Automobiles - Vehicle_Dynamics

S

CIMANYD ELCIHEVFACHHOCHSCHULEREGENSBURGUNIVERSITYOFAPPLIEDSCIENCES

HOCHSCHULEFÜR

TECHNIKWIRTSCHAFT

SOZIALES

LECTURENOTESProf.Dr.GeorgRill©October

2003

download:http://homepages.fh-regensburg.de/%7Erig39165/

Contents

Contents

1Introduction

1.1Terminology.............................

.......1.1.1VehicleDynamics.............................1.1.2Driver....................................1.1.3Vehicle...................................1.1.4Load....................................1.1.5Environment................................1.2Wheel/AxleSuspensionSystems..................

.......1.2.1GeneralRemarks.............................1.2.2MultiPurposeSuspensionSystems....................1.2.3Speci cSuspensionSystems.......................1.3SteeringSystems..........................

.......1.3.1Requirements...............................1.3.2RackandPinionSteering.........................1.3.3LeverArmSteeringSystem........................1.3.4DragLinkSteeringSystem........................1.3.5BusSteerSystem.............................1.4De nitions..............................

.......1.4.1CoordinateSystems............................1.4.2ToeandCamberAngle....................

.......1.4.2.1De nitionsaccordingtoDIN70000...............1.4.2.2Calculation.....................

.......1.4.3SteeringGeometry......................

.......1.4.3.1Kingpin..............................1.4.3.2CasterandKingpinAngle....................1.4.3.3DisturbingForceLever,CasterandKingpinOffset.......

2TheTire

2.1Introduction......................

...............2.1.1TireDevelopment.............................2.1.2TireComposites..............................2.1.3ForcesandTorquesintheTireContactArea...............2.2ContactGeometry..................

...............

I111223344455566778899910101112131313131415

I

2.2.1DynamicRollingRadius..........................2.2.2ContactPoint................................2.2.3LocalTrackPlane.............................2.2.4ContactPointVelocity...........................2.3WheelLoad....................................2.4LongitudinalForceandLongitudinalSlip.....................2.5LateralSlip,LateralForceandSelfAligningTorque................2.6CamberIn uence.................................2.7BoreTorque....................................2.8

TypicalTireCharacteristics............................3LongitudinalDynamics

3.1DynamicWheelLoads.................

..............3.1.1SimpleVehicleModel...........................3.1.2In uenceofGrade.............................3.1.3AerodynamicForces............................3.2MaximumAcceleration.................

..............3.2.1TiltingLimits................................3.2.2FrictionLimits................................3.3DrivingandBraking..................

..............3.3.1SingleAxleDrive..............................3.3.2BrakingatSingleAxle...........................3.3.3OptimalDistributionofDriveandBrakeForces..............3.3.4DifferentDistributionsofBrakeForces...................3.3.5Anti-Lock-Systems.............................3.4DriveandBrakePitch.................

..............3.4.1VehicleModel...............................3.4.2EquationsofMotion............................3.4.3Equilibrium.................................3.4.4DrivingandBraking............................3.4.5BrakePitchPole..............................

4LateralDynamics

4.1KinematicApproach.........

.......................4.1.1KinematicTireModel............................4.1.2AckermannGeometry...........................4.1.3SpaceRequirement............................4.1.4VehicleModelwithTrailer..

.......................4.1.4.1Position..............................4.1.4.2Vehicle..............................4.1.4.3EnteringaCurve.........................4.1.4.4Trailer...............................4.1.4.5CourseCalculations

.......................4.2SteadyStateCornering.......

.......................

II

151618192020232527283030303132333333343435363838393941424344454545454648484951515253

4.2.1OverturningLimit..............................4.2.2RollSupportandCamberCompensation.................4.2.3RollCenterandRollAxis.........................4.2.4WheelLoads................................4.2.5CorneringResistance...........................4.3SimpleHandlingModel.................

.............4.3.1Forces...................................4.3.2Kinematics.................................4.3.3LateralSlips................................4.3.4EquationsofMotion............................4.3.5Stability......................

.............4.3.5.1Eigenvalues...........................4.3.5.2LowSpeedApproximation....................4.3.5.3HighSpeedApproximation.......

.............4.3.6SteadyStateSolution..............

.............4.3.6.1SideSlipAngleandYawVelocity................4.3.6.2SteeringTendency........................4.3.6.3SlipAngles...............

.............4.3.7In uenceofWheelLoadonCorneringStiffness.............5VerticalDynamics

5.1Goals........................................5.2BasicTuning.....................

...............5.2.1SimpleModels...............................5.2.2Track....................................5.2.3SpringPreload...............................5.2.4Eigenvalues................................5.2.5FreeVibrations...............................5.3NonlinearForceElements..............

...............5.3.1QuarterCarModel.............................5.3.2RandomRoadPro le...........................5.3.3VehicleData................................5.3.4QualityCriteria...............................5.3.5OptimalParameter..............

...............5.3.5.1LinearCharacteristics......................5.3.5.2NonlinearCharacteristics....................5.3.5.3LimitedSpringTravel........

...............5.4DynamicForceElements..............

...............5.4.1SystemResponseintheFrequencyDomain

...............5.4.1.1FirstHarmonicOscillation....................5.4.1.2Sweep-SineExcitation.......

...............5.4.2Hydro-Mount.................

...............5.4.2.1PrincipleandModel.......................5.4.2.2DynamicForceCharacteristics..

...............

53565859606262626364656566666767697070737373737474757678787980808181818384848486878789

III

5.5DifferentIn uencesonComfortandSafety....................90

5.5.1VehicleModel...............................905.5.2SimulationResults.............................916DrivingBehaviorofSingleVehicles

6.1StandardDrivingManeuvers...........

6.1.1SteadyStateCornering..........6.1.2StepSteerInput..............6.1.3DrivingStraightAhead...........

6.1.3.1RandomRoadPro le......6.1.3.2SteeringActivity.........

6.2CoachwithdifferentLoadingConditions.....

6.2.1Data....................6.2.2RollSteerBehavior.............6.2.3SteadyStateCornering..........6.2.4StepSteerInput..............6.3DifferentRearAxleConceptsforaPassengerCar

939393949595979898989999100

................................................................................................................................................................................................

IV

1Introduction

1.1Terminology

1.1.1VehicleDynamics

TheExpression’VehicleDynamics’encompassestheinteractionof

driver, vehicle loadand environment

Vehicledynamicsmainlydealswith

theimprovementofactivesafetyanddrivingcomfortaswellas thereductionofroaddestruction.Invehicledynamics

computercalculations testrigmeasurementsand eldtestsareemployed.

Theinteractionsbetweenthesinglesystemsandtheproblemswithcomputercalculationsand/ormeasurementsshallbediscussedinthefollowing.

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VehicleDynamicsFHRegensburg,UniversityofAppliedSciences

1.1.2Driver

Byvariousmeansofinterferencethedrivercaninterferewiththevehicle:

steeringwheel gaspedal brakepedaldriver clutch

gearshift

lateraldynamics

longitudinaldynamics

→vehicle

Thevehicleprovidesthedriverwithsomeinformation:

longitudinal,lateral,vertical

sound:motor,aerodynamics,tiresvehicle →driver instruments:velocity,externaltemperature,...

Theenvironmentalsoin uencesthedriver:

environment

vibrations:

climate

traf cdensity →driver

track

Adriver’sreactionisverycomplex.Toachieveobjectiveresults,an”ideal”driverisusedincomputersimulationsandindrivingexperimentsautomateddrivers(e.g.steeringmachines)areemployed.

Transferringresultstonormaldriversisoftendif cult,if eldtestsaremadewithtestdrivers.Fieldtestswithnormaldrivershavetobeevaluatedstatistically.Inalltests,thedriver’ssecuritymusthaveabsolutepriority.

Drivingsimulatorsprovideanexcellentmeansofanalyzingthebehaviorofdriverseveninlimitsituationswithoutdanger.

Forsomeyearsithasbeentriedtoanalyzetheinteractionbetweendriverandvehiclewithcomplexdrivermodels.

1.1.3Vehicle

ThefollowingvehiclesarelistedintheISO3833directive:

Motorcycles, PassengerCars, Busses, Trucks

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AgriculturalTractors, PassengerCarswithTrailer TruckTrailer/Semitrailer, RoadTrains.

Prof.Dr.-Ing.G.Rill

Forcomputercalculationsthesevehicleshavetobedepictedinmathematicallydescribablesubstitutesystems.Thegenerationoftheequationsofmotionsandthenumericsolutionaswellastheacquisitionofdatarequiregreatexpenses.

IntimesofPCsandworkstationscomputingcostshardlymatteranymore.

Atanearlystageofdevelopmentoftenonlyprototypesareavailablefor eldand/orlaboratorytests.

Resultscanbefalsi edbysafetydevices,e.g.jockeywheelsontrucks.

1.1.4Load

Trucksareconceivedfortakingupload.Thustheirdrivingbehaviorchanges.

Load

mass,inertia,centerofgravitydynamicbehaviour(liquidload)

Incomputercalculationsproblemsoccurwiththedeterminationoftheinertiasandthemod-ellingofliquidloads.

Eventheloadingandunloadingprocessofexperimentalvehiclestakessomeeffort.Whenmakingexperimentswithtanktrucks, ammableliquidshavetobesubstitutedwithwater.Theresultsthusachievedcannotbesimplytransferredtorealloads.

1.1.5Environment

TheEnvironmentin uencesprimarilythevehicle:

Environment

Road:irregularities,coef cientoffrictionAir:resistance,crosswind

→vehicle

butalsoin uencesthedriver

Environment

climate

visibility

→driver

Throughtheinteractionsbetweenvehicleandroad,roadscanquicklybedestroyed.

Thegreatestproblemin eldtestandlaboratoryexperimentsisthevirtualimpossibilityofreproducingenvironmentalin uences.

Themainproblemsincomputersimulationarethedescriptionofrandomroadirregularitiesandtheinteractionoftiresandroadaswellasthecalculationofaerodynamicforcesandtorques.

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VehicleDynamicsFHRegensburg,UniversityofAppliedSciences

1.2Wheel/AxleSuspensionSystems

1.2.1GeneralRemarks

TheAutomotiveIndustryusesdifferentkindsofwheel/axlesuspensionsystems.Importantcriteriaarecosts,spacerequirements,kinematicpropertiesandcomplianceattributes.

1.2.2MultiPurposeSuspensionSystems

TheDoubleWishboneSuspension,theMcPhersonSuspensionandtheMulti-LinkSuspensionaremultipurposewheelsuspensionsystems,

Fig.1.1.

Figure1.1:DoubleWishbone,McPhersonandMulti-LinkSuspension

Theyareusedassteeredfrontornonsteeredrearaxlesuspensionsystems.Thesesuspen-sionsystemsarealsosuitablefordrivenaxles.

InaMcPhersonsuspensionthespringismountedwithaninclinationtothestrutaxis.Thusbendingtorquesatthestrutwhichcausehighfrictionforcescanbereduced.

Atpickups,trucksandbussesoftensolidaxlesareused.Solidaxlesareguidedeitherbyleaf

1

Figure1.2:SolidAxles

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springsorbyrigidlinks,

Fig.1.2.Solidaxlestendtotramponroughroad.

Prof.Dr.-Ing.G.Rill

Leafspringguidedsolidaxlesuspensionsystemsareveryrobust.Dryfrictionbetweentheleafsleadstolockingeffectsinthesuspension.Althoughtheleafspringsprovideaxleguidanceonsomesolidaxlesuspensionsystemsadditionallinksinlongitudinalandlateraldirectionareused.Thusthetypicalwindupeffectonbrakingcanbeavoided.

Solidaxlessuspendedbyairspringsneedatleastfourlinksforguidance.Inadditiontoagooddrivingcomfortairspringsallowlevelcontroltoo.

1.2.3Speci cSuspensionSystems

TheSemi-TrailingArm,theSLAandtheTwistBeamaxlesuspensionaresuitableonlyfornonsteeredaxles,Fig.1.3.

Figure1.3:Speci cWheel/AxlesSuspensionSystems

Thesemi-trailingarmisasimpleandcheapdesignwhichrequiresonlyfewspace.Itismostlyusedfordrivenrearaxles.

TheSLAaxledesignallowsanearlyindependentlayoutoflongitudinalandlateralaxlemo-tions.ItissimilartotheCentralControlArmaxlesuspension,wherethetrailingarmiscom-pletelyrigidandhenceonlytwolaterallinksareneeded.

Thetwistbeamaxlesuspensionexhibitseitheratrailingarmorasemi-trailingarmcharacter-istic.Itisusedfornondrivenrearaxlesonly.Thetwistbeamaxleprovidesenoughspaceforsparetireandfueltank.

1.3SteeringSystems

1.3.1Requirements

Thesteeringsystemmustguaranteeeasyandsafesteeringofthevehicle.Theentiretyofthemechanicaltransmissiondevicesmustbeabletocopewithallloadsandstressesoccurringinoperation.

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VehicleDynamicsFHRegensburg,UniversityofAppliedSciences

Inordertoachieveagoodmaneuverabilityamaximumsteerangleofapprox.30 mustbeprovidedatthefrontwheelsofpassengercars.Dependingonthewheelbasebussesandtrucksneedmaximumsteeranglesupto55 atthefrontwheels.

Recentlysomecompanieshavestartedinvestigationson’steerbywire’techniques.

1.3.2RackandPinionSteering

Rackandpinionisthemostcommonsteeringsystemonpassengercars,Fig.

1.4.Therackmaybelocatedeitherinfrontoforbehindtheaxle.TherotationsofthesteeringwheelδLare

Figure1.4:RackandPinionSteering

rstlytransformedbythesteeringboxtotheracktraveluZ=uZ(δL)andthenviathedraglinkstransmittedtothewheelrotationsδ1=δ1(uZ),δ2=δ2(uZ).Hencetheoverallsteeringratiodependsontheratioofthesteerboxandonthekinematicsofthesteerlinkage.

1.3.3LeverArmSteeringSystem

UsingaleverarmsteeringsystemFig.1.5,largesteeranglesatthewheelsarepossible.This

Figure1.5:LeverArmSteeringSystem

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FHRegensburg,UniversityofAppliedSciencesProf.Dr.-Ing.G.Rill

steeringsystemisusedontruckswithlargewheelbasesandindependentwheelsuspensionatthefrontaxle.Herethesteeringboxcanbeplacedoutsideoftheaxlecenter.

TherotationsofthesteeringwheelδLare rstlytransformedbythesteeringboxtothero-tationofthesteerleversδG=δG(δL).Thedraglinkstransmitthisrotationtothewheelδ1=δ1(δG),δ2=δ2(δG).Hence,againtheoverallsteeringratiodependsontheratioofthesteerboxandonthekinematicsofthesteerlinkage.

1.3.4DragLinkSteeringSystem

Atsolidaxlesthedraglinksteeringsystemisused,

Fig.1.6.

Figure1.6:DragLinkSteeringSystem

TherotationsofthesteeringwheelδLaretransformedbythesteeringboxtotherotationofthesteerleverarmδH=δH(δL)andfurtherontotherotationoftheleftwheel,δ1=δ1(δH).Thedraglinktransmitstherotationoftheleftwheeltotherightwheel,δ2=δ2(δ1).Thesteeringratioisde nedbytheratioofthesteerboxandthekinematicsofthesteerlink.Heretheratioδ2=δ2(δ1)givenbythekinematicsofthedraglinkcanbechangedseparately.

1.3.5BusSteerSystem

Inbussesthedriversitsmorethan2minfrontofthefrontaxle.Here,sophisticatedsteersystemsareneeded,Fig.1.7.

TherotationsofthesteeringwheelδLaretransformedbythesteeringboxtotherotationofthesteerleverarmδH=δH(δL).Viathesteerlinktheleftleverarmismoved,δH=δH(δG).Thismotionistransferredbyacouplinglinktotherightleverarm.Viathedraglinkstheleftandrightwheelarerotated,δ1=δ1(δH)andδ2=δ2(δH).

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VehicleDynamics

FHRegensburg,UniversityofAppliedSciences

Figure1.7:BusSteerSystem

1.4De nitions

1.4.1CoordinateSystems

Invehicledynamicsseveraldifferentcoordinatesystemsareused,Fig1.8.

Figure1.8:CoordinateSystems

Theinertialsystemwiththeaxesx0,y0,z0is xedtothetrack.Withinthevehicle xedsystemthexF-axisispointingforward,theyF-axisleftandthezF-axisupward.TheorientationofthewheelisgivenbytheunitvectoreyRindirectionofthewheelrotationaxis.

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FHRegensburg,UniversityofAppliedSciencesProf.Dr.-Ing.G.Rill

Theunitvectorsinthedirectionsofcircumferentialandlateralforcesexandeyaswellasthetracknormalenfollowfromthecontactgeometry.

1.4.2ToeandCamberAngle

1.4.2.1De nitionsaccordingtoDIN70000

Theanglebetweenthevehiclecenterplaneinlongitudinaldirectionandtheintersectionlineofthetirecenterplanewiththetrackplaneisnamedtoeangle.Itispositive,ifthefrontpartofthewheelisorientedtowardsthevehiclecenterplane,

Fig.1.9.

front

F

F

rear

Figure1.9:PositiveToeAngle

Thecamberangleistheanglebetweenthewheelcenterplaneandthetracknormal.Itispositive,iftheupperpartofthewheelisinclinedoutwards,Fig.1.10.

topF

F

bottom

Figure1.10:PositiveCamberAngle

1.4.2.2Calculation

Thecalculationofthetoeangleisdonefortheleftwheel.TheunitvectoreyRindirectionofthewheelrotationaxisisdescribedinthevehicle xedcoordinatesystemF,Fig.1.11

eyR,F=

(1)eyR,F(2)eyR,F(3)eyR,F

T

,(1.1)

wheretheaxisxFandzFspanthevehiclecenterplane.ThexF-axispointsforwardandthezF-axispointsupward.Thetoeangleδcanthenbecalculatedfrom

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VehicleDynamicsFHRegensburg,UniversityofAppliedSciences

Figure1.11:ToeAngle

tanδ=

eyR,F

(2)

eyR,F

(1)

.(1.2)

Therealcamberangleγfollowsfromthescalarproductbetweentheunitvectorsinthedirec-tionofthewheelrotationaxiseyRandinthedirectionofthetracknormalen,

sinγ= eTneyR.

Thewheelcamberanglecanbecalculatedby

(1.3)

sinγ= eyR,F.

Ona athorizontalroadbothde nitionsareequal.

(3)

(1.4)

1.4.3SteeringGeometry

1.4.3.1Kingpin

AtthesteeredfrontaxletheMcPherson-damperstrutaxis,thedoublewishboneaxisand

multi-linkwheelsuspensionordissolveddoublewishboneaxisarefrequentlyemployedinpassengercars,Fig.1.12andFig.1.13.

Thewheelbodyrotatesaroundthekingpinatsteeringmovements.

Atthedoublewishboneaxis,theballjointsAandB,whichdeterminethekingpin,are xedtothewheelbody.

TheballjointpointAisalso xedtothewheelbodyattheclassicMcPhersonwheelsuspen-sion,butthepointBis xedtothevehiclebody.

Atamulti-linkaxle,thekingpinisnolongerde nedbyreallinkpoints.Here,aswellaswiththeMcPhersonwheelsuspension,thekingpinchangesitspositionagainstthewheelbodyatwheeltravelandsteermotions.

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FHRegensburg,UniversityofAppliedSciencesProf.Dr.-Ing.G.Rill

Figure1.12:DoubleWishboneWheel

Suspension

Figure1.13:McPhersonandMulti-LinkWheelSuspensions

1.4.3.2CasterandKingpinAngle

Thecurrentdirectionofthekingpincanbede nedbytwoangleswithinthevehicle xedcoordinatesystem,Fig.1.14.

IfthekingpinisprojectedintotheyF-,zF-plane,thekingpininclinationangleσcanbereadastheanglebetweenthezF-axisandtheprojectionofthekingpin.

TheprojectionofthekingpinintothexF-,zF-planedeliversthecasterangleνwiththeanglebetweenthezF-axisandtheprojectionofthekingpin.

Withmanyaxlesthekingpinandcasteranglecannolongerbedetermineddirectly.

Thecurrentrotationaxisatsteeringmovements,thatcanbetakenfromkinematiccalculationsheredeliversavirtualkingpin.Thecurrentvaluesofthecasterangleνandthekingpinincli-nationangleσcanbecalculatedfromthecomponentsoftheunitvectorinthedirectionofthe

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VehicleDynamicsFHRegensburg,UniversityofAppliedSciences

Figure1.14:KingpinandCasterAngle

kingpin,describedinthevehicle xedcoordinatesystem

tanν=

with

eS,F

(3)eS,F

(1)

andtanσ=

T

eS,F

(3)eS,F

(2)

(1.5)

eS,F=

(1)eS,F(2)eS,F(3)eS,F

.(1.6)

1.4.3.3DisturbingForceLever,CasterandKingpinOffset

Thedistancedbetweenthewheelcenterandthekingpinaxisiscalleddisturbingforcelever.Itisanimportantquantityinevaluatingtheoverallsteerbehavior.

Ingeneral,thepointSwherethekingpinrunsthroughthetrackplanedoesnotcoincide

withthecontactpointP,Fig.1.15.

Figure1.15:CasterandKingpinOffset

Ifthekingpinpenetratesthetrackplanebeforethecontactpoint,thekinematickingpinoffsetispositive,nK>0.

Thecasteroffsetispositive,rS>0,ifthecontactpointPliesoutwardsofS.

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2TheTire

2.1Introduction

2.1.1TireDevelopment

Thefollowingtableshowssomeimportantmilestonesinthedevelopmentoftires.

1839CharlesGoodyear:vulcanization

1845

RobertWilliamThompson: rstpneumatictire(severalthinin atedtubesinsidealeathercover)

1888JohnBoydDunlop:patentforbicycle(pneumatic)tires1893TheDunlopPneumaticandTyreCo.GmbH,Hanau,Germany1895

AndréandEdouardMichelin:pneumatictiresforPeugeotParis-Bordeaux-Paris(720Miles):50tirede ations,

22completeinnertubechanges1899Continental:longerlifetires(approx.500Kilometer)1904Carbonadded:blacktires.

1908FrankSeiberling:groovedtireswithimprovedroadtraction1922Dunlop:steelcordthreadinthetirebead1943Continental:patentfortubelesstires1946RadialTire

...

Table2.1:MileStonesintheDevelopmentofTires

2.1.2TireComposites

Amoderntireisamixtureofsteel,fabric,andrubber.

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Reinforcements:steel,rayon,nylonRubber:natural/synthetic

Compounds:carbon,silica,chalk,...Softener:oil,resin

Vulcanization:sulfur,zincoxide,...MiscellaneousTireMass

16%38%30%10%4%2%8.5kg

Table2.2:TireComposites:195/65R15ContiEcoContact,Datafromwww.felge.de

2.1.3ForcesandTorquesintheTireContactArea

Inanypointofcontactbetweentireandtracknormalandfrictionforcesaredelivered.Accord-ingtothetire’spro ledesignthecontactareaformsanotnecessarilycoherentarea.Theeffectofthecontactforcescanbefullydescribedbyavectorofforceandatorqueinreferencetoapointinthecontactpatch.Thevectorsaredescribedinatrack- xedcoordinatesystem.Thez-axisisnormaltothetrack,thex-axisisperpendiculartothez-axisandperpen-diculartothewheelrotationaxiseyR.Thedemandforaright-handedcoordinatesystemthenalso xesthey-axis.

FxFyFzMxMyMz

longitudinalorcircumferentialforcelateralforce

verticalforceorwheelloadtiltingtorque

rollingresistancetorque

selfaligningandbore

torque

Figure2.1:ContactForcesandTorques

Thecomponentsofthecontactforcearenamedaccordingtothedirectionoftheaxes,Fig.2.1.Nonsymmetricdistributionsofforceinthecontactpatchcausetorquesaroundthexandyaxes.ThetiltingtorqueMxoccurswhenthetireiscambered.Myalsocontainstherollingresistanceofthetire.Inparticularthetorquearoundthez-axisisrelevantinvehicledynamics.Itconsistsoftwoparts,

Mz=MB+MS.(2.1)Rotationofthetirearoundthez-axiscausestheboretorqueMB.TheselfaligningtorqueMS

respectsthefactthatingeneraltheresultinglateralforceisnotappliedinthecenterofthecontactpatch.

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FHRegensburg,UniversityofAppliedSciencesProf.Dr.-Ing.G.Rill

2.2ContactGeometry

2.2.1DynamicRollingRadius

Atanangularrotationof ,assumingthetreadparticlessticktothetrack,thede ectedtiremovesonadistanceofx,

Fig.2.2.

deflected tirerigid wheelFigure2.2:DynamicRollingRadius

Withr0asunloadedandrS=r0 rasloadedorstatictireradius

r0sin =x

and

(2.2)

r0cos =rS.

hold.

(2.3)

Ifthemovementofatireiscomparedtotherollingofarigidwheel,itsradiusrDthenhastobechosenso,thatatanangularrotationof thetiremovesthedistance

r0sin =x=rD .

Hence,thedynamictireradiusisgivenby

(2.4)

rD=

r0sin

.

(2.5)

For →0onegetsthetrivialsolutionrD=r0.

Atsmall,yet niteangularrotationsthesine-functioncanbeapproximatedbythe rsttermsofitsTaylor-Expansion.Then,(2.5)readsas

rD=r0

1

3=r0

11 2

6

.(2.6)

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VehicleDynamicsFHRegensburg,UniversityofAppliedSciences

Withtheaccordingapproximationforthecosine-function

rS1

=cos =1 2or 2=2r02

one nallygets

rS

1

r0

(2.7)

rD=r0

remains.

11

3

rS1

r0

=

21r0+rS33

(2.8)

TheradiusrDdependsonthewheelloadFzbecauseofrS=rS(Fz)andthusisnamed

dynamictireradius.Withthis rstapproximationitcanbecalculatedfromtheundeformedradiusr0andthesteadystateradiusrS.By

vt=rD

(2.9)

theaveragevelocityisgivenwithwhichtreadparticlesaretransportedthroughthecontactarea.

2.2.2ContactPoint

Thecurrentpositionofawheelinrelationtothe xedx0-,y0-z0-systemisgivenbythewheelcenterMandtheunitvectoreyRinthedirectionofthewheelrotationaxis,

Fig.2.3.

Figure2.3:ContactGeometry

Theirregularitiesofthetrackcanbedescribedbyanarbitraryfunctionoftwospatialcoordi-nates

z=z(x,y).(2.10)

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