research article an adaptive metamodel-based...

Research ArticleAn Adaptive Metamodel-Based Optimization Approach forVehicle Suspension System Design

Qinwen Yang1 Jin Huang2 Gang Wang1 and Hamid Reza Karimi3

1 College of Mechanical and Vehicle Engineering Hunan University Changsha Hunan 410082 China2 School of Software Tsinghua University Beijing 100084 China3Department of Engineering Faculty of Engineering and Science University of Agder 4898 Grimstad Norway

Correspondence should be addressed to Jin Huang huangjintsinghuaeducn

Received 8 October 2013 Accepted 21 November 2013 Published 4 February 2014

Academic Editor Hui Zhang

Copyright copy 2014 Qinwen Yang et al This 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 index of a suspension system is a function of the maximum and minimum values over the parameter intervalThus metamodel-based techniques can be used for designing suspension system hardpoints locations In this study an adaptivemetamodel-based optimization approach is used to find the proper locations of the hardpoints with the objectives considering thekinematic performance of the suspensionThe adaptive optimizationmethod helps to find the optimum locations of the hardpointsefficiently as it may be unachievable through manually adjusting For each iteration in the process of adaptive optimizationprediction uncertainty is considered and themultiobjective optimizationmethod is applied to optimize all the performance indexessimultaneously It is shown that the proposed optimization method is effective while being applied in the kinematic performanceoptimization of a McPherson suspension system

1 Introduction

The suspension KampC characteristics have directly effects onvehicle handling and riding performances and thus gainmuch effort and are of great importance in vehicle develop-ment With bush uncertainty and mechanical flexibility it isvery difficult to predict sensitivity of hardpoints locations inthe kinematic performance of a suspension system as theyare highly nonlinear and coupled [1 2] Traditional chassisdeveloping which benefits from the development of modernvirtual prototyping technology can now do system designeffectively through some techniques like the DOE (designof experiment) technique as well as other experience-basedattempts [3ndash5] However the mechanism of the suspensionsystem is designed by trial and error based on the designerrsquosexperiences and intuition which will be time-consuming infinding a sufficiently good solution since a lot of attemptsmay be needed in doing virtual prototyping simulationsA featured optimization technique may be useful to giveguidance through the design process

The performance index of a suspension system is afunction of the maximum and minimum values over theparameter interval [6 7] Thus it is impossible to applydirectly a well-developed optimization algorithm based ongradient information It can be very difficult to evaluatethe analytical design sensitivity of the hardpoints locationsbecause the deviation is defined by using the maximum andminimum values over the parameter interval Metamodelingtechniques which were initially developed as ldquosurrogatesrdquo ofthe expensive simulation process for improving the overallcomputation efficiency and quality [8] are useful in sucha field Metamodel-based methods in vehicle design areamainly focused on FEM related problems [9] Much workhas been done in suspension design area most of whichfocused on complex structural related area The authors in[10] studied a mechanical analysis of a suspension optimaldesign for suspension system based on reliability analysestaking into consideration tolerances and grafting a reliabilityanalysis that applied the mean-value first order methodwith tolerance optimization Choi et al recently studied

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 965157 9 pageshttpdxdoiorg1011552014965157

2 Mathematical Problems in Engineering

optimal design for automotive suspension systems based onreliability analyses for enhancing kinematics and compliancecharacteristics they performed reliability optimization withthe single-loop single-variable method by using the resultsfrom a deterministic optimization as initial valuesThe robustdesign problem was solved with 1700 analyses for 15 designvariables and four randomconstants [11] Recently researcheson applying metamodel-based optimization techniques tothe KampC performance design of vehicle suspension systemswere tried and gained reasonable good results Kang et alintroduced a robust suspension system design approachwhich takes into account the kinematic behaviors influ-enced by bush compliance uncertainty using a sequentialapproximation optimization technique The robust designproblem has 18 design variables and 18 random constantswith uncertainty [12] After then the team proposed aso-called target cascading method for the robust designoptimization process of suspension system for improvingvehicle dynamic performances [13] The design target of thesystem is cascaded from a vehicle level to a suspension systemlevel The design problem structure of suspension systemis defined as a hierarchical multilevel design optimizationand the design problem for each level is solved using therobust design optimization technique based on ametamodelThe researches above opened the way for doing suspensionsystem optimization by using metamodel-based techniqueswith their effectiveness tested which motivated us to furtherresearch along the direction However the former researchneither considered optimizing several objectives simulta-neously to make them stable in comparable intervals norgave the searching guidance for each design parameter inaccelerating the convergence of objective parameters

In this paper we employ a new adaptive metamodel-based optimization approach for guiding suspension systemdesign in determining appropriate hardpoints locations Thefollowing characteristics distinguish the approach from othermetamodel-based applications in suspension system design(1) Aswe have several vehicle performance related parametersto be optimized adaptive weighting factors are used formultiobjective optimization to ensure that all the objectivesare optimized simultaneously (2) For each iteration ofthe adaptive optimization both predicted mean value andprediction standard deviation are considered in case of thesystem converging to the wrong optimum values (3) Weselect kriging method as it is more accurate and efficient thanother metamodeling methods in solving highly nonlinearproblems (4) The optimization approach provides the possi-ble trends in selecting hardpoints locations for optimizing thesystem performancesWe organize the paper in the followingmanner Section 2 introduces the engineering requirementsfor the optimization problem in suspension system designSection 3 presents the methodology of adaptive metamodel-based optimization considering modeling uncertainty Theapplication of adaptive optimization method in suspen-sion system is given in Section 4 Section 5 compares theresults of proposed adaptive metamodel-based optimizationconsidering modeling uncertainty and the regular adaptivemetamodeling approach At last Section 6 summarizes thecontents of this research

12

3

5

4

6

7

Figure 1 The kinematic structural model of a McPherson suspen-sion system

2 Optimization Problem inSuspension System

In multibody dynamics point of view a suspension sys-tem can be classified into several groups according to themechanical joints In this study we take McPherson typesuspension system in consideration which is sensitive tothe kinematic performance Figure 1 shows a kinematicstructure model of a McPherson suspension system Majorcomponents include the strut the lower control arm thetie rod and the knuckle Connections between individualcomponents are spherical revolute and universal joints aswell as compliance elements such as springs dampers andbushingsThe design purpose of this study is to determine thelocations of the hardpoints according to the system kinematicperformances without considering the elastic deformationsof the rigid components except for compliance elementsThe commercial software ADAMS which can easily get thesuspension system performance is employed for modelingand analyzing the suspension system

The kinematic characteristics of the system include thepositions of the fixed points of the suspension system Toachieve optimal solution for suspension system designinga good kinematic performance is the first step and wethus carry out kinematic optimization of the suspensionsystem in this study For that purpose the chosen suspensionperformance indexes are the deviations in the camber anglethe caster angle the kingpin incline angle and the toe angleduring the wheel stroke Camber angle is the angle betweenthe vertical axis of the wheels used for steering and thevertical axis of the vehicle when viewed from the front orrear It is defined as positive when the top of the wheelmoves to the outside The camber angle alters the handling

Mathematical Problems in Engineering 3

qualities of a suspension system in particular a negativecamber angle improves grip when cornering Caster angleis the angular displacement from the vertical axis of thesuspension of a steered wheel in a vehicle measured in thelongitudinal direction It is the angle between the pivot line(in a car an imaginary line that runs through the center ofthe upper ball joint to the center of the lower ball joint) andvertical line On most modern designs the kingpin is set atan angle relative to the true vertical line which is the kingpininclination angle as viewed from the front or back of thevehicle The angle has an important effect on the steeringmaking it tend to return to the straight ahead or centerposition Toe angle is the symmetric angle that each wheelmakes with the longitudinal axis of the vehicle as a functionof static geometry and kinematic and compliant effects Thetoe angle change plays an important role in determining theapparent transient oversteer or understeer Positive toe is thefront of the wheel pointing in towards the centerline of thevehicle Negative toe is the front of the wheel pointing awayfrom the centerline of the vehicle Large errors will have anegative effect on the behavior of the chassis when brakingor accelerating

In suspension system design the locations of hardpointsare the most important influencing factors that determinethe system kinematic performances Although engineersobtained some experiences on adjusting the locations of thehardpoints much effort is still needed on trying differenttrials Furthermore all the geometric parameters of suspen-sion system are coupled which make finding the influenceof the hardpoints locations on the system performances amore toughwork notmentioning there are several objectivesto be determined In this study while determining thelocations of the key hardpoints we select the importantcharacteristics that is the variations of the camber angle thecaster angle the kingpin incline angle and the toe angle forthe objectives since they are related much to the kinematicperformance An adaptive metamodel-based optimizationapproach will be proposed in the next section consideringmodeling uncertainties

3 Adaptive Metamodel-BasedOptimization Method

31 Metamodeling Method In metamodeling the relation-ship between a vector of input parameters x and an outputparameter 119884 can be formulated as

119884 = 119891 (x) + 120576 (1)

where119884 is a randomoutput variable119891(sdot) is the approximatedrelationship and 120576 is the error of metamodel due to theuncertainty introduced by the metamodeling method Manydifferentmetamodels such asmultivariate polynomial radialbasis function (RBF) and kriging can be used to build theapproximation relationship 119891(sdot) In metamodeling first 119898sample data (x

119894 119910119894) (119894 = 1 2 119898) are collected to build

119891(sdot)When an input point x0 is given themetamodel can thenbe used to predict the output 119884

0using

1198840= 119891 (x

0) (2)

Among various metamodeling schemes kriging methodis often selected due to its high accuracy and efficiencyfor solving nonlinear problems [14] Kriging method wasoriginated from the geostatistics community [15] and used bySacks et al [16] for modeling computer experiments Krigingmethod is based on the assumption that the true systemresponse 119884 can be modeled by

119884 =

119898

sum

119894=0

120573119894119891119894 (x) + 119885 (x) (3)

where 119891119894(sdot) is a regression function 120573

119894is the coefficient for

119891119894(sdot)119898+ 1 is the number of regression functions and 119885(sdot) is

the stochastic process with zeromean and covariance definedby

Cov (Z (x119895) Z (x

119896)) = 120590

2119877119895119896(120579 x119895 x119896) (4)

where 1205902 is the process variance 119877119895119896(sdot) is the correlation

function and 120579 is a vector with coefficients to be determinedFor ordinary kriging the linear part of (3) is usually

assumed to be a constant whereas the correlation function119877119895119896(120579 x119895 x119896) is generally formulated as

119877119895119896(120579 x119895 x119896) =

119901

prod

119894=1

119876(120579119894 119909119895119894 119909119896119894) (5)

where 119901 is the dimension of x 119909119895119894is the 119894th component of x

119895

119909119896119894is the 119894th component of x

119896 and119876(sdot) is usually assumed to

be Gaussian as

119876(120579119894 119909119895119894 119909119896119894) = exp (minus120579

1198941198892

119894)

119889119894=10038161003816100381610038161003816119909119895119894minus 119909119896119894

10038161003816100381610038161003816

(6)

The linear predictor of kriging method can be formulated as

119892 (x) = c119879 (x) y (7)

where c119879(sdot) is the coefficient vector and y is the vector of theobservations at the sample sites (x

1 x

119899)

y = [119910 (x1) sdot sdot sdot 119910 (x119899)]119879 (8)

Through minimizing the prediction variance 1205902119905

1205902

119905= 119864 [(119892 (x) minus 119884)2] (9)

concerning the coefficient vector c119879(x) the best linear unbi-ased predictor (BLUP) is solved as [17]

119892 (x) = r119879Rminus1y minus (F119879Rminus1r minus f)119879

(F119879Rminus1F)minus1

(F119879Rminus1y) (10)


where

r = [119877 (120579 x1 x) sdot sdot sdot 119877 (120579 x119899 x)]119879

R = [[

119877 (120579 x1 x1) sdot sdot sdot 119877 (120579 x

1 x119899)

sdot sdot sdot sdot sdot sdot sdot sdot sdot


119899 x119899)

]

]

F = [[

1198910(x1) sdot sdot sdot 119891

0(xn)


119891119898(x1) sdot sdot sdot 119891119898 (xn)

]

]

119879

f = [1198910 (x) sdot sdot sdot 119891119898 (x)]119879

(11)

The coefficients in 120579 can be obtained by using maximumlikelihood estimation as [17]

min120579

120595 (120579) = |R|11198991205902 (12)

where |R| is the determinant of R and 120590 is obtained bygeneralized least squares fit as [17]

2=

1

119899 minus 119898 minus 1(y minus F120573lowast)119879Rminus1 (y minus F120573lowast) (13)

where 120573lowast is the vector with coefficients achieved fromgeneralized least squares fit and is calculated by

120573lowast= (F119879Rminus1F)

minus1

F119879Rminus1y (14)

When metamodeling is used for solving specific prob-lems collection of sample data in specific parameter spacerather than the whole parameter space is then required toimprove the quality and efficiency This issue is critical whenexpensive or extensive experimentssimulations are requiredto collect the sample data Since the relationship is unknownat the beginning initial samples are usually collected to buildthe initial metamodel This developed metamodel is thenused to identify the input parameters that have the bestpotential to lead to the expected output result Due to theerrors of the metamodel the actual output obtained fromexperimentsimulation is usually different from the expectedone The previously obtained metamodel is subsequentlyupdated to improve its quality using the new pair of input-output dataThemethod to iteratively modify themetamodelthrough an iterative sampling process is called adaptivemetamodeling

32 Adaptive Metamodel-Based Optimization When adap-tive metamodeling is used for optimization the optimizationprocess can be referred to as adaptive metamodel-based opti-mization [8]The detailed algorithm for adaptivemetamodel-based optimization is introduced here and we call it adaptiveoptimization in later parts of the paper First the 119898 initialsamples with input parameters x

119894(119894 = 1 2 119898) and output

parameter 119884119894(119894 = 1 2 119898) are collected to build the

metamodel

119884 = 119891119898 (x) (15)

Based on the metamodel relationship 119891119898 we can identify

the potential input parameters xlowast that lead to the minimumoutput parameter through optimization

min119909119891119898 (x) (16)

The optimization result of xlowast is then selected as the vectorof input parameters for the (119898+1)th sample xm+1The output119884119898+1

corresponding to the xm+1 is subsequently obtainedthrough experiment or simulation The new pair of data(xm+1 119884119898+1) togetherwith all the previously collected sampledata are used to update themetamodel into a new relationship119891119898+1

119884 = 119891119898+1 (x) (17)

The process of identifying the potential optimal inputparameters obtaining the output parameter through exper-imentsimulation and updating the metamodel is continuediteratively until the optimization criteria are satisfied

33 Adaptive Optimization considering Modeling UncertaintyIn the process of adaptive optimization the predictionuncertainty of the developed metamodel would influencethe accuracy of the predicted optimum Minimizing the 119891

119898

directly to find the optimal input parameters may not lead togood convergence for output parameters The dual responsesurface methodology is a powerful tool for simultaneouslyoptimizing the mean and the variance of responses to tacklethe problem of misleading [18] Lin and Tu [19] gave adual response surface method using the mean squared error(MSE) approach as follows

min119909

MSE = (120583minus 119879)2

+ 2

120590 (18)

where 120583is the predicted response value 119879 is the target

value and 120590is the prediction standard deviation Following

the format of (18) the objective function for our adaptiveoptimization can be defined as follows

min119909

MSE = (119891119898 (x) minus 119879)

2+ 120590(x)2 (19)

The optimization result of xlowast is then selected as the vector ofinput parameters for the (119898 + 1)th sample xm+1 The output119884119898+1


as shown in (17)We will formulate the specific problem for McPherson

suspension system using the adaptivemetamodel-based opti-mization considering prediction uncertainty in the comingsection

4 Adaptive Optimizationconsidering Modeling Uncertaintyin McPherson Suspension

To investigate the performance of the designated suspensionsystem a classic McPherson suspension system is modeled


in ADAMS as shown in Figure 2 The classic parallel wheeltravel suspension system analysis can be employed to dosystem analysis Vertical bound and rebound of 50mm areused By changing the locations of each hardpoint thecorresponding change of kinematic characteristics can beobtained

In this study we take the locations of the 7 key hardpointslabeled in Figure 1 as variables 3 variables at each hardpointthat is the coordinates of the hardpoint along axes 119909 119910 and 119911of the vehicle coordinate Therefore we have 21 variables (v1ndashv21) for the designated suspension system Table 1 listed the21 variables versus the hardpoint coordinate

For the kinematic characteristics as stated in Section 2we choose the camber angle the caster angle the kingpinincline angle and the toe angle as the objective parametersto be designed and optimized For McPherson suspensionsystem small variations of the four estimate angles surelyare more acceptable Thus we choose the four optimizationobjectives to be the minimum deviations of the four anglesversus wheel travel from rebound minus50mm to bound 50mm

We thus have 21 design variables as input parameters and4 optimumobjectives as output parametersThe optimizationproblem is clearly highly nonlinear and coupled on the inputvariables We thus employ the metamodeling method forthe optimal design We clearly have several optimizationobjectives the usual way that people deal with multiobjectiveoptimization problems is assigning weighting factors to eachobjective and then adding them to build a single objectiveHowever the predefined weighting factors may not be properfor the whole process of optimization In our work we onlyassign the same value to the weighting factors for the outputparameters in the initial samples Adaptive weighting factorsare used in the process of adaptive optimization which iscarried out as follows

119884119894= 11990811198841119894+ 11990821198842119894+ sdot sdot sdot + 119908

119903119884119903119894

119894 = 1 119898

(20)

where 119903 is the number of output parameters and119898 is numberof initial samplesThe119898 initial sampleswith input parametersx119894(119894 = 1 2 119898) and output parameter 119884

119894(119894 = 1 2 119898)

are collected to build the metamodel and obtain the newgroup of input parameters using (19)

The initial metamodel is built on the data generatedfrom Latin hypercube sampling [20] which has the followingadvantages (1) its sample mean has a relatively smallervariance compared with simple random sampling (2) it canbe used for generating design points when the number ofinput variables is large and a great many runs are requiredand (3) it is cheap in computing and easy for implementationcompared with other more complex sampling methods Byusing ADAMS batch processing tools the initial sets samplescan easily be obtained With the built initial metamodel theadaptive optimization method introduced in Section 33 canbe used

The 4 output parameters corresponding to the newgroup of input parameters are subsequently obtained throughsimulations The weighting factors 119908

119894are adjusted using (21)

and the value of 119884119872+1

is calculated using (22) The larger the

y

zx

Figure 2 ADAMS model of the McPherson suspension system

Table 1 The key hardpoints and corresponding variables

Number Hardpoint name Coordinates Variables1 LCA front 119909 119910 119911 V1 V2 V32 LCA outer 119909 119910 119911 V4 V5 V63 LCA rear 119909 119910 119911 V7 V8 V94 Struct lwr 119909 119910 119911 V10 V11 V125 Tierod inner 119909 119910 119911 V13 V14 V156 Tierod outer 119909 119910 119911 V16 V17 V187 Top mount 119909 119910 119911 V19 V20 V21

119884119894119872

is the larger weighting factor will be assigned to it inorder tominimize all the output parameters at the same timeConsider

119908119872+1

119894=

10038161003816100381610038161198841198941198721003816100381610038161003816

100381610038161003816100381611988411198721003816100381610038161003816 +10038161003816100381610038161198842119872

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119884119903119872

1003816100381610038161003816

119894 = 1 119903 119872 = 50 119873 minus 1

(21)

where119872+1 is the number of samples for each iteration in theprocess of adaptive optimization and 119873 is the total samplesize Consider

119884119872+1

= 119908119872+1

11198841119872+ 119908119872+1

21198841119872+ sdot sdot sdot + 119908

119872+1

119903119884119903119872 (22)

The new pair of data (xM+1 119884119872+1) together with allthe previously collected sample data are used to update themetamodel into a new relationshipThe optimization processis stopped when the change of the objective function inseveral consecutive iterations is less than a predefined valueor the maximum number of iterations is reached

As there are 4 output parameters as optimizing objectivesfor our suspension system (23) is used to transform the mul-tiobjective optimization into single objective optimization


Initial sampling for design parameters Multioutput parameters

Single output parameters

ADAMS simulation

Metamodel construction

Optimization

New group of design parameters Multioutput parametersADAMS simulation

Adaptive weightingTerminal condition satisfied

New sample End

Yes

No

Adaptive weighting

Single output parameter

Figure 3 Flowchart of the proposed adaptive multiobjective optimization approach

The adaptive weighting factors are obtained from (21) Thedesign objective is to minimize 119884 as in

119884 = 1199081119884camber + 1199082119884caster + 1199083119884kingpin + 1199084119884toe (23)

The flowchart for the adaptive optimization process is givenin Figure 3

5 Comparisons and Analysis of theOptimization Results

The number of the initial sample is selected as 50 to build theinitial metamodel 50 initial samples may not be enough fora metamodeling problem with 21 variables however due tothe cost consideration in engineering design we just initiallychoose 50 samples and gradually add new samples duringthe design process Based on the initial metamodel moretrails are sampled sequentially and adaptively to approachthe optimum value When the total sample size reaches 100the values of the output parameters are generally stableWe analyze each of the 4 output parameters individuallyevery time while we update the metamodel For this highdimensional (21D input) problem 100 samples may notbe enough to reach the optimum value However we canevaluate its effectiveness through its convergent trend Theoptimization results are shown in Figure 4 The value of theobjective function is well converged and the variations of the4 output parameters are generally stable

The 4 types of curves with ldquo∙rdquo ldquotimesrdquo ldquo+rdquo and ldquoordquo indicate thecamber angle variation the caster angle variation the kingpinincline angle variation and the toe angle variation The trail

0 10 20 30 40 50 60 70 80 90 10002468

Trail number

0 10 20 30 40 50 60 70 80 90 100012345

Trail number

Varia

tions

of

Valu

e of o

bjec

tive

func

tion

outp

ut p

aram

eter

s

YcamberYcaster

YkingpinYtoe

Figure 4 Adaptive optimization results

distributions from 1 to 50 are obtained from Latin hypercubesampling in the whole design space for building the initialmetamodel From Figure 4 we can see that correspondingto the same design interval (minus50 50) the variation intervalfor the camber angle the caster angle the kingpin inclineangle and the toe angle are (0 25) (0 45) (0 15) and(0 45) While adaptive optimization is applied for the trailsfrom 51 to 100 the variation intervals decrease and thecurves become smooth The adaptive weighting factors help


Table 2 The initial and optimized hardpoints coordinates

Variables Initial OptimizedV1 V2 V3 160 minus410 265 1957 minus381 234V4 V5 V6 minus80 minus710 165 minus95 minus7106 213V7 V8 V9 190 minus395 185 1906 minus366 166V10 V11 V12 50 minus630 530 646 minus622 5313V13 V14 V15 155 minus440 355 130 minus3912 305V16 V17 V18 210 minus700 240 2195 minus749 290V19 V20 V21 75 minus6038 850 352 minus642 8736

the output parameters with different variation intervals con-verge to the same variation interval (0 033) It can be seenfrom Figure 4 that our method succeeds in optimizing the4 output parameters simultaneously while the value of theobjective function is well converged Table 2 listed the initialand the optimized hardpoints locations for the suspensionsystem We have chosen the 100th sample as the optimizedresult

As to the regular adaptive optimization prediction stan-dard deviation is not considered in the process of theoptimization and the objective function is usually definedas in (16) We used the same 50 initial samples to build theinitialmetamodel and then apply the objective function givenin (16) to search the subsequent 50 samples We comparethe mean output values of the last 5 samples for each of the4 output parameters from optimization objective functionsdefined by (19) and (16) as shown inTable 3Themean outputvalues are compared by a ratio 120572 representing how muchthe adaptive optimization method considering uncertainty isbetter than the regular adaptive optimization method Here

120572 =Varmean minus Varmean(un)

Varmean(un) (24)

where Varmean indicates the mean output value in the last5 iterations from regular adaptive optimization methodand Varmean(un) gives the mean output value in the last 5iterations from adaptive optimization method consideringthe modeling uncertainty

From Table 3 we could see that the adaptive optimiza-tion method with consideration of modeling uncertaintyperforms better The adaptive optimization performs betterup to 8111 for all the four output parameters The reasonshould be that the number of sample points is far fromsufficient to build an accurate metamodel especially at thevery beginning Thus the optimization result based on thebuilt metamodel only considering predicted mean value maynot converge to the right direction With the predictionstandard deviation being considered at the same time theinaccuracy of the metamodel can be compensated to someextent to approach the right direction

What is interesting is that the approach can illustratea suggested trend that the hardpoints should be movedover to Figure 5 shows the approximate trends that thehardpoints tierod inner and tierod outer stay around during

0 10 20 30 40 50 60 70 80 90 100Trail number

0 10 20 30 40 50 60 70 80 90 100Trail number

minus50

0

50

Offs

et fr

om in

itial

minus50

0

50

Offs

et fr

om in

itial

Tierod-inner xTierod-inner yTierod-inner z

Tierod-outer xTierod-outer yTierod-outer z

Figure 5 The evolution trend of the hardpoint tierod inner andtierod outer

the optimization procedure and other hardpoints performsimilarly From this figure it can be seen that the 3 coordinatevalues for each input parameter tend to change in a smallerinterval rather than change in the original design interval(minus50 50) This result from our research can help to reducethe design space in the process of adaptive optimizationin order to greatly improve the optimization efficiencywhich would be especially significant for higher dimensionalproblems

The apparent result that the optimization procedure canachieve is reducing the variation of the design objectivesFigure 6 gives the variations of the camber angle the casterangle the kingpin incline angle and the toe angle versus thewheel travel for the optimized trail number 100 and the initialtrail during the parallel suspension travel analysisWe can seethat the optimized data significantly reduced the variation ofthe camber angle the kingpin incline angle and the toe anglewhile also slightly reducing the caster angle This shows theeffectiveness of the proposed approach on guiding to searchfor the optimized solution for kinematic performance designof vehicle suspension systems Other related parameters canbe considered in a similar way

6 Conclusion

This study introduced adaptive metamodel-based optimiza-tion considering modeling uncertainty to optimize the kine-matic performance of a McPherson suspension system Theoptimization design problem is of 21 input parameters and4 output parameters The multioutput optimization in theMcPherson suspension system is transformed to a single out-put optimization problem using adaptive weighting factors


Table 3 Comparison between adaptive optimization considering modeling uncertainty and regular adaptive optimization

Optimization objective function Camber anglevariation

Caster anglevariation

Kingpin anglevariation

Toe anglevariation

Adaptive optimization considering modeling uncertainty 01056 00916 03044 00577Regular adaptive optimization 01616 01585 04717 01045120572 5303 7303 5496 8111

25

2

15

1

05

0

minus05

minus1

minus15

50403020100minus10minus20minus30minus40minus50

Wheel travel (mm)

Cam

ber a

ngle

(deg

)

(a)


Wheel travel (mm)

115

98333

81667

65

Caste

r ang

le (d

eg)

(b)


Wheel travel (mm)InitialMetamodel oriented

135

12625

1175

10875

10

9125

825

7375

65

King

pin

incli

ne an

gle (

deg)

(c)


Wheel travel (mm)

6

43333

26667

1

0minus06667

minus23333

minus4

Toe a

ngle

(deg

)

InitialMetamodel oriented

(d)

Figure 6The variation of camber angle caster angle kingpin incline angle and toe angle versuswheel travel before and after the optimization

50 initial samples and 50 sequential trails are generated toanalyze the convergent trend of the objective parameters Itshows that the proposed optimization method provided a setof relatively good results of the 4 output objective parameterssimultaneously for the 50 sequential trails Possible optimal

design trends of the hardpoints are given as the trail goeson Comparisons showed that the adaptive metamodel-based optimization method considering modeling uncer-tainty worked better than general adaptive metamodel-basedoptimization for the suspension design problem


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by NSFC Grant no 11302074

References

[1] M Blundell and D Harty The Multibody Systems Approach toVehicle Dynamics Maple-Vail Kirkwood NY USA 2004

[2] M Zapateiro F Pozo H R Karimi and N Luo ldquoSemiactivecontrol methodologies for suspension control with magne-torheological dampersrdquo IEEEASME Transactions on Mecha-tronics vol 17 no 2 pp 370ndash380 2012

[3] H Li X Jing and H R Karimi ldquoOutput-feedback-based 119867infin

control for active suspension systems with control delayrdquo IEEETransactions on Industrial Electronics vol 61 no 1 pp 436ndash4462013

[4] H Zhang Y Shi and A S Mehr ldquoRobust 119867infin

PID con-trol for multivariable networked control systems with distur-bancenoise attenuationrdquo International Journal of Robust andNonlinear Control vol 22 no 2 pp 183ndash204 2012

[5] R Wang H Zhang and J Wang ldquoLinear parameter-varyingcontroller design for four-wheel independently actuated electricground vehicleswith active steering systemsrdquo IEEETransactionson Control Systems Technology 2013

[6] Z Shuai H Zhang JWang and J Li ldquoCombinedAFS andDYCcontrol of four-wheel-independent-drive electric vehicles overCAN network with time-varying delaysrdquo IEEE Transactions onVehicular Technology 2013

[7] H Zhang J Wang and Y Shi ldquoRobust 119867infin

sliding-modecontrol for Markovian jump systems subject to intermittentobservations and partially known transition probabilitiesrdquo Sys-tems amp Control Letters vol 62 no 12 pp 1114ndash1124 2013

[8] G G Wang and S Shan ldquoReview of metamodeling techniquesin support of engineering design optimizationrdquo Journal ofMechanical Design Transactions of the ASME vol 129 no 4 pp370ndash380 2007

[9] P Zhu Y Zhang and G-L Chen ldquoMetamodel-basedlightweight design of an automotive front-body structureusing robust optimizationrdquo Proceedings of the Institution ofMechanical Engineers D vol 223 no 9 pp 1133ndash1147 2009

[10] H H Chun S J Kwon and T Tak ldquoReliability based designoptimization of automotive suspension systemsrdquo InternationalJournal of Automotive Technology vol 8 no 6 pp 713ndash7222007

[11] 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 2005

[12] D O Kang S J Heo and M S Kim ldquoRobust design optimiza-tion of theMcPherson suspension systemwith consideration ofa bush compliance uncertaintyrdquo Proceedings of the Institution ofMechanical Engineers D vol 224 no 6 pp 705ndash716 2010

[13] D O Kang S J Heo M S Kim W C Choi and I H KimldquoRobust design optimization of suspension system by using

target cascading methodrdquo International Journal of AutomotiveTechnology vol 13 no 1 pp 109ndash122 2012

[14] D Zhao and D Xue ldquoA comparative study of metamodelingmethods considering sample quality meritsrdquo Structural andMultidisciplinary Optimization vol 42 no 6 pp 923ndash938 2010

[15] G Matheron ldquoPrincipals of geostatisticsrdquo Economic Geologyvol 58 no 8 pp 1246ndash1266 1963

[16] J SacksW JWelch T J Mitchell andH PWynn ldquoDesign andanalysis of computer experimentsrdquo Statistical Science vol 4 no4 pp 409ndash435 1989

[17] S N Lophaven H B Nielsen and J Soslashndergaard ldquoDACE AMatlab Kriging Toolbox Version 20rdquo Tech Rep IMMREP-2002-12 Technical University of Denmark Kongens LyngbyDenmark 2002

[18] R Ding D K J Lin and D Wei ldquoDual-response surfaceoptimization a weighted MSE approachrdquo Quality Engineeringvol 16 no 3 pp 377ndash385 2004

[19] D K J Lin and W Tu ldquoDual response surface optimizationrdquoJournal of Quality Technology vol 27 no 1 pp 34ndash39 1995

[20] M D McKay R J Beckman andW J Conover ldquoA comparisonof three methods for selecting values of input variables in theanalysis of output from a computer coderdquo Technometrics vol21 no 2 pp 239ndash245 1979

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


optimal design for automotive suspension systems based onreliability analyses for enhancing kinematics and compliancecharacteristics they performed reliability optimization withthe single-loop single-variable method by using the resultsfrom a deterministic optimization as initial valuesThe robustdesign problem was solved with 1700 analyses for 15 designvariables and four randomconstants [11] Recently researcheson applying metamodel-based optimization techniques tothe KampC performance design of vehicle suspension systemswere tried and gained reasonable good results Kang et alintroduced a robust suspension system design approachwhich takes into account the kinematic behaviors influ-enced by bush compliance uncertainty using a sequentialapproximation optimization technique The robust designproblem has 18 design variables and 18 random constantswith uncertainty [12] After then the team proposed aso-called target cascading method for the robust designoptimization process of suspension system for improvingvehicle dynamic performances [13] The design target of thesystem is cascaded from a vehicle level to a suspension systemlevel The design problem structure of suspension systemis defined as a hierarchical multilevel design optimizationand the design problem for each level is solved using therobust design optimization technique based on ametamodelThe researches above opened the way for doing suspensionsystem optimization by using metamodel-based techniqueswith their effectiveness tested which motivated us to furtherresearch along the direction However the former researchneither considered optimizing several objectives simulta-neously to make them stable in comparable intervals norgave the searching guidance for each design parameter inaccelerating the convergence of objective parameters

In this paper we employ a new adaptive metamodel-based optimization approach for guiding suspension systemdesign in determining appropriate hardpoints locations Thefollowing characteristics distinguish the approach from othermetamodel-based applications in suspension system design(1) Aswe have several vehicle performance related parametersto be optimized adaptive weighting factors are used formultiobjective optimization to ensure that all the objectivesare optimized simultaneously (2) For each iteration ofthe adaptive optimization both predicted mean value andprediction standard deviation are considered in case of thesystem converging to the wrong optimum values (3) Weselect kriging method as it is more accurate and efficient thanother metamodeling methods in solving highly nonlinearproblems (4) The optimization approach provides the possi-ble trends in selecting hardpoints locations for optimizing thesystem performancesWe organize the paper in the followingmanner Section 2 introduces the engineering requirementsfor the optimization problem in suspension system designSection 3 presents the methodology of adaptive metamodel-based optimization considering modeling uncertainty Theapplication of adaptive optimization method in suspen-sion system is given in Section 4 Section 5 compares theresults of proposed adaptive metamodel-based optimizationconsidering modeling uncertainty and the regular adaptivemetamodeling approach At last Section 6 summarizes thecontents of this research

12

3

5

4

6

7

Figure 1 The kinematic structural model of a McPherson suspen-sion system

2 Optimization Problem inSuspension System

In multibody dynamics point of view a suspension sys-tem can be classified into several groups according to themechanical joints In this study we take McPherson typesuspension system in consideration which is sensitive tothe kinematic performance Figure 1 shows a kinematicstructure model of a McPherson suspension system Majorcomponents include the strut the lower control arm thetie rod and the knuckle Connections between individualcomponents are spherical revolute and universal joints aswell as compliance elements such as springs dampers andbushingsThe design purpose of this study is to determine thelocations of the hardpoints according to the system kinematicperformances without considering the elastic deformationsof the rigid components except for compliance elementsThe commercial software ADAMS which can easily get thesuspension system performance is employed for modelingand analyzing the suspension system

The kinematic characteristics of the system include thepositions of the fixed points of the suspension system Toachieve optimal solution for suspension system designinga good kinematic performance is the first step and wethus carry out kinematic optimization of the suspensionsystem in this study For that purpose the chosen suspensionperformance indexes are the deviations in the camber anglethe caster angle the kingpin incline angle and the toe angleduring the wheel stroke Camber angle is the angle betweenthe vertical axis of the wheels used for steering and thevertical axis of the vehicle when viewed from the front orrear It is defined as positive when the top of the wheelmoves to the outside The camber angle alters the handling






119884 = 119891 (x) + 120576 (1)




0using

1198840= 119891 (x

0) (2)


119884 =

119898

sum

119894=0

120573119894119891119894 (x) + 119885 (x) (3)





Cov (Z (x119895) Z (x

119896)) = 120590

2119877119895119896(120579 x119895 x119896) (4)




119877119895119896(120579 x119895 x119896) =

119901

prod

119894=1

119876(120579119894 119909119895119894 119909119896119894) (5)


119895



be Gaussian as

119876(120579119894 119909119895119894 119909119896119894) = exp (minus120579

1198941198892

119894)

119889119894=10038161003816100381610038161003816119909119895119894minus 119909119896119894

10038161003816100381610038161003816

(6)


119892 (x) = c119879 (x) y (7)


1 x

119899)



1205902

119905= 119864 [(119892 (x) minus 119884)2] (9)




(F119879Rminus1y) (10)


where


R = [[


1 x119899)



119899 x119899)

]

]

F = [[

1198910(x1) sdot sdot sdot 119891

0(xn)



]

]

119879


(11)


min120579

120595 (120579) = |R|11198991205902 (12)


2=

1




minus1




119894(119894 = 1 2 119898) and output


metamodel

119884 = 119891119898 (x) (15)



min119909119891119898 (x) (16)



119884 = 119891119898+1 (x) (17)



119898


min119909

MSE = (120583minus 119879)2

+ 2

120590 (18)




min119909

MSE = (119891119898 (x) minus 119879)

2+ 120590(x)2 (19)












119884119894= 11990811198841119894+ 11990821198842119894+ sdot sdot sdot + 119908

119903119884119903119894

119894 = 1 119898

(20)


119894(119894 = 1 2 119898)







y

zx




119884119894119872


119908119872+1

119894=

10038161003816100381610038161198841198941198721003816100381610038161003816

100381610038161003816100381611988411198721003816100381610038161003816 +10038161003816100381610038161198842119872

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119884119903119872

1003816100381610038161003816

119894 = 1 119903 119872 = 50 119873 minus 1

(21)


119884119872+1

= 119908119872+1

11198841119872+ 119908119872+1

21198841119872+ sdot sdot sdot + 119908

119872+1

119903119884119903119872 (22)






ADAMS simulation


Optimization



New sample End

Yes

No

Adaptive weighting









0 10 20 30 40 50 60 70 80 90 10002468

Trail number

0 10 20 30 40 50 60 70 80 90 100012345

Trail number

Varia

tions

of

Valu

e of o

bjec

tive

func

tion

outp

ut p

aram

eter

s

YcamberYcaster

YkingpinYtoe









Varmean(un) (24)




0 10 20 30 40 50 60 70 80 90 100Trail number

0 10 20 30 40 50 60 70 80 90 100Trail number

minus50

0

50

Offs

et fr

om in

itial

minus50

0

50

Offs

et fr

om in

itial






6 Conclusion







Toe anglevariation


25

2

15

1

05

0

minus05

minus1

minus15


Wheel travel (mm)

Cam

ber a

ngle

(deg

)

(a)


Wheel travel (mm)

115

98333

81667

65

Caste

r ang

le (d

eg)

(b)



135

12625

1175

10875

10

9125

825

7375

65

King

pin

incli

ne an

gle (

deg)

(c)


Wheel travel (mm)

6

43333

26667

1

0minus06667

minus23333

minus4

Toe a

ngle

(deg

)


(d)







Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119884 = 119891 (x) + 120576 (1)




0using

1198840= 119891 (x

0) (2)


119884 =

119898

sum

119894=0

120573119894119891119894 (x) + 119885 (x) (3)





Cov (Z (x119895) Z (x

119896)) = 120590

2119877119895119896(120579 x119895 x119896) (4)




119877119895119896(120579 x119895 x119896) =

119901

prod

119894=1

119876(120579119894 119909119895119894 119909119896119894) (5)


119895



be Gaussian as

119876(120579119894 119909119895119894 119909119896119894) = exp (minus120579

1198941198892

119894)

119889119894=10038161003816100381610038161003816119909119895119894minus 119909119896119894

10038161003816100381610038161003816

(6)


119892 (x) = c119879 (x) y (7)


1 x

119899)



1205902

119905= 119864 [(119892 (x) minus 119884)2] (9)




(F119879Rminus1y) (10)


where


R = [[


1 x119899)



119899 x119899)

]

]

F = [[

1198910(x1) sdot sdot sdot 119891

0(xn)



]

]

119879


(11)


min120579

120595 (120579) = |R|11198991205902 (12)


2=

1




minus1




119894(119894 = 1 2 119898) and output


metamodel

119884 = 119891119898 (x) (15)



min119909119891119898 (x) (16)



119884 = 119891119898+1 (x) (17)



119898


min119909

MSE = (120583minus 119879)2

+ 2

120590 (18)




min119909

MSE = (119891119898 (x) minus 119879)

2+ 120590(x)2 (19)












119884119894= 11990811198841119894+ 11990821198842119894+ sdot sdot sdot + 119908

119903119884119903119894

119894 = 1 119898

(20)


119894(119894 = 1 2 119898)







y

zx




119884119894119872


119908119872+1

119894=

10038161003816100381610038161198841198941198721003816100381610038161003816

100381610038161003816100381611988411198721003816100381610038161003816 +10038161003816100381610038161198842119872

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119884119903119872

1003816100381610038161003816

119894 = 1 119903 119872 = 50 119873 minus 1

(21)


119884119872+1

= 119908119872+1

11198841119872+ 119908119872+1

21198841119872+ sdot sdot sdot + 119908

119872+1

119903119884119903119872 (22)






ADAMS simulation


Optimization



New sample End

Yes

No

Adaptive weighting









0 10 20 30 40 50 60 70 80 90 10002468

Trail number

0 10 20 30 40 50 60 70 80 90 100012345

Trail number

Varia

tions

of

Valu

e of o

bjec

tive

func

tion

outp

ut p

aram

eter

s

YcamberYcaster

YkingpinYtoe









Varmean(un) (24)




0 10 20 30 40 50 60 70 80 90 100Trail number

0 10 20 30 40 50 60 70 80 90 100Trail number

minus50

0

50

Offs

et fr

om in

itial

minus50

0

50

Offs

et fr

om in

itial






6 Conclusion







Toe anglevariation


25

2

15

1

05

0

minus05

minus1

minus15


Wheel travel (mm)

Cam

ber a

ngle

(deg

)

(a)


Wheel travel (mm)

115

98333

81667

65

Caste

r ang

le (d

eg)

(b)



135

12625

1175

10875

10

9125

825

7375

65

King

pin

incli

ne an

gle (

deg)

(c)


Wheel travel (mm)

6

43333

26667

1

0minus06667

minus23333

minus4

Toe a

ngle

(deg

)


(d)







Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where


R = [[


1 x119899)



119899 x119899)

]

]

F = [[

1198910(x1) sdot sdot sdot 119891

0(xn)



]

]

119879


(11)


min120579

120595 (120579) = |R|11198991205902 (12)


2=

1




minus1




119894(119894 = 1 2 119898) and output


metamodel

119884 = 119891119898 (x) (15)



min119909119891119898 (x) (16)



119884 = 119891119898+1 (x) (17)



119898


min119909

MSE = (120583minus 119879)2

+ 2

120590 (18)




min119909

MSE = (119891119898 (x) minus 119879)

2+ 120590(x)2 (19)












119884119894= 11990811198841119894+ 11990821198842119894+ sdot sdot sdot + 119908

119903119884119903119894

119894 = 1 119898

(20)


119894(119894 = 1 2 119898)







y

zx




119884119894119872


119908119872+1

119894=

10038161003816100381610038161198841198941198721003816100381610038161003816

100381610038161003816100381611988411198721003816100381610038161003816 +10038161003816100381610038161198842119872

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119884119903119872

1003816100381610038161003816

119894 = 1 119903 119872 = 50 119873 minus 1

(21)


119884119872+1

= 119908119872+1

11198841119872+ 119908119872+1

21198841119872+ sdot sdot sdot + 119908

119872+1

119903119884119903119872 (22)






ADAMS simulation


Optimization



New sample End

Yes

No

Adaptive weighting









0 10 20 30 40 50 60 70 80 90 10002468

Trail number

0 10 20 30 40 50 60 70 80 90 100012345

Trail number

Varia

tions

of

Valu

e of o

bjec

tive

func

tion

outp

ut p

aram

eter

s

YcamberYcaster

YkingpinYtoe









Varmean(un) (24)




0 10 20 30 40 50 60 70 80 90 100Trail number

0 10 20 30 40 50 60 70 80 90 100Trail number

minus50

0

50

Offs

et fr

om in

itial

minus50

0

50

Offs

et fr

om in

itial






6 Conclusion







Toe anglevariation


25

2

15

1

05

0

minus05

minus1

minus15


Wheel travel (mm)

Cam

ber a

ngle

(deg

)

(a)


Wheel travel (mm)

115

98333

81667

65

Caste

r ang

le (d

eg)

(b)



135

12625

1175

10875

10

9125

825

7375

65

King

pin

incli

ne an

gle (

deg)

(c)


Wheel travel (mm)

6

43333

26667

1

0minus06667

minus23333

minus4

Toe a

ngle

(deg

)


(d)







Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119884119894= 11990811198841119894+ 11990821198842119894+ sdot sdot sdot + 119908

119903119884119903119894

119894 = 1 119898

(20)


119894(119894 = 1 2 119898)







y

zx




119884119894119872


119908119872+1

119894=

10038161003816100381610038161198841198941198721003816100381610038161003816

100381610038161003816100381611988411198721003816100381610038161003816 +10038161003816100381610038161198842119872

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119884119903119872

1003816100381610038161003816

119894 = 1 119903 119872 = 50 119873 minus 1

(21)


119884119872+1

= 119908119872+1

11198841119872+ 119908119872+1

21198841119872+ sdot sdot sdot + 119908

119872+1

119903119884119903119872 (22)






ADAMS simulation


Optimization



New sample End

Yes

No

Adaptive weighting









0 10 20 30 40 50 60 70 80 90 10002468

Trail number

0 10 20 30 40 50 60 70 80 90 100012345

Trail number

Varia

tions

of

Valu

e of o

bjec

tive

func

tion

outp

ut p

aram

eter

s

YcamberYcaster

YkingpinYtoe









Varmean(un) (24)




0 10 20 30 40 50 60 70 80 90 100Trail number

0 10 20 30 40 50 60 70 80 90 100Trail number

minus50

0

50

Offs

et fr

om in

itial

minus50

0

50

Offs

et fr

om in

itial






6 Conclusion







Toe anglevariation


25

2

15

1

05

0

minus05

minus1

minus15


Wheel travel (mm)

Cam

ber a

ngle

(deg

)

(a)


Wheel travel (mm)

115

98333

81667

65

Caste

r ang

le (d

eg)

(b)



135

12625

1175

10875

10

9125

825

7375

65

King

pin

incli

ne an

gle (

deg)

(c)


Wheel travel (mm)

6

43333

26667

1

0minus06667

minus23333

minus4

Toe a

ngle

(deg

)


(d)







Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












ADAMS simulation


Optimization



New sample End

Yes

No

Adaptive weighting









0 10 20 30 40 50 60 70 80 90 10002468

Trail number

0 10 20 30 40 50 60 70 80 90 100012345

Trail number

Varia

tions

of

Valu

e of o

bjec

tive

func

tion

outp

ut p

aram

eter

s

YcamberYcaster

YkingpinYtoe









Varmean(un) (24)




0 10 20 30 40 50 60 70 80 90 100Trail number

0 10 20 30 40 50 60 70 80 90 100Trail number

minus50

0

50

Offs

et fr

om in

itial

minus50

0

50

Offs

et fr

om in

itial






6 Conclusion







Toe anglevariation


25

2

15

1

05

0

minus05

minus1

minus15


Wheel travel (mm)

Cam

ber a

ngle

(deg

)

(a)


Wheel travel (mm)

115

98333

81667

65

Caste

r ang

le (d

eg)

(b)



135

12625

1175

10875

10

9125

825

7375

65

King

pin

incli

ne an

gle (

deg)

(c)


Wheel travel (mm)

6

43333

26667

1

0minus06667

minus23333

minus4

Toe a

ngle

(deg

)


(d)







Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Varmean(un) (24)




0 10 20 30 40 50 60 70 80 90 100Trail number

0 10 20 30 40 50 60 70 80 90 100Trail number

minus50

0

50

Offs

et fr

om in

itial

minus50

0

50

Offs

et fr

om in

itial






6 Conclusion







Toe anglevariation


25

2

15

1

05

0

minus05

minus1

minus15


Wheel travel (mm)

Cam

ber a

ngle

(deg

)

(a)


Wheel travel (mm)

115

98333

81667

65

Caste

r ang

le (d

eg)

(b)



135

12625

1175

10875

10

9125

825

7375

65

King

pin

incli

ne an

gle (

deg)

(c)


Wheel travel (mm)

6

43333

26667

1

0minus06667

minus23333

minus4

Toe a

ngle

(deg

)


(d)







Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Toe anglevariation


25

2

15

1

05

0

minus05

minus1

minus15


Wheel travel (mm)

Cam

ber a

ngle

(deg

)

(a)


Wheel travel (mm)

115

98333

81667

65

Caste

r ang

le (d

eg)

(b)



135

12625

1175

10875

10

9125

825

7375

65

King

pin

incli

ne an

gle (

deg)

(c)


Wheel travel (mm)

6

43333

26667

1

0minus06667

minus23333

minus4

Toe a

ngle

(deg

)


(d)







Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article an adaptive metamodel-based...

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