2014/568451 Zhou LR and Ou JP Finite element based on the model updating of long-span cable-stayed bridge based on the response surface method of radial basis function China Railw Sci 2012; 33: 8–15 (in Chinese) ... ‘? ?Fatigue life analysis based on six sigma robust optimization for CHS’’ carries out fatigue life analysis based on six sigma robust optimization for CHS A brief discussion and conclusion closes... 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... 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