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Vehicle Planar Motion Stability Study for Tyres Working in Extremely Nonlinear Region

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Vehicle Planar Motion Stability Study for Tyres Working in Extremely Nonlinear Regionin,Study,for,Tyres,study

    Vehicle Planar Motion Stability Study for

    Tyres Working in Extremely Nonlinear

    Region

    CHINESEJOURNALOFMECHANICALENGINEERG

    Vo1.23,No.2,2010?l85?

    DOI:10.3901/CJME.2010.02.185,availableonlineatwww.cjmenet.com;www.cjmenet.com.cn

    VehiclePlanarMotionStabilityStudyforTyresWorking

    inExtremelyNonlinearRegion

    LIULi,SHIShuming,SHENShuiwen,andCHUJiangwei

    1TransportationCollege,JilinUniversity,Changchun130022,China

    2RicardoUKLtd.CambridgeCB41YG,UK

    ReceivedApril21,2009;revisedNovember30,2009;acceptedJanuary11,2010;publishedelectronicallyJanuary25,2010

    Abstract:Manyresearchesonvehicleplanarmotionstabilityfocusontwodegreesoffreedom(2DOF)vehiclemodel,andonlythe

    lateralvelocity(orsideslipangle)andyawrateareconsideredasthestatevariables.Thestabilityanalysismethods,suchasphaseplane

    analysis,equilibriumsana1)rsisandbifurcationanalysis,areallusedtodrawmanyclassicalconclusions.Itisconcludedfromthese

    researchesthatunboundedgrowthofthevehiclemotionduringunstableoperationisuntrueinrealitythusonelimitationofthe2DOF

    mode1.Thefundamenta1assumptionofthe2DOFmode1isthatthelongitudinalvelocityistreatedasaconstant.butthisisintrinsically

    incorrect.Whentyresworkinextremelynonlinearregion,thecouplingbetweenthevehiclelongitudinalandlateralmotionbecomes

    significant.Forthepurposeofsolvingtheaboveproblem,theeffectofvehiclelongitudinalvelocityonthestabilityofthevehicleplanar

    motionwhentyresworkinextremelynonlinearregionisinvestigated.Tothisend,a3DOFmodelwhichintroducingthevehicular

    longitudinaldynamicsisproposedandthe3Dphasespaceportraitmethodisemployedforvisualizationofvehicledynamics.Through

    thecomparisonsofthe2DOFand3D0Fmodels.itisdiscoveredthatthevehiclelongitudinalvelocitygreatlyaffectsthevehicleplanar

    motion,andthevehicledynamicsrepresentedinphasespaceportraitarefundamentallydifferentfromthatofthe2DOFmode1.The

    vehicleplanarmotionwithdifferentfrontwheelsteeringanglesisfurtherrepresentedbythecorrespondingvehicleroute,yawrateand

    yawangle.Theseresearchresultsenhancetheunderstandingofthestabilityofthevehiclesystemparticularlyduringnonlinearregion,

    andprovidetheinsightintoanalyzingtheattractiveregionanddesigningthevehiclestabilitycontroller,whichwillbethetopicsof

    futureworks.

    Keywords:vehicledynamics,steeringstability,nonlineardynamics,phasespace lIntrOducti0n

    Itiscrystalclearthatthelostofthevehiclestability

    duringicyandwetroadconditionsiscausedbythe

    nonlinearityoftyres.Yet,thedevelopmentofthevehicle

    motionafterlosingitsstabilityislargelyunknown.

    Preventingtyresfromnonlinearregionbywhateverthe

    meansplaysamajorpartforastablyvehiclehandling,but

    toregainthemaneuverabilityisnotbyallmeansless

    critica1.Thelaterbecomesviableonlyprovidedwith

    necessaryplanarmonitioncharacteristics.whichjsthe

    topicofthispaper.

    Uptodate,manyresearchersfocusontwodegreesof

    freedom(2DOF)vehicleplanarmotions.Theworksof INAGA.etaland0N0.etal1edtoaconclusionthat

    thevehiclehadthreeequilibriums,onebeingstablefocus whiletheothertwounstablesaddles,andthatthestable regionwasrathernarrowintheso.called?phaseplane

    spannedbythebodysideslipangleandyawrateco.The Correspondingauthor.Email:shishuming?lu.edu.en

    ThisprojectissupposedbyNationalNaturalScienceFoundationof China(GrantNo.50775094)

    similarwasalsoconeludedbvKO,eta1andVINCENTt, buttheconelusionswereonlyvalidatelocally.SHEN. etaladoptedageometricalmethodtoexplorethesystem frommoreglobalpointofviewforthefirsttime.The complicateddynamicswithbifurcationand1imit.cycles werediscovered.SHI.etal【….discoveredthestableregion

    ofvehiclecorneringsystemspannedbythebodyslipangle andtheyawratethroughtheanalysisofLyapunovpotential energyfunctionsurface.

    6oundedgrowthofthevehiclemotionduringunstable operationisuntrueinrealitythusonelimitationofthe 2D0Fmode1.Theimpactfromthevehicle

    longitudinaImotionisinsignificantthusthevehicle longitudinalvelocityistreatedasaconstant.Thisisthe fundamentalassumptionofthe2DOFmodel,butthisis intrinsicallyincorrectiftyresworkinextremelynonlinear region.Thecouplingbetweenthevehiclelongitudinaland lateralmotionbecomessignificant,J.Consequently.the

    longitudinalvelocitycannotbeeatedasaconstant.

    Therefore.a3DOF(1ongitudinalvelocity,lateralvelocity andyawrate)modelisintroducedandanalyzed

numericallyinthisPaDer.Thecomparisonbetween2DOF

    and3D0Fmode1intermsofthevehiclesystemdynamics issummarizedandillustrated.

    ?186?LIULi,etal:VehiclePlanarMotionStabilityStudyforTyresWorkinginEx~emelyNon

    linearRegion

    23DOFVehicleSteeringModel

    Thebicyclemodelofvehiclesteeringsystemisshownin

    Fig.1.Theequationsof3DOFvehiclemodelareshownas

    follows[:

    fm(vxv/o)=ZFx,

    {m(+VxCO)=?,

    l,2=ZM,

    wherem一眙hiclemassm=l500kg;

    I,Yawmomentofnertia.I3000kg

    vx--Longitudinalvelocity;

    ,Lateralvelocity:

    ?一Yawrate:

    --

    Longitudinalforce;

    

    Lateralforce;

    ^Yawmoment.

    AndEq.(1)canbewrittenasfollows: (1)

    m(Lv/o)=fCOSfsin+COS

    sin8rCait~2

    ,

    +VxCO)=fCOS+fsin+(2)

    Fstcos6t+Fsin6t.

西=(fIfsin+fIfcos8f)

    (frsin8r+ffCOSSr),

    wheref_Frontwheelsteeringangle;

    

    Rearwheelsteeringangle;

    If--Distancefromfrontaxletothemasscenter, lf=1.2m:

    lr--Distancefromrearaxletothemasscenter, lr=1.3m:

    F:f--Longitudinaltyreforceoffrontwheel; f_Latera1tyreforceoffrontwheel;

    --

    LongitudinalWeforceofrearwheel;

    rLateraltyreforceofrearwheel;

    irAirresistanceefficient;

    

    _Vehiclefrontalarea;

    Pirdensity.

    Fig.1.Vehiclebicyclemodel

    Intheclassical2DOFmodel,boththelongitudinaltyre forcesandairresistantforceareneglected.The longitudinalvelocityistreatedasaconstant.Consbquentl~

    the2DOFvehiclesteeringmodelisasimplifiedversionof

    Eq.(1)andEq.(2)asfollows:

    cos8f+cos6r

    

    ,

    fIfCOSFslrCOS

    I.

    (3)

    Forthevehicleplanarmotion.ifthesystemisnot cons~ained,allthestatevariablesofthesystemarefreeto developwithtime.Thisallowstheinvestigationofthe dynamica1couplingbetweenthelateralandlongitudinal directions.However,asaconsequenceofthelongitudinal velocitybeingconstrainedasaconstantin2DOFmodel, thedynamicsrepresentedby2DOFcouldonlybepartially true.Itisthereforereasonabletoin~oducethe3DOFmodel InthisDapeLthefocusisgiventotheinteractionbetween longitudinalandlateralmotions.AIthoughthelongitudinal tyreforcesandairresistantforceareneglected,the 1ongitudinalvelocityisnownotaconstant.Theproposed 3D0Fsteeringmodelisshownasfollows:

    Vy

    (sin6f+sinfir)

    FsfCOS8f+cosdr

    

    ?,(4)

    :!!!!

    Iz

    Thelateraltyreforcedependingonthesideslipangleis representedbythemagicformulamodel:

    F=Dsin(Carctan(BaE(BaarctanBa),(5)

    whereB,c,D,Earecoefficients,Fisthelateraltyreforce, ando[isthesideslipangle.

    ThecoefficientsarelistedinTablewhileFig.2 graphicallyshowstherelationshipbetweenthelateraltyre forcesandsideslipanglesfortherespectivefrontandrear tyres.

    TableBicyclemodeltyreparameters

    Thesideslipanglesofbothfrontandreartyresaregiven asbelow:

    =-arctanfV+?If(6)

    ==

    .

    西

    ,??????,,,,??????L

    CHINESEJ0URNALOFMECHANICALENGINEERING?l87?

    .

    2

    

    ;

    

    

    (].

    Sideslipanglea/rad

    Fig.2.Lateraltyreforceversussideslipangle forfrontandreartyresinbicyclemode1.

    3ComparisonofPhasePortraitAnalysis

    betweenthe2DOFand3DOFModels

    Vehiclemotioncanbevisualizedinaphasespace spannedbylongitudinalvelocity,lateralvelocityandyaw rate(Vx,Vy,andco).However,conventionalstudyof2D0F modelismorecommontoemployaphaseplanespanned onlybythetwoofjustmentionedstatevariables.

    Fig.3andFig.4showthetypicalVy一?portraitsofthe

    2D0Fmode1.Fig.3iStherepresentationofthevehicle motioninalocalregionofVyandO9.Theregionsforvyand

    ?are(_10,10)m/sand(1,1)rad/s,respectively. Simulationasts20S.Notethat,thered"O"indicatesthe endpointsofthesimulation.

    

    864202468

    Lateralvelocityvyl(m?S1

    Fig.3.LocalviewOfVy一?phaseplanetr~ectories Fig.5andFig.6areothersetsofvehicletrajectoriesin

    thephaseplaneportraitresultingfrom2DOFmode1. SimilartoFig.3,Fig.5istherepresentationofthevehicle

    motioninalocalregionoffland?.

    

    ?

    

    ?

    Lateralvelocityvyl(m?S1)

    Fig.4.GlobalviewOfVy-09phaseplanetrajectories

    .

    

    ?

    -

    0.3-0.2O.100.10.20.3

    Sideslipanglefl/rad

    phaseplanetrajectories Fig.5.Localviewoffl一?

    

    

    

    =

0?1.51.0-0.500.51.01.52.O

    Sideslipanglefl/rad

    Fig.6.Globalviewoffl-COphaseplanetrajectories TheresultsgiveninFig.3andFig.5arealsopresented bymanyresearchesl'4J,whichareusedtodistinguishthe stableandunstableregionfor2DOFmode1.Nevertheless, thesetwofiguersonlyreflecttheloca1portraitofthestate variables.ItiSstilllargelyunkownthedevelopmentofthe statevariablesifthevehicle1osesitsstablility.Fig.4and Fig.6showthemoreglobalviewofvehiclemotionsin vr(oand8-COphaseplane.Itcouldbeconcludedfrom Fig.4andFig.6thatoncethevehiclelosesitsstablilicV.the bodysideslipangleincreasestohi2rapidly,andthatboth theyawrateandlateralvelocitygrowexponentially. BecausethelongitudinalvelocityiStreatedasaconstant theenergyoflateralandyawmotionbecomesunbounded. m8642Om

    (_TBJ)/0BJB^

    ?

    188?LIULi.etal:VehiclePlanarMotionStabilityStudyf0rTyresWorkinginExtremelyNonl

    inearRegion

    whichiscontradictorytothevehiclemotioninreality. Therefore,theglobalphaseportraitderivedfromthe2DOF vehiclemodelrevealsthelimitationofthe2DOFmode1. Fig.7andFig.8showtheglobalviewinVxVyphase

    spaceofthe3DOFwithVyandcobeingrespectivelyregion of(l0,10)m/sand(1,1)rad/s.Theinitia1vlauesofis

    20m/s,andtheintervalsofv,andcoare4m/sand0.4rad/s, respectively.Fig.8istheprojectionoftheportraitontothe

vycoplane.

    ?Initia1value

    

    Phasetrajectory

    Fig.7.Vy09phasespacetrajectories

    ?Initia1value

    Lateralvelocity,/(m?s)

    Fig.8Vx--09phasespacetrajectoriesonto09plane Aspreviouslymentioned,thestatevanaNegrows exponentiallyonceitcomeintotheunstableregionforthe 2D0Fmode1.However,thiswouldnothappentothe3DOF mode1.Sincevehiclelongitudinalvelocityisnottreatedas aconstantanymore,partofthelateralandyawmotion convertsintolongitudinalmotionandthetrajectorieswhich forillclosedandboundedorbitspresentmultiplicity. 4PhaseSpacePortraitsof3DOF

    4.1Phasespaceportraitswiththevariationofinitial longitudinalvelocity

    Fig.9showsthevehicletrajectorieswith,and()taking

    theirinitialvaluefromtheintervalof(_10,10)m/sand (-1,1)rad/sandtheinitiallongitudinalvelocitybeing respectively20m/sand40m/s.Fig.10andFig.11show theprojectionsoftheportraitontovycoplanewhenis

    20m/sand40m/s,respectively.Itcouldbeconcludedfrom Fig.9thattheorbitsformedbythevehicletrajectories growlargerin3Dphasespacewheninitiallongitudinal velocityisincreased.Howerer,Itisalsoindicatedfrom Fig.10andFig.11thattheattractiveregionofequilibrium becomesnarrowwiththeincreaseofinitiallongitudinal velocity.Thistrendencyoflosingstabilitycouldalsobe

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