research article structural optimization of a knuckle with...
TRANSCRIPT
Research ArticleStructural Optimization of a Knuckle with Consideration ofStiffness and Durability Requirements
Geun-Yeon Kim Seung-Ho Han and Kwon-Hee Lee
Department of Mechanical Engineering Dong-A University Busan 604-714 Republic of Korea
Correspondence should be addressed to Kwon-Hee Lee leekhdauackr
Received 27 January 2014 Accepted 9 February 2014 Published 8 June 2014
Academic Editors N Barsoum V N Dieu P Vasant and G-W Weber
Copyright copy 2014 Geun-Yeon Kim et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The automobilersquos knuckle is connected to the parts of the steering system and the suspension system and it is used for adjusting thedirection of a rotation through its attachment to thewheelThis study changes the existingmaterialmade ofGCD45 toAl6082Mandrecommends the lightweight design of the knuckle as the optimal design technique to be installed in small cars Six shape designvariables were selected for the optimization of the knuckle and the criteria relevant to stiffness and durability were consideredas the design requirements during the optimization process The metamodel-based optimization method that uses the kriginginterpolation method as the optimization technique was applied The result shows that all constraints for stiffness and durabilityare satisfied using A16082M while reducing the weight of the knuckle by 60 compared to that of the existing GCD450
1 Introduction
The linking parts of the steering system and the suspen-sion system of automobiles have a direct impact on theperformance of the vehiclersquos ride durability and steerabilityTherefore the performance of these parts is directly related tothe quality of the vehicle This paper examines the structuraldesign of the knuckle which can adjust the directionalrotation due to its connection to the parts of the suspensionsystem and steering system as well as the wheel Whendesigning the structure of the knuckle it is common toconsider durability and stiffness [1ndash5]
For this purpose the strength of the knuckle underthe vehiclersquos service loads is calculated and the durabilityis examined Under the same load condition the strain ordeformation is also calculated and it is verified that this valueis within the allowable value If the design requirement forstiffness is not satisfied it can be considered that the qualitytargets such as ride and steerability are not satisfied In thisstudy the finite element analysis was used to examine thevehiclersquos performance of stiffness and durability
Because the knuckle arm has a greater weight comparedto the outer tie rod inner tie rod control arm and ball jointit can have a greater impact in reducing the overall weight
of the steering parts compared to the other parts To lessenthe weight of the knuckle a material less dense than steel isused and the design methodology is applied In this studyaluminum was used as an alternate material to steel and thestructural optimization was applied as a design technique
The shape of the knuckle arm is complex compared tothe other parts Additionally casting and forging are mainlyused in the production process of the knuckle arm Thusthe knuckle is modeled as a solid element in the finiteelement analysis corresponding to the shape optimizationin the structural optimization category The iteration can bestopped due to the mesh distortion during the optimizationprocess in the structural optimization with complex shape Inthis study the metamodel-based optimization technique wasapplied to solve this problem The optimization techniqueusing the metamodel is suitable for the following designproblems (1) the long and extensive calculations required bythe analysis (2) the difficulty to mathematically define theshape parameter due to the presence of the complex surfacesor (3) the severe or broken distortion of the finite elementduring the process of optimization [3 4]
The following is a recent research trend related to theknuckle Triantafyllidis et al revealed that the fracture mech-anismof the knuckle ismainly due to bending fatigue through
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 763692 7 pageshttpdxdoiorg1011552014763692
2 The Scientific World Journal
Ball joint
Wheel center
Caliper
Strut
OTR
Knuckle arm
Figure 1 Structure of steering parts
scanning electronmicroscopy (SEM) images [1] DrsquoIppolito etal suggested that the response surfacemodel for the knucklersquosfatigue life was built and then an optimization scheme wasintroduced considering structural reliability [2] In [3] thefatigue life of the knuckle was predicted and the reliabilitywas calculated by considering uncertainties Vijayarangan etal suggested that a new material made of aluminum alloyis reinforced with titanium carbide particulate to replacespheroidal graphite iron [4] The aforementioned papers arethe studies that mostly dealt with the development of a newmaterial or the reliability of fatigue analysis In addition Parket al [5] have proposed the shape optimization that appliesthe design of experiments using the orthogonal array to thestructural design of the knuckle However this method hassevere limitations in selecting the optimum design variablesamong very limited values
In this study the lightweight design method that can beapplied was suggested at the early stage in the developmentof the knuckle First the base design was completed and sixshape design variables were defined During this process theshapes of the parts such as the strut OTR and spring werefixed Al6082M which was developed in the existing Refer-ence [6] was used as the material of the knuckle In additionthe kriging interpolation method [7ndash10] was applied as themetamodel technique MSCNastran and MSCFatigue wereused for durability analysis and Abaqus was used for stiffnessanalysis
2 Initial Finite Element Analysis of a Knuckle
The knucklersquos shape designed at the early stage of develop-ment is shown in Figure 1Thematerial used was GCD450 atype of spherical graphite cast iron The finite element modelconsisted of the tetrahedron element was created by usingHypermesh The number of the nodes of the finite elementmodel is 67128 and the number of the elements is 38837Theouter tie rod strut caliper ball joint and hard point of thewheel center which are the peripheral parts of the knuckle
Fixd dofs
Fixd dofs
Fixd dofs
Fixd dofs
Fixd dofs
Figure 2 FE model of knuckle
0
200
400
600
800
01 03
True
stre
ss (M
Pa)
Plastic strain
Figure 3 Plastic strain and true stress curve
Table 1 Material properties of GCD450
Property ValueElasticity (MPa) 170000Density (gcm3) 79Poissonrsquos ratio 029Tensile strength (MPa) 450Yield strength (MPa) 370
were modeled such that they were connected by a rigid barwith the joint part of the knuckle This modeling is shownin Figure 2The material properties of GCD450 are shown inTable 1 and Figure 3
21 Stiffness Analysis The load that is delivered from the roadsurface to a car during the operation is transferred to theparts of the suspension and steering systems through the tireand wheel and this load affects the stiffness and strength ofthe vehicle If the stiffness of each part is reduced then itinduces the excessive deformation that has negative effecton the ride handling and NVH performance Thus eachautomobilemaker sets its own allowable value for the amountof deformation on each car model In this study the loadingcondition and the design criterion that CompanyA uses were
The Scientific World Journal 3
Force and moment
ForceForce
Force and m
ForForce
Figure 4 FE model for stiffness analysis of forward-braking case
Table 2 Loading condition and case for stiffness
Number Loading condition Number of load case1 Braking 22 Pothole 63 Wheel bump 24 Cornering 2Total 12
applied The number of entire loading cases is 12 and theequivalent plastic strain for each number is calculated andchecked that it is within the allowable value
The equivalent plastic strain 120576pl is defined by the follow-ing equation [11]
120576pl = 120576pl100381610038161003816100381610038160
+ int119905
0
radic2
3120576pl 120576pl119889119905 (1)
where the initial equivalent plastic strain which is the firstterm of the right-hand side was set to 0
As the boundary condition for the stiffness analysis alldegrees of freedom of the wheel center which is the centerpart of the knuckle hole were constrained Twelve load casesfor the stiffness analysis can be determined by consideringthe ultimate load that can be received when the car operatesThese load cases are shown in Table 2 A force in each loadingcase is acted on the hard point of outer tie rod strut andball joint Additionally the moment is acted on the strutThefinite element model for the forward-braking load among theloading cases for the stiffness analysis is shown in Figure 4
The severe result of the stiffness analysis of the knucklearm came from within the Pothole loading case The maxi-mum 120576pl occurred at the connection area with the strut andits value is 00034 This is shown in Figure 5 The calculatedmaximum 120576pl value fully satisfies the acceptance criterion
22 Durability Analysis Automobile pats can fail by fatigue ifrepeated loading is appliedTherefore it is essential to reviewthe durability of an entire car unit or parts unit when a newcar is developed In this study the fatigue life caused by the
120576pl = 00034
Figure 5 Stiffness analysis result at pothole loading condition
repeated loads acting on the knuckle parts was calculatedand it is concluded that this value was less than the allowablevalue
Stress-lifemethod and strain-lifemethod are themethodsthat calculate the fatigue lifeThe stress-lifemethod is suitableonly when stress and strain exist in the elastic area On theother hand the strain-lifemethod is suitable for the problemsthat occur due to stress concentration causing plastic strainIn this study the strain-life method was used to predict thefatigue life of the knuckle [6 12]
The loading case for determining the fatigue life of theknuckle arm is 23 loading cases as suggested by the CompanyA These loading cases are divided into the nonbrakingloading condition and the braking loading condition Theboundary condition of the nonbraking loading condition isthe same as the case of the stiffness analysis On the otherhand in the case of the braking loading condition all degreesof freedom except for the rotational degree of freedom of 119884direction in the wheel center of the knuckle arm were fixedAlso the element between the caliper and wheel center wasmodeled as a springThe FEmodel of the nonbraking loadingcondition and the applied load as well as the FE analysisresults of this particular case are shown in Figure 6 As theresult analyzing the durability by considering all loadingcases the minimum lifetime was calculated to be 635000cycles This lifetime exists within the allowable value
3 Shape Optimization of a KnuckleUsing Krigng Metamodel
The results of the stiffness and durability analyses of theknuckle made of material GCD450 show that both resultssatisfy the criteria and have the marginal safety of about 10times In this study the material of the knuckle was replacedwith GCD450 fromA16082M and the lightweight design wasimplemented by applying themetamodel-based optimizationusing the krigingmodelThe initial design of the optimization
4 The Scientific World Journal
Force and moment
Force
Force
Moment
Force and mo
F
Force
Moment
(a) FE model
635000 cycles
(b) Fatigue life
Figure 6 FE analysis for durability analysis of nonbraking condition
is the shape of a knuckle made of GCD450 material Alsoduring the optimization process the most vulnerable loadingcase for the stiffness analysis and the durability analysis wasincluded on each one The material property of A16082M isthe same as shown in Table 3
31 Shape Design Variables and Formulation The areas thatare themost vulnerable in the stiffness and durability analysesare the joint of the outer tie rod and knuckle and the joint ofthe strut and knuckle Design variables 119905
1and 1199052were defined
respectively to include the joint between the outer tie rod andknuckle to the shape design variables and design variables1199053 1199054 1199055 and 119905
6were defined respectively to include the
joint between the strut and knuckle to the variables Thesevariables are shown in Figure 7
The formulation for the structural optimization of aknuckle is expressed as follows
Minimize 119882(t)
Subject to 120576pl (t) le 120576pl119886
SF (t) ge SF119886
t119871le t le t
119880
(2)
where 119882(t) means the weight of a knuckle and 120576pl(t) meansthe allowable equivalent plastic strain and SF(t) means thesafety factor for fatigue life and SF
119886means the allowable
safety factor In addition t = [1199051 1199052 1199053 1199054 1199055 1199056] is the
design variable vector and t119871and t
119880are the lower and
upper bounds of the design variable vectors respectivelythat are set as t
119871= [18 16 20 20 20 20]mm and t
119880=
[28 24 32 32 32 32]mmThe lower and upper bounds of each design variable
are established by considering the quality of the geometricalshape and the mesh for the finite element analysis of aknuckle The first inequality equation in (2) is the constraint
Table 3 Material properties of Al6082M
Property ValueElasticity (MPa) 72000Density (gcm3) 271Poissonrsquos ratio 029Tensile strength (MPa) 380Yield strength (MPa) 340
t1
t2
t3t4
t5t6
Figure 7 Definition of shape design variables
function related to the stiffness and the second inequalityequation is the constraint function for fatigue life In orderto solve (2) by using the metamodel-based optimizationmethod the surrogate model for119882 120576pl and SF in (2) shouldbe generated Then because (2) was expressed as simpleexpressions of design variables obtaining the optimum
The Scientific World Journal 5
design becomes a very easy process In this study119882 120576pl andSF were approximated as the kriging metamodel
32 Optimization Using Kriging Interpolation Method Forglobal optimization the kriging interpolation method isintroduced Kriging is an interpolation method named aftera South African mining engineer named D G Krige whodeveloped the method while trying to increase the accuracyin predicting the ore reserves Kriging interpolation forapproximation model is well explained in [6ndash12] In generalthe response function 119891(t) is represented as
119891 (t) = 120573 + 119911 (t) (3)
where 120573 is a constant and 119911(t) is the realization of astochastic process with mean zero and variance 120590
2 followingthe Gaussian distribution
Ifand
119891(t) is designated as the approximation model and the
mean squared errors of119891(t) andand
119891(t) areminimized to satisfythe unbiased condition 119891(t) can be estimated as
and
119891(t) =and
120573 + r119879 (t)Rminus1 (y minusand
120573 i) (4)
where R is the correlation matrix r is the correlation vectory is the observed data and i is the unit vector
In this research 119882 120576pl and SF are considered as 119891(t)respectively Correlation matrix and correlation vector aredefined as
R (t119895 t119896) = Exp[minus
119899
sum119894=1
120579119894
10038161003816100381610038161003816119905119894
119895minus 119905119894
11989610038161003816100381610038161003816
2
]
(119895 = 1 119899s 119896 = 1 119899s)
(5)
r (t) = [119877 (t t(1)) 119877 (t t(2)) 119877 (t t(119899119904))]119879
(6)
The unknown parameters 1205791 1205792 120579
119899are obtained by max-
imizing the following equation
minus
[119899119904ln(and
1205902) + ln |R|]
2
(7)
where 120579119894(119894 = 1 2 119899) gt 0
To assess the kriging model a fewmetrics can be utilizedIn this study the CV called the cross validation is used TheCV is defined as
CV = radic1
119899119904
119899119904
sum119894=1
(119891119894minusand
119891minus119894)
2
(8)
whereand
119891minus119894is the 119894th estimator of kriging model constructed
without the 119894th observationThe optimization process applied in this study is shown
in Figure 8 First the base design of a knuckle was completedthrough CATIA and then the finite element modeling was
CAD and FE modelings
FE analysis (stiffness durability)
Kriging model building VisualDOC and FORTRAN programs
Optimization
DOE using LHD FE analysis (stiffness and durability)
Figure 8 Flowchart of the suggested design process
performed It was verified that each criterion was met aftercarrying out the stiffness and durability analyses for the initialmodel of steelmaterialThe optimizationwas performed afterchanging the material into aluminum in order to reduce theweight of the knuckle First the shape design variables weredefined and then each design variable from the CAD modelwas parameterized The Latin hypercube design using thecommandof ldquolhsdesignrdquo built-in inMATLABwas used as thesampling method At this time the sampling points n
119904were
30 This number is to be empirically determined from pre-vious studies [6 7 12] Then for each experiment point thefinite element analysis for the stiffness and durability analysesis performed Each approximated function forW 120576pl and SFis generated by using the kriging interpolation method basedon the results of finite element analysis The optimizationproblem of (2) was solved by the algorithm of the methodof modified feasible direction built in VisualDOC
33 Optimization Results Table 4 shows the results of thefinite element analysis of the stiffness analysis and durabilityanalysis for the sampling point generated by the ldquolhsdesignrdquocommand The responses of W 120576pl and SF with respect toeach sampling point are listed The kriging model for W120576pl and SF were generated based on those values and theparameter values for each kriging model are summarized inTable 5 including the CV value of (8) for each response Inaddition the optimum design obtained from (2) is shown inTable 6 where W 120576pl and SF values are the values predictedfrom the kriging model The results of the confirmationanalysis through the finite element analysis for the optimalsolution were calculated as 1140 kg for weight 002 for 120576pland 304 for SF If it is assumed that the result of the finiteelement analysis in the optimum design is the true valuethen the kriging predicted values of 119882 120576pl and SF havethe errors of 02 05 and 73 The active constraint oftwo constraint functions for the stiffness and fatigue of (2)is the constraint for stiffness and the constraint for fatiguewhich was confirmed as an inactive constraint In otherwords the constraint function for stiffness played a big rolein determining this optimumdesign andwasmore influentialthan the constraint for fatigue
As shown in Table 6 the constraint function for stiffnessis not satisfied in the initial design Therefore the valueof the weight in the initial design is meaningless In thisstudy a design that satisfies all of the stiffness and durability
6 The Scientific World Journal
Table 4 Design of experiments using LHD
Number Design variable (mm) Response1199051
1199052
1199053
1199054
1199055
1199056
119882 (kg) PEEQ SF1 215 2253 2624 2765 2733 2624 1189 00097 2672 2317 2173 287 2969 2706 2596 1197 00122 3383 2117 22 2596 2663 2569 2405 1176 00100 1944 2783 2387 2897 2527 2542 2487 1196 00095 2945 225 1613 2487 2595 2214 235 1147 00215 3096 2017 1773 2815 2323 2924 2268 1167 00184 2827 2417 1933 3006 2459 2432 287 1175 00100 3188 2283 1827 2569 2697 2678 2432 1173 00357 2679 2517 2093 2432 2901 287 2569 1197 00089 28210 205 1747 2924 2357 3006 2651 1175 00335 31611 2617 2333 276 2935 2815 2788 1211 00149 30912 245 1987 2405 2153 2487 2842 1165 00141 29913 2383 1667 2979 2391 2842 3006 118 00272 32314 2083 236 2678 2561 2241 2678 117 00250 31815 185 1853 2241 3003 2378 2706 1161 00120 2516 1983 164 2952 2833 276 2296 1171 00187 29117 2717 1693 2296 2255 2624 2924 1169 00242 30418 1917 2013 246 2799 2897 246 1179 00198 28219 1817 2147 2378 2867 2514 2979 1174 00351 30420 2683 172 2514 2051 2979 2542 1174 00242 31321 1883 2067 2214 2119 2296 2241 1139 00228 28922 255 212 2268 2493 235 2897 1173 00155 31823 2483 228 2788 2629 2952 2733 1203 00103 27424 2217 18 2542 2085 246 276 1153 00367 29425 2183 196 2733 2731 2323 2323 1165 00091 27526 195 204 2706 2289 2405 2214 1153 00206 29927 265 2307 2842 2017 2596 2815 1183 00357 30928 235 188 235 2425 2268 2952 1159 00257 33329 275 1907 2323 2187 2788 2378 1174 00136 28930 2583 2227 2651 2221 2651 2514 1181 00249 299
Table 5 Optimum parameter and validation of kriging model
Responses Optimum parameterlowast Validation120573lowast
1205791
lowast1205792
lowast1205793
lowast1205794
lowast1205795
lowast1205796
lowast CV119882 11749 44769 79588 00341 02377 22620 51316 0014PEEQ 00170 20487 19495 03696 00483 02500 02140 0084SF 28378 20705 28048 00006 57071 00021 00003 3129
requirements of a knuckle with the minimum weight isproposed using an optimization process The final shapedetermined through this study is shown in Figure 9 Thoughthis study focused on the specific car the proposed designprocess can be applicable for all kinds of knuckle
4 Conclusion and Future Work
In this study of a lightweight design of the knuckle mountedto a small car the material was changed from steel to alu-minum and the metamodel-based optimization was appliedThe results are as follows
(1) When aluminum was first adapted to the existingdesign the constraint function for stiffness was not met Inthe proposed design the shape of the knuckle was redesignedand the optimal minimum weight was calculated achievinga weight reduction of about 60 (as compared to theinitial design) without sacrificing the stiffness and durabilityrequirements The weight reduction of the knuckle can makea direct contribution to improved fuel efficiency and reducedemissions
(2) It could be seen that the approximated optimizationthat uses kriging for the lightweight design of a knuckle isvery effective for the shape optimization that is difficult to
The Scientific World Journal 7
Table 6 Responses at initial and optimum designs
Design Deign variable (mm) Response1199051
1199052
1199053
1199054
1199055
1199056
119882 (kg) 120576pl SFInitial (steel) 180 220 230 240 215 275 34 00034 127Initial (Al) 180 220 230 240 215 275 1160 0025 31Optimum 193 206 200 218 206 262 1142 00199 28
Figure 9 Suggested optimum design of a knuckle
implement in existing commercial software In addition thevalidation of the kriging model was carried out through thecross validation and CV index and the predicted values ofthe kriging model for the weight equivalent plastic strainand safety factor of fatigue life had no significant differencescompared to the results from the finite element analysis Forfuture work the forging of the optimum shape of the knuckleproposed is scheduled to be reviewed
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was financially supported by the Ministryof Education Science and Technology (MEST) and theNational Research Foundation of Korea (NRF) through theHuman Resource Training Project for Regional Innovation(2012H1B8A2026078)
References
[1] G K Triantafyllidis A Antonopoulos A Spiliotis S Fedonosand D Repanis ldquoFracture characteristics of fatigue failure ofa vehicles ductile iron steering knucklerdquo Journal of FailureAnalysis and Prevention vol 9 no 4 pp 323ndash328 2009
[2] R dIppolitoMHack S Donders L Hermans N Tzannetakisand D Vandepitte ldquoImproving the fatigue life of a vehicle
knuckle with a reliability-based design optimization approachrdquoJournal of Statistical Planning and Inference vol 139 no 5 pp1619ndash1632 2009
[3] E A Azrulhisham Y M Asri A W Dzuraidah N M NikAbdullah A Shahrum and C H Che Hassan ldquoEvaluationof fatigue life reliability of steering knuckle using pearsonparametric distributionmodelrdquo International Journal ofQualityStatistics and Reliability vol 2010 Article ID 816407 8 pages2010
[4] S Vijayarangan N Rajamanickam and V Sivananth ldquoEvalua-tion of metal matrix composite to replace spheroidal graphiteiron for a critical component steering knucklerdquo Materials ampDesign vol 43 pp 532ndash541 2013
[5] Y C Park K H Lee D H Lee and K Y Lee ldquoShapeoptimization design of the knuckle using orthogonal array andthe finite element analysisrdquo Transactions of the Korean Society ofAutomotive Engineers vol 11 pp 138ndash144 2003 (Korean)
[6] B-C Song Y-C Park S-W Kang and K-H Lee ldquoStructuraloptimization of an upper control arm considering the strengthrdquoProceedings of the Institution of Mechanical Engineers D Journalof Automobile Engineering vol 223 no 6 pp 727ndash735 2009
[7] X G Song J H Jung H J Son J H Park K H Lee andY C Park ldquoMetamodel-based optimization of a control armconsidering strength and durability performancerdquo Computersand Mathematics with Applications vol 60 no 4 pp 976ndash9802010
[8] J SacksW JWelch T J Mitchell andH PWynn ldquoDesign andanalysis of computer experimentsrdquo Statistical Science vol 4 pp409ndash435 1989
[9] A Guinta and L Watson ldquoA comparison of approximationmodeling techniques polynomial versus interpolating modelsrdquoin Proceedings of the 7th AIAAUSAFNASAISSMO Symp onMultid Anal and Optim (AIAA rsquo98) vol 2 pp 392ndash440 1998
[10] K T Fang R Li and A Sudjianto Design and Modeling forComputer Experiments Chapman amp HallCRC 2006
[11] Simulia ldquoAbaqus 610 analysis userrsquos manual materialsrdquo vol 3[12] J K Kim Y J Kim W H Yang Y C Park and K-H Lee
ldquoStructural design of an outer tie rod for a passenger carrdquoInternational Journal of Automotive Technology vol 12 no 3pp 375ndash381 2011
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2 The Scientific World Journal
Ball joint
Wheel center
Caliper
Strut
OTR
Knuckle arm
Figure 1 Structure of steering parts
scanning electronmicroscopy (SEM) images [1] DrsquoIppolito etal suggested that the response surfacemodel for the knucklersquosfatigue life was built and then an optimization scheme wasintroduced considering structural reliability [2] In [3] thefatigue life of the knuckle was predicted and the reliabilitywas calculated by considering uncertainties Vijayarangan etal suggested that a new material made of aluminum alloyis reinforced with titanium carbide particulate to replacespheroidal graphite iron [4] The aforementioned papers arethe studies that mostly dealt with the development of a newmaterial or the reliability of fatigue analysis In addition Parket al [5] have proposed the shape optimization that appliesthe design of experiments using the orthogonal array to thestructural design of the knuckle However this method hassevere limitations in selecting the optimum design variablesamong very limited values
In this study the lightweight design method that can beapplied was suggested at the early stage in the developmentof the knuckle First the base design was completed and sixshape design variables were defined During this process theshapes of the parts such as the strut OTR and spring werefixed Al6082M which was developed in the existing Refer-ence [6] was used as the material of the knuckle In additionthe kriging interpolation method [7ndash10] was applied as themetamodel technique MSCNastran and MSCFatigue wereused for durability analysis and Abaqus was used for stiffnessanalysis
2 Initial Finite Element Analysis of a Knuckle
The knucklersquos shape designed at the early stage of develop-ment is shown in Figure 1Thematerial used was GCD450 atype of spherical graphite cast iron The finite element modelconsisted of the tetrahedron element was created by usingHypermesh The number of the nodes of the finite elementmodel is 67128 and the number of the elements is 38837Theouter tie rod strut caliper ball joint and hard point of thewheel center which are the peripheral parts of the knuckle
Fixd dofs
Fixd dofs
Fixd dofs
Fixd dofs
Fixd dofs
Figure 2 FE model of knuckle
0
200
400
600
800
01 03
True
stre
ss (M
Pa)
Plastic strain
Figure 3 Plastic strain and true stress curve
Table 1 Material properties of GCD450
Property ValueElasticity (MPa) 170000Density (gcm3) 79Poissonrsquos ratio 029Tensile strength (MPa) 450Yield strength (MPa) 370
were modeled such that they were connected by a rigid barwith the joint part of the knuckle This modeling is shownin Figure 2The material properties of GCD450 are shown inTable 1 and Figure 3
21 Stiffness Analysis The load that is delivered from the roadsurface to a car during the operation is transferred to theparts of the suspension and steering systems through the tireand wheel and this load affects the stiffness and strength ofthe vehicle If the stiffness of each part is reduced then itinduces the excessive deformation that has negative effecton the ride handling and NVH performance Thus eachautomobilemaker sets its own allowable value for the amountof deformation on each car model In this study the loadingcondition and the design criterion that CompanyA uses were
The Scientific World Journal 3
Force and moment
ForceForce
Force and m
ForForce
Figure 4 FE model for stiffness analysis of forward-braking case
Table 2 Loading condition and case for stiffness
Number Loading condition Number of load case1 Braking 22 Pothole 63 Wheel bump 24 Cornering 2Total 12
applied The number of entire loading cases is 12 and theequivalent plastic strain for each number is calculated andchecked that it is within the allowable value
The equivalent plastic strain 120576pl is defined by the follow-ing equation [11]
120576pl = 120576pl100381610038161003816100381610038160
+ int119905
0
radic2
3120576pl 120576pl119889119905 (1)
where the initial equivalent plastic strain which is the firstterm of the right-hand side was set to 0
As the boundary condition for the stiffness analysis alldegrees of freedom of the wheel center which is the centerpart of the knuckle hole were constrained Twelve load casesfor the stiffness analysis can be determined by consideringthe ultimate load that can be received when the car operatesThese load cases are shown in Table 2 A force in each loadingcase is acted on the hard point of outer tie rod strut andball joint Additionally the moment is acted on the strutThefinite element model for the forward-braking load among theloading cases for the stiffness analysis is shown in Figure 4
The severe result of the stiffness analysis of the knucklearm came from within the Pothole loading case The maxi-mum 120576pl occurred at the connection area with the strut andits value is 00034 This is shown in Figure 5 The calculatedmaximum 120576pl value fully satisfies the acceptance criterion
22 Durability Analysis Automobile pats can fail by fatigue ifrepeated loading is appliedTherefore it is essential to reviewthe durability of an entire car unit or parts unit when a newcar is developed In this study the fatigue life caused by the
120576pl = 00034
Figure 5 Stiffness analysis result at pothole loading condition
repeated loads acting on the knuckle parts was calculatedand it is concluded that this value was less than the allowablevalue
Stress-lifemethod and strain-lifemethod are themethodsthat calculate the fatigue lifeThe stress-lifemethod is suitableonly when stress and strain exist in the elastic area On theother hand the strain-lifemethod is suitable for the problemsthat occur due to stress concentration causing plastic strainIn this study the strain-life method was used to predict thefatigue life of the knuckle [6 12]
The loading case for determining the fatigue life of theknuckle arm is 23 loading cases as suggested by the CompanyA These loading cases are divided into the nonbrakingloading condition and the braking loading condition Theboundary condition of the nonbraking loading condition isthe same as the case of the stiffness analysis On the otherhand in the case of the braking loading condition all degreesof freedom except for the rotational degree of freedom of 119884direction in the wheel center of the knuckle arm were fixedAlso the element between the caliper and wheel center wasmodeled as a springThe FEmodel of the nonbraking loadingcondition and the applied load as well as the FE analysisresults of this particular case are shown in Figure 6 As theresult analyzing the durability by considering all loadingcases the minimum lifetime was calculated to be 635000cycles This lifetime exists within the allowable value
3 Shape Optimization of a KnuckleUsing Krigng Metamodel
The results of the stiffness and durability analyses of theknuckle made of material GCD450 show that both resultssatisfy the criteria and have the marginal safety of about 10times In this study the material of the knuckle was replacedwith GCD450 fromA16082M and the lightweight design wasimplemented by applying themetamodel-based optimizationusing the krigingmodelThe initial design of the optimization
4 The Scientific World Journal
Force and moment
Force
Force
Moment
Force and mo
F
Force
Moment
(a) FE model
635000 cycles
(b) Fatigue life
Figure 6 FE analysis for durability analysis of nonbraking condition
is the shape of a knuckle made of GCD450 material Alsoduring the optimization process the most vulnerable loadingcase for the stiffness analysis and the durability analysis wasincluded on each one The material property of A16082M isthe same as shown in Table 3
31 Shape Design Variables and Formulation The areas thatare themost vulnerable in the stiffness and durability analysesare the joint of the outer tie rod and knuckle and the joint ofthe strut and knuckle Design variables 119905
1and 1199052were defined
respectively to include the joint between the outer tie rod andknuckle to the shape design variables and design variables1199053 1199054 1199055 and 119905
6were defined respectively to include the
joint between the strut and knuckle to the variables Thesevariables are shown in Figure 7
The formulation for the structural optimization of aknuckle is expressed as follows
Minimize 119882(t)
Subject to 120576pl (t) le 120576pl119886
SF (t) ge SF119886
t119871le t le t
119880
(2)
where 119882(t) means the weight of a knuckle and 120576pl(t) meansthe allowable equivalent plastic strain and SF(t) means thesafety factor for fatigue life and SF
119886means the allowable
safety factor In addition t = [1199051 1199052 1199053 1199054 1199055 1199056] is the
design variable vector and t119871and t
119880are the lower and
upper bounds of the design variable vectors respectivelythat are set as t
119871= [18 16 20 20 20 20]mm and t
119880=
[28 24 32 32 32 32]mmThe lower and upper bounds of each design variable
are established by considering the quality of the geometricalshape and the mesh for the finite element analysis of aknuckle The first inequality equation in (2) is the constraint
Table 3 Material properties of Al6082M
Property ValueElasticity (MPa) 72000Density (gcm3) 271Poissonrsquos ratio 029Tensile strength (MPa) 380Yield strength (MPa) 340
t1
t2
t3t4
t5t6
Figure 7 Definition of shape design variables
function related to the stiffness and the second inequalityequation is the constraint function for fatigue life In orderto solve (2) by using the metamodel-based optimizationmethod the surrogate model for119882 120576pl and SF in (2) shouldbe generated Then because (2) was expressed as simpleexpressions of design variables obtaining the optimum
The Scientific World Journal 5
design becomes a very easy process In this study119882 120576pl andSF were approximated as the kriging metamodel
32 Optimization Using Kriging Interpolation Method Forglobal optimization the kriging interpolation method isintroduced Kriging is an interpolation method named aftera South African mining engineer named D G Krige whodeveloped the method while trying to increase the accuracyin predicting the ore reserves Kriging interpolation forapproximation model is well explained in [6ndash12] In generalthe response function 119891(t) is represented as
119891 (t) = 120573 + 119911 (t) (3)
where 120573 is a constant and 119911(t) is the realization of astochastic process with mean zero and variance 120590
2 followingthe Gaussian distribution
Ifand
119891(t) is designated as the approximation model and the
mean squared errors of119891(t) andand
119891(t) areminimized to satisfythe unbiased condition 119891(t) can be estimated as
and
119891(t) =and
120573 + r119879 (t)Rminus1 (y minusand
120573 i) (4)
where R is the correlation matrix r is the correlation vectory is the observed data and i is the unit vector
In this research 119882 120576pl and SF are considered as 119891(t)respectively Correlation matrix and correlation vector aredefined as
R (t119895 t119896) = Exp[minus
119899
sum119894=1
120579119894
10038161003816100381610038161003816119905119894
119895minus 119905119894
11989610038161003816100381610038161003816
2
]
(119895 = 1 119899s 119896 = 1 119899s)
(5)
r (t) = [119877 (t t(1)) 119877 (t t(2)) 119877 (t t(119899119904))]119879
(6)
The unknown parameters 1205791 1205792 120579
119899are obtained by max-
imizing the following equation
minus
[119899119904ln(and
1205902) + ln |R|]
2
(7)
where 120579119894(119894 = 1 2 119899) gt 0
To assess the kriging model a fewmetrics can be utilizedIn this study the CV called the cross validation is used TheCV is defined as
CV = radic1
119899119904
119899119904
sum119894=1
(119891119894minusand
119891minus119894)
2
(8)
whereand
119891minus119894is the 119894th estimator of kriging model constructed
without the 119894th observationThe optimization process applied in this study is shown
in Figure 8 First the base design of a knuckle was completedthrough CATIA and then the finite element modeling was
CAD and FE modelings
FE analysis (stiffness durability)
Kriging model building VisualDOC and FORTRAN programs
Optimization
DOE using LHD FE analysis (stiffness and durability)
Figure 8 Flowchart of the suggested design process
performed It was verified that each criterion was met aftercarrying out the stiffness and durability analyses for the initialmodel of steelmaterialThe optimizationwas performed afterchanging the material into aluminum in order to reduce theweight of the knuckle First the shape design variables weredefined and then each design variable from the CAD modelwas parameterized The Latin hypercube design using thecommandof ldquolhsdesignrdquo built-in inMATLABwas used as thesampling method At this time the sampling points n
119904were
30 This number is to be empirically determined from pre-vious studies [6 7 12] Then for each experiment point thefinite element analysis for the stiffness and durability analysesis performed Each approximated function forW 120576pl and SFis generated by using the kriging interpolation method basedon the results of finite element analysis The optimizationproblem of (2) was solved by the algorithm of the methodof modified feasible direction built in VisualDOC
33 Optimization Results Table 4 shows the results of thefinite element analysis of the stiffness analysis and durabilityanalysis for the sampling point generated by the ldquolhsdesignrdquocommand The responses of W 120576pl and SF with respect toeach sampling point are listed The kriging model for W120576pl and SF were generated based on those values and theparameter values for each kriging model are summarized inTable 5 including the CV value of (8) for each response Inaddition the optimum design obtained from (2) is shown inTable 6 where W 120576pl and SF values are the values predictedfrom the kriging model The results of the confirmationanalysis through the finite element analysis for the optimalsolution were calculated as 1140 kg for weight 002 for 120576pland 304 for SF If it is assumed that the result of the finiteelement analysis in the optimum design is the true valuethen the kriging predicted values of 119882 120576pl and SF havethe errors of 02 05 and 73 The active constraint oftwo constraint functions for the stiffness and fatigue of (2)is the constraint for stiffness and the constraint for fatiguewhich was confirmed as an inactive constraint In otherwords the constraint function for stiffness played a big rolein determining this optimumdesign andwasmore influentialthan the constraint for fatigue
As shown in Table 6 the constraint function for stiffnessis not satisfied in the initial design Therefore the valueof the weight in the initial design is meaningless In thisstudy a design that satisfies all of the stiffness and durability
6 The Scientific World Journal
Table 4 Design of experiments using LHD
Number Design variable (mm) Response1199051
1199052
1199053
1199054
1199055
1199056
119882 (kg) PEEQ SF1 215 2253 2624 2765 2733 2624 1189 00097 2672 2317 2173 287 2969 2706 2596 1197 00122 3383 2117 22 2596 2663 2569 2405 1176 00100 1944 2783 2387 2897 2527 2542 2487 1196 00095 2945 225 1613 2487 2595 2214 235 1147 00215 3096 2017 1773 2815 2323 2924 2268 1167 00184 2827 2417 1933 3006 2459 2432 287 1175 00100 3188 2283 1827 2569 2697 2678 2432 1173 00357 2679 2517 2093 2432 2901 287 2569 1197 00089 28210 205 1747 2924 2357 3006 2651 1175 00335 31611 2617 2333 276 2935 2815 2788 1211 00149 30912 245 1987 2405 2153 2487 2842 1165 00141 29913 2383 1667 2979 2391 2842 3006 118 00272 32314 2083 236 2678 2561 2241 2678 117 00250 31815 185 1853 2241 3003 2378 2706 1161 00120 2516 1983 164 2952 2833 276 2296 1171 00187 29117 2717 1693 2296 2255 2624 2924 1169 00242 30418 1917 2013 246 2799 2897 246 1179 00198 28219 1817 2147 2378 2867 2514 2979 1174 00351 30420 2683 172 2514 2051 2979 2542 1174 00242 31321 1883 2067 2214 2119 2296 2241 1139 00228 28922 255 212 2268 2493 235 2897 1173 00155 31823 2483 228 2788 2629 2952 2733 1203 00103 27424 2217 18 2542 2085 246 276 1153 00367 29425 2183 196 2733 2731 2323 2323 1165 00091 27526 195 204 2706 2289 2405 2214 1153 00206 29927 265 2307 2842 2017 2596 2815 1183 00357 30928 235 188 235 2425 2268 2952 1159 00257 33329 275 1907 2323 2187 2788 2378 1174 00136 28930 2583 2227 2651 2221 2651 2514 1181 00249 299
Table 5 Optimum parameter and validation of kriging model
Responses Optimum parameterlowast Validation120573lowast
1205791
lowast1205792
lowast1205793
lowast1205794
lowast1205795
lowast1205796
lowast CV119882 11749 44769 79588 00341 02377 22620 51316 0014PEEQ 00170 20487 19495 03696 00483 02500 02140 0084SF 28378 20705 28048 00006 57071 00021 00003 3129
requirements of a knuckle with the minimum weight isproposed using an optimization process The final shapedetermined through this study is shown in Figure 9 Thoughthis study focused on the specific car the proposed designprocess can be applicable for all kinds of knuckle
4 Conclusion and Future Work
In this study of a lightweight design of the knuckle mountedto a small car the material was changed from steel to alu-minum and the metamodel-based optimization was appliedThe results are as follows
(1) When aluminum was first adapted to the existingdesign the constraint function for stiffness was not met Inthe proposed design the shape of the knuckle was redesignedand the optimal minimum weight was calculated achievinga weight reduction of about 60 (as compared to theinitial design) without sacrificing the stiffness and durabilityrequirements The weight reduction of the knuckle can makea direct contribution to improved fuel efficiency and reducedemissions
(2) It could be seen that the approximated optimizationthat uses kriging for the lightweight design of a knuckle isvery effective for the shape optimization that is difficult to
The Scientific World Journal 7
Table 6 Responses at initial and optimum designs
Design Deign variable (mm) Response1199051
1199052
1199053
1199054
1199055
1199056
119882 (kg) 120576pl SFInitial (steel) 180 220 230 240 215 275 34 00034 127Initial (Al) 180 220 230 240 215 275 1160 0025 31Optimum 193 206 200 218 206 262 1142 00199 28
Figure 9 Suggested optimum design of a knuckle
implement in existing commercial software In addition thevalidation of the kriging model was carried out through thecross validation and CV index and the predicted values ofthe kriging model for the weight equivalent plastic strainand safety factor of fatigue life had no significant differencescompared to the results from the finite element analysis Forfuture work the forging of the optimum shape of the knuckleproposed is scheduled to be reviewed
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was financially supported by the Ministryof Education Science and Technology (MEST) and theNational Research Foundation of Korea (NRF) through theHuman Resource Training Project for Regional Innovation(2012H1B8A2026078)
References
[1] G K Triantafyllidis A Antonopoulos A Spiliotis S Fedonosand D Repanis ldquoFracture characteristics of fatigue failure ofa vehicles ductile iron steering knucklerdquo Journal of FailureAnalysis and Prevention vol 9 no 4 pp 323ndash328 2009
[2] R dIppolitoMHack S Donders L Hermans N Tzannetakisand D Vandepitte ldquoImproving the fatigue life of a vehicle
knuckle with a reliability-based design optimization approachrdquoJournal of Statistical Planning and Inference vol 139 no 5 pp1619ndash1632 2009
[3] E A Azrulhisham Y M Asri A W Dzuraidah N M NikAbdullah A Shahrum and C H Che Hassan ldquoEvaluationof fatigue life reliability of steering knuckle using pearsonparametric distributionmodelrdquo International Journal ofQualityStatistics and Reliability vol 2010 Article ID 816407 8 pages2010
[4] S Vijayarangan N Rajamanickam and V Sivananth ldquoEvalua-tion of metal matrix composite to replace spheroidal graphiteiron for a critical component steering knucklerdquo Materials ampDesign vol 43 pp 532ndash541 2013
[5] Y C Park K H Lee D H Lee and K Y Lee ldquoShapeoptimization design of the knuckle using orthogonal array andthe finite element analysisrdquo Transactions of the Korean Society ofAutomotive Engineers vol 11 pp 138ndash144 2003 (Korean)
[6] B-C Song Y-C Park S-W Kang and K-H Lee ldquoStructuraloptimization of an upper control arm considering the strengthrdquoProceedings of the Institution of Mechanical Engineers D Journalof Automobile Engineering vol 223 no 6 pp 727ndash735 2009
[7] X G Song J H Jung H J Son J H Park K H Lee andY C Park ldquoMetamodel-based optimization of a control armconsidering strength and durability performancerdquo Computersand Mathematics with Applications vol 60 no 4 pp 976ndash9802010
[8] J SacksW JWelch T J Mitchell andH PWynn ldquoDesign andanalysis of computer experimentsrdquo Statistical Science vol 4 pp409ndash435 1989
[9] A Guinta and L Watson ldquoA comparison of approximationmodeling techniques polynomial versus interpolating modelsrdquoin Proceedings of the 7th AIAAUSAFNASAISSMO Symp onMultid Anal and Optim (AIAA rsquo98) vol 2 pp 392ndash440 1998
[10] K T Fang R Li and A Sudjianto Design and Modeling forComputer Experiments Chapman amp HallCRC 2006
[11] Simulia ldquoAbaqus 610 analysis userrsquos manual materialsrdquo vol 3[12] J K Kim Y J Kim W H Yang Y C Park and K-H Lee
ldquoStructural design of an outer tie rod for a passenger carrdquoInternational Journal of Automotive Technology vol 12 no 3pp 375ndash381 2011
Submit your manuscripts athttpwwwhindawicom
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The Scientific World Journal 3
Force and moment
ForceForce
Force and m
ForForce
Figure 4 FE model for stiffness analysis of forward-braking case
Table 2 Loading condition and case for stiffness
Number Loading condition Number of load case1 Braking 22 Pothole 63 Wheel bump 24 Cornering 2Total 12
applied The number of entire loading cases is 12 and theequivalent plastic strain for each number is calculated andchecked that it is within the allowable value
The equivalent plastic strain 120576pl is defined by the follow-ing equation [11]
120576pl = 120576pl100381610038161003816100381610038160
+ int119905
0
radic2
3120576pl 120576pl119889119905 (1)
where the initial equivalent plastic strain which is the firstterm of the right-hand side was set to 0
As the boundary condition for the stiffness analysis alldegrees of freedom of the wheel center which is the centerpart of the knuckle hole were constrained Twelve load casesfor the stiffness analysis can be determined by consideringthe ultimate load that can be received when the car operatesThese load cases are shown in Table 2 A force in each loadingcase is acted on the hard point of outer tie rod strut andball joint Additionally the moment is acted on the strutThefinite element model for the forward-braking load among theloading cases for the stiffness analysis is shown in Figure 4
The severe result of the stiffness analysis of the knucklearm came from within the Pothole loading case The maxi-mum 120576pl occurred at the connection area with the strut andits value is 00034 This is shown in Figure 5 The calculatedmaximum 120576pl value fully satisfies the acceptance criterion
22 Durability Analysis Automobile pats can fail by fatigue ifrepeated loading is appliedTherefore it is essential to reviewthe durability of an entire car unit or parts unit when a newcar is developed In this study the fatigue life caused by the
120576pl = 00034
Figure 5 Stiffness analysis result at pothole loading condition
repeated loads acting on the knuckle parts was calculatedand it is concluded that this value was less than the allowablevalue
Stress-lifemethod and strain-lifemethod are themethodsthat calculate the fatigue lifeThe stress-lifemethod is suitableonly when stress and strain exist in the elastic area On theother hand the strain-lifemethod is suitable for the problemsthat occur due to stress concentration causing plastic strainIn this study the strain-life method was used to predict thefatigue life of the knuckle [6 12]
The loading case for determining the fatigue life of theknuckle arm is 23 loading cases as suggested by the CompanyA These loading cases are divided into the nonbrakingloading condition and the braking loading condition Theboundary condition of the nonbraking loading condition isthe same as the case of the stiffness analysis On the otherhand in the case of the braking loading condition all degreesof freedom except for the rotational degree of freedom of 119884direction in the wheel center of the knuckle arm were fixedAlso the element between the caliper and wheel center wasmodeled as a springThe FEmodel of the nonbraking loadingcondition and the applied load as well as the FE analysisresults of this particular case are shown in Figure 6 As theresult analyzing the durability by considering all loadingcases the minimum lifetime was calculated to be 635000cycles This lifetime exists within the allowable value
3 Shape Optimization of a KnuckleUsing Krigng Metamodel
The results of the stiffness and durability analyses of theknuckle made of material GCD450 show that both resultssatisfy the criteria and have the marginal safety of about 10times In this study the material of the knuckle was replacedwith GCD450 fromA16082M and the lightweight design wasimplemented by applying themetamodel-based optimizationusing the krigingmodelThe initial design of the optimization
4 The Scientific World Journal
Force and moment
Force
Force
Moment
Force and mo
F
Force
Moment
(a) FE model
635000 cycles
(b) Fatigue life
Figure 6 FE analysis for durability analysis of nonbraking condition
is the shape of a knuckle made of GCD450 material Alsoduring the optimization process the most vulnerable loadingcase for the stiffness analysis and the durability analysis wasincluded on each one The material property of A16082M isthe same as shown in Table 3
31 Shape Design Variables and Formulation The areas thatare themost vulnerable in the stiffness and durability analysesare the joint of the outer tie rod and knuckle and the joint ofthe strut and knuckle Design variables 119905
1and 1199052were defined
respectively to include the joint between the outer tie rod andknuckle to the shape design variables and design variables1199053 1199054 1199055 and 119905
6were defined respectively to include the
joint between the strut and knuckle to the variables Thesevariables are shown in Figure 7
The formulation for the structural optimization of aknuckle is expressed as follows
Minimize 119882(t)
Subject to 120576pl (t) le 120576pl119886
SF (t) ge SF119886
t119871le t le t
119880
(2)
where 119882(t) means the weight of a knuckle and 120576pl(t) meansthe allowable equivalent plastic strain and SF(t) means thesafety factor for fatigue life and SF
119886means the allowable
safety factor In addition t = [1199051 1199052 1199053 1199054 1199055 1199056] is the
design variable vector and t119871and t
119880are the lower and
upper bounds of the design variable vectors respectivelythat are set as t
119871= [18 16 20 20 20 20]mm and t
119880=
[28 24 32 32 32 32]mmThe lower and upper bounds of each design variable
are established by considering the quality of the geometricalshape and the mesh for the finite element analysis of aknuckle The first inequality equation in (2) is the constraint
Table 3 Material properties of Al6082M
Property ValueElasticity (MPa) 72000Density (gcm3) 271Poissonrsquos ratio 029Tensile strength (MPa) 380Yield strength (MPa) 340
t1
t2
t3t4
t5t6
Figure 7 Definition of shape design variables
function related to the stiffness and the second inequalityequation is the constraint function for fatigue life In orderto solve (2) by using the metamodel-based optimizationmethod the surrogate model for119882 120576pl and SF in (2) shouldbe generated Then because (2) was expressed as simpleexpressions of design variables obtaining the optimum
The Scientific World Journal 5
design becomes a very easy process In this study119882 120576pl andSF were approximated as the kriging metamodel
32 Optimization Using Kriging Interpolation Method Forglobal optimization the kriging interpolation method isintroduced Kriging is an interpolation method named aftera South African mining engineer named D G Krige whodeveloped the method while trying to increase the accuracyin predicting the ore reserves Kriging interpolation forapproximation model is well explained in [6ndash12] In generalthe response function 119891(t) is represented as
119891 (t) = 120573 + 119911 (t) (3)
where 120573 is a constant and 119911(t) is the realization of astochastic process with mean zero and variance 120590
2 followingthe Gaussian distribution
Ifand
119891(t) is designated as the approximation model and the
mean squared errors of119891(t) andand
119891(t) areminimized to satisfythe unbiased condition 119891(t) can be estimated as
and
119891(t) =and
120573 + r119879 (t)Rminus1 (y minusand
120573 i) (4)
where R is the correlation matrix r is the correlation vectory is the observed data and i is the unit vector
In this research 119882 120576pl and SF are considered as 119891(t)respectively Correlation matrix and correlation vector aredefined as
R (t119895 t119896) = Exp[minus
119899
sum119894=1
120579119894
10038161003816100381610038161003816119905119894
119895minus 119905119894
11989610038161003816100381610038161003816
2
]
(119895 = 1 119899s 119896 = 1 119899s)
(5)
r (t) = [119877 (t t(1)) 119877 (t t(2)) 119877 (t t(119899119904))]119879
(6)
The unknown parameters 1205791 1205792 120579
119899are obtained by max-
imizing the following equation
minus
[119899119904ln(and
1205902) + ln |R|]
2
(7)
where 120579119894(119894 = 1 2 119899) gt 0
To assess the kriging model a fewmetrics can be utilizedIn this study the CV called the cross validation is used TheCV is defined as
CV = radic1
119899119904
119899119904
sum119894=1
(119891119894minusand
119891minus119894)
2
(8)
whereand
119891minus119894is the 119894th estimator of kriging model constructed
without the 119894th observationThe optimization process applied in this study is shown
in Figure 8 First the base design of a knuckle was completedthrough CATIA and then the finite element modeling was
CAD and FE modelings
FE analysis (stiffness durability)
Kriging model building VisualDOC and FORTRAN programs
Optimization
DOE using LHD FE analysis (stiffness and durability)
Figure 8 Flowchart of the suggested design process
performed It was verified that each criterion was met aftercarrying out the stiffness and durability analyses for the initialmodel of steelmaterialThe optimizationwas performed afterchanging the material into aluminum in order to reduce theweight of the knuckle First the shape design variables weredefined and then each design variable from the CAD modelwas parameterized The Latin hypercube design using thecommandof ldquolhsdesignrdquo built-in inMATLABwas used as thesampling method At this time the sampling points n
119904were
30 This number is to be empirically determined from pre-vious studies [6 7 12] Then for each experiment point thefinite element analysis for the stiffness and durability analysesis performed Each approximated function forW 120576pl and SFis generated by using the kriging interpolation method basedon the results of finite element analysis The optimizationproblem of (2) was solved by the algorithm of the methodof modified feasible direction built in VisualDOC
33 Optimization Results Table 4 shows the results of thefinite element analysis of the stiffness analysis and durabilityanalysis for the sampling point generated by the ldquolhsdesignrdquocommand The responses of W 120576pl and SF with respect toeach sampling point are listed The kriging model for W120576pl and SF were generated based on those values and theparameter values for each kriging model are summarized inTable 5 including the CV value of (8) for each response Inaddition the optimum design obtained from (2) is shown inTable 6 where W 120576pl and SF values are the values predictedfrom the kriging model The results of the confirmationanalysis through the finite element analysis for the optimalsolution were calculated as 1140 kg for weight 002 for 120576pland 304 for SF If it is assumed that the result of the finiteelement analysis in the optimum design is the true valuethen the kriging predicted values of 119882 120576pl and SF havethe errors of 02 05 and 73 The active constraint oftwo constraint functions for the stiffness and fatigue of (2)is the constraint for stiffness and the constraint for fatiguewhich was confirmed as an inactive constraint In otherwords the constraint function for stiffness played a big rolein determining this optimumdesign andwasmore influentialthan the constraint for fatigue
As shown in Table 6 the constraint function for stiffnessis not satisfied in the initial design Therefore the valueof the weight in the initial design is meaningless In thisstudy a design that satisfies all of the stiffness and durability
6 The Scientific World Journal
Table 4 Design of experiments using LHD
Number Design variable (mm) Response1199051
1199052
1199053
1199054
1199055
1199056
119882 (kg) PEEQ SF1 215 2253 2624 2765 2733 2624 1189 00097 2672 2317 2173 287 2969 2706 2596 1197 00122 3383 2117 22 2596 2663 2569 2405 1176 00100 1944 2783 2387 2897 2527 2542 2487 1196 00095 2945 225 1613 2487 2595 2214 235 1147 00215 3096 2017 1773 2815 2323 2924 2268 1167 00184 2827 2417 1933 3006 2459 2432 287 1175 00100 3188 2283 1827 2569 2697 2678 2432 1173 00357 2679 2517 2093 2432 2901 287 2569 1197 00089 28210 205 1747 2924 2357 3006 2651 1175 00335 31611 2617 2333 276 2935 2815 2788 1211 00149 30912 245 1987 2405 2153 2487 2842 1165 00141 29913 2383 1667 2979 2391 2842 3006 118 00272 32314 2083 236 2678 2561 2241 2678 117 00250 31815 185 1853 2241 3003 2378 2706 1161 00120 2516 1983 164 2952 2833 276 2296 1171 00187 29117 2717 1693 2296 2255 2624 2924 1169 00242 30418 1917 2013 246 2799 2897 246 1179 00198 28219 1817 2147 2378 2867 2514 2979 1174 00351 30420 2683 172 2514 2051 2979 2542 1174 00242 31321 1883 2067 2214 2119 2296 2241 1139 00228 28922 255 212 2268 2493 235 2897 1173 00155 31823 2483 228 2788 2629 2952 2733 1203 00103 27424 2217 18 2542 2085 246 276 1153 00367 29425 2183 196 2733 2731 2323 2323 1165 00091 27526 195 204 2706 2289 2405 2214 1153 00206 29927 265 2307 2842 2017 2596 2815 1183 00357 30928 235 188 235 2425 2268 2952 1159 00257 33329 275 1907 2323 2187 2788 2378 1174 00136 28930 2583 2227 2651 2221 2651 2514 1181 00249 299
Table 5 Optimum parameter and validation of kriging model
Responses Optimum parameterlowast Validation120573lowast
1205791
lowast1205792
lowast1205793
lowast1205794
lowast1205795
lowast1205796
lowast CV119882 11749 44769 79588 00341 02377 22620 51316 0014PEEQ 00170 20487 19495 03696 00483 02500 02140 0084SF 28378 20705 28048 00006 57071 00021 00003 3129
requirements of a knuckle with the minimum weight isproposed using an optimization process The final shapedetermined through this study is shown in Figure 9 Thoughthis study focused on the specific car the proposed designprocess can be applicable for all kinds of knuckle
4 Conclusion and Future Work
In this study of a lightweight design of the knuckle mountedto a small car the material was changed from steel to alu-minum and the metamodel-based optimization was appliedThe results are as follows
(1) When aluminum was first adapted to the existingdesign the constraint function for stiffness was not met Inthe proposed design the shape of the knuckle was redesignedand the optimal minimum weight was calculated achievinga weight reduction of about 60 (as compared to theinitial design) without sacrificing the stiffness and durabilityrequirements The weight reduction of the knuckle can makea direct contribution to improved fuel efficiency and reducedemissions
(2) It could be seen that the approximated optimizationthat uses kriging for the lightweight design of a knuckle isvery effective for the shape optimization that is difficult to
The Scientific World Journal 7
Table 6 Responses at initial and optimum designs
Design Deign variable (mm) Response1199051
1199052
1199053
1199054
1199055
1199056
119882 (kg) 120576pl SFInitial (steel) 180 220 230 240 215 275 34 00034 127Initial (Al) 180 220 230 240 215 275 1160 0025 31Optimum 193 206 200 218 206 262 1142 00199 28
Figure 9 Suggested optimum design of a knuckle
implement in existing commercial software In addition thevalidation of the kriging model was carried out through thecross validation and CV index and the predicted values ofthe kriging model for the weight equivalent plastic strainand safety factor of fatigue life had no significant differencescompared to the results from the finite element analysis Forfuture work the forging of the optimum shape of the knuckleproposed is scheduled to be reviewed
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was financially supported by the Ministryof Education Science and Technology (MEST) and theNational Research Foundation of Korea (NRF) through theHuman Resource Training Project for Regional Innovation(2012H1B8A2026078)
References
[1] G K Triantafyllidis A Antonopoulos A Spiliotis S Fedonosand D Repanis ldquoFracture characteristics of fatigue failure ofa vehicles ductile iron steering knucklerdquo Journal of FailureAnalysis and Prevention vol 9 no 4 pp 323ndash328 2009
[2] R dIppolitoMHack S Donders L Hermans N Tzannetakisand D Vandepitte ldquoImproving the fatigue life of a vehicle
knuckle with a reliability-based design optimization approachrdquoJournal of Statistical Planning and Inference vol 139 no 5 pp1619ndash1632 2009
[3] E A Azrulhisham Y M Asri A W Dzuraidah N M NikAbdullah A Shahrum and C H Che Hassan ldquoEvaluationof fatigue life reliability of steering knuckle using pearsonparametric distributionmodelrdquo International Journal ofQualityStatistics and Reliability vol 2010 Article ID 816407 8 pages2010
[4] S Vijayarangan N Rajamanickam and V Sivananth ldquoEvalua-tion of metal matrix composite to replace spheroidal graphiteiron for a critical component steering knucklerdquo Materials ampDesign vol 43 pp 532ndash541 2013
[5] Y C Park K H Lee D H Lee and K Y Lee ldquoShapeoptimization design of the knuckle using orthogonal array andthe finite element analysisrdquo Transactions of the Korean Society ofAutomotive Engineers vol 11 pp 138ndash144 2003 (Korean)
[6] B-C Song Y-C Park S-W Kang and K-H Lee ldquoStructuraloptimization of an upper control arm considering the strengthrdquoProceedings of the Institution of Mechanical Engineers D Journalof Automobile Engineering vol 223 no 6 pp 727ndash735 2009
[7] X G Song J H Jung H J Son J H Park K H Lee andY C Park ldquoMetamodel-based optimization of a control armconsidering strength and durability performancerdquo Computersand Mathematics with Applications vol 60 no 4 pp 976ndash9802010
[8] J SacksW JWelch T J Mitchell andH PWynn ldquoDesign andanalysis of computer experimentsrdquo Statistical Science vol 4 pp409ndash435 1989
[9] A Guinta and L Watson ldquoA comparison of approximationmodeling techniques polynomial versus interpolating modelsrdquoin Proceedings of the 7th AIAAUSAFNASAISSMO Symp onMultid Anal and Optim (AIAA rsquo98) vol 2 pp 392ndash440 1998
[10] K T Fang R Li and A Sudjianto Design and Modeling forComputer Experiments Chapman amp HallCRC 2006
[11] Simulia ldquoAbaqus 610 analysis userrsquos manual materialsrdquo vol 3[12] J K Kim Y J Kim W H Yang Y C Park and K-H Lee
ldquoStructural design of an outer tie rod for a passenger carrdquoInternational Journal of Automotive Technology vol 12 no 3pp 375ndash381 2011
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
4 The Scientific World Journal
Force and moment
Force
Force
Moment
Force and mo
F
Force
Moment
(a) FE model
635000 cycles
(b) Fatigue life
Figure 6 FE analysis for durability analysis of nonbraking condition
is the shape of a knuckle made of GCD450 material Alsoduring the optimization process the most vulnerable loadingcase for the stiffness analysis and the durability analysis wasincluded on each one The material property of A16082M isthe same as shown in Table 3
31 Shape Design Variables and Formulation The areas thatare themost vulnerable in the stiffness and durability analysesare the joint of the outer tie rod and knuckle and the joint ofthe strut and knuckle Design variables 119905
1and 1199052were defined
respectively to include the joint between the outer tie rod andknuckle to the shape design variables and design variables1199053 1199054 1199055 and 119905
6were defined respectively to include the
joint between the strut and knuckle to the variables Thesevariables are shown in Figure 7
The formulation for the structural optimization of aknuckle is expressed as follows
Minimize 119882(t)
Subject to 120576pl (t) le 120576pl119886
SF (t) ge SF119886
t119871le t le t
119880
(2)
where 119882(t) means the weight of a knuckle and 120576pl(t) meansthe allowable equivalent plastic strain and SF(t) means thesafety factor for fatigue life and SF
119886means the allowable
safety factor In addition t = [1199051 1199052 1199053 1199054 1199055 1199056] is the
design variable vector and t119871and t
119880are the lower and
upper bounds of the design variable vectors respectivelythat are set as t
119871= [18 16 20 20 20 20]mm and t
119880=
[28 24 32 32 32 32]mmThe lower and upper bounds of each design variable
are established by considering the quality of the geometricalshape and the mesh for the finite element analysis of aknuckle The first inequality equation in (2) is the constraint
Table 3 Material properties of Al6082M
Property ValueElasticity (MPa) 72000Density (gcm3) 271Poissonrsquos ratio 029Tensile strength (MPa) 380Yield strength (MPa) 340
t1
t2
t3t4
t5t6
Figure 7 Definition of shape design variables
function related to the stiffness and the second inequalityequation is the constraint function for fatigue life In orderto solve (2) by using the metamodel-based optimizationmethod the surrogate model for119882 120576pl and SF in (2) shouldbe generated Then because (2) was expressed as simpleexpressions of design variables obtaining the optimum
The Scientific World Journal 5
design becomes a very easy process In this study119882 120576pl andSF were approximated as the kriging metamodel
32 Optimization Using Kriging Interpolation Method Forglobal optimization the kriging interpolation method isintroduced Kriging is an interpolation method named aftera South African mining engineer named D G Krige whodeveloped the method while trying to increase the accuracyin predicting the ore reserves Kriging interpolation forapproximation model is well explained in [6ndash12] In generalthe response function 119891(t) is represented as
119891 (t) = 120573 + 119911 (t) (3)
where 120573 is a constant and 119911(t) is the realization of astochastic process with mean zero and variance 120590
2 followingthe Gaussian distribution
Ifand
119891(t) is designated as the approximation model and the
mean squared errors of119891(t) andand
119891(t) areminimized to satisfythe unbiased condition 119891(t) can be estimated as
and
119891(t) =and
120573 + r119879 (t)Rminus1 (y minusand
120573 i) (4)
where R is the correlation matrix r is the correlation vectory is the observed data and i is the unit vector
In this research 119882 120576pl and SF are considered as 119891(t)respectively Correlation matrix and correlation vector aredefined as
R (t119895 t119896) = Exp[minus
119899
sum119894=1
120579119894
10038161003816100381610038161003816119905119894
119895minus 119905119894
11989610038161003816100381610038161003816
2
]
(119895 = 1 119899s 119896 = 1 119899s)
(5)
r (t) = [119877 (t t(1)) 119877 (t t(2)) 119877 (t t(119899119904))]119879
(6)
The unknown parameters 1205791 1205792 120579
119899are obtained by max-
imizing the following equation
minus
[119899119904ln(and
1205902) + ln |R|]
2
(7)
where 120579119894(119894 = 1 2 119899) gt 0
To assess the kriging model a fewmetrics can be utilizedIn this study the CV called the cross validation is used TheCV is defined as
CV = radic1
119899119904
119899119904
sum119894=1
(119891119894minusand
119891minus119894)
2
(8)
whereand
119891minus119894is the 119894th estimator of kriging model constructed
without the 119894th observationThe optimization process applied in this study is shown
in Figure 8 First the base design of a knuckle was completedthrough CATIA and then the finite element modeling was
CAD and FE modelings
FE analysis (stiffness durability)
Kriging model building VisualDOC and FORTRAN programs
Optimization
DOE using LHD FE analysis (stiffness and durability)
Figure 8 Flowchart of the suggested design process
performed It was verified that each criterion was met aftercarrying out the stiffness and durability analyses for the initialmodel of steelmaterialThe optimizationwas performed afterchanging the material into aluminum in order to reduce theweight of the knuckle First the shape design variables weredefined and then each design variable from the CAD modelwas parameterized The Latin hypercube design using thecommandof ldquolhsdesignrdquo built-in inMATLABwas used as thesampling method At this time the sampling points n
119904were
30 This number is to be empirically determined from pre-vious studies [6 7 12] Then for each experiment point thefinite element analysis for the stiffness and durability analysesis performed Each approximated function forW 120576pl and SFis generated by using the kriging interpolation method basedon the results of finite element analysis The optimizationproblem of (2) was solved by the algorithm of the methodof modified feasible direction built in VisualDOC
33 Optimization Results Table 4 shows the results of thefinite element analysis of the stiffness analysis and durabilityanalysis for the sampling point generated by the ldquolhsdesignrdquocommand The responses of W 120576pl and SF with respect toeach sampling point are listed The kriging model for W120576pl and SF were generated based on those values and theparameter values for each kriging model are summarized inTable 5 including the CV value of (8) for each response Inaddition the optimum design obtained from (2) is shown inTable 6 where W 120576pl and SF values are the values predictedfrom the kriging model The results of the confirmationanalysis through the finite element analysis for the optimalsolution were calculated as 1140 kg for weight 002 for 120576pland 304 for SF If it is assumed that the result of the finiteelement analysis in the optimum design is the true valuethen the kriging predicted values of 119882 120576pl and SF havethe errors of 02 05 and 73 The active constraint oftwo constraint functions for the stiffness and fatigue of (2)is the constraint for stiffness and the constraint for fatiguewhich was confirmed as an inactive constraint In otherwords the constraint function for stiffness played a big rolein determining this optimumdesign andwasmore influentialthan the constraint for fatigue
As shown in Table 6 the constraint function for stiffnessis not satisfied in the initial design Therefore the valueof the weight in the initial design is meaningless In thisstudy a design that satisfies all of the stiffness and durability
6 The Scientific World Journal
Table 4 Design of experiments using LHD
Number Design variable (mm) Response1199051
1199052
1199053
1199054
1199055
1199056
119882 (kg) PEEQ SF1 215 2253 2624 2765 2733 2624 1189 00097 2672 2317 2173 287 2969 2706 2596 1197 00122 3383 2117 22 2596 2663 2569 2405 1176 00100 1944 2783 2387 2897 2527 2542 2487 1196 00095 2945 225 1613 2487 2595 2214 235 1147 00215 3096 2017 1773 2815 2323 2924 2268 1167 00184 2827 2417 1933 3006 2459 2432 287 1175 00100 3188 2283 1827 2569 2697 2678 2432 1173 00357 2679 2517 2093 2432 2901 287 2569 1197 00089 28210 205 1747 2924 2357 3006 2651 1175 00335 31611 2617 2333 276 2935 2815 2788 1211 00149 30912 245 1987 2405 2153 2487 2842 1165 00141 29913 2383 1667 2979 2391 2842 3006 118 00272 32314 2083 236 2678 2561 2241 2678 117 00250 31815 185 1853 2241 3003 2378 2706 1161 00120 2516 1983 164 2952 2833 276 2296 1171 00187 29117 2717 1693 2296 2255 2624 2924 1169 00242 30418 1917 2013 246 2799 2897 246 1179 00198 28219 1817 2147 2378 2867 2514 2979 1174 00351 30420 2683 172 2514 2051 2979 2542 1174 00242 31321 1883 2067 2214 2119 2296 2241 1139 00228 28922 255 212 2268 2493 235 2897 1173 00155 31823 2483 228 2788 2629 2952 2733 1203 00103 27424 2217 18 2542 2085 246 276 1153 00367 29425 2183 196 2733 2731 2323 2323 1165 00091 27526 195 204 2706 2289 2405 2214 1153 00206 29927 265 2307 2842 2017 2596 2815 1183 00357 30928 235 188 235 2425 2268 2952 1159 00257 33329 275 1907 2323 2187 2788 2378 1174 00136 28930 2583 2227 2651 2221 2651 2514 1181 00249 299
Table 5 Optimum parameter and validation of kriging model
Responses Optimum parameterlowast Validation120573lowast
1205791
lowast1205792
lowast1205793
lowast1205794
lowast1205795
lowast1205796
lowast CV119882 11749 44769 79588 00341 02377 22620 51316 0014PEEQ 00170 20487 19495 03696 00483 02500 02140 0084SF 28378 20705 28048 00006 57071 00021 00003 3129
requirements of a knuckle with the minimum weight isproposed using an optimization process The final shapedetermined through this study is shown in Figure 9 Thoughthis study focused on the specific car the proposed designprocess can be applicable for all kinds of knuckle
4 Conclusion and Future Work
In this study of a lightweight design of the knuckle mountedto a small car the material was changed from steel to alu-minum and the metamodel-based optimization was appliedThe results are as follows
(1) When aluminum was first adapted to the existingdesign the constraint function for stiffness was not met Inthe proposed design the shape of the knuckle was redesignedand the optimal minimum weight was calculated achievinga weight reduction of about 60 (as compared to theinitial design) without sacrificing the stiffness and durabilityrequirements The weight reduction of the knuckle can makea direct contribution to improved fuel efficiency and reducedemissions
(2) It could be seen that the approximated optimizationthat uses kriging for the lightweight design of a knuckle isvery effective for the shape optimization that is difficult to
The Scientific World Journal 7
Table 6 Responses at initial and optimum designs
Design Deign variable (mm) Response1199051
1199052
1199053
1199054
1199055
1199056
119882 (kg) 120576pl SFInitial (steel) 180 220 230 240 215 275 34 00034 127Initial (Al) 180 220 230 240 215 275 1160 0025 31Optimum 193 206 200 218 206 262 1142 00199 28
Figure 9 Suggested optimum design of a knuckle
implement in existing commercial software In addition thevalidation of the kriging model was carried out through thecross validation and CV index and the predicted values ofthe kriging model for the weight equivalent plastic strainand safety factor of fatigue life had no significant differencescompared to the results from the finite element analysis Forfuture work the forging of the optimum shape of the knuckleproposed is scheduled to be reviewed
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was financially supported by the Ministryof Education Science and Technology (MEST) and theNational Research Foundation of Korea (NRF) through theHuman Resource Training Project for Regional Innovation(2012H1B8A2026078)
References
[1] G K Triantafyllidis A Antonopoulos A Spiliotis S Fedonosand D Repanis ldquoFracture characteristics of fatigue failure ofa vehicles ductile iron steering knucklerdquo Journal of FailureAnalysis and Prevention vol 9 no 4 pp 323ndash328 2009
[2] R dIppolitoMHack S Donders L Hermans N Tzannetakisand D Vandepitte ldquoImproving the fatigue life of a vehicle
knuckle with a reliability-based design optimization approachrdquoJournal of Statistical Planning and Inference vol 139 no 5 pp1619ndash1632 2009
[3] E A Azrulhisham Y M Asri A W Dzuraidah N M NikAbdullah A Shahrum and C H Che Hassan ldquoEvaluationof fatigue life reliability of steering knuckle using pearsonparametric distributionmodelrdquo International Journal ofQualityStatistics and Reliability vol 2010 Article ID 816407 8 pages2010
[4] S Vijayarangan N Rajamanickam and V Sivananth ldquoEvalua-tion of metal matrix composite to replace spheroidal graphiteiron for a critical component steering knucklerdquo Materials ampDesign vol 43 pp 532ndash541 2013
[5] Y C Park K H Lee D H Lee and K Y Lee ldquoShapeoptimization design of the knuckle using orthogonal array andthe finite element analysisrdquo Transactions of the Korean Society ofAutomotive Engineers vol 11 pp 138ndash144 2003 (Korean)
[6] B-C Song Y-C Park S-W Kang and K-H Lee ldquoStructuraloptimization of an upper control arm considering the strengthrdquoProceedings of the Institution of Mechanical Engineers D Journalof Automobile Engineering vol 223 no 6 pp 727ndash735 2009
[7] X G Song J H Jung H J Son J H Park K H Lee andY C Park ldquoMetamodel-based optimization of a control armconsidering strength and durability performancerdquo Computersand Mathematics with Applications vol 60 no 4 pp 976ndash9802010
[8] J SacksW JWelch T J Mitchell andH PWynn ldquoDesign andanalysis of computer experimentsrdquo Statistical Science vol 4 pp409ndash435 1989
[9] A Guinta and L Watson ldquoA comparison of approximationmodeling techniques polynomial versus interpolating modelsrdquoin Proceedings of the 7th AIAAUSAFNASAISSMO Symp onMultid Anal and Optim (AIAA rsquo98) vol 2 pp 392ndash440 1998
[10] K T Fang R Li and A Sudjianto Design and Modeling forComputer Experiments Chapman amp HallCRC 2006
[11] Simulia ldquoAbaqus 610 analysis userrsquos manual materialsrdquo vol 3[12] J K Kim Y J Kim W H Yang Y C Park and K-H Lee
ldquoStructural design of an outer tie rod for a passenger carrdquoInternational Journal of Automotive Technology vol 12 no 3pp 375ndash381 2011
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World Journal 5
design becomes a very easy process In this study119882 120576pl andSF were approximated as the kriging metamodel
32 Optimization Using Kriging Interpolation Method Forglobal optimization the kriging interpolation method isintroduced Kriging is an interpolation method named aftera South African mining engineer named D G Krige whodeveloped the method while trying to increase the accuracyin predicting the ore reserves Kriging interpolation forapproximation model is well explained in [6ndash12] In generalthe response function 119891(t) is represented as
119891 (t) = 120573 + 119911 (t) (3)
where 120573 is a constant and 119911(t) is the realization of astochastic process with mean zero and variance 120590
2 followingthe Gaussian distribution
Ifand
119891(t) is designated as the approximation model and the
mean squared errors of119891(t) andand
119891(t) areminimized to satisfythe unbiased condition 119891(t) can be estimated as
and
119891(t) =and
120573 + r119879 (t)Rminus1 (y minusand
120573 i) (4)
where R is the correlation matrix r is the correlation vectory is the observed data and i is the unit vector
In this research 119882 120576pl and SF are considered as 119891(t)respectively Correlation matrix and correlation vector aredefined as
R (t119895 t119896) = Exp[minus
119899
sum119894=1
120579119894
10038161003816100381610038161003816119905119894
119895minus 119905119894
11989610038161003816100381610038161003816
2
]
(119895 = 1 119899s 119896 = 1 119899s)
(5)
r (t) = [119877 (t t(1)) 119877 (t t(2)) 119877 (t t(119899119904))]119879
(6)
The unknown parameters 1205791 1205792 120579
119899are obtained by max-
imizing the following equation
minus
[119899119904ln(and
1205902) + ln |R|]
2
(7)
where 120579119894(119894 = 1 2 119899) gt 0
To assess the kriging model a fewmetrics can be utilizedIn this study the CV called the cross validation is used TheCV is defined as
CV = radic1
119899119904
119899119904
sum119894=1
(119891119894minusand
119891minus119894)
2
(8)
whereand
119891minus119894is the 119894th estimator of kriging model constructed
without the 119894th observationThe optimization process applied in this study is shown
in Figure 8 First the base design of a knuckle was completedthrough CATIA and then the finite element modeling was
CAD and FE modelings
FE analysis (stiffness durability)
Kriging model building VisualDOC and FORTRAN programs
Optimization
DOE using LHD FE analysis (stiffness and durability)
Figure 8 Flowchart of the suggested design process
performed It was verified that each criterion was met aftercarrying out the stiffness and durability analyses for the initialmodel of steelmaterialThe optimizationwas performed afterchanging the material into aluminum in order to reduce theweight of the knuckle First the shape design variables weredefined and then each design variable from the CAD modelwas parameterized The Latin hypercube design using thecommandof ldquolhsdesignrdquo built-in inMATLABwas used as thesampling method At this time the sampling points n
119904were
30 This number is to be empirically determined from pre-vious studies [6 7 12] Then for each experiment point thefinite element analysis for the stiffness and durability analysesis performed Each approximated function forW 120576pl and SFis generated by using the kriging interpolation method basedon the results of finite element analysis The optimizationproblem of (2) was solved by the algorithm of the methodof modified feasible direction built in VisualDOC
33 Optimization Results Table 4 shows the results of thefinite element analysis of the stiffness analysis and durabilityanalysis for the sampling point generated by the ldquolhsdesignrdquocommand The responses of W 120576pl and SF with respect toeach sampling point are listed The kriging model for W120576pl and SF were generated based on those values and theparameter values for each kriging model are summarized inTable 5 including the CV value of (8) for each response Inaddition the optimum design obtained from (2) is shown inTable 6 where W 120576pl and SF values are the values predictedfrom the kriging model The results of the confirmationanalysis through the finite element analysis for the optimalsolution were calculated as 1140 kg for weight 002 for 120576pland 304 for SF If it is assumed that the result of the finiteelement analysis in the optimum design is the true valuethen the kriging predicted values of 119882 120576pl and SF havethe errors of 02 05 and 73 The active constraint oftwo constraint functions for the stiffness and fatigue of (2)is the constraint for stiffness and the constraint for fatiguewhich was confirmed as an inactive constraint In otherwords the constraint function for stiffness played a big rolein determining this optimumdesign andwasmore influentialthan the constraint for fatigue
As shown in Table 6 the constraint function for stiffnessis not satisfied in the initial design Therefore the valueof the weight in the initial design is meaningless In thisstudy a design that satisfies all of the stiffness and durability
6 The Scientific World Journal
Table 4 Design of experiments using LHD
Number Design variable (mm) Response1199051
1199052
1199053
1199054
1199055
1199056
119882 (kg) PEEQ SF1 215 2253 2624 2765 2733 2624 1189 00097 2672 2317 2173 287 2969 2706 2596 1197 00122 3383 2117 22 2596 2663 2569 2405 1176 00100 1944 2783 2387 2897 2527 2542 2487 1196 00095 2945 225 1613 2487 2595 2214 235 1147 00215 3096 2017 1773 2815 2323 2924 2268 1167 00184 2827 2417 1933 3006 2459 2432 287 1175 00100 3188 2283 1827 2569 2697 2678 2432 1173 00357 2679 2517 2093 2432 2901 287 2569 1197 00089 28210 205 1747 2924 2357 3006 2651 1175 00335 31611 2617 2333 276 2935 2815 2788 1211 00149 30912 245 1987 2405 2153 2487 2842 1165 00141 29913 2383 1667 2979 2391 2842 3006 118 00272 32314 2083 236 2678 2561 2241 2678 117 00250 31815 185 1853 2241 3003 2378 2706 1161 00120 2516 1983 164 2952 2833 276 2296 1171 00187 29117 2717 1693 2296 2255 2624 2924 1169 00242 30418 1917 2013 246 2799 2897 246 1179 00198 28219 1817 2147 2378 2867 2514 2979 1174 00351 30420 2683 172 2514 2051 2979 2542 1174 00242 31321 1883 2067 2214 2119 2296 2241 1139 00228 28922 255 212 2268 2493 235 2897 1173 00155 31823 2483 228 2788 2629 2952 2733 1203 00103 27424 2217 18 2542 2085 246 276 1153 00367 29425 2183 196 2733 2731 2323 2323 1165 00091 27526 195 204 2706 2289 2405 2214 1153 00206 29927 265 2307 2842 2017 2596 2815 1183 00357 30928 235 188 235 2425 2268 2952 1159 00257 33329 275 1907 2323 2187 2788 2378 1174 00136 28930 2583 2227 2651 2221 2651 2514 1181 00249 299
Table 5 Optimum parameter and validation of kriging model
Responses Optimum parameterlowast Validation120573lowast
1205791
lowast1205792
lowast1205793
lowast1205794
lowast1205795
lowast1205796
lowast CV119882 11749 44769 79588 00341 02377 22620 51316 0014PEEQ 00170 20487 19495 03696 00483 02500 02140 0084SF 28378 20705 28048 00006 57071 00021 00003 3129
requirements of a knuckle with the minimum weight isproposed using an optimization process The final shapedetermined through this study is shown in Figure 9 Thoughthis study focused on the specific car the proposed designprocess can be applicable for all kinds of knuckle
4 Conclusion and Future Work
In this study of a lightweight design of the knuckle mountedto a small car the material was changed from steel to alu-minum and the metamodel-based optimization was appliedThe results are as follows
(1) When aluminum was first adapted to the existingdesign the constraint function for stiffness was not met Inthe proposed design the shape of the knuckle was redesignedand the optimal minimum weight was calculated achievinga weight reduction of about 60 (as compared to theinitial design) without sacrificing the stiffness and durabilityrequirements The weight reduction of the knuckle can makea direct contribution to improved fuel efficiency and reducedemissions
(2) It could be seen that the approximated optimizationthat uses kriging for the lightweight design of a knuckle isvery effective for the shape optimization that is difficult to
The Scientific World Journal 7
Table 6 Responses at initial and optimum designs
Design Deign variable (mm) Response1199051
1199052
1199053
1199054
1199055
1199056
119882 (kg) 120576pl SFInitial (steel) 180 220 230 240 215 275 34 00034 127Initial (Al) 180 220 230 240 215 275 1160 0025 31Optimum 193 206 200 218 206 262 1142 00199 28
Figure 9 Suggested optimum design of a knuckle
implement in existing commercial software In addition thevalidation of the kriging model was carried out through thecross validation and CV index and the predicted values ofthe kriging model for the weight equivalent plastic strainand safety factor of fatigue life had no significant differencescompared to the results from the finite element analysis Forfuture work the forging of the optimum shape of the knuckleproposed is scheduled to be reviewed
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was financially supported by the Ministryof Education Science and Technology (MEST) and theNational Research Foundation of Korea (NRF) through theHuman Resource Training Project for Regional Innovation(2012H1B8A2026078)
References
[1] G K Triantafyllidis A Antonopoulos A Spiliotis S Fedonosand D Repanis ldquoFracture characteristics of fatigue failure ofa vehicles ductile iron steering knucklerdquo Journal of FailureAnalysis and Prevention vol 9 no 4 pp 323ndash328 2009
[2] R dIppolitoMHack S Donders L Hermans N Tzannetakisand D Vandepitte ldquoImproving the fatigue life of a vehicle
knuckle with a reliability-based design optimization approachrdquoJournal of Statistical Planning and Inference vol 139 no 5 pp1619ndash1632 2009
[3] E A Azrulhisham Y M Asri A W Dzuraidah N M NikAbdullah A Shahrum and C H Che Hassan ldquoEvaluationof fatigue life reliability of steering knuckle using pearsonparametric distributionmodelrdquo International Journal ofQualityStatistics and Reliability vol 2010 Article ID 816407 8 pages2010
[4] S Vijayarangan N Rajamanickam and V Sivananth ldquoEvalua-tion of metal matrix composite to replace spheroidal graphiteiron for a critical component steering knucklerdquo Materials ampDesign vol 43 pp 532ndash541 2013
[5] Y C Park K H Lee D H Lee and K Y Lee ldquoShapeoptimization design of the knuckle using orthogonal array andthe finite element analysisrdquo Transactions of the Korean Society ofAutomotive Engineers vol 11 pp 138ndash144 2003 (Korean)
[6] B-C Song Y-C Park S-W Kang and K-H Lee ldquoStructuraloptimization of an upper control arm considering the strengthrdquoProceedings of the Institution of Mechanical Engineers D Journalof Automobile Engineering vol 223 no 6 pp 727ndash735 2009
[7] X G Song J H Jung H J Son J H Park K H Lee andY C Park ldquoMetamodel-based optimization of a control armconsidering strength and durability performancerdquo Computersand Mathematics with Applications vol 60 no 4 pp 976ndash9802010
[8] J SacksW JWelch T J Mitchell andH PWynn ldquoDesign andanalysis of computer experimentsrdquo Statistical Science vol 4 pp409ndash435 1989
[9] A Guinta and L Watson ldquoA comparison of approximationmodeling techniques polynomial versus interpolating modelsrdquoin Proceedings of the 7th AIAAUSAFNASAISSMO Symp onMultid Anal and Optim (AIAA rsquo98) vol 2 pp 392ndash440 1998
[10] K T Fang R Li and A Sudjianto Design and Modeling forComputer Experiments Chapman amp HallCRC 2006
[11] Simulia ldquoAbaqus 610 analysis userrsquos manual materialsrdquo vol 3[12] J K Kim Y J Kim W H Yang Y C Park and K-H Lee
ldquoStructural design of an outer tie rod for a passenger carrdquoInternational Journal of Automotive Technology vol 12 no 3pp 375ndash381 2011
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
6 The Scientific World Journal
Table 4 Design of experiments using LHD
Number Design variable (mm) Response1199051
1199052
1199053
1199054
1199055
1199056
119882 (kg) PEEQ SF1 215 2253 2624 2765 2733 2624 1189 00097 2672 2317 2173 287 2969 2706 2596 1197 00122 3383 2117 22 2596 2663 2569 2405 1176 00100 1944 2783 2387 2897 2527 2542 2487 1196 00095 2945 225 1613 2487 2595 2214 235 1147 00215 3096 2017 1773 2815 2323 2924 2268 1167 00184 2827 2417 1933 3006 2459 2432 287 1175 00100 3188 2283 1827 2569 2697 2678 2432 1173 00357 2679 2517 2093 2432 2901 287 2569 1197 00089 28210 205 1747 2924 2357 3006 2651 1175 00335 31611 2617 2333 276 2935 2815 2788 1211 00149 30912 245 1987 2405 2153 2487 2842 1165 00141 29913 2383 1667 2979 2391 2842 3006 118 00272 32314 2083 236 2678 2561 2241 2678 117 00250 31815 185 1853 2241 3003 2378 2706 1161 00120 2516 1983 164 2952 2833 276 2296 1171 00187 29117 2717 1693 2296 2255 2624 2924 1169 00242 30418 1917 2013 246 2799 2897 246 1179 00198 28219 1817 2147 2378 2867 2514 2979 1174 00351 30420 2683 172 2514 2051 2979 2542 1174 00242 31321 1883 2067 2214 2119 2296 2241 1139 00228 28922 255 212 2268 2493 235 2897 1173 00155 31823 2483 228 2788 2629 2952 2733 1203 00103 27424 2217 18 2542 2085 246 276 1153 00367 29425 2183 196 2733 2731 2323 2323 1165 00091 27526 195 204 2706 2289 2405 2214 1153 00206 29927 265 2307 2842 2017 2596 2815 1183 00357 30928 235 188 235 2425 2268 2952 1159 00257 33329 275 1907 2323 2187 2788 2378 1174 00136 28930 2583 2227 2651 2221 2651 2514 1181 00249 299
Table 5 Optimum parameter and validation of kriging model
Responses Optimum parameterlowast Validation120573lowast
1205791
lowast1205792
lowast1205793
lowast1205794
lowast1205795
lowast1205796
lowast CV119882 11749 44769 79588 00341 02377 22620 51316 0014PEEQ 00170 20487 19495 03696 00483 02500 02140 0084SF 28378 20705 28048 00006 57071 00021 00003 3129
requirements of a knuckle with the minimum weight isproposed using an optimization process The final shapedetermined through this study is shown in Figure 9 Thoughthis study focused on the specific car the proposed designprocess can be applicable for all kinds of knuckle
4 Conclusion and Future Work
In this study of a lightweight design of the knuckle mountedto a small car the material was changed from steel to alu-minum and the metamodel-based optimization was appliedThe results are as follows
(1) When aluminum was first adapted to the existingdesign the constraint function for stiffness was not met Inthe proposed design the shape of the knuckle was redesignedand the optimal minimum weight was calculated achievinga weight reduction of about 60 (as compared to theinitial design) without sacrificing the stiffness and durabilityrequirements The weight reduction of the knuckle can makea direct contribution to improved fuel efficiency and reducedemissions
(2) It could be seen that the approximated optimizationthat uses kriging for the lightweight design of a knuckle isvery effective for the shape optimization that is difficult to
The Scientific World Journal 7
Table 6 Responses at initial and optimum designs
Design Deign variable (mm) Response1199051
1199052
1199053
1199054
1199055
1199056
119882 (kg) 120576pl SFInitial (steel) 180 220 230 240 215 275 34 00034 127Initial (Al) 180 220 230 240 215 275 1160 0025 31Optimum 193 206 200 218 206 262 1142 00199 28
Figure 9 Suggested optimum design of a knuckle
implement in existing commercial software In addition thevalidation of the kriging model was carried out through thecross validation and CV index and the predicted values ofthe kriging model for the weight equivalent plastic strainand safety factor of fatigue life had no significant differencescompared to the results from the finite element analysis Forfuture work the forging of the optimum shape of the knuckleproposed is scheduled to be reviewed
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was financially supported by the Ministryof Education Science and Technology (MEST) and theNational Research Foundation of Korea (NRF) through theHuman Resource Training Project for Regional Innovation(2012H1B8A2026078)
References
[1] G K Triantafyllidis A Antonopoulos A Spiliotis S Fedonosand D Repanis ldquoFracture characteristics of fatigue failure ofa vehicles ductile iron steering knucklerdquo Journal of FailureAnalysis and Prevention vol 9 no 4 pp 323ndash328 2009
[2] R dIppolitoMHack S Donders L Hermans N Tzannetakisand D Vandepitte ldquoImproving the fatigue life of a vehicle
knuckle with a reliability-based design optimization approachrdquoJournal of Statistical Planning and Inference vol 139 no 5 pp1619ndash1632 2009
[3] E A Azrulhisham Y M Asri A W Dzuraidah N M NikAbdullah A Shahrum and C H Che Hassan ldquoEvaluationof fatigue life reliability of steering knuckle using pearsonparametric distributionmodelrdquo International Journal ofQualityStatistics and Reliability vol 2010 Article ID 816407 8 pages2010
[4] S Vijayarangan N Rajamanickam and V Sivananth ldquoEvalua-tion of metal matrix composite to replace spheroidal graphiteiron for a critical component steering knucklerdquo Materials ampDesign vol 43 pp 532ndash541 2013
[5] Y C Park K H Lee D H Lee and K Y Lee ldquoShapeoptimization design of the knuckle using orthogonal array andthe finite element analysisrdquo Transactions of the Korean Society ofAutomotive Engineers vol 11 pp 138ndash144 2003 (Korean)
[6] B-C Song Y-C Park S-W Kang and K-H Lee ldquoStructuraloptimization of an upper control arm considering the strengthrdquoProceedings of the Institution of Mechanical Engineers D Journalof Automobile Engineering vol 223 no 6 pp 727ndash735 2009
[7] X G Song J H Jung H J Son J H Park K H Lee andY C Park ldquoMetamodel-based optimization of a control armconsidering strength and durability performancerdquo Computersand Mathematics with Applications vol 60 no 4 pp 976ndash9802010
[8] J SacksW JWelch T J Mitchell andH PWynn ldquoDesign andanalysis of computer experimentsrdquo Statistical Science vol 4 pp409ndash435 1989
[9] A Guinta and L Watson ldquoA comparison of approximationmodeling techniques polynomial versus interpolating modelsrdquoin Proceedings of the 7th AIAAUSAFNASAISSMO Symp onMultid Anal and Optim (AIAA rsquo98) vol 2 pp 392ndash440 1998
[10] K T Fang R Li and A Sudjianto Design and Modeling forComputer Experiments Chapman amp HallCRC 2006
[11] Simulia ldquoAbaqus 610 analysis userrsquos manual materialsrdquo vol 3[12] J K Kim Y J Kim W H Yang Y C Park and K-H Lee
ldquoStructural design of an outer tie rod for a passenger carrdquoInternational Journal of Automotive Technology vol 12 no 3pp 375ndash381 2011
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World Journal 7
Table 6 Responses at initial and optimum designs
Design Deign variable (mm) Response1199051
1199052
1199053
1199054
1199055
1199056
119882 (kg) 120576pl SFInitial (steel) 180 220 230 240 215 275 34 00034 127Initial (Al) 180 220 230 240 215 275 1160 0025 31Optimum 193 206 200 218 206 262 1142 00199 28
Figure 9 Suggested optimum design of a knuckle
implement in existing commercial software In addition thevalidation of the kriging model was carried out through thecross validation and CV index and the predicted values ofthe kriging model for the weight equivalent plastic strainand safety factor of fatigue life had no significant differencescompared to the results from the finite element analysis Forfuture work the forging of the optimum shape of the knuckleproposed is scheduled to be reviewed
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was financially supported by the Ministryof Education Science and Technology (MEST) and theNational Research Foundation of Korea (NRF) through theHuman Resource Training Project for Regional Innovation(2012H1B8A2026078)
References
[1] G K Triantafyllidis A Antonopoulos A Spiliotis S Fedonosand D Repanis ldquoFracture characteristics of fatigue failure ofa vehicles ductile iron steering knucklerdquo Journal of FailureAnalysis and Prevention vol 9 no 4 pp 323ndash328 2009
[2] R dIppolitoMHack S Donders L Hermans N Tzannetakisand D Vandepitte ldquoImproving the fatigue life of a vehicle
knuckle with a reliability-based design optimization approachrdquoJournal of Statistical Planning and Inference vol 139 no 5 pp1619ndash1632 2009
[3] E A Azrulhisham Y M Asri A W Dzuraidah N M NikAbdullah A Shahrum and C H Che Hassan ldquoEvaluationof fatigue life reliability of steering knuckle using pearsonparametric distributionmodelrdquo International Journal ofQualityStatistics and Reliability vol 2010 Article ID 816407 8 pages2010
[4] S Vijayarangan N Rajamanickam and V Sivananth ldquoEvalua-tion of metal matrix composite to replace spheroidal graphiteiron for a critical component steering knucklerdquo Materials ampDesign vol 43 pp 532ndash541 2013
[5] Y C Park K H Lee D H Lee and K Y Lee ldquoShapeoptimization design of the knuckle using orthogonal array andthe finite element analysisrdquo Transactions of the Korean Society ofAutomotive Engineers vol 11 pp 138ndash144 2003 (Korean)
[6] B-C Song Y-C Park S-W Kang and K-H Lee ldquoStructuraloptimization of an upper control arm considering the strengthrdquoProceedings of the Institution of Mechanical Engineers D Journalof Automobile Engineering vol 223 no 6 pp 727ndash735 2009
[7] X G Song J H Jung H J Son J H Park K H Lee andY C Park ldquoMetamodel-based optimization of a control armconsidering strength and durability performancerdquo Computersand Mathematics with Applications vol 60 no 4 pp 976ndash9802010
[8] J SacksW JWelch T J Mitchell andH PWynn ldquoDesign andanalysis of computer experimentsrdquo Statistical Science vol 4 pp409ndash435 1989
[9] A Guinta and L Watson ldquoA comparison of approximationmodeling techniques polynomial versus interpolating modelsrdquoin Proceedings of the 7th AIAAUSAFNASAISSMO Symp onMultid Anal and Optim (AIAA rsquo98) vol 2 pp 392ndash440 1998
[10] K T Fang R Li and A Sudjianto Design and Modeling forComputer Experiments Chapman amp HallCRC 2006
[11] Simulia ldquoAbaqus 610 analysis userrsquos manual materialsrdquo vol 3[12] J K Kim Y J Kim W H Yang Y C Park and K-H Lee
ldquoStructural design of an outer tie rod for a passenger carrdquoInternational Journal of Automotive Technology vol 12 no 3pp 375ndash381 2011
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014