MarkusSchmid

TireModelingforMultibodyDynamicsApplications

Simulation‐BasedEngineeringLaboratory

UniversityofWisconsin‐Madison

2011

i

Abstract

Invehicledynamics,tiresareoneofthemostimportantfactorsthatgovernthebehaviorofa

movingvehicle.Theyaretheonlylinkbetweenthevehiclechassisandtheroadandhaveto

transmitvertical,longitudinalandlateralforces.Inordertobeabletodescribeavehicle’s

movementproperly,itisnecessarytounderstandthetires’characteristicsandtheireffecton

certaindrivingsituations.

Whileempiricaldatagainedfromon‐the‐roaddrivingexperimentswithrealtiresis

obviouslythemostaccurate(giventhatconditionsaresimilartothedesiredsimulation),thereare

reasonstochoosedynamicalsimulationsinstead.Sinceproductcyclesanddevelopmenttimesare

continuouslyreducedduetocompetitionandprogressoftechnology,buildingphysicalprototypes

inanearlystageofdevelopmentistime‐consumingandveryexpensive.Furthermore,ifthedesign

ischanged,theprototypehastobeupdatedagainandagain.Clearly,withlargermachinerylike

constructionvehicles,thisbecomestimeandcostprohibitive.Thustendingtowardssimulationsin

earlydevelopmentstagescanbehighlybeneficialtothefinaldesign.

Tiresareverycomplexproductsandnottrivialtodescribe.Theircharacteristicsare

influencedbyseveralfactorslikedesignofthetireitself(e.g.material,treadpattern,carcass

stiffness),airpressureinsidethetireandtextureoftheroadsurface,tonameafew.

Severalmodelingapproacheshavebeendevelopedtodescribethephysicsoftiressothat

theycanbeanalyzedandevaluatedmathematically.Inthefirstpartofthiswork,acombinationof

thesteady‐stateMagicFormulaapproach(describedin[1],[2]and[3])andtheSingleContactPoint

transienttiremodel[3]isusedtoimplementatiremodelthatcanhandletranisentdriving

situationsintothemultibodydynamicsengineChrono::Engine[4].Forvalidationandexperimental

purposes,thetiremodelisthenintegratedwithavehiclemodelanddifferentdrivingsituationsare

analyzed.Thesecondpartofthisresearchintroducesabeamelementtypephysicalmodelthat

permitsinteractionwithflexibleterrainforoff‐roadapplications.

ii

Acknowledgements

Firstofall,IwouldliketogreatlythankmyadvisorattheDepartmentofMechanical

Engineering,ProfessorDanNegrut,forallthesupport,thevaluableadviceandneverrunningoutof

M&M’stokeepupthemotivation.ThankyouDanformakingmepartoftheteam!

Also,thankyouverymuchtoallthepeopleattheSimulation‐BasedEngineeringLaboratory

(SBEL)forthegreatatmosphereinthelab,helpingoutwheneverpossibleandmakingthisagreat

year!

Finally,IwouldliketothanktheGermanAcademicExchangeService(DAAD),theInstitute

forMachineTools(IfW)attheUniversityofStuttgartandtheDepartmentofMechanical

EngineeringattheUniversityofWisconsin‐MadisonformakingmystudiesintheUnitedStatesand

thereforethisworkpossible.

iii

iv

TableofContents

Abstract.......................................................................................................................................................................i

Acknowledgements...............................................................................................................................................ii

TableofContents..................................................................................................................................................iv

ListofFigures.........................................................................................................................................................vi

1 Introduction....................................................................................................................................................1

1.1 TireDynamics.........................................................................................................................................................1

1.1.1 TireForcesandTorques...........................................................................................................................1

1.1.2 LongitudinalandLateralSlip..................................................................................................................2

1.1.3 TurnSpin.........................................................................................................................................................3

1.2 TireModeling..........................................................................................................................................................5

1.2.1 TheMagicFormulaTireModel..............................................................................................................5

1.2.2 TheSingleContactPointTransientTireModel...............................................................................8

2 ImplementationoftheMagicFormulaTireModel.........................................................................14

2.1 MATLABimplementation................................................................................................................................14

2.1.1 Tirepropertyfile(*.tire)........................................................................................................................15

2.1.2 Transienttiremodel................................................................................................................................16

2.1.3 MagicFormulasteady‐statemodel....................................................................................................16

2.1.4 Modelverification.....................................................................................................................................18

2.2 Chrono::Engineimplementation..................................................................................................................24

2.2.1 ThemultibodydynamicssimulationengineChrono::Engine.................................................24

v

2.2.2 TranslationoftheMATLABimplementationtoC++..................................................................24

2.2.3 VehicleandtiremodelusedinChrono::Engine...........................................................................25

2.2.4 TheStandardTyreInterface(STI).....................................................................................................26

2.2.5 OrganizationoftheMagicFormulatiremodelinChrono::Engine.......................................28

2.2.6 Obtainingreal‐timevehicledata.........................................................................................................29

2.2.7 Applicationoftheforcesandmomentstothevehiclemodel.................................................31

2.2.8 Resultsandconclusions.........................................................................................................................32

3 Beamtiremodelforinteractionwithdeformableterrain...........................................................35

3.1 Tiretreadmodel..................................................................................................................................................35

3.2 Forcesinthetiretreadandcontactpatch................................................................................................37

3.3 Modelverification...............................................................................................................................................41

3.3.1 Verticalmotionofthewheel................................................................................................................42

3.3.2 Longitudinalmotionofthewheel......................................................................................................44

3.4 Resultsandfuturework...................................................................................................................................50

4 Summary........................................................................................................................................................51

5 References.....................................................................................................................................................52

A Appendix........................................................................................................................................................54

A.1 MagicFormulaequationsandfactors[3].................................................................................................54

A.2 .tirepropertyfileusedintheMagicFormulaimplementations.....................................................62

vi

ListofFigures

Fig.1.1:Signconventionandtirereferenceframe[3]..............................................................................................1

Fig.1.2:Rotationalslipresultingfrompathcurvatureandwheelcamber[3]..............................................4

Fig.1.3:Thetireasanonlinearfunctionwithmultipleinputsandoutputs(steady‐state)[8]...............5

Fig.1.4:MagicFormulafactors[9]....................................................................................................................................7

Fig.1.5:Sideforcecharacteristicsofa315/80R22.5trucktireforvaryingnormalloads.Comparison

ofMagicFormulacomputedresultswithmeasureddata[3].................................................................................8

Fig.1.6:SingleContactPointTireModel(topview)[3]...........................................................................................9

Fig.1.7:EffectiverollingradiusanddefinitionofslippointS[7].....................................................................10

Fig.1.8:Proceduretocalculateforceandmomentvariationsatthecontactpatch.................................13

Fig.2.1:FlowchartshowingtheconnectionofthetransientmodelandtheMagicFormulamodelin

theMATLABimplementation...........................................................................................................................................15

Fig.2.2:SubroutinesusedintheMATLABimplementationofthetransientmodelandtheMagic

Formula......................................................................................................................................................................................17

Fig.2.3:VelocityprofilesfortestingscenarioA........................................................................................................18

Fig.2.4:TransientslipquantitiesforscenarioA......................................................................................................19

Fig.2.5:LongitudinalforceforscenarioA...................................................................................................................20

Fig.2.6:RollingresistancemomentforscenarioA..................................................................................................20

Fig.2.7:SideslipangleandlateralslipvelocityforscenarioB..........................................................................21

Fig.2.8:TransientslipquantitiesforscenarioB.......................................................................................................22

Fig.2.9:LongitudinalforceforscenarioB...................................................................................................................22

Fig.2.10:LateralforceforscenarioB............................................................................................................................23

Fig.2.11:VisualizationofthevehiclemodelusedinChrono::Engineasspecifiedin

demo_suspension.cpp(modified)...................................................................................................................................25

Fig.2.12:SchematicviewoftheSTI[8]........................................................................................................................27

vii

Fig.2.13:Vectorsusedtodescribethemovementofthevehicle......................................................................29

Fig.2.14:CodesnippetofthegetVehicleData()subroutine......................................................................30

Fig.2.15:Longitudinalvelocitiesfortheleftreartire............................................................................................32

Fig.2.16:Longitudinalslipandlongitudinalforcefortheleftreartire..........................................................32

Fig.2.17:Sideslipangleandlateralslipvelocityfortheleftfronttire...........................................................33

Fig.2.18:Transientslipangleandlateralforcefortheleftfronttire..............................................................33

Fig.3.1:Beamelementsetupforthetiretread:anetofbeamsconnectedtotherim..............................36

Fig.3.2:Circumferentialandradialforcesatelement(i,j)...................................................................................37

Fig.3.3:Setofforcesactingononeelement(i,j).......................................................................................................38

Fig.3.4:Deflectedtireandresultingnormalforcesduetocontactwithground........................................42

Fig.3.5:Deflectedtireandresultingnormalforcesduetocontactwithground........................................43

Fig.3.6:ResultingverticalForcedevelopedinthetirecontactpatch.............................................................43

Fig.3.7:Verticalmovementofthewheelhubcenter(integrationstepsize:0.001s)...............................44

Fig.3.8:Kinematic‐basedfrictionmodel[17]...........................................................................................................45

Fig.3.9:ScenarioA:angularposition,velocityandaccelerationofthewheel.............................................47

Fig.3.10:Slipandtotallongitudinalforce...................................................................................................................48

Fig.3.11:ScenarioB:xvelocityandaccelerationofthewheelcenter............................................................49

Fig.3.12:ScenarioB:Slipandtotallongitudinalforce...........................................................................................50

Fig.A.5.1:Positivedirectionsofforcesandmoments[3]....................................................................................56

1 Int

B

Chrono:

theMagi

1.1 Ti

T

Thesequ

1.1.1 T

T

Thisisfo

adapted

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Themostim

uantitiesallo

TireForce

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orpracticalr

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owustodes

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ateframean

reasons,sin

theSAEstan

Fig.1

implementa

erthebasic

hesinglecon

torsintired

scribethech

ques

ndsignconv

cethetirem

ndardcoordi

1.1:Signconv

1

tionoftheM

ideasoftire

ntactpointt

dynamicsare

haracteristic

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modelsprop

inateframe

ventionandti

MagicFormu

edynamicsa

transientmo

etireforces

csofatirein

dinthiswor

osedin[3]a

(SAEJ670e

irereference

ulatiremode

andtiremod

odel).

andtorques

nalldrivings

kisadopted

alsousethis

1976)andis

frame[3]

elinto

deling(inpa

s,slipandtu

situations.

dfromPacej

sasareferen

sshowninF

articular

urnspin.

ka[3].

nce.Itisan

Fig.1.1.

2

Inthisreferenceframe,

isthespeedoftravelofthewheelcenter.

isthesideslipangle(theanglebetweenthexz‐planeand ).

isthecamberangle(theanglebetweenthexz‐planeandthemeanwheelplane).

istheyawrateorturnslipvelocity.

isthelongitudinalforceactingalongthex‐axis( 0foracceleration, 0for

braking).

isthelateralorsideforce.Itisappliedatapointinadistance behindthecenter

ofthecontactzoneoftireandroad[5].

istheloadornormalforce.

istheself‐aligningtorque.Itiscausedbythenon‐centralapplicationof and

forcesthemeanplaneofthewheeltowardsthedirectionof . canbecalculated

as ∗ ,wheretisthepneumatictrail[5].

Ingeneral, , and arefunctionsoflongitudinalslip ,load ,sideslipangle and

camberangle [3].

1.1.2 LongitudinalandLateralSlip

Unlessthewheelisrollingfreely(withnodrivingorbrakingtorqueappliedand 0),

longitudinalandlateralslipquantitieshavetobeconsidered.Morespecific,atireneedsslip(dueto

elasticdeformationsorsliding)totransmitforces[6].

3

Ingeneral,whenatorqueisappliedtothewheel,wehavetodistinguishtheactualforward

speedofthetireovertheroadsurface(x‐componentofthespeedoftravel: )andthevelocity

thatisclassifiedbythetire’sangularvelocityΩanditseffectiverollingradius [3]:

x0 Ω eV r (1.1)

Toquantifythedifferenceof and ,longitudinalslip isdefinedas[3]:

x0 Ω x x e

x x

V V V r

V V

(1.2)

Thisresultsinnegative forbrakingandpositive foracceleration(positivelongitudinal

slipmeanspositivelongitudinalforce ).

Bythesametoken,lateralorsideslip iscalculatedfromtheratioofthelateralvelocity

andtheforwardvelocity ofthewheel:

tan y

x

V

V (1.3)

Again,positivesideslip meanspositivesideforce .Theslipangle denotestheangle

betweenthetravellingdirectionandtherollingdirectionofthetire.

1.1.3 TurnSpin

Theyawvelocity(orturnslipvelocity)ΨdisplayedinFig.1.2denotestheangularvelocityof

thewheelaboutthenormaltotheroad[7].

I

canbed

where

axleand

T

Inthisconte

escribedas

∗ istheforw

dlocatedatd

Turnslip

t

Fig.1.2

ext,spinslip

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*z

cxV

wardrunnin

distance fr

(thatis,spin

*

1

cxV R

2:Rotationals

(thecomp

ofyawratea

*

sin

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ngspeedoft

romthewhe

nonlydueto

1

R

4

slipresultingwheelcambe

ponent

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,

theimaginar

eelcenterbe

opathcurva

frompathcuer[3]

oftheabsol

angle:

rypoint ∗t

elowroadle

ature,notwh

urvatureand

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hatisperpe

evel[3].

heelcamber

frotationve

endiculartot

r )isdefine

ector )

(1.4)

thewheel

edas:

(1.5)

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M

quantitie

shownin

T

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propose

compare

formulas

ireModel

Mostgenera

es,anglesan

nFig.1.3.

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nditions,sev

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saren’tderi

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3:Thetireas

the“blackb

veralapproa

.

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differenttir

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mentaldata

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bemodeled

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ox”between

acheshaveb

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remodelsth

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.Theapproa

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asanonline

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hatareavaila

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ted,includin

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teralforcesa

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rsteady‐stat

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ormulaTire

veryaccurat

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6

mathematicalapproximationsofcurvesthatwererecordedinexperiments.Forthispurpose,

scalingfactorshavetobeobtainedfrommeasurements.

ThegeneralformoftheMagicFormulais[3]:

sin[ arctan arctan( ) ]y D C Bx E Bx Bx , (1.6)

whereyrepresentsatireforceortorqueandxistheslipquantitythisforceortorquedependson

(i.e.longitudinalorlateralslip).B,C,D,Earefactorstodefinethecurve’sshapeinordertogetan

appearancesimilartotherecordedcurve.Specifically,

B isastiffnessfactor

C isashapefactor

D isthepeakvalue

E isacurvaturefactor

arctan istheslopeofthecurveattheorigin.

Eachofthesefactorshastobeapproximatedfrommeasureddatafromexperimentsforthe

respectivetireandenvironment.Itisalsopossibletoapplyanoffsetin x and y withrespecttothe

origintothisgeneralformula.Anoffsetcanariseduetoply‐steerandconicityeffectsaswellas

wheelcamber[3].Theshiftin x and y canbeperformedbyusingthemodifiedcoordinates

VY X y x S with beingtheverticalshiftand

Hx X S with beingthehorizontalshift.

F

A

thosepa

describe

torque),

A

where

T

showscu

Fig.1.4show

Asinputvari

arametersal

edbythefor

depending

Asanexamp

yoF

istheslip

Todisplayth

urvesobtain

wsaninterpr

iable wec

sodependo

rmulamight

ontheprobl

ple,thesidef

sin[y yD C

angle.Thef

heagreemen

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retationoft

Fig.1.4:

canusetan

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be (longi

lem.

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arctany yB

fullsetoffac

ntoftheMag

easurements

7

thefactorsu

MagicFormu

( beingth

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nbedescrib

(y y yE B

ctorsinvolve

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scompared

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arctan(y B

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tionsthatare

ffects,tirew

ractionwith

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si)steady‐st

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orenotsuffic

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cientto

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springs(

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currentt

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by and

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nswhencom

emodelislim

Thebasicide

point that

(representin

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acanthenbe

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Figure1.6sh

d duetoth

ontactpatch

mbinedwith

mitedtolow

eaofthemo

tisconnecte

ngtheflexib

itudinaland

ongitudinala

eputintoth

rceandtorq

Fig.1

howstheset

hedifferentv

hspeed ′ r

thesteady‐

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odelistocon

edtothewh

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andsideforc

esteady‐sta

quevariation

.6:SingleCon

tupofthem

velocitiesof

respectively

9

stateMagic

(excludinge

ncentratethe

heelrimwith

carcass)allow

ection.Thus

ceaswellas

ateMagicFo

ns[3].

ntactPointTi

odelfromth

slippoint

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.g.ABSbrak

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hlongitudina

wthecontac

theslipofth

self‐alignin

rmulamode

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hetopview.

andsinglec

uations.How

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alandlatera

ctpointtosl

hecontactp

ngtorque.Lo

eldescribed

pview)[3]

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contactpoin

wever,assh

dtireinasi

alsprings.Th

lipwithresp

pointrelative

ongitudinala

in1.2.1toc

sspringsare

nt ′(wheels

hownin

ngle

hese

pectto

eto

andlateral

calculate

edeflected

slipspeed

T

imaginar

istheeff

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freely,i.e

direction

when m

I

displaye

T

compon

Todescribet

rypointisa

fectiverollin

Obviously,th

e.slipisnot

nwiththelo

movessidew

Inthefollow

ed.Theyhav

Thechangeo

entsofthev

du

dt

dv

dt

Fig.1.7:E

themotiono

ttachedtoth

ngradius t

heslippoint

equaltozer

ongitudinals

ways[7].

wing,theequ

vebeenderiv

oflongitudin

velocitiesof

'( sx sV V

'( sy sV V

ffectiverollin

of ′theslip

hewheelrim

thathasbee

tmovesrela

ro.Whenthe

slipvelocity

uationsforth

vedin[3].

nalandlater

S and 'S ,re

)sx

)sy

10

ngradiusand

ppoint isin

mperpendic

endefinedin

ativetothew

ewheelisbr

.Bythe

heSingleCo

raldeflection

espectively,

definitionof

ntroduceda

culartothew

n(1.1).

wheelaxisw

raked,forex

sametoken,

ntactPointt

ns and o

asfollows:

f slippointS[

sareferenc

wheelcenter

whenthewhe

xample, mo

,alateralsli

transienttir

overtimeisr

7]

e(Fig.1.7).

r,wherethe

eelisnotrol

ovesinafor

ipvelocity

remodelare

relatedtoth

This

edistance

lling

rward

arises

e

hexandy

(1.8)

(1.9)

11

Afterseveralconversions,weobtainthefollowingdifferentialequationthatdescribesthe

lateraldeflectionduetosideslip :

1x x sy

dvV v V V

dt

, (1.10)

where istherelaxationlengthforsideslip / ( isthecorneringstiffness, isthe

lateraltirestiffnessatroadlevel)and isthewheelslipangle /| |.

Similarly,weobtainthedifferentialequationthatdescribesthelongitudinaldeflection :

1x x sx

duV u V V

dt

, (1.11)

where / istherelaxationlengthforlongitudinalslip(with beingthelongitudinal

tirestiffnessatroadleveland beingthelongitudinalslipstiffness),and /| |isthe

longitudinalwheelslipratio.

Ifthewheelistiltedsothatthecamberangle 0,anadditionalsideforceapplies.Thisis

partlybecauseofwheelpathcurvature,butalsobecauseofthetransientsideslipangle thatis

developedimmediatelyandcausesadditionalcarcassdeflection .

Thedifferentialequationforthiscaseis:

1| |F

x xF

dv CV v V

dt C

(withsideslipkeptequaltozero) , (1.12)

where isthecamberstiffnessforsideforce.

12

Bythesametoken,regardingtotalspin (thatis,includingturnslipandcamber)yieldsthe

followingdifferentialequation:

1| |F

x xF

dv CV v V

dt C

, (1.13)

where isthespinstiffnessforsideforceandtotalspin isdefinedas( isareductionfactor):

1 1 sin

xV . (1.14)

Usingtheseequations,wearenowabletocalculatethetransientslipquantities , ′and ′

forthetire.First,thecontactpointdeflections and areobtainedbysolvingtheinitialvalue

problemdefinedbythedifferentialequations(1.10),(1.11),(1.12)and (1.13)andusing

velocitiesandtirestiffnessesasinput.Inasecondstep, , ′and ′canbecomputedusingthe

followingequations:

'v

(1.15)

'u

(1.16)

' F

F

C v

C

(1.17)

Withthetransientmodelequationsintroducedabove,itisnowpossibletocalculatefirstthe

transientslipquantitiesandthen,usingtheMagicFormulaequationsshowninappendixA.1,

transientforceandmomentvariationsatthecontactpatch.ThisprocedureisshowninFig.1.8.

13

Fig.1.8:Proceduretocalculateforceandmomentvariationsatthecontactpatch

', ', '

,u v

14

2 ImplementationoftheMagicFormulaTireModel

ThisresearchaimsforanimplementationoftheMagicFormulatiremodelintothe

multibodydynamicsengineChrono::Engine[4].However,fordebuggingandtestingpurposes,the

transienttiremodelandtheMagicFormulasteady‐statemodelareimplementedinMATLABfirst.

Thismakesiteasiertooptimizethestructureofthecodewhileatthesametimethetranslationto

C++(thelanguageinwhichChrono::Engineiswritten)isofreasonableeffort.Also,MATLABoffers

fairlygoodvisualizationtoevaluatemeasureddata.

2.1 MATLABimplementation

Theflowoftheforcesandmomentscalculationprocessperformedateachtimestepis

showninFig.2.1.Specialattentionwaspaidtoaclear,easytounderstandmodularstructure.Thisis

especiallyimportantbecauseofthecomplexityoftheMagicFormulamodelwiththeseveral

subroutinesandthelargeamountofparametersusedtocharacterizethetire.

Thevehiclemodelprovidesvelocities(suchas| |,thelongitudinalvelocityofthewheel

center),wheelspeedofrevolution ,sideslipangle ,camberangle andturnslipvelocityΨ.For

codetestingwithoutneedforanactualvehiclemodel,thosequantitiesareassumedtobegivenat

eachtimestepwhendifferenttestingmaneuversareexamined.

15

Fig.2.1:FlowchartshowingtheconnectionofthetransientmodelandtheMagicFormulamodelintheMATLABimplementation

2.1.1 Tirepropertyfile(*.tire)

Touseavailabletiredataspecifiedforthecommercialmultibodydynamicssimulation

softwareMSCADAMS,itwouldbeappreciatedtouseADAMS’*.tirtiredatafilestoinputtiredata.

However,therearesomecompatibilityissuesbetweenthe2006versionoftheMagicFormulaused

inthisworkandthe2002versionusedinADAMS.Forthisreason,anewyetsimilartireproperty

file(*.tire)isintroduced.Thisfilecontainsalltheinformationtofullycharacterizeaspecifictire

underspecificenvironmentalconditionsusingtheMagicFormulacoefficients.Asmentionedbefore,

thecoefficientscanbeobtainedthroughparameterfittingofmeasurementdata(usuallycarriedout

bythetiresupplier).

Transient tire model- Solve ODEs -

Velocities,

, , , MF parametersfor tire of interest

Magic Formula steady-state model

Transient slip quantities

VelocitiesMF parameters

', ', ', '

Transient tire forces and moments

, , , , ,x y z x y zF F F M M M

Time loopTire property file(.tire) Vehicle model

16

2.1.2 Transienttiremodel

Sincesolvingthedifferentialequationsofthetransientsinglecontactpointmodelrequires

initialvaluesfordeflections and ,initiallystartingfromstandstill(where 0)is

suggested.Inthefollowingtimesteps,thedeflectionsfromtheprevioustimestepareusedasinitial

values.Theinitialvalueproblemsaresolvednumericallyusinga4thorderRunge‐Kuttamethod.

Specialhandlingforverylowvelocities| | hasbeenimplementedtoavoidhigh

deflectionsthatarebeyondwhatisphysicallypossibleandintroducesomeartificialdampingto

reduceoscillationsthatwouldotherwisebeundamped[3].

2.1.3 MagicFormulasteady‐statemodel

IntheMATLABimplementation,thetransientmodelsubroutinecallstheMagicFormula

subroutine,usingthetransientslipquantitiesaswellasvelocitiesandMagicFormulaparametersas

input.TheMagicFormulasubroutineitselfconsistsofsubroutinesforcalculationofeachforceand

momentofinterest.Atfirst,longitudinalforce,lateralforceandself‐aligningmomentarecalculated

forpurelongitudinalandsideslipconditions,respectively.Thesequantitiesarethenusedto

calculatetheactualforcesandmomentsduetocombinedlongitudinalandsideslip.Adetailed

structureofsubroutinesandparametersinvolvedisshowninFig.2.2.

17

Fig.2.2:SubroutinesusedintheMATLABimplementationofthetransientmodelandtheMagicFormula.

readTIRE.m

tireData

alphaPrimegammaPrimekappaPrimephiPrimephi_t

tireDataV_cV_s

getRelaxationLengths.m

can turn slip phi_t

be neglected?no yes

zeta_i = 1(i = 0,…,8)zeta_i ≠ 1

longitudinalForcePure.m lateralForcePure.m aligningTorquePure.m

longitudinalForceCombined.m lateralForceCombined.m aligningTorqueCombined.m

overturningCouple.mrollingResistanceMoment.m

205_60_R15_91V_2-2bar.tire

turnSlip.m

normalLoad.m

F_z

F_y0

F_yF_x

F_x0 M_z0

magicFormula.m

transientMF.m

2.1.4 M

A

andtoid

I

accelera

I

reaches

resulting

Thelong

Modelveri

Asetoftesti

dentifytheir

InscenarioA

ateandbrake

Inthisscena

avelocityof

gfromwhee

gitudinalslip

ification

ngscenario

rlimits.The

A,apurelylo

einastraigh

F

ario,aftersta

fabout

elspeedofre

pvelocity

s,AandB,h

tiretypeuse

ongitudinalm

htline(cf.Fi

ig.2.3:Veloci

artingfroms

13.3 att

evolutionΩ

isthediffe

18

havebeenca

edhereis20

maneuveris

ig.2.3).

ityprofilesfo

standstill,th

=9s.Dueto

andeffectiv

erenceoftho

arriedoutto

05/60R159

sexaminedw

rtestingscen

hevehicleac

oincreasing

verollingrad

osevelocitie

verifytheim

91V.

wherethev

narioA

cceleratesco

longitudinal

dius ishig

s:

mplemented

ehicleisass

onstantlyun

lslip,theve

gher(about1

Ω .

dmodels

umedto

ntilit

elocity

14.6 ).

W

isthenr

asbrake

t=19.9s

F

modelin

builtup

brakesli

T

thisman

braking

appliedf

Whilemaint

ollingfreely

eslipincreas

s.

Figure2.4sh

nconjunctio

whenstarti

ipisdevelop

Theresulting

neuveraresh

theforcetur

for 0.

tainingavelo

y.Att=14s,

ses.Finally,a

Fi

howsthetra

nwiththeM

ngfromstan

ped(negativ

glongitudin

howninFig

rnsnegative

.15.

ocityof13.3

brakingbeg

att=18s,th

ig.2.4:Transi

ansientslipq

MagicFormu

ndstill.Atfre

ve′)andrea

nalforceasw

.2.6.Aposit

e.Asitcanb

19

3 ,theclut

gins.Theslip

hewheelislo

ientslipquan

quantitiesfo

ulamodel.As

eerolling, ′

aches

wellasrollin

tivelongitud

beseeninth

tchisdiseng

pvelocity

ockedandth

ntitiesforscen

orscenarioA

sexpected,p

′dropstoze

1asthewh

ngresistance

dinalforceis

efigure,am

gagedatt=1

becomesp

hevehicleco

narioA

Afoundthro

positivelong

ero.Whenth

heelislocke

emomentap

sappliedfor

maximumbra

11s,sothatt

positiveand

omestoast

oughthetran

gitudinalslip

hewheelisb

ed.

ppliedtothe

racceleratio

akingforce

thewheel

increases

andstillat

nsient

p ′is

braked,

ewheelin

on,for

is

20

Fig.2.5:LongitudinalforceforscenarioA

Fig.2.6:RollingresistancemomentforscenarioA

0 2 4 6 8 10 12 14 16 18 20-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

time [s]

Long

itudi

nal F

orce

Fx [

N]

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

10

time [s]

Rol

ling

Res

ista

nce

Mom

ent

My [

Nm

]

standstil

turntot

T

constant

cornerin

inFig.1.

I

Howeve

Kamm’s

thatcan

andlater

ScenarioBd

lltoaconsta

theright.Fig

Theresulting

tvalueof0.0

ngtotheleft

.1.

Inthiscorne

r,lateraland

Circle[11].

beapplieda

ralforcesfo

describesal

antlongitud

g.2.7shows

Fig.2.7:S

gtransients

05forconsta

tandpositiv

eringmaneu

dlongitudin

Briefly,alat

andvicever

rscenarioB

laterallanec

dinalspeedo

theimposed

Sideslipangl

slipquantiti

antspeedof

veforcorner

uver,lateralf

nalforcesde

teralforcea

rsa.Thisrela

BshowninF

21

changeman

of10 andt

dsideslipan

leandlateral

esareshow

ftravel,whil

ringtotheri

forcesareof

pendoneac

actingonthe

ationshipcan

Fig.2.9andF

euver.Thev

thenperform

ngle andla

slipvelocityf

wninFig.2.8

letransient

ight.Thiscoi

fmoreinter

chother,asd

etirereduce

nbeverified

Fig.2.10.

vehicleisacc

msaturnto

ateralslipve

forscenarioB

.Longitudin

lateralslip

incideswith

estthanlon

describedin

esthemaxim

dintheplots

celeratedfro

theleft,foll

elocity .

B

nalslip ′att

′isnegativ

hthesignco

gitudinalfor

nasimplified

mumlongitu

softhelong

om

lowedbya

tainsa

efor

nvention

rces.

dwayby

dinalforce

gitudinal

Fiig.2.8:Transi

Fig.2.9:Lon

22

ientslipquan

ngitudinalfor

ntitiesforscen

rceforscenar

narioB

rioB

23

Fig.2.10:LateralforceforscenarioB

Asaconclusionfromtheverificationscenarios,theresultsobtainedthroughtheMagic

FormulaMATLABimplementationareplausibleandmatchwhatwouldbeexpectedasbehaviorof

thephysicaltireintherespectivedrivingmaneuvers.Themodelisthereforereadytobetranslated

toC++inordertouseitinChrono::Engineandintegrateitwithavehiclemodel.

0 2 4 6 8 10 12 14 16 18 20-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

time [s]

Late

ral F

orce

Fy [

N]

24

2.2 Chrono::Engineimplementation

AftersuccessfulverificationusingtheMATLABmodel,thenextstepistoswitchfrom

MATLABtoC++andthemultibodydynamicssoftwareChrono::Engine.

2.2.1 ThemultibodydynamicssimulationengineChrono::Engine

Chrono::Engineismiddleware,thatisready‐to‐useC++librarieswithmultibodysimulation

methodsthatcanbeusedforhigh‐performancedynamics,kinematicsandstaticssimulations[12].

Complexrigidbodymechanismscanbemodeledusingalargesetofpre‐definedroutines,jointsand

constraints,motorsandactuators.TheChrono::Enginemiddlewarealsoincludesmodulesforlinear

algebra,advancednumericalmethodsandmethodsforcollisiondetection.

ThearchitectureisbasedonC++headerfilesalongwithpre‐compiledcodeinstaticand/or

dynamiclibraries(.dll)andisstructuredinclasses.Visualizationofthesimulationismadepossible

throughtheopensource3DgraphicsengineIrrlicht.ItsefficiencyandspeedmakeChrono::Engine

suitableforreal‐timesimulations;verylargeproblemscanbesolvedbyleveragingGPUparallel

computingandNVIDIACUDAtechnology.Chrono::EngineisdevelopedbyProfessorAlessandro

Tasora[4]attheUniversityofParma,Italy,andiswidelyusedformultibodydynamicssimulations

intheSimulation‐BasedEngineeringLaboratoryattheUniversityofWisconsin‐Madison.

2.2.2 TranslationoftheMATLABimplementationtoC++

InordertoimplementtheMagicFormulatiremodelintoChrono::Engine,atranslationofthe

existingMATLABcodetoC++needstobeperformed.Whilethebasicstructureofthecodeaswell

asthetirepropertyfileisthesameasintheMATLABimplementation(cf.Fig.2.2),some

optimizationsofdetailshavebeencarriedout,e.g.reorganizationofthenearly300variablesin

structuresforimprovedclarityanddataexchangeandintroductionofaclass“PacejkaTire”for

easierinclusionandbetterinterchangeabilityofthetiremodel.Inaddition,thecodewasmodified

toallow

calculate

2.2.3 V

T

spring‐d

simplifie

ofinertia

demopr

graphica

forinteract

edforcesto

Vehiclean

Thevehiclem

damperwhe

edlineareng

aandareco

rovided,the

aluserinterf

Fig.2.1

tionbetween

thetires).

dtiremod

modelthati

elsuspensio

ginemodel.A

onnectedusi

usercancon

face(GUI).A

11:Visualizat

nthetireand

delusedin

susedinCh

onusingmas

Allbodiesar

ngrevolute

ntrolthrottl

A3Dvisualiz

tionofthevehdemo_s

25

dthevehicle

nChrono::E

hrono::Engin

sslessrods,

reassumed

joints,sphe

le,steeringa

zationofthe

hiclemodeluuspension.cp

emodel(ob

Engine

neisarather

rearwheeld

toberigid,h

ricaljointsa

andspring‐d

evehicleiss

usedinChronpp(modified)

tainingvehi

rsimpleone

drivewitha

haveconsta

anddistance

damperchar

howninFig

no::Engineas

icledataand

e,basicallyc

differential

ntmassand

econstraints

racteristicsi

g.2.11.

specifiedin

dapplying

ontaining

anda

dmoments

s.Inthe

na

26

Thegeometryofthevehicleaswellasthetransmissionandenginesetupandthesuspension

systemaredefinedinthe“MySimpleCar”class.Thismakesiteasiertomanagedataflowsuchas

calculationandapplicationofthewheeltorque.Thesimpletiremodelusedinthismodelisfriction‐

based,sotimeevolutionofthevehicleiscalculatedbysolvingthecontactproblembetweenrigid

bodies(tiresandground)atgivenfrictioncoefficientsandwheeltorque.

Thewheeltorqueiscomputedateverytimestepofthesimulationusinginformationsuchas

throttle(userinput),wheelspeed,finaldriveratioandgearratio.Thetorquecurveoftheengineis

assumedtobelineartokeepthemodelsimple.

2.2.4 TheStandardTyreInterface(STI)

In1997,theso‐calledStandardTyreInterface(STI)wasintroducedbytheinternational

TYDEXworkshopasastandardFORTRAN‐77interfacebetweenvehicle,tireandroadmodel[8].

Theuseofastandardizedinterfacemakesitpossibletoswitchbetweendifferenttiremodels

withouttheneedtomodifythevehicleorroadmodels[13].STIfeaturesanexchangeofallthe

necessaryinformationtoconnectroad,tireandvehiclemodel(cf.Fig.2.12)andalsodefines

coordinatesystemsandphysicalunits(SI).TheTYDEXworkshopalsodefinedtheTYDEXfileformat

tostandardizestorageoftiredataobtainedfromexperimentalmeasurements.

T

thetirem

1

2

3

4

5

6

7

F

between

approac

canbefo

Thefollowin

modelwhen

1) Current

2) Kinemat

3) Forcesan

4) Statevar

5) Signalsfo

6) Tirespec

7) Workarr

Furthermore

nroad,tirea

htheroadis

oundin[14]

nginformatio

nusingSTI[8

simulationt

ticinformati

ndtorquesa

riables,state

forpost‐proc

cificparame

raytostore

e,roadinfor

andvehiclem

sassumedto

].

Fig.2.12:S

onistransfe

8]:

time

ononmotio

appliedtoth

ederivatives

cessing

etersandnam

internalcom

rmation(like

modelusing

obeplanar.

27

Schematicvie

erredbetwe

onofthewh

hewheelcen

sandinitial

meofthetir

mputationre

ealtitude a

STI.Howev

Thefulldes

ewoftheSTI

enthemulti

eelcenter

nter

conditions

repropertyf

esultsofthe

anditsderiv

ver,inthissi

scriptionofv

[8]

ibodysystem

file

etiremodel

vatives)can

mplifiedimp

variablesan

m(MBS)solv

beexchange

plementatio

ndarraysuse

verand

ed

on

edinSTI

28

2.2.5 OrganizationoftheMagicFormulatiremodelinChrono::Engine

InsideChrono::Engine,theMagicFormulatiremodelisorganizedasaclass,“PacejkaTire”,

whichincludesallthenecessarysubroutines,structuresandvariablesthatareneededtodefinethe

tireproperties,theStandardTyreInterfaceandtheconnectiontothevehiclemodel,inthissimple

casethedemovehicle“MySimpleCar”.Twofilesdescribethemodel:“pacejka_definitions.h”,a

headerfilethatcontainsclassdefinitionandfileincludes,and“pacejka_functions.cpp”,which

incorporatestheactualsubroutinesusedbythe“PacejkaTire“class.Uponexecutingthemain

functionofthesimulation,fourtireobjectsarebeinginitialized,eachwithanindependentMagic

Formulamodel(whichalsoallowsfordifferent.tirepropertyfilesforeachtire)andauniquetireID.

29

2.2.6 Obtainingreal‐timevehicledata

Ahighlyaccuratetiremodelmakesitimperativetousereal‐timedatafromthevehicle

model.Forthispurpose,thegetVehicleData()subroutineusesvectorsandtransformation

matricesprovidedbytheChrono::Engineenvironmenttocalculatevehicleandtireinformationsuch

aswheelspeedofrevolution orsideslipangle ,whichistheanglebetweenthez‐axisofthe

wheelspindleanditsabsolutevelocity spindle wheel xV V V .Calculationofthesequantitiestakes

placeateveryintegrationstepofthedynamicssolverandisperformedseparatelyforeachtire.

Fig.2.13:Vectorsusedtodescribethemovementofthevehicle

Toavoidsingularitieswhichcouldhavenegativeimpactonthequalityofslipdataandforce

calculationsofthetiremodel,specialstepsaretakenforcriticalsituations,e.g.whenawheelisata

standstillandtherefore 0spindleV ,whichwouldresultinunrealistic,arbitraryslipangles and

potentiallydisturbthesimulation.

wheelV

chassisV

spindlezvehicle path

wheel path

wheel

30

Asanexampleofinteractionofvehiclemodelandtiremodel,acodesnippetofthe

getVehicleData()subroutineisshowninFig.2.14.Inthissectionofthecode,thewheelangular

velocity andspindlevelocityareusedtocalculatelongitudinalvelocity wheel xV V ofthewheel,

whichisanelementaryinputtotheMagicFormula.

Fig.2.14:CodesnippetofthegetVehicleData()subroutine

FollowingthesamestructureasintheMATLABimplementation,thevehicledatacanthen

beusedtodetermineforces ,,x y zF F F andmoments , ,x y zM M M usingcomputeForces()(called

transientMF.mintheMATLABcodeofFig.2.2)anditssubroutines.

switch (stiParameters.IDTYRE)case1:

// left front tire (tireID 1)

// wheel position in global RF ------------------------------------------------------------stiParameters .DIS[1] = mWheelLF ‐>get_ptr ()‐>GetPos().x;stiParameters .DIS[2] = mWheelLF ‐>get_ptr ()‐>GetPos().y;

stiParameters .DIS[3] = mWheelLF ‐>get_ptr ()‐>GetPos().z;

// wheel rotation in local RF --------------------------------------------------------------ChVector <>wheelRotationLRF = mWheelLF ‐>get_ptr ()‐>GetWvel _loc();stiParameters .OMEGAR = ‐wheelRotationLRF .y;

if (stiParameters.OMEGAR > 0)vehicleData.travelDirection = 1.0;

else

vehicleData.travelDirection = ‐ 1.0;

// -------------------------------------------------------------------------------------------------// transform spindle orientation to global reference frameChMatrix 33<>*spindleMatA = mSpindleLF ‐>get_ptr()‐>GetA ();ChVector <>spindleOrientation = spindleMatA ‐>Matr_x_Vect(VECT_Z);

ChVector <>spindleVelocity = mSpindleLF ‐>get_ptr()‐>GetPos _dt();

// calculate V _x -------------------------------------------------------------------------------vehicleData.V_x = vehicleData .travelDirection *sqrt(spindleVelocity .x *

spindleVelocity .x + spindleVelocity .z * spindleVelocity .z);

31

2.2.7 Applicationoftheforcesandmomentstothevehiclemodel

Inordertoapplythetireforcesandmomentstothevehicle,anewreferenceframeis

attachedtothecontactpatchofeachtire.Theprimarypurposeofthistirereferenceframeisto

transformtheforcesandmomentsobtainedthroughtheMagicFormulasubroutines(andexpressed

inthelocaltirereferenceframe)totheglobalreferenceframeusingthetransformationmatrix A .

Thustheforcesandmomentscaneasilybeappliedtothewheelobjectsinglobalcoordinatesusing

Chrono::Engine’sAccumulate_force() function.

TheverticalforceoftheChrono::Engineimplementationismodeledasaspringforce,

linearlydependingonthedeflectionofthetire.Therefore,thecollisioncontactbetweenwheelsand

groundhasbeendeactivatedtoallowthetires(rigidbodiesinthismodel)tosinkintotheground,

simulatingthetires’deflectionunderload.Inthisway,thevehicleisfloatinginanequilibriumof

normalload(staticanddynamic)andnormalforcesproducedbythetires.

TheverticalforceintheMagicFormulamodelcanbeobtainedusing

01

0

'zz z z Cz

FF p

R and (2.1)

0max(( )cos( ) (1 cos( )),0)z l cr r r , (2.2)

where lr istheloadedtireradius, 0r istheunloadedtireradiusand cr istheradiusofthe

approximatelycirculartirecontour[3].Theotherparametersarespecificfortherespectivetire.

2.2.8 R

I

theonee

Resultsan

Inalongitud

examinedin

0

0

1

` [

-]

0

dconclusi

dinalacceler

n2.1.4,thefo

Fig.2

Fig.2.16:Lon

11

`

Fx

ions

rationofthe

ollowingdat

2.15:Longitu

ngitudinalslip

2

` a

2

32

vehiclefollo

tawerereco

udinalvelociti

pandlongitu

3time [s]

and longitudinal for

3

owedbyala

orded(Fig.2

iesfortheleft

udinalforcefo

4

rce, tire ID 3

4

anechangem

2.15‐Fig.2.1

ftreartire

ortheleftrea

55

maneuversi

18):

rtire

6

6

0

5

milarto

0

5000

Fx [

N]

33

Fig.2.17:Sideslipangleandlateralslipvelocityfortheleftfronttire

Fig.2.18:Transientslipangleandlateralforcefortheleftfronttire

0 1 2 3 4 5 6-4

-3

-2

-1

0

1

2

3

4

time [s]

[-]

[m/s

]Side slip and lateral slip velocity, tire ID 1

V

sy

0 1 2 3 4 5 6-0.5

0

0.5

time [s]

`

[-]

` and lateral force, tire ID 1

0 1 2 3 4 5 6-5000

0

5000

Fy [

N]

`

Fy

34

AsitcanbeseeninFig.2.15andFig.2.16,thelongitudinalaccelerationoftherear‐wheel

drivenvehiclecauseslongitudinalslip(duetothedifferencebetween xV and eR )andtherefore

alongitudinalforce, xF .Thepeaksin eR and sxV arecausedbywheelspinduringacceleration.

Theactuallanechangeisbestexaminedforthesteeredfronttires(Fig.2.17‐Fig.2.18).In

Fig.2.17,thesideslipangle andlateralslipvelocity syV areshownfortheleftfronttire.The

correspondingtransientslipangle ' andthelateralforce yF areshowninFig.2.18.

InFig.2.16andFig.2.18,itcanbeseenthatespeciallythecalculatedlongitudinalforceson

thedrivenwheels,butalsothelateralforcesaresubjecttoheavyoscillations.Thiscouldbefor

severalreasons,likenoiseandalackofdampingintheinputparameters.However,asmentionedin

1.2.2,thesimplecontactpointmodelusedtocalculatetransientslipquantitiesislimitedtolow

slippages.Toavoidsuchproblems,switchingtoamoreadvanced,butalsomorecomplexand

computationallymoreexpensivetransientmodeltoproduceinputdatafortheMagicFormula

modelmightbeconsidered.

35

3 Beamtiremodelforinteractionwithdeformableterrain

Thesemi‐empiricalPacejkaMagicFormulatiremodeldescribedinthepreviouschaptersis

veryaccurateonrigidterrainandpowerfulforreal‐timeapplications.Itis,however,alsorelianton

precedingprecisemeasurementsoftherespectivetire.FurtherpracticalissuesoftheMagic

Formulatiremodelindailyworkhavebeendescribedin[15].Also,itthemodelisnotcapableof

interactingwithdeformable,undulatedterrainsuchassoil,sandorsnow.Foroff‐roadvehicle

applicationsthisinteractionisanimportantfactorthathastobeconsideredandthetiremodel

shouldbeabletohandle.Allowingfordeformableterrainalsotakesintoaccounttherolling

resistanceduetoelastichysteresisofthegroundandbulldozingresistancewhengroundmaterialis

pushedinfrontofthecontactpatchofthetire[5].Thereforeinthischapteraphysics‐basedtire

modelusingabeam‐basedapproachissuggested.

3.1 Tiretreadmodel

Inthismodel,thetiretreadisrepresentedbyasetofmasslessthinbeamswithequivalent

springstiffnesses /axialk AE L (axial)and 3 3 /lateralk EI L (lateral)[16].Theradialbeamsare

connectedtothewheelrimatonesideandarelinkeduptoeachotherattheothersidewithshort

connectingbeamstoformanet,simulatingthetiretread(Fig.3.1).Thisconfiguration,also

consideringfrictionbetweenthebeamtippoints(furtherreferredtoas“elements”)andground,

allowsforlongitudinalaswellaslateralflexibilityandthereforeslipinxandydirectionstoinduce

longitudinalandlateralforcesinthecontactpatch.Itisimportanttonotethatthetireisnottreated

asadynamicsystemwithmassesandequationsofmotionassociatedwithit,butasaforceelement.

Thisisjustifiedbytheobservationthat,whenconnectedtoavehiclemodel,nottheforcesinsidethe

tireareofinterest,buttheforcesthatareappliedtothewheelhub.Consequently,itisassumedto

36

besufficienttoobtaintheequationsofmotionforthewheel,therebyconsideringtheforcescreated

bythedeflectionofthebeams.

rz

Fig.3.1:Beamelementsetupforthetiretread:anetofbeamsconnectedtotherim

Forsimplicityandtolimitcomputationalcostofthesimulation,bendingandtorsional

momentsinthebeamsarebeingneglectedandsmalldisplacementsandrotationsareassumed.

37

3.2 Forcesinthetiretreadandcontactpatch

Sincethecontactpatchismodeledasacollectionofbeamsorientedinlongitudinal,lateral

andradialdirection,numerousforcesbetweenthebeamshavetobeconsidered.Theseforcesarea

resultoftherelativedisplacementoftheelementswithrespecttoeachotheraswellasthewheel

rimduetoloadonthetireandinteractionwithground(friction,normalforce).Asideviewofthe

tirewithforcesin and r directionisshowninFig.3.2.

,i jr, 1i jr

, 2i jr

, 1i jr ,i jrF

,i jF

,i jF

Fig.3.2:Circumferentialandradialforcesatelement(i,j)

Theradialforces rF representthetirecarcassstiffnessandtheairpressureinsidethetire

andcausedeformationoftheterrainunderthecontactpatchifthegroundisnotrigid.Atopview

schematicoffiveelementsofthetiretreadandthecorrespondingforcesisshowninFig.3.3.

38

1F

2F

1zF 2zFrF

r z

1,i j

, 1i j

, 1i j

1,i j

,i j

Fig.3.3:Setofforcesactingononeelement(i,j)

Theforcesactingonanelement ,i j andresultingfromtherelativedeflectionofthebeams

toeachotheraswellastheradialdeflectionwithrespecttotheunloadedtireradiuscanbe

calculatedasfollows(cf.Fig.3.3):

I. Forcesin direction:

Force ,1F duetolongitudinaldeflectionofbeam1:

,1 , , , 1 ,( )axial i j i j i j refF k r (3.1)

Force ,2F duetolateraldeflectionofbeam2:

,2 , , 1, ,( )z lateral i j i j i jF k r (3.2)

39

Force ,3F duetolateraldeflectionofbeam3:

,3 , , 1, ,( )z lateral i j i j i jF k r (3.3)

Force ,4F duetolongitudinaldeflectionofbeam4:

,4 , , , , 1( )axial i j i j i j refF k r (3.4)

Force ,5F duetolateraldeflectionoftheradialbeam:

,,5 , , , 0 0( )i jr lateral i j i jF k r r

(3.5)

Thesumofforcesinthe directionforoneelement ,i j istherefore:

,

, , , , 1 , , , 1, ,

, , 1, , , , , , 1

, , , 0 0

( ) ( )

( ) ( )

( )i j

i axial i j i j i j ref z lateral i j i j i ji

z lateral i j i j i j axial i j i j i j ref

r lateral i j i j

F F k r k r

k r k r

k r r

(3.6)

II. Forcesin z direction:

Force ,1zF duetolateraldeflectionofbeam1:

,1 , , 1 ,( )z lateral i j i jF k z z (3.7)

Force ,2zF duetolongitudinaldeflectionofbeam2:

,2 , , 1,( )z z axial i j i j refF k z z z (3.8)

40

Force ,3zF duetolongitudinaldeflectionofbeam3:

,3 , 1, ,( )z z axial i j i j refF k z z z (3.9)

Force ,4zF duetolateraldeflectionofbeam4:

,4 , , 1 ,( )z lateral i j i jF k z z (3.10)

Force ,5zF duetolateraldeflectionoftheradialbeam:

,,5 , , 0( )i jz r lateral i jF k z z (3.11)

Thus,thesumofforcesinthe z directionforoneelement ,i j is:

,

, , , 1 , , , 1,

, 1, , , , 1 ,

, , 0

( ) ( )

( ) ( )

( )i j

z z i lateral i j i j z axial i j i j refi

z axial i j i j ref lateral i j i j

r lateral i j

F F k z z k z z z

k z z z k z z

k z z

(3.12)

III. Forcesin r direction:

Force ,1rF duetolateraldeflectionofbeam1:

,1 , , 1 ,( )r lateral i j i jF k r r (3.13)

Force ,2rF duetolateraldeflectionofbeam2:

,2 , 1, ,( )r z lateral i j i jF k r r (3.14)

Force ,3rF duetolateraldeflectionofbeam3:

41

,3 , 1, ,( )r z lateral i j i jF k r r (3.15)

Force ,4rF duetolateraldeflectionofbeam4:

,4 , , 1 ,( )r lateral i j i jF k r r (3.16)

Force ,5rF duetoradialdeflectionofthecenterelement(thisincorporatestireinflation

pressureandcarcassstiffness:

,5 , ,( )r r axial ref i jF k r r (3.17)

Theresultingradialforceis

, , , 1 , , 1, ,

, 1, , , , 1 ,

, ,

( ) ( )

( ) ( )

( )

r r i lateral i j i j z lateral i j i ji

z lateral i j i j lateral i j i j

r axial ref i j

F F k r r k r r

k r r k r r

k r r

. (3.18)

Bysumminguptheforcesofalltheelementsthatarepartofthecontactpatchin , r and

z direction,wecanthenobtaintheoverallforces(andmoments)actingontherim.

3.3 Modelverification

Thebeammodelisimplementedintoasimulationenvironmentandverifiedusingaself‐

writtenMATLAB2DsimulationenginecalledSimEngine2D,developedinProfessorDanNegrut’s

“ME451:Kinematics&DynamicsofMachineSystems”classattheUniversityofWisconsin‐Madison.

SimEngine2Dfeaturesstandardizeddatainput(twodatafilestocharacterizetheanalysisandthe

desiredmodel)andusesaNewmarkintegratortoperformdynamicsanalysis.Theadvantageofthis

simulationpackageisitsversatileandhighlycustomizablecode,facilitatingtheimplementationof

thetiremodelasasetofforceelementsaswellasitsattachmenttotherigidbodyrim.However,

42

sincethesoftwareislimitedto2‐dimensions,onlyverticalandlongitudinalmotionofthewheelcan

beexamined.

3.3.1 Verticalmotionofthewheel

Thenormalforceisverifiedby“dropping”thetire(with240elementsin4layers)tothe

groundfromaheightof0.35m(theunloadedtireradiusbeing0.305m).Sinceaflexibleterrain

modelhasn’tbeenimplementedyet,thegroundisassumedtobenon‐deformable.However,the

respectiveelementscouldinteractwiththeterrainaswell.

Wewouldexpectthetiretodeflectradiallyinthecontactpatchzoneandtherebydevelopa

resultingverticalforcethatcounteractsthegravitationalinertiaofthetire.Indeed,thetire

successivelydeformsasithitstheground(cf.Fig.3.4andFig.3.5)andaverticalforce(thesumof

theforcesatalloftheelements,Fig.3.6)pushesthewheelbackupsothatitstartsbouncingupand

downcontinuously(Fig.3.7).

Fig.3.4:Deflectedtireandresultingnormalforcesduetocontactwithground

(t=0.125s‐2Dview)

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

43

InFig.3.6andFig.3.7,itcanbeobservedthatdampingoccurs(theamplitudedecreases)–

however,thisisduetonumericalintegrationdampingsincethebeamtiremodelitselfdoesn’t

includeatthistimeanymechanicaldamping.

Fig.3.5:Deflectedtireandresultingnormalforcesduetocontactwithground

(t=0.125s‐3Dviewofcontactpatch)

Fig.3.6:ResultingverticalForcedevelopedinthetirecontactpatch

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

200

400

600

800

1000

1200

1400

1600

1800

time [s]

Fr [

N]

3.3.2 L

F

contactp

direction

between

relatedt

I

Fig.3.7

Longitudin

Forlongitud

patchisnece

nandthusin

ntheelemen

totheonede

Inthecaseo

z

7:Verticalmo

nalmotion

dinal(andlat

essaryinord

nducelongit

ntsandgrou

escribedin[

ofbraking( v

1 0

0

.

ovementofth

nofthewh

teral)motio

dertodeflec

tudinalandl

ndinlongitu

[17]isused

v r ),the

44

hewheelhub

eel

nofthetire

cttheeleme

lateralforce

udinaldirec

(Fig.3.8):

normalized

center(integ

,frictionbet

ntsincircum

es,respectiv

ction,akinem

dlongitudina

grationsteps

tweentirea

mferential(

ely.Tomode

matic‐based

alslip z isde

ize:0.001s)

ndgroundin

)orlatera

elfrictional

frictionmo

efinedas

nthe

al( z )

contact

del

(3.19)

F

yields

forbraki

fordrivi

T

motionb

Forsimplicit

bs

ing( ,v r

ds

ng( ,v r

Toobtainth

betweenthe

,i j

F

ty,however,

1r

v

,

, 0v )and

1v

r ,

0 ).

edeflection

ebeamelem

,0 ,i j i js

Fig.3.8:Kinem

,insteadwe

s(cf.Fig.3.2

entsandgro

,

45

matic‐basedf

usethedefi

2and 1

ound,thefol

frictionmode

initionofsli

0 inFig.3.8)

llowingemp

el[17]

pasalready

resultingfr

piricalrelatio

ydefinedin(

romtherelat

onisused:

(1.2)‐this

(3.20)

(3.21)

tive

(3.22)

46

where ,i j istheactualangularcoordinate(polarcoordinatesystem)oftheelement ( , )i j ,,0i j

is

thereferenceangularcoordinateoftheundeformedelement(free‐rollingtire), s isslipforbraking

ordriving, isthekinematicfrictioncoefficientbetweentireandground,and isacorrection

factor.

Thus,theprocessofdevelopinglongitudinalforcesinthetirecontactpatchisthefollowing:

Withthewheelspinning(forexamplebyapplyingatorquetothewheelhub)orbypullingthetire,

relativemotionbetweenthetireandground(i.e.slip)results,causingthebeamelementsinthe

contactpatchtodeformin directionasdepictedinFig.3.8.Thisgivesrisetoalongitudinal

reactionforcethatcanbecalculatedusingtheequationsintroducedin3.2andthateventually

contributestothegeneralizedforcesappliedtothewheelinSimEngine2D.

Toverifythisprocess,thefollowingscenarioshavebeenimplementedinSimEngine2D:

A.Pullingthe(non‐rotating)wheeloverground,thewheelshouldstartturning;

B.Rotatingthewheelhub,alongitudinalmotionshouldresult.

Foreachscenario,thefollowingvaluesareused:

kinematicfriction(constant) 0.4

correctionfactor 0.25

unloadedtireradius 0.305refr m

loadedtireradius 0.295r m

radiusoftherim 0.2rimr m

tirewidth 0.2b m

Theequivalentspringstiffnessesareasfollows:

beamsin direction:

axial , 100axialNk m

lateral , 100lateralNk m

beamsin z direction:

axial , 100z axialNk m

lateral , 100z lateralNk m

beamsin r direction:

axial , 5000r axialNk m

lateral , 100r lateralNk m

I

speedof

thewhee

InscenarioA

f 0.5v m

eltostartro

Fig

A,thewheel

s .Through

otating(cf.F

g.3.9:Scenari

(1080elem

themechan

Fig.3.9):

oA:angularp

47

mentsin3lay

ismsdescrib

position,velo

yers)ispulle

bedabove,t

ocityandacce

edhorizonta

thetireelem

elerationofth

allyatacons

mentdeflectio

hewheel

stant

onscause

T

coincide

F

correspo

zero,the

I

standstil

therotat

(Fig.3.1

Theangular

eswiththea

Figure3.10s

ondingsum

elongitudina

InscenarioB

llandarota

tionistrans

1):

velocityapp

ngularveloc

showsthed

oflongitudi

alforcealso

B,thewheel

ation(

formedtoa

proachesan

cityofafree

ecreaseofb

nalforcesof

decreasesa

Fig.3.10:Sl

(withtirem

0.4 rad s

longitudina

48

asymptotic

e‐rollingwhe

brakeslipas

falltheelem

andeventua

ipandtotallo

modelspecifi

s )isapplied

almotionby

maximumo

eel:

thewheelst

mentsinthe

llybecomes

ongitudinalfo

icationssim

dtothewhee

thetireelem

of 1.695

0.5 0v r

tartsrotatin

contactpatc

zeroforthe

orce

milartoscena

elhub.Thes

mentsalmos

rad s ,whic

0.295 1.695

ngandthe

ch.Asslipap

efree‐rolling

arioA)isini

simulations

stinstantane

ch

5rad s .

pproaches

gwheel.

tiallyat

howsthat

eously

S

accelera

velocity

Similartosc

ationofthew

plots,tracti

Fig.3.11:Sce

enarioA,th

wheeldecrea

onisbuiltu

enarioB:xve

etotallongi

asesasslipd

pveryquick

49

elocityandac

tudinalforc

decreases(F

kly.

ccelerationof

e F andthe

Fig.3.12).As

fthewheelce

ereforethel

sitcanbese

enter

longitudinal

eeninthesli

ipand

50

Fig.3.12:ScenarioB:Slipandtotallongitudinalforce

3.4 Resultsandfuturework

Thebeamtiremodel’sfunctionalityhasbeenshownintheprecedingchapter.However,this

rathersimplemodelstillneedsvalidation–e.g.sincetheelementdeflectionsandthereforeforce

calculationsarelinear,thismodelmightberestrictedtosmallslipconditions,wherethefriction

coefficientandthelongitudinalforcearenearlylinearfunctionsofslip.

Anoff‐roadimplementationofthetiremodelfurthermorerequiresanappropriateinterface

betweentireandflexibleterraintoobtaintheradial,circumferentialandlateraldeflectionsofthe

beamelements.Foroff‐roaduse,factorssuchassinkagealsohavetobeconsidered,resultingin

higherrollingresistance.Thisofcoursedependsontheterrainmodelused.

Also,sincethesimulationpackageusedispurely2D,portingtheimplementationofthetire

modeltoa3DengineorcommercialsoftwaresuchasMSCADAMScanbeconsidered.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

500

time [s]

Fal

pha [

N]

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.5

1

slip

[-]

Falpha

slip

51

4 Summary

Inthiswork,thefocushasbeenonimplementingtwoverydifferenttiremodelsforusein

multibodydynamicssimulations.Thesemi‐empiricalPacejkaMagicFormulatiremodelhasshown

tobeaveryaccurateapproachonflat,rigidterrainbuthassomedifficultieswhenitcomestohighly

dynamicdrivingsituations.Also,itisnotsuitableforoff‐roadapplicationsthatrequireinteraction

withflexibleterrainorundulatedsurfaceswithshortwavelengths.

Forthisreason,asecondmodelhasbeenintroducedthatusesbeamstosimulatethetire.

Thisdiscretemodelhasaphysicalbackgroundbutisalsosubjecttolimitationsthatcouldbesolved

byextendingandfine‐tuningthemodel.Thismodelcertainlyisn’tsuitableforhighlydynamicon‐

roadapplicationsbuthasbeendesignedtosimulatee.g.construction‐typevehiclesmovingover

flexibleterrainsuchassoil,gravelorsand.

52

5 References

[1]EgbertBakker,LarsNyborg,andHansB.Pacejka,"TyreModellingforUseinVehicleDynamicsStudies,"SAEPaperNo.8704211987.

[2]EgbertBakker,HansB.Pacejka,andLarsLidner,"ANewTireModelwithanApplicationinVehicleDynamicsStudies,"SAEPaperNo.8900871989.

[3]HansB.Pacejka,TireandVehicleDynamics.Warrendale:SAEInternational,2006.

[4]AlessandroTasora.(2011)DeltaKnowledgeWebsite.[Online].http://www.deltaknowledge.com/chronoengine/

[5]GiancarloGentaandLorenzoMorello,TheAutomotiveChassis,Volume1:ComponentsDesign.Berlin:Springer,2009.

[6]Hans‐HermannBraessandUlrichSeiffert,HandbookofAutomotiveEngineering.Warrendale:SAEInternational,2005.

[7]AkiraHiguchi,"TransientResponseofTyresatLargeWheelSlipandCamber,"TU‐Delft,Dissertation1997.

[8]IgoJ.M.Besselink,"ExperienceswiththeTYDEXstandardtyreinterfaceandfileformat,"inTyreModelsforVehicleDynamicsAnalysis.London:Taylor&Francis,2005,vol.43Supplement1,pp.63‐75.

[9]DieterSchramm,ManfredHiller,andRobertoBardini,ModellbildungundSimulationderDynamikvonKraftfahrzeugen.Berlin:Springer,2010.

[10]FrancescoBraghinandEdoardoSabbioni,"ADynamicTireModelforABSManeuverSimulations,"TireScienceandTechnology,no.Vol.38,No.2,pp.137‐154,2010.

[11]JochenWiedemann,SkriptKraftfahrzeugeI(Wintersemester2009/2010).Stuttgart:InstitutfürVerbrennungsmotorenundKraftfahrwesen,UniversitätStuttgart,2009.

[12]AlessandroTasora,MarcoSilvestri,andPaoloRighettini,"ArchitectureoftheChrono:EnginePhysicsSimulationMiddleware,"inProceedingsofMultibodyDynamics.ECCOMASThematicConference,Milano,Italy,2007.

[13]J.J.M.VanOosten,H.‐J.Unrau,A.Riedel,andE.Bakker,"TYDEXWorkshop:StandardisationofDataExchangeinTyreTestingandTyreModelling,"VehicleSystemDynamics,no.27S1,pp.272‐288,1997.

[14]AndreasRiedelandUweWurster,"STI‐StandardizedInterfaceTyreModel‐VehicleModel(Release1.4),"Karlsruhe,1996.

53

[15]JochenRauhandMonikaMössner‐Beigel,"Tyresimulationchallenges,"VehicleSystemDynamics,no.46S1,pp.49‐62,2008.

[16]HorstIrretier,GrundlagenderSchwingungstechnikI.Braunschweig:Vieweg,2000.

[17]CarlosCanudas‐de‐Wit,PanagiotisTsiotras,EfstathiosVelenis,MichelBasset,andGerardGissinger,"DynamicFrictionModelsforRoad/TireLongitudinalInteraction,"VehicleSystemDynamics,no.39Issue3,pp.189‐226,2003.

54

A Appendix

A.1 MagicFormulaequationsandfactors[3]

Parametersused

g accelerationduetogravity

cV magnitudeofthevelocityofthewheelcontactcenterC

,cx yV componentsofthevelocityofthewheelcontactcenterC

,sx yV componentsofslipvelocity sV (ofpointS)with s cyV V

rV ( )e cx sxR V V forwardspeedofrolling

oV referencevelocity(= ogR orotherspecifiedvalue)

oR unloadedtireradius ( )or

eR effectiverollingradius ( )er

wheelspeedofrevolution

z tireradialdeflection( 0 ifcompression)

zoF nominal(rated)load( 0 )

'zoF adaptednominalload: 'zo Fzo zoF F

Otherquantities

'

'z zo

zzo

F Fdf

F

normalizedchangeinverticalload

* tan( )sgn( )| |

cycx

cx

VV

V tangentoftheslipangle(forverylargeslipangles)

* sin spinduetocamberangle

| |sx

cx

V

V longitudinalslipratio

cos'( )'

cx cx

c c V

V V

V V

avoidsingularitiesat 0cV byaddingasmall 0.1V

1 ( 0,..,8)i i factors i canbesetequaltounitywhenturnslipmay

beneglected(pathradiusR )andcamber remainssmall

55

Userscalingfactors (defaultvalueofthesefactorsissetequaltooneifnotused)pureslip

Fzo nominal(rated)load

,x y peakfrictioncoefficient(lessthanoneforlowfrictionroadsurface)

V withslipspeed sV decayingfriction(defaultvalueisequaltozero!)

Kx brakeslipstiffness

Ky corneringstiffness

,Cx y shapefactor

,Ex y curvaturefactor

,Hx y horizontalshift

,Vx y verticalshift

Ky camberforcestiffness

Kz cambertorquestiffness

t pneumatictrail(effectingself‐aligningtorquestiffness)

Mr residualtorque

,*,

1

x yx y

sV

o

V

V

,

*,

, *,

'1 ( 1)

x yx y

x y

A

A

(suggestion: 10A )

combinedslip

x influenceon ( )xF

y influenceon ( )yF

Vy induced‘ply‐steer’ yF

s zM momentarmof xF

other

Cz radialtirestiffness

', ', ', '

,u v

overturningcouplestiffness

My rollingresistancemoment

L

F

C

D

E

K

B

S

S

L

F

α

C

Longitudinal

sinxo xF D

Hxκ Sx

1 x Cx CC p

x x zD F

1x Dxp

1x ExE p

x z KK F p

/ (x xB K C

1Hx HxS p

Vx z VS F p

LateralForce

yo sinyF D

*y Hα α S

1 y Cy CC p

lForce

n[ arctanxC

( 0)Cx

1 ( 0)

2 Dx zp df

2 Ex zp df p

2 2Kx K zp df

)x x xC D

2 Hx zp df

1 2 Vx Vx zp df

e(pureside

yn[ arctanC

Hy

( 0)Cy

Fig.A.5.1:

x x xB E B

*x

23 z f 1Exp d

3 exp Kp d

Hx

cx

Vx cx

V

V

eslip)

y y α yB E

56

Positivedireandmoment

arctax xB

4 Exp sgn

z Kxdf B

1 ' Vx x

y yα arctaB

ectionsofforcts[3]

an ]x xB

Ex λ (x

x x xB C D at

y yan α ]B

ces

VxS

1)

0 (x C

Vy] S

)FC

(A.5.1)

(A.5.2)

(A.5.3)

(A.5.4)

(A.5.5)

(A.5.6)

(A.5.7)

(A.5.8)

(A.5.9)

(A.5.10)

(A.5.11)

(A.5.12)

(A.5.13)

57

2 y y zD F (A.5.14)

Dy1 Dy2 z *μy*2

Dy3

p p df λ ( 0)

1 p γy

(A.5.15)

*2 *21 2 5 3 4 1 signy Ey Ey z Ey Ey Ey y EyE p p df p p p (A.5.16)

' *2z

1 Ky4 3 3*2 'Ky2 Ky5 z0

Fsin p arctan / (1 )

p p γ Fy Ky zo Ky KyK p F p

(A.5.17)

yy y y

yKB

C D

(A.5.18)

*0 0

1 2 4

1

y Vy

Hy Hy Hy z Hy

y K

K SS p p df

K

(A.5.19)

*3 4 2 ' Vy z Vy Vy z Ky yS F p p df (A.5.20)

1 2 2 ' Vy z Vy Vy z Vy y VyS F p p df S (A.5.21)

6 7y o z Ky Ky z KyK F p p df (A.5.22)

Self‐aligningTorque(puresideslip)

'zo zo zroM M M (A.5.23)

'zo o yoM t F (A.5.24)

( ) cos[ arctan ( arctan( ))]cos'( )o t t t t t t t t t tt t D C B E B B (A.5.25)

*t HtS (A.5.26)

*1 2 3 4( )Ht Hz Hz z Hz Hz zS q q df q q df (A.5.27)

( ) cos[ arctan( )]zro zr r r r r rM M D C B (A.5.28)

* ( )r Hf fS (A.5.29)

'Vy

Hf Hyy

SS S

K

(A.5.30)

'y y KK K (A.5.31)

58

2 * *21 2 3 5 6 *

( )(1 | | ) ( 0)Kyt Bz Bz z Bz z Bz Bz

y

B q q df q df q q

(A.5.32)

1( 0)t CzC q (A.5.33)

1 2( ) sgn'o

to z Dz Dz z t cxzo

RD F q q df V

F (A.5.34)

* *23 4 5(1 | | )t to Dz DzD D q q (A.5.35)

2 *1 2 3 4 5

2( )1 ( ) arctan( )( 1)t Ez Ez z Ez z Ez Ez t t tE q q df q df q q B C

(A.5.36)

9 10 6 9*( ) ( : 0)Ky

r Bz Bz y y Bzy

B q q B C preferred q

(A.5.37)

7rC (A.5.38)

*6 7 2 8 9 0

* * *10 11 0 8

( ) ( ) ...

... ( ) | | cos'( ) sgn 1

r z o Dz Dz z Mr Dz Dz z Kz

Dz Dz z y cx

D F R q q df q q df

q q df V

(A.5.39)

, 0 ( ~ , 0)( )zoz o to y y M

y

MK D K C

(A.5.40)

8 9( ) ( ~ , 0)( )zoz o z o Dz Dz z Kz to y o y M

MK F R q q df D K C

(A.5.41)

LongitudinalForce(combinedslip)

x x xoF G F (A.5.42)

cos[ arctan ( arctan( )) /x x x s x x s x s x oG C B E B B G (A.5.43)

cos[ arctan ( arctan( ))]x o x x Hxa x x Hxa x HxaG C B S E B S B S (A.5.44)

*s HxS (A.5.45)

*21 3 2( ) cos[arctan( )] ( 0)x Bx Bx Bx xB r r r (A.5.46)

1x CxC r (A.5.47)

1 2x Ex Ex zE r r df (A.5.48)

1Hx HxS r (A.5.49)

59

LateralForce(combinedslip)

y y yo VyF G F S (A.5.50)

cos[ arctan ( arctan( ))] /y y y s y y s y s y oG C B E B B G (A.5.51)

cos[ arctan ( arctan( ))]y o y y Hy y y Hy y HyG C B S E B S B S (A.5.52)

s HyS (A.5.53)

*2 *1 4 2 3( ) cos[arctan ( )]y By By By By yB r r r r (A.5.54)

1y CyC r (A.5.55)

1 2 ( 1)y Ey Ey zE r r df (A.5.56)

1 2Hy Hy Hy zS r r df (A.5.57)

5 6sin[ arctan( )]Vy Vy Vy Vy VyS D r r (A.5.58)

* *1 2 3 4 2( ) cos[arctan( )]Vy y z Vy Vy z Vy VyD F r r df r r (A.5.59)

NormalLoad

01

0

'zz z z Cz

FF p

R (A.5.60)

0max(( ) cos( ) (1 cos( )),0)z l cr r r (A.5.61)

( lr :loadedtireradius, cr :radiusofcirculartirecontour(approx.))

OverturningCouple

*1 2 3( )

'y

x z o sx sx sx Mxzo

FM F R q q q

F (A.5.62)

RollingResistanceMoment

1 2 arctan 'xr

y z o sy sy Myo zo

FVM F R q q

V F (A.5.63)

Self‐aligningTorque(combinedslip)

'z z zr xM M M sF (A.5.64)

60

' 'z yM tF (A.5.65)

, , , ,( ) cos[ arctan ( arctan( ))]cos'( )t eq t t t t eq t t t eq t t eqt t D C B E B B (A.5.66)

'y y VyF F S (A.5.67)

, ,( ) cos[ arctan( )]zr zr r eq r r r r eqM M D C B (A.5.68)

*1 2 3 4 ( )

'y

o sz sz sz sz z szo

Fs R s s s s df

F (A.5.69)

2 2 2, ( ) sgn( )

'x

t eq t ty

K

K

(A.5.70)

2 2 2, ( ) sgn( )

'x

r eq r ry

K

K

(A.5.71)

ExtensionofthemodelforTurnSlip(if t islarge,otherwise 1, 1,...,8i i )

2 0 4 0cos[arctan ( | | | |)]y t Dy tB R p R (A.5.72)

1 2 3(1 ) cos[arctan( tan )]y Dy Dy z DyB p p df p (A.5.73)

2 23 1 0cos[arctan( )]Ky tp R (A.5.74)

0 0 0sin[ arctan ( arctan( ))]Hy Hy Hy Hy Hy Hy HyS D C B R E B R B R (A.5.75)

1 2( )'

VyHy Hy Hy z Hy Hy

y

SS p p df S

K

(A..5.76)

*3 4 2( ) 'Vy z Vy Vy z Ky yS F p p df (A.5.77)

0 0 (A.5.78)

4 1'

VyHy

y

SS

K

(A.5.79)

1Hy HyC p (A.5.80)

2 3( ) sgn( )Hy Hy Hy z cxD p p df V (A.5.81)

4Hy HyE p (A.5.82)

0

0( )yR

HyHy Hy y K

KB

C D K

(A.5.83)

61

00 1

yyR

y

KK

(A.5.84)

1 2(1 )zp p df (A.5.85)

5 1 0cos[arctan( )]Dt tq R (A.5.86)

8 1 rD (A.5.87)

sin[ arctan ( arctan( ))]r Dr Dr Dr o Dr Dr o Dr oD D C B R E B R B R (A.5.88)

sin(0.5 )z

DrDr

MD

C

(A.5.89)

1 00'

zz Cr y z M

z

FM q R F

F (A.5.90)

1Dr DrC q (A.5.91)

2Dr DrE q (A.5.92)

0

( )(1 )z r

DrDr Dr r

KB

C D

(A.5.93)

0 0 8 9( )z r z Dz Dz z KzK F R q q df (A.5.94)

0 0 0 0z z r t yK K D K (A.5.95)

00 1

zzR

KK

(A.5.96)

6 1 0cos[arctan( )]Brq R (A.5.97)

90 2 0

2arctan( | |) ( )z z Cr t yM M q R G

(A.5.98)

907

2arccos[ ]

| |z

r r

M

D

(A.5.99)

1 0cos[arctan( )]xB R (A.5.100)

1 2 3(1 ) cos[arctan( )]x Dx Dx z DxB p p df p (A.5.101)

62

A.2 .tirepropertyfileusedintheMagicFormulaimplementations

%========================================================================== % Magic Formula parameters % Tire: 205/60R15 91V, 2.2bar (Pacejka, 2006) % Comments: %========================================================================== %-------------------------------------------------------------------------- % general parameters %-------------------------------------------------------------------------- % free unloaded tire radius R_0 = 0.313 % effective rolling radius (R_e = V_x / Omega_0) R_e = 0.305 % radius of the circular tire contour R_c = 0.15 % nominal (rated) load F_z0 = 4000 % reference velocity V_0 = 16.67 % tire stiffnesses C_Fx = 435000 % (taken from ADAMS file) C_Fy = 166500 % (taken from ADAMS file) %========================================================================== %-------------------------------------------------------------------------- % user scaling factors / default values %-------------------------------------------------------------------------- % pure slip lambda_Fz0 = 1.0 % 1.0 nominal (rated) load lambda_mux = 1.0 % 1.0 peak friction coefficient (x) lambda_muy = 1.0 % 1.0 peak friction coefficient (y) lambda_muV = 0.0 % 0.0 with slip speed decaying friction lambda_KxKap = 1.0 % 1.0 brake slip stiffness lambda_KyAlp = 1.0 % 1.0 cornering stiffness lambda_Cx = 1.0 % 1.0 shape factor (x) lambda_Cy = 1.0 % 1.0 shape factor (y) lambda_Ex = 1.0 % 1.0 curvature factor (x) lambda_Ey = 1.0 % 1.0 curvature factor (y) lambda_Hx = 0.0 % 1.0 horizontal shift (x) lambda_Hy = 0.0 % 1.0 horizontal shift (y) lambda_Vx = 0.0 % 1.0 vertical shift (x) lambda_Vy = 0.0 % 1.0 vertical shift (y) lambda_KyGam = 1.0 % 1.0 camber force stiffness lambda_KzGam = 1.0 % 1.0 camber torque stiffness lambda_t = 1.0 % 1.0 pneumatic trail lambda_Mr = 1.0 % 1.0 residual torque

63

% combined slip lambda_xAlp = 1.0 % 1.0 alpha influence on F_x(kappa) lambda_yKap = 1.0 % 1.0 kappa influence on F_y(alpha) lambda_VyKap = 1.0 % 1.0 kappa induces ply-steer F_y lambda_s = 1.0 % 1.0 M_z moment arm of F_x % other lambda_Cz = 1.0 lambda_Mx = 1.0 lambda_My = 1.0 lambda_MPhi = 1.0 %========================================================================== %-------------------------------------------------------------------------- % parameters for longitudinal force at pure longitudinal slip %-------------------------------------------------------------------------- % shape factor p_Cx1 = 1.685 % peak value p_Dx1 = 1.210 p_Dx2 = -0.037 % curvature factors p_Ex1 = 0.344 p_Ex2 = 0.095 p_Ex3 = -0.020 p_Ex4 = 0 % horizontal shift p_Hx1 = -0.002 p_Hx2 = 0.002 % slip stiffness p_Kx1 = 21.51 p_Kx2 = -0.163 p_Kx3 = 0.245 % vertical shift p_Vx1 = 0 p_Vx2 = 0 %-------------------------------------------------------------------------- % parameters for overturning couple %-------------------------------------------------------------------------- q_sx1 = 0 q_sx2 = 0 q_sx3 = 0 %-------------------------------------------------------------------------- % parameters for longitudinal force at combined slip %-------------------------------------------------------------------------- % stiffness factors r_Bx1 = 12.35 r_Bx2 = -10.77 r_Bx3 = 0

64

% shape factor r_Cx1 = 1.092 % curvature factors r_Ex1 = 1.644; r_Ex2 = -0.0064359; % horizontal shift r_Hx1 = 0.007 %-------------------------------------------------------------------------- % parameters for lateral force at pure side slip %-------------------------------------------------------------------------- % shape factor p_Cy1 = 1.193 % peak values p_Dy1 = -0.990 p_Dy2 = 0.145 p_Dy3 = -11.23 % curvature factors p_Ey1 = -1.003 p_Ey2 = -0.537 p_Ey3 = -0.083 p_Ey4 = -4.787 p_Ey5 = 0 % slip stiffness p_Ky1 = -14.95 p_Ky2 = 2.130 p_Ky3 = -0.028 p_Ky4 = 2 p_Ky5 = 0 p_Ky6 = -0.92 p_Ky7 = -0.24 % horizontal shift p_Hy1 = 0.009 p_Hy2 = -0.001 p_Hy3 = 0 % vertical shift p_Vy1 = 0.045 p_Vy2 = -0.024 p_Vy3 = -0.532 p_Vy4 = 0.039 %-------------------------------------------------------------------------- % parameters for lateral force at combined slip %-------------------------------------------------------------------------- % stiffness factors r_By1 = 6.461 r_By2 = 4.196 r_By3 = -0.015 r_By4 = 0

65

% shape factor r_Cy1 = 1.081 % curvature factors r_Ey1 = 0 % (taken from ADAMS file) r_Ey2 = 0 % (taken from ADAMS file) % horizontal shift r_Hy1 = 0.009 r_Hy2 = 0 % (taken from ADAMS file) % vertical shift r_Vy1 = 0.053 r_Vy2 = -0.073 r_Vy3 = 0.517 r_Vy4 = 35.44 r_Vy5 = 1.9 r_Vy6 = -10.71 %-------------------------------------------------------------------------- % parameters for self-aligning moment at pure side slip %-------------------------------------------------------------------------- % stiffness factors q_Bz1 = 8.964 q_Bz2 = -1.106 q_Bz3 = -0.842 q_Bz5 = -0.227 q_Bz6 = 0 q_Bz9 = 18.47 q_Bz10 = 0 % shape factor q_Cz1 = 1.180 % peak values q_Dz1 = 0.100 q_Dz2 = -0.001 q_Dz3 = 0.007 q_Dz4 = 13.05 q_Dz6 = -0.008 q_Dz7 = 0.000 q_Dz8 = -0.296 q_Dz9 = -0.009 q_Dz10 = 0 q_Dz11 = 0 % curvature factors q_Ez1 = -1.609 q_Ez2 = -0.359 q_Ez3 = 0 q_Ez4 = 0.174 q_Ez5 = -0.896 % horizontal shift q_Hz1 = 0.007 q_Hz2 = -0.002 q_Hz3 = 0.147 q_Hz4 = 0.004

66

%-------------------------------------------------------------------------- % parameters for self-aligning moment at combined slip %-------------------------------------------------------------------------- s_sz1 = 0.043 s_sz2 = 0.001 s_sz3 = 0.731 s_sz4 = -0.238 %-------------------------------------------------------------------------- % parameters for normal load %-------------------------------------------------------------------------- p_z1 = 15.0 %-------------------------------------------------------------------------- % parameters for turn slip %-------------------------------------------------------------------------- p_DxPhi1 = 0.4 % (taken from ADAMS file) p_DxPhi2 = 0 % (taken from ADAMS file) p_DxPhi3 = 0 % (taken from ADAMS file) p_DyPhi1 = 0.4 % (taken from ADAMS file) p_DyPhi2 = 0 % (taken from ADAMS file) p_DyPhi3 = 0 % (taken from ADAMS file) p_DyPhi4 = 0 % (taken from ADAMS file) p_epsGamPhi1 = 0.7 % (taken from ADAMS file) p_epsGamPhi2 = 0 % (taken from ADAMS file) p_HyPhi1 = 1.0 % (taken from ADAMS file) p_HyPhi2 = 0.15 % (taken from ADAMS file) p_HyPhi3 = 0 % (taken from ADAMS file) p_HyPhi4 = -4.0 % (taken from ADAMS file) p_KyPhi1 = 1 % (taken from ADAMS file) % q_BrPhi1 = 0.1 % (taken from ADAMS file) q_CrPhi1 = 0.2 % (taken from ADAMS file) q_CrPhi2 = 0.1 % (taken from ADAMS file) q_DrPhi1 = 1.0 % (taken from ADAMS file) q_DrPhi2 = -1.5 % (taken from ADAMS file) q_DtPhi1 = 10.0 % (taken from ADAMS file) %-------------------------------------------------------------------------- % parameters for rolling resistance moment %-------------------------------------------------------------------------- q_sy1 = 0.01 % (taken from ADAMS file) q_sy2 = 0 % (taken from ADAMS file) %==========================================================================