Special Issue Article
Multi-objective reliability design optimization of electric multiple unit pantograph geometric parameters
Advances in Mechanical Engineering 2016, Vol. 8(10) 1–10 Ó The Author(s) 2016 DOI: 10.1177/1687814016675078 aime.sagepub.com
Bingzhi Chen, Yonghua Li, Shaodi Dong and Jian Wang
Abstract In order to reduce the design variable fluctuations in the multi-objective optimization of the pantograph geometric parameters of electric multiple units, a multi-objective reliability design optimization method was proposed based on the dual response surface method in this article. The corresponding model of the pantograph was built by integrating the dual response surface method, the reliability indexes, and fuzzy preference method. Considering different working conditions and reliability index, the reliability optimal design target was realized by optimizing the established model. A case was studied to verify the feasibility and effectiveness of the proposed method. Keywords Pantograph, multi-objective optimization, reliability, dual response surface method, fuzzy preference
Date received: 7 June 2016; accepted: 26 September 2016 Academic Editor: Yongming Liu
Introduction The pantograph is one of the important parts of electric multiple unit (EMU). Its good dynamic performance can reduce the impact of pantograph catenary and the contact force effectively. Once the off-line rate was reduced, EMU can operate with few failures.1 To obtain a better performance of the EMU pantograph, many investigations on the pantograph system structure,2 dynamic model,3 contact network,4 gas dynamic noise,5 and mechanical properties6,7 have been carried out to solve system optimization problems. However, most of these studies ignore design variables’ fluctuations generally, which may lead to the unreasonable design solutions which cannot satisfy the requirements of actual situations.8–11 Reliability design optimization is one of the modern design methods.12–15 It can reduce the sensitivity of objective function to design variables. At present, the studies of reliability design optimization mainly focus on the multidisciplinary design optimization,12 collaborative optimization,13 interaction balance
optimization,14 interaction prediction optimization,15 and providing the thought for the multi-objective robust reliability optimization. However, such studies just considered the uncertainty analysis of the design, and the variable fluctuations of the design goals have not been investigated. In optimization of the pantograph geometric parameters, the reliability design method was introduced to objective function and constraints, which can reduce the system interference and improve the electrical performance of bow reliability. The robust reliability optimal design was developed by Ben Haim since 1990s. Later, many corresponding studies have been proposed.16–19 The most robust reliability design optimization model was established based School of Traffic and Transportation Engineering, Dalian Jiaotong University, Dalian, China Corresponding author: Yonghua Li, School of Traffic and Transportation Engineering, Dalian Jiaotong University, Dalian 116028, China. Email:
[email protected]
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).
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on reliability analysis method,16,17 stochastic model,18 the reliability index,19 and so on. However, the robust reliability optimization method research in the multiparameter, multi-objective optimization problem is rarely researched in these studies, and the practical application in the vehicle track is rarely too. The Box– Behnken design (BBD) experimental design is a kind of experimental design method that can consider many design variables and their interactions at the same time. In this article, the BBD experimental design is combined with the dual response surface method, and then the fluctuation variance can be reduced effectively and the sensitivity of the design parameters can be reduced too. Therefore, the multi-objective robustness and reliability of the pantograph can be realized. In this study, an EMU pantograph is chosen as the engineering object. A reliability design optimization method was proposed, which was based on the dual response surface method. Then, a multi-objective reliability design optimization model of the pantograph geometric parameters was established. Different levels of design variables’ values were obtained using the BBD experimental design. The target reliability indexes were calculated using the dual response surface method. Then the comprehensive design objective can be obtained by adopting fuzzy preference method. The optimization problem was solved in the ISIGHT 5.6 platform. Finally, the reliability optimization solutions were obtained, which improved the performance of the EMU pantograph.
Multi-objective reliability design optimization model Response surface methodology is based on the experimental design, which is one of the statistical processing techniques. It is used to model and analyze multivariable issues.20 The dual response surface methodology is based on the results of the physical experiment and simulation test. The main response surface model and the sub-response surface model can be fitted by the mean m and the variance s2 of the product quality characteristics. Their formulas are as follows21 ym = b 0 +
ys = g 0 +
n X
bii xi +
n X
bii x2i +
n X
i=1
i=1
p\i
n X
n X
n X
i=1
g ii xi +
i=1
g ii x2i +
bi xp xi + em ð1Þ
In this study, the multi-objective reliability design optimization is based on the dual response surface model of the mean value and variance. Then, the geometric parameters of reliability can be measured with the application of reliability index, which improves the reliability of performance. In reliability analysis, both the structural resistance R and the comprehensive effect S were subjected to the normal distribution. Thus, the structural function Z = R S was also subjected to the normal distribution. Then, the mean value of the function can be obtained by uZ = uR uS , and the standard deviation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi can be obtained by sZ = s2R + s2S . Finally, the reliability index b of objective function can be expressed as b=
uZ uR uS = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sZ s2R + s2S
ð3Þ
where uR , s2R , uS , and s2S are the average values and the variances of the structural resistance and the comprehensive effect, respectively. In the design period, the reliability index of the structure b is not less than the target reliability index bT , as b bT and bT = 3:0. Therefore, based on equation (3), the target reliability of the product should be made as high as possible. And the average and variance of the objective function can be concluded using the dual response surface method, then the reliability index can be obtained based on the theory of reliability, and finally, the robust reliability of the pantograph was realized. The anti-interference ability of the product can also be enhanced. Its specific implementation steps are illustrated as shown in Figure 1. As shown in Figure 1, a general multi-objective reliability design optimization process was illustrated in detail. The dual response surface method and fuzzy preference method were applied to construct the total objective function, and the working constraints’ robustness was combined. Then the corresponding optimization model was established as max F(X ) = w1 b1 (X ) + w2 b2 (X ) + + wk bi (X ) 8 > < gj (X ) 0 s:t: bi bT > : L X X XU ð4Þ
gpi xp xi + es ð2Þ
p\i
where ym and ys are the average and variance of the system response, respectively; xi , xp are the random variables; b0 , bi and g0 , gi are the coefficients of each polynomial; and em and es are the random errors.
where wk is the weight coefficient of each target reliability; bi (X ) is the reliability index of each goal; k = 1, 2, 3; i = 1, 2, 3; gj (X ) is the pantograph’s constraints; bT is the target reliability index, and its value is generally equal to 3; X is the design vector; X U is upper limit of the design vector; and X L is the lower limit of the design vector.
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Figure 2. Geometric parameters’ model of the EMU pantograph mechanism.
balance rod, and other parts.23 In Figure 2, section SP is the bow head swing lever and section AC is the active pole. The design of the pantograph contains 11 design parameters, that is, x1 , x2 , x3 , . . . , x11 . Furthermore, Ex , Ey , and brad can be obtained by analyzing their geometric relationships, which can be expressed as
Ex = lce cos g x1 cos a Ey = lce sin g x1 sin a b = p arctan c
c=
Figure 1. The process of multi-objective reliability design optimization.
Multi-objective reliability design optimization of pantograph geometric parameters There are many different EMU pantograph models in China, such as DSA250 series, SSS400 series, TSG19 series, and CX-PG series.22 In this study, the multiobject reliability optimization problem of SSS400 series pantograph of EMU was discussed.
Geometric parameter model of pantograph mechanism The arch frame part of the pantograph is a four-bar mechanism. Its transmission mechanism is connected with a driving rod to realize the bow down of the pantograph. Its model of the pantograph frame mechanism is as shown in Figure 2. The pantograph is mainly composed of bow head, swing arm, upper arm rod, lower arm rod, support rod,
ð5Þ ð6Þ
lce sin g + x1 sin a (x8 sin x11 + x3 sin a + x6 cos u) lce cos g x1 cos a (x8 cos x11 x3 cos a + x6 cos u)
where lce is the length of the rod CE; g, a, and u can be seen in Figure 2. The descriptions of the factors are given in Table 1.
The design variables and level values To test design, the reliability design optimization based on the dual response surface method was applied in the pantograph by setting design variables. There are 11 design variables in the optimization problem. Thus, the BBD method is chosen which is suitable for three factors or more. The BBD experimental design. The BBD is an experimental design method which evaluates the nonlinear relationship between the indicators and factors. This method utilizes an approximate rotational symmetry. It is also composed of a plurality orthogonal cube and a center point.24 The BBD experimental design of the three levels is shown in Figure 3. The advantage of this method is that it can evaluate the main factors and the factor interactions. Moreover, it can arrange the experiment combinations of factors and avoid the emergence of extreme point.
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Table 1. The descriptions of the factors. Factors
Unit
The descriptions of design variables
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 Ex Ey brad
mm mm mm mm mm mm mm mm rad rad rad mm mm rad
The length of the upper arm pole AC The length of the upper arm pole CD The length of the push rod BG The length of the push rod BD The length of the upper arm rod DE period The length of the lever GH on framework The length of the bow the head balance beam EH The center distance of two fixed hinge support AB The angle of CD rod and DE rod on the framework The angle of BG rod with BD rod on the push rod The angle of A,B center line and the x axis The horizontal coordinates of the point E The longitudinal coordinates of the point E The deflection angle of the bow head balancing bar EF
Table 2. The level of design variables. Factors
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
Unit
mm mm mm mm mm mm mm mm rad rad rad
Level value 21
0
+1
1188.00 310.00 1560.00 1598.00 1730.00 1698.00 85.00 725.00 2.90 0.021 0.20
1190.00 313.00 1565.00 1600.00 1735.00 1700.00 97.50 730.00 2.95 0.025 0.204
1192.00 316.00 1570.00 1602.00 1740.00 1702.00 110.00 735.00 3.00 0.028 0.21
The objective function
Figure 3. Box–Behnken design.
During the operation process of EMU, to ensure normal work, the geometric parameters of the pantograph are required to satisfy certain requirements. First, to ensure the pantograph to close bow down normally, the trajectory of the bow head point E within working height range and the maximum deviation of the transverse direction (x-direction) were not too large. Second, to prevent the slide force of the bow head from subjecting to uneven stress, the bow head balancing rod EF tries to keep the translational motion within the working height range. A general calculating formula for the degree of freedom of a planar mechanism is F = 3m 2pi ph
The design variables and level values. All the 11 design factors of the EMU pantograph were applied in BBD experimental design using Design-Expert software. The test number was 204 times; the setting levels of the design variables are given in Table 2.
ð7Þ
where m is the number of active members, pi is the number of low pairs, and ph is the number of high pairs. From Figure 2, it can be seen that the pantograph has five active members, seven low pairs, and zero high pairs; the degree of freedom of the pantographs is
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F = 3 3 5 2 3 7 = 1. So only one freedom was considered in this study. When the arm AC rotates around fixed hinge support A, the angle brad between the trajectories of the bow head point E, the head balance rod EF, and the horizontal direction can only be determined. Ex , Ey , and brad are the functions of the design variables x1 , x2 , . . . , x11 . In order to achieve high reliability of these three objectives, the target design steps were designed as follows: 1. Establishing the dual response surface model The mean and the variance of the three goals were fitted through the BBD, respectively. The target average response surface model and the target variance response surface model can be obtained as shown in Figure 4. The mean and variance response surface models were fitted the relationship of target and variables by testing the deterministic coefficient R2. The R2a value of the objective mean and variance function was greater than 0.90 in Design-Expert software. So the proposed response surface model can be used to approximate the actual models and solve the real problem. There existed a large difference between the value of the objective means and the value of objective variance function. So it can be controlled within the range of [0, 1] through the linear transformation method.25,26 Such transformation was expressed as yTi (x) =
yi (x) , i = uxE , sxE , uyE , syEs , ubm , sbs yi max
ð8Þ
ub , and s2b represent the mean and the variance values of the lateral, longitudinal, and partial corner on point E, respectively, which can be viewed as the corresponding comprehensive effect in reliability index calculation. 2. Determining the comprehensive weight coefficient It is difficult to determine the weight of each objective function involving the mean and variance in engineering optimization problems. In this article, the method based on fuzzy preference27 can be used in the multiobjective reliability optimization problems of the EMU pantograph. For the three target reliability indexes b1 , b2 , and b3 in the reliability optimization design of the pantograph, the important preference degree of various objective functions can be obtained according to the EMU pantograph working conditions: b1 \b2 , b1 b3 , and b2 b3 . The preference relation matrix can be determined by determining and evaluating the function in equation (12) 8 The working height of the pantograph should satisfy the requirements of the falling bow. So the point E ordinate should be within the scope of 300–2250 mm, and its conditions are as follows 300\Ey \2250
4.
ð16Þ
i = 1, 2, 3
ð19Þ
To make pantograph mechanism move normally, and satisfy the pantograph height requirements, the design variable boundary conditions should be met as follows
Figure 5. The reliability optimization history.
8 1100\x1 \1300 > > > > > 205\x2 \350 > > > > > 1500\x > 3 \1650 > > > > 1550\x4 \1650 > > > > > > < 1950\x5 \2100 1650\x6 \1750 > > > 60\x7 \130 > > > > > > 650\x8 \800 > > > > > 0\x9 \0:3 > > > > 0\x \0:03 > 10 > > : 0\x11 \0:5
ð20Þ
Optimization calculations and the results’ analysis The design solution was obtained by solving the optimization problem using the ISIGHT 5.6 software. The iteration number is 500 times. The optimization history is also shown in Figure 5. The reliability design optimization solutions, the initial value, and the conventional optimization solutions29 are given in Table 3. The EMU pantograph’s initial value, the conventional optimization value, and the reliability optimization solutions were similar as shown in Table 3. But the reliability was improved after the reliability optimization. According to different solutions, the diagram of rising bow angle a and the bow rising height Ey can be obtained as shown in Figure 6, and the diagram of the rising bow angle a and the deflection angle brad could be obtained as shown in Figure 7. The relationship
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Table 3. The comparison between conventional design solutions and reliability design optimization solutions.
Design variables x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 Reliability index b1 b2 b3
Unit
Initial value
Conventional optimization
Reliability optimization
mm mm mm mm mm mm mm mm rad rad rad
1190 313 1565 1600 1735 1700 95 730 2.95 0.022 0.204
1190 310 1570 1599 1733 1703 108 732 2.95 0.0256 0.204
1189 314 1571 1598 1733 1703 109 729 2.93 0.026 0.203
3.12 2.99 3.08
3.06 3.04 3.23
3.22 3.16 3.25
Figure 6. The relationships of the rising bow angle a and bow head height Ey .
between bow displacement Ey and the deflection angle brad could be derived as shown in Figure 8. From Figure 6 and Table 4, it can be seen that the pantograph height range of three kinds of parameter design will be between 300 and 1400 mm when the pantograph movement is between 5° and 60°. Therefore, the bow could normally work within the scope of 300– 2250 mm. From Figure 7 and Table 4, it can be seen that the maximum deviation angle Db of the bow head balancing bar EF was reduced to 2.5° from 9° when the diagram of rising bow angle varied between 5° and 60°. The fluctuation range of the deflection angle Db was greatly reduced, which ensures the normal operation of the pantograph. From Figure 8 and Table 4, it can be seen that the working height Ey = 500–1300 mm, and the maximum
Figure 7. The relationships of the rising bow angle a and the deflection angle brad .
deviation angle Db of the bow head balancing bar was reduced to 2.7° from 7°. Therefore, it can be concluded that the bow head balancing EF rod is almost translational. From Figures 6–8 and Table 4, the fluctuation range caused by the interference factors was reduced, and a higher reliability of the performance was achieved.
Conclusion In order to realize the EMU pantograph multi-objective reliability, the BBD was applied to consider a number of design variable parameters. The dual response surface method was introduced to calculate the reliability index of the multiple targets, and a multi-objective reliability design optimization model was built by combining with fuzzy preference method.
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Table 4. The comparison between conventional design solutions and reliability design optimization solutions. a (5°–55°)
Ey (500–1300 mm)
300–1300 295–1288 290–1200
– – –
9.0 4.0 2.5
5.0 3.0 2.7
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.
Ey Initial value Conventional optimization Reliability optimization Db Initial value Conventional optimization Reliability optimization
Figure 8. The relationships of displacement Ey and the deflection angle brad .
To guarantee the longitudinal migration and transverse offset in the direction of the pantograph head as small as possible, the pantograph reliability optimization model was established based on the pantograph under the normal working condition to obtain the reliability design solutions. The reliability of the performance was improved; the influence of the interference factors on the pantograph work was also reduced. The study case indicates that the proposed method is available for EMU pantograph optimization. This method can also be extended to the reliability design optimization of other EMU components. 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.
Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by the program of National Natural Foundation under contract number 11272070, the program of Educational Commission of Liaoning Province under contract number JDL2016001, the
References 1. Ma GL, Zhang WH and Mei GM. Analysis of whole structure characteristics of high-speed pantograph. J Mech Strength 2010; 2: 158–164 (in Chinese). 2. Liu HJ and Zhang WH. Optimization of geometric parameters for high-speed pantograph mechanism. J China Rail Soc 2001; 23: 16–20 (in Chinese). 3. Guo GL and Wang JY. Kinematics analysis for highspeed pantograph. Electr Driv Locomot 2013; 5: 13–16 (in Chinese). 4. Pombo J and Ambro´sio J. Environmental and track perturbations on multiple pantograph interaction with catenaries in high-speed trains. Comput Struct 2013; 124: 88–101. 5. Raghunathan RS, Kim HD and Setoguchi T. Aerodynamics of high-speed railway train. Prog Aerosp Sci 2014; 38: 496–514. 6. Li RP, Zhou N, Zhang WH, et al. Calculation and analysis of pantograph aerodynamic uplift force. J China Rail Soc 2012; 34: 26–32 (in Chinese). 7. Jiang YN, Zhang WH, Zhou D, et al. Static strength analysis of high speed pantograph considering aerodynamic load. Rail Locomot Car 2014; 34: 120–125 (in Chinese). 8. 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. 9. 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 and Rapid Transit 2015; 229: 23–33. 10. Zhu SP, Huang HZ, Smith R, et al. Bayesian framework for probabilistic low cycle fatigue life prediction and uncertainty modeling of aircraft turbine disk alloys. Probabilist Eng Mech 2013; 34: 114–122. 11. 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. 12. Meng D, Li YF, Huang HZ, et al. Reliability-based multidisciplinary design optimization using subset simulation analysis and its application in the hydraulic transmission mechanism design. J Mech Design 2015; 137: 051402. 13. Meng D, Huang HZ, Wang Z, et al. Mean-value firstorder saddlepoint approximation based collaborative optimization for multidisciplinary problems under aleatory uncertainty. J Mech Sci Technol 2014; 28: 3925–3935. 14. Meng D, Zhang X, Yang YJ, et al. Interaction balance optimization in multidisciplinary design optimization problems. Concurrent Eng: Res A 2016; 24: 48–57. 15. Meng D, Zhang X, Huang HZ, et al. Interaction prediction optimization in multidisciplinary design optimization problems. Sci World J 2014; 2014: 1–7.
10 16. Ben-Haim Y. Robust reliability in the mechanical sciences. Berlin: Springer, 1996. 17. Dong SD and Li YH. Multi-response robust optimization of anti-rolling torsion bar based on the stochastic model. J Rail Sci Eng 2015; 12: 900–904 (in Chinese). 18. Jensen HA, Vergara C, Papadimitriou C, et al. The use of updated robust reliability measures in stochastic dynamical systems. Comput Method Appl M 2013; 267: 293–317. 19. Cheng CL, Lv ZZ, Luo ZQ, et al. A general robust reliability index. J Mech Eng 2011; 47: 192–198 (in Chinese). 20. Suhail AH, Ismail N, Wong S, et al. Response surface methodology. J Appl Polym Sci 2011; 11: 308–315. 21. Taguchi G, Chowdhury S and Wu Y. Taguchi’s quality engineering handbook. New York: Wiley-Interscience, 2005. 22. Zhang KK, Mao Z, Jiang B, et al. Adaptive and robust fault-tolerant tracking control of contact force of pantograph-catenary for high-speed trains. Int Feder Autom Contr 2015; 48: 740–745. 23. Tohidi E, Bhrawy AH and Erfani K. A collocation method based on Bernoulli operational matrix for
Advances in Mechanical Engineering
24.
25.
26.
27.
28.
29.
numerical solution of generalized pantograph equation. Appl Math Model 2013; 37: 4283–4294. Ferreira SLC, Bruns RE, Ferreira HS, et al. Box-Behnken design: an alternative for the optimization of analytical methods. Anal Chim Acta 2007; 597: 179–186. Cheng XF, Yuan JP, Wu ZQ, et al. Robust design of the vehicle suspension system based on dual response surface methodology and BBD. J East China Jiaotong Univ 2012; 29: 1–6 (in Chinese). Zhou XJ, Ma YZ, Tu YL, et al. Ensemble of surrogates for dual response surface modeling in robust parameter design. Qual Reliab Eng Int 2013; 29: 173–197. Guan ZH and Kou JS. An evolutionary multi-objective optimization algorithm based on fuzzy preferences. J Tianjin Univ 2002; 35: 275–280 (in Chinese). Wang Y. Robust constraints on dark energy and gravity from galaxy clustering data. Mon Not R Astron Soc 2012; 432: 3631–3637. Sun YT. Structure parameters optimization design of highreach pantograph. Beijing, China: Beijing Jiaotong University, 2014 (in Chinese).
