Research ArticleMultiobjective Robust Design of the Double WishboneSuspension System Based on Particle Swarm Optimization
Xianfu Cheng and Yuqun Lin
School of Mechanical and Electronical Engineering East China Jiaotong University Nanchang 330013 China
Correspondence should be addressed to Xianfu Cheng chxf xnsinacom
Received 22 October 2013 Accepted 19 December 2013 Published 11 February 2014
Academic Editors T Chen Q Cheng and J Yang
Copyright copy 2014 X Cheng and Y LinThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The performance of the suspension system is one of the most important factors in the vehicle design For the double wishbonesuspension system the conventional deterministic optimization does not consider any deviations of design parameters so designsensitivity analysis and robust optimization design are proposed In this study the design parameters of the robust optimization arethe positions of the key points and the random factors are the uncertainties in manufacturing A simplified model of the doublewishbone suspension is established by software ADAMS The sensitivity analysis is utilized to determine main design variablesThen the simulation experiment is arranged and the Latin hypercube design is adopted to find the initial pointsTheKrigingmodelis employed for fitting the mean and variance of the quality characteristics according to the simulation results Further a particleswarm optimization method based on simple PSO is applied and the tradeoff between the mean and deviation of performance ismade to solve the robust optimization problem of the double wishbone suspension system
1 Introduction
Suspension used in an automobile is a system mediatingthe interface between the vehicle and the road and thefunctions of it are related to a wide range of drivabilitysuch as handing ability stability and comfortability [1]Thereare many different structures of vehicle suspension systemaccording to the mechanical jointing pattern the type ofsprings the independence of the left and right wheels and soforth of which the independent double wishbone suspensionis extensively used
With reference to automobile suspension system a num-ber of researches have devoted considerable efforts to designoptimization Many important relationships have been high-lighted among vehicle suspension parameters and suspensionperformance indices [2] These researches can be classedinto several aspects (1) a single-objective optimization ofseparately only considering reducing the dynamic load ofthe tire on the road or smoothness [3] (2) transforming thetraditional multiobjective optimization problem into single-objective optimization problem through a mathematical
transformation [4] (3) using multiobjective and multide-cision optimization of true sense of the decision makingafter the first optimization [5] (4) carrying out the analysesof displacement velocity and acceleration for McPhersonstrut suspension system using displacement matrix [6] Theoptimal design is a balance of the kinematics and com-pliance characteristics of the suspension system [7] Butthese approaches are based on conventional deterministicoptimization and do not consider any deviations of designparameters such asmanufacturing errors of parts whichmayresult in unreliability of design objectives and constraintsand the computation time of these approaches is enormousRobust design is powerful and effective in helping manu-facturers to design their products and process as well as tosolve troublesome quality problems ultimately leading tohigher customer satisfaction and operational performance[7] However a comprehensive multi-objective and robustapproach seems to little be addressed Chun et al studiedoptimal designs for suspension systems based on reliabilityanalyses [8] Choi et al performed a reliability optimizationwith the single-loop single-variable method by using results
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 354857 7 pageshttpdxdoiorg1011552014354857
Figure 1 Robust design based on particle swarm optimization
of a deterministic optimization as initial values of reliability-based optimization using the finite difference design sensitiv-ity [9]
Robust optimization design is essentially multiple objec-tives (1) optimizing the mean of performance and (2)
minimizing the variation of performance Since performancevariation is often minimized at the cost of sacrificing perfor-mance a tradeoff between the aforementioned two aims isgenerally presented The particle swarm optimization (PSO)approach has demonstrated its strength in various typesof multiobjective optimization design including vehiclesaircrafts and manufacturing facilities [10] So this studypresents a new optimal methodmdashbased on robust designand PSO for suspension system The objectives are the toeangle and the lateral slip of the wheel grounding pointand their variations of the double wishbone suspensionsystemThemean and variance models are established by theKriging model and then PSO is used to analyze the robustperformance of the system This may help the designers toidentify layout of the suspension system and to develop theoptimum design system of suspension
This paper is organized as follows in Section 2 thevirtual model of the double wishbone suspension system isestablished in Section 3 the model of multi-objective robustoptimization is built the robust designs based on particleswarm optimization are described and analyzed in Section 4and conclusions are presented in Section 5The process of therobust design based on particle swarm optimization is shownin Figure 1
Table 1 Key point
Key point 119909 119910 119911
Lca front minus200 minus400 150Lca back 200 minus450 155Lca outer 0 minus750 100Uca front 100 minus450 525Uca back 250 minus490 530Uca outer 40 minus675 525Shaft inner 0 minus200 225Spring lower 0 minus600 150Subframe front minus400 minus450 150Subframe back 400 minus450 150Tie rod inner 200 minus400 300Tie rod outer 150 minus750 300Spring upper 40 minus500 650The center of the wheel 0 minus800 300
2 The Virtual Model of the Double WishboneSuspension System
The double-wishbone suspension system is a group ofspace RSSR (revolute jointmdashspherical joinmdashspherical joinmdashrevolute joint) four-bar linkage mechanisms Its kinematicsrelations are complicated kinematics visualization analysis isdifficult and its performance is poor Thus rational settingsof the position parameters of the guiding mechanism arecrucial to assuring good performance of the independentdouble-wishbone suspension
The vehiclersquos right and left suspensions are symmetricalso choose the left or the right part of the suspension systemwhich is studied to simulate the entire mechanism excludingthe variation of wheel centre distance (WCD) which is advis-ableThe key design parameters are the coordinates of the keypoints (see Table 1) and the assembly relationship betweenevery member A model of the left half of an independentdouble wishbone suspension system is established as shownin Figure 2Major components include the upper control arm(UCA) lower control arm (LCA) tie rod knuckle springand absorberThedesign purpose of this study is to determinethe positions of the joints A commercial program ADAMSis employed for modelling and analysing the suspensionsystem
Make the following assumptions on double wishbonesuspension
(1) The compositionmembers of the suspension are rigidbody and the elastic deformation is ignored
(2) Rigid connection between the various components isused and ignores the internal clearance and friction
(3) Only consider the ground roughness without regard-ing to the dynamic factors
(4) Add an incentive on the test platform to simulatethe unevenness of the ground the tires are always incontact with the test bench
The Scientific World Journal 3
Wheel
UCAAbsorber
Tie rod
Knuckle
LCA
Test platform
Figure 2 Double wishbone suspension model
Add an excitation source on the test platform 119910 =
50 sin(2120587119905) and then taking numerical simulation the resultsare shown in Figures 3 and 4
As shown in Figure 3 the wheel sideways displacementchanges with time The change of sideways displacement iscalculated according to the variation of the wheel travel Asshown in Figure 4 the toe angle changes with time Thechange of toe angle is calculated according to the variationof the wheel travel too
3 Model of the Robust Design andApproximation Model
In this section the model of the robust design is built A full-factor test and sensitivity analysis are utilized to determinemain design variablesThe Latin hypercube design is adoptedto find the initial point and the database is created for fittingthe kriging model of the robust design
31 RobustDesign Robust design has become a powerful toolto aid designers in making judicious selection and controlof variation The fundamental principle of robust designis to improve the quality of a product by eliminating thevariation of controllable factors (ie dimension assemblegap material properties etc) and uncontrollable factors (ieapplied loadings environment aging etc) Consequentlycomparedwith traditional optimization design robust designcan make the product maintain good performance [11]
A standard engineering optimization problem is nor-mally formulated as follows
min 119891 (119909)
st 119892119895 (119909) le 0 119895 = 1 2 119869
119909119871 lt 119909 lt 119909119880
(1)
where 119891(119909) is the objective function and 119892119895(119909) is the 119895thconstraint function 119909 119909119871 and 119909119880 are vectors of designvariables their lower bounds and upper bounds respectivelyIf the design variable 119909 follows a statistical distribution
Figure 3 Sideways displacement
Figure 4 Toe angle
a robust design problem can be stated as a biobjective robustdesign problem as follows
119909119871 + Δ119909 le 119909 le 119909 119880 minusΔ119909
(2)
where 120583119891 and 120590119891 are the mean and deviation of the objectivefunction 119891(119909) respectively Their values can be obtainedthrough Monte Carlo simulation or the first order Taylorexpansion if the design deviation of 119909119894 is small When usingTaylor expansions 120583119891 and 120590119891 can be represented by thefollowing equations
120583119891 = 119891 (119909)
1205902119891 =
119899
sum119894=1
(120597119891
120597119909119894
)
2
1205972119909119894
(3)
where 120590119909119894 is the standard deviation of the 119894th 119909 component
32 Sensitivity Analysis There are 12 key points and each oneof them has 3 coordinate values So there are 36 coordinateparameters If every one of the coordinate is selected asdesign variables it needs much iteration In order to reducetime of analysis and save resources the full-factor test isutilized to determine main design variable and the impactof every dependent variable is in Table 2 There are threelevels 1ndash3 and the larger the value the greater the impactof the dependent variable As shown in Table 2 Lca front119909 Lca outer 119909 Uca front 119909 Uca front 119910 Uca back 119909and Uca outer 119909 have made a minimal impact on sideways
4 The Scientific World Journal
Table 2 Impact of each variable
Coordinates of key points Impact
Toe angle Sidewaysdisplacement
Lca front 119909 1 1Lca front 119910 2 2Lca front 119911 3 3Lca back 119909 1 2Lca back 119910 1 3Lca back 119911 2 3Lca outer 119909 1 1Lca outer 119910 1 3Lca outer 119911 3 2Uca front 119909 1 1Uca front 119910 1 1Uca front 119911 3 3Uca back 119909 1 1Uca back 119910 1 2Uca back 119911 2 3Uca outer 119909 1 1Uca outer 119910 1 3Uca outer 119911 3 3
displacement and toe angle Other coordinates of key pointshave made a great impact on sideways displacement and toeangle Based on the test results 12 main design variables areselected as controllable factors and the variable name and itscorresponding physical quantities are shown in Table 3
33 KrigingModel Engineering optimization problems oftenneed enormous computation time for several programsrunning at the same time We cannot provide the evaluationof the objective function and constraints to execute such largescale of exact analysis So the application of approximationis necessary In this paper the Kriging model is adopted tobuild the approximation Kriging model one of the responsesurface models (RSM) has such advantages as unbiasedestimator at the training sample point desirably strong non-linear approximating ability and flexible parameter selectionof the model and thus it is quite suitable for approximatemodels [12] Krigingmodels have a great promise for buildingaccurate global approximations of a design space Thesemodels are extremely flexible because of the wide range ofspatial correlation functions that can be chosen for buildingthe approximation provided that sufficient sample data areavailable to capture the trends in the system responses asa result Kriging models can approach linear and nonlinearfunctions equally well In addition Krigingmodels can eitherldquohonor the datardquo by providing an exact interpolation of thedata or ldquosmooth the datardquo by providing an inexact interpola-tion One of the defects of using RSM in optimization is thatit is apt to miss the global optimum because estimation valueobtained with RSM includes errors at an unknown point [13]
Table 3 Controllable factors
Key point Level 1 Level 2 Level 3Lca front 119910(1199091) minus405 minus400 minus395Lca front 119911(1199092) 145 150 155Lca back 119909(1199093) 195 200 205Lca back 119910(1199094) minus455 minus450 minus445Lca back 119911(1199095) 150 155 160Lca outer 119910(1199096) minus755 minus750 minus745Lca outer 119911(1199097) 95 100 105Uca front 119911(1199098) 520 525 530Uca back 119910(1199099) minus495 minus490 minus485Uca back 119911(11990910) 525 530 535Uca outer 119910(11990911) minus680 minus675 minus670Uca outer 119911(11990912) 520 525 530
In this paper the Kriging model is introduced intothe robust design In the conventional Kriging model theperformance 119910(119909) is modelled as follows
where 120573119879ℎ(119909) is the regression component (eg a polyno-mial) which captures global trends 119885(119909) is assumed to bea Gaussian process indexed by input variables 119909 with zeromean and stationary covariance
From a Bayesian perspective the prior knowledge of theperformance 119910(119909) is specified by a Gaussian process whichis characterized by the prior mean (ie the global trend)and prior covariance Given the observations the posteriorprocess is also a Gaussian process (treating the covarianceparameters as known and assuming a Gaussian prior distri-bution for 120573)The prediction of 119910(119909) is usually taken to be theposterior mean and the prediction uncertainty is quantifiedby the posterior covariance
The conventional Kriging model assumes that the Gaus-sian process has a stationary covariance with the covariancefunction defined as follows
The variance 1205902 provides the overall vertical scale rela-tive to the mean of Gaussian process in the output space120579 = 120579
(119897) (119897 = 1 2 119871) are the correlation parameters(scaling factors) associated with each input variable 119909(119897)which reflects the smoothness of the true performance Thestationary covariance indicates that the correlation function120588st(119909119898 119909119899 120579) between any two sites 119909119898 and 119909119899 depends ononly the distance (scaled by 120579) between 119909119898 and 119909119899 in (5) and(6) the subscript ldquostrdquo means ldquostationaryrdquo
The Scientific World Journal 5
In order to innovate or improve and develop a newproduct and confirm a new technical parameter experimentsusually need to be done repeatedly in the process of produc-tion and scientific research It is very important to reasonablyarrange experimental procedures to reduce the times ofexperiments and shorten the time of each experiment andavoid blindness It requires two aspects of works to be done inorder to solve the problemmentioned above One is to designan experiment that can fully reflect the effect of all factorswhich can reduce time of experiments and save resourcesAnother is to analyze the experimental results in order toacquire reasonable conclusions and the error analysis
DOE can analyze a design space and provide a roughestimate of an optimal design which can be used as astarting point for numerical optimization The Latin hyper-cube design could cover the design space more evenly thanother DOE methods and generate more evenly distributedpoints Therefore in this paper the Latin hypercube designis adopted to find the initial point and created the databasefor approximation model For this problem the inputs arethe 12 main design variables the outputs are the mean andtheir variance of the toe angle and sideways displacementand 200 sample points from an LHS design are used to fitthe kriging model A set of 393 verification points randomlyselected across the domain is used to evaluate the RMSEfor each kriging model Kriging model is established to fitthe multiobjective robust design The r-square of the krigingmodel is 087 so it can fit the virtual model
4 Robust Design Based on ParticleSwarm Optimization
The particle swarm optimization (PSO) is one of the evo-lutionary computation techniques introduced by Kennedyand Eberhart in 1995 [14] It is a population-based searchalgorithm and is initialized with a population of randomsolutions named particles PSOmakes use of a velocity vectorto update the current position of each particle in the swarm[15 16]
Particle swarm optimization is usually used as a tradi-tional optimization method which is inspired from the socialbehaviour of flocks of birds It is more competitive in variousaspects for example due to its simplicity Particle swarmoptimization genetic algorithms and other evolutionaryalgorithms are all artificial life calculated But particle swarmoptimization is different from other evolutionary algorithmsusing group iterative solution of cooperation mechanisms togenerate the optimal solution instead of using group iterativesolution of competing mechanisms In PSO algorithm eachindividual is called ldquoparticlerdquo which represents a potentialsolution The algorithm achieves the best solution by thevariability of some particles in the tracing spaceThe particlessearch in the solution space following the best particleby changing their positions and the fitness frequently theflying direction and velocity are determined by the objectivefunction
Update optimum position
Update particle position
Start
Initialize
Evaluate particle
Termination conditions
No
End
Yes
Figure 5 Particle swarm optimization
The procedure of PSO is as follows
(1) initialize the original position and velocity of particleswarm
(2) calculate the fitness value of each particle(3) for each particle compare the fitness value with the
fitness value of pbest if current value is better thenrenew the position with current position and updatethe fitness value simultaneously
(4) determine the best particle of group with the bestfitness value if the fitness value is better than thefitness value of gbest then update the gbest and itsfitness value with the position
(5) check the finalizing criterion if it has been satisfiedquit the iteration otherwise return to step (2)
It can be shown as Figure 5Assuming 119883119894 = (1199091198941 1199091198942 119909119894119863) is the position of 119894th
particle in D-dimension 119881119894 = (V1198941 V1198942 V119894119863) is its velocitywhich represents its direction of searching In iterationprocess each particle keeps the best position pbest found byitself besides it also knows the best position gbest searchedby the group particles and changes its velocity accordingto the two best positions The PSO is described in vectornotation as to the follows
where 119904 is the swarm size 1198881 and 1198882 are the nonnegative accel-eration coefficients these two constants make the particleshave the ability of self-summary and learn from the excellentindividuals of the groups so the particles can close to thepersonal best solution of its own history and the global bestsolution within population or field Typically value of 1198881 and1198882 is 2 120596 is the inertia weight 119903119897(119905) and 1199032(119905) sim 119880(0 1) 119909119894(119905)
is the position of particle 119894 at time 119905 ]119894(119905) is the velocity ofparticle 119894 at time 119905119901119894(119905) is the personal best solution of particle119894 at time 119905 and 119901119892(119905) is the global best solution at time 119905
The first term of (8) is the previous velocity of the particlevector The second and third terms are used to change thevelocity of the particle Without the second and third termsthe particle will keep on ldquoflyingrdquo in the same direction untilit hits the boundary The particle position 119909(119905 + 119897) is updatedusing its current value and the newly computed velocity V119894(119905+119897) which is determined by the values of V119894(119905) 119909119894(119905) 119901119894(119905) and119901119892(119905) and coefficients 120596 1198881 and 1198882 [17]
In experiment the population of group particle is 40 1198881and 1198882 are set to 2 the maximum time of iteration is 10000It is acceptable if the difference between the best solutionobtained by the optimization algorithm and the true solutionis less than 1119890 minus 6 The inertia weight is linear decreasinginertia all which is determined by the following equation
119908 = 119908max minus119908max minus 119908min
itermaxtimes 119896 (8)
where119908max is the start of inertia weight which is set to 09 and119908min is the end of inertia weight which is set to 005 itermaxis the maximum times of iteration 119896 is the current iterationtimes In order to reflect the universality of experiment theoriginal position and velocity are randomly generated
Particle swarm optimization was used to search the opti-mal solution A particle swarm optimization is created themaximum iterations are set to 50 the number of particles isset to 15 and the objectives are the values and their variationsof the toe angle and sideways displacement V Pareto theFrench economist who studied themulti-objective optimiza-tion problem of economics first proposed the concept ofPareto solution setThere are 27 Pareto solutions in the resultsof the optimization
In multiobjective optimization each optimization objec-tive is often conflicting which requires coordination betweenthe optimal solutions of each target Considering the impor-tance of each target choose one Pareto optimal solutionand the design values of it are shown in Table 4 Using theresults of the robust design to have a test in the ADAMS thesimulation results are shown in Figures 6 and 7
It can be seen that the maximum deviation in the toeangle for the optimal design has been reduced by 52 percentcompared with the base design As the discussion of theresults the most concerned factor is the relationship betweenobjective function and design parameter By comparing theexperimental results the robust design based on particleswarm significantly improved the robust of the toe angle andthe sideways displacement ensuring the reasonable of thedesign performance
Table 4 Robust results
Key point Initial value Robust resultsLca front 119910(1199091) minus400 minus400Lca front 119911(1199092) 150 15033Lca back 119909(1199093) 200 205Lca back 119910(1199094) minus450 minus450Lca back 119911(1199095) 155 15167Lca outer 119910(1199096) minus750 minus7525Lca outer 119911(1199097) 100 10133Uca front 119911(1199098) 525 51967Uca back 119910(1199099) minus490 minus48967Uca back 119911(11990910) 530 53217Uca outer 119910(11990911) minus675 minus67983Uca outer 119911(11990912) 525 52333
Figure 6 Sideways displacement
5 Conclusion
In this study a robust design based on bioinspired compu-tation is presented and illustrated by the design of a doublewishbone suspension system in order to reduce the effectof variations due to uncertainties in fabrication As theyare directly related to fabrication errors the coordinates ofkey points were taken as design variables and at the sametime are considered as random variables So the robustdesign optimization problem had 13 design variables (jointpositions) and 13 random constants (fabrication errors ofjoint positions) In this paper the Latin hypercube designis adopted to make DOE design matrix of the 13 designvariablesTheKrigingmodel is built according to the result ofDOE and then the particle swarm is used to search optimalsolution of the robust design Particle swarm is implementedin a test case and the results show that the method candecrease the solutionrsquos time The robustness of solution isimproved The improvement in robustness became larger asthe amount of fabrication errors increases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
The Scientific World Journal 7
Figure 7 Toe angle
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under the Grant no 51165007 and bytheNatural Science Foundation of Jiangxi Province under theGrant no 20132BAB206025 The authors would like to thankthe reviewers for their valuable comments and suggestions
References
[1] J C Dixon Tires Suspension and Handlin Society of Automo-tive Engineers 2nd edition 1996
[2] G Massimiliano and L Francesco ldquoMulti-objective robustdesign of the suspension systemof road vehiclesrdquoVehicle SystemDynamics vol 41 pp 537ndash546 2004
[3] F-C Wu and C-C Chyu ldquoOptimization of robust designfor multiple quality characteristicsrdquo International Journal ofProduction Research vol 42 no 2 pp 337ndash354 2004
[4] W-J Yin Y Han and S-P Yang ldquoDynamics analysis ofair spring suspension system under forced vibrationrdquo ChinaJournal of Highway and Transport vol 19 no 3 pp 117ndash1212006
[5] P-M Lu L-M He and J-M You ldquoOptimization of vehiclesuspension parameters based on comfort and tire dynamicloadrdquo China Journal of Highway and Transport vol 20 no 1pp 112ndash117 2007
[6] H Y Kang and C H Suh ldquoSynthesis and analysis of spherical-cylindrical (SC) link in theMcPherson strut suspensionmecha-nismrdquo Journal of Mechanical Design vol 116 no 2 pp 599ndash6061994
[7] TWang ldquoMulti-objective andmulti-criteria decision optimiza-tion of automobile suspension parametersrdquo Transactions of theChinese Society of Agricultural Machinery vol 28 no 11 pp 27ndash32 2009
[8] H H Chun S J Kwon and T Tak ldquoReliabilitybased designoptimization of automotive suspension systemsrdquo InternationalJournal of Automotive Technology vol 8 no 6 pp 713ndash7222007
[9] B-L Choi J-H Choi and D-H Choi ldquoReliability-baseddesign optimization of an automotive suspension system forenhancing kinematic and compliance characteristicsrdquo Interna-tional Journal of Automotive Technology vol 6 no 3 pp 235ndash242 2004
[10] S R Singiresu and K A Kiran ldquoParticle swarmmethodologiesfor engineering design optimizationrdquo in Proceedings of theInternational Design Engineering Technical Conferences andComputers and Information in Engineering Conference (ASMErsquo09) pp 507ndash516 San Diego Calif USA August 2009
[11] J-J Chen R Xiao Y Zhong and G Dou ldquoMultidisciplinaryrobust optimization designrdquo Chinese Journal of MechanicalEngineering vol 18 no 1 pp 46ndash50 2005
[12] A Giunta and L T Watson A Comparison of ApproximationModeling Technique Polynomial Versus Interpolating ModelsAIAAUSAF NASAISSMO 7th edition 1998
[13] R J Donald S Matthias and J W William ldquoEfficient globaloptimization of expensive black-box functionsrdquo Journal ofGlobal Optimization vol 13 no 4 pp 455ndash492 1998
[14] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork pp 1942ndash1948 Perth Australia December 1995
[15] S Mandal R Kar D Mandal and S P Ghoshal ldquoSwarmintelligence based optimal linear phaseFIR high pass filterdesign using particle swarm optimization with constrictionfactor and inertia weight approachrdquoWorld Academy of ScienceEngineering and Technology vol 5 no 8 pp 1155ndash1161 2011
[16] W-M Zhongi and S-J Li ldquoFeng QIAN 120579-PSO a new strategyof particle swarm optimizationrdquo Journal of Zhejiang Universityvol 9 no 6 pp 786ndash790 2008
[17] S Mandal and S P Ghshal ldquoSwarm intelligence based optimallinear fir high pass filter design using particle swarm optimiza-tion with constriction factor and inertia weight approachrdquo inProceedings of the IEEE Student Conference on Research andDevelopment (SCOReD rsquo11) pp 352ndash357 Cyberjaya MalaysiaDecember 2011
Figure 1 Robust design based on particle swarm optimization
of a deterministic optimization as initial values of reliability-based optimization using the finite difference design sensitiv-ity [9]
Robust optimization design is essentially multiple objec-tives (1) optimizing the mean of performance and (2)
minimizing the variation of performance Since performancevariation is often minimized at the cost of sacrificing perfor-mance a tradeoff between the aforementioned two aims isgenerally presented The particle swarm optimization (PSO)approach has demonstrated its strength in various typesof multiobjective optimization design including vehiclesaircrafts and manufacturing facilities [10] So this studypresents a new optimal methodmdashbased on robust designand PSO for suspension system The objectives are the toeangle and the lateral slip of the wheel grounding pointand their variations of the double wishbone suspensionsystemThemean and variance models are established by theKriging model and then PSO is used to analyze the robustperformance of the system This may help the designers toidentify layout of the suspension system and to develop theoptimum design system of suspension
This paper is organized as follows in Section 2 thevirtual model of the double wishbone suspension system isestablished in Section 3 the model of multi-objective robustoptimization is built the robust designs based on particleswarm optimization are described and analyzed in Section 4and conclusions are presented in Section 5The process of therobust design based on particle swarm optimization is shownin Figure 1
Table 1 Key point
Key point 119909 119910 119911
Lca front minus200 minus400 150Lca back 200 minus450 155Lca outer 0 minus750 100Uca front 100 minus450 525Uca back 250 minus490 530Uca outer 40 minus675 525Shaft inner 0 minus200 225Spring lower 0 minus600 150Subframe front minus400 minus450 150Subframe back 400 minus450 150Tie rod inner 200 minus400 300Tie rod outer 150 minus750 300Spring upper 40 minus500 650The center of the wheel 0 minus800 300
2 The Virtual Model of the Double WishboneSuspension System
The double-wishbone suspension system is a group ofspace RSSR (revolute jointmdashspherical joinmdashspherical joinmdashrevolute joint) four-bar linkage mechanisms Its kinematicsrelations are complicated kinematics visualization analysis isdifficult and its performance is poor Thus rational settingsof the position parameters of the guiding mechanism arecrucial to assuring good performance of the independentdouble-wishbone suspension
The vehiclersquos right and left suspensions are symmetricalso choose the left or the right part of the suspension systemwhich is studied to simulate the entire mechanism excludingthe variation of wheel centre distance (WCD) which is advis-ableThe key design parameters are the coordinates of the keypoints (see Table 1) and the assembly relationship betweenevery member A model of the left half of an independentdouble wishbone suspension system is established as shownin Figure 2Major components include the upper control arm(UCA) lower control arm (LCA) tie rod knuckle springand absorberThedesign purpose of this study is to determinethe positions of the joints A commercial program ADAMSis employed for modelling and analysing the suspensionsystem
Make the following assumptions on double wishbonesuspension
(1) The compositionmembers of the suspension are rigidbody and the elastic deformation is ignored
(2) Rigid connection between the various components isused and ignores the internal clearance and friction
(3) Only consider the ground roughness without regard-ing to the dynamic factors
(4) Add an incentive on the test platform to simulatethe unevenness of the ground the tires are always incontact with the test bench
The Scientific World Journal 3
Wheel
UCAAbsorber
Tie rod
Knuckle
LCA
Test platform
Figure 2 Double wishbone suspension model
Add an excitation source on the test platform 119910 =
50 sin(2120587119905) and then taking numerical simulation the resultsare shown in Figures 3 and 4
As shown in Figure 3 the wheel sideways displacementchanges with time The change of sideways displacement iscalculated according to the variation of the wheel travel Asshown in Figure 4 the toe angle changes with time Thechange of toe angle is calculated according to the variationof the wheel travel too
3 Model of the Robust Design andApproximation Model
In this section the model of the robust design is built A full-factor test and sensitivity analysis are utilized to determinemain design variablesThe Latin hypercube design is adoptedto find the initial point and the database is created for fittingthe kriging model of the robust design
31 RobustDesign Robust design has become a powerful toolto aid designers in making judicious selection and controlof variation The fundamental principle of robust designis to improve the quality of a product by eliminating thevariation of controllable factors (ie dimension assemblegap material properties etc) and uncontrollable factors (ieapplied loadings environment aging etc) Consequentlycomparedwith traditional optimization design robust designcan make the product maintain good performance [11]
A standard engineering optimization problem is nor-mally formulated as follows
min 119891 (119909)
st 119892119895 (119909) le 0 119895 = 1 2 119869
119909119871 lt 119909 lt 119909119880
(1)
where 119891(119909) is the objective function and 119892119895(119909) is the 119895thconstraint function 119909 119909119871 and 119909119880 are vectors of designvariables their lower bounds and upper bounds respectivelyIf the design variable 119909 follows a statistical distribution
Figure 3 Sideways displacement
Figure 4 Toe angle
a robust design problem can be stated as a biobjective robustdesign problem as follows
119909119871 + Δ119909 le 119909 le 119909 119880 minusΔ119909
(2)
where 120583119891 and 120590119891 are the mean and deviation of the objectivefunction 119891(119909) respectively Their values can be obtainedthrough Monte Carlo simulation or the first order Taylorexpansion if the design deviation of 119909119894 is small When usingTaylor expansions 120583119891 and 120590119891 can be represented by thefollowing equations
120583119891 = 119891 (119909)
1205902119891 =
119899
sum119894=1
(120597119891
120597119909119894
)
2
1205972119909119894
(3)
where 120590119909119894 is the standard deviation of the 119894th 119909 component
32 Sensitivity Analysis There are 12 key points and each oneof them has 3 coordinate values So there are 36 coordinateparameters If every one of the coordinate is selected asdesign variables it needs much iteration In order to reducetime of analysis and save resources the full-factor test isutilized to determine main design variable and the impactof every dependent variable is in Table 2 There are threelevels 1ndash3 and the larger the value the greater the impactof the dependent variable As shown in Table 2 Lca front119909 Lca outer 119909 Uca front 119909 Uca front 119910 Uca back 119909and Uca outer 119909 have made a minimal impact on sideways
4 The Scientific World Journal
Table 2 Impact of each variable
Coordinates of key points Impact
Toe angle Sidewaysdisplacement
Lca front 119909 1 1Lca front 119910 2 2Lca front 119911 3 3Lca back 119909 1 2Lca back 119910 1 3Lca back 119911 2 3Lca outer 119909 1 1Lca outer 119910 1 3Lca outer 119911 3 2Uca front 119909 1 1Uca front 119910 1 1Uca front 119911 3 3Uca back 119909 1 1Uca back 119910 1 2Uca back 119911 2 3Uca outer 119909 1 1Uca outer 119910 1 3Uca outer 119911 3 3
displacement and toe angle Other coordinates of key pointshave made a great impact on sideways displacement and toeangle Based on the test results 12 main design variables areselected as controllable factors and the variable name and itscorresponding physical quantities are shown in Table 3
33 KrigingModel Engineering optimization problems oftenneed enormous computation time for several programsrunning at the same time We cannot provide the evaluationof the objective function and constraints to execute such largescale of exact analysis So the application of approximationis necessary In this paper the Kriging model is adopted tobuild the approximation Kriging model one of the responsesurface models (RSM) has such advantages as unbiasedestimator at the training sample point desirably strong non-linear approximating ability and flexible parameter selectionof the model and thus it is quite suitable for approximatemodels [12] Krigingmodels have a great promise for buildingaccurate global approximations of a design space Thesemodels are extremely flexible because of the wide range ofspatial correlation functions that can be chosen for buildingthe approximation provided that sufficient sample data areavailable to capture the trends in the system responses asa result Kriging models can approach linear and nonlinearfunctions equally well In addition Krigingmodels can eitherldquohonor the datardquo by providing an exact interpolation of thedata or ldquosmooth the datardquo by providing an inexact interpola-tion One of the defects of using RSM in optimization is thatit is apt to miss the global optimum because estimation valueobtained with RSM includes errors at an unknown point [13]
Table 3 Controllable factors
Key point Level 1 Level 2 Level 3Lca front 119910(1199091) minus405 minus400 minus395Lca front 119911(1199092) 145 150 155Lca back 119909(1199093) 195 200 205Lca back 119910(1199094) minus455 minus450 minus445Lca back 119911(1199095) 150 155 160Lca outer 119910(1199096) minus755 minus750 minus745Lca outer 119911(1199097) 95 100 105Uca front 119911(1199098) 520 525 530Uca back 119910(1199099) minus495 minus490 minus485Uca back 119911(11990910) 525 530 535Uca outer 119910(11990911) minus680 minus675 minus670Uca outer 119911(11990912) 520 525 530
In this paper the Kriging model is introduced intothe robust design In the conventional Kriging model theperformance 119910(119909) is modelled as follows
where 120573119879ℎ(119909) is the regression component (eg a polyno-mial) which captures global trends 119885(119909) is assumed to bea Gaussian process indexed by input variables 119909 with zeromean and stationary covariance
From a Bayesian perspective the prior knowledge of theperformance 119910(119909) is specified by a Gaussian process whichis characterized by the prior mean (ie the global trend)and prior covariance Given the observations the posteriorprocess is also a Gaussian process (treating the covarianceparameters as known and assuming a Gaussian prior distri-bution for 120573)The prediction of 119910(119909) is usually taken to be theposterior mean and the prediction uncertainty is quantifiedby the posterior covariance
The conventional Kriging model assumes that the Gaus-sian process has a stationary covariance with the covariancefunction defined as follows
The variance 1205902 provides the overall vertical scale rela-tive to the mean of Gaussian process in the output space120579 = 120579
(119897) (119897 = 1 2 119871) are the correlation parameters(scaling factors) associated with each input variable 119909(119897)which reflects the smoothness of the true performance Thestationary covariance indicates that the correlation function120588st(119909119898 119909119899 120579) between any two sites 119909119898 and 119909119899 depends ononly the distance (scaled by 120579) between 119909119898 and 119909119899 in (5) and(6) the subscript ldquostrdquo means ldquostationaryrdquo
The Scientific World Journal 5
In order to innovate or improve and develop a newproduct and confirm a new technical parameter experimentsusually need to be done repeatedly in the process of produc-tion and scientific research It is very important to reasonablyarrange experimental procedures to reduce the times ofexperiments and shorten the time of each experiment andavoid blindness It requires two aspects of works to be done inorder to solve the problemmentioned above One is to designan experiment that can fully reflect the effect of all factorswhich can reduce time of experiments and save resourcesAnother is to analyze the experimental results in order toacquire reasonable conclusions and the error analysis
DOE can analyze a design space and provide a roughestimate of an optimal design which can be used as astarting point for numerical optimization The Latin hyper-cube design could cover the design space more evenly thanother DOE methods and generate more evenly distributedpoints Therefore in this paper the Latin hypercube designis adopted to find the initial point and created the databasefor approximation model For this problem the inputs arethe 12 main design variables the outputs are the mean andtheir variance of the toe angle and sideways displacementand 200 sample points from an LHS design are used to fitthe kriging model A set of 393 verification points randomlyselected across the domain is used to evaluate the RMSEfor each kriging model Kriging model is established to fitthe multiobjective robust design The r-square of the krigingmodel is 087 so it can fit the virtual model
4 Robust Design Based on ParticleSwarm Optimization
The particle swarm optimization (PSO) is one of the evo-lutionary computation techniques introduced by Kennedyand Eberhart in 1995 [14] It is a population-based searchalgorithm and is initialized with a population of randomsolutions named particles PSOmakes use of a velocity vectorto update the current position of each particle in the swarm[15 16]
Particle swarm optimization is usually used as a tradi-tional optimization method which is inspired from the socialbehaviour of flocks of birds It is more competitive in variousaspects for example due to its simplicity Particle swarmoptimization genetic algorithms and other evolutionaryalgorithms are all artificial life calculated But particle swarmoptimization is different from other evolutionary algorithmsusing group iterative solution of cooperation mechanisms togenerate the optimal solution instead of using group iterativesolution of competing mechanisms In PSO algorithm eachindividual is called ldquoparticlerdquo which represents a potentialsolution The algorithm achieves the best solution by thevariability of some particles in the tracing spaceThe particlessearch in the solution space following the best particleby changing their positions and the fitness frequently theflying direction and velocity are determined by the objectivefunction
Update optimum position
Update particle position
Start
Initialize
Evaluate particle
Termination conditions
No
End
Yes
Figure 5 Particle swarm optimization
The procedure of PSO is as follows
(1) initialize the original position and velocity of particleswarm
(2) calculate the fitness value of each particle(3) for each particle compare the fitness value with the
fitness value of pbest if current value is better thenrenew the position with current position and updatethe fitness value simultaneously
(4) determine the best particle of group with the bestfitness value if the fitness value is better than thefitness value of gbest then update the gbest and itsfitness value with the position
(5) check the finalizing criterion if it has been satisfiedquit the iteration otherwise return to step (2)
It can be shown as Figure 5Assuming 119883119894 = (1199091198941 1199091198942 119909119894119863) is the position of 119894th
particle in D-dimension 119881119894 = (V1198941 V1198942 V119894119863) is its velocitywhich represents its direction of searching In iterationprocess each particle keeps the best position pbest found byitself besides it also knows the best position gbest searchedby the group particles and changes its velocity accordingto the two best positions The PSO is described in vectornotation as to the follows
where 119904 is the swarm size 1198881 and 1198882 are the nonnegative accel-eration coefficients these two constants make the particleshave the ability of self-summary and learn from the excellentindividuals of the groups so the particles can close to thepersonal best solution of its own history and the global bestsolution within population or field Typically value of 1198881 and1198882 is 2 120596 is the inertia weight 119903119897(119905) and 1199032(119905) sim 119880(0 1) 119909119894(119905)
is the position of particle 119894 at time 119905 ]119894(119905) is the velocity ofparticle 119894 at time 119905119901119894(119905) is the personal best solution of particle119894 at time 119905 and 119901119892(119905) is the global best solution at time 119905
The first term of (8) is the previous velocity of the particlevector The second and third terms are used to change thevelocity of the particle Without the second and third termsthe particle will keep on ldquoflyingrdquo in the same direction untilit hits the boundary The particle position 119909(119905 + 119897) is updatedusing its current value and the newly computed velocity V119894(119905+119897) which is determined by the values of V119894(119905) 119909119894(119905) 119901119894(119905) and119901119892(119905) and coefficients 120596 1198881 and 1198882 [17]
In experiment the population of group particle is 40 1198881and 1198882 are set to 2 the maximum time of iteration is 10000It is acceptable if the difference between the best solutionobtained by the optimization algorithm and the true solutionis less than 1119890 minus 6 The inertia weight is linear decreasinginertia all which is determined by the following equation
119908 = 119908max minus119908max minus 119908min
itermaxtimes 119896 (8)
where119908max is the start of inertia weight which is set to 09 and119908min is the end of inertia weight which is set to 005 itermaxis the maximum times of iteration 119896 is the current iterationtimes In order to reflect the universality of experiment theoriginal position and velocity are randomly generated
Particle swarm optimization was used to search the opti-mal solution A particle swarm optimization is created themaximum iterations are set to 50 the number of particles isset to 15 and the objectives are the values and their variationsof the toe angle and sideways displacement V Pareto theFrench economist who studied themulti-objective optimiza-tion problem of economics first proposed the concept ofPareto solution setThere are 27 Pareto solutions in the resultsof the optimization
In multiobjective optimization each optimization objec-tive is often conflicting which requires coordination betweenthe optimal solutions of each target Considering the impor-tance of each target choose one Pareto optimal solutionand the design values of it are shown in Table 4 Using theresults of the robust design to have a test in the ADAMS thesimulation results are shown in Figures 6 and 7
It can be seen that the maximum deviation in the toeangle for the optimal design has been reduced by 52 percentcompared with the base design As the discussion of theresults the most concerned factor is the relationship betweenobjective function and design parameter By comparing theexperimental results the robust design based on particleswarm significantly improved the robust of the toe angle andthe sideways displacement ensuring the reasonable of thedesign performance
Table 4 Robust results
Key point Initial value Robust resultsLca front 119910(1199091) minus400 minus400Lca front 119911(1199092) 150 15033Lca back 119909(1199093) 200 205Lca back 119910(1199094) minus450 minus450Lca back 119911(1199095) 155 15167Lca outer 119910(1199096) minus750 minus7525Lca outer 119911(1199097) 100 10133Uca front 119911(1199098) 525 51967Uca back 119910(1199099) minus490 minus48967Uca back 119911(11990910) 530 53217Uca outer 119910(11990911) minus675 minus67983Uca outer 119911(11990912) 525 52333
Figure 6 Sideways displacement
5 Conclusion
In this study a robust design based on bioinspired compu-tation is presented and illustrated by the design of a doublewishbone suspension system in order to reduce the effectof variations due to uncertainties in fabrication As theyare directly related to fabrication errors the coordinates ofkey points were taken as design variables and at the sametime are considered as random variables So the robustdesign optimization problem had 13 design variables (jointpositions) and 13 random constants (fabrication errors ofjoint positions) In this paper the Latin hypercube designis adopted to make DOE design matrix of the 13 designvariablesTheKrigingmodel is built according to the result ofDOE and then the particle swarm is used to search optimalsolution of the robust design Particle swarm is implementedin a test case and the results show that the method candecrease the solutionrsquos time The robustness of solution isimproved The improvement in robustness became larger asthe amount of fabrication errors increases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
The Scientific World Journal 7
Figure 7 Toe angle
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under the Grant no 51165007 and bytheNatural Science Foundation of Jiangxi Province under theGrant no 20132BAB206025 The authors would like to thankthe reviewers for their valuable comments and suggestions
References
[1] J C Dixon Tires Suspension and Handlin Society of Automo-tive Engineers 2nd edition 1996
[2] G Massimiliano and L Francesco ldquoMulti-objective robustdesign of the suspension systemof road vehiclesrdquoVehicle SystemDynamics vol 41 pp 537ndash546 2004
[3] F-C Wu and C-C Chyu ldquoOptimization of robust designfor multiple quality characteristicsrdquo International Journal ofProduction Research vol 42 no 2 pp 337ndash354 2004
[4] W-J Yin Y Han and S-P Yang ldquoDynamics analysis ofair spring suspension system under forced vibrationrdquo ChinaJournal of Highway and Transport vol 19 no 3 pp 117ndash1212006
[5] P-M Lu L-M He and J-M You ldquoOptimization of vehiclesuspension parameters based on comfort and tire dynamicloadrdquo China Journal of Highway and Transport vol 20 no 1pp 112ndash117 2007
[6] H Y Kang and C H Suh ldquoSynthesis and analysis of spherical-cylindrical (SC) link in theMcPherson strut suspensionmecha-nismrdquo Journal of Mechanical Design vol 116 no 2 pp 599ndash6061994
[7] TWang ldquoMulti-objective andmulti-criteria decision optimiza-tion of automobile suspension parametersrdquo Transactions of theChinese Society of Agricultural Machinery vol 28 no 11 pp 27ndash32 2009
[8] H H Chun S J Kwon and T Tak ldquoReliabilitybased designoptimization of automotive suspension systemsrdquo InternationalJournal of Automotive Technology vol 8 no 6 pp 713ndash7222007
[9] B-L Choi J-H Choi and D-H Choi ldquoReliability-baseddesign optimization of an automotive suspension system forenhancing kinematic and compliance characteristicsrdquo Interna-tional Journal of Automotive Technology vol 6 no 3 pp 235ndash242 2004
[10] S R Singiresu and K A Kiran ldquoParticle swarmmethodologiesfor engineering design optimizationrdquo in Proceedings of theInternational Design Engineering Technical Conferences andComputers and Information in Engineering Conference (ASMErsquo09) pp 507ndash516 San Diego Calif USA August 2009
[11] J-J Chen R Xiao Y Zhong and G Dou ldquoMultidisciplinaryrobust optimization designrdquo Chinese Journal of MechanicalEngineering vol 18 no 1 pp 46ndash50 2005
[12] A Giunta and L T Watson A Comparison of ApproximationModeling Technique Polynomial Versus Interpolating ModelsAIAAUSAF NASAISSMO 7th edition 1998
[13] R J Donald S Matthias and J W William ldquoEfficient globaloptimization of expensive black-box functionsrdquo Journal ofGlobal Optimization vol 13 no 4 pp 455ndash492 1998
[14] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork pp 1942ndash1948 Perth Australia December 1995
[15] S Mandal R Kar D Mandal and S P Ghoshal ldquoSwarmintelligence based optimal linear phaseFIR high pass filterdesign using particle swarm optimization with constrictionfactor and inertia weight approachrdquoWorld Academy of ScienceEngineering and Technology vol 5 no 8 pp 1155ndash1161 2011
[16] W-M Zhongi and S-J Li ldquoFeng QIAN 120579-PSO a new strategyof particle swarm optimizationrdquo Journal of Zhejiang Universityvol 9 no 6 pp 786ndash790 2008
[17] S Mandal and S P Ghshal ldquoSwarm intelligence based optimallinear fir high pass filter design using particle swarm optimiza-tion with constriction factor and inertia weight approachrdquo inProceedings of the IEEE Student Conference on Research andDevelopment (SCOReD rsquo11) pp 352ndash357 Cyberjaya MalaysiaDecember 2011
Add an excitation source on the test platform 119910 =
50 sin(2120587119905) and then taking numerical simulation the resultsare shown in Figures 3 and 4
As shown in Figure 3 the wheel sideways displacementchanges with time The change of sideways displacement iscalculated according to the variation of the wheel travel Asshown in Figure 4 the toe angle changes with time Thechange of toe angle is calculated according to the variationof the wheel travel too
3 Model of the Robust Design andApproximation Model
In this section the model of the robust design is built A full-factor test and sensitivity analysis are utilized to determinemain design variablesThe Latin hypercube design is adoptedto find the initial point and the database is created for fittingthe kriging model of the robust design
31 RobustDesign Robust design has become a powerful toolto aid designers in making judicious selection and controlof variation The fundamental principle of robust designis to improve the quality of a product by eliminating thevariation of controllable factors (ie dimension assemblegap material properties etc) and uncontrollable factors (ieapplied loadings environment aging etc) Consequentlycomparedwith traditional optimization design robust designcan make the product maintain good performance [11]
A standard engineering optimization problem is nor-mally formulated as follows
min 119891 (119909)
st 119892119895 (119909) le 0 119895 = 1 2 119869
119909119871 lt 119909 lt 119909119880
(1)
where 119891(119909) is the objective function and 119892119895(119909) is the 119895thconstraint function 119909 119909119871 and 119909119880 are vectors of designvariables their lower bounds and upper bounds respectivelyIf the design variable 119909 follows a statistical distribution
Figure 3 Sideways displacement
Figure 4 Toe angle
a robust design problem can be stated as a biobjective robustdesign problem as follows
119909119871 + Δ119909 le 119909 le 119909 119880 minusΔ119909
(2)
where 120583119891 and 120590119891 are the mean and deviation of the objectivefunction 119891(119909) respectively Their values can be obtainedthrough Monte Carlo simulation or the first order Taylorexpansion if the design deviation of 119909119894 is small When usingTaylor expansions 120583119891 and 120590119891 can be represented by thefollowing equations
120583119891 = 119891 (119909)
1205902119891 =
119899
sum119894=1
(120597119891
120597119909119894
)
2
1205972119909119894
(3)
where 120590119909119894 is the standard deviation of the 119894th 119909 component
32 Sensitivity Analysis There are 12 key points and each oneof them has 3 coordinate values So there are 36 coordinateparameters If every one of the coordinate is selected asdesign variables it needs much iteration In order to reducetime of analysis and save resources the full-factor test isutilized to determine main design variable and the impactof every dependent variable is in Table 2 There are threelevels 1ndash3 and the larger the value the greater the impactof the dependent variable As shown in Table 2 Lca front119909 Lca outer 119909 Uca front 119909 Uca front 119910 Uca back 119909and Uca outer 119909 have made a minimal impact on sideways
4 The Scientific World Journal
Table 2 Impact of each variable
Coordinates of key points Impact
Toe angle Sidewaysdisplacement
Lca front 119909 1 1Lca front 119910 2 2Lca front 119911 3 3Lca back 119909 1 2Lca back 119910 1 3Lca back 119911 2 3Lca outer 119909 1 1Lca outer 119910 1 3Lca outer 119911 3 2Uca front 119909 1 1Uca front 119910 1 1Uca front 119911 3 3Uca back 119909 1 1Uca back 119910 1 2Uca back 119911 2 3Uca outer 119909 1 1Uca outer 119910 1 3Uca outer 119911 3 3
displacement and toe angle Other coordinates of key pointshave made a great impact on sideways displacement and toeangle Based on the test results 12 main design variables areselected as controllable factors and the variable name and itscorresponding physical quantities are shown in Table 3
33 KrigingModel Engineering optimization problems oftenneed enormous computation time for several programsrunning at the same time We cannot provide the evaluationof the objective function and constraints to execute such largescale of exact analysis So the application of approximationis necessary In this paper the Kriging model is adopted tobuild the approximation Kriging model one of the responsesurface models (RSM) has such advantages as unbiasedestimator at the training sample point desirably strong non-linear approximating ability and flexible parameter selectionof the model and thus it is quite suitable for approximatemodels [12] Krigingmodels have a great promise for buildingaccurate global approximations of a design space Thesemodels are extremely flexible because of the wide range ofspatial correlation functions that can be chosen for buildingthe approximation provided that sufficient sample data areavailable to capture the trends in the system responses asa result Kriging models can approach linear and nonlinearfunctions equally well In addition Krigingmodels can eitherldquohonor the datardquo by providing an exact interpolation of thedata or ldquosmooth the datardquo by providing an inexact interpola-tion One of the defects of using RSM in optimization is thatit is apt to miss the global optimum because estimation valueobtained with RSM includes errors at an unknown point [13]
Table 3 Controllable factors
Key point Level 1 Level 2 Level 3Lca front 119910(1199091) minus405 minus400 minus395Lca front 119911(1199092) 145 150 155Lca back 119909(1199093) 195 200 205Lca back 119910(1199094) minus455 minus450 minus445Lca back 119911(1199095) 150 155 160Lca outer 119910(1199096) minus755 minus750 minus745Lca outer 119911(1199097) 95 100 105Uca front 119911(1199098) 520 525 530Uca back 119910(1199099) minus495 minus490 minus485Uca back 119911(11990910) 525 530 535Uca outer 119910(11990911) minus680 minus675 minus670Uca outer 119911(11990912) 520 525 530
In this paper the Kriging model is introduced intothe robust design In the conventional Kriging model theperformance 119910(119909) is modelled as follows
where 120573119879ℎ(119909) is the regression component (eg a polyno-mial) which captures global trends 119885(119909) is assumed to bea Gaussian process indexed by input variables 119909 with zeromean and stationary covariance
From a Bayesian perspective the prior knowledge of theperformance 119910(119909) is specified by a Gaussian process whichis characterized by the prior mean (ie the global trend)and prior covariance Given the observations the posteriorprocess is also a Gaussian process (treating the covarianceparameters as known and assuming a Gaussian prior distri-bution for 120573)The prediction of 119910(119909) is usually taken to be theposterior mean and the prediction uncertainty is quantifiedby the posterior covariance
The conventional Kriging model assumes that the Gaus-sian process has a stationary covariance with the covariancefunction defined as follows
The variance 1205902 provides the overall vertical scale rela-tive to the mean of Gaussian process in the output space120579 = 120579
(119897) (119897 = 1 2 119871) are the correlation parameters(scaling factors) associated with each input variable 119909(119897)which reflects the smoothness of the true performance Thestationary covariance indicates that the correlation function120588st(119909119898 119909119899 120579) between any two sites 119909119898 and 119909119899 depends ononly the distance (scaled by 120579) between 119909119898 and 119909119899 in (5) and(6) the subscript ldquostrdquo means ldquostationaryrdquo
The Scientific World Journal 5
In order to innovate or improve and develop a newproduct and confirm a new technical parameter experimentsusually need to be done repeatedly in the process of produc-tion and scientific research It is very important to reasonablyarrange experimental procedures to reduce the times ofexperiments and shorten the time of each experiment andavoid blindness It requires two aspects of works to be done inorder to solve the problemmentioned above One is to designan experiment that can fully reflect the effect of all factorswhich can reduce time of experiments and save resourcesAnother is to analyze the experimental results in order toacquire reasonable conclusions and the error analysis
DOE can analyze a design space and provide a roughestimate of an optimal design which can be used as astarting point for numerical optimization The Latin hyper-cube design could cover the design space more evenly thanother DOE methods and generate more evenly distributedpoints Therefore in this paper the Latin hypercube designis adopted to find the initial point and created the databasefor approximation model For this problem the inputs arethe 12 main design variables the outputs are the mean andtheir variance of the toe angle and sideways displacementand 200 sample points from an LHS design are used to fitthe kriging model A set of 393 verification points randomlyselected across the domain is used to evaluate the RMSEfor each kriging model Kriging model is established to fitthe multiobjective robust design The r-square of the krigingmodel is 087 so it can fit the virtual model
4 Robust Design Based on ParticleSwarm Optimization
The particle swarm optimization (PSO) is one of the evo-lutionary computation techniques introduced by Kennedyand Eberhart in 1995 [14] It is a population-based searchalgorithm and is initialized with a population of randomsolutions named particles PSOmakes use of a velocity vectorto update the current position of each particle in the swarm[15 16]
Particle swarm optimization is usually used as a tradi-tional optimization method which is inspired from the socialbehaviour of flocks of birds It is more competitive in variousaspects for example due to its simplicity Particle swarmoptimization genetic algorithms and other evolutionaryalgorithms are all artificial life calculated But particle swarmoptimization is different from other evolutionary algorithmsusing group iterative solution of cooperation mechanisms togenerate the optimal solution instead of using group iterativesolution of competing mechanisms In PSO algorithm eachindividual is called ldquoparticlerdquo which represents a potentialsolution The algorithm achieves the best solution by thevariability of some particles in the tracing spaceThe particlessearch in the solution space following the best particleby changing their positions and the fitness frequently theflying direction and velocity are determined by the objectivefunction
Update optimum position
Update particle position
Start
Initialize
Evaluate particle
Termination conditions
No
End
Yes
Figure 5 Particle swarm optimization
The procedure of PSO is as follows
(1) initialize the original position and velocity of particleswarm
(2) calculate the fitness value of each particle(3) for each particle compare the fitness value with the
fitness value of pbest if current value is better thenrenew the position with current position and updatethe fitness value simultaneously
(4) determine the best particle of group with the bestfitness value if the fitness value is better than thefitness value of gbest then update the gbest and itsfitness value with the position
(5) check the finalizing criterion if it has been satisfiedquit the iteration otherwise return to step (2)
It can be shown as Figure 5Assuming 119883119894 = (1199091198941 1199091198942 119909119894119863) is the position of 119894th
particle in D-dimension 119881119894 = (V1198941 V1198942 V119894119863) is its velocitywhich represents its direction of searching In iterationprocess each particle keeps the best position pbest found byitself besides it also knows the best position gbest searchedby the group particles and changes its velocity accordingto the two best positions The PSO is described in vectornotation as to the follows
where 119904 is the swarm size 1198881 and 1198882 are the nonnegative accel-eration coefficients these two constants make the particleshave the ability of self-summary and learn from the excellentindividuals of the groups so the particles can close to thepersonal best solution of its own history and the global bestsolution within population or field Typically value of 1198881 and1198882 is 2 120596 is the inertia weight 119903119897(119905) and 1199032(119905) sim 119880(0 1) 119909119894(119905)
is the position of particle 119894 at time 119905 ]119894(119905) is the velocity ofparticle 119894 at time 119905119901119894(119905) is the personal best solution of particle119894 at time 119905 and 119901119892(119905) is the global best solution at time 119905
The first term of (8) is the previous velocity of the particlevector The second and third terms are used to change thevelocity of the particle Without the second and third termsthe particle will keep on ldquoflyingrdquo in the same direction untilit hits the boundary The particle position 119909(119905 + 119897) is updatedusing its current value and the newly computed velocity V119894(119905+119897) which is determined by the values of V119894(119905) 119909119894(119905) 119901119894(119905) and119901119892(119905) and coefficients 120596 1198881 and 1198882 [17]
In experiment the population of group particle is 40 1198881and 1198882 are set to 2 the maximum time of iteration is 10000It is acceptable if the difference between the best solutionobtained by the optimization algorithm and the true solutionis less than 1119890 minus 6 The inertia weight is linear decreasinginertia all which is determined by the following equation
119908 = 119908max minus119908max minus 119908min
itermaxtimes 119896 (8)
where119908max is the start of inertia weight which is set to 09 and119908min is the end of inertia weight which is set to 005 itermaxis the maximum times of iteration 119896 is the current iterationtimes In order to reflect the universality of experiment theoriginal position and velocity are randomly generated
Particle swarm optimization was used to search the opti-mal solution A particle swarm optimization is created themaximum iterations are set to 50 the number of particles isset to 15 and the objectives are the values and their variationsof the toe angle and sideways displacement V Pareto theFrench economist who studied themulti-objective optimiza-tion problem of economics first proposed the concept ofPareto solution setThere are 27 Pareto solutions in the resultsof the optimization
In multiobjective optimization each optimization objec-tive is often conflicting which requires coordination betweenthe optimal solutions of each target Considering the impor-tance of each target choose one Pareto optimal solutionand the design values of it are shown in Table 4 Using theresults of the robust design to have a test in the ADAMS thesimulation results are shown in Figures 6 and 7
It can be seen that the maximum deviation in the toeangle for the optimal design has been reduced by 52 percentcompared with the base design As the discussion of theresults the most concerned factor is the relationship betweenobjective function and design parameter By comparing theexperimental results the robust design based on particleswarm significantly improved the robust of the toe angle andthe sideways displacement ensuring the reasonable of thedesign performance
Table 4 Robust results
Key point Initial value Robust resultsLca front 119910(1199091) minus400 minus400Lca front 119911(1199092) 150 15033Lca back 119909(1199093) 200 205Lca back 119910(1199094) minus450 minus450Lca back 119911(1199095) 155 15167Lca outer 119910(1199096) minus750 minus7525Lca outer 119911(1199097) 100 10133Uca front 119911(1199098) 525 51967Uca back 119910(1199099) minus490 minus48967Uca back 119911(11990910) 530 53217Uca outer 119910(11990911) minus675 minus67983Uca outer 119911(11990912) 525 52333
Figure 6 Sideways displacement
5 Conclusion
In this study a robust design based on bioinspired compu-tation is presented and illustrated by the design of a doublewishbone suspension system in order to reduce the effectof variations due to uncertainties in fabrication As theyare directly related to fabrication errors the coordinates ofkey points were taken as design variables and at the sametime are considered as random variables So the robustdesign optimization problem had 13 design variables (jointpositions) and 13 random constants (fabrication errors ofjoint positions) In this paper the Latin hypercube designis adopted to make DOE design matrix of the 13 designvariablesTheKrigingmodel is built according to the result ofDOE and then the particle swarm is used to search optimalsolution of the robust design Particle swarm is implementedin a test case and the results show that the method candecrease the solutionrsquos time The robustness of solution isimproved The improvement in robustness became larger asthe amount of fabrication errors increases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
The Scientific World Journal 7
Figure 7 Toe angle
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under the Grant no 51165007 and bytheNatural Science Foundation of Jiangxi Province under theGrant no 20132BAB206025 The authors would like to thankthe reviewers for their valuable comments and suggestions
References
[1] J C Dixon Tires Suspension and Handlin Society of Automo-tive Engineers 2nd edition 1996
[2] G Massimiliano and L Francesco ldquoMulti-objective robustdesign of the suspension systemof road vehiclesrdquoVehicle SystemDynamics vol 41 pp 537ndash546 2004
[3] F-C Wu and C-C Chyu ldquoOptimization of robust designfor multiple quality characteristicsrdquo International Journal ofProduction Research vol 42 no 2 pp 337ndash354 2004
[4] W-J Yin Y Han and S-P Yang ldquoDynamics analysis ofair spring suspension system under forced vibrationrdquo ChinaJournal of Highway and Transport vol 19 no 3 pp 117ndash1212006
[5] P-M Lu L-M He and J-M You ldquoOptimization of vehiclesuspension parameters based on comfort and tire dynamicloadrdquo China Journal of Highway and Transport vol 20 no 1pp 112ndash117 2007
[6] H Y Kang and C H Suh ldquoSynthesis and analysis of spherical-cylindrical (SC) link in theMcPherson strut suspensionmecha-nismrdquo Journal of Mechanical Design vol 116 no 2 pp 599ndash6061994
[7] TWang ldquoMulti-objective andmulti-criteria decision optimiza-tion of automobile suspension parametersrdquo Transactions of theChinese Society of Agricultural Machinery vol 28 no 11 pp 27ndash32 2009
[8] H H Chun S J Kwon and T Tak ldquoReliabilitybased designoptimization of automotive suspension systemsrdquo InternationalJournal of Automotive Technology vol 8 no 6 pp 713ndash7222007
[9] B-L Choi J-H Choi and D-H Choi ldquoReliability-baseddesign optimization of an automotive suspension system forenhancing kinematic and compliance characteristicsrdquo Interna-tional Journal of Automotive Technology vol 6 no 3 pp 235ndash242 2004
[10] S R Singiresu and K A Kiran ldquoParticle swarmmethodologiesfor engineering design optimizationrdquo in Proceedings of theInternational Design Engineering Technical Conferences andComputers and Information in Engineering Conference (ASMErsquo09) pp 507ndash516 San Diego Calif USA August 2009
[11] J-J Chen R Xiao Y Zhong and G Dou ldquoMultidisciplinaryrobust optimization designrdquo Chinese Journal of MechanicalEngineering vol 18 no 1 pp 46ndash50 2005
[12] A Giunta and L T Watson A Comparison of ApproximationModeling Technique Polynomial Versus Interpolating ModelsAIAAUSAF NASAISSMO 7th edition 1998
[13] R J Donald S Matthias and J W William ldquoEfficient globaloptimization of expensive black-box functionsrdquo Journal ofGlobal Optimization vol 13 no 4 pp 455ndash492 1998
[14] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork pp 1942ndash1948 Perth Australia December 1995
[15] S Mandal R Kar D Mandal and S P Ghoshal ldquoSwarmintelligence based optimal linear phaseFIR high pass filterdesign using particle swarm optimization with constrictionfactor and inertia weight approachrdquoWorld Academy of ScienceEngineering and Technology vol 5 no 8 pp 1155ndash1161 2011
[16] W-M Zhongi and S-J Li ldquoFeng QIAN 120579-PSO a new strategyof particle swarm optimizationrdquo Journal of Zhejiang Universityvol 9 no 6 pp 786ndash790 2008
[17] S Mandal and S P Ghshal ldquoSwarm intelligence based optimallinear fir high pass filter design using particle swarm optimiza-tion with constriction factor and inertia weight approachrdquo inProceedings of the IEEE Student Conference on Research andDevelopment (SCOReD rsquo11) pp 352ndash357 Cyberjaya MalaysiaDecember 2011
Lca front 119909 1 1Lca front 119910 2 2Lca front 119911 3 3Lca back 119909 1 2Lca back 119910 1 3Lca back 119911 2 3Lca outer 119909 1 1Lca outer 119910 1 3Lca outer 119911 3 2Uca front 119909 1 1Uca front 119910 1 1Uca front 119911 3 3Uca back 119909 1 1Uca back 119910 1 2Uca back 119911 2 3Uca outer 119909 1 1Uca outer 119910 1 3Uca outer 119911 3 3
displacement and toe angle Other coordinates of key pointshave made a great impact on sideways displacement and toeangle Based on the test results 12 main design variables areselected as controllable factors and the variable name and itscorresponding physical quantities are shown in Table 3
33 KrigingModel Engineering optimization problems oftenneed enormous computation time for several programsrunning at the same time We cannot provide the evaluationof the objective function and constraints to execute such largescale of exact analysis So the application of approximationis necessary In this paper the Kriging model is adopted tobuild the approximation Kriging model one of the responsesurface models (RSM) has such advantages as unbiasedestimator at the training sample point desirably strong non-linear approximating ability and flexible parameter selectionof the model and thus it is quite suitable for approximatemodels [12] Krigingmodels have a great promise for buildingaccurate global approximations of a design space Thesemodels are extremely flexible because of the wide range ofspatial correlation functions that can be chosen for buildingthe approximation provided that sufficient sample data areavailable to capture the trends in the system responses asa result Kriging models can approach linear and nonlinearfunctions equally well In addition Krigingmodels can eitherldquohonor the datardquo by providing an exact interpolation of thedata or ldquosmooth the datardquo by providing an inexact interpola-tion One of the defects of using RSM in optimization is thatit is apt to miss the global optimum because estimation valueobtained with RSM includes errors at an unknown point [13]
Table 3 Controllable factors
Key point Level 1 Level 2 Level 3Lca front 119910(1199091) minus405 minus400 minus395Lca front 119911(1199092) 145 150 155Lca back 119909(1199093) 195 200 205Lca back 119910(1199094) minus455 minus450 minus445Lca back 119911(1199095) 150 155 160Lca outer 119910(1199096) minus755 minus750 minus745Lca outer 119911(1199097) 95 100 105Uca front 119911(1199098) 520 525 530Uca back 119910(1199099) minus495 minus490 minus485Uca back 119911(11990910) 525 530 535Uca outer 119910(11990911) minus680 minus675 minus670Uca outer 119911(11990912) 520 525 530
In this paper the Kriging model is introduced intothe robust design In the conventional Kriging model theperformance 119910(119909) is modelled as follows
where 120573119879ℎ(119909) is the regression component (eg a polyno-mial) which captures global trends 119885(119909) is assumed to bea Gaussian process indexed by input variables 119909 with zeromean and stationary covariance
From a Bayesian perspective the prior knowledge of theperformance 119910(119909) is specified by a Gaussian process whichis characterized by the prior mean (ie the global trend)and prior covariance Given the observations the posteriorprocess is also a Gaussian process (treating the covarianceparameters as known and assuming a Gaussian prior distri-bution for 120573)The prediction of 119910(119909) is usually taken to be theposterior mean and the prediction uncertainty is quantifiedby the posterior covariance
The conventional Kriging model assumes that the Gaus-sian process has a stationary covariance with the covariancefunction defined as follows
The variance 1205902 provides the overall vertical scale rela-tive to the mean of Gaussian process in the output space120579 = 120579
(119897) (119897 = 1 2 119871) are the correlation parameters(scaling factors) associated with each input variable 119909(119897)which reflects the smoothness of the true performance Thestationary covariance indicates that the correlation function120588st(119909119898 119909119899 120579) between any two sites 119909119898 and 119909119899 depends ononly the distance (scaled by 120579) between 119909119898 and 119909119899 in (5) and(6) the subscript ldquostrdquo means ldquostationaryrdquo
The Scientific World Journal 5
In order to innovate or improve and develop a newproduct and confirm a new technical parameter experimentsusually need to be done repeatedly in the process of produc-tion and scientific research It is very important to reasonablyarrange experimental procedures to reduce the times ofexperiments and shorten the time of each experiment andavoid blindness It requires two aspects of works to be done inorder to solve the problemmentioned above One is to designan experiment that can fully reflect the effect of all factorswhich can reduce time of experiments and save resourcesAnother is to analyze the experimental results in order toacquire reasonable conclusions and the error analysis
DOE can analyze a design space and provide a roughestimate of an optimal design which can be used as astarting point for numerical optimization The Latin hyper-cube design could cover the design space more evenly thanother DOE methods and generate more evenly distributedpoints Therefore in this paper the Latin hypercube designis adopted to find the initial point and created the databasefor approximation model For this problem the inputs arethe 12 main design variables the outputs are the mean andtheir variance of the toe angle and sideways displacementand 200 sample points from an LHS design are used to fitthe kriging model A set of 393 verification points randomlyselected across the domain is used to evaluate the RMSEfor each kriging model Kriging model is established to fitthe multiobjective robust design The r-square of the krigingmodel is 087 so it can fit the virtual model
4 Robust Design Based on ParticleSwarm Optimization
The particle swarm optimization (PSO) is one of the evo-lutionary computation techniques introduced by Kennedyand Eberhart in 1995 [14] It is a population-based searchalgorithm and is initialized with a population of randomsolutions named particles PSOmakes use of a velocity vectorto update the current position of each particle in the swarm[15 16]
Particle swarm optimization is usually used as a tradi-tional optimization method which is inspired from the socialbehaviour of flocks of birds It is more competitive in variousaspects for example due to its simplicity Particle swarmoptimization genetic algorithms and other evolutionaryalgorithms are all artificial life calculated But particle swarmoptimization is different from other evolutionary algorithmsusing group iterative solution of cooperation mechanisms togenerate the optimal solution instead of using group iterativesolution of competing mechanisms In PSO algorithm eachindividual is called ldquoparticlerdquo which represents a potentialsolution The algorithm achieves the best solution by thevariability of some particles in the tracing spaceThe particlessearch in the solution space following the best particleby changing their positions and the fitness frequently theflying direction and velocity are determined by the objectivefunction
Update optimum position
Update particle position
Start
Initialize
Evaluate particle
Termination conditions
No
End
Yes
Figure 5 Particle swarm optimization
The procedure of PSO is as follows
(1) initialize the original position and velocity of particleswarm
(2) calculate the fitness value of each particle(3) for each particle compare the fitness value with the
fitness value of pbest if current value is better thenrenew the position with current position and updatethe fitness value simultaneously
(4) determine the best particle of group with the bestfitness value if the fitness value is better than thefitness value of gbest then update the gbest and itsfitness value with the position
(5) check the finalizing criterion if it has been satisfiedquit the iteration otherwise return to step (2)
It can be shown as Figure 5Assuming 119883119894 = (1199091198941 1199091198942 119909119894119863) is the position of 119894th
particle in D-dimension 119881119894 = (V1198941 V1198942 V119894119863) is its velocitywhich represents its direction of searching In iterationprocess each particle keeps the best position pbest found byitself besides it also knows the best position gbest searchedby the group particles and changes its velocity accordingto the two best positions The PSO is described in vectornotation as to the follows
where 119904 is the swarm size 1198881 and 1198882 are the nonnegative accel-eration coefficients these two constants make the particleshave the ability of self-summary and learn from the excellentindividuals of the groups so the particles can close to thepersonal best solution of its own history and the global bestsolution within population or field Typically value of 1198881 and1198882 is 2 120596 is the inertia weight 119903119897(119905) and 1199032(119905) sim 119880(0 1) 119909119894(119905)
is the position of particle 119894 at time 119905 ]119894(119905) is the velocity ofparticle 119894 at time 119905119901119894(119905) is the personal best solution of particle119894 at time 119905 and 119901119892(119905) is the global best solution at time 119905
The first term of (8) is the previous velocity of the particlevector The second and third terms are used to change thevelocity of the particle Without the second and third termsthe particle will keep on ldquoflyingrdquo in the same direction untilit hits the boundary The particle position 119909(119905 + 119897) is updatedusing its current value and the newly computed velocity V119894(119905+119897) which is determined by the values of V119894(119905) 119909119894(119905) 119901119894(119905) and119901119892(119905) and coefficients 120596 1198881 and 1198882 [17]
In experiment the population of group particle is 40 1198881and 1198882 are set to 2 the maximum time of iteration is 10000It is acceptable if the difference between the best solutionobtained by the optimization algorithm and the true solutionis less than 1119890 minus 6 The inertia weight is linear decreasinginertia all which is determined by the following equation
119908 = 119908max minus119908max minus 119908min
itermaxtimes 119896 (8)
where119908max is the start of inertia weight which is set to 09 and119908min is the end of inertia weight which is set to 005 itermaxis the maximum times of iteration 119896 is the current iterationtimes In order to reflect the universality of experiment theoriginal position and velocity are randomly generated
Particle swarm optimization was used to search the opti-mal solution A particle swarm optimization is created themaximum iterations are set to 50 the number of particles isset to 15 and the objectives are the values and their variationsof the toe angle and sideways displacement V Pareto theFrench economist who studied themulti-objective optimiza-tion problem of economics first proposed the concept ofPareto solution setThere are 27 Pareto solutions in the resultsof the optimization
In multiobjective optimization each optimization objec-tive is often conflicting which requires coordination betweenthe optimal solutions of each target Considering the impor-tance of each target choose one Pareto optimal solutionand the design values of it are shown in Table 4 Using theresults of the robust design to have a test in the ADAMS thesimulation results are shown in Figures 6 and 7
It can be seen that the maximum deviation in the toeangle for the optimal design has been reduced by 52 percentcompared with the base design As the discussion of theresults the most concerned factor is the relationship betweenobjective function and design parameter By comparing theexperimental results the robust design based on particleswarm significantly improved the robust of the toe angle andthe sideways displacement ensuring the reasonable of thedesign performance
Table 4 Robust results
Key point Initial value Robust resultsLca front 119910(1199091) minus400 minus400Lca front 119911(1199092) 150 15033Lca back 119909(1199093) 200 205Lca back 119910(1199094) minus450 minus450Lca back 119911(1199095) 155 15167Lca outer 119910(1199096) minus750 minus7525Lca outer 119911(1199097) 100 10133Uca front 119911(1199098) 525 51967Uca back 119910(1199099) minus490 minus48967Uca back 119911(11990910) 530 53217Uca outer 119910(11990911) minus675 minus67983Uca outer 119911(11990912) 525 52333
Figure 6 Sideways displacement
5 Conclusion
In this study a robust design based on bioinspired compu-tation is presented and illustrated by the design of a doublewishbone suspension system in order to reduce the effectof variations due to uncertainties in fabrication As theyare directly related to fabrication errors the coordinates ofkey points were taken as design variables and at the sametime are considered as random variables So the robustdesign optimization problem had 13 design variables (jointpositions) and 13 random constants (fabrication errors ofjoint positions) In this paper the Latin hypercube designis adopted to make DOE design matrix of the 13 designvariablesTheKrigingmodel is built according to the result ofDOE and then the particle swarm is used to search optimalsolution of the robust design Particle swarm is implementedin a test case and the results show that the method candecrease the solutionrsquos time The robustness of solution isimproved The improvement in robustness became larger asthe amount of fabrication errors increases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
The Scientific World Journal 7
Figure 7 Toe angle
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under the Grant no 51165007 and bytheNatural Science Foundation of Jiangxi Province under theGrant no 20132BAB206025 The authors would like to thankthe reviewers for their valuable comments and suggestions
References
[1] J C Dixon Tires Suspension and Handlin Society of Automo-tive Engineers 2nd edition 1996
[2] G Massimiliano and L Francesco ldquoMulti-objective robustdesign of the suspension systemof road vehiclesrdquoVehicle SystemDynamics vol 41 pp 537ndash546 2004
[3] F-C Wu and C-C Chyu ldquoOptimization of robust designfor multiple quality characteristicsrdquo International Journal ofProduction Research vol 42 no 2 pp 337ndash354 2004
[4] W-J Yin Y Han and S-P Yang ldquoDynamics analysis ofair spring suspension system under forced vibrationrdquo ChinaJournal of Highway and Transport vol 19 no 3 pp 117ndash1212006
[5] P-M Lu L-M He and J-M You ldquoOptimization of vehiclesuspension parameters based on comfort and tire dynamicloadrdquo China Journal of Highway and Transport vol 20 no 1pp 112ndash117 2007
[6] H Y Kang and C H Suh ldquoSynthesis and analysis of spherical-cylindrical (SC) link in theMcPherson strut suspensionmecha-nismrdquo Journal of Mechanical Design vol 116 no 2 pp 599ndash6061994
[7] TWang ldquoMulti-objective andmulti-criteria decision optimiza-tion of automobile suspension parametersrdquo Transactions of theChinese Society of Agricultural Machinery vol 28 no 11 pp 27ndash32 2009
[8] H H Chun S J Kwon and T Tak ldquoReliabilitybased designoptimization of automotive suspension systemsrdquo InternationalJournal of Automotive Technology vol 8 no 6 pp 713ndash7222007
[9] B-L Choi J-H Choi and D-H Choi ldquoReliability-baseddesign optimization of an automotive suspension system forenhancing kinematic and compliance characteristicsrdquo Interna-tional Journal of Automotive Technology vol 6 no 3 pp 235ndash242 2004
[10] S R Singiresu and K A Kiran ldquoParticle swarmmethodologiesfor engineering design optimizationrdquo in Proceedings of theInternational Design Engineering Technical Conferences andComputers and Information in Engineering Conference (ASMErsquo09) pp 507ndash516 San Diego Calif USA August 2009
[11] J-J Chen R Xiao Y Zhong and G Dou ldquoMultidisciplinaryrobust optimization designrdquo Chinese Journal of MechanicalEngineering vol 18 no 1 pp 46ndash50 2005
[12] A Giunta and L T Watson A Comparison of ApproximationModeling Technique Polynomial Versus Interpolating ModelsAIAAUSAF NASAISSMO 7th edition 1998
[13] R J Donald S Matthias and J W William ldquoEfficient globaloptimization of expensive black-box functionsrdquo Journal ofGlobal Optimization vol 13 no 4 pp 455ndash492 1998
[14] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork pp 1942ndash1948 Perth Australia December 1995
[15] S Mandal R Kar D Mandal and S P Ghoshal ldquoSwarmintelligence based optimal linear phaseFIR high pass filterdesign using particle swarm optimization with constrictionfactor and inertia weight approachrdquoWorld Academy of ScienceEngineering and Technology vol 5 no 8 pp 1155ndash1161 2011
[16] W-M Zhongi and S-J Li ldquoFeng QIAN 120579-PSO a new strategyof particle swarm optimizationrdquo Journal of Zhejiang Universityvol 9 no 6 pp 786ndash790 2008
[17] S Mandal and S P Ghshal ldquoSwarm intelligence based optimallinear fir high pass filter design using particle swarm optimiza-tion with constriction factor and inertia weight approachrdquo inProceedings of the IEEE Student Conference on Research andDevelopment (SCOReD rsquo11) pp 352ndash357 Cyberjaya MalaysiaDecember 2011
In order to innovate or improve and develop a newproduct and confirm a new technical parameter experimentsusually need to be done repeatedly in the process of produc-tion and scientific research It is very important to reasonablyarrange experimental procedures to reduce the times ofexperiments and shorten the time of each experiment andavoid blindness It requires two aspects of works to be done inorder to solve the problemmentioned above One is to designan experiment that can fully reflect the effect of all factorswhich can reduce time of experiments and save resourcesAnother is to analyze the experimental results in order toacquire reasonable conclusions and the error analysis
DOE can analyze a design space and provide a roughestimate of an optimal design which can be used as astarting point for numerical optimization The Latin hyper-cube design could cover the design space more evenly thanother DOE methods and generate more evenly distributedpoints Therefore in this paper the Latin hypercube designis adopted to find the initial point and created the databasefor approximation model For this problem the inputs arethe 12 main design variables the outputs are the mean andtheir variance of the toe angle and sideways displacementand 200 sample points from an LHS design are used to fitthe kriging model A set of 393 verification points randomlyselected across the domain is used to evaluate the RMSEfor each kriging model Kriging model is established to fitthe multiobjective robust design The r-square of the krigingmodel is 087 so it can fit the virtual model
4 Robust Design Based on ParticleSwarm Optimization
The particle swarm optimization (PSO) is one of the evo-lutionary computation techniques introduced by Kennedyand Eberhart in 1995 [14] It is a population-based searchalgorithm and is initialized with a population of randomsolutions named particles PSOmakes use of a velocity vectorto update the current position of each particle in the swarm[15 16]
Particle swarm optimization is usually used as a tradi-tional optimization method which is inspired from the socialbehaviour of flocks of birds It is more competitive in variousaspects for example due to its simplicity Particle swarmoptimization genetic algorithms and other evolutionaryalgorithms are all artificial life calculated But particle swarmoptimization is different from other evolutionary algorithmsusing group iterative solution of cooperation mechanisms togenerate the optimal solution instead of using group iterativesolution of competing mechanisms In PSO algorithm eachindividual is called ldquoparticlerdquo which represents a potentialsolution The algorithm achieves the best solution by thevariability of some particles in the tracing spaceThe particlessearch in the solution space following the best particleby changing their positions and the fitness frequently theflying direction and velocity are determined by the objectivefunction
Update optimum position
Update particle position
Start
Initialize
Evaluate particle
Termination conditions
No
End
Yes
Figure 5 Particle swarm optimization
The procedure of PSO is as follows
(1) initialize the original position and velocity of particleswarm
(2) calculate the fitness value of each particle(3) for each particle compare the fitness value with the
fitness value of pbest if current value is better thenrenew the position with current position and updatethe fitness value simultaneously
(4) determine the best particle of group with the bestfitness value if the fitness value is better than thefitness value of gbest then update the gbest and itsfitness value with the position
(5) check the finalizing criterion if it has been satisfiedquit the iteration otherwise return to step (2)
It can be shown as Figure 5Assuming 119883119894 = (1199091198941 1199091198942 119909119894119863) is the position of 119894th
particle in D-dimension 119881119894 = (V1198941 V1198942 V119894119863) is its velocitywhich represents its direction of searching In iterationprocess each particle keeps the best position pbest found byitself besides it also knows the best position gbest searchedby the group particles and changes its velocity accordingto the two best positions The PSO is described in vectornotation as to the follows
where 119904 is the swarm size 1198881 and 1198882 are the nonnegative accel-eration coefficients these two constants make the particleshave the ability of self-summary and learn from the excellentindividuals of the groups so the particles can close to thepersonal best solution of its own history and the global bestsolution within population or field Typically value of 1198881 and1198882 is 2 120596 is the inertia weight 119903119897(119905) and 1199032(119905) sim 119880(0 1) 119909119894(119905)
is the position of particle 119894 at time 119905 ]119894(119905) is the velocity ofparticle 119894 at time 119905119901119894(119905) is the personal best solution of particle119894 at time 119905 and 119901119892(119905) is the global best solution at time 119905
The first term of (8) is the previous velocity of the particlevector The second and third terms are used to change thevelocity of the particle Without the second and third termsthe particle will keep on ldquoflyingrdquo in the same direction untilit hits the boundary The particle position 119909(119905 + 119897) is updatedusing its current value and the newly computed velocity V119894(119905+119897) which is determined by the values of V119894(119905) 119909119894(119905) 119901119894(119905) and119901119892(119905) and coefficients 120596 1198881 and 1198882 [17]
In experiment the population of group particle is 40 1198881and 1198882 are set to 2 the maximum time of iteration is 10000It is acceptable if the difference between the best solutionobtained by the optimization algorithm and the true solutionis less than 1119890 minus 6 The inertia weight is linear decreasinginertia all which is determined by the following equation
119908 = 119908max minus119908max minus 119908min
itermaxtimes 119896 (8)
where119908max is the start of inertia weight which is set to 09 and119908min is the end of inertia weight which is set to 005 itermaxis the maximum times of iteration 119896 is the current iterationtimes In order to reflect the universality of experiment theoriginal position and velocity are randomly generated
Particle swarm optimization was used to search the opti-mal solution A particle swarm optimization is created themaximum iterations are set to 50 the number of particles isset to 15 and the objectives are the values and their variationsof the toe angle and sideways displacement V Pareto theFrench economist who studied themulti-objective optimiza-tion problem of economics first proposed the concept ofPareto solution setThere are 27 Pareto solutions in the resultsof the optimization
In multiobjective optimization each optimization objec-tive is often conflicting which requires coordination betweenthe optimal solutions of each target Considering the impor-tance of each target choose one Pareto optimal solutionand the design values of it are shown in Table 4 Using theresults of the robust design to have a test in the ADAMS thesimulation results are shown in Figures 6 and 7
It can be seen that the maximum deviation in the toeangle for the optimal design has been reduced by 52 percentcompared with the base design As the discussion of theresults the most concerned factor is the relationship betweenobjective function and design parameter By comparing theexperimental results the robust design based on particleswarm significantly improved the robust of the toe angle andthe sideways displacement ensuring the reasonable of thedesign performance
Table 4 Robust results
Key point Initial value Robust resultsLca front 119910(1199091) minus400 minus400Lca front 119911(1199092) 150 15033Lca back 119909(1199093) 200 205Lca back 119910(1199094) minus450 minus450Lca back 119911(1199095) 155 15167Lca outer 119910(1199096) minus750 minus7525Lca outer 119911(1199097) 100 10133Uca front 119911(1199098) 525 51967Uca back 119910(1199099) minus490 minus48967Uca back 119911(11990910) 530 53217Uca outer 119910(11990911) minus675 minus67983Uca outer 119911(11990912) 525 52333
Figure 6 Sideways displacement
5 Conclusion
In this study a robust design based on bioinspired compu-tation is presented and illustrated by the design of a doublewishbone suspension system in order to reduce the effectof variations due to uncertainties in fabrication As theyare directly related to fabrication errors the coordinates ofkey points were taken as design variables and at the sametime are considered as random variables So the robustdesign optimization problem had 13 design variables (jointpositions) and 13 random constants (fabrication errors ofjoint positions) In this paper the Latin hypercube designis adopted to make DOE design matrix of the 13 designvariablesTheKrigingmodel is built according to the result ofDOE and then the particle swarm is used to search optimalsolution of the robust design Particle swarm is implementedin a test case and the results show that the method candecrease the solutionrsquos time The robustness of solution isimproved The improvement in robustness became larger asthe amount of fabrication errors increases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
The Scientific World Journal 7
Figure 7 Toe angle
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under the Grant no 51165007 and bytheNatural Science Foundation of Jiangxi Province under theGrant no 20132BAB206025 The authors would like to thankthe reviewers for their valuable comments and suggestions
References
[1] J C Dixon Tires Suspension and Handlin Society of Automo-tive Engineers 2nd edition 1996
[2] G Massimiliano and L Francesco ldquoMulti-objective robustdesign of the suspension systemof road vehiclesrdquoVehicle SystemDynamics vol 41 pp 537ndash546 2004
[3] F-C Wu and C-C Chyu ldquoOptimization of robust designfor multiple quality characteristicsrdquo International Journal ofProduction Research vol 42 no 2 pp 337ndash354 2004
[4] W-J Yin Y Han and S-P Yang ldquoDynamics analysis ofair spring suspension system under forced vibrationrdquo ChinaJournal of Highway and Transport vol 19 no 3 pp 117ndash1212006
[5] P-M Lu L-M He and J-M You ldquoOptimization of vehiclesuspension parameters based on comfort and tire dynamicloadrdquo China Journal of Highway and Transport vol 20 no 1pp 112ndash117 2007
[6] H Y Kang and C H Suh ldquoSynthesis and analysis of spherical-cylindrical (SC) link in theMcPherson strut suspensionmecha-nismrdquo Journal of Mechanical Design vol 116 no 2 pp 599ndash6061994
[7] TWang ldquoMulti-objective andmulti-criteria decision optimiza-tion of automobile suspension parametersrdquo Transactions of theChinese Society of Agricultural Machinery vol 28 no 11 pp 27ndash32 2009
[8] H H Chun S J Kwon and T Tak ldquoReliabilitybased designoptimization of automotive suspension systemsrdquo InternationalJournal of Automotive Technology vol 8 no 6 pp 713ndash7222007
[9] B-L Choi J-H Choi and D-H Choi ldquoReliability-baseddesign optimization of an automotive suspension system forenhancing kinematic and compliance characteristicsrdquo Interna-tional Journal of Automotive Technology vol 6 no 3 pp 235ndash242 2004
[10] S R Singiresu and K A Kiran ldquoParticle swarmmethodologiesfor engineering design optimizationrdquo in Proceedings of theInternational Design Engineering Technical Conferences andComputers and Information in Engineering Conference (ASMErsquo09) pp 507ndash516 San Diego Calif USA August 2009
[11] J-J Chen R Xiao Y Zhong and G Dou ldquoMultidisciplinaryrobust optimization designrdquo Chinese Journal of MechanicalEngineering vol 18 no 1 pp 46ndash50 2005
[12] A Giunta and L T Watson A Comparison of ApproximationModeling Technique Polynomial Versus Interpolating ModelsAIAAUSAF NASAISSMO 7th edition 1998
[13] R J Donald S Matthias and J W William ldquoEfficient globaloptimization of expensive black-box functionsrdquo Journal ofGlobal Optimization vol 13 no 4 pp 455ndash492 1998
[14] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork pp 1942ndash1948 Perth Australia December 1995
[15] S Mandal R Kar D Mandal and S P Ghoshal ldquoSwarmintelligence based optimal linear phaseFIR high pass filterdesign using particle swarm optimization with constrictionfactor and inertia weight approachrdquoWorld Academy of ScienceEngineering and Technology vol 5 no 8 pp 1155ndash1161 2011
[16] W-M Zhongi and S-J Li ldquoFeng QIAN 120579-PSO a new strategyof particle swarm optimizationrdquo Journal of Zhejiang Universityvol 9 no 6 pp 786ndash790 2008
[17] S Mandal and S P Ghshal ldquoSwarm intelligence based optimallinear fir high pass filter design using particle swarm optimiza-tion with constriction factor and inertia weight approachrdquo inProceedings of the IEEE Student Conference on Research andDevelopment (SCOReD rsquo11) pp 352ndash357 Cyberjaya MalaysiaDecember 2011
where 119904 is the swarm size 1198881 and 1198882 are the nonnegative accel-eration coefficients these two constants make the particleshave the ability of self-summary and learn from the excellentindividuals of the groups so the particles can close to thepersonal best solution of its own history and the global bestsolution within population or field Typically value of 1198881 and1198882 is 2 120596 is the inertia weight 119903119897(119905) and 1199032(119905) sim 119880(0 1) 119909119894(119905)
is the position of particle 119894 at time 119905 ]119894(119905) is the velocity ofparticle 119894 at time 119905119901119894(119905) is the personal best solution of particle119894 at time 119905 and 119901119892(119905) is the global best solution at time 119905
The first term of (8) is the previous velocity of the particlevector The second and third terms are used to change thevelocity of the particle Without the second and third termsthe particle will keep on ldquoflyingrdquo in the same direction untilit hits the boundary The particle position 119909(119905 + 119897) is updatedusing its current value and the newly computed velocity V119894(119905+119897) which is determined by the values of V119894(119905) 119909119894(119905) 119901119894(119905) and119901119892(119905) and coefficients 120596 1198881 and 1198882 [17]
In experiment the population of group particle is 40 1198881and 1198882 are set to 2 the maximum time of iteration is 10000It is acceptable if the difference between the best solutionobtained by the optimization algorithm and the true solutionis less than 1119890 minus 6 The inertia weight is linear decreasinginertia all which is determined by the following equation
119908 = 119908max minus119908max minus 119908min
itermaxtimes 119896 (8)
where119908max is the start of inertia weight which is set to 09 and119908min is the end of inertia weight which is set to 005 itermaxis the maximum times of iteration 119896 is the current iterationtimes In order to reflect the universality of experiment theoriginal position and velocity are randomly generated
Particle swarm optimization was used to search the opti-mal solution A particle swarm optimization is created themaximum iterations are set to 50 the number of particles isset to 15 and the objectives are the values and their variationsof the toe angle and sideways displacement V Pareto theFrench economist who studied themulti-objective optimiza-tion problem of economics first proposed the concept ofPareto solution setThere are 27 Pareto solutions in the resultsof the optimization
In multiobjective optimization each optimization objec-tive is often conflicting which requires coordination betweenthe optimal solutions of each target Considering the impor-tance of each target choose one Pareto optimal solutionand the design values of it are shown in Table 4 Using theresults of the robust design to have a test in the ADAMS thesimulation results are shown in Figures 6 and 7
It can be seen that the maximum deviation in the toeangle for the optimal design has been reduced by 52 percentcompared with the base design As the discussion of theresults the most concerned factor is the relationship betweenobjective function and design parameter By comparing theexperimental results the robust design based on particleswarm significantly improved the robust of the toe angle andthe sideways displacement ensuring the reasonable of thedesign performance
Table 4 Robust results
Key point Initial value Robust resultsLca front 119910(1199091) minus400 minus400Lca front 119911(1199092) 150 15033Lca back 119909(1199093) 200 205Lca back 119910(1199094) minus450 minus450Lca back 119911(1199095) 155 15167Lca outer 119910(1199096) minus750 minus7525Lca outer 119911(1199097) 100 10133Uca front 119911(1199098) 525 51967Uca back 119910(1199099) minus490 minus48967Uca back 119911(11990910) 530 53217Uca outer 119910(11990911) minus675 minus67983Uca outer 119911(11990912) 525 52333
Figure 6 Sideways displacement
5 Conclusion
In this study a robust design based on bioinspired compu-tation is presented and illustrated by the design of a doublewishbone suspension system in order to reduce the effectof variations due to uncertainties in fabrication As theyare directly related to fabrication errors the coordinates ofkey points were taken as design variables and at the sametime are considered as random variables So the robustdesign optimization problem had 13 design variables (jointpositions) and 13 random constants (fabrication errors ofjoint positions) In this paper the Latin hypercube designis adopted to make DOE design matrix of the 13 designvariablesTheKrigingmodel is built according to the result ofDOE and then the particle swarm is used to search optimalsolution of the robust design Particle swarm is implementedin a test case and the results show that the method candecrease the solutionrsquos time The robustness of solution isimproved The improvement in robustness became larger asthe amount of fabrication errors increases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
The Scientific World Journal 7
Figure 7 Toe angle
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under the Grant no 51165007 and bytheNatural Science Foundation of Jiangxi Province under theGrant no 20132BAB206025 The authors would like to thankthe reviewers for their valuable comments and suggestions
References
[1] J C Dixon Tires Suspension and Handlin Society of Automo-tive Engineers 2nd edition 1996
[2] G Massimiliano and L Francesco ldquoMulti-objective robustdesign of the suspension systemof road vehiclesrdquoVehicle SystemDynamics vol 41 pp 537ndash546 2004
[3] F-C Wu and C-C Chyu ldquoOptimization of robust designfor multiple quality characteristicsrdquo International Journal ofProduction Research vol 42 no 2 pp 337ndash354 2004
[4] W-J Yin Y Han and S-P Yang ldquoDynamics analysis ofair spring suspension system under forced vibrationrdquo ChinaJournal of Highway and Transport vol 19 no 3 pp 117ndash1212006
[5] P-M Lu L-M He and J-M You ldquoOptimization of vehiclesuspension parameters based on comfort and tire dynamicloadrdquo China Journal of Highway and Transport vol 20 no 1pp 112ndash117 2007
[6] H Y Kang and C H Suh ldquoSynthesis and analysis of spherical-cylindrical (SC) link in theMcPherson strut suspensionmecha-nismrdquo Journal of Mechanical Design vol 116 no 2 pp 599ndash6061994
[7] TWang ldquoMulti-objective andmulti-criteria decision optimiza-tion of automobile suspension parametersrdquo Transactions of theChinese Society of Agricultural Machinery vol 28 no 11 pp 27ndash32 2009
[8] H H Chun S J Kwon and T Tak ldquoReliabilitybased designoptimization of automotive suspension systemsrdquo InternationalJournal of Automotive Technology vol 8 no 6 pp 713ndash7222007
[9] B-L Choi J-H Choi and D-H Choi ldquoReliability-baseddesign optimization of an automotive suspension system forenhancing kinematic and compliance characteristicsrdquo Interna-tional Journal of Automotive Technology vol 6 no 3 pp 235ndash242 2004
[10] S R Singiresu and K A Kiran ldquoParticle swarmmethodologiesfor engineering design optimizationrdquo in Proceedings of theInternational Design Engineering Technical Conferences andComputers and Information in Engineering Conference (ASMErsquo09) pp 507ndash516 San Diego Calif USA August 2009
[11] J-J Chen R Xiao Y Zhong and G Dou ldquoMultidisciplinaryrobust optimization designrdquo Chinese Journal of MechanicalEngineering vol 18 no 1 pp 46ndash50 2005
[12] A Giunta and L T Watson A Comparison of ApproximationModeling Technique Polynomial Versus Interpolating ModelsAIAAUSAF NASAISSMO 7th edition 1998
[13] R J Donald S Matthias and J W William ldquoEfficient globaloptimization of expensive black-box functionsrdquo Journal ofGlobal Optimization vol 13 no 4 pp 455ndash492 1998
[14] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork pp 1942ndash1948 Perth Australia December 1995
[15] S Mandal R Kar D Mandal and S P Ghoshal ldquoSwarmintelligence based optimal linear phaseFIR high pass filterdesign using particle swarm optimization with constrictionfactor and inertia weight approachrdquoWorld Academy of ScienceEngineering and Technology vol 5 no 8 pp 1155ndash1161 2011
[16] W-M Zhongi and S-J Li ldquoFeng QIAN 120579-PSO a new strategyof particle swarm optimizationrdquo Journal of Zhejiang Universityvol 9 no 6 pp 786ndash790 2008
[17] S Mandal and S P Ghshal ldquoSwarm intelligence based optimallinear fir high pass filter design using particle swarm optimiza-tion with constriction factor and inertia weight approachrdquo inProceedings of the IEEE Student Conference on Research andDevelopment (SCOReD rsquo11) pp 352ndash357 Cyberjaya MalaysiaDecember 2011
This work was supported by the National Natural ScienceFoundation of China under the Grant no 51165007 and bytheNatural Science Foundation of Jiangxi Province under theGrant no 20132BAB206025 The authors would like to thankthe reviewers for their valuable comments and suggestions
References
[1] J C Dixon Tires Suspension and Handlin Society of Automo-tive Engineers 2nd edition 1996
[2] G Massimiliano and L Francesco ldquoMulti-objective robustdesign of the suspension systemof road vehiclesrdquoVehicle SystemDynamics vol 41 pp 537ndash546 2004
[3] F-C Wu and C-C Chyu ldquoOptimization of robust designfor multiple quality characteristicsrdquo International Journal ofProduction Research vol 42 no 2 pp 337ndash354 2004
[4] W-J Yin Y Han and S-P Yang ldquoDynamics analysis ofair spring suspension system under forced vibrationrdquo ChinaJournal of Highway and Transport vol 19 no 3 pp 117ndash1212006
[5] P-M Lu L-M He and J-M You ldquoOptimization of vehiclesuspension parameters based on comfort and tire dynamicloadrdquo China Journal of Highway and Transport vol 20 no 1pp 112ndash117 2007
[6] H Y Kang and C H Suh ldquoSynthesis and analysis of spherical-cylindrical (SC) link in theMcPherson strut suspensionmecha-nismrdquo Journal of Mechanical Design vol 116 no 2 pp 599ndash6061994
[7] TWang ldquoMulti-objective andmulti-criteria decision optimiza-tion of automobile suspension parametersrdquo Transactions of theChinese Society of Agricultural Machinery vol 28 no 11 pp 27ndash32 2009
[8] H H Chun S J Kwon and T Tak ldquoReliabilitybased designoptimization of automotive suspension systemsrdquo InternationalJournal of Automotive Technology vol 8 no 6 pp 713ndash7222007
[9] B-L Choi J-H Choi and D-H Choi ldquoReliability-baseddesign optimization of an automotive suspension system forenhancing kinematic and compliance characteristicsrdquo Interna-tional Journal of Automotive Technology vol 6 no 3 pp 235ndash242 2004
[10] S R Singiresu and K A Kiran ldquoParticle swarmmethodologiesfor engineering design optimizationrdquo in Proceedings of theInternational Design Engineering Technical Conferences andComputers and Information in Engineering Conference (ASMErsquo09) pp 507ndash516 San Diego Calif USA August 2009
[11] J-J Chen R Xiao Y Zhong and G Dou ldquoMultidisciplinaryrobust optimization designrdquo Chinese Journal of MechanicalEngineering vol 18 no 1 pp 46ndash50 2005
[12] A Giunta and L T Watson A Comparison of ApproximationModeling Technique Polynomial Versus Interpolating ModelsAIAAUSAF NASAISSMO 7th edition 1998
[13] R J Donald S Matthias and J W William ldquoEfficient globaloptimization of expensive black-box functionsrdquo Journal ofGlobal Optimization vol 13 no 4 pp 455ndash492 1998
[14] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork pp 1942ndash1948 Perth Australia December 1995
[15] S Mandal R Kar D Mandal and S P Ghoshal ldquoSwarmintelligence based optimal linear phaseFIR high pass filterdesign using particle swarm optimization with constrictionfactor and inertia weight approachrdquoWorld Academy of ScienceEngineering and Technology vol 5 no 8 pp 1155ndash1161 2011
[16] W-M Zhongi and S-J Li ldquoFeng QIAN 120579-PSO a new strategyof particle swarm optimizationrdquo Journal of Zhejiang Universityvol 9 no 6 pp 786ndash790 2008
[17] S Mandal and S P Ghshal ldquoSwarm intelligence based optimallinear fir high pass filter design using particle swarm optimiza-tion with constriction factor and inertia weight approachrdquo inProceedings of the IEEE Student Conference on Research andDevelopment (SCOReD rsquo11) pp 352ndash357 Cyberjaya MalaysiaDecember 2011
