research article structural optimization of a knuckle with...

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


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


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


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


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


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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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


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


Force and moment

ForceForce

Force and m

ForForce






120576pl = 120576pl100381610038161003816100381610038160

+ int119905

0

radic2

3120576pl 120576pl119889119905 (1)





120576pl = 00034








Force and moment

Force

Force

Moment

Force and mo

F

Force

Moment

(a) FE model

635000 cycles

(b) Fatigue life









Minimize 119882(t)


SF (t) ge SF119886

t119871le t le t

119880

(2)







119871= [18 16 20 20 20 20]mm and t

119880=





t1

t2

t3t4

t5t6






119891 (t) = 120573 + 119911 (t) (3)



Ifand




and

119891(t) =and


120573 i) (4)



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)




minus

[119899119904ln(and

1205902) + ln |R|]

2

(7)

where 120579119894(119894 = 1 2 119899) gt 0


CV = radic1

119899119904

119899119904

sum119894=1

(119891119894minusand

119891minus119894)

2

(8)

whereand







Optimization




119904were







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



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









1199052

1199053

1199054

1199055

1199056






Acknowledgments


References





















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Force and moment

ForceForce

Force and m

ForForce






120576pl = 120576pl100381610038161003816100381610038160

+ int119905

0

radic2

3120576pl 120576pl119889119905 (1)





120576pl = 00034








Force and moment

Force

Force

Moment

Force and mo

F

Force

Moment

(a) FE model

635000 cycles

(b) Fatigue life









Minimize 119882(t)


SF (t) ge SF119886

t119871le t le t

119880

(2)







119871= [18 16 20 20 20 20]mm and t

119880=





t1

t2

t3t4

t5t6






119891 (t) = 120573 + 119911 (t) (3)



Ifand




and

119891(t) =and


120573 i) (4)



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)




minus

[119899119904ln(and

1205902) + ln |R|]

2

(7)

where 120579119894(119894 = 1 2 119899) gt 0


CV = radic1

119899119904

119899119904

sum119894=1

(119891119894minusand

119891minus119894)

2

(8)

whereand







Optimization




119904were







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



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









1199052

1199053

1199054

1199055

1199056






Acknowledgments


References





















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Force and moment

Force

Force

Moment

Force and mo

F

Force

Moment

(a) FE model

635000 cycles

(b) Fatigue life









Minimize 119882(t)


SF (t) ge SF119886

t119871le t le t

119880

(2)







119871= [18 16 20 20 20 20]mm and t

119880=





t1

t2

t3t4

t5t6






119891 (t) = 120573 + 119911 (t) (3)



Ifand




and

119891(t) =and


120573 i) (4)



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)




minus

[119899119904ln(and

1205902) + ln |R|]

2

(7)

where 120579119894(119894 = 1 2 119899) gt 0


CV = radic1

119899119904

119899119904

sum119894=1

(119891119894minusand

119891minus119894)

2

(8)

whereand







Optimization




119904were







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



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









1199052

1199053

1199054

1199055

1199056






Acknowledgments


References





















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






119891 (t) = 120573 + 119911 (t) (3)



Ifand




and

119891(t) =and


120573 i) (4)



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)




minus

[119899119904ln(and

1205902) + ln |R|]

2

(7)

where 120579119894(119894 = 1 2 119899) gt 0


CV = radic1

119899119904

119899119904

sum119894=1

(119891119894minusand

119891minus119894)

2

(8)

whereand







Optimization




119904were







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



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









1199052

1199053

1199054

1199055

1199056






Acknowledgments


References





















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






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



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









1199052

1199053

1199054

1199055

1199056






Acknowledgments


References





















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






1199052

1199053

1199054

1199055

1199056






Acknowledgments


References





















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



research article structural optimization of a knuckle with...

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