MotorcycleDynamics

SecondEdition

VittoreCossalter

Importantenotice

Thisbookshouldnotbeseenasaguideformodifying,designingormanufacturingamotorcycle.Anyonewhousesitassuchdoessoathisownriskandperil.Streettestingmotorcyclescanbedangerous.Theauthorandpublisherarenotresponsibleforanydamagecausedbytheuseofanyinformationcontainedinthisbook.

Allrightsreserved.Noparttothisbookmaybereproducedortransmittedinanyformorbyanymeans,electronicormechanical,includingphotocopying,recording,orbyanyinformationstorageandretrievalsystem,withoutpermissioninwritingfromtheauthor.

2ndEnglishedition,2006

Copyright©2006byVittoreCossalter

9781447532767

Designandillustrationsbytheauthor.

ForAnnalia,Fabrizio,Flavio

Acknowledgment

IamdeeplyindebtedparticularlytoRobertoLot,MauroDaLio,AlbertoDoria,whohelpedtomakethisbookpossible.

ThisbookwaswrittenthankstotheenthusiasticparticipationofPhDstudentsoftheMotorcyclesEngineeringCourse:AlessandroBellati,RobertoBerritta,FrancescoBiral,DanieleBortoluzzi,MarioDallaFontana,GiovanniDallaTorre,DavideFabris,PasqualeDeLuca,StefanoGarbin,GiuseppeLisciani,FabianoMaggio,MassimoMaso,MatteoMassaro,LuigiMitolo,MartinoPeretto,NicolayRuffo,JimSadauckas,MauroSalvador,RiccardoToazza………

Foreword

Intoday’sglobalizedandhyper-technologicalworldallyouhavetodotobuyamotorcycleisgoon-lineandgiveyourcreditcardnumberandyou’vegotamotorcycle.However,thistypeoftransactiontakesplacewithouttheemotionsandspecialrelationshipwhichhavealwaysboundmetomotorcycles.

IrememberwatchingmyneighborgethisParillaready.Dressedinblackleather,hewouldslowlyputhisgloveson,pushdownonthepedalandfinallydriveoff.AsthemotorcycledisappearedintothedistanceIcouldhearthesymphonycreatedbyitsengineslowlyfadeawayamongtheclouds:minewastruepassion.

ItwastheSixties.ThereweretheelegantandrefinedModswiththeirshiningscooters,andtheRockers,bothfearedandrespected,withtheirmotorcycles.Englandwasthehomelandofmotorcycles.WhenIgotofftheshipinDover,Irememberseeingagroupofmotorcyclesnexttosomescooters.TheywerethewonderfulEnglishmotorcyclesoftheSixtiesandSeventiesthatleftthesignoftheirpassingwithdropsofmotoroilontheroadwherevertheywent.

IimmediatelyknewthatIwasamotorcyclist.AssoonasIgothome,ImanagedtobuyanoldGuzziFalcone500.Iworkedallwinter,everyevening,toperfectlyrestoreit.Mydesiretoheartheengineroar,tosmelltheairandtofeelthewindblowacrossmycheeksdrovemeinmymissionuntilonedayinearlyspringeverythingwasready.MyFalconeneverbetrayedme,italwaysgavemeincomparableemotions.AsIrodeit,curveaftercurve,theenginepushedonalmostasifitwereahammerstrongenoughtoforgeanytypeofsteelwithviolentblowsofmetalonmetal.

Thisbookistheresultofthispastandpresentpassionofmineformotorcycles.Ihavetriedtoofferanewapproachtotechnical-scientificwritingbycombiningtheexactandoftenasepticnatureofscientificdiscoursewithmypassionforthisperfectvehicle.Irealizethatthisisnosmallchallenge,butitisthisverypassion,ofamanwhofeelsmoreateaseonamotorcyclethanbehindadesk,whichhasmotivatedmyresearchinthefieldofmotorcycles.Togetherwithitsthoroughtechnicaldiscussion,thisbookalsotakesintoaccountthefascinatinghistoryofthemotorcycleandmotorcyclists.Nobusinesswilleverbeabletotakeawaytheadventuresome,andsomewhatcrazy,natureofthemotorcycle.

VittoreCossalter

Padova,spring2002

Tableof Contents

TitlePageImportantenoticeCopyrightPageDedicationAcknowledgmentForeword1KinematicsofMotorcycles2MotorcycleTires3RectilinearMotionofMotorcycles4SteadyTurning5In-PlaneDynamics6MotorcycleTrim7MotorcycleVibration-ModesandStability8MotorcycleManeuverabilityandHandlingListofsymbolsReferencesIndex

MotorcycleBianchi“FrecciaCeleste”350ccof1924

1Kinematicsof Motorcycles

Thekinematicstudyofmotorcyclesisimportant,especiallyinrelationtoitseffectsonthedynamicbehaviorofmotorcycles.Therefore,inthischapter,inadditiontothekinematicstudy,somesimpleexamplesofthedynamicbehaviorofmotorcyclesarereportedinordertoshowhowkinematicpeculiaritiesinfluencethedirectionalstabilityandmaneuverabilityofmotorcycles.

1.1DefinitionofmotorcyclesAlthoughmotorcyclesarecomposedofagreatvarietyofmechanicalparts,includingsome

complexones,fromastrictlykinematicpointofview,byconsideringthesuspensionstoberigid,amotorcyclecanbedefinedassimplyaspatialmechanismcomposedoffourrigidbodies:

therearassembly(frame,saddle,tankandmοtοr-transmissiondrivetraingroup),thefrontassembly(thefork,thesteeringheadandthehandlebars),thefrontwheel,therearwheel.

Theserigidbodiesareconnectedbythreerevolutejoints(thesteeringaxisandthetwowheelaxles)andareincontactwiththegroundattwowheel/groundcontactpointsasshowninFig.1-1.

Eachrevolutejointinhibitsfivedegreesoffreedominthespatialmechanism,whileeachwheel-groundcontactpointleavesthreedegreesoffreedomfree.Ifweconsiderthehypothesisofthepurerollingoftiresontheroadtobevalid,itiseasytoascertainthateachwheel,withrespecttothefixedroad,canonlyrotatearound:

thecontactpointonthewheelplane(forwardmotion),theintersectionaxisofthemotorcycleandroadplanes(rollmotion),

theaxispassingthroughthecontactpointandthecenterofthewheel(spin).

Fig.1-1Kinematicstructureofamotorcycle.

Inconclusion,amotorcycle’snumberofdegreesoffreedomisequalto3,giventhatthe15degreesoffreedominhibitedbythe3revolutejointsandthe6degreesoffreedomeliminatedbythe2wheel-groundcontactpointsmustbesubtractedfromthe4rigidbodies’24degreesoffreedom,assummarizedinFig.1-2.

Amotorcycle’sthreedegreesoffreedommaybeassociatedwiththreeprincipalmotions:forwardmotionofthemotorcycle(representedbytherearwheelrotation);rollmotionaroundthestraightlinewhichjoinsthetirecontactpointsontheroadplane;steeringrotation.

Whilehedrives,theridermanagesallthreemajormovements,accordingtohispersonalstyleandskill:theresultingmovementofthemotorcycleandthecorrespondingtrajectory(e.g.acurve)dependonacombination,inthetimedomain,ofthethreemotionsrelatedtothethreedegreesoffreedom.Thisgeneratesonemaneuver,amongthethousandspossible,whichrepresentsthepersonalstyleofthedriver.

Theseconsiderationshavebeenformulatedassumingthatthetiresmovewithoutslippage.However,inreality,thetiremovementisnotjustarollingprocess.

Thegenerationoflongitudinalforces(drivingandbrakingforces)andlateralforcesrequiressomedegreeofslippageinbothdirections,longitudinallyandlaterally,dependingontheroadconditions.Thenumberofdegreesoffreedomisthereforeseven:

forwardmotionofthemotorcycle,rollingmotion,handlebarrotation,

longitudinalslippageofthefrontwheel(braking),longitudinalslippageoftherearwheel(thrustorbraking),lateralslippageofthefrontwheel,lateralslippageoftherearwheel.

Fig.1-2Degreesoffreedomofamotorcycle.

1.2The geometryofmotorcyclesThiskinematicstudyreferstoarigidmotorcycle,i.e.onewithoutsuspensionswiththewheels

fittedtonondeformabletires,andschematizedastwotoroidalsolidbodieswithcircularsections(Fig.1-3).

Motorcyclescanbedescribedusingthefollowinggeometricparameters:pwheelbase;dforkoffset:perpendiculardistancebetweentheaxisofthesteeringheadandthecenterofthefrontwheel;εcasterangle;Rrradiusoftherearwheel;Rfradiusofthefrontwheel;trradiusofthereartirecrosssection;tfradiusofthefronttirecrosssection.

Someimportantgeometricparameterscanbeexpressedintermsofthesevariables:ρr=(Rr−tr)radiusofthefronttoruscentercircle;ρf=(Rf−tf)radiusofthereartoruscentercircle;an=Rfsinε−dnormaltrail;a=an/cosε=Rftanε−d/cosεmechanicaltrail.

Thegeometricparametersusuallyusedtodescribemotorcyclesarethefollowing:thewheelbasep;thecasterangleε;thetraila.

Theseparametersaremeasuredwiththemotorcycleinaverticalpositionandthesteeringangleofthehandlebarssettozero.

Fig.1.3Geometryofamotorcycle.

Thewheelbasepisthedistancebetweenthecontactpointsofthetiresontheroad.Thecasterangleεistheanglebetweentheverticalaxisandtherotationaxisofthefrontsection(theaxisofthesteeringhead).Thetrailaisthedistancebetweenthecontactpointofthefrontwheelandtheintersectionpointofthesteeringheadaxiswiththeroadmeasuredinthegroundplane.

Togethertheseparametersareimportantindefiningthemaneuverabilityofthemotorcycleasperceivedbytherider.Itisnotpractical,however,toexaminetheeffectsproducedbyonlyonegeometricparameter,independentlyoftheothers,becauseofthestronginteractionbetweenthem.Herewewillpresentsomeconsiderationsregardingthewayinwhichtheseparametersinfluencethekinematicanddynamicbehaviorofmotorcycles.

Thevalueofthewheelbasevariesaccordingtothetypeofmotorcycle.Itrangesfrom1200mminthecaseofsmallscootersto1300mmforlightmotorcycles(125ccdisplacement)to1350mmformediumdisplacementmotorcycles(250cc)upto1600mm,andbeyond,fortouringmotorcycleswithgreaterdisplacement.

Ingeneral,anincreaseinthewheelbase,assumingthattheotherparametersremainconstant,leadsto:

anunfavorableincreaseintheflexionalandtorsionaldeformabilityoftheframe.These

parametersareveryimportantformaneuverability(framesthataremoredeformablemakethemotorcyclelessmaneuverable),anunfavorableincreaseintheminimumcurvatureradius,sinceitmakesitmoredifficulttoturnonapaththathasasmallcurvatureradius,inordertoturn,theremustbeanunfavorableincreaseinthetorqueappliedtothehandlebars,afavorabledecreaseintransferringtheloadbetweenthetwowheelsduringtheaccelerationandbrakingphases,witharesultingdecreaseinthepitchingmotion;thismakesforwardandrearwardflip-overmoredifficult,afavorablereductioninthepitchingmovementgeneratedbyroadunevenness,afavorableincreaseinthedirectionalstabilityofthemotorcycle.

Thetrailandcasterangleareespeciallyimportantinasmuchastheydefinethegeometriccharacteristicsofthesteeringhead.Thedefinitionofthepropertiesofmaneuverabilityanddirectionalstabilityofmotorcyclesdependonthem,amongothers.

Thecasteranglevariesaccordingtothetypeofmotorcycle:from19°(speedway)to21-24°forcompetitionorsportmotorcycles,upto27-34°fortouringmotorcycles.Fromastructuralpointofview,averysmallanglecausesnotablestressontheforkduringbraking.Sincethefrontforkisratherdeformable,bothflexionallyandtorsionally,smallvaluesoftheanglewillleadtogreaterstressandthereforegreaterdeformations,whichcancausedangerousvibrationsinthefrontassembly(oscillationofthefrontassemblyaroundtheaxisofthesteeringhead,calledwobble).

Thevalueofthecasterangleiscloselyrelatedtothevalueofthetrail.Ingeneral,inordertohaveagoodfeelingforthemotorcycle’smaneuverability,anincreaseinthecasteranglemustbecoupledwithacorrespondingincreaseinthetrail.

Thevalueofthetraildependsonthetypeofmotorcycleanditswheelbase.Itrangesfromvaluesof75to90mmincompetitionmotorcyclestovaluesof90to100mmintouringandsportmotorcycles,uptovaluesof120mmandbeyondinpurelytouringmotorcycles.

1.3The importance oftrai lOneofthepeculiaritiesofmotorcyclesisthesteeringsystem,whosefunctionisessentiallyto

produceavariationinthelateralforceneeded,forexample,tochangethemotorcycle’sdirectionorassureequilibrium.

Accordingtothispointofview,thesteeringsystemcouldhypotheticallybemadeupoftwolittlerocketsplacedperpendiculartothefrontwheelwhich,whenappropriatelyactivated,could,althoughnotwithoutsignificantifnotinsurmountabledifficultiesfortherider,generatelateralthrusts,thatis,performthesamefunctionasthesteeringsystem.

Fromageometricalpointofview,theclassicsteeringmechanismisdescribedbythreeparameters:thecasterangleε;theforkoffsetd;theradiusofthewheelRf.

Theseparametersmakeitpossibletocalculatethevalueofthenormaltrailan,whichistheperpendiculardistancebetweenthecontactpointandtheaxisofthemotorcycle’ssteeringhead.Thisisconsideredpositivewhenthefrontwheel’scontactpointwiththeroadplaneisbehindthepointoftheaxisintersectionofthesteeringheadwiththeroaditself,aspresentedinFig.1-4.Aswehave

previouslyseen,thetrailmeasuredontheroadisrelatedtothenormaltrailbytheequation:

a=an/cosε

Thevalueofthetrailismostimportantforthestabilityofthemotorcycle,especiallyinrectilinearmotion.

Fig.1-4Stabilizingeffectofthepositivetrailduringforwardmovement.

Todevelopthisconcept,letusconsideramotorcycledrivingstraightahead,atconstantvelocityV,andletussupposethatanexternaldisturbance(forexample,anirregularityintheroadsurfaceoralateralgustofwind)causesaslightrotationofthefrontwheeltotheleft.Forthetimebeing,letusignorethefactthatthemotorcyclestartstoturntotheleftandthatbecauseofcentrifugalforces,beginsatthesametimetoleantotheright,concentratingourattentioninsteadonthelateralfrictionforceFgeneratedbythecontactofthetirewiththeground.

Inotherwords,letussupposethatthemotorcycleisdrivingatconstantvelocityVandthatthefrontwheelcontactpointalsohasvelocityVinthesamedirection.ThevectorVmaybedividedintotwoorthogonalcomponents:

thecomponentωfRf,whichrepresentsthevelocityduetorolling:itisplacedintheplaneofthewheelandrotatedtotheleftatananglewhichdependsonthesteeringangle;thecomponentVslide,whichrepresentstheslidingvelocityofthecontactpointwithrespecttotheroadplane.

Africtionalforce,F,thereforeactsonthefronttire.Fisparalleltothevelocityofslippagebuthastheoppositesense,asillustratedinFig.1-4.Sincethetrailispositive,frictionforceFgeneratesamomentthattendstoalignthefrontwheel.Thestraighteningmomentisproportionaltothevalueofthenormaltrail.

Fig.1-5Destabilizingeffectofthenegativetrailduringforwardmovement.

Ifthevalueofthetrailwerenegative(thecontactpointinfrontoftheintersectionpointofthesteeringheadaxiswiththeroadplane)andconsideringthatfrictionforceFisalwaysintheoppositedirectionofthevelocityofslippage,amomentaroundthesteeringheadaxisthatwouldtendtoincreasetherotationtotheleftwouldbegenerated.InFig.1-5onecanobservehowfrictionforceFwouldamplifythedisturbingeffect,seriouslycompromisingthemotorcycle’sequilibrium.Figure1-5demonstratesthattheroadprofilecanmakethetrailnegative,forexample,whenthewheelgoesoverasteporbump.

Fig.1-6Motorcyclewithahighvalueoftrail.

Smalltrailvaluesgeneratesmallaligningmomentsofthelateralfrictionforce.Eveniftheriderhastheimpressionthatthesteeringmovementiseasy,thesteeringmechanismisverysensitivetoirregularitiesintheroad.Highervaluesofthetrail(obtainedwithhighvaluesofthecasterangleasshowninFig.1-6)increasethestabilityofthemotorcycle’srectilinearmotion,buttheydrasticallyreducemaneuverability.

Consider,forexample,“chopper”typemotorcycleswhichbecameverypopularfollowingthesuccessofthewell-knownfilm,“EasyRider”.Thesemotorcycleshavecasteranglevaluesupto40°,

makingthemmoreadaptabletostraighthighwaysthantocurvingroads.

Fig1-7Summaryoftheeffectoftrailduringforwardmovement.

Duringcurvilinearmotion,roadgrippingisassuredbythelateralfrictionalforces,whichareperpendiculartothelineofintersectionofthewheelplanewiththeroad.

Thefrontandrearlateralforcescreatemomentsaroundthesteeringheadaxisthatareproportionalrespectivelytodistancesanandbn,whicharerelatedtothewheelbaseandthetrailbytheequations:

an=acosε

bn=(p+a)cosε

whereanrepresentsthenormalfronttrailandbnmaybeconsideredthenormaltrailoftherearwheel.

Thissimpleconsiderationshowshowthewheelbaseandthetrailareintimatelyconnectedtoeachotherandshouldthereforebeconsideredtogether.Itisnotentirelycorrecttodefineatrailassmallorlargewithoutreferencetothemotorcycle’swheelbase.Asacomparisonparameter,wecouldusetheratiobetweenthefrontandrearnormaltrail:

Rn=an/bn

Ingeneralthenormalfronttrailisapproximately4-8%ofthevalueoftherearone.Thevalueofthisratioforracingmotorcyclesisapproximately6%;forsportandsupersportmotorcyclesitisfrom6to6.5%;andfortouringmotorcycles,whicharemoreorlesssimilartosportmotorcycles,itvariesfrom6to8%.

“Cruiser”motorcycles(heavy,slowermotorcycles)arecharacterizedbyvaluesof5-6%andhavetrailsthataremodestincomparisonwiththewheelbase.Thisisprobablyduetothenecessityof

makingthemotorcyclesmaneuverableatlowvelocities.Sincetheloadonthefrontwheelsishighduetotheweightofthemotorcycle,thechoiceofasmalltraillowersthevalueofthetorquethattheridermustapplytothehandlebarstoexecuteagivenmaneuver.Inaddition,itisworthpointingoutthatthesemotorcyclesarenormallyusedatratherlowvelocities,andtheydonotthereforeneedlongtrails,which,aspreviouslynoted,assuresahighdirectionalstabilityathighvelocities.

Thisratioisalsolowforscooterssincetheyareused(orshouldbeused)atlowvelocitiesandthereforemaneuverabilityhasahigherprioritythandirectionalstability.

Strictlyspeaking,theratioshouldtakeintoaccountthedistributionoftheloadonthewheels.Amotorcyclethathasaheavyloadonthefrontwheelneedsashortertrail.Infact,heavierloadsonthefrontwheelgenerategreaterlateralfrictionalforcesinproportiontothelateralmotionofthewheel.Therefore,forthesamealigningtorqueactingaroundtheaxisofthesteeringheadasmallertrailissufficient.

Thecorrectratioonthebasisoftheloaddistribution,isexpressedbytheequation:

Rn=(an/bn)(Nf/Nr)

whereNfistheloadonthefrontwheelandNrtheloadontherearone.

1.4Kinematics ofthe s teeringmechanismItisclearthatwhenturningthehandlebars,keepingthemotorcycleperfectlyvertical,thesteering

headlowersandonlybeginstoriseforveryhighvaluesofthesteeringangle.Wewilldemonstratethisstatementbyconsideringthefollowingcases:

steeringmechanismwithnoforkoffset,d=0;steeringmechanismwithanon-zeroforkoffset,d≠0.

1.4.1Steeringmechanismwithzeroforkoffs e t

Inthecaseoftheforkwithnooffsetthecenterofthewheelisontheaxisofthesteeringhead.Letusaddthefollowingassumptions:

therollangleofthemotorcycleiszero;thewheelshavezerothickness.

AsshowninFig.1-8,whenthesteeringangleδiszero,thewheelisperfectlyverticalandliesinthexzplane.

Thecasterangleε,thesteeringangleδ,thecamberangleofthefrontwheelβ,thekinematicsteeringangleΔ(projectionoftheangleofrotationδontotheroadplane)andtheangleαarerelatedtoeachotherthroughthefollowingtrigonometricequations:

tanα=tanε⋅cosδtanΔ=tanδ⋅cosεsinβ=sinα⋅sinδ

Itispossibletoderivesinαandcosαasfunctionsofδandεfromtheprecedingequations:

Wenowassumethatthewheelcenter(pointO)canneitherrisenorfall.Theδrotationofthefrontwheelcausesittoinclinewithrespecttotheverticalpositionandtodetachitselffromthehorizontalplanexy.ThedistanceODofthewheelcenterfromtheroadplaneisgreaterthantheradiusofthewheelOP.

Actually,thewheelisnotraisedfromthegroundbutratherlowered.Supposingthatwekeeptheaxisofthesteeringheadimmobile,thecenterofthewheelmovesalongthesteeringheadaxistothepointO1.Consequently,thecontactpointP1movesforward,asshowninFig.1-8.InthefinalpositionthedistanceOPisobviouslyequaltotheradiusofthewheelOP.

Fig.1-8Geometryofthesteeringmechanism,withthemotorcycleinverticalpositionandnoforkoffset.

Whenthesteeringangleiszero(Fig.1-8,left),thenormaltrailandthetrailmeasuredontheroadplaneare:

an=EP=Rfsinε

a=CP=Rftanε

HereRfindicatestheradiusofthefrontwheel.Whenthesteeringangleδisnotzero,thenormal

traila=P1E1=Rfsinαbecomes,

Thetrailmeasuredontheroadplaneisrelatedtothenormaltrailandsteeringangleδbytheequation:

Theverticaldisplacementofthewheelcenterisgivenbythedifference:

Expressingtheangleαintermsofδandε,wehave:

1.4.2Steeringwithnon-zeroforkoffs e t

Letusnowconsidertheeffectofoffsetd,i.e.thedistancebetweenthecenterofthewheelandthesteeringheadaxis.Theconsiderationsthathaveallowedustoexpresstheloweringofthesteeringheadasafunctionoftheanglesδandεinthecaseofzero-offsetremainvalid.However,thezero-offsetformulamustbecorrectedsincetheoffsetcausesthecenterOofthewheeltomovetoO*,asisshowninFig.1-9.

Fig.1-9Loweringwheelcenterwithnon-zerooffset.

Withazerosteeringanglethetrailis:

Withanon-zerosteeringangleδthetrailis:

Thepresenceoftheforkoffsetleadstoareduction,d⋅sinε⋅(1−cosδ),intheloweringofthewheel.Thisvaluemustbesubtractedfromtheloweringofthefrontaxle,calculatedwithouttheoffset.

Inconclusion,withoffset,thereislessloweringofthefrontwheelcenterthanwithzerooffset.

ExampleI

Letusconsideramotorcyclecharacterizedbythefollowingsteeringparameters:

•radiusofthefrontwheel: Rf=0.3m;

•offset: d=0.05m;

•casterangle: ε=27°.

Nowcalculatetheeffectsoftwodifferentsteeringangles,9°and45°,onsteeringheadloweringwithandwithoutforkoffset.

Withasteeringangleδ=9°,theloweringis:

•withzerooffset: Δh=0.75mm;

•withoffset: Δh=0.478mm.

Byincreasingthesteeringangletoδ=45°theloweringis:

•withzerooffset: Δh=1.59mm;

•withoffset: Δh=0.92mm.

Theexampleshowsthatignoringtheoffsetcausesasignificanterrorincalculatingtheloweringofthefrontwheelcenter.Itmustbepointedoutthattherangeofsteeringisgenerallylessthan±35°.

Example2

Letusconsidertwomotorcyclesinaverticalposition,withthesamemechanicaltrail(a=101mm),thesameradius(Rf=0.3m)anddifferentcasterangles(ε1=27°,ε2=20°).

Ifthesteeringangleischangedfromδ=0°toδ=9°,calculatetheloweringofthefrontwheelcenterforeachofthecasterangle:

•withε1=27°andδ=9°: Δh=0.50mm;

•withε2=20°andδ=9°: h=0.40mm.

Lowercasteranglesreducetheloweringofthefrontwheelcenter.

Ifthesteeringangleδisequalto9°,calculatethechangeintrailforeachcasterangle.

Thetrailisreducedfromthe101mmto:

•withε1=27°andδ=9°: a=99.5mm;

•withε2=20°andδ=9°: a=99.8mm.

Thevalueofthetrailslightlydependsonthesteeringangle.

Thepreviousconsiderationshaveallowedustofindanalyticalequationsthatexpresstheloweringofthesteeringheadandthevaluesofthetrailintermsoftheanglesδandεandwiththelimitinghypothesesofzerorollangleandzerowheelthickness.Inthefollowingsection,amorecomplicatedkinematicmodelisusedtakingintoaccountboththerollangleandtheradiusofthefronttirecrosssection.

1.5Rol l motionands teeringNotonlyisthekinematicsofatwo-wheeledvehiclesignificantlymorecomplexthanofafour-

wheeledvehicle,butitalsopresentssomeuniqueaspects.

Forexample,letusconsideramotorcycleinrectilinearmotionatvelocityV,whichatacertainpointentersintoacurve.Themotorcyclepassesfromaverticalposition,inwhichthesteeringanglewaszero,toantiltedpositionwitharollangleϕ.Inordertostaybalanced,thehandlebar ’sangleofrotationwilldeviatefromzerodependingontheradiusofthecurveandthevelocity.

Fig.1-10Motorcycleinacurve.

Wehaveseenthattherotationofthesteering,consideringzerowheelthickness,generatesasmallloweringofthesteeringhead,whichcausesasmallforwardrotationoftherearframearoundtheaxisoftherearwheel(pitchrotation).

Wewillnowseehow,inreality,followingtherollmotion,thecontactpointoftherearwheelwiththeroadplaneisdisplaced.Twotriadscanbedefinedasfollows:

amobiletriad(Pr,x,y,z),definedasspecifiedbytheSocietyofAutomotiveEngineers(SAE).TheoriginisestablishedatthecontactpointProftherearwheelwiththeroadplane.Theaxisxishorizontalandparalleltotherearwheelplane.Thezaxisisverticalanddirecteddownwardwhiletheyaxisliesontheroadplane.Theroadsurfaceisthereforerepresentedbytheplanez=0;atriadfixedtotherearframe(Ar,Xr,Yr,Zr)whichissuperimposedontheSAEtriadwhenthemotorcycleisperfectlyverticalandthesteeringangleδzero.

Letusnowsupposethatonlytherearwheelistiltedattherollangleϕ.Consequentlythetriadfixedtotherearaxle(Ar,Xr,Yr,Zr)rotatesatthesameanglearoundthex-axis.Therefore,thetriad’soriginAristranslatedwithrespecttoPr,asillustratedinFig.1-11b.

Fig1-11Rearwheelinacurve:displacementofthecontactpoint.

Itcanbeseenthattheloweringofthesteeringheadcausesasmallpitchingrotationoftherearframeor,inotherwords,anotherrotationofthetriadfixedtotherearframe,asshowninFig.1-11c.ItisimportanttopointoutthattheoriginArofthetriadisfixedtotherearframeandnottotherearwheel:ArcoincideswithPronlywhentherollangleϕandthepitchangleμarebothzero.

Thebehaviorofthefrontwheelisevenmorecomplicated,since,inadditiontotherollingandpitchingmotion,thefrontwheelisalsosubjecttorotationaroundtheaxisofthesteeringhead.Thechangefromaverticaltoatiltedpositionwasassumedtobeapurerollmotionasiftheslippagebetweenthetireandtheroadplanewerezero.Inreality,thebehaviorismorecomplex.Inordertoproducethelateralreactionforcesonthecurve,alateralslippage,whichisexpressedintermsofthesideslipangleλ,mightbenecessary.(Thenextchapter,ontires,willshowthatslippagecanbeeitherpositiveornegativedependingonthevalueoftheforcegeneratedbythecamberangleofthewheel.)

Fig.1-12illustratesthecasesofpurerollingmotionandofmotionwithlateralslippage.Theabsenceofslippagemeansthatthevelocityvectoroftheforwardmotionofthewheel’scontactpointliesinaplaneparalleltothewheelitself,evenwhenthemotorcycleistravelinginacurve.

Fig.1-12Purerollingmotionandwithlateralslippage.

1.6Motorcycle pi tchWehaveshownthatwhenamotorcycleisperfectlyvertical(ϕ=0),therotationofthehandlebars

causesaloweringofthefrontwheelcenterand,therefore,arotationoftherearframearoundtherearwheelaxis.Inotherwords,themovementofthehandlebarscausesapitchingmotion.Nowwewouldliketostudymotorcyclepitchinamoregeneralcase,consideringarollangleϕotherthanzeroandtakingintoaccountthesizeofthetirecrosssections(seeFig.1-10).

Thepitchangleoftheframe,indicatedbyμ,isassumedtobepositiveinacounterclockwisedirection.Therefore,loweringthefrontwheelcenterleadstoanegativevalueofthepitchangle.Akinematicanalysisofmotorcyclesallowsustodetermineanon-linearequation,whichconnectstheunknownpitchangleμtoaseriesofknownquantities:therollangleϕ,thesteeringangleδ,thewheelbasep,theradiiofthecrosssectionsofthetires,tfandtr,theradiiofthetoruscentercirclesρrandρf,andthecasterangleε.

where:c1=dsinε(1−cosδ)+tr−tfc2=ρf[cosεcos(β’−ε)−cosδsinεsin(β−ε)−1]c3=dsinδ+ρfsinδsin(β’−ε)c4=p−dcosε(1−cosδ)c5=ρf[sinεcos(β’−ε)+cosδcosεsin(β’−ε)]

Thephysicalmeaningoftheangleβ’isshownlaterinFig.1-22ofSection1.7.2.

Thepreviousequationwasdeterminedbyignoringthepitchangleμwithrespecttoε,sinceitsvalueisonlyafewdegreescomparedtothecasterangleε,whosevaluenormallyvariesfrom20°to35°.

Oncethevalueofthepitchisknown,itiseasytocalculatetheresultingloweringofthefrontwheelcenter,whichismeasuredintheplaneofthemotorcycle.Agoodapproximationoftheloweringcanbederivedfromtheproductofthepitchangleandthewheelbase.

Fig.1-13Idealtrail.

Theprecedingequationscanbesignificantlysimplifiedifweconsidersmallrotationsofthesteeringangleδ(sinδ≅δ).Theexpressionforthepitchthenbecomes:

Thepitchisproportionaltothegeometricparameter(an−tfsinε),whichcorrespondstoanidealnormaltrail,measuredincorrespondencewiththecircularaxisofthetoruscentercircle,asshowninFig.1-13.Thepitchalsodependsonthedifferencebetweentheradiiofthetiresections(tr−tf):theneedtomountalargertireonthereartoimprovetheadherenceduringthrusting,increasestheeffectofloweringthesteeringhead.Itisworthnotingthatthesecondtermdoesnotdependonthesteeringangleδ,butonlyontherollangleϕ.

Ifwealsoignoretirethickness,i.e.ifweconsiderzerothicknesswheels(tr=tf=0),weobtainthesimpleequation:

Thislatterexpressionshowsthatthenormaltrailistheparameterthathasthegreatestinfluenceonthepitchingmotion.

1.6.1Pitchinterms ofs teeringandrol l angles

Figure1-14showstheeffectofthesteeringangleδandrollangleϕonthepitchangleμ.Itisimportanttostressthatnegativevaluesofthepitchangleμcorrespondtodownwardrotationsofthevehiclearoundtherearwheelaxle.Therefore,anegativevalueofthepitchangleμcausesamotorcycle’scenterofgravitytolower.

Fig.1-14Thepitchangleμasafunctionofsteeringangleδatdifferentrollanglesϕ.[p=1.4m,an=0.1m,ε=30°,Rr=Rf=0.36m,tr=tf=0.06m]

Whentherollangleϕvaluesarenothigh(0°and15°inFig.1-14),anincreaseinthesteeringangleδleadstoacontinuousloweringofthemotorcycle’scenterofgravityG.Sincetheloweringcorrespondstoareductioninpotentialenergy,theincreaseinthesteeringangletakesplacenaturally,evenwithoutapplyingtorquetothehandlebars.

Itiseasytoverifythisbehavior,especiallywithalighttwo-wheeledvehicle,suchasabicycle.Whenthebicycleistilted,therollangleimposeddeterminestheangleatwhichthehandlebarsnaturallyrotate.Forhighvaluesoftherollangle(30°and45°inFig.1-14),thevariationinthepitchangleμintermsofthesteeringangleδstopsdecreasingandpresentsaminimum.Atthispoint,alimitingvalueofthesteeringangleδisreached,beyondwhichthepitchslopereversesitssign.

Letusnowconsidertheminimumconditionofthepitchangleμ.Thiscorrespondstotheminimum

potentialenergy(centerofgravityatitslowestpoint).Fromaphysicalpointofview,thismeansthat,ifadeterminedrollangleϕisimposedandnoexternaltorquesareappliedtothehandlebars,thefrontframetendstorotatenaturallytowardsthevalueofthesteeringangleδwhichcorrespondstotheminimumvalueofthepitchangleμ.

Inconclusion,astheroIlangleϕgraduallyincreases,theminimumvalueofthepitchangleμcorrespondstoalowersteeringangleδ.

1.6.2Pitchas a functionofthe cas terangle

Figure1-15showstheinfluenceofthesteeringangleδandsteeringheadangleε,onthepitchangleμ,forafixedvalueoftherollangleϕ.

Thepitchangleμbecomesmorenegativeasthesteeringangleδincreases.Theinfluenceofthecasterangleismodest.

Fig.1-15Thepitchangleμversusthesteeringangleδforvariousvaluesofthecasterangleεandwitharollangleequalto30°.

1.6.3Pitchas a functionofthe normal trai l

Figure1-16showsthatthenormaltrailistheparameterwhichmostinfluencesmotorcyclepitch.Forexample,whenthesteeringangleδis10°andtheoffsetismodifiedtoobtaina20%variationinthenormaltrail,thevariationinthepitchangleisapproximately35%.

Fig.1-16Thepitchangleμasafunctionofthesteeringangleδforvariousvaluesofnormaltrail.

AsisshowninFig.1-16,whenthenormaltrailincreases,theminimumconditionofthepitchangleμcorrespondstoincreasingvaluesofthesteeringangle.Thisistheoppositeofwhathappenswhentherollangleincreases,asisclearbycomparingFig.1-16withFig.1-14.

1.7The rearwhee l contactpoint

1.7.1The e ffectofcamberandtire cros s s ection

Letusconsideramotorcyclethatisinitiallyinaverticalposition.Thecrosssectionofthereartireislargerthanthatofthefront.Therearframetilts,assumingthatthereislateralrollwithoutslippageontheroadplane,asillustratedinFig.1-17b.

Fig.1-17Lateraldisplacementofthecontactpointswithoutlateralslippage.

Thecontactpointofthereartiremoveslaterally,intheydirection,overadistancetr⋅ϕ,whichisproportionaltotheradiusofthetirecrosssectionandtherollangleoftherearframe.

Letussupposethattherollmotionoftherearframetakesplacewhilethesteeringangleiskeptatzero,andthatthereisnopitchofthemotorcyclearoundtheaxisoftherearwheel.Sincethefrontwheelhasasmallersectionthantherearone,thefrontwheelwouldberaisedfromtheroadplanefollowingtherollmotion.However,contactofthefrontwheelwiththeroadisassuredbythesimultaneouspitchrotationoftheentiremotorcyclearoundtheaxisoftherearwheel.

Oncetherollandpitchrotationshavetakenplace,thefrontwheelcontactpointmovestotheleftoftherearwheelcontactpointbythequantity(tr−tf)tanϕ,asisshowninFig.1-17c.Itisclearthatifthetireshaveequalsections,thelateraldisplacementofthetwocontactpointshavethesamevalue.

1.7.2The combinede ffectof rol l ands teering

Therotationofthehandlebargenerateslateralandlongitudinaldisplacementsofthefrontwheel’scontactpoint.

Letusconsideramotorcyclethatisinitiallyinaverticalposition.Themotorcycleistiltedthroughtherollangleϕandthenthehandlebarsrotatedthroughangleδ.Followingthismaneuver,thefrontwheel’scontactpointPfmovesawayfromtheplaneoftherearframe.

Fig.1-18LateraldisplacementofthecontactpointPf.

ThecoordinatesofthepointPfintheSAEreferencesystem,areexpressedinthefollowing

equations:

xPf=(c1+c2)sinμ+(c4+c5)cosμ

yPf=[−(c1+c2)cosμ+(c4+c5)sinμ]sinϕ+c3cosϕ−(tf−tr)

Thequantitiesc1...c5havebeendefinedpreviouslyinthesection1.6.

Figure1-19showsthelateralandlongitudinaldisplacementsofthefrontcontactpointforfourvaluesoftherollangleandcorrespondingsteeringrotation.

Thepointmovesforwardasthesteeringangleδandpresetrollangleϕincrease.Figure1-19showshowthexcoordinateofPfincreasesfromtheinitialvalue,whichisequaltothewheelbase.

Thepointmovesinitiallytotheleft(theycoordinateofPfisinitiallynegative)andthenreturnstotheright,passingoverthexaxis(theycoordinateofPfchangesitssign).Atthepointwherethepass-overtakesplace,thesteeringangleδdecreasesasthepresetrollangleϕincreases.

Fig.1-19PositionofthefrontcontactpointPf.

Fig.1-20LateralpositionofthefrontcontactpointPfasafunctionofthesteeringangleδ,forrollangleϕ=30°,andforvariousvaluesofthecasterangleε.

ThelateraldisplacementofthecontactpointPfisnotgreatlyaffectedbythecasterangleε,asisseeninFig.1-20,whileitisverysensitivetothevalueofthenormaltrailan,ascanbeobservedinFig.1-21.

Fig.1-21LateralpositionofPfversusthesteeringangleδ,forangleϕ=30°andforvariousvaluesofthenormaltrailan[ε=30°].

Itisinterestingtostudythedisplacementofthefrontcontactpointinatriadfixedtothefrontframe.Forthispurpose,weconsiderazerothicknesswheel(Fig.1-22).Whenthemotorcycleisperfectlyvertical(therollandsteeringanglesbeingzero),thecontactpointislocatedatA,asisshowninFig.1-22.WhileincreasingtherollandthesteeringanglesthecontactpointPfmovesalongthearcACuptoitslimitingpositionC,.ThepointPfreachesthepointConlywhentherollangleϕis

equalto90°,i.e.,ifthemotorcycleishorizontal.

Fig.1-22Thegeometryofsteering(zerothicknesswheel).

Fig.1-23Angularpositionβ’ofthefrontcontactpointofazerothicknesswheelversusthesteeringangleδ,forvariousvaluesoftherollangleϕ.

Asisclear,thefrontcontactpointneverreachespointCwiththevaluesoftherollandsteeringanglesnormallyusedindriving.InfactthecontactpointPfmoveswithinthearcAPo,Pobeingtheintersectionpointofthesteeringaxiswiththeprofileofthewheel.

ThefrontcontactpointPfreachespointPodependingonthesteeringandrollangles,asisshown

inFig.1-22.Whencarryingoutasteeringmaneuvertotherightwithasetrollangle,thecontactpointPfmovesforwardalongthearcAPo,whileitstracemovestotheleftandforwardontheroadsurface.TheeffectivetrailiszerowhenthecontactpointisexactlyatPo.FurtherincreasesintherollandsteeringanglesmovethecontactpointPftowardsthepositionC,whileitstraceontheroadplanemovestotherightandthetrailbecomesnegative.

Theprecedingequations,thatgivethepositionofthefrontcontactpoint,canbesignificantlysimplifiedbyassumingtherotationsofthesteeringangleδtobesmall(sinδ≅δ).TheexpressionforthecontactpointPfthenbecomes:

Ifwealsoignoretirethickness,theseequationsarefurthersimplified:

1.7.3The influence ofcontactpointlateral displacementonrol lmotion

ItisclearfromtheprecedingsectionthattheleftwarddisplacementofthecontactpointPf,followingasteeringmaneuvertotheright,favorsroll.ThisstatementcanbeexplainedbyFig.1-24,whichrepresentsthemotorcycle,schematizedasarigidbodyofmassm,inequilibriumonacurvewitharollangleϕequalto30°.

Assumingthatwemaintainaconstantrollangle,thefrontcontactpointPfmovestotheoutsideofthecurveasthesteeringangleδincreases.Therefore,theweightmomentincreaseswiththeincreaseinthesteeringangleδ.Thismomenttendstotiltthemotorcycleevenmore.Theincreaseintheweightarm,asshowninFig.1-24,isproportionaltotheleftwardlateraldisplacementofthefrontcontactpoint.ThelateraldisplacementΔybeginstodecreasewhenacertainsteeringangleδisreached.

Thecontactpointreachesitsmaximumexternaldisplacementatacertainsteeringangleδ.Thisvalueofδdoesnotcorrespondtotheδvaluethatminimizesthepitchangleμ.Forexample,witharollangleϕequalto30°,themaximumlateraldisplacementΔyoccurswithasteeringangleδequalto12.5°,whilethepitchangleμisataminimumwhenδisequalto22.5°,asshowninFigs.1-19and1-14respectively.

Fig.1-24Lateraldisplacementofthefrontcontactpoint.

1.8Frontwhee l camberangleThecamberangleβofthefrontwheelisdifferentfromtherollangleϕoftherearframe,whenthe

steeringangleδisotherthanzero.Ashasalreadybeenshown,thefrontandrearframerollanglescoincideonlyforzerosteeringangle.

Fig.1-25Pitchofthemotorcycleandcamberanglesofthefrontandrearwheels.

Thecamberangleofthefrontwheelβdependsontherearframerollangleϕ,thesteeringangleδ,thecasterangleεandthepitchangleμ:

Thefrontframeisalwaysmoretiltedwithrespecttotherearframewhensteeringangleisotherthanzero(samesignasrollangle).Asthesteeringangleδincreases,sodoesthecamberangleβ.

Ifthepitchμisignoredwithrespecttothecasterangleε,weobtain:

β=arcsin(cosδsinϕ+cosϕsinδ⋅sinε)

Ifthesteeringandtherollanglesaresmallenoughthefrontcamberanglecanbeapproximatedas:

β=ϕ+δ⋅sinε

Theequationshowsthat,forrollandsteeringangles“inphase”,e.g.withtherollangletotherightandthehandlebarsalsoturnedtotheright,thefrontframerollangleisalwaysgreaterthantherearframerollangle.Thisaspectisimportantbecausethetirelateralforce,aswillbeseeninthenext

chapter,dependsheavilyonthecamberangle.

Fig.1-26Frontwheelcamberangleversustherearframerollangleϕforvariousvaluesofthesteeringangle.

1.9The kinematics teeringangleThekinematicsteeringangleΔdependsontherearframerollangleϕ,steeringangleδ,caster

angleεandpitchangleμ:

Fig.1-27KinematicsteeringangleΔ.

Fromastrictlygeometricpointofview,thesteeringangleδistheanglebetweentherearandfrontwheelplanes,whilethekinematicsteeringangleΔrepresentstheintersectionofthisactualanglewiththeroadplanez=0.

Figure1-28showsthevariationinthekinematicsteeringangleΔasafunctionofthesteeringangleδforfourdifferentvaluesoftherollangleϕ.ThedottedlinerepresentstheconditionΔ=δ.Therefore,itappearsimmediatelyevidentthatthereisatransitionvalueoftherollangle,belowwhichthekinematicsteeringangleΔremainslowerthanthesetvalueδ,andabovewhichΔismorethanthesetvalueδ.Inthespecificcaseexamined,thetransitionvalueisapproximately27.5°.

InFig.1-29thevariationinthekinematicsteeringangleΔisshown,thistimeintermsoftherollangleϕforfourtypicalvaluesofthesteeringangleδ.ThehorizontaldottedlinesrepresenttheconditionΔ=δforeachofthefoursetvaluesofδ.Clearly,itcanbeobservedthatforlowervaluesoftherollangleϕ(25°to30°)thesteeringmechanismis“attenuated”(Δ<δ).Inthiscase,thesteeringmechanismislesssensitivetotherotationofthehandlebarsandthemotorcyclecanbemoreeasilysteered.Theriderexperiencesthesameeaseofsteeringofferedbywidehandlebarseveniftheonesbeingusedarenot.Ontheotherhand,forhighervaluesoftherollangleϕ,thesteeringmechanismis“amplified”(Δ>δ)makingthemotorcyclemoresensitivetochangesindirection.

ThekinematicsteeringangleΔalsodependsonthegeometryofthesteeringmechanism.Figure1-30iscarriedoutwiththerollanglesettoϕ=30°.Thefigure,inwhichthedottedlinerepresentstheconditionΔ=δ,showsthatdecreasingthecasteranglesmakethesteeringmechanismmoresensitive(Δ>δ).Thissensitivityispracticallyindependentofthevalueofthenormaltrail.Infact,itiswellknownthatsmallcasteranglesareneededformotorcyclestobeverysensitivetorapidsteeringandthathighcasteranglesvaluesmakesteeringmorecontrollable.

Fig.1-28KinematicsteeringangleΔasafunctionofthesteeringangleδfοrdifferentvaluesoftherollangleϕ[ε=30°].

Fig.1-29KinematicsteeringangleΔasafunctionoftherollangleϕforvariousvaluesofthesteeringangleδ[ε=30°].

Ifweignorethepitchμwithrespecttothecasterangleεandthetermsinϕsinδsinεwithrespectcosϕcosδ,theapproximateequationforthekinematicsteeringanglethenbecomes:

Thisequationcanalsobeobtainedonthebasisofsimplegeometricconsiderations.

Let’sconsideramotorcycleinaverticalpositionandsupposethatthehandlebarrotates,whilelockingthecontactpointPfofthefrontwheel.

Fig.1-30Kinematicsteeringangleversusthesteeringangleδ,forvariousvaluesofthecasterangleε[ϕ=30°].

TherearcontactpointPrmovesbackslightly,whiletheintersectionpointofthesteeringheadaxiswiththeroadplanemoveslaterally.Thismovementdescribesanapproximatelycirculartrajectory,asshowninFig.1-31.

Therearframerotationangleδpdependsonthesteeringangleδnasthefollowingequationshows:

(p+a)tanδp=atan(δp+δn)

Thesteeringangleδ(whichbydefinitionismeasuredinaplaneorthogonaltothesteeringheadaxis)isrelatedtotheangleδn(whichismeasuredinaplaneorthogonaltothemotorcycleplane),bytheexpression(seealsosection1.4):

tanδn=tanδ⋅cosε

Assumingthattherotationsaresmall,thefollowingsimplifiedequationisobtained:

Therefore,thedisplacementoftherearplanefromthefrontcontactpoint,keepingzerorollangle,is:

yPf≅aδcosε=anδ

ThedisplacementyPfisproportionaltothevalueofthetrailanddecreaseswithanincreaseinthecasterangleasdemonstratedpreviouslyonpage24.

Fig.1-31Steeringgeometry.

Nowthatthesteeringangleδhasbeenfixed,letustiltthemotorcyclethroughasetrollangleϕ,asinFig.1-31and1-32.ThekinematicsteeringangleΔisrepresentedbytheangleformedbythedirectionofforwardmotionofthefrontandrearwheels.Theapproximateequationofthisangleisgivenbytheratioofthewheelbaseandtheradiusofcurvature:

Thesteeringangleδn,measuredintheplanenormaltotherearframeplane,is:

ThekinematicsteeringangleΔ,intermsoftherollangleϕ,casterangleεandsteeringangleδ,isthen:

whichwasshownpreviouslyonpage28.

Fig.1-32Kinematicsteeringangle.

Onthebasisofthisequationwecandrawthefollowingconclusions:onlywhentherollangleϕisequaltothecasterangleε,canthekinematicsteeringangleΔbeequaltotherotationangleofthehandlebarsδ;anattenuation,Δ<δ,occursforlowvaluesoftherollangle,whileanamplification,Δ>δ,occursforlargerollangles:withhighvaluesofthecasterangleε(likechoppers),agreaterrotationofthehandlebaris

neededtoproducethesamevalueofthekinematicsteeringangle.

1.10The pathcurvatureThekinematicstudyofthepathtracedbyamotorcycleiscarriedoutassumingthatthereisno

lateralslippagebetweenthewheelsandtheroadplane(“kinematicsteering”).ThecurvatureCofthepath(theinverseofthepathradius)dependsonthepositionofthefrontcontactpointPfandthekinematicsteeringangleΔ:

Forsmallsteeringangles(sinδ≅δ),thecurvatureCcanbeexpressedintermsoftherollangleϕandthesteeringangleδ:

Fig.1-33Radiusofpath.

SincethedisplacementofthecontactpointPfofthefrontwheelissmallwithrespecttothewheelbase,thecurvaturecanbecomputedwiththefollowingsimplifiedformula:

Itcanbeobservedthatthepath’sradiusisdirectlyproportionaltothewheelbase.Fig.1-34showshowthecurvatureCvarieswiththesteeringangleδ,forvariousvaluesoftherollangleϕ.Themaximumerrorusingtheapproximateformulaisequaltoabout2%.

Fig.1-34CurvatureCversusthesteeringangleδforvariousvaluesoftherollangleϕ.

1.11The e ffective trai l inacurveThetrailisthedistancebetweenthefrontcontactpointandtheintersectionpointofthesteering

headaxiswiththeroadplane.Ontheotherhand,thenormaltrailistheperpendiculardistancebetweenthefrontcontactpointandthesteeringheadaxis(Fig.1-35).

Fig.1-35Effectivetrailwiththefrontwheeltiltedandsteered.

Incorneringconditionsthenormalandmechanicaltraildependonthewheelbase,thecasterangle,theoffsetofthefrontwheel,thegeometricalpropertiesofthetires,thepositionofthefrontcontactpointandthepitchangle.Thenormaltrailincorneringcondition is:

Themechanicaltrailincorneringconditiona*is:

Itisworthhighlightingthatthetraildependsonthegeometryofthefronttirebecausethepitchangleandthefrontcontactpointpositiondependonρfandtf.

Theimportanceofthenormaltrailderivesfromthefactthatthemomentsgeneratedbythetirereactionforces(verticalloadandlateralforce)actingaroundthesteeringheadaxisareproportionaltothevalueofthenormaltrail.

Fig.1-36Forcesactingatthecontactpoint.

Fig.1-37Componentsofthereactionforcesthatgenerateamomentaroundthesteeringheadaxis.

Let’sconsiderthelateralforceandnormalloadappliedatthefrontcontactpoint(Fig.1-36).Eachforcecanbesplitintocomponentsthatactperpendiculartothesteeringaxisandnormaltrail(thereforeinapositiontoproduceamomentaroundtheaxis)andcomponentsparalleltoorintersectingthesteeringaxis(whichdonotproduceamoment).ThisisshownschematicallyinFigs.1-36and1-37.

Thenormaltrailrepresentsthearmoftheusefulcomponents.Theusefulcomponentofthelateralforce,tendstoalignthewheeltotheforwardvelocity,whiletheusefulcomponentoftheverticalload,hasamisaligningeffect,i.e.,ittendstocausethewheeltorotatetowardstheinsideofthecurve.Thevaluesofthemomentsaroundthesteeringhead,generatedbythesetwousefulcomponents,areimportantfortheequilibriumofthefrontsection(aroundthesteeringheadaxis).Thetorquetheridermustapplytomaintainequilibriumdependsonthem.

Fig.1-38Normaltrail[radiusoffrontcrosssectiontf=50mm].

Fig.1-39Normaltrail[radiusoffrontcrosssectiontf=80mm].

Fig.1-40Trail[radiusoffrontcrosssectiontf=50mm].

Fig.1-41Trail[radiusoffrontcrosssectiontf=80mm].

Wewillnowtrytounderstandwhetherandhowthenormaltrailvarieswhenthemotorcycleisinacurve.Figure1-38showsthenormaltrailasafunctionoftherollangleandthesteeringangle.Itcanbenotedthatthenormaltraildiminisheswhentherollangleincreasesandevenmoresowhenthesteeringangleincreases.However,ifweconsidersteeringanglesbelow5°androllanglesbelow40°,thetrailvariationsremainbelow20%.

Fig.1-39showsthatanincreaseintheradiusofthefronttirecrosssectionfrom50mmto80mmfurtherreducesthesevariations.Withasteeringangleof5°andarollangleof40°thereductionofthetrailgoesfrom20%to10%.

Changingthetypeoffronttirecancauseavariationinthecrosssectionradius.Therefore,thenormaleffectivetrailinturning,i.e.thearmofthereactiveforces,varies.Sincetherider“feels”thebehaviorofthefrontsectionthroughthetorqueappliedonthehandlebars,itisclearthatavariationinthecrosssectionradiuscanproduceadifferentfeeling.Ifthecasterangleisvaried,verysimilargraphsareobtained.However,thecasteranglevalueinfluencesthevariationinthenormaltrailtoalesserextent.

Inconclusion,itcanbestatedthatwhencomparingdifferentmotorcyclesitisimportanttorefertothenormaltrailsinceithasaprecisephysicalmeaning.InFig.1-40andFig.1-41thetrailispresentedasafunctionoftherollangleandsteeringangle.Itcanbeobservedthatanincreaseintherollangle,withsmallvaluesofthesteeringangle,producesanincreaseinthetrail.Thisisdifferentfromwhathappenswiththenormaltrail.

Tosummarizewecansaythat:bothtrailandnormaltraildiminishwithanincreaseinthesteeringangleδ,thevalueofthetrail,whethernormalormeasuredontheroadplane,alsodependsontherollangleϕ,areductioninthetrail,withtheincreaseinthesteeringangleδ,isattenuatedasthefronttirecrosssectionradiusandtheexternaltireradiusincrease.

1.12The e ffectoft ire s ize onthe rearframeyawWewouldnowliketostudyanotherparticularaspectofmotorcycleswhichoccurswhenthe

motorcycletiltstotheside:theyawingeffectcausedbydifferentcrosssectionsizesofthetires.

Consideramotorcycleinitiallyinaverticalpositionandwithazerosteeringangle(Fig.1-42a).Ashasalreadybeenstated,accordingtotheSAEreferencesystem,thetriad’soriginisintherearwheelcontactpointPrandthex-axisrepresentsthemotorcycle’sforwardmotion.

Let’ssupposethatthemotorcycletiltswhilethesteeringangleisheldatzero,asshowninFig.1-42b.Ifthetires’cross-sectionshavethesameradii(tr=tf),theintersectionoftherearframeplanewiththeroadplanecoincideswiththedirectionoftheforwardmotion.

Inthiscase,therearplanedoesnotyaw,butrathermoveslaterallyduetothelateralrollingofthetires.Thelateraldisplacement,trϕ=tfϕ,isequaltotheproductoftherollangleϕtimestheradiusofthetire’scross-section.

Iftheradiiofthecrosssectionshavedifferentvalues(tr>tf),alsoshowninFig.1-42b,thetiltingmotion,withthesteeringanglefixedatzero,producesarotationψoftherearframeplane,i.e.,ayawmotionwhosevalueisgivenby:

Fig.1-42Motorcycleyawcausedbytireswithdifferentcrosssections.

Example3

Consideramotorcyclewiththefollowingcharacteristics:wheelbasep=1400mm,theradiiofthetirecrosssectionsaretr=100mmandtf=40mm.

Ifthemotorcyclechangesfromstraightrunning(verticalmotorcycle,ϕ=0)toturning(rollangleϕ=45°),calculatetheyawangleoftherearframe.

Theyawangleoftherearframeplane,duetothedifferenceintheradiiofthecrosssections,isequaltoψ=0.53°.

MotoGuzzi500ccof1924

2MotorcycleTires

Thetireisoneofthemotorcycle’smostimportantcomponents.Itsfundamentalcharacteristicisitsdeformability,whichallowscontactbetweenthewheelandtheroadtobemaintainedevenwhensmallobstaclesareencountered.

Inadditiontoimprovingthecomfortoftheride,thetireimprovesadherence,animportantcharacteristicbothforthetransferoflargedrivingandbrakingforcestotheground,andforthegenerationoflateralforces.Theperformanceofamotorcycleislargelyinfluencedbythecharacteristicsofitstires.Inordertounderstandtheirimportance,onemustconsiderthatcontrolofthevehicle’sequilibriumandmotionoccursthroughthegenerationoflongitudinalandlateralforcesactingbetweenthecontactpatchesofthetireswiththeroadplane.Theforcesoriginateasaresultofactiontakenbytheriderthroughthesteeringmechanism,theacceleratorandthebrakingsystem.

2.1Contactforces betweenthe t ire andthe roadFromthedynamicviewofthemotorcycle,itisfundamentaltoportraytheoverallbehaviorofthe

tireinvariousconditionsofusethroughamodelcapableofrepresentingtheforcesandmomentsofcontactintermsofforwardvelocity,camberangle,longitudinalslip,lateralslipandloadactingonthetireitself.

Fromamacroscopicviewpoint,theinteractionofthetirewiththeroadcanberepresentedbyasystemcomposedofthreeforcesandthreemoments,asinFig.2-1:

alongitudinalforceactingalongtheaxisparalleltotheintersectionofthewheelplanewiththeroadplane,andpassingthroughthecontactpoint(assumedpositiveifdrivingandnegativeifbraking),inxdirection;averticalforceorthogonaltotheroadplane(averticalloadthatactsonthewheel,assumedpositiveinanupwarddirection),alongthezaxis;

alateralforce,intheroadplane,orthogonaltothelongitudinalforce,inydirection;anoverturningmomentaroundthex-axis,arollingresistancemomentaroundthey-axis,ayawingmomentaroundthez-axis.

Fig.2-1Forcesandtorquesofcontactbetweenthetireandtheroadplane.

Fig.2-2Longitudinalandlateralforces.

InFig.2-2typicallongitudinalandlateralforceshavebeendepictedintheconditionofpureslip.Puresliprepresentsthesituationwheneitherlongitudinalorlateralslipoccursinisolation.Thelongitudinalforcedependsonthelongitudinalslipandshowsaclearpeakwhilethelateralforceisafunctionbothofthecamberangleandofthesideslipangle.CurveswhichexhibitashapeliketheforcesdepictedinFig.2-2canberepresentedbyamathematicalformulanamedthe

“MagicFormula”.

2.2The “MagicFormula”forrepresentingexperimental resultsThemodelproposedbyΡacejka(1993)isverymuchinuse.Theapproachissubstantially

empiricalandtheresultsreproducetherealbehaviorofthetireverywell.Theentiremodelrevolvesaroundwhatiscalledthe“magicformula,”thatis,asingleexpressionthatcanbeusedtorepresentthelongitudinaldrivingorbrakingforce,thelateralforceorthemomentaroundthezaxis.Theexpressionisasfollows:

Y(x)=y(x)+Svy(x)=D⋅sin{C⋅arctan[Bx−E(Bx−arctanBx)]}X=x+Sh

whereB,C,DandEarefourparameters,Svindicatesthetranslationofthecurvealongtheyaxis,andShindicatesthetranslationofthecurvealongthexaxis.

Fig.2-3Meaningoftheparametersinthe“magicformula”.

Themagnitudeycanrepresenteitherthelongitudinalthrustorthelateralforce,whilexrepresentsthecorrespondingslipquantity.Figure2-3reproducesthetypicalvariationofthePacejkacurveandiseffectiveinvisualizingthemeaningofthefourparametersappearingthere.

ParameterDrepresentsthepeakvalue(onlywithE<1andC≥1)anddependsontheverticalload.ParameterCcontrolstheasymptoticvalueassumedbythecurveasthesliptendstoinfinity,andinthiswaydeterminestheresultingformofthecurve.ParameterBdeterminestheslopeofthecurvefromtheorigin.ParameterEcharacterizesthecurvaturenearthepeak,andatthesametimedeterminesthepositionofthepeakitself.

ItcanbeshownthatthegradientattheoriginisgivenbytheproductBCD.

2.3Rol l ingres is tance

Considerawheelthatrotateswithoutslippageonaflatsurface.Therollingradiusisdefinedbytheratiooftheforwardvelocitytoitsangularspeed:

Theeffectiverollingradiusinfreemotionis,asshowninFig.2-4,smallerthantheradiusoftheunloadedtirebecauseofthedeformationofthetire.Itsvaluedependsonthetypeoftire,itsradialstiffness,theload,theinflationpressureandtheforwardvelocity.Itcanbedemonstratedthatitsvalueinfreemotionissmallerthanthatoftheradiusoftheunloadedtirebutgreaterthanthedistancefromthecenterofthetiretotheroadplane.Anapproximatevalueisgivenbytheequation:

R0=R−(R−h)/3

Fig.2-4Effectiverollingradiusofthetire.

Duringthetire’srolling,theportionofthecircumferencethatpassesoverthetrackundergoesadeflection.Inthecontactareastressesaregenerated,whicharebothnormal(duetotheload)andshearduetothedifferenceinlengthofthearcofcircumferenceanditstreadchord(thatrepresentsthelengthofthecontacttread).Becauseofthehysteresisofthetirematerial,partoftheenergythatwasspentindeformingthetirecarcassisnotrestoredinthefollowingphaseofrelaxation,orisrestoredlate.Thiscausesachangeinthedistributionofthecontactpressures,whichthereforearenotsymmetric,butarehigherintheareasinfrontofthewheel’saxis.

AsshowninFig.2-5,theresultantofthenormalcontactpressuresisdisplacedforwardwithrespecttothecenterofthewheelbythedistanced.Theforwarddisplacementiscalledtherollingfrictionparameter.Hence,tomovethewheelwithconstantforwardvelocityitisnecessarytoovercomearollingresistancemomentequalto:

Mw=dN

Theresistancetorollingisexpressedviaaresistanceforcethatopposestheforwardmotion,andwhosevalueisgivenbytheproductoftherollingresistancecoefficientfwandtheverticalload.

Inadditiontothetypeoftire(eitherradialorbias-ply),itsdimensions,thecharacteristicsofthetire,thetemperatureandtheconditionsofusetherollingresistancecoefficientdependsprincipallyontheforwardvelocityandontheinflationpressure.Therollingresistancecoefficientincreaseswiththecamberangle.Typicalvaluesareontheorderof0.02.

Fig.2-5Descriptionofcontactpressuresandforcesactingonarollingwheel.

KevinCooper(see[J.Bradley,1996])hasproposedthefollowingempiricalformulaforcalculatinglossesthroughresistanceduetotherollingofthemotorcycletires.Theformulatakesinflationpressureandforwardvelocityintoaccount:

Velocityisexpressedinkilometersperhourandthetirepressurepinbar(1bar≅1atm).Figure2-6showsthevariationoftherollingresistancecoefficientversusthevariationofvelocityatcertainvaluesoftirepressure.Itcanbeobservedthatanincreaseinpressurediminishestheresistancetorolling.

Thepowerthatisdissipatedbecauseoftherollingresistanceforce,isgivenbytheproductoftheresistanceforceandtheforwardvelocity:

HereNrepresentstheloadonthewheel(expressedinNewtons);thepowerdissipatedPisexpressedinkilowatts.

Tosummarizeitmaybesaidthattherollingresistanceforcedependson:inflationpressurethedeformationofthetire(inviewofthehysteresisofthematerial),therelativeslipbetweenthetireandtheroad,theaerodynamicresistanceduetotheventilation.

Fig.2.6Therollingresistancecoefficientversustheforwardvelocityforvariousvaluesoftirepressure.

Ofthesethreecauses,thefirstoneisbyfarthemostimportant.Lossesthroughventilationarecausedbytheinteractionbetweenthewheelandthecirculatingair,whichinturndependsontheformofthewheelitself(armorspokes),theprofileofthetireandtherotationalvelocity.

Example1

Consideramotorcyclewithamassof200kgandtwodifferentvelocities:100km/hand250km/h.Assumingthetirepressureis2.25bar,determinethepowerdissipatedtoovercomerollingresistance.

Thepowerdissipatedinordertoovercometherollingresistanceforcesatavelocityof100km/hisonly1.1kW,whileatavelocityof250km/hthepowerrisesto12kW.

2.4Longitudinal force (driving-braking)Thepresenceofdrivingorbrakingforcesgeneratesfurtherlongitudinalshearstressesalongthe

areaofcontact.Thecircumferentialstress,inthecaseofdrivingforce,compressesthefibersinthecontactarea(Fig.2-7);inthecaseofbrakingforces,thefibersareengagedintension(Fig.2-8).

Theforwardvelocityofthecontactpointisthereforeless,inthecaseoftraction,thanthetire’speripheralvelocity.Alternatively,inthecaseofbraking,itisgreaterthanthetire’speripheralvelocity.Thisisexpressedbythelongitudinalslip,definedbytheratiobetweentheslipvelocity(V−ωR)andtheforwardvelocityV:

Thelongitudinalslipispositiveinthecaseoftractionandnegativeinthecaseofbraking.Inthelattercase,longitudinalshearstresseshavetheoppositesignoftheforwardvelocity.

Inthecaseofdrivingwheel,somelongitudinalshearstressesaregeneratedinthecontactareahavingthesamesignastheforwardvelocityandthereforethetiretreadinthecontactpatchiscompressed.Inthefirstpartofthepatchthecontactisoneofadhesion,butinthesecondpartthecontactoccurswithsliding(Fig.2.7).

Fig.2-7Longitudinalshearstressinthecontactareaandforcesactingonadrivingwheel.

Inbraking,theinstantaneousrollingradius,whichinconditionsofpurerollingislessthantheperipheralradiusofthewheel,increaseswithanincreaseofthebrakingforceuntilitbecomesgreaterthanthewheel’sradius(inasuddenstopthatincludeslockingthewheel,thisradiusisinfinite).Inthefirstpartofthepatch,thecontactisoneofadhesion.Atacertainpoint,thedifferencebetweentheforwardvelocityandtheperipheralvelocityproducesshearstressesgreaterthanthosethatcanbegeneratedinconditionsofadhesion,andforthisreasonaslidingzoneisgenerated.Thelengthoftheslidingzoneisapproximatelyproportionaltothebrakingforce(Fig.2.8).

Fig.2-8Longitudinalshearstressinthecontactareaandforcesactingonabrakingwheel.

2.4.1Non-l inearmodel

Thelongitudinalforceofbothtractionandbrakingisproportionalinafirstapproximationtotheloadapplied;theratioμbetweenthelongitudinalforceandtheload(normalizedlongitudinalforce)iscalledlongitudinalbraking/drivingforcecoefficient.

ThelongitudinalforceatnominalloadNcanbedescribedbymeansoftheMagicFormula:

ThecoefficientDκ=μprepresentsthepeakofthebraking/drivingforcecoefficientwhiletheproductDκCκBκisthelongitudinalslipstiffness.

Fig.2-9showsinqualitativetermstheratioofthelongitudinalforcetothenormalload,versusvariationofthevalueofthelongitudinalslip.Themaximumvalue(braking/drivingtractioncoefficient)dependsstronglyonroadconditions.

2.4.2Linearmodel

Theforce,inbrakingandthrustingphasesrespectively,canbeexpressedbyalinearequation,such

as:

F=Kκκ=(kκN)κS=Kκκ=(kκN)κ

where,

indicatesthedimensionalstiffness(N)oflongitudinalslip,and

thenon-dimensionallongitudinalslipstiffness.

Theorderofmagnitudeofthevalueoflongitudinalstiffnesskκ(gradientofthecurveofzeroslippage)rangesfrom12-30(non-dimensionalvalue).

Fig.2-9Qualitativevariationofthebraking/drivingforcecoefficientversusslip.

Example2

Supposethatavehicleof81kWpower(110HP)attainsamaximumvelocityof270km/h(75m/s).Determinethedrivingthrustrequiredtoattainthisvelocity.Nextassumingmaximumthrustwiththeentireloadonthemotorcycleontherearwheel,determinethedrivingforcecoefficient.

Thenecessarydrivingthrustisequaltotheratiobetweenpowerandvelocity:

S=81*1000/75=1079.5N

Thedrivingforcecoefficientisequalto:

Theslipnecessarytoproducethisnormalizedlongitudinalforcecanbedeterminedifweknowthevariationofthelongitudinalforcecoefficientintermsoftheslip.Thevalueofthenecessaryslipdependsonthetypeoftire.InthetwocurvesgiveninFig.2-10,thevalueofthefrictioncoefficient0.72isobtained,with3.6%slippageinthecaseoftireAand8%withtireB.ItisclearthattireBissubjecttomorerapidwearthantireA,becauseofthegreaterlongitudinalslipnecessaryforgeneratingthesamethrustforce.

Fig.2-10Longitudinalforcecoefficientversuslongitudinalslipfortwodifferenttires.

2.5Lateral forceThelateralforce,whichthetireexertsontheground,dependsonboththesideslipangleλandthe

camberangleϕ.Thesideslipangleisdefinedastheanglemeasuredintheroadplanebetweenthedirectionoftravelandtheintersectionofthewheelplanewiththeroadplane,ascanbeseeninFig.2-11.Sideslipforcesdependontirecarcassdistortionwhilecamberforcesdependprimarilyon

geometry.

Thetireisdeformedoncontactwiththeground,producingapatchofvariableshapeanddimensionsaccordingtothecharacteristicsofthetire,therollangle,thesideslipangle,aswellasexternalfactorssuchastheload,theinflationpressure,etc.Anypresenceoflateralforcesandbrakingordrivingtorquesintroducesfurtherdeformationstothecontactpatch.Ingeneral,thepatchisnotsymmetricalwithrespecttothexandy-axes.

Fig.2-11Thesideslipangle.

2.5.1Lateral force generatedbythe camberangle

Firstletusconsiderthecaseofatireinclinedtoasetcamberangle,whichmovesforwardinthedirectionofitsplaneandhasazerosideslipangle(Fig.2-12).Inthecaseofanundeformabletirecarcass,thepatchisdot-shapedandthegenericpointP,situatedontheexternalsurfaceofthetorusofthewheel,describesacirculartrajectoryinspacewhoseprojectionontheroadplaneisacurveintheformofanellipse.ItthereforetouchestheroadatthesinglecontactpointA.Thereisnolateraldeformationofthetire;therefore,itgeneratesnocamberforce.

Inthecaseinwhichthetire’scarcassisdeformable,thecontactzoneisextended,andpointPatthemomentwhenitenterstheareaofcontactwiththegroundisobligedtoabandonthetheoreticalellipticaltrajectoryandtomovealongarectilineartrajectoryinthedirectionofthewheel’sforwardmotion;thisdirectionisindicatedwiththelinea−ainFig.2-12.

WecanimaginethatthedeformationofthetirecarcassPP”willoccurintwodistinctphases:first,theverticalloadgeneratestheverticaldeformationPP,thenthelateralforceofthecamberthrustgeneratesthedeformationP’P”.Thelateralforceduetocamberisimportant,especiallyatsmallslipangles.

Fig.2-12Originofthecamberthrust.

2.5.2Lateral force generatedbylateral s l ip

Considerawheelthatrotatesandatthesametimeslipslaterally.InthiscasetheformofthecontactpatchisdistortedasshowninFig.2-13.

ConsiderapointPsituatedonatreadthatreachescontactwiththegroundatpointA.WhenpointPmovestoadeterminedpointindicatedwithB,itdescribesarectilineartrajectory.ItsvelocityhasthedirectionoftheforwardvelocityV.WhenitreachespointB,theelasticrestoringshearstress,duetothedeformationofthecarcassandoftherubberelementsinthetiretread,becomesgreaterthantheadhesionforcesandthereforebecomesuchastomakeitdeviateintheoppositedirection,causingittoslideonthegrounduntilthetrailingedgeC.

Twozonesarethereforetobedistinguishedinthecontactarea:afrontzonewhereadhesiontakesplace;arearzoneinwhichthereissliding.

Theslidingzoneismoreextendedthegreatertheslipangleis.Oncealimitingvalueofthelateralforcehasbeenreached,theentirecontactzonebecomesaslidingarea.

Fig.2-13Thepatchofamotorcycletireinthepresenceoflateralslip.

2.5.3Non-l inearmodel

Figure2-14showsinqualitativeterms,thenormalizedlateralforceversustheslipangleandversusthecamberangle.Themaximumvalueoftheforcethatcanbeobtained,givenacertaintire,isstronglydependentonroadconditions.

Theforcesweremeasuredbymeanstherotatingdisktestmachinedescribedin[Cossalteretal.,2003]andshowninFig.2-15.Thediskrotatesaroundaverticalaxisandisequippedwithasafetywalktrack.Thewheelundertestingrollsonthetrackandisplacedinpositionbyanarticulatedarmthatmakesitpossibletosetthecamberandsideslipanglesatassignedvalues.

Fig.2.14Measuredvaluesoflateralforceasafunctionofthesideslipangleλandforvariousvaluesofthecamberangleϕ(left)andasafunctionofthecamberangleϕforvariousvaluesofthe

sideslipangleλ(right)[fronttire120/70/17].

Fig.2.15Therotatingdisktiretestmachine.

Asshowninthefigurethelateralforceisafunctionoftheverticalload,sideslipandcamberangle.Thecouplingbetweenthecorneringandthecambercomponentscanbeexpressedwiththeequivalentforceapproachbymeansofthefollowingexpression:

ThisapproachisthemostrecentofPacejka’sformulationsformotorcycletires[Pacejka,2005].

Ds=μyisthepeakofthelateralforcecoefficient,DλCλBλ=kλisthecorneringstiffnesscoefficientandDϕCϕBϕ=kϕisthecamberstiffnesscoefficient.

2.5.4Linearmodel

Thecontactforcesbetweenthetireandtheroadplanedependontheslipangleandthecamber

angle.Itcanbeseenthatforsmallslipangles,thedependenceontheslipangleisnearlylinearwhilethecambercomponentisalmostalinearfunctionofthecamberangle.

Thelateralforceforsmallslipangleandlimitedcamberanglecanbeexpressedbymeansofthelinearexpression:

F=Kλ⋅λ+Kϕ⋅ϕ=(kλ⋅λ+kϕ⋅ϕ)N

Figure2-16shows,ontheleft,thetypicalvariationofthenormalizedlateralforcewithrespecttotheverticalforce,versustheslipangleforvariousvaluesofthecamberangleand,ontheright,versusthecamberangleforvariousvaluesoftheslipangle.

Fig.2-16Geometricinterpretationofthecorneringkλandcamberkϕstiffness.

Thecharacteristicsofthetire,asfarasthelateralforceisconcerned,aredefinedbythecorneringandcamberdimensionalstiffnesses(N/rad):

Thecorneringstiffnesscoefficientkλvarieswiththevariationofthecharacteristicsofthetires.Itsfieldofvariabilityrangesfromapproximately10rad−1uptovaluesof25rad−1.

Thecamberstiffnesscoefficientkϕisoftheorderofmagnitudeof0.7to1.5rad-

Theratiobetweenthemaximumlateralforceandtheverticalload,canreachvaluesof1.3to1.6whentheroadsurfaceiscleananddry.

2.5.5Lateral force neededformotorcycle equi l ibrium

Consideramotorcycleinacurveinsteadystate.Theequilibriumofthemomentsoftheforcesactingonthecenterofmassshowsthatthenormalizedlateralforcenecessarytoassurethemotorcycle’sequilibriumisequaltothetangentoftherollangle,asrepresentedinFig.2-17.

Fig.2-17Equilibriumofthemotorcycleinacurve.

ConsiderFig.2-18.Therightgraphshows,foracertaintypeoftire,thecomponentoflateralthrustduetocamberalone(thestraightline)andthelateralforcenecessaryfortheequilibriumofthemotorcycleonacurve(dottedline).TheforcesarenormalizedwithrespecttotheverticalloadN.Thegraphontheleftrepresentsthenormalizedlateralforceversusthelateralslipangle.

Itcanbeobservedthatthestraightline,whichapproximatesthevariationofthecamberthrust,intersectsthecurvetanϕincorrespondencewitharollangleof28°.Thismeansthatwithintherange0to28°,thelateralforceneededforequilibriumislessthanthethrustforcegeneratedbycamberalone.Sincethelateralforcegeneratedmustbeexactlyequaltothatneededforequilibrium,thediminutionofthelateralforceisobtainedthroughanegativesideslipangle.Thatis,thewheelpresentsalateralvelocitycomponenttowardstheinteriorofthecurve.

Figure2-18shows,forexample,thatinthecaseofacamberangleof10°,theequalityoftheforcegeneratedwiththatneededisobtainedwhenthereisanegativeslipangleof0.3°(PointA).Withacamberangleof28°thelateralslipiszero(PointB).Forvaluesofthecamberanglegreaterthan28°thelateralforceproducedbycamberaloneisnotsufficientfortheequilibriumofthemotorcycleandthereforetheincreaseinthelateralforceisobtainedwiththelateralslipofthetire(positiveslip).

Thisbehaviorisacharacteristicofmotorcycletiresinwhichthelateralforcegeneratedis,upto

determinedrollangles,almostentirelyduetothecambercomponent.Sincethiscomponentappearsmorerapidlywithrespecttothecomponentduetoslip,itplaysafundamentalroleinsafety.Thecambercomponentappearsmorerapidlybecauseitdependsonthecarcassdeformationwhilethecorneringcomponentdependsontheslipanglewhichneedssometimetooccur.

Fig.2-18Componentsofthelateralforcegeneratedbycamberandslip.TireA.

Fig.2-19Componentsoflateralforcegeneratedbycamberandslip.TireB.

NowconsiderthegraphinFig.2-19,whichreferstoadifferenttypeoftire.Inthiscasethecamberthrustisalwaysinferiortothelateralforceneededforequilibrium.Thismeansthatitisalwaysnecessarytohavelateralslipinordertogeneratetheadditionallateralforcerequiredforequilibrium.Thelateralforces,thewaytheyareproducedandtheirdependenceonthecamberangleandtheslipangle,playafundamentalroleinthemotorcycle’sunder-steeringorover-steeringbehavior.

Ifthegenerationoflateralfrontforcerequiresaslipanglelargerthanthatneededforthegenerationoflateralrearforce,themotorcyclewilltend,astherollangleincreases,toskidmorewiththefrontwheel.Thisbehaviorcausesthevehicletounder-steer.Ontheotherhand,iftheslipintherearwheelisgreaterthanthatofthefrontone,thebehaviorwillbeover-steering.Neutralbehavioroccurswhentheslipanglesareequal.Onthebasisoftheseconsiderations,thetire’sidealbehavioroccurswhentheslipangleiszero,thatis,whenthelateralforcenecessaryforequilibriumisproducedbycamberalone.

2.5.6Dependence of lateral force onload,pres sure ,temperature

Thetire’scapabilityforlateralgripiswellrepresentedbytheplotsoflateralforceversuscamberangleandsideslipangle,asdepictedinFig.2.20forseveralverydifferenttypesoffrontandreartires.

Fig.2-20Anexampleoflateralforceasafunctionofthecamberangleϕ(left)andofthesideslipangleλ(right)fordifferentfront(top)andrear(bottom)tires.

Fig.2-21Anexampleoflateralforceasafunctionofthecamberangleϕ(left)andsideslipangleλ

(right)forvariousvaluesofpressure,loadandtemperature.

Figure2-21showsasanexamplethemeasurednormalizedlateralforceversusboththeslipangleandthecamberangleforvariousvaluesofinflationpressure,ofverticalloadandtemperatureofthecarcass.

Corneringstiffnessisnottobeconfusedwiththetire’slateralorradialstiffness.Lateralstiffnessistheratiobetweenappliedlateralforceandtheresultinglateraldeformationofthetire’scarcass.Itdependsonthetire’sconstructioncharacteristics.Theorderofmagnitudeofitsvalueisof100to200kN/m.Theradialstiffnessofthetireistherelationbetweentheverticalloadandtheverticaldeformationandhasvaluesintherangeof100to350kN/m.Inflationpressureandforwardvelocityinfluencebothofthesestructuralstiffnesses.

Anincreaseinverticalloaddecreasesthecorneringstiffnesscoefficientwhereasthecamberstiffnesscoefficientisalmostnotinfluenced.Anincreaseininflationpressure,decreasesthecorneringstiffnesscoefficientand,toalesserextent,decreasesthecamberstiffnesscoefficient.Tireswithlargersections,orgreatercrosssectionradii,usuallyhavealargercorneringstiffnesscoefficient.Anincreaseintemperaturedecreasesbothcorneringandcamberstiffnesscoefficientsbutincreasesthemaximumvalueoftheratiobetweenthemaximumlateralforceandtheverticalload.

2.5.7Lateral force intrans ients tate

Wehavestatedthatthetirelateralforcedoesnotariseinstantaneously.Toappear,thewheelneedstorollacertaindistance,whichdependsonthetire’scorneringcharacteristicsandlateralstiffness.

Supposethatthemotorcycleisinitiallyinastateofverticalequilibrium.Therollandslipanglesarezerowithcorrespondingzerovaluesofthelateralcontactforces.Ifweinstantaneouslyassignnon-zerovaluestotherollandslipangles,thelateralcontactforcesincreaseexponentially,fromzerotothesteadystatevaluecorrespondingtotheassignedrollandsideslipangles;thecontactforcesthereforefollow,withadelay,thevariationoftheanglesonwhichtheydepend.Thisisduetothefactthatthecarcassdistortiontakessometimetoestablishitself.Thecomponentduetocamber,dependingprimarilyontiregeometry,hasalesserdelaythanthatofthecomponentduetolateralslip.

Thetire’sbehaviorintransientstatecanberepresentedbythemodel(Fig2-22),whichiscomposedofaspring(withstiffnessksexpressedinN/m)anddamperinseries(withdampingcoefficientcexpressedinkg/s).Thespringksrepresentsthetire’slateralstiffnessanddependsmainlyontheformandcharacteristicsofthetire’scarcasswhilethedampercdescribesthebehaviorofthetireunderconditionsoflateralslip.Ifweignoretheinertiaofthetire’scarcass,thecorneringforceisequalandoppositetotheelasticforcegeneratedbythedeformationofthetire:

Fs=cẏ=cλ’V=Kλλ’=−ks(y−yi)

Hereλ’=ẏ/Vrepresentsthetransientslipangle,thatistheslipangleofthecontactpatch(pointP),ythelateraldisplacementofthecontactpatchandyithedisplacementimposedonthewheel.

Fig.2-22Springanddamperconnectedinseriestorepresentthelateralbehaviorofthetire.

Takingintoaccountthedefinitionoftheslipangleandtheapparentdampingcoefficientc=Kλ/V,theequationcanbeexpressedintheform:

Thedifferentialequationcanalsobeexpressedintermsofthelateralforceproduced.Aftersomemanipulation,thefollowingexpression,withtheimposedwheelslipangleλ=ẏi/V,isobtained:

HereL=Kλ/ksrepresentstherelaxationlength.

Supposethatthewheelissuddenlysubjected,atinstantt=0,toalateralmotionwithconstantslipangleλo.Asaresultthelateralforceincreasesexponentially:

Keepinginmindthatthedistancextraversedbythewheelisgivenbytheproductofvelocityandtime,wehave:

Figure2-23showsthevariationofthelateralforce,normalizedwithrespecttothevalueKλλo,whichitassumesinasteadystate,asafunctionofthedistancexcoveredbythewheel.Thetangentconstructedthroughtheoriginisequaltotheratioks/Kλ.Theinverseofthetangentiscalledrelaxationlength:

Therelaxationlengthrepresentsthedistancethewheelhastocoverinorderforthelateralforcetoreach63%ofthesteadystateforce.Integratingthedifferentialequationgivesusthelateralforceoncewehaveassignedatemporalvariationtotheslipangle.

Thevaluesoftherelaxationlengthofthecorneringforcerangesbetween0.12-0.45m.Thesmallvaluescorrespondtolowvelocity(20km/h),thehighervaluestoveryhighvelocity(250km/h).Itincreasesslightlywiththeload.ItisinterestingtohighlightthattherelaxationlengthisalmostconstantwithrespecttotheratiobetweenthefrequencyνofthesideslipangleoscillationandtheforwardvelocityV.Thisratioiscalledthepathfrequencyandrepresentsthenumberofcyclespermeterofforwardmotion.

Ontheotherhandthevaluesoftherelaxationlengthofthecamberforceinsomeexperimentaltestshasbeenfoundtobealmostnegligible.However,furtherexperimentalresultsareneededtoverifythisbehavior.

Fig.2-23Lateralforceasafunctionofthedistancecoveredbythewheel.

2.6Moments actingbetweenthe t ire andthe road

2.6.1Se l f-al ignmentmoment

Thedistributionofthelateralshearstressgeneratedbythelateralslipofthetireisnotsymmetric.Theresultingforceisthereforeappliedatapointsituatedatacertaindistancefromthecenterofthepatch,acenterwhich,inafirstapproximation,canbeassumedtocoincidewiththetheoreticalcontactpointoftherigidtoroidwiththeroadplane.Thedistanceatisdesignatedthetrailofthetireorpneumatictrail.ItisclearfromFig.2-24thatthelateralforcegeneratesamomentthattendstorotatethetireinsuchawayastodiminishtheslipangle.Forthisreasonthismomentiscalledtheself-aligningmomentofthetire.

Fig.2-24Trailofthetire.

Theself-aligningmomentMzisexpressedastheproductofthecorneringforceFsandthetrailofthetireat.

Mz=−atFsExperimentalresultsshowthatthetrailisatamaximumwhentheslipangleiszero;thatitdecreaseswithanincreaseintheslipangleuntilitreacheszero,andthatitincreaseswithincreasesintheverticalload.Itcanbeapproximatelyexpressedintermsoftheslipanglebythefollowinglinearequation:

where atorepresentsthemaximumvalueofthetiretrail( rangesfrom1.5to5cm)andλmaxtheslipangleatwhichthetiretrailbecomeszero.

Figure2-25representsthetypicalvariationofthesideslipforce,ofthepneumatictrailandoftheself-aligningmomentasafunctionoftheslip.Whentheslipanglereachesthevalueλmax(about15°)themomentiszerosincethelateralforcepassesthroughthecenterofthepatchbecausetheslidingzonecoversthewholepatch.

Fig.2-25Exampleofmeasuredvariationofthelateralforce,pneumatictrailandself-aligningmomentversussideslipangle.Camberangle=0°,Frontnominalload=1300N,rearnominal

load=1400N.

2.6.2Twis tingmoment

Consideraninclinedwheelthatrollsovertheroadplanewithangularvelocityωaboutthewheelaxis(Fig.2-26).

Fig.2-26Originofthetwistingmoment.

WeindicatewithCothepointofintersectionofthewheel’saxiswiththeroadplane.Iftheturn

centerpointCofthecirculartrajectorydescribedbythewheelcoincideswiththepointCo,motionoccurswithoutlongitudinalslippage(underkinematicconditions).Infact,theperipheralvelocitiesofthetwopointsAandBofthetire,whicharepartofthepatch,areequaltotheforwardvelocitiesduetotherotationofthewheelaroundthepointCwithangularvelocityΩ.

ωrA=ΩRAωrB=ΩRB

Inreality,atfreerollingthecenterofcurvatureCisalwayslocatedexternallywithrespecttothepointCo.

Supposethatatthemidpointofthepatchtheperipheralvelocityisequaltotheforwardvelocity:

Inthemostexternalareaofthepatchtheperipheralvelocityisgreaterthantheforwardvelocity,whileintheinteriorareaofthepatchthecontraryistrue.Motionthereforeoccurswithslip,andtwozonescanbedistinguishedinthepatch:onewithpositivelongitudinalslipvelocity,andtheotherwithnegativelongitudinalslipvelocity.Therefore,thereareforwarddirectedshearstressintheexternalzoneandbackwarddirectedshearstressintheinternalzone.

Theseshearstressesgenerateatwistingmomentthattendstomovethewheelalongatrajectorywithasmallercurvatureradius,therebyactingtotwistthewheeloutofalignment.Thetwistingmomentisapproximatelyproportionaltothecamberangle.AtypicalvariationisrepresentedinFig.2-27.

Fig.2-27Exampleofmeasuredvariationoftwistingmoment.

Wehaveseenthattwomomentsofoppositesignactonthetire:theself-aligningmomentandthetwistingmoment.Theirsumdefinestheyawingmomentofthetire,whosequalitativevariation

againstthesideslipangle,isshowninFig.2-28.TheyawingmomentMziszerowhentheslipandrollanglesarezero;itincreaseswithincreasesintherollangleandhasaminimumcorrespondingtoaslipangleofλ=2°to6°.

Fig.2-28Exampleoftheyawingmoment.

2.6.3Torque generatedbythe drivingorbrakingforce

Thedrivingforcegeneratesamomentthattendstoaligntheplaneofthetireinthedirectionofvelocity,whilethebrakingforcegeneratesamomentofoppositesignwhichthereforemovesitoutofalignment.Thearmofthelongitudinalforcedependsonthelateraldeformationofthetire.

whereksindicatesthelateralstiffnessofthetire’scarcass.

Fig.2-29Momentsgeneratedbydrivingorbrakingforce.

WithrespecttopointQthearmofthelongitudinalforcealsodependsontheradiusofthecrosssectionandthecamberangle.

wheretindicatestheradiusofthecrosssectionofthetire.

Ingeneral,thelateraldeformationsp,hasanegligiblevaluewithrespecttothelateraldisplacementsofthecontactpointofthetire.

2.7Combinedlateral andlongitudinal forces :the frictione l l ipseThelongitudinalforceFx,eitherdriving(positivevalueofFx)orbraking(negativevalueofFx),is

assumedtobeassignedsinceitiscontrolledbytherider.

ThelateralforceFythatcanbeexercisedisreducedbythesimultaneouspresenceofthelongitudinalforce.

Fig.2-30Thefrictionellipse.

Theirresultantmustbewithinthefrictionellipsethathasthemaximumvalues longitudinal,and,lateralrespectively,whentheyactalone:

where isthelongitudinaltractioncoefficientand isthelateraltractioncoefficient.Forthisreasontheformulathatyieldsthelateralforceismultipliedbyacorrectioncoefficientthatdependsonthelongitudinalforceapplied:

Figure2-31showsthevariationofthenormalizedlateralforcecurveswiththevariationofthelongitudinalforceapplied.

Fig.2-31Variationofthenormalizedlateralforceforvariousvaluesofthelongitudinalforce(rollangle=0°).

Fig.2-32Lateralandlongitudinalforcesforvariousvaluesoflongitudinalslipκandsideslipλ(rollangleϕ=0°).

TheinteractionbetweenthelongitudinalandlateralforcescanbeshownbyrepresentingtheconstantlateralslipcurvesandconstantlongitudinalslipcurvesinadiagramthathasasitsordinatethenormalizedlateralforceFy/NandasitsabscissathenormalizedlongitudinalforceFx/N.

ThecurvesforconstantlongitudinalslippageandconstantlateralsliparerepresentedinFig.2-32.

Example3

Consideramotorcyclebrakingasitentersacurve.Supposethattherearwheelhasanormalizedlongitudinalforceequalto0.75andanormalizedlateralforceequalto0.53(pointA),whichcorrespondtoasideslipangleof3.5°.

Fig.2-33Forceactingonthemotorcycle.

Ifthebrakingissuddenlystopped,thelateralforcecorrespondingtoa3.5°slipangleisincreasedsharply,thereismotionfrompointAtopointC.Thelateralforcegeneratedbytheslipisnowequalto0.78,whichisgreaterthanthe0.53neededforequilibrium.Sincethemotorcycleistilted,thesuddenincreaseinthelateralforcegeneratesanaccelerationofthevehiclethattendstobereturnedtotheverticalpositionandtoprojecttheriderupward(high-sidefall).Thelateralslipdiminishesuntilitreachesthevalueneededforequilibrium(PointB).

2.8The e las tici tyofthe carcas sWhenlateralandverticalforcesareappliedtothetire,bothlateralandradialelasticdeformation

ofthecarcassarise.Additionallythedriving/brakingforcegenerates,inthelongitudinalplane,adeformationthatmainlyconsistsinarelativerotationbetweentherimandthecarcass.Becauseoftiredeformation,thecontactisnolongerdotshaped,butinvolvesacontactpatchsurfacewhoseformdependsonthecamberangle,ontheloadandontheinflationpressure.

Thelengthandwidthofthecontactpatchofmotorcycletireschangeinaratherregularmannerwiththeverticalloadandcamberangleaslongasthecontactpatchisnotverylarge(largeloads)andthecamberangledoesnotapproach40°-45°.Theeffectofinflationpressureoncontactpatchisimportantifitislowerthanthenominalvalue2-2.5bar

Fig.2-34Theeffectofcamberangleoncontactpatchshape.[ReartireFz=2000Nandp=2bar].

IntheroadplanethetirechangesfootprintandthecontactpointmoveslaterallydependingonthegeometryofthecarcassasshowninFig.2-34.

Inthepresenceofcamber,apureverticalloadinducedbothhorizontalandverticaldeflectionofthecarcass.However,byexpressingresultswithrespecttothewheelcamberedreferenceframe,therelationshipsbetweenforcesanddeformationsaresimpler.InfacttheelasticpropertiesofthecarcasscanbeeffectivelydescribedbymeansofapairofspringswhichactintheradialdirectionZandthelateraldirectionY,asshownintheFig.2-35.

Fig.2-35Elasticityofthetirecarcass.

Typicalvaluesofstructurallateralstiffnessesrangesfrom100kN/mto250kN/mwhileradialstiffnessesrangefrom100kN/mto200kN/m.

2.9Model ofthe motorcycle t ire

Nowadaysmulti-bodycodesmakeitpossibletocalculatethepointsofcontactbetweentheroadandthemotorcycleequippedwithrigidorelastictoroidaltires.Henceinthetiremodeltheforcescanbeappliedintheareaaroundthepointofcontactbetweentheroadandthetoroidaltire.

Fig.2-36Forcesandmomentsactingonthetire.

Themodelofthemotorcycletiretakesintoconsiderationtheforcesactingonpointsnearthetheoreticalcontactpointdefinedbythetire’sgeometry.

Theforcesunderconsiderationareasfollows:

Normalforce.Normalforceisappliedatapointthatprecedes,bythedistanced,thepositionofthetheoreticalcontactpoint.Thedistanceddependsontherollingresistancecoefficientandthetire’sradius:

d=fwR

Lateralforce.Lateralforceactsinthedirectionorthogonaltotheintersectionofthewheelplanewiththeroadplane.Theapplicationpointisdisplacedbackwardswithrespecttothetheoreticalcontactpointbyadistanceatthatrepresentsthetire’strail,whichvarieswiththesideslipangle.

Longitudinalforce.Theforceisappliedtoapointdisplacedlaterallyfromthetheoreticalcontactpointbecauseofthetire’slateraldeformability.Thelateraldisplacementsp,dependingonthelateralstiffnessofthetire,isgenerallynegligiblewithrespecttothegeometricdisplacementsderivingfromtherollinclinationofthewheel.

Themomentsactingaroundthex,yandz-axesaregeneratedbytheforcesdescribedaboveandbythetwistingmoment.

Overturningmoment.TheoverturningmomentMxisgeneratedbytheverticalloadNwhosearmisthelateraldeformationsp.

Mx=−spN

Rollingresistancemoment.Rollingresistancemomentisgeneratedbytheasymmetricdistributionofnormalstressesthatcausesaforwarddisplacementofverticalload.Thetire’srollingresistancemomentis:

My=dN

Yawingmoment.Yawingmomentincludestwocontributions.Thefirstterm,duetothelateralforce,tendstoaligntheplaneofthetireinthedirectionofvelocity.Thesecondtermincreaseswiththecamberangleandworksagainstalignment.

Mz=−atFs+Mt

Fig.2-37showsamotorcycletiremodeledwithvirtualprototyping.Thetire,launchedontheroadatacertainvelocityandwithaninitialinclinedorientationdescribesatrajectorythatdependsonthetire’scharacteristicsandinparticular:

onthecomponentofthelateralforceduetocamber(functionofthecamberangle),onthetire’strail(functionofthelateralslip),onthetwistingmoment(functionofthecamberangle).

Fig.2-37Pathofthedifferenttires.

ThegraphinFig.2-37showsthevarioustrajectoriesdescribedbythetire.Ifthecomponentofthelateralforceduetocamberiszero,equilibriumisassuredonlybythecomponentgeneratedbylateral

slip.Lateralslipisthereforealwaysgreaterthanthatpresentinthereferencecase.Becauseofthegreaterlateralslip,thetrajectorycoveredismoreexternalthanthatofthereferencecase.Thereductionofthetire’strailtozeroalsoreducestozerotheself-aligningmomentgeneratedbythelateralforce.Thetrajectorydescribedisthereforemoreinnerwithrespecttothatofthereferencecase.Thetwistingmoment,dependingonthecamberangle,hasasignificantinfluenceonthetire’sbehavior.Sinceitseffectsworkagainstalignment,thatis,ittendstocausethetiretoyawmore,itszeroingcausesalargechangeinthetrajectorycovered.Thetiremovesalongapathcharacterizedbyanotablylargercurvatureradius.

2.10Vibrationmodes ofthe t iresThedynamicpropertiesoftireshaveanimportantinfluenceonseveralfeaturesofmotorcycle

behavioursuchascomfort,shock-absorptionandbraking,whicharerelatedtoin-planedynamics,alongwithstabilityandhandling,whicharerelatedtoout-of-planedynamics.

Motorcycletires’modesofvibrationcanbedividedintoin-planemodes,out-of-planemodesandmixedmodes.In-planemodesarecharacterisedbyradialand/orcircumferentialdisplacementofthepointslocatedinthesymmetryplaneofthewheel.Out-of-planemodesaredominatedbylateraldisplacementofthepointslocatedinthesymmetryplaneofthewheel.Mixedmodesexhibitcombinationsofradial,circumferentialandlateraldisplacement.

Inplanemodescanbeclassifiedaccordingtonumbernofcircumferentialwaves:then=0modedoesnotexhibitanycircumferentialwaveandisabreathmode;then=1modeexhibitsonecircumferentialwaveandisessentiallyadisplacementoftiretreadwithrespecttotherim;then=2modeexhibitstwocircumferentialwavesandthetreadhasanovalshape.

In-planemodesarethemostexcitedmodes,sinceinsteadystateconditions(rectilinearpath)theresultantoftireforces(loadN,brakingforceFanddrivingforceS)staysapproximatelyinthesymmetryplaneofthewheel.Ifacoordinatesystemfixedtothewheelisconsidered,theresultantoftireforcesrotatesaroundthewheelwithangularvelocityω=V/Ro,whereRoisthetirerollingradiusandVisforwardspeed.Therotatingforcemayexciteinresonanceconditionsthecircumferentialmodethatexhibitsncircumferentialwavesifthefollowingconditionissatisfied:

ω=2πνj/n

whereνjisthenaturalfrequencyofthemode.

Thepresenceofroadunevennessandgroovesonthetiresurfaceareothersourcesofexcitationinthehighfrequencyrange.Finally,thetransientmaneuvers(e.g.braking,changinglanes),whichcorrespondtosuddenvariationsintireforcesandtorques,mayexcitebothin-planeandout-of-planemodes.

Generally,thefirstnaturalfrequenciesoftiresareintherange100÷200Hzandcorrespondtoout-of-planemodes,thenthereissometimesabandoffrequencywithmixedmodes,whenthefrequencyishigherthan300÷400Hz,themodeswithlargein-planedisplacementdominate.

Fig.2-38Carcassdeformationsofthetire.

Fig.2-39Exampleofvibrationmodesofthetires(withgroundcontact).

Naturalfrequencies,lossfactorsandmodeshapesstronglydependontiresize,construction(radial-ply,bias-ply)andmaterial.Thecomparisonbetweenthemodalpropertiesofradial-plyandbias-plymotorcycletiresshowsthatnaturalfrequenciesofradial-plytiresarehigherthantheonesofthesimilarmodesofbias-plytires;thedifferenceislargeespeciallyinthecaseofin-planemodes.

Thepresenceofcontactwiththegroundincreasesthecomplexityofmodesand,betweenthemodesthatweremeasuredinfreeconditions,newmodeshavingintermediateshapeandfrequencyappear.Therangeofnaturalfrequenciesandlossfactorsofthemodesincontactwiththegroundarenotverydifferentfromtheonesmeasuredinfreeconditions.

Figure2-39showsthenaturalfrequenciesandmodesofa120/65R17fronttireinflatedto2.2bars.Thistireisahighperformanceradial-plytirewith0°steelbelts.Thefirstout-of-planemodeisthelateraldisplacementofthetiretreadwithrespecttotherim.Thenthereisabanana-shapedmode(with1.5wavesinthelateraldirectionandminordisplacementsintheotherdirections).Thefirstin-planemode(410Hz)isessentiallyanin-planedisplacementoftiretreadandderivesfromthen=1modemeasuredinfreecondition(withoutcontact).

Thefollowingmodes(422and447Hz)derivefromthemodeswithtwocircumferentialwaves(n=2)andthreecircumferentialwaves(n=3)measuredinfreeconditions.Becauseofthehighcircumferentialrigidityofthe0°steelbelt,thebreathmodeisnotidentifiedinthe0-500Hzrangeoffrequencies.Thenaturalfrequenciesarehigherthanthoseofcartires.Lossfactorsofin-planemodesaresimilartothosemeasuredincartires.Lossfactorsofout-of-planemodesarehigherthantheonesofin-planemodes,becausetheyaremainlyinfluencedbythetire’sside-walls.

Benelli250ccof1938

3Recti l inearMotionof Motorcycles

Thebehaviorofmotorcyclesduringrectilinearmotiondependsonthelongitudinalforcesexchangedbetweenthetiresandtheroad,theaerodynamicforcesinducedthroughthismotion,andtheslopeoftheroadplane.Thestudyofrectilinearmotionhighlightscertaindynamicaspectsthatarealsoimportantforsafety,suchasthemotorcycle’sbehaviorduringbrakingwithpossibleforwardoverturning,andinacceleration,withpossiblewheeling.

3.1Res is tance forces actingonmotorcyclesDuringsteadystatemotion,thethrustproducedbytheengineisequatedtotheforcesthatoppose

forwardmotionanddependessentiallyonthreephenomena(Fig.3-1):resistancetotirerolling;aerodynamicresistancetoforwardmotion;thecomponentoftheweightforcecausedbytheslopeoftheroadplane.

Resistancetotirerolling,Fw,wasamplydiscussedinthepreviouschapter.Itwasseenthatitcouldgenerallybeconsideredequaltoabout2%oftheweightforce.

Fig.3.1Resistanceforcesactingonthemotorcycle.

3.1.1Aerodynamicres is tance forces

Alltheaerodynamicinfluencesthatactonthemotorcyclecanberepresentedbythreeforces,whichareassumedtobeappliedonthecenterofgravity,andbythreemomentsactingaroundthecenterofgravityaxesx,y,z,asshowninFig.3-2:

thedragforce,inoppositiontoforwardmotion;theliftforcethattendstoraisethemotorcycle;thelateralforcethatpushesthemotorcyclesideways;thepitchingmoment;theyawingmoment;therollingmoment.

Fig.3-2Aerodynamicforcesandmoments.

Themostimportantcomponentsarethedragandliftforces.Theyareappliedatapoint,calledthepressurecenter,whichdoesnotcoincidewiththecenterofgravity,butratherisgenerallylocatedaboveit.Theresultantofthetwoaerodynamicforcesthereforegeneratesapitchingmomentaroundthey-axis.

Thedragforceinfluencesboththemaximumattainablevelocityandperformanceinacceleration.ThedragforceFDisapproximatelyproportionaltothesquareofthemotorcycle’sforwardvelocity:

ρrepresentsthedensityoftheair(equalto1.167kg/m3atanatmosphericpressureof987mbarandatemperatureof20°C);Aisthefrontalareaofthemotorcycle;CDrepresentsthecoefficientofaerodynamicresistance(dragcoefficient);Vistheforwardvelocityofthemotorcycle.

ThevalueofthecoefficientCDisstronglyinfluencedbytheshapeofthemotorcycle,inparticularbythepresenceorlackofafairing.Ingeneral,itcanbestatedthatthereisasignificantincreaseinaerodynamicresistancewhenvortexwakesareformedandtheboundarylayerbreaksfromthesurfaceofthefairing.

TheinteractionofthemotorcyclewithairalsogeneratesaliftforceFLproportionaltothesquareofthevelocity,whichreducestheloadonthefrontandinsomecasestherearwheel:

whereCLrepresentstheliftcoefficient.

Motorcycleliftisdangeroussinceitreducestheloadonthewheelsand,thus,tireadherence.Thisisespeciallytrueregardingthefronttiresincethecenterofpressureisgenerallyinfrontofandabovethecenterofgravity.Typicalmotorcyclesgeneratepositive(upward)liftforce,however,inordertocounteractthisphenomenonandincreaseloadonthewheels,itwouldbenecessaryaffixsomesortofwingatthefrontofthemotorcycleasinthecaseofracingcars.Tolessentheundesiredlifteffects,modernfairingsaredesignedtoreducelifttoaminimum.

TheaerodynamiccharacteristicsofmotorcyclesaregivenbythedragareaCDA(dragcoefficienttimesthefrontalarea)andbytheliftareaCLA(liftcoefficienttimesthefrontalarea).

ThevalueoftheproductCDAcanvaryfrom0.18m2forspeedrecordcontendersthatarecompletelyfairedto0.7m2formotorcycleswithnofairingandtheriderinanerectposition.Αtypicalvaluefor“superbike”motorcyclesis0.30to0.35m2,while“GrandPrix”motorcyclesreach0.22m2orevensmallervalues.Touringand/orsportingmotorcycleswithasmallfrontfairinghavevaluesaround0.4to0.5m2.ThechangefromanerecttoaproneridingpositionleadstoareductioninthevalueoftheproductCDAthatvariesfrom5to20%,dependingonthetypeofmotorcycleandtherider ’sbodystructure.

Theresistancetoforwardmotionisinfluencedindifferentwaysbythevariousmotorcyclecomponents.Forexample,thefollowingaretheeffectsofsomecomponentsontheproductCDA:

frontfairingsproduceanimprovementrangingfrom0.02to0.08m2;sidefairingsdecreasetheCDAbyaquantityofapproximately0.15m2;sidemirrorsincreasethedragareafrom0.012to0.025m2;thepresenceofarearfairingimprovesitbyafactorof0.015m2;thesaddlebags,ifappropriatelydesigned,improveitby0.02m2;alowerspoilerimprovesitbyafactorthatvariesfrom0.01to0.02m2.

ThefrontalareaAdiffersaccordingtothetypeofmotorcycleandisstronglyinfluencedbythebodyoftheriderandhis/herpositionduringtravel.Referencevaluesmayvaryfrom0.6to0.9m2forlargedisplacementtouringmotorcycles,from0.40to0.6m2forsportingmodelsandfrom0.4to0.5m2forGrandPrixmotorcycles.SmalldisplacementGrandPrixclassmotorcycles(125cc)reachvaluesaround0.32m2.IfthefrontalareaAandtheproductCDAvaluesareknown,theresistancecoefficientCD,whichisusuallyontheorderof0.4to0.5,canbeevaluated.TheproductoftheliftcoefficienttimesthefrontalareaofthesectionCLArangesfrom0.06to0.12m2.

Fig.3-3Dragforceversusvelocity.

Thepitchingmomentcausedbytheaforementionedforcescanbedangerous,sinceitleadstoadecreaseintheloadonthefrontwheelandanincreaseintherearone.Thesevariationscansignificantlymodifythedynamicbehaviorofthemotorcycle.

Inrectilinearmotion,ifthereisnocrosswind,thex-zplaneofthemotorcyclewithrideristheplaneofsymmetryandtheforwardvelocityofthemotorcycleliesinthatplane.Thelateralaerodynamicforceandtherollingandyawingmomentsarezero.However,theyarenotzeroiftheridermovesfromasymmetricposition,ifthereislateralwind,orifthesideslipanglesofthetiresarenotzero.Inparticular,whentheridermovesintothecurve,displacinghisorherbodyandkneetowardstheinsideofthecurve,anaerodynamicyawingmomentisgeneratedhelpingthemotorcyclemoveintothecurve.Duringthecurveiftheriderstaysinthisleanedpositionthelateralaerodynamicforcepersists.

Sincethepowerdissipatedbytheaerodynamicforcesdependsonthecubeofthevelocity,alotofpowerisneededtoattainhighvelocities.Figure3-3showsthevariationofthedragforceagainstvelocityforvariousvaluesofthedragarea.

Theforcesandaerodynamicmomentscanbemeasuredinawindtunnelbymountingthemotorcycleonaforcebalance.Thewindtunneltestsmakeitpossibletoidentifythepresenceofvortices,ifany,andthecurrentlinessurroundingthemotorcycle.Smokeisusedtovisualizethecurrentlines,ascanbeseeninFig.3-4.

Fig.3-4Currentlinessurroundingthewindscreenandrider(UniversityofPerugiawindtunnel).

Ifnowindtunnelisavailable,thedragareaCDAcanbedeterminedinthefollowingways.ThemotorcyclecanbedrivenatitsmaximumvelocityonastraightroadrecordingtheengineRPM(revolutionsperminute)andthemaximumvelocity.Thepower,correspondingtothenumberofrevolutionsmeasured,isdeterminedbythedynamometercurve.Theproductofthedragcoefficienttimesthefrontalareais:

withoutconsideringtherollingresistance.Therecanbesignificanterrorsifthemaximumvelocityisnotdeterminedcorrectlyoriftheactualpowerofthemotorcycledoesnotcorrespondtothedynamometercurveusedforthecalculation..

Asecondapproachisasfollows.Themotorcyclecanbedrivenonaflatroadatasustainedvelocityandthenplacedinneutral.ThetimeΔtthatthemotorcycleneedstoslowdownfromaninitialvelocity(Vinitial)toalowerone(Vend)ismeasuredandthedragareaisgivenby:

Therecanbeacertainoperativedifficultytryingtoidleatasustainedvelocity.Furthermore,the

massmshould,strictlyspeaking,alsotakeintoaccounttherotatinginertia.

Example1

Whatpowerisrequiredtopushasportmotorcycle(CDA=0.35)toavelocityof250km/hand275km/h,takingintoaccountboththerollingresistanceforceandthedragforce?

Thepoweronthewheelforamaximumvelocityof250km/his71.1kW.Toincreasethemaximumvelocityby10%(275km/h),a32%increaseinpowerisnecessary(94.0kW).

3.1.2Res is tantforce causedbyroads lope

TheresistantforceFPcausedbytheslopeoftheroadplaneisequaltothecomponentoftheweightforceinthemotorcycle’sdirection:

FP=mg⋅sinα

whereαrepresentstheslopeoftheroadplane.

ThegraphinFig.3-5displaysthecurvesofdifferentpowerlevelsatthewheelversusthevelocityandtheroadslope.

Fig.3-5Poweratthewheel(kW)asafunctionoftheforwardvelocityandroadslope.

Example2

Consideramotorcyclewiththefollowingcharacteristics:

•mass: m=200kg;

•frontalarea: A=0.6m2;

•dragcoefficient: CD=0.7.

Whatisthedrivingforceandpowernecessarytosustainthevelocityundertheconditionsgiven?

Case1:Flatroadtravelingat200km/h.

Thedrivingforcenecessarytomaintainthemotorcycleataconstantvelocityof200km/hor55.6m/salongahorizontalroad(α=0)mustbeequaltothesumoftheaerodynamicresistanceforceFDandtherollingresistanceforceFw:

•forceofaerodynamicresistance: FD=0.5•1.167•0.6•0.7•55.62=756.4N

•forceofrollingresistance: Fw=0.02•200•9.8=39.2N

Thereforetherequireddrivingforceis: S=FD+Fw=795.6N

Forthisvalueofthedrivingforce,thepoweratthewheel(P)isequalto:

P=796.8N•55.6m/s=44.20kW

Case2:Uphillroadat200km/h.

Ifthemotorcycletravelsatthesamevelocity,butalongaroadwithaconstantslopeof12%(angleα=6.84°)theresistantforcecausedbytheslopeoftheroadplanemustalsobetakenintoaccount.

•resistantforcecausedbytheroadslope: FP=sin(6.84)•200•9.8=233.5N

Inthiscasetherequireddrivingforceis: S=FD+Fw+FP=1029.1N

Forthisvalueofthedrivingforce,thepoweratthewheel(P)isequalto:

P=1029.1N•55.6m/s=57.2kW

3.2The centerofgravityandthe moments of inertia

3.2.1Motorcycle centerofgravity

Thepositionofamotorcycle’scenterofgravityhasasignificantinfluenceonthemotorcycle’sdynamicbehavior.Itspositiondependsonthedistributionandquantityofthemassesoftheindividualcomponentsofthemotorcycle(engine,tank,battery,exhaustpipes,radiators,wheels,fork,frame,etc.).Sincetheengineistheheaviestcomponent(about25%ofthetotalmass),itslocationgreatlyinfluencesthelocationofthemotorcycle’scenterofgravity.

Fig.3-6Thelongitudinalpositionofthecenterofgravity.

Thelongitudinaldistancebbetweenthecontactpointoftherearwheelandthecenterofgravitycaneasilybedeterminedbymeasuringthetotalmassofthemotorcycleandtheloadsonthewheelsunderstaticconditions(frontload ,rearloadNsr):

Ingeneral,amotorcycleischaracterizedbythestaticloadsthatactonthewheels,expressedinapercentageformula:

Thedistributionoftheloadonthetwowheelsunderstaticconditionsisgenerallygreateronthefrontwheelforracingmotorcycles(50-57%front,43-50%rear);andconversely,itisgreaterontherearwheelinthecaseoftouringorsportmotorcycles(43-50%front,50-57%rear).

Whenthecenterofgravityismoreforward(frontload>50%),wheelingthemotorcyclebecomesmoredifficult,orinotherwords,thereisaneasiertransferofthepowertotheground.Thisisonereasonracingmotorcyclesaremoreheavilyloadedinfront.Inaddition,thegreaterloadinthefrontpartiallycompensatesfortheaerodynamiceffectsthatunloadthefrontwheel;thisfactbecomesimportantathighvelocities.Whenthepositionofthecenterofgravityismoretowardstherearofthemotorcycle,brakingcapacityisincreasedreducingthedangerofa“stoppie”orevenforwardflipoverduringasuddenstopwiththefrontbrake.

Modernsportmotorcyclestendtohavea50÷50%distributionsoastoperformwellinboth

accelerationandbrakingphases.Itisimportanttokeepinmindthatitispreferable,asaquestionofsafety,tohavelongitudinalslipoftherearwheelinanaccelerationphase,ratherthanlongitudinalslipofthefrontwheelinabrakingphase.Theratiob/pwithoutridervariesfrom0.35to0.51:thesmallestvaluesforthescooterandthehighestforracingmotorcycles.

Ingeneral,thepositionoftheridermovestheoverallcenterofgravitytowardstherear(Fig.3-7),andtherefore,hisorherpresenceincreasestheloadontherearwheeltherebydiminishingthepercentageofloadonthefrontwheel(forexampletheratiob/pgoesfrom0.53to0.50).

Fig.3-7Thepositionofthecenterofgravityofthemotorcycleandtherider.

Oncethelongitudinalpositionofthecenterofgravityhasbeenfound,itsheightcanbedeterminedbymeasuringtheloadononlyonewheel,forexample,therearonewiththefrontwheelraisedbyaknownamountasinFig.3-8.

Theheightofthecenterofgravityhasasignificantinfluenceonthedynamicbehaviorofamotorcycle,especiallyintheaccelerationandbrakingphases.Ahighcenterofgravity,duringtheaccelerationphase,leadstoalargerloadtransferfromthefronttotherearwheel.Thegreaterloadontherearwheelincreasesthedrivingforcethatcanbeappliedontheground,butthelesserloadonthefrontwheelmakeswheelingmoreprobable.

Inbraking,ahighercenterofgravitycausesagreaterloadonthefrontwheelandaresultinglowerloadontherear.Thegreaterloadonthefrontwheelimprovesbrakingbutitalsomakestheforwardflip-overmorelikely,whichoccurswhentherearwheeliscompletelyunloaded.

Fig.3-8Measureoftheheightofthecenterofgravity.

Theoptimalheightofthecenterofgravityalsodependsonthedriving/brakingtractioncoefficientbetweenthetiresandroadplane.Withlowvaluesofthedriving/brakingtractioncoefficient(whentheroadiswetand/ordirty)itisgoodtohaveahighcenterofgravitytoimproveboththeaccelerationandbrakingcapacities.Withhighvaluesofthedriving/brakingtractioncoefficientitisgoodtohavealowercenterofgravityinordertoavoidthelimitconditionsofwheelingandforwardflipover.

Itisclearthatthechoiceoftheheightofthecenterofgravityanditslongitudinalpositionisacompromisethatmusttakeintoaccounttheintendeduseandpowerofthemotorcycle.All-terrainmotorcyclesarecharacterizedbyratherhighcentersofgravity,whileverypowerfulmotorcyclestypicallyhavealowercenterofgravity.Themaineffectsofthelocationofthecenterofgravitymaybesummarizedinthefollowingdiagram:

Forwardcenterofgravity

Themotorcycletendstoover-steer(incurvestherearwheelslipslaterallytoagreaterextent).

Rearcenterofgravity

Themotorcycletendstounder-steer(incurvesthefrontwheelslipslaterallytoagreaterextent).

Highcenterofgravity

Thefrontwheeltendstoliftinacceleration.Therearwheelmayliftinbraking.

Lowcenterofgravity

Therearwheeltendstoslipinacceleration.Thefrontwheeltendstoslipinbraking.

Theheightofthecenterofgravityofthemotorcyclealonehasvaluesvaryingfrom0.4to0.55m,butthepresenceoftheriderraisesthecenterofgravitytovaluesrangingfrom0.5to0.7m.Obviously,thedisplacementofthecenterofgravityduethepresenceoftheriderdependsonthe

relationbetweenthemassoftheriderandthatofthemotorcycle.

Theratioh/pwithoutriderandwithfullyextendedsuspensionvariesintherange0.3-0.4;thesmallestvaluesforthecruiserandscooterandthehighestfordualsportandendurotypemotorcycles.

Example3

Amotorcyclewithamassof196kghasastaticweightdistribution(50%to50%)anda1390mmwheelbase.A77kgridermasshashisowncenterofgravityat600mmfromthecenteroftherearwheel.Howdoestheriderchangetheoverallpercentageweightdistribution?

Thepercentageloadontherearwheelwithriderincreasesto52%,whilethatonthefrontwheelisreducedto48%.

3.2.2The moments of inertia

Thedynamicbehaviorofamotorcyclealsodependsontheinertiaofthemotorcycleandtherider.Themeasurementofthemomentsofinertiaisbasedoncomplexidentificationmethodologies,whichareoutsidethepurposeofthisbook.Themostimportantmomentsofinertiaaretheroll,pitchandyawmomentsofthemainframe,themomentofinertiaofthefrontframewithrespecttothesteeringaxis,themomentsofthewheelsandtheinertiamomentoftheengine.Inthefollowingtable,thevaluesofthegyrationradiiofthemotorcycleandrider,withrespecttothecenterofgravity,arepresented(themomentofinertiaisgivenbytheproductofthemasstimesthesquareoftheradiusofgyration).

Theyawmomentofinertiainfluencesthemaneuverabilityofthemotorcycle.Inparticular,highvaluesoftheyawmoment(obtained,forexamplebyheavybaggageplacedontheluggagerack)reducehandling.Therollmomentofinertiainfluencesthespeedofthemotorcycleinrollmotion.Highvaluesoftherollinertia,maintainingthesameheightofthecenterofgravity,slowdowntherollmotioninbothentryandexitofacurve.

Fig.3-9Momentsofinertia.

Table3-1.

3.3Motorcycle equi l ibriumins teadys tate recti l inearmotionWewillintroducethefollowingthreehypothesesregardingthemodelofthemotorcycle-rider

systemdepictedinFig.3-10.therollingresistanceforceiszero(Fw=0);theaerodynamicliftforceFLisalsoconsideredzero;sincetheroadsurfaceisflat,theforceresistingtheforwardmotionofthemotorcycleisreducedtojusttheaerodynamicdragforceFD.

Thepressurecenterofthemotorcycle(inwhichthedragforceisapplied)coincideswithitscenterofgravity.

Inadditiontothedragforce,thefollowingforcesactonamotorcycle:theweightmgthatactsatitscenterofgravity;thedrivingforceS,whichthegroundappliestothemotorcycleatthecontactpointoftherearwheel;theverticalreactionforcesNfandNrexchangedbetweenthetiresandtheroadplane.

TheequationsofequilibriumofamotorcycleenableustodeterminetheunknownvaluesofthereactionforcesNfandNr,oncetheweightforcemg,drivingforceS,anddragforceFD_areknown.

(⇒)Equilibriumofhorizontalforces: S−FD=0

(⇑)Equilibriumofverticalforces: mg−Nr−Nf=0

(∩)Equilibriumofmomentswithrespecttothecenterofgravity:

Sh−Nrb+Nf(p−b)=0

Fig.3-10Forcesactingonamotorcycle.

Theverticalforcesexchangedbetweenthetiresandtheroadplanearetherefore:dynamicloadonthefrontwheel:

dynamicloadontherearwheel:

Thesereactionforcesarecomposedoftwoelements.

Thefirstterm(staticloadonthewheel),dependsonthedistributionoftheweightforce.

Thesecondterm(loadtransfer),isdirectlyproportionaltothedrivingforceSandtheheighthofthecenterofgravity,andinverselyproportionaltothemotorcycle’swheelbasep.

Wewillnowfocusonthesecondterm.“Loadtransfer”referstothefactthatthereisadecreaseintheloadonthefrontwheelandacorrespondingincreaseintheloadontherearwheel;“loadistransferredfromthefronttotherearwheel,”hencethedesignation.

Theratiobetweentheheightofthecenterofgravityandthewheelbaseismuchhigherinmotorcyclesthanincars,whereh/pisusuallyintheinterval0.3to0.45.

Theloadsonthewheelscanberepresentedinnon-dimensionalformwithrespecttotheweight:Normalizedloadonthefrontwheel:

Normalizedloadontherearwheel:

whereSaindicatestheratiobetweenthedrivingforceSandthetotalweightmg(non-dimensionaldrivingforce).

Figure3-11illustratesthephenomenonofloadtransfer.Thevariationsinthenormalizedloadsareindicatedasafunctionofthenormalizeddrivingforcefortwomotorcycleswiththefollowingcharacteristics.

The1stmotorcycleb/p=0.45andh/p=0.3.The2ndmotorcycleb/p=0.45andh/p=0.43.

Itcanbeobservedthatthevariationsintheloadsonthewheelsaregreaterforthemotorcyclehavingthehigherh/pvalue.

Fig.3-11Normalizedloadsonthewheelsasafunctionofnormalizeddrivingforce(b/p=0.45).

Nowlet’sconsidertheforcesactingonthemotorcycle,illustratedinFig.3-12.Theweightmgisequaltothesumofthestaticloadsactingonthewheels and .ThedrivingforceSandtheforcecausedbythetransferoftheloadNtr,turnedupwardbecauseithasapositivesign,areappliedattherearwheelcontactpoint.

Fig.3-12Loadtransferangleτ.

Thedirectionoftheresultantofthesetwoforcesisinclinedwithrespecttotheroadbytheangle:

whichisthereforecalledtheloadtransferangle.

Inorderforamotorcycletomaintainequilibrium,thisresultantforcemustbeequaltoandoppositeinsigntotheresultantofthedragforceFDandtheloadtransferNtr,whichactsonthefrontwheel(directeddownwardbecauseithasnegativesign).

Example4

Whatmaximumvelocitycanbereachedbya200kgmotorcycle(CDA=0.3)withadrivingtractioncoefficientequalto1.0,ignoringtheliftandrollingresistanceforces?

Themaximumhypotheticalforwardvelocityofthemotorcycledependsontheloadtransferfromthefronttotherearwheel.Whenthefrontwheeliscompletelyunloaded,andthusthewholeloadmovestotherearwheel,thelimitingconditionisreachedwhenmaximumvelocityisattained.Undertheseconditions,themaximumdrivingforcethatcanbeappliedwithadrivingforcecoefficientequal

to1.0isequalto1962Ν.Thisforceisequaltothedragforceandgeneratesavelocityof381km/h.Thepowerontherearwheelis208kW.

Example5

Calculatethemaximumvelocityatamotorcycle’slimitconditionrepresentedbythewheelingphenomena,supposingthatthecenterofthepressurescoincideswiththecenterofgravity.

•mass: m=200kg;

•dragarea: CDA=0.3m2;

•liftarea: CLA=0.1m2;

•aerodynamicpitchingmoment=0;

•longitudinaldistanceofthecenterofgravity: b=0.7m;

•heightofthecenterofgravity: h=0.65m;

•wheelbase: p=1.40m.

Inconditionsapproachingthelimitofwheelingphenomena,thefrontverticalloadbecomeszero.Thesumoftheloadtransfergeneratedbythedragforceandthefrontcomponentoftheliftforceareequaltothefrontstaticload.Ifweconsiderthatthecenterofpressurecoincideswiththecenterofgravity,theliftforceisdistributedequallybetweenthetwowheelsinthemotorcycleunderconsideration.Wethereforehave:

Thevelocity,correspondingtothisequation,is:

ThemaximumvelocityisthereforeV=339km/h.Thepowerattherearwheel,ignoringtherollingresistanceforceshouldbeatleast147kW.Atthisvelocitythefrontwheelistotallyunloaded,sothatitbecomesimpossibletocontrolthemotorcycle.Thisvalueshouldthereforebeconsidereda

maximumlimitnevertobereached.

3.4Motorcycles intrans i toryrecti l inearmotionWewouldliketoconsideramotorcycleintransitoryrectilinearmotionassumingthehypotheses

presentedintheprecedingparagraphtobevalid.Themotorcycle’sequilibriumequationswhichwerewrittenforsteadystatemotioncanstillbeconsideredvalidforverticaltranslationandrotation.

(⇑)Equilibriumoftheverticalforces: mg−Nr−Nf=0

(∩)Equilibriumofthemomentswithrespecttothecenterofgravity:

Sh−Nrb+Nf(p−b)=0

whereSindicatesthedrivingforceduringacceleration(+)orthebrakingforceduringdeceleration(-).

Theequationofequilibriumforamotorcycleinhorizontalmotiontakesoncertaincharacteristicsaccordingtowhetherthemotorcycleisinanaccelerationorbrakingphase.

3.4.1Acce leration

Inthiscase,thedrivingforceisequaltothesumoftheinertialandresistanceforces.

(⇒)Equilibriumofthehorizontalforces:S*=FD+m*ẍ

whereS*=T(ωm/V)istheequivalentdrivingforceobtainedbymultiplyingT,theenginetorque,byωm/V,theratiobetweentheenginespeedandtheforwardvelocity,andm*indicatestheequivalentmassofamotorcycle,whichalsotakesintoaccounttheelementsofrotationalinertia.Thelatteriscalculatedbyequatingthetotalkineticenergyofthemotorcycle(thesumofrotationalkineticenergyoftherotatingpartsandthekineticenergyoftranslation)tothekineticenergyofan

equivalentsystemconstitutedbyonemassm*(equivalentmass).Fromtheviewpointofdynamics,themotionlawoftheequivalentmassisequaltothatofarealmotorcycle(Fig.3-13).

Fig.3-13Rotatingpartsofamotorcycle.

Equatingthekineticenergies,wehavethefollowingexpression.

Where:

•m isthemassofthemotorcycle;

• istheinertiaoftherearwheel;

• istheinertiaofthefrontwheel;

• istheinertiaoftheprimaryshaft(includingclutch);

• istheinertiaofthesecondaryshaft;

•τ isthevelocityratio.

istheinertiaoftheengine(crankshaft,counter-rotationshafts),reducedtothecrankshaft.

• Thisinertiacanbeconsideredconstantinaninitialapproximationifweignorethefluctuatingtermsofthemasseshavingreciprocatingmotion.

Theratiooftheangularvelocityoftherearwheelandamotorcycle’sforwardvelocityis:

Theratiobetweentheangularvelocityofthefrontwheelandamotorcycle’sforwardvelocityis:

Theratiobetweentheangularvelocityofthesecondaryshaftandamotorcycle’sforwardvelocityis:

whereτs,rindicatesthetransmissionratiobetweenthepinionandrearsprockets.

Theratiobetweenthevelocityoftheprimarygearshaftandthevelocityofamotorcycleis:

whereτp,sisthetransmissionratiobetweentheprimaryandsecondarygearshafts.Thedrivesprocketiskeyedonthesecondaryshaft.

Theratiobetweenthevelocityoftheengineshaftandamotorcycle’sforwardvelocityis:

whereτm,pindicatesthetransmissionratiobetweentheenginecrankshaftandprimaryshaft.

Asisthecaseofsteadystatemotion,thedynamicloadsonthewheelsaregivenbytheequations:dynamicloadonthefrontwheel:

dynamicloadontherearwheel:

Example6

Consideraracingmotorcyclewiththefollowingcharacteristics.Whatistheequivalentmass?

• totalmass(motorcycle+rider): m=205kg;

• frontwheelradius: Rf=0.30m;

• rearwheelradius: Rr=0.32m;

• frontwheelmomentofinertia: Iwf=0.6kgm2;

• rearwheelmomentofinertia: Iwr=0.8kgm2;

• enginemomentofinertia: Iwm=0.05kgm2;

• primaryshaftmomentofinertia: Iwp=0.005kgm2;

• secondaryshaftmomentofinertia: Iws=0.007kgm2;

• transmissionratioforthedriving-wheelsprockets: τs,r=2.6;

• transmissionratiofortheprimary-secondarygearshafts: τp,s=0.9(inIVgear);

• transmissionratiofortheengine-primaryshafts: τm,p=2.

Thevelocityratiosthusbecome:τr=3.125;τf=3.33;

Oncethevelocityratiosareknown,itispossibletocalculatetheequivalentmassm*asfollows:

Itisworthpointingoutthattheengineplaysaveryimportantrolesincethevelocityratioishigh,evenifthevalueinthemomentofinertiaislow.

Theequivalentmassobviouslydependsonthegearengaged.Thetransmissionratioofthegearshiftvariesfromvaluesequaltoabout3forthefirstgear(thegearshiftfunctionsasareductiongear),tovaluesthatcanapproachorbeslightlylowerthanunity(downtoabout0.7)inthetallestgear.Themaximumvalueofreducedinertiaisreachedwhenfirstgearisengaged.

Figure3-14showsthevariationofthedrivingforceonthewheelversusvelocity,inthevariousgears,foraracingmotorcycle.Thecurvetracedalsoshowsthevariationintheresistanceforce(thesumofaerodynamicandrollingresistanceforces)intermsofthemotorcycle’svelocity.

Fig.3-14Drivingforceasafunctionofvelocity.

Let’ssupposethatthemotorcycleproceedsinthirdgearatadeterminedvelocity,asgiveninthefigure.Thedrivingforce,thoughlowerthanthemaximumavailableinthatgear,isgreaterthantheresistanceforce,sothattheremainingdrivingforcecanbeusedtoaccelerateorgoupaslopeatthesamevelocity.Ifweconsiderthesamesetvelocitywithhigherratios,weseethatthereislessdrivingforceavailableforaccelerating.Asthevelocitygraduallyincreases,thepassagetothehigherratiomakesalowerquotaavailable.Maximumvelocityisobviouslyreachedwhentheresistantforceisthesameasthedrivingforceinthehighestgear.

Thecomparisonbetweenthedrivingforcecurvesandtheresistancecurvecanalsobemadeintermsofpower.Adiagramofusefulpowertothewheelcanbeobtainedforeachratiobymultiplyingeachcurvebyitscorrespondingforwardvelocity.Fig.3-15showsthecurvesofusefulpowertothe

wheelandtheresistancepowerintermsofforwardvelocity.Inthiscaseaswell,theintersectionpointoftheresistancepowercurvewiththeusefulpowercurve,inthehighestgear,determinesthemaximumvelocitythatcanbereached.

Let’snowsupposethattheenginepowerremainsconstantunderanincreaseinvelocity.Thisisanidealcaseinwhichtheefficienciesareindependentofthevelocities,representedinFigure3-15byahorizontalline.Maximummotorcycleaccelerationcanbedeterminedbyintegratingthefollowingdifferentialequation:

wherePmaxindicatesthemaximumpoweroftheengine.Bycarryingoutthenumericalintegrationoftheprecedingdifferentialequation,wecancalculatethemaximumaccelerationthemotorcycleiscapableofreaching.Thisisclearlyanidealvalue,sincethemaximumaccelerationofthemotorcyclecanactuallybelimitedbothbythereartire’sadherence,thepossiblewheelingofthemotorcycleandthefinitenumberofgearsetc.

Fig.3-15Poweratthewheelasafunctionofvelocity.

Example7

Consideramotorcyclewiththefollowingcharacteristics.Letuseexaminehowchangingthemassaffectsthevelocityandacceleration.

•Equivalentmass: m*=230.9kg;

•frontalarea: A=0.6m2;

•dragcoefficient: CD=0.7.

ThemaximumpoweroftheengineisequaltoPmax=70kWwhilethemaximumtransmissibledrivingforcetotheroadisequalto4000Ν.

Thecurveofthevelocityversustime,obtainedbycarryingoutanumericalintegrationofthedifferentialequationofmotion,isgiveninFig.3-16.A30%lightermotorcyclepresentsagreateraccelerationbutthemaximumvelocityremainsthesame.

Fig.3-16Anexampleofacceleration.

Inrealitytheaccelerations(thegradientofthecurve)areactuallylowerbecauseofthetimeintervalsneededtochangegears,duringwhichtheusefuldrivingforceiszero.Furthermore,intheinitialphase,aspreviouslyanticipated,itisnotalwayspossibletoapplytheentiredrivingforcebecauseofreartireslippageand/orpossiblewheeling.

3.4.2Traction-l imitedacce leration

IfwetakeintoconsiderationamotorcycleacceleratingasinFig.3-17andassumeitispossibletoignoretherollingresistanceforceFw,thenthemotionequationcanbewrittenasfollows:

S=mẍ+FD

whereSindicatesthedrivingforceonthewheelandFDthedragforce.Presumingthattheenginehasadequatepower,thedrivingforcemustbelower,oratmost,equaltothemaximumforcegivenbytheproductofthedrivingtractioncoefficientμpwiththeverticalloadNr.

S≤μp⋅NrIfwerememberthat

wenowhave:

Fig.3-17Accelerationlimitedbythedrivingtractioncoefficient.

MaximumaccelerationisreachedwhentheresistanceforceFDiszero,i.e.startingfromlowspeed.Asthevelocityincreases,theaccelerationunderlimitingfrictionconditionsdiminishes.Thishappensbecausepartofthedrivingforceisequatedtotheresistanceforceandthereforecannotbeusedtoaccelerate.

3.4.3Wheel ing-l imitedacce leration

Thelimitingconditionattheonsetofwheelingisachievedwhentheloadonthefrontwheelisreducedtozero,asseeninFig.3-18.Thissituationisexpressedbythefollowingrelation:

fromwhichwehave

Accelerationwhichcorrespondstoimpendingwheeliethereforedependsontheratiob/h.

Astheforwardvelocitygraduallyincreases,theaccelerationatwhichthewheelingphenomenonbegins,decreases.ThisisthecasesincethemotionofwheelingisalsofavoredbythedragforceFD,thevalueofwhichincreaseswithvelocity.

Fig.3-18Accelerationlimitedbywheeling.

Example8

Consideramotorcyclewiththefollowingproperties.Determinethewheelinglimitedaccelerationatinitialspeed0km/hand100km/h.

•totalmass: m=200kg;

•frontalarea: A=0.7m2;

•dragcoefficient: CD=0.6;

•liftcoefficient: CL=0;

•longitudinaldistanceofthecenterofgravity: b=0.58m;

•heightofthecenterofgravity: h=0.62m;

•wheelbase: p=1.35m.

Themaximumaccelerationofthemotorcycle,attherearwheelfrictionlimit,isrepresentedinthegraphinFig.3-19intermsofthedrivingforcecoefficientbetweenthetireandtheroadatavelocityofzeroand100km/h.Asthevelocityincreases,themaximumaccelerationdecreases,sincepartofthedrivingforceisequatedtotheresistanceforceandcannotbeusedtoaccelerate.

Fig.3-19Maximumaccelerationatlimitconditions.

Theaccelerationatthemotorcycle’swheelinglimitatzerokm/hand100km/hisalsoshowninthegraph.Thehorizontallinerepresentingwheeling-limitedaccelerationisexplainedbythefactthataccelerationdoesnotdependonthedrivingforcecoefficient.

Thewheeling-limitedaccelerationisequaltothetraction-limitedaccelerationwhenthedrivingtractioncoefficientis:

Accelerationis:

Forvaluesofthecoefficientbelowtheratiob/h,themotorcycle,initsaccelerationmaneuver,willnotliftthefrontwheel,sincerearwheelslippagepreventsitfromreachingwheelingacceleration.Analogously,forvaluesofthecoefficientabovetheratiob/h,themotorcyclecannotreachmaximumaccelerationatthedrivingtractioncoefficientsincethefrontwheelrisesbeforereachingthatlimitingvalue.Theseconsiderationssuggestthatitisappropriatetolimitthemaximumtorquetheenginecandeliver,iftheintentionistoavoidmotorcyclewheelingandrearwheelslippage.

Inthecaseunderconsideration,theaccelerationasshowninFig.3-19isequalto9.18m/s2.Thisvalueisobtainedbyapplyingadrivingforceequalto1835Νtothemotorcycle.Wheneverthemotorcycle’sengineisunabletodeliverausefulforcetothewheelofthatmagnitude,wheelingcannotoccurnaturally(nonetheless,theridercouldcausethefrontwheeltorisebymovingandmakinguseofthepitchingmotion).

3.4.4Braking

Drivingsafetyrequires,inadditiontoanefficientbrakingsystem,thattheriderbeabletojudgethestoppingdistancerequiredundervariousconditionsandbrakeinthebestway,usingallofthebrakingsystem’spossibilitiesandinparticularthoseoftherearbrake.Infact,manymotorcycleriderstendtoforgettherearbrake,whichincertaincircumstancesprovidesausefulcontribution.Itscorrectuseisimportantbothinbrakingwhenenteringacurveandinbrakingduringrectilinearmotionwhenanunforeseenobstacleappearsinfrontofthemotorcycle(especiallywhenroadadherenceisprecarious).

Role ofthe rearbrake insuddens tops

Duringcurveentrytheuseoftherearbrakecanbequiteuseful.Expertridersusetherearbrakenotonlytodeceleratethemotorcyclebutalsotocontroltheyawmotion.Rearbrakinginenteringthecurveincreasesthesideslipangleandthereforetheyawmotionofthemotorcycle.

Insuddendecelerationadangerousconditioncouldariseespeciallywhentheloadontherearwheeldiminishestowardzeroduetoloadtransfer.

Ifthemotorcycleisnotinperfectlystraightrunningtheforceofthefrontbrakeandtheinertialforceofdecelerationgenerateamomentthattendstocausethemotorcycletoyaw.ThisisillustratedinFig.3-20.

Fig.3-20Motorcycleinacurvewithabrakingforceappliedonlyinthefront.

AsshownintheinFig.3-20,thetorquegeneratedbythefrontbrakingforceandtheinertialforcetendstoyawthevehicle.Onthecontrary,thepresenceofarearbrakingforcegeneratesatorquewhichtendstoalignandstabilizethevehicleascanbeseenintuitivelyinFig.3-21.

Fig.3-21Motorcycleinacurvewithabrakingforceappliedonlyintherear.

Thesesimpleconsiderationssuggestthatproperutilizationoffrontandtherearbrakeshasapositiveeffectonvehiclestability.

Loadtrans ferduringbraking

Inordertoevaluatetheroleoftherearbrakeduringabrakingeventatthelimitofslippage,weneedtobringupsomepointsregardingtheforcesactingonamotorcycle.Duringdeceleration,theloadonthefrontwheelincreases,whilethatontherearwheeldecreasesandthusthereisaloadtransferfromthereartothefrontwheel.Ifweconsideramotorcycleinabrakingphase(Fig.3-22)andapplyNewton’slawtothemotorcycle,wecancalculatetheloadtransferfromthereartothefrontwheel.

(⇒) Equilibriumofthehorizontalforces: mẍẍ=−Ff−Fr

(⇑) Equilibriumoftheverticalforces: mg−Nr−Nf=0

(∩) Equilibriumofthemomentsaroundthecenterofgravity: −Fh−Nrb+Nf(p−b)=0

whereF(overallbrakingforce)indicatesthesumofthefrontbrakingforceFfandtherearbrakingforceFr.Thedynamicloadonthefrontwheelisequaltothesumofthestaticloadandtheloadtransfer:

whilethedynamicloadontherearwheelisequaltothedifferencebetweenthestaticloadandtheloadtransfer:

ItcanbeseenthattheloadtransferFh/pisproportionaltotheoverallbrakingforce,andtotheheightofthecenterofgravity,andisinverselyproportionaltothewheelbase.Topreventatirefromslippingduringbraking,thevalueofthebrakingforceappliedtoitmustnotexceedtheproductofthedynamicloadactingonthattiretimesthelocalbrakingtractioncoefficient.Thislatterproductrepresentsthemaximumbrakingforceapplicabletothetire,thatis,thebrakingforceatthelimitofslippage.

Fig.3-22Αmotorcycleunderbraking.

Given and ,thebrakingtractioncoefficientsrelative,respectively,tothefrontwheelandtherearwheel,theoverallbrakingforceatthelimitofslippageisgivenbythefollowingexpression:

Thelimitsofslippagearenotusuallyattainedduringbrakingandthereforethebrakingforcedependsonthebrakingforcecoefficientsused(indicatedbyμanddefinedastheratiosofthelongitudinalforceandthecorrespondingverticalload)ofthefrontandrearwheels.

F=Ff+Fr=μfNf+μrNrFigure3-23showsthevariationofdynamicloadsonthewheelsintermsofthebrakingforce.Both

theloadsonthewheelsandthebrakingforcehavebeenreducedtonon-dimensionalstatuswithrespecttotheweight.Themotorcycleunderconsiderationhasa50%to50%staticloaddistributiononitstwowheels,i.e.thecenterofgravityfallsinthecenterlineofthewheelbase.

Let’ssupposethatthebrakingforcecoefficientusedisverylow,μ=0.2forbothwheels.Wecannotefromthegraphthatthedynamicloadsonthewheelsareapproximatelyequalto0.4ontherearwheeland0.6onthefrontone.Undertheseconditions,iftherearbrakeisnotused,40%ofthemaximumattainablebrakingforceisnotused.However,ifthebrakingforcecoefficientusedisveryhigh,forexampleμ=0.9,asshowninFig.3-23,theloadonthefrontwheelis0.95,whiletheloadontherearwheelisonly0.05.Inthiscase,thepossiblecontributionoftherearbrakingforceisnearly

negligible.

Inconclusion,thefollowinggeneralprinciplescanbestated.Theoptimaldistributionofthebrakingforcevariesaccordingtothebrakingtractioncoefficient.Therearbrakeisoflittleuseonoptimalroadsandwithhighgriptires(highcoefficientoffriction),butbecomesindispensableonslipperysurfaces(reducedcoefficientoffriction).

Fig.3-23Non-dimensionalloadsonthewheelsversustheoverallbrakingforcecoefficient.

3.4.5Forwardfl ipoverofthe motorcycle

Fig.3.23showsthat,withanincreaseintheoverallbrakingforce,theloadontherearwheelbecomeszero.Thislimitingconditionrepresentstheforwardflipoverofthemotorcyclewhenthedynamicloadontherearwheelgoestozero.

Inthissituation,thedynamicloadonthefrontwheelisequaltotheweightofthemotorcycleandthedirectionoftheresultantofthedynamicloadandbrakingforcepassesthroughthemotorcycle’scenterofgravity.Theequationofequilibriumofthemomentswithrespecttothecenterofgravityprovidestheexpressionforthebrakingforceatthepointofturnover:

Alowvalueofthislimitbrakingforceindicatesanincreasedpropensityforaforwardflipover.Itcanthereforebeconcludedthatforwardfallisfavoredwhenamotorcycleislightandwhenithasahighandforwardpositionofthecenterofgravity.

Themotionequation,inconditionswhereafallisimminent,ignoringtheaerodynamicresistance,is:

Themaximumdeceleration,expresseding’s,isproportionaltothelongitudinaldistancefromthecenterofgravitytothecontactpointofthefrontwheel,andisinverselyproportionaltotheheightofthecenterofgravity.

Fig.3-24Motorcycleatthepointoffallingforward.

Itisimportanttonotethatthedecelerationattheflipoverlimitdependsonlyonthepositionofthecenterofgravity,andnotontheweightofamotorcycle.Toincreasethevaluethislimit,itisnecessarytoloweramotorcycle’scenterofgravityandplaceitasfarbackaspossible.Takingintoaccounttheaerodynamicresistantforce:

mẍ=−F−FD

themaximumdecelerationdependsonboththemassandthevelocity:

Example9

Whatisthemaximumdecelerationinbrakingtothelimitofflipover,witha50%to50%distributionoftheloadsonthewheels,awheelbaseof1400mmandaheightofthecenterofgravityof700mm?

Itiseasytoverifythatthemaximumdecelerationisequaltogravity.Ifthevelocityisalsotakenintoaccount,decelerationincreasesasthevelocityincreasesduetotheeffectoftheaerodynamicresistanceforce.Withavelocityof100km/h,adragareaequalto0.4m2andamassof200kg,themaximumdecelerationisequalto1.26gwhileatavelocityof200km/hthemaximumdecelerationincreasesto1.54g.Obviouslyitisverydifficult,ifnotimpossible,tobrakeattheflipoverlimitwithazeroloadontherearwheel.Inthisconditionnearingthelimit,thebestridersareabletoattaindecelerationsequalto1.1to1.2g’s.

3.4.6Optimal braking

Theequilibriumequationofthehorizontalforces:mẍ=−Ff−Frandoftheverticalforces.mg−Nr−Nf=0:allowsustoexpressthebrakingdecelerationasafunctionofthefrontandrearbrakingforcecoefficientsusedduringtheevent:

Itcanbeobservedthatdecelerationdependsonthegeometriccharacteristics(wheelbasep,heightofthecenterofgravityh,longitudinaldistanceofthecenterofgravityb)andthebrakingforcecoefficientsused,anddoesnotdependonthemotorcycle’smass.

Thebrakingforceofthefrontandrearwheels,withrespecttothetotalbrakingforce,alsodependsonlyonthegeometricmagnitudes,andthebrakingforcecoefficientsofthetwowheels:

Inthelimitingconditionsoffriction,withequalbrakingtractioncoefficientsforthetwowheels,thevalueofthemaximumpossibledecelerationbecomes:

ẍmax=μg

Therelationbetweenthebrakingforces,toattainthelimitconditionatbothwheels(equalbrakingtractioncoefficients),simultaneouslymustbeequalto:

Thisrelationindicateshowtodistributethebrakingforcesinordertohaveoptimalbraking,giventhevalueofthebrakingforcecoefficientμ.

ThedistributioncurvesforbrakinganddecelerationareshowninFig.3-25(consideringtheaccelerationofgravitytobeg=9.81m/s2)intermsofthebrakingforcecoefficientsusedoneachwheel.InFig.3-25wecanseethatdecelerationincreasesasthebrakingforcecoefficientsincrease,especiallywithregardtothefrontwheel.Thisbehaviorisunderstandablesince,ashasalreadybeenexplained,duringbrakingthereisaloadtransferfromthereartothefrontwheel.Thesolidlinesrepresentthedistributionofbrakingbetweenthefrontandrearwheels.

Thehorizontalaxiscorrespondstobrakingwiththerearwheelalone(0/100)whiletheverticalaxisrepresentsthecaseofbrakingwiththefrontwheelalone(100/0).Thefiguresshowtheutilityofusingtherearbrake,especiallywhenthebrakingtractioncoefficientislow.Itsusefulnessdiminishesuntilitbecomesalmostnegligibleinthepresenceofveryhighbrakingtractioncoefficients.

Fig.3-25Curvesofdecelerationanddistributionofbraking.

[p=1.4m;h=0.7m;b=0.7m]

Consideringthisdata,itcanbeseenthatthelimitingconditionofaflipoveroccurswhendecelerationiscloseto1.0g.Inthiscase,thecurve(1.0g)is,thereforethemaximumattainabledeceleration.

Supposewewantedtobrakethemotorcyclewithadecelerationequalto0.5g.Thepossiblecombinationsofuseofthefrontandrearbrakesthatcouldprovidethedesireddecelerationareinfinite.Forexample,brakingonlywiththefrontbrake,thedecelerationof0.5gisobtainedbyusingabrakingforcecoefficientinfrontequalto0.68(pointA).Ontheotherhand,withadistributionofthebrakingforcesof80%frontand20%rear,abrakingforcecoefficientof0.55infrontand0.4inbackmustbeused(pointB).AnotherpossibilityisgivenbypointCwhichshowsadistributionofthebrakingforceof60%frontand40%rear,inwhichthereisagreateruseofthereartireandacorrespondinglesseruseofthefrontone.

Let’snowsupposethatthebrakingtractioncoefficientsofthefrontandreartiresarethesame.Figure3-26showsthatbyusingthesamebrakingcoefficientforthetwotires,weobtainthemaximumpossibledeceleration.Forexample,ifthebrakingforcecoefficientisequalto0.8forboththefrontandrearwheels,themaximumdeceleration(equalto0.8g)isobtainedwitha90/10brakingdistribution.Themaximumuseofthetwotiresisattainedwiththisdistribution.Thefigurefurthershowsthatusingonlythefrontbrakegivesadecelerationthatislowerat0.67gandthatusingonlytherearbrakeyieldsonly0.29g.Iftheroadismoreslipperyandthebrakingtractioncoefficientofboththewheelsis0.4,theoptimalbrakingoccurswithadifferentdistribution(70/30)andgivesadecelerationof0.4g.

Fig.3-26Exampleofbrakingondryandwetsurfaces.

Thisexampleshowsthatoptimalbrakingrequiresadifferentdistributionofbrakingbetweenthetwowheelswhenvaryingthedesireddeceleration.Infact,the45°linecorrespondingtoμf=μr,whichrepresentstheconditionforoptimalbraking,intersectsdifferentcurvesofbrakingdistributionwhenvaryingthedesireddeceleration.Thismeansthatthedevicesforautomaticdistributionofbrakingthatarepresentonsomemotorcycles,shouldadaptthedistributiontotheconditionsoftheroad.

Furthermore,itisworthpointingoutthatintheexampleconsidereditisnotagoodideatousearearbrakingforcegreaterthanthefrontone.Figure3-27showsthattheoptimaldistributionofbraking(dottedline)istangentatthepointoforigintothecurveof50%to50%distribution;butitdoesnotintersectthecurvesofdistributioncharacterizedbygreaterforcetotherear.

Thisisalsovalidformotorcycleswithadifferentdistributionofthestaticloadonthetwowheels,forexample45%onthefrontand55%ontherearwheel.Theoptimallineofbrakingisalwaystangentattheorigintothebrakingdistributionlinehavingthesamedistributionbetweenthestaticloadsonthetwowheels.Forexample,withaloaddistributed45%tothefrontand55%totherear,theoptimalbrakinglineistangenttothedistributioncurveofthebrakingforce45%tothefrontand55%totherear.

Fig.3-27Optimalbraking.

[p=1.4m;h=0.7m;b=0.7m]

Parilla250ccof1946

4Steady Turning

Duringsteadyturningmotionthemotorcyclecanhaveneutral,underorover-steeringbehavior.Tomaintainequilibriumtheriderappliesatorquetothehandlebarsthatcanbezero,positive,inthesamedirectionofthehandlebarrotation,ornegative,i.e.,appliedinthedirectionoppositetotherotationofthehandlebar.Thesecharacteristicsareimportantandconcurtodefinethesensationofthemotorcycle’shandling.

4.1The motorcycle rol l ins teadyturning

4.1.1Ideal rol l angle

Themotorcycle,insteadyturning,issubjecttobotharestoringmoment,generatedbythecentrifugalforcethattendstoreturnthemotorcycletoaverticalposition,andtoatiltingmoment,generatedbytheweightforce,thattendstoincreasethemotorcycle’sinclination,orrollangle(Fig.4-1).

Weintroducethefollowingsimplifyinghypotheses:themotorcyclerunsalongaturnofconstantradiusatconstantvelocity(steadystateconditions);thegyroscopiceffectisnegligible.

Consideringthecrosssectionthicknessofthetirestobezero,theequilibriumofthemomentsallowsustoderivetherollangleintermsoftheforwardvelocityVandtheradiusoftheturnRc(theradiusoftheturninthiscaseismeasuredfromthecenterofgravitytotheturningaxis):

whereΩindicatestheangularyawrate,whileV=ΩRcindicatesthevehicle’sforwardvelocity.

Fig.4-1Steadyturning:rollangleofthemotorcycleequippedwithzerothicknesstires.

Inconditionsofequilibriumtheresultantofthecentrifugalforceandtheweightforcepassesthroughthelinejoiningthecontactpointsofthetiresontheroadplane.Thislineliesinthemotorcycleplaneifthewheelshavezerothicknessandthesteeringangleisverysmall.

Inreality,ifanon-zerosteeringangleisassigned,thefrontcontactpointisdisplacedlaterallywithrespecttothex-axisoftherearframeandthelinejoiningthecontactpointsofthetiresisnotcontainedintheplaneoftherearframe.

4.1.2Effective rol l angle

Nowconsideramotorcyclewithtiresofthickness2twhichdescribesthesameturnradiusRcatthesameyawvelocityΩ.Sincethethicknessofthetiresisnotzero,therollangleϕthatisnecessaryfortheequilibriumofthemomentsexertedbytheweightforceandthecentrifugalforce,isgreaterthantheidealoneϕi(Fig.4-2):

ϕ=ϕi+Δϕ

TheincreaseΔϕoftherollangleisgivenbytheequation:

Theeffectiverollangleis:

TheprecedingequationshowsthatΔϕincreasesbothastherollangleandthecrosssectionradiusincreaseandastheheightofthecenterofgravityhdecreases.Therefore,theuseofwidetiresforcestheridertousegreaterrollangleswithrespecttotheanglenecessarywithamotorcycleequippedwithtiresthathavesmallercrosssections.Furthermore,withequalcrosssectionsofthetires,todescribethesameturnwiththesameforwardvelocity,amotorcyclewithalowcenterofgravityneedstobetiltedmorethanamotorcyclewithahighercenterofgravity.

Fig.4-2Steadyturning:rollangleofthemotorcycleequippedwithrealtires.

Themotorcyclerollangleonaturnisinfluenced,inasignificantway,bytherider ’sdrivingstyle.Byleaningwithrespecttothevehicle,theriderchangesthepositionofthehiscenterofgravitywithrespecttothemotorcycle.Figure4-3illustratesthepossiblesituations.

Iftheriderremainsimmobilewithrespecttothechassis(Fig.4-3a),thecenterofgravityofthemotorcycle-ridersystemremainsinthemotorcycleplane.Undoubtedly,thisisanelegantwayof

handlingtheturns,butnotthebest.Infact,inthiscase(andonlyinthiscase),theactualrollanglecorrespondsexactlytothetheoreticalrollangleϕthatwaspreviouslycalculated.

Iftheriderleanstowardstheexterioroftheturn(Fig.4-3b),thecenterofgravityisalsomovedtotheexterioroftheturnwithrespecttothemotorcycle.Asaresult,heneedstoinclinethemotorcyclefurthersothatthetires,beingmoreinclinedthannecessary,operateunderlessfavorableconditions.Certainlythisriderisnotanexpert.

Iftheriderleanshistorsotowardstheinterioroftheturnandatthesametimerotateshislegsoastonearlytouchthegroundwithhisknee,hemanagestoreducetherollangleofthemotorcycleplane(Fig.4-3c).

Whenracing,theridersmovetheirentirebodiestotheinterioroftheturn,bothtoreducetherollangleofthemotorcycleandtobettercontrolthevehicleontheturn.Thedisplacementofthemotorcycle-ridersystem’scenterofgravitytowardstheinterioriscarriedoutbothbymovingthelegandbythemovementofthebodyinthesaddle(Fig.4-3d).Thedisplacementofthebodytowardstheinteriorandinparticular,therotationofthelegcauseanaerodynamicyawingmomentthatfacilitatesenteringandroundingtheturn.

Fig.4-3Influenceofdrivingstyleontherollangle.

4.1.3Wheel ve loci tyinaturn

Thevelocityofthevehicleisrepresentedbytheforwardvelocityofthecontactpointoftherearwheel.Therefore,theyawvelocityΩis:

Ifwesupposenolongitudinalslippagebetweenthetiresandtheroadsurface(intheforwarddirectionofthewheels),thespinvelocityofthewheels,intermsofthevehicleforwardvelocity,rollangleandkinematicsteeringangle,isthen:

Inreality,itmustbeobservedthatduringthethrustandbrakingphasesthereisalwaysalongitudinalslippagebetweentherearwheelandtheroadplane.Inthefrontwheelthereislongitudinalslippageinthebrakingphase,whileundersteadystateconditionstheslippageisnegligiblebecauseitisonlyduetorollingresistance.

Itisimportanttonotethat,withthesameforwardvelocity,theangularvelocityofthewheelsincreasesduringturningwithrespecttotheangularvelocityofthewheelsinstraightrunning,becausecontactdoesnotoccuronthelargestcircumferenceofthewheels.

4.2Directional behaviorofthe motorcycle inaturnLetusnowconsideramotorcycleinasteadyturningcondition.Ifeachwheeladvancesideallywith

apurerollingmotion,thevelocityvectorofthewheel’scenterwouldbecontainedintheplaneofthewheel.

Thelateralslipisexpressedbythesideslipangleλ,definedastheangleformedbythedirectionofforwardmotionandtheplaneofthewheel.Whensideslipanglesapproachzero,steeringiscalledkinematicsteering.

Thelateralreactionforcesdependonthesideslipanglesofthetires,rollangleandverticalloads.Theforcescanbeexpressedbythefollowinglinearexpressions,whenslipandrollanglesaresmall:

Theconstantk(expressedinradians-1)representthestiffnesscoefficientsofthetires:

kϕ=camberstiffnesscoefficient;kλ=corneringstiffnesscoefficient.

Thelargerthesideslipandcamberstiffnessesare,thesmallerthesideslipanglenecessarytogeneratethelateralforceonthetireis.

4.2.1Effective s teeringangle andpathradius

Theeffectivesteeringangleofamotorcycle(Fig.4-4)alsodependsonsideslipangles;itsvalueisgivenbytheequation:

whereΔindicatesthekinematicsteeringanglethatdependsonthesteeringangleδ,casterangleεandrollangleϕ.

Theturningradiusofthetrajectorydescribedbytherearwheelisalsoafunctionofthesideslipanglesandofthekinematicsteeringangle:

Ifthesideslipanglesandthekinematicsteeringanglearesmall,theradiuscanbecalculatedwiththefollowingapproximateformula:

wherepindicatesthemotorcyclewheelbase(Fig.4-4).

Fig.4-4Pathradius,steeringangleandsideslipangles.

4.2.2Steeringratio

Themotorcyclesteeringbehaviordependsonvariousgeometricparameters(wheelbase,offset,casterangle,wheelradiiandcrosssectionradii),onthemassdistributionandtireproperties.Tireproperties,inparticular,areveryimportantbecausetheeffectivesteeringangledependsonthedifferencebetweenthesideslipangles.

TheeffectivesteeringangleΔ*isonlyequaltothekinematicsteeringangleΔchosenbytherideriftheslipanglesofbothwheelsareequal.Inthiscasethesteeringsystemhas“neutral”behavior.Otherwise,theeffectivesteeringangleissmallerorlargerthanwhatisexpectedbytheriderandthevehiclehasunderorover-steeringbehavior.Itisworthpointingoutthatwheel’ssteeringangleΔmaybesmallerthanthesideslipangleswhenthesteadyturningradiusislargeandthespeedhigh.

Thesteeringbehaviorcanbeexpressedbymeansofthesteeringratioξ:

where isthekinematicradiusofcurvature.

Thevehicle’sbehavioris:neutralifξ=1:thesideslipanglesareequal(λf=λr);over-steeringifξ>1:thedifferenceofthesideslipanglesispositive(λr>λf);under-steeringifξ<1:thedifferenceisnegative(λr<λf).

Neutralbehavior

Figure4-5showsavehicleinaturn,inthespecialcasewherethesideslipanglesofthetwowheelsareequal(λf=λr).Ifthesideslipangleswerezero,theturncenter(pointCo)wouldbedeterminedbytheintersectionofthelinesperpendiculartotheplanesofthewheelsandpassingthroughthecontactpoints.Sincethesideslipanglesofthetwotiresareequal,theeffectivesteeringangleΔremainsconstantasthevaluesofthesideslipanglesvary,whilethecenterofrotationCmovestowardsthefrontandalongapathpassingthroughthepointCo.Theradiusofcurvatureremainsapproximatelyconstantandequaltothatrelatingtokinematicsteering.Thisbehaviorisdefinedasneutral,sincethecurvatureradiusdependsonlyonthesteeringangleselectedbytheriderandnotonthevalueofthesideslipangles.

Under-steering

Ifthesideslipangleofthereartireislessthanthefronttire(λr<λf)thecenterofcurvatureCisdisplacedtotheexteriorofthepathoftheneutralcenter(Fig.4-6a).Heretheradiusofcurvature isgreaterthantheidealone associatedwiththekinematicsteering.Thevehicle’sbehavioristhereforedefinedasunder-steering.

Fig.4-5Neutralbehaviorofthemotorcycleinaturn(λr=λf).

Fig.4-6Under-steeringandover-steeringbehaviorofthemotorcycleinaturn.

Over-steering

Ifthesideslipangleofthereartireisgreaterthanthatofthefronttire(λr>λf),thecenterofcurvatureCisinsidethepathoftheneutralcenter,sothatthecurvatureradius islessthantheidealradius associatedwiththekinematicsteering(Fig.4-6b).Thevehicleinthiscasehasanover-steeringbehavior.

Nowconsideramotorcyclethatisunder-steeringwhileitisroundingaturn.Sincethevehicletendstoexpandtheturn,inordertocorrectthetrajectorytheriderisobligedtoincreasethesteeringanglesothatthelateralreactionforceofthefrontwheelwillbeincreased.

Whentherotationofthehandlebarsbecomesconsiderable,theforceneededforequilibriumcanovercomethemaximumfrictionforcebetweenthefronttireandtheroadplane,withtheresultthatthewheelslipsandtheriderfalls.

Αmotorcyclethatisunder-steeringisthereforedangerous,sincetheridercannotcontrolthevehicleoncethefrontwheelhaslostadherence.

Ontheotherhand,withanover-steeringmotorcycle,iftheforceneededforequilibriumovercomesthemaximumfrictionforcebetweenthetireandtheroadplane,therearwheelslips,buttheexpertrider,withacountersteeringmaneuver,hasabetterchanceofcontrollingthevehicleequilibriumandavoidingafall.

4.3CorneringforcesFigure4-7showsthemotorcycleinkinematicturningwithoutdrivingforce,rotatingaboutthe

idealturncenterCo.Rollingresistanceandaerodynamicforcesareneglected.Thetirelateralforcesareperpendiculartothewheelsandtheirresultantisequaltothedesiredcentripetalforcedirectedtowardstheturncenterpoint.

Figure4-8showsamotorcyclewithsideslipanglesnotequaltozeroandwiththedrivingforcerequiredtogivesteadytangentialvelocity.Frontrollingresistanceisconsideredwhereasaerodynamicforcesareneglected.TheturncenterCistheintersectionofthelinesperpendiculartothedirectionsoftheforwardvelocityofthetwowheels.ForequilibriumtheintersectionofthetotalfrontandrearforcesmustintersectthelineGC.TheresultantofthetwotireforcesgivesthecentripetalcomponentdirectedtowardstheturncenterC.

Iftheaerodynamicforceisincludedthedrivingforcenecessaryfortheequilibriumwillincrease.

Theaerodynamicforcealsoinfluencestheverticalloadsonthewheels:

Thefrontverticalloadisequaltothestaticloadlesstheloadtransferduetotheaerodynamicforce.Alternatively,therearloadisequaltothesumofthestaticloadandloadtransfer.

Fig.4-7Planview:Forcesactingonthemotorcyclewithzerosideslipangles.

Fig.4-8Planview:Forcesactingonthemotorcyclewithnon-zerosideslipangles.

Theverticalloadonthefrontwheelincreasesslightlyinthepassagefromrectilinearmotiontotheturn,whileitdecreasesontherearwheel,duetothedependenceoftheloadtransferontherollangle.

Itisworthhighlightingthat,evenifrollingresistanceisneglected,adrivingforceisnecessaryforequilibriuminsteadyturningwhensideslipanglesarepresent.

4.4Linearizedmodel ofthe motorcycle inaturnNowconsideramotorcycleroundingaturnwithlargeradius,withrespecttothemotorcycle’s

wheelbase,andsupposethattheaerodynamicforceisnegligiblesothattheloadtransferisnegligiblewithrespecttotheverticalloadsonthewheels.

Ifweconsidersmallroll,steeringandsideslipanglesthelateralforcesactingonthewheelsareequalto(seeFig.4-7):

Thelateralforcesdependonthedistributionofthestaticloadsonthetwowheels,whilethefrontlateralforcealsodependsontheeffectivesteeringangle.Theratiosbetweenthelateralforcesactingonthewheels,andtheverticalloadsareequalto:

Keepinginmindtheexpressionsforthelateralforcesintermsofthesideslipandrollangles,thesideslipanglescanbewritteninthefollowingway:

Itcanbenotedthatthesideslipanglesdependontwotermsofoppositesign.Thenegativetermisproportionaltothecamberstiffnessanditsincreasebringsaboutareductioninthesideslipanglethatcanalsobecomenegative,aswehaveseeninthechapterontires.Thefirst,remainingtermdependsonthelateralforce(numerator)andtheverticalload(denominator).

Thesideslipanglescanthereforebeexpressedintheform:

Itcanbenotedthatthesideslipanglesareinverselyproportionaltothecorneringstiffnessandthattheydependonboththecamberstiffnessandtherollangle(andthereforeontheforwardvelocityandtheradiusofcurvature).Observethatifthecamberstiffnesscoefficientisgreaterthanonethesideslipangleisnegative.

Nowletusconsiderthesteeringratioξthatcharacterizesthedirectionalbehaviorofthevehicle:

Takingintoaccountthesideslipexpressionsweobtain:

Fig.4-9Steeringratioξasthecamberstiffnessvaries .

Thedirectionalbehaviorisneutralifthefollowingrelationissatisfied:

Inthiscase,thevehicle’sresponseatanyvelocitycoincideswiththatpresentunderidealconditionsofkinematicsteering.

Thedirectionalbehaviorisover-steeringiftheratioξisgreaterthanone:

Inthiscase,thecurvatureradiusdiminisheswithanincreaseinvelocity.

Thedirectionalbehaviorisunder-steeringiftheratioξislessthanone:

Inthiscase,asthevelocitygraduallyincreases,theradiusofcurvaturealsoincreasesandthereby,increasinglygreatersteeringanglesarerequiredtogothroughthesametrajectory.

Figure4-9showsthesteeringratioundervariationinthecamberstiffnesses,inthecaseinwhich

thecorneringstiffnessesareequal.Thegraphshowsthatforneutralbehaviorthetwotiresmusthaveequalcamberstiffnesscoefficientvalues.

Example1

Consideramotorcyclewithawheelbasep=1.4m,equippedwithdifferenttires.Thetireshavethefollowingthreecombinationsofthecorneringstiffnesscoefficientwhilethecamberstiffnesscoefficientisassumedtobeconstant

Motorcycle1:

Motorcycle2:

Motorcycle3:

Figure4-10showsthesteeringratioversustheforwardvelocity.Itcanbeobservedthatthebehaviorofthevehicleis:

over-steeringwhenthefrontcorneringstiffnesscoefficientisgreaterthanthatofthereartire();

neutralwhenthestiffnessesareequal( );under-steeringwhentherearcorneringstiffnesscoefficientisgreaterthanthatofthefronttire(

).

Thisisduetothefactthatthesideslipangleofatireisgreatertotheextentthatitsstiffnessisless;withhighvaluesofthecorneringstiffness,thesideslipanglescouldbezero(kinematicsteering).

Itisinterestingtoobservehow,onthebasisofthislinearmodel(expressedintermsofstiffnesscoefficients),neitherthesteeringanglenorthelongitudinalpositionofthecenterofgravityinfluencethevehicle’sdirectionalbehavior.

Actually,asthesideslipangleincreases,thelateralforcesincreaseatanincreasinglylowerratethanthatpredictedbythelinearlawformingthebasisofthetiremodel.

Furthermore,inrealitythedirectionalbehaviorofthemotorcycleisalsoinfluencedbythelongitudinalpositionofthecenterofgravityandbythevalueofthedrivingforce.

Fig.4-10Steeringratioξversusthevelocity[ ].

4.4.1Critical ve loci ty

Thepreviousexamplehasshownthattheunder/over-steerbehaviorinaturndependsmainlyonthecamberandcorneringstiffnesses.Theirinfluenceisimportantespeciallyifthestiffnessvaluesofthefrontandreartiresaredifferent.

Consideragainamotorcyclethatroundsaturnwithalargeradiuswithrespecttoitswheelbase.

Whenthebehaviorofthemotorcycleisover-steering,thesteeringratioξapproaches∞atacertainvalueofthevelocity,calledcriticalvelocity:

Thecontrolofthemotorcycleoverthecriticalvelocityispossiblebyperformingcounter-steeringmaneuvers.Thisstrategyisadoptedbyridersinspeedwayandmotardracing.

Example2

Consideramotorcyclewithawheelbasep=1.4m,characterizedbythefollowingtireproperties:

Themotorcycleisover-steeringbecausethesteeringratioisgreaterthanone.Thecriticalvelocityisequalto50.6m/s.

Figure4-11givestheprogressoftheratioξasafunctionofthevelocity.Itcanbeobservedhere

thatthevalueofξincreasesrapidly.Thismeansthatathighvelocitiesevensmallvaluesofthesteeringanglesufficetoturnthevehicle.

Fig.4-11Over-steeringmotorcycleandcriticalvelocity.

Figure4-12illustratesthevariationofcriticalvelocityintermsofthecorneringstiffnesscoefficientofthetires.Itcanbenotedintheleftplotthatif ,inthefieldofvalues( )criticalvelocitydoesnotexist;theequationshowsthatcriticalvelocityisimaginary.

Therightplotreferstoamotorcyclewithcamberstiffnesscoefficientinthefronttiregreaterthanthatofthereartire.Inthiscasethevehicleisalwaysinover-steering.Criticalvelocityincreasesbyincreasingthereartirecorneringstiffnessanddecreasingthefronttirecorneringstiffness.

Fig.4-12Criticalvelocityasafunctionofthestiffnessesofthetires.

4.5Multi -bodymodel ofmotorcycles ins teadyturningAmotorcyclecanbedescribedasasystemofsixrigidbodies:sprungsteeringcomponents,

unsprungsteeringcomponents,rearframe(includingframe,engine,tankanddriver),rearswingingarmandthetwowheels.

Thedriverisconsideredtobearigidbodyfirmlyattachedtotheframe.ThefollowingFig.4.13showsasketchofamotorcycleinsteadyturningmotion.

ThespeedoftravelVisthespeedofthecontactpointoftherearwheel.Whennoslipispresent,Visdirectlyproportionaltotheangularvelocityoftherearwheelandisdirectedalongthewheelsymmetryplane.

Thedistributedaerodynamicforceswhichairexertsonthemotorcyclearetakenintoaccountbyconsideringdrag,liftandlateralforcesactingatthecenterofmassoftherearframe(FD,FL,FS)andthreeaerodynamictorques( ).

Theinteractionbetweeneachtireandtheroadisrepresentedbythreeforces(verticalload,longitudinalandlateralforces)actingatthegeometriccontactpointandbythreetorques(overturning,rollingresistanceandyawtorque),actingalongthethreeindependentaxes.Thetireforcesandtorquesarenon-linearlydependentontherollangleandslipquantities.

4.5.1Mathematical model ofmotorcycle

Theequationsofmotionofthemotorcycleinsteadyturningmotionaredescribedinthepaper[Cossalteretal.,1999].

Theequilibriumconditionsgivesixequations:threeforceequilibriumequations;threemomentequilibriumequations:equilibriumaroundtheX-axis(roll),the

Y-axis(pitch)andtheZ-axis(yaw).

Inadditionwehavetwoequationsthatgivethelateralforcesasfunctionsofsideslipandcamberangles.

Oncetherollangleϕandthesteeringangleδareassigned,theeightequationsallowustoobtaintheeightunknowns:

forwardvelocityV;verticalforcesNfandNrappliedrespectivelytothefrontandrearwheels;lateralforces and appliedrespectivelytothefrontandrearwheels;sideslipanglesλf,λr,thedrivingforceS.

Finally,theequilibriumofthefrontand/ortherearframearoundthesteeringaxisgivesthetorqueexertedbytheriderandappliedtothehandlebars,whichprovidesandequalandoppositereactionontherearframe.

Fig.4-13Forcesandmomentsactingonthemotorcycle.

TheinertialandgeometricalpropertiesaredefinedwithrespecttothecoordinatesystemsrepresentedinFig.4-14.

Letusexaminetherearframe:ithasmassMr;itischaracterizedbythecenterofgravityGrhavingcoordinates(br,0,−hr)withrespecttotherearcoordinatesystem(Ar,Xr,Yr,Zr);itisconsideredsymmetricalwithrespecttothex−zplane,henceitsinertialcharacteristicsarerepresentedbythefollowingfourterms:− =masscentermomentofinertiaaboutxraxis(rollmomentofinertia);− =masscentermomentofinertiaaboutyraxis(pitchmomentofinertia);

− =masscentermomentofinertiaaboutzraxis(yawmomentofinertia);− =masscenterinertiaproductaboutxr-zr-axes.

Letusexaminethefrontframe:ithasmassMf;itischaracterizedbythecenterofgravityGfhavingcoordinates(bf,0,−hf)withrespecttothefrontcoordinatesystem(Af,Xf,Yf,Zf);

Thecoordinatesystemaxes(Af,Xf,Yf,Zf)areassumedtobeprincipalaxesofinertiasothattheinertiatensorisdiagonal.

Fig.4-14Sketchofthemotorcycle.

4.5.2Simpli fiedmodel ofmotorcycles

Ignoringthesmalldisplacementsofthewheels’contactpoints(withrespecttotheradiusofcurvature)which,aswehaveseeninthechapteronkinematics,dependontheanglesofpitch,rollandsteeringaswellasthegeometryofthewheels,thesixequationsofequilibriumforthemotorcycleinaturncanbeeasilyderived(Fig-4-15):

(⇒)equilibriumoftheforcesalongtheXaxis:

(⇒)equilibriumoftheforcesalongtheYaxis:

(⇑)equilibriumoftheforcesalongtheZaxis:

−Nf−Nr+mg=0

equilibriumofthemoments:

(∩)aroundtheXaxis:

(∩)aroundtheYaxis:

−IXZΩ2−Nf(p+Xr)+mg⋅XG+FA⋅ZG−NrXr=0

(∩)aroundtheZaxis:

Themeaningofthesymbolsasthefollows:

•S thethrustwhichisnecessaryforholdingthemotorcyclestationaryinaturn;

•FA theaerodynamicresistantforceassumedtobeappliedtothecenterofgravity;

• thelateralforcesappliedtothetiresbytheroad;

•Nf,Nr theverticalloads;

• spinmomentsofinertiaofthewheels;

•ωf,ωr angularvelocitiesofthewheels;

•Ω yawvelocity;

•Δ kinematicsteeringanglemeasuredontheroadplane.

•XG,YG,

coordinatesofthemotorcyclecenterofgravitywithrespecttothereferencesystem(C,X,Y,Z):

ZG

ZG=−hcosϕ

Xr,Yr

coordinatesofthecontactpointoftherearwheelwithrespecttothereferencesystem(C,X,Y,Z);

Xr=−Rcrsinλr

Yr=−Rcrcosλr

IXZ,IYZproductsofinertiaofthemotorcyclewithrespecttotheaxesX−ZandY−Z.Theseproductsofinertiadependonthemasscentermomentsofthemotorcycle, ,massm,rollangleϕ,andonthecoordinatesXG,YG,ZGofthemotorcyclemasscenter:

Thesixequationsconstituteanon-linearsystem.ExpressingtherollangleϕasafunctionoftheyawvelocityΩandoftheradius ,andexpressingthelateralforcesofthetiresaslinearfunctionsofthesideslipanglesλf,λrandrollangleϕ,wecancalculatethesixunknowns.

Fig.4-15Motorcycleinsteadyturning.

ForexamplesettingthesteeringangleδandtheyawvelocityΩthesixunknownsare:thesideslipanglesλf,λr;theradius :theverticalloadsNf,Nr;thethrustSnecessaryforassuringmotionataconstantvelocity.

Ifthesideslipanglesλf,λr,therollangleϕandtheeffectivesteeringangleΔareknown,itispossibletocalculatetheradiusofthecirculartrajectorycoveredbytherearwheel .

4.6Rol l , s teeringands ides l ipanglesWewillshowhowamotorcycleinaturn,withassignedforwardvelocity,rollandsteeringangles,

describesatrajectorywhosepathradiusdependsonthesideslipanglesofthetires.Ifthesideslipanglesarezero,thetrajectorycoincideswiththekinematicone.

Theconditionsofstationaryequilibriumofamotorcycleinaturn,inthecurvature-forward

velocitydiagram,arerepresentedbymeansofthecontourlinesofthesteeringangleδandtherollangleϕ.Thesecurvesprovidethenecessaryvaluesforequilibriumonaturn,oncetheradiusofthetrajectoryandthevelocityofthemotorcyclehavebeenset(Fig4-16).

Therollanglecontourlinesshowhowthevelocityandtheturningradiushavetovarytoassurevehicleequilibrium,maintainingtherollangleconstant.Thesteeringangleneededundervariousstationaryequilibriumconditionsisrepresentedbytheintersectionoftherollcurvewiththesteeringcurve.

Inthesamewaywecanconsideramotorcyclethatroundsaturnholdingthesteeringangleconstant.Thesteeringcontourlinesshowhowtheforwardspeedandtheradiusoftheturnneedtovaryinordertoassurethevehicle’sequilibrium.Therollanglenecessaryundervariousequilibriumconditionsisrepresentedbytheintersectionofthesteeringcontourlinewiththerollline.

Threemotorcycleshavingthesamegeometryandinertialpropertiesbutequippedwithdifferenttiresareconsidered.Thedifferentbehaviorofthethreemotorcyclesdependsonthevarioussideslipanglesofthetiresrequiredtogeneratethelateralforcesthatarenecessaryforequilibrium.Itshouldbenotedthattheradiusoftheturndependsonthedifferenceinthesideslipanglesaswellasonthesteeringangle.

4.6.1Case 1:re ference motorcycle

Rollandsteeringangles

Figure4-16representsthecaseofamotorcyclewithtireshavingequalcamberandcorneringstiffness.

Thehorizontalstraightlinesrepresenttheequilibriumconditionsofamotorcycleroundingturnsofincreasingradius,atconstantvelocity.Thegraphshowshowtherollandsteeringanglesvaryintermsofthecurvature.

Theverticallinesrepresentmotorcyclesroundingturnsofconstantradiuswithvariablevelocity.Thegraphshowshowtherollandsteeringanglesneedtovaryintermsofvelocity.Toroundaturnwithconstantradius,thesteeringanglemustdiminishasvelocityincreases.Thisphenomenonderivesfromthefactthattheeffectivesteeringanglealsodependsontherollangle(seechapter1).

Itmustberecalledthattherangeofsteeringanglesusedisnormallymuchmorerestrictedthanthatindicatedinthefigure,especiallyathighvelocities.

Wecanseeinthefigurethatevenifϕ=0asteadystateturningmotion,atlowvelocity,ispossible.Ifweconsideraperfectlyverticalmotorcycle(rollangleϕ=0)withthehandlebarsturnedtotheright,weareledtosupposethatitcannotattainanequilibriumvelocitysinceweimaginethatthecentrifugalforcegeneratedastheturnisroundedtotheright,tendstomakethemotorcyclefalloutsidetheturn,i.e.,totheleft.

Fig.4-16Rollandsteeringanglesasfunctionsofvelocityandcurvature.

Fig.4-17Sideslipanglesintermsofvelocityandcurvature.Referencemotorcycle.

Actually,itmustberecalledthatthepresenceofthemotorcycletrailcausesadisplacementtotheleftofthefrontwheel’scontactpointandthat,furthermore,therollangleofthefrontsectionisnotzero,butincreaseswiththesteeringangleeveniftherolloftherearsectioniszero.Itfollowsintuitivelythatboththecenterofgravityofthefrontsectionandthatoftherearsectionaredisplacedtotherightofthestraightlinejoiningthecontactpoints.Non-zeroequilibriumvelocitiesarethereforepossible,sincetheoverturningmomentduetothecentrifugalforceisbalancedoutbythemomentsgeneratedbytheweightforces.

Sideslip angles.

ThesideslipanglesrequiredforamotorcyclewithtiresofequalstiffnessarerepresentedinFig.4-17.Notethatthevehiclehasnearlyneutralbehavior,withnearlyequalfrontandrearsideslipangles(λr−λf=0.25°foravelocityof15m/sandturnradiusof30m).

4.6.2Case 2:frontt ire s ti ffnes s (+10%),reart ire s ti ffnes s (-10%)

Rollandsteeringangles

Ifthevaluesofboththecamberandcorneringstiffnessesofthetwotiresarechanged,thevehiclebehavesdifferentlyintheturn.Ifthefronttirehaslarger(+10%)stiffnessvaluesandthereartiresmaller(-10%)thantheprecedingcase,thebehaviorwillbedifferent,ascanbeobservedinFig.4-18.

Fig.4-18Rollandsteeringanglesasfunctionsofvelocityandcurvature.

Atthesamevelocityandcurvatureradius,thesteeringanglenecessaryforequilibriumonaturnisnotablysmaller(forexample,atavelocityofabout15m/sandwithacurvatureratioof30m,thesteeringanglediminishesfromapproximately2°toapproximately1°).

Furthermore,asinthepreviouscase,toroundaturnwithconstantradius,thesteeringanglemustdiminishasthevelocityincreases.

Sideslip angles.

Withafronttirehavinghighercamberandcorneringstiffnesses,i.e.,withafronttireperforming

morethantherearone,thesideslipangleschangesignificantly.Thefrontsideslipangleisverysmallorevennegative,whiletherearsideslipangleincreases(Fig.4-19).Thedifferencebetweenthesideslipangleshaspositivesignandbecomes:λr−λf=1.9°.

Fig.4-19Sideslipanglesasfunctionsofvelocityandcurvature.Over-steeringmotorcycle.

Fig4-20Motorcyclewithnegativefrontsideslipangle.

Fig.4-20showstheplanviewofamotorcyclewithanegativevalueofthefrontsideslipangle.Inthiscasethefrontlateralforceisthesumofapositivecomponentduetothecamberangleandanegativecomponentduetothesideslipangle.Suchaconditionwouldexistbelowtheλf=0curveforthefronttireinFig.4-19.

4.6.3Case 3:frontt ire s ti ffnes s (-10%),reart ire s ti ffnes s (+10%)

Rollandsteeringangles

Considerathirdvehiclewithlargerstiffnessvaluesofthereartire(+10%)andsmallerstiffnessvaluesofthefronttire(-10%).Inthiscase(Fig.4-21)thevehiclebehavesverydifferentlyfromtheprevioustwocases:toroundaturnofequalradius,greatersteeringanglesarenecessary.Forexampletoroundaturnofradius30matavelocityof15m/s,requiresasteeringangleabout3°greaterthanthatofthepreviouscases.

Thegreaterdifferencebetweenthiscaseandtheprevioustwoishighlightedbytheplotofthesteeringanglecontourlines.Considerforexamplethevehiclewhileitroundsaturnof50mwithincreasingvelocity.Withanincreaseinvelocityofuptoapproximately15m/s,thesteeringanglenecessaryforequilibriummustincreaseslightly,while,forgreatervelocitiesthesteeringanglehastodiminish.

Sideslip angles.

Inthiscasethereartirehasbettercharacteristicsthanthefrontone,thefrontsideslipangleincreaseswhilethereardiminishessignificantly(Fig.4-22).Thedifferencebetweenthesideslipangleshasanegativesignandbecomes:λr−λf=-1.8°.

Fig.4-21Case3:rollandsteeringangleintermsofvelocityandcurvature.

Fig.4-22Sideslipanglesintermsofvelocityandcurvature.Under-steeringmotorcycle.

4.7SteeringratioCase1.

Letusexamineamotorcycleinsteadyturningequippedwithfrontandreartireswhichhavethesamecorneringstiffnesscoefficient( )andcamberstiffnesscoefficient().

Fig.4-23Steeringratio:frontandreartireswithcorneringstiffnesskλ=15rad-1andcamberstiffnesskϕ=0.8rad-1.

Thiscaseisnotrealisticbecausethefrontandreartiresusuallyhavedifferentproperties,butithelpsusunderstandtheeffectoftirepropertiesonsteeringbehavior.Thecontourplotofξ,representedinFig.4-23,showsthatthevehicle’sbehaviorisalmost“neutral”whenthespeedisverylowandbecomesover-steeringwhenthespeedincreases.

Thesideslipangleofthefrontwheelissmallerthatoftherearwheelforatleasttworeasons.Firstofall,thedrivingforcethatactsonthereartirenecessitatesalargersideslipangletogeneratethelateralforce.Thiseffectbecomesmoreimportantwhenthespeedincreasesbecausetheaerodynamicforce(andthedrivingforce)increases.Secondly,frontwheelcamberangleβislargerthanrearwheelcamberangleϕ,hence,theeffectofcamberangleonthelateralforceismoresignificantonthefrontwheel.

Case2.

Inthiscasethemotorcycleisequippedwiththesamefronttire( and )butwiththereartirehavingdifferentproperties:therearcamberstiffnesscoefficientisincreased(

)whilethecorneringstiffnesscoefficientisthesameasthefronttire.

Fig.4-24Steeringratio:reartirewithincreasedcamberstiffnesscoefficient( ).

Figure4-24showsthat,whenthespeedishigherthan15m/s,under-steeringoccursforawiderangeofvaluesofthesteadyturningradius.Under-steeringbehaviordoesnottakeplacewhenthespeedislowforatleasttworeasons.First,ifthesteadyturningradiusislarge,therollangleissmall(lessthan5°)andtheincreasedcamberstiffnessofthereartireisunabletosignificantlyinfluencethesideslipangle.Then,ifthesteadyturningradiusissmall,bothrollangleϕandsteeringangleδareratherlarge,butthecamberangleβofthefrontwheelislargerthanϕandthiseffectcompensatesfortheincreasedcamberstiffnessofthereartire.

Case3.

Finally,Fig.4-25dealswithareartirewhichhasincreasedbothcorneringstiffnesscoefficientandcamberstiffnesscoefficient( and ).Inthiscaseunder-steeringbehaviortakesplaceevenwhenthespeedislowerthan15m/s.

Fig.4-25Steeringratio:reartirewithincreasedcorneringstiffnesscoefficient( andcamberstiffnesscoefficient ).

4.8The torque appl iedtos teering

Theequilibriumofmomentsaroundthesteeringaxisenablestheevaluationofthetorqueτthattheridermustapplytothehandlebarstoassurethemotorcycle’sequilibriuminaturn(Fig.4-26).Itmustbespecifiedthatthisreferstosteadyturning,i.e.,toamotorcycleataconstantvelocityandturnradius.Intransitorymovement,inaturnwithvariablevelocityandcurvatureradius,thetorquetheridermustexercisewillbesubstantiallydifferentfromthatcalculatedinasteadystate,especiallyifthevariationsinvelocityandtrajectoryoccursuddenly.

Thetorqueappliedbytheriderisequal,butofoppositesign,totheresultantofallthemomentsgeneratedbytheforcesactingonthefrontsection.Theresultanttorqueiscomposedofsixterms:

disaligninginfluence(sign+)duetotheweightforceofthefrontsection,

aligninginfluence(sign-)duetothecentrifugalforceofthefrontsection,

disaligninginfluence(sign+)duetothenormalloadonthefrontwheel,

aligninginfluence(sign-)duetothelateralforceonthefrontwheel,

aligninginfluence(sign-)duetothegyroscopiceffectofthefrontwheel(Fig.4-27),

disaligninginfluence(sign+)duetothetwistingtorqueofthefronttire,

τM=Mzfcosεcosϕ

Fig.4.26Equilibriumofthefrontframe.

PointAindicatestheintersectionofthesteeringaxiswiththenormallinepassingthroughthefronttirecontactpoint.ThedistancebetweenpointAandcontactpointPfrepresentstheeffectivetrailofthetire.

Fig.4.27Genesisofthegyroscopicmomentonthefrontsection.

4.8.1Torque components

InFig.4-28thevariationsofthetorqueappliedbytheridertothehandlebarsofthesamplemotorcycleisshown.Itisusefultorecallthatthetorqueexercisedbytherideris,bydefinitionpositiveifittendstoincreasethesteeringangleintotheturn.

Fig.4-28Torquesappliedtothehandlebarsversustheturnradiusandthevelocity.

Thismeansthatthereareessentiallytwopossiblesituations:atlowvelocitiesthesteeringtorqueisnegative.Therefore,insteering,theridermustblockthehandlebars,whichotherwisetendtorotatefurther.Whenthevaluesofthesteeringtorquebecomestronglynegative,theinclinationandtheentryintotheturnbecomeeasier;withanincreaseinvelocity,thetorquetobeappliedtothehandlebarsbecomespositive.Thiscircumstance,ifthevalueofthetorqueremainshigh,generatesintheridertheunpleasantsensationofdrivingamotorcyclethatishardtoinclineandtoinsertintotightturns.

Figure4-29showsthevariouscomponentscombiningtodefinetheresultingcouplethatactsonthesteeringaxis,intermsoftherollangle.Thevariouscontributionshavethefollowingeffect:

verticalload:theverticalreactiveforcegeneratesapositivemomentofhighvalue;lateralforce:thelateralreactiveforcegeneratesahighvaluenegativemomentofthesameorderofmagnitudegeneratedbytheverticalload;frontweightforce:themomentispositive;centrifugalforce:themomentisnegative,ofthesameorderofmagnitudeasthatgeneratedbytheweightforce;gyroscopicmoment:itgeneratesanaligningeffect;twistingmoment:itgeneratesadisaligningeffectthatincreaseswiththerollangle.

Fig.4-29Momentsexercisedaroundthesteeringaxis.

Fig.4-30Torqueappliedbytheriderandmomentsexercisedaroundthesteeringaxis.

Itshouldbeobservedthatboththetwoforcesappliedtothecenterofgravityofthefrontsection(weightforceandcentrifugalforce)andthetworeactionforcesappliedatthecontactpoint(lateralforceandverticalload)eachcontaininfluencesofoppositesign.

Figure4-30showsthevariationinthetorqueappliedbytheriderastherollanglevaries.Thetorqueisequaltothesum,withsignschanged,ofthefollowinginfluences:

massforces(vectorsumofthemomentsofweightforceandcentrifugalforce);reactionforces(vectorsumofthemomentsofverticalloadandlateralforce);gyroscopicmoment;

twistingtorqueofthetire.

Figure4-30alsoshowsanexampleinwhichforsmallrollanglestheriderneedstoexerciseanegativecouple,whileforlargerollangles,hemustapplyapositivecouple.

Themaximummaneuverabilityisobtainedwhenthecouplenecessaryforassuringequilibriumiszeroornearlyso.Infactundertheseconditions,iftheriderletsgoofthehandlebarsthemotorcyclecontinuestoroundthesetturn.

4.8.2The influence ofmotorcycle geometryonthe s teeringtorque

Thesteadyturningbehaviorofamotorcycleisafunctionofvehiclegeometry,inertiaandtireproperties.

Fig.4-31Steeringtorqueagainstturnradiusandvelocity.

Normaltrail

Fig.4-31showsthereferencecase,whileFig.4-32showstheeffectofapositiveincrementinthenormaltrail:thesteeringtorquecontourplotshiftstowardslowervaluesinthewholeareaunderconsideration,whereasthereisnosignificantchangeintheshapeofthecurves.

Thisresultcanbeexplainedbyconsideringthefactthatwhenthetrailincreases,thedisaligningeffectduetothefronttireverticalloadincreasesmorethanthealigningeffectduetothelateralforce.Theresultisamorestablesteeringbehavior,intheareaofinterest.

Steeringheadangle

Onthecontrary,theincreaseinthecasterangle(Fig.4-33)hasanaligningeffect,sincethesteeringtorqueincreases(asithasnegativevalues,itsmagnitudedecreases).Itisworthnotingthattheeffectofthesteeringheadangleisrelevant.Consideringthattherealsteeringheadangleisinfluencedbythemotorcycle’sattitude,dependingonspeed,massdistributionandsuspensionbehavior,particularattentionshouldbepaidtothisparameter.

Fig.4-32Influenceofanincreaseinthenormaltrail.

Fig.4-33Influenceofanincreaseinthesteeringheadangle.

Fronttirecrosssectionradius

Thepositiveincrementinfronttiresectionradiushasastrongaligninginfluence(Fig.4-34):thiseffectiscausedbythedisplacementofthefrontwheelcontactpointduetotherollangle.Itcanbeseenthatthezerosteeringtorquecurveshiftstowardslowervaluesoftheforwardspeed,andtheresultingbehaviorisquitedifferentfromthereferencecase.

Fig.4-34Influenceofanincreaseofthecrosssectionradius.

Riderposition

Aforwarddisplacementoftherider ’scenterofmasshasaslightself-steeringeffect.Theverticalpositionoftherider ’scenterofmasshasaverysmallaligningeffect.Theresultisthatiftheridermoves,alwaysremainingintheplaneofsymmetryofthemotorcycle,thesteeringbehaviordoesn’tchangesignificantly.Onthecontrary,alateraldisplacementoftheridertowardtheinsideofthecurvehasastrongaligningeffect.Consideringthatsportridersusuallymovesidewaysconsiderablyitisobviousthatthesteeringcharacteristicsofthemotorcyclearestronglyinfluencedbydrivingstyle.Anexpertridercantakeadvantageofthisphenomenonandshiftthezoneoflowsteeringtorquetomatchthecurrentsteeringconditionsandthusgainbetter,easiersteeringcontrol.

Thepresenceofapassengeraltersmassdistributionofthemotorcycle.Theresultingeffectisslightlyaligning,butthesteadyturningsteeringtorqueisnotsubstantiallychanged.

Theinfluenceoftherider ’slateralpositiononthesteeringbehaviorofthemotorcycleis

representedinFig.4-35.A0.05mlateraldisplacementoftherider ’scenterofmassintowardsthecurvewasconsidered,thiscorrespondstoadecreaseintherollangleofabout1°.Thefigureshowstheaccelerationindex(ratiobetweenthetorqueandthelateralacceleration)bothinthepresenceoflateraldisplacementandundernormalconditions(withoutlateraldisplacement).

Inthepresenceoflateraldisplacementtheaccelerationindexispositiveforeveryvalueofforwardspeedifthesteadyturningradiusislargerthan25m.Foreachvalueofthesteadyturningradiusthecurve,whichiscalculatedbytakingintoaccountthelateraldisplacementoftherider,liesabovethecurvecalculatedinnominalconditions.Thedifferencebetweenthetwocurvesbecomesverylargewhenlateralacceleration(V2/R)islow.

Thisbehaviorcanbeexplainedbytakingintoaccounttheeffectduetothedecreaseinrollanglecausedbytherider ’slateraldisplacement.Thefirsteffectisthedecreaseofthedisaligningeffectofthetiretwistingtorque.Thesecondisthevariationinthemomentoftireforcesaboutthesteeringaxis.Inparticularthedisaligningeffectofthetireload,whichtendstorotatethewheeltowardstheinsideofthecurve,decreases.

Fig.4-35Influenceofrider’slateraldisplacementontheaccelerationindex.

Tireproperties

Alldriversknowthattireshaveanimportanteffectonthebehaviorofthemotorcycle.Thesamemotorcycleequippedwithdifferenttiressometimesbehavesasacompletelydifferentmotorcycle.Anincreaseinthecorneringstiffnessorinthecamberstiffnessofthefronttirecausesonlysmallvariationsintheaccelerationindex.

Themoreimportanttireparameteristheyawtorqueofthefronttirewhich,aswehaveseeninthesecondchapter,includestwoterms:

atermwhichtendstoalignthewheelwiththeforwardspeedandisduetothelateralforceandpneumatictrail(whichdependsonsideslipangleλ);

aterm,whichisnamedtwistingtorque,thattendstodisalign.

Figure4-36showstheyawmomentofthefronttireinthereferencecase(ontheleft)andwithatirehavingadecreasedtrail(-20%)andanincreasedtwistingtorque(+11%).

Figure4-37highlightsthatthedecreaseinthetiretrailandtheincreaseinthetwistingtorquemakethesteeringtorquenegativeinthewholerangeofvelocitiesandsteadyturningradii.Neverthelessthetwofamiliesofcurvesshowthesamegeneraltrends,liketheincrementofaccelerationindexwhenforwardspeedincreasesandsteadyturningradiusdecreases.

Fig.4-36Yawtorquecharacteristics.

Fig.4-37Influenceoftheyawtorqueontheaccelerationindex.

Wehaveseenthatthetorquetobeappliedtothehandlebarscanhavezerovaluetoassureequilibrium,positive(atorqueinaccordancewiththesteeringangle)ornegative(atorquenotinaccordancewiththesteeringangle).

Anymodificationstothemotorcyclewillbringvariationsinthetorquetobeappliedtothehandlebars;theinfluenceofthemaingeometricandinertialparametersonthesteeringtorqueisbroughttolightinFig.4-38.

Itcanbeobservedthatthesteeringheadangle,thefronttirecrosssectionradius,theheightofthecenterofgravityandthenormaltrailaretheparametersthatmostinfluencethevalueoftorque.

Fig.4-38Influenceofsomeparametersonthesteeringtorque.

Aligningeffect:thesteeringangletendstodecrease.Theridermuststeerintotheturn(+)tocounteractthiseffect.Ifthetorquewasnegativeitmustbecomeslessnegative.

Disaligningeffect:thesteeringangletendstoincrease.Theridermuststeeroutoftheturn(-)tocounteract.Ifthetorquewasnegativeitbecomesmorenegative.

GileraSaturno“Sanremo”500ccversion1947

5In-PlaneDynamics

Amotorcyclewithoutsuspensionmovingoverunevengroundpresentsdifficultiesinsteeringbecauseofthelossofwheelgripontheroad,andbecauseofriderdiscomfort.Smallbumpsonthegroundareeasilyabsorbedbythetires,butforadequateabsorptionoflargerbumps,themotorcycleneedsappropriatesuspension.

Amotorcyclewithsuspension,fromadynamicspointofview,canbeconsideredasarigidbodyconnectedtothewheelswithelasticsystems(frontandrearsuspension).Therigidbodyconstitutesthesprungmass(chassis,engine,steeringhead,rider),whilethemassesattachedtothewheelsarecalledunsprungmasses.

Suspensionhastosatisfythefollowingthreepurposes:allowthewheelstofollowtheprofileoftheroadwithouttransmittingexcessivevibrationtotherider.Thispurposeconcernsridercomfort,thatistheisolationofthesprungmassfromthevibrationgeneratedbytheinteractionofthewheelswithroadirregularities;ensurewheelgripontheroadplaneinordertotransmittherequireddriving,brakingandlateralforces;ensurethedesiredtrimofthevehicleundervariousoperatingconditions(acceleration,braking,enteringandexitingturns).

Thedegreeofrequiredcomfortvariesaccordingtotheuseofthevehicle.Forexample,withracingvehicles,comfortislessimportantthanthemotorcycle’scapacitytokeepthewheelsincontactwiththegroundandtoassumethedesiredtrim.

However,inothervehiclesthesuspensionisexpectedtoserveotherpurposes.Forexample,inoff-

roadvehiclesthesuspensionservestoisolatethesprungmassfromcontinuousimpactgeneratedbyvehiclejumps.Forthisreason,suspensioninoff-roadvehicleshasgreaterwheeltravelthanintouringvehicles,andmoresothaninracingvehicles.

Asforthetrim,itshouldbehighlightedthatitdependsonthestiffnessofthesuspensionandontheloads.Theloadcanbequitevariableinmotorcycles(oneortwopassengers,possiblywithbaggage);andfurthermore,loadtransferbetweenthefrontandrearwheeloccursinbothaccelerationandbraking.

5.1Pre l iminarycons iderationsInthestudyofin-planedynamics,themotorcycleisconsideredasanelasticallysuspendedrigid

body.Ithasthreedegreesoffreedom:onedegreeoffreedomisassociatedwiththevehicle’sforwardmotion,whiletheothertwoareassociatedwithtwovibratingmodesandare,therefore,characterizedbytheirrespectivenaturalfrequencies.

Thecombinationofdistancebetweenbumpsontheroadplaneandforwardvelocitycausesexcitationsofthevehicleinarangeoffrequencythatcanbeevaluatedfrom0.25Hzto20Hz.Sincethetireshaveradialstiffnessmuchgreaterthanthatofthesuspension(6-12timesgreater),theirinfluenceatlowfrequencies(belowapproximately3Hz)becomesnegligible.

Fig.5-1Wavelengthofadisturbance.

Nowletusseewhenresonanceconditionscanbegeneratedasaresultofirregularitiesintheroadsurface.SupposethemotorcycleadvanceswithconstantvelocityVonaroadprofilepresentingequidistantirregularities-forexample,thebaysofaviaduct(Fig.5-1).ThetimerequiredforthemotorcycletocoverthedistanceLwavebetweenthetwoirregularities(lengthofthebay)isequalto:

Tthereforerepresentstheperiodofexternalexcitationofthemotorcycle.

Theresonanceconditionoccurswhentheexcitationfrequencyisequaltothenaturalfrequencyofoneofthevibrationmodesofthevehicleintheplane.

Criticalforwardmotorcyclevelocityisdefinedastheforwardvelocityatwhichthemotionimposedbytheroaddisturbancehasthesamefrequencyasoneofthevibrationmodesofthevehicle

intheplane.

IfLwaveisthewavelengthofthedisturbanceandνnisthefrequencyofoneofthemotorcyclemodes,(Tnisthenaturalperiod),thecriticalforwardvelocityisgivenbythefollowingexpression:

Forexample,withanaturalfrequencyνnequalto2HzandaperturbationwithwavelengthLwaveequalto6m,theresonanceconditionoccursatavelocityof12m/s(criticalvelocity).

Ifthemotorcycleproceedsatavelocitybelowcriticalvelocity,thefrequencyνofthemotionimposed:

islowerthanthenaturalfrequencyνn.Alternatively,itisabovethevehicle’scriticalvelocityforvelocitiesgreaterthanthecriticalvelocity.

Itisalsopossibletofollowadifferentapproach.Assumingthatthemotorcycleforwardvelocityis50m/s.Giventhenaturalfrequencyνn,atthisvelocitytheresonanceconditionoccurswhentheperturbationhasa(critical)wavelengthequalto:

which,undertheassumptionmade(νn=2Hz,V=50m/s),correspondstoacriticalwavelengthof25m.Criticalwavelengthsthereforediminishinproportiontotheforwardvelocity.

5.2Suspens ionoverviewSuspensionsystemswereintroducedonmotorcyclesinthe1930sand1940sandnumerous

architecturesandkinematicmodelshavebeenproposed.Wewillbrieflyanalyzethekinematicschemesofthefrontandrearsuspensionthatarenowmostcommon.

5.2.1Frontsuspens ion

Themostwidespreadfrontsuspensionis,undoubtedly,thetelescopicfork.Itismadeupoftwotelescopicsliderswhichrunalongtheinnertubeoftheforkandformaprismaticjointbetweentheunsprungmassofthefrontwheelandthesprungmassofthechassis.

Theconstructivesolutionwiththetwotelescopicslidersattachedtothesteeringheadisreferredtoas“conventional,”andiscurrentlythemostcommonconstructionforstreetmotorcycles(Fig.5-2).

Thesolutionwiththetwoforktubesfixedtothesteeringheadandthetwoslidertubesonthelower

end,called“upsidedown”,istheonemostcommonlyusedinsportmotorcycles,especiallysinceithasmorebendingandtorsionalstiffness.

Fig.5-2Schemesofclassicandupside-downtelescopicforks.

Thetelescopicforkischaracterizedbylowinertiaaroundtheaxisofthesteeringhead.Itsgreatestdisadvantageisrepresentedbythehighfrictionforcesencounteredwhenforcesareappliedorthogonaltotheaxisalongwhichtheslidersrun-forexample,inbrakingandoncurves.

Inbraking,becauseoftheloadtransfer,thetelescopicforkcompressesastherearsuspensionisunloaded;thus,thevehiclepitchesforward.Thepitchingchangesthetrimofthevehicleandfurtherdiminishesthesteeringheadangle.Asmallerangleofinclinationoftheforkcausesareductionofthevalueofthetrail.

Twolimitationsofthetelescopicforkaretheimpossibilityofachievingprogressiveforce/displacementandtheratherhighvaluesoftheunsprungmassthatisanintegralpartofthewheel.

Toovercomethetypicaldefectsofthetelescopicfork,differentsuspensionsystemshavebeenused.Thesecanbeclassifiedfromakinematicpointofviewas:

pusharm;trailingarm;four-barlinkage.

Inanarmfrontsuspension,thearmcanbe“pushed”(Earles-typefork)orpulledback(aschemeusedbythePiaggioVespa),asillustratedinFig.5-3.

Thefour-barlinkagecanalsobeusedinafrontsuspension.Inthiscasetheaxisofthesteeringheadcanbeattachedtothechassis,ortotheconnectinglink,asshowninFig.5-4.

Thefrontarmsuspensionandfour-barlinkagesuspensioncanbedesignedsoastoprovidetotalorpartialanti-divebehaviorinbraking.Inaddition,nothavingprismaticjointsthedryfrictionproblemstypicaloftelescopicforksareeliminatedfromthestart.

Thetorsionalstiffness(withrespecttothesteeringaxisandanaxisnormaltoit)ofthese

suspensionsystemsdependsonthedesignbutingeneralitiseasiertoobtaingreatervalueswiththetelescopicfork.Furthermore,theseelaboratedesignscanalsoreducetheunsprungmass.Anappropriatepositionofthespringorsprings,especiallyinthecaseofthefour-barlinkagesuspension,makesprogressivesuspensionpossible.

Fig.5-3Schemesoffrontsuspensionwithpushedandpulledwishbones.

Fig.5-4Schemesoffrontfour-barlinkagesuspension.

Fig.5-5Schemesoffour-barlinkagefrontsuspensionwithprismaticpairs.

Avariantofthefour-barlinkagefrontsuspensionisobtainedbysubstitutingarevolutejointwithaprismaticjoint,asillustratedinthediagramontheleftinFig.5-5.Thiskinematicdesignhasthedisadvantagethattheverticalmovementofthewheelinrelationtothechassiscausesrotationinthehandlebarsaroundtheupperrevolutejointfastenedtothechassis.ThisdisadvantageisgreatlyreducedinthedesignontherightinFig.5-5,whichisobtainedbymovingtherevolutejointfastenedtothechassisinrelationtothesteeringheadaxis.ThisdesignsolutionisusedintheBMW“Telelever”suspension.

5.2.2Rearsuspens ion

Theclassicrearsuspensioniscomposedofalargeforkmadeupoftwotrailing-ingarmswithtwospring-damperunits,oneoneachside,inclinedatacertainanglewithrespecttotheswingingarm.(Fig.5-6).

Theprincipaladvantagesofthetraditionalrearsuspensionare:simplicityofconstruction;easeofdissipationoftheheatproducedbytheshockabsorbers;largeamplitudeofthemotionofspring-damperunitswhichisnearlyequaltotheverticalamplitudeofthewheelmotionandwhichthereforecauseshighcompressionandextensionvelocitiesoftheshockabsorbers;thelowreactionforcestransmittedtothechassis.

Thegreatestdisadvantagesare:limitationoftheverticaloscillationamplitudeofthewheel;notveryprogressiveforce-displacementcharacteristic;possibilitythatthetwospring-damperunitsgeneratedifferentforcesduetodifferencesinthespringpreloadsorthecharacteristicsoftheshockabsorbers,withconsequentmalfunctioningofthesuspension,duetothegenerationofmomentsthattorsionallystresstheswingingarm.

Onevariantofthedual-strutsuspensionisthecantilevermono-shocksystem,characterizedbyonlyonespring-damperunit.Ithasthefollowingadvantagesoverthetwinshockarm:

easeofadjustmentsincethereisonlyoneshockabsorber;

smallerunsprungmass;hightorsionalandbendingstiffnesses;highverticalwheelamplitude.

Thissuspensiondoesnotenableaprogressiveforce-displacementcharacteristicandthepositioningofthespring-shockabsorberunitaboveorbehindtheenginecancauseheatdissipationproblemsfortheshockabsorber.

Intheclassicandcantileversystemtheintroductionofalinkageintherearsuspensionmakesiteasiertoobtainthedesiredstiffnesscurves.Thesedesignsaregenerallybasedonthefour-barlinkage.Theyaredistinguishedonlybythedifferentattachmentpointsofthespring-damperunit,whichcanbeinsertedbetweenthechassisandtherocker(UnitrakdesignofKawasaki)orbetweentheconnectinglinkandthechassis(Pro-LinkdesignofHonda)orbetweentheswingingarmandtherocker(FullFloaterdesignofSuzuki)asshowninFig.5-7.Modestunsprungmassesareobtained,aswellaslargewheelamplitude,butgreatreactionforcesareexchangedamongthevariouspartsofthefour-barlinkage.

Fig.5-6Schemesofrearsuspensionwithswingingarm.

Fig.5-7Schemesofrearsuspensionwithswingingarmandfour-barlinkage.

Thefour-barlinkageisalsocommonlyusedtoaccommodateashaft-drivewithuniversaljoints.Thewheelisattachedtotheconnectingrodofthefour-barlinkage.Itscenterofrotationwithrespecttothechassisisthereforethepointofintersectionoftheaxesofthetworockers.Thepositionoftherotationcenterdependsontheanglesofinclinationofthetworockers.Thesuspensionactsasifitwerecomposedofaverylongforkfastenedtothechassisinthecenterofrotation.ThiskinematicdesignisusedintheBΜW“Paralever”suspensionandintheGuzzimotorcyclesmodifiedbyMagni(withparallelrockers).

Αsuspensionbasedonasix-barlinkagehasalsobeentried(Morbidelli500GP).Thiscanpotentiallygeneratecurveswithmoreuniqueprogressionofsuspensionstiffness.Thispotentialadvantage,however,doesnotjustifythehighlycomplexconstruction.

Fig.5-8Schemesofrearsuspensionswithfour-barandsix-barlinkages.

5.3Reducedsuspens ions ti ffnes sThechoiceoffrontandrearsuspensioncharacteristics(stiffness,damping,preload)dependson

manyparameters:theweightoftheriderandthemotorcycle,thepositionofthecenterofgravityorthedistributionoftheloadsonthewheels,thecharacteristicsofstiffnessandverticaldampingofthetires,thegeometryofthemotorcycle,theconditionsofuse,theroadsurface,thebrakingperformance,themotorpower,thedrivingtechnique,etc..

Fig.5-9Equivalentfrontandrearsuspension.

Forthestudyofin-planedynamics,itisappropriatetoreducetherealsuspensiontoequivalentsuspension,representedbytwoverticalspring-damperunitsthatconnecttheunsprungmassestothesprungmass.

Theparametersdefiningequivalentsuspensionare:

reducedstiffness,reduceddamping,dependenceofthereducedstiffnessontheverticaldisplacement(progressive/degressivesuspension),maximumtravelandpreload.

5.3.1Reducedfrontsuspens ions ti ffnes s

NowconsiderthefrontforksuspensionasdepictedinFig.5-10.

Fig.5-10Reducedfrontsuspensionstiffness.

Ifsubjectedtothesameverticalload,therealfrontforkandtheequivalentverticalsuspensionhaveequalverticaldisplacements.Hence,thereducedstiffnesskfandtherealstiffnesskhavetosatisfythefollowingequation:

Considernowtheforkandtheequivalentsuspensionwithjustthedampingdevices.Ifthesameverticalvelocityonthewheelhubisimposed,theviscousverticalforcesareequal.Then,theequivalentdampingofthefrontsuspensioncfmustsatisfytheequation:

wherecrepresentsthedampingconstantofthefork.Sincetherearetwogroupsofspringdampersarrangedinparallelinthefork,thestiffnesskisequaltothesumofthestiffnessesofthetwosprings,andthedampingcisequaltothesumofthedampingofthetwodampers.Itshouldbenotedthattheincreaseinthesteeringheadangleofinclinationcausesareductioninthestiffnessanddampingcoefficientsofthereducedsuspension.

Example1

Αfrontsuspensionisrequiredwithreducedverticalstiffnesskf=14N/mm.Theangleofinclinationoftheforkisequalto30°.Determinetheactualstiffnessofeachforkspring.

Theoverallstiffnessoftheforkisequalto:

k=kfcos2ε=10.5N/mm

Thestiffnessofthesinglespringmustthereforebeequalto5.25N/mm.Ifthecasterangleisless,andequalto24°,thestiffnessofthesinglespringmustbegreaterorequalto5.84N/mm.

5.3.2Reducedrearsuspens ions ti ffnes s

NowconsidertheclassicrearsuspensionrepresentedinFig.5-11.TheelasticforceFeis

proportionaltothedeformationofthespring: wherekindicatesthestiffnessofthespring, itsinitiallengthandLmthelengthofthedeformedspring(itisafunctionoftheswingingarmangleofinclinationϑ).

TheelasticmomentMeexertedontheswingingarmisgivenbytheproductoftheforceandthevelocityratioτm,ϑ.

Me=Feτm,ϑτm,ϑistheratiobetweenthespring’sdeformationvelocityandtheswingingarmangularvelocity:

Wecanconsiderthattheswingingarmhas,inplaceoftheeffectivespring,atorsionalspringthatgeneratesamomentequaltotheonegeneratedbytheeffectivespring.Thederivativeoftheelasticmoment,withrespecttotheangleofrotationoftheswingingarm,representsthereducedstiffnessofthetorsionalspring:

fromwhich,throughsubstitution:

Thesecondtermislessimportantthanthefirst,and,inafirstapproximation,itcanbeignored.

Fig.5-11Reducedstiffnessoftherearsuspension.

Therearsuspensioncanalsobesubstitutedbyaverticalspringattachedtothewheelhub,ratherthanatorsionalspring.

ThereducedelasticforceFisequaltotheproductoftheelasticforceexertedbythespringandthevelocityratio :

where representstheratiobetweenthedeformationvelocityofthespring(whichisobviouslyequaltothevelocityofthedamper)andtheverticalvelocityofthewheel.

Inthiscase,thereducedverticalstiffnessisequaltothederivativeoftheverticalforceappliedtothewheelpinwithrespecttotheverticaldisplacementofthewheel:

Thevelocityratiodependsonthegeometriccharacteristicsoftherearsuspensionmechanismandvarieswiththeverticalwheeltravel.

Inthecaseoftheclassicswingingarm,thevelocityratiois:

wherexT,yTindicatethecoordinatesofthespring-damperpinattachedtothechassis.

Nowconsiderthedampingforce:

cindicatesthedampingconstantofthedamper.

Thedampingmomentactingontheswingingarmisgivenbytheproductoftheforceandthevelocityratioτm,ϑ:

Ms=Fsτm,ϑ

Thereduceddampingofthetorsionaldamperisobtainedbytheratiobetweenthedampingmomentandtheangularvelocity:

Substituting,weobtain:

Thereducedverticalforceis:

Thereducedverticaldamping,ontheotherhand,is:

Itshouldbenotedthattheverticalforcedependsonthevelocityratio,whilethereducedstiffnessanddampingdependonthesquareofthevelocityratio.Thevelocityratio insuspensionsystemsbasedonthefour-barlinkagevariesfrom0.25to0.5.Thismeansthatthestiffnessandthedampingofthespring-shockabsorbergroupmustbebetween4and16timesgreaterthanthevaluesofthereducedspring-shockabsorbergroup.

5.3.3Sti ffnes s curve

Thecurverepresentingtheelasticforceagainsttheverticaldisplacementofthewheelcanhavealineartrace,oraprogressivelyincreasingordecreasingone,towhichaconstant,increasingordecreasingreducedstiffnesscorresponds,asshowninFig.5-12.Thesecasesarereferredto,respectively,aslinear,progressiveordegressivesuspension.

Fig.5-12Elasticforceandstiffnessofthesuspensionversustheverticalwheeltravel.

Forthesakeofcomfortinmotion,itwouldbeappropriateforthestiffnesstobeaslowaspossible,soastominimizenaturalfrequenciesofthemotorcyclevibrationmodes,inrelationtotheexcitationfrequenciesofthemotionimposedonthewheelsbyirregularitiesintheroadplane.Verysoftsprings,however,causewidevariationsinvehicleheightastheloadvaries,aswellassignificantvariationsintrim,inthepassagefromrectilineartocurvedmotion,andduringtheaccelerationand

brakingphases.

Ontheotherhand,withirregularitiesintheroadsurface,veryhardspringscancause,besidesdrasticallyreducedcomfort,tireadherenceproblemsintherearsectionduringaccelerationandinthefrontsectioninbraking.

Toavoidthesedifficulties,moreorlessprogressivesuspensionsystemsareemployedinaccordancewiththetypeofuseofthevehicle.Substantially,progressivesuspensionprovidestwoimportantadvantages:

anincreaseinstiffnesstogetherwithanincreaseindeformation,whichenablesthemaintenanceofmoreorlessconstantfrequencyofthemodesofvibrationintheplaneasthevehiclemassincreases(anincreasecaused,forexample,bythepassengerortheluggage);thesuspensionissoftinthecaseofsmalldisturbancesandthusinthecaseofsmallwheeltravel,whileitisrigidinthecaseofhighwheeltravelduetomoreseveredisturbances.Ridingcomfortistherebyincreased.

5.3.4Pre load

Toregulatethetrimofthemotorcycle,forexample,undervariationoftheload,preloadingofthespringscanbeused.Preloadingconsistsofapre-compressionofthespring.Ifthespringisstressedwithforcesthatarelowerthanorequaltothatofpreloading,itisnotdeformed.

Theforceexertedbythespring,withpreload,is:

F=kΔy+ky

whereΔyindicatesthedeformationduetopreload.

Preloadingalsomakesitpossibletolimitdeformationincompressionofthespring-shockabsorbergroup.ThegraphofFig.5-13showsthatwithpreloading,inordertoobtainmaximumamplitudegreaterforcesmustbeappliedorconversely,thatwiththesameforceappliedtheamplitudewillbeless.

Fig.5-13Characteristicsofthesuspensionasthepreloadvaries.

Figure5-14ashowsasuspensionwithaspringthatisnotpreloaded.Thestaticloadofthesprungmasscompressesthespring-shockabsorbergroupbyanamountthatdependsonthestiffnessofthespring,assumingthatduringforwardmotionthesprungmassisnotdisplacedintheverticaldirection,i.e.itideallydoesnotencounterirregularitiesintheroadsurface.

Inorderforthewheeltofollowtheprofilewhenpassingoverahole,thespring-shockabsorbergroupneedstobeabletoextendbyaquantityequaltothedepthofthehole.Inthecaseofsuspensionwithoutpreload,theextension,orbetterthehole’sdepth,canatmostbeequaltotheratiobetweentheweightforceofthesprungmassandthestiffnessofthesuspension.

However,inthecaseofsuspensionwithapreloadedspring,illustratedinFig.5-14b,themaximumextensionofthespring-shockabsorbergroupisless,inthisexample,byaquantityequaltothepreload.

Thepreloadthereforegovernsthemaximumvalueofthewheeltravelinextension.Thecapacityofthesuspensiontofollowtheirregularitiesbelowtheroadplanedependsonthisvalue.Theseirregularitiesarecalled“negative”.Forexample,ifapreloadisappliedthatisequaltothestaticload,thewheelwillnotbeabletofollowthenegativeirregularities.Infact,inFig.5-15itcanbeobservedthat,withanincreaseintheforceofthepreload,thefieldoftheamplitudesofthesuspensionforthenegativeirregularitiesdiminishes.

Fig.5-14Suspensionwithpreloadedspring.

Anapproximatevalueofthereducedstiffnessofasuspensioncanbedeterminedonthebasisofsimplestaticconsiderations.Themaximumloadoneitherthefrontorrearwheelscanbeequaltothetotalweightofthemotorcycleplustherider.Suchcircumstancescanoccurunderthelimitingconditionofawheelingorforwardfallofthemotorcycle,respectively.Stiffnessdependsonthevalueofthewheeltravelrequiredinthiscondition:

Fig.5-15Characteristicsofthesuspensionasthepreloadvaries.

Ifitisrequiredthatforasetload(forexample,thestaticloadactingonawheelduetotheweightofthemotorcycleplustherider)thesamepreloadamplitudeapplies,thendifferentpreloadsmustbeadoptedbyvaryingthevaluesofthesuspensionstiffness,asillustratedinFig.5-16.

Fig.5-16Characteristicsofthesuspensionwithvaryingstiffness.

5.3.5Frontsuspens ions ti ffnes s

Thefrontforkshaveaslightlyprogressivebehaviorbecauseoftheinfluenceofaircontainedinthesleeves,whichactslikeapneumaticspringpositionedinparallelwiththehelicoidalspring.

Theelasticforce,withtheinfluenceofthepneumaticspring,isgivenbytheequation:

where:

kΔyrepresentstheforcegeneratedbythepreload;

kyrepresentsthelinearforceofthemetalspring.

Thethirdtermrepresentstheinfluenceofthepneumaticspring;p1indicatestheinitialairpressurecontainedinthesleeveofthefork;V1theinitialvolumeofairandAtheareaofthesectionofthecylindricalchamberthatcontainstheair.Theeffectofthecompressedairincreaseswiththedecreaseoftheinitialairvolumecontainedintheforkandtheincreaseinwheeltravely.

Thevalueofthereducedforkstiffnessvariesaccordingtotheweightofthemotorcycleanditsuse.Values(foroneforkleg)rangefromabout10N/mmforlightmotorcyclestovaluesofabout20N/mmforheavymotorcycles.Theprogressivebehaviorduetotheuseofvariablesprings,ortotheuseofseveralspringsplacedinseriesandhavingdifferentrigidities,and/ortotheinfluenceofair,causesanincreaseinthestiffnessattheendofthestroke,whichcanbeevaluatedas30to50%.

Example2

Consideramotorcyclewithatotalmass(includingtherider)of200kgandwithadistributionoftheloadsat50%-50%.Lettheunsprungmassinfrontbe18kgandtheangleofinclinationofthesteeringhead24°.

Calculatethestiffnessofthereducedspringsothatwithaloadequaltotheweightofthemotorcycletheforkiscompressedby80mm.

Themaximumweightonthespringisequaltotheweightforceofthemotorcycleminustheweightforceofthefrontunsprungmass.Theequivalentverticalstiffnessis:

Theoverallstiffnessoftheforkmustthereforebeequalto:

k=kfcos2ε=9.18N/mm

Finally,letuscalculatethedeflectionofthespring,underconditionsofstaticequilibrium,supposingthatthespringoftheforkispreloadedby20mm:

Example3

Nowletusevaluatethestiffnessofaforktakingintoaccounttheinfluenceofair.

•initialvolumeofair:casea)

•initialvolumeofair:caseb)

•crosssectionareaofthecylindricalchamber: A=10cm2;

•initialpressure: p1=1bar;

•linearstiffnessofthemetalspring: k=8N/mm;

•preload: Δy=10mm.

Figure5-17showstheincreaseinelasticforcesandstiffness,underthevariationofthedeformationofthespringfortwodifferentinitialvolumesofair.

Fig.5-17Influenceofthepneumaticspringonoverallstiffness.

Theinfluenceofthepneumaticspringbecomesimportantwhentheairvolumeisstronglycompressed.Forhighwheeltravelthevalueofthestiffnesscanevenbedoubled.Theinitialvolume

ofairinthesleevedependsonthequantityofoilcontainedinthesleeve.Alargequantityofoilcorrespondstoalowerinitialvolumeandthereforetoanincreaseintheinfluenceofthepneumaticspring.StiffnessvaluesandtheloadspresentedinFig.5-17refertoonlyonespring;thereforetheforkasawholewillhaveastiffnesstwicethatindicated.

5.3.6Rearsuspens ions ti ffnes s

Havingthecurveofdesiredprogressivebehavior,thesynthesisofthemechanismscanbecarriedoutbymeansofnumericaloptimizationalgorithms.Generally,theratiobetweentheverticalvelocityofthewheelandthedeformationvelocityofthespring-shockabsorbergroup(theinverseoftheratio

)hasvaluesvaryingfrom2to4.Thehighervalues,withequalwheeltravelrate,causesmalldeformationvelocitiesoftheshockabsorberandthereistheneedtouseshockabsorbersoflargerdimensions.

Figure5-18showsthekinematicschemeandthemovementofarearfour-barlinkagesuspension,withthespring-shockabsorbergroupfastenedbetweenthechassisandtherocker.Thismechanismmakesitpossibletogeneratecurveswithasignificantprogressiverate.Asillustratedinthecurvesofthethreeexamplesshownontherightinthefigure,thedegreeofthestiffnessrising-ratedependsonthepositionsofthespringattachmentpoint.Thegreatestvalueofthevelocityratio,andthereforethegreatestraisingrate,isattainedwhenthespringisorthogonaltotherocker.

Figure5-19representsakinematicdiagramwiththespringfastenedbetweenthechassisandtheconnectinglink.Withthisarrangementweattainverydifferentcurvesofstiffness,asthepositionofthepointtowhichthespringisattachedvaries.Theright-handfigureshowshowitispossibletoobtaindegressive,progressiveandapproximatelyconstantstiffness.

Fig.5-18Rearsuspensionwiththespringattachedtotherockerarm.

Fig.5-19Rearsuspensionwiththespringattachedtotheconnectinglink.

Example4

Consideraclassicrearsuspensionwithaswing-armhavingthefollowingcharacteristics:

•lengthofswingingarm: L=0.6m;

•distancefrompivottospring: L1=0.4m;

•lengthofundeformedspring:

Intheinitialposition,correspondingtoy=0theaxisofthewheelisloweredby100mminrelationtothepivotoftheswingingarm.

InthegraphinFig.5-20,thevariationofthereducedverticalstiffness(madedimensionlesswithrespecttotheinitialvaluecalculatedincorrespondencetoy=0)undervariationoftheverticaldisplacementofthewheelpiny,isshownforvariousvaluesoftheinitialinclinationangleofthespring.

Fig.5-20Stiffnesscurveforvariousinclinationvaluesofthespring-shockabsorbergroup.

Fig.5-21Stiffnesscurveforvariousvaluesoftheratiobetweenthespringdistanceandthelengthoftheswingingarm.

Onecanobservethatwhenincreasingtheinitialinclinationofthespring,thecurvesobtainedarecharacterizedbyagreaterstiffnessrising-rate.

Figure5-21showstheinfluenceofthepositionofthespringattachmentpoint,expressedbytheratiobetweenthearmL1andthelengthoftheswingingarmL.Itcanbeobservedthatpositioningthespringattachmentpointclosetotheswingingarmpivotcausesanincreaseinthestiffnessrate.Theprogressivebehaviorattainablewiththeclassicsuspension,withlongspringsthataregreatlyinclinedandfastenedneartheswingingarmpivot,canreach50-60%atmost.

Example5

Considerthesuspensionintheprecedingexampleandevaluatethetorsionalstiffnesswhenthe

springsareinclined135°tothehorizontal.Takethestiffnessofthespringask=80kN/m.

Thereducedtorsionalstiffnessisequalto:kϑ=4.30kΝm/radfory=0:kϑ=5.67kNm/radfory=150mm.

Inthelattercase,theprincipaltermk givesavalueof5.31kNm/rad,whilethesecondarytermgivesalowervalueof0.46kNm/rad.

5.4Cons iderations oncl imbingas tepConsideramotorcycletravelingatconstantvelocityVwhichatacertaininstantencountersastep

withheighth.Toclimbthestep,thewheelmustadvancebyadistances:

Considerfirstthecaseofamotorcyclewithnosuspensionsystems.Thetimethewheeltakestoclimbthestepisequaltotheratio:

Inthatcase,theadvancementsofthewheelcorrespondsexactlytotheadvancementofthemotorcycle.Thevelocityandverticalaccelerationofthewheelare:

Example6

Atavelocityof10m/s,awheelofradiusR=0.307mtravelsoveranobstacle50mmhighinanintervalequalto0.0168s.Inthattimethewheelclimbstoaheightequaltothestepsothattheaverageverticalliftspeedis3.3m/s.Themaximumvalueofthevelocityisreachedattheinitialinstantandisequalto6.5m/s.

Themaximumverticalaccelerationalsooccursattheinitialinstant(assumingthatthetireisinfinitelyrigid),andis555m/s2,whichis55timestheaccelerationduetogravity.Actually,theaccelerationismuchlower,sincethetireisdeformedandpartiallyabsorbstheimpulse.

Thissimpleexampleshowstheimportanceofthesuspension.

Inthecaseofamotorcyclewithsuspension,thetimetakenbythewheeltoclimbthestepdependsonthetypeofsuspensionandnolongerjustonthedimensionsoftheobstacle.

Whenthefrontwheelprovidedwithsuspensionclimbsasteptheforcetransmittedtothechassis

dependsonboththeelasticandthedampingcharacteristicsofthesuspension(Fig.5-22).Atlowvelocities,theelasticforceprevailswhileatmediumandhighvelocitiesthedampingforceisdominant.

Fig.5-22Thefrontwheelatthebeginningandendofclimbingthestep.

Example7

Letusconsiderafrontsuspensionwiththefollowingproperties:k=8000N/mandc=500Ns/m,andsupposethatwhileclimbingthesamestepofexample6thechassis(sprungmass)doesnotraise.Determinetheelasticanddampingforcesataforwardvelocityof1m/sand10m/s.

Case1:ε=27°

Theelasticforcedependsonlyonthedeformationofthespring.Itsmaximumvalueis356N.Iftheforwardvelocityislow(1m/s),theelasticforceisgreaterthanthedampingforcewhichisequalto291m/s.Atavelocityof10m/sthedampingforceprevails(2916N)andisatamaximumatthebeginningofclimbingthestep.

Case2:ε=33°.

Astheangleofinclinationoftheforkincreases,boththemaximumvalueoftheelasticforce(335N)andthatoftheviscousforce(2740N)diminish.

Letusnowconsidertherearwheelprovidedwithasuspension.Letusassumethattheverticalpositionofthevehiclechassisdoesnotchangewhileovercomingtheobstacle,orthattheswingingarmpivotmovesalongthex-axis(Fig.5-23).

Thetimenecessaryfortherearwheeltopassfromtheinitialtothefinalpositiondependsonthevelocityofthevehicleandthegeometryofthesuspension(lengthofthefork,radiusofthewheel,initialangleofinclinationofthefork).Thetimeneededtorunovertheobstacleisgivenbythefollowingexpression:

where:

ϑ1indicatestheinitialangleofinclination;ϑ2indicatesthefinalangleofinclination;sindicatesthedistancebetweenthepinofthewheelandthestep.

ThetimeΔtemployedbythewheeltoclimbthestepcandifferfromthetimeusedbythevehicletotravelthedistances:whenthewheelclimbsthestepandadvancesbydistancesinthesametimeusedbythevehicletoadvancebythesamedistances,thebehaviorofthesuspensionisdefinedasneutral.InthecaseofclimbingtimesΔtthatexceedthoseoftheneutralsuspension,thebehaviorofthesuspensionisreferredtoaspositive.

Fig.5-23Therearwheelatthebeginningandendofclimbingthestep.

Thesuspensionhaspositivebehaviorwhenthefork,inboththeinitialandfinalpositions,remainsbelowthexaxis(Fig.5-24);inthecaseofclimbingtimesΔtthatarelowerthantheneutralsuspension,thebehaviorofthesuspensionisreferredtoasnegative.Thesuspensionbehavesnegativelywhentheforkremainsabovethexaxisinboththeinitialandthefinalposition(Fig.5-25).

Thecirculartrajectorydescribedbythecenterofthewheelinclimbingthestepisthereforecoveredindifferingtimesaccordingtothetypeofbehaviorofthesuspension.Inthecaseofsuspensionwithpositivebehavior,thetrajectoryistraversedinagreatertimeinterval.Thissubjectsthewheeltolowerverticalaccelerations.Inparticular,thedeformationlawofthespring-shockabsorbergroupshowsloweracceleration.

Fig.5-24Rearsuspensionwithpositivebehavior(s<VΔt).

Fig.5-25Rearsuspensionwithnegativebehavior(s>VΔt).

Example8

Letusconsiderarearwheelwiththefollowingproperties:

•wheelradius: Rr=0.307m;

•swingingarmlength: L=0.6m;

•obstacleheight: h=0.05m;

•velocity: V=10m/s.

Thepeakvalueofverticalaccelerationdependsonthebehavioroftherearsuspension:

Asmentionedearlier,peakaccelerationislowerinthecaseofpositivesuspension.Thepeakvalueincreasesastheradiusofthewheeldecreases.Adiminutionof10%inthewheelradiuscausesa5%increaseinthepeakaccelerationvalue.

Adiminutionofthedeformationvelocityoftheshockabsorberisobtainedbyincreasingtheinclinationofthespring-shockabsorberunit.Forexample,passingfromtheverticalpositiontoanangleof45°reducestheviscousforcemaximumby35%.

Thesameconsiderationscanalsobeextendedtothefrontsuspension.Theclassictelescopicforksuspensionispositivewhilethepulledorpushedarmsuspensioncanbepositiveornegativeaccordingtothearm’sangle.

Fig.5-26Frontsuspensionwithforwardarmswithnegativeandpositivebehavior.

5.5Sl ippingofthe rearwhee l contactpointConsideramotorcycletravelingatconstantvelocityunderathrustthatisalsoconstant.

Actually,thetransmissionofthethrustforcealwaysoccursinthepresenceofarelativeslipbetweenthetireandtheroad(theperipheralvelocityofthetireisgreaterthantheforwardvelocityofthemotorcycle).Supposethethrustforceisconstantandthereforetherelativeslipisalsoconstant.

Ifthereisrelativemotionbetweenthechassisandtheswingingarm(forexample,becauseofirregularitiesintheroadplane),therewillbeextraslipthatwillbeaddedto,orsubtractedfrom,thenecessaryslipduetothethrust.

Toevaluatetheamountofslipduetotheoscillationsoftheswingingarm,weneedtoconcentrateourattentionontherelativemotionofthereararmwithrespecttothechassis.

Themodelcanbesimplifiedbyassumingthatthedrivingsprocketconcentricwiththeaxisoftheswingingarmandthatthemotorcycleisstoppedwithitsengineoff(Fig.5-27).

Letussupposethatinitiallythesprocketislocktogroundwiththeswingingarmandwheelfreetorotate(Fig5-27b).IftheswingingarmisrotatedbytheangleΔϑ,thepointofcontactofthechainonthedrivesprocketisrotatedbythesameanglewhileaportionofthechainoflengthΔϑrp,equaltotheproductoftheangleofrotationbytheradiusofthesprocket,doesnotrotate.

Inthissituation,thewheelissubjectto:acounterclockwiserotationΔϑ,duetotherotationoftheswingingarm(Fig.5-27b),aclockwiserotationΔβ,causedbytheextensionofaportionofthechainfromthelargerwheelsprocket(Fig.5-27c).

TheclockwiserotationΔβisequalto:

indicatesthetransmissionratiobetweenthedrivingsprocketandtherearsprocket.

Fig.5-27Slippageofthecontactpointoftherearwheel.

LetΡbethepointbelongingtothetirewhichattheinitialinstantrepresentsthecontactpointbetweenthetireandtheground.Followingtheoscillationofthefork,itisrotatedthroughtheangle:

Δα=(1−τcp)Δϑ

αthereforerepresentstheslideangleduetotherotationoftheswingingarm.ΑslidelengthofαRrcorrespondstotheslippageangle(Rrindicatestheradiusofthewheel).

Inconclusion,givingacounterclockwiserotationontheswingingarm,i.e.,compressingtherearsuspension,causesaslipthatisaddedtothatneededtotransmitthethrust.Therefore,itiscorrecttostatethattheslipbetweenthewheelandthegroundincreasesduringthecompressionofthesuspensionanddiminishesduringitsextension.

Ifwesupposethattheswingingarmissubjecttoanoscillatorymotion:

Δϑ(t)=ϑo+Δϑsinωt

theangularvelocityofslippageofthecontactpointis:

ThismeansthatduringforwardmotionatconstantvelocityV,afluctuatingcomponentissuperimposedontheconstantangularvelocityofrotationofthewheelΩ=V/Rr.Actually,theslippageduetotheswingingarmoscillationstransmitsfluctuationstothesprocketmotionaswell,sothattheirregularityofthemotor ’srotationisincreased.Inthesameway,thepullonthechainiscomposedofaconstanttension,towhichafluctuatingtensiongeneratedbyswingingarmoscillationisadded.

Inthemoregeneralcaseinwhichthesprocketisnotconcentricwiththeswingingarmpivot(Fig.5-28),theexpressionfortheslippageofpointPrisslightlymodified.

Fig.5-28Theslippageofthecontactpointoftherearwheelwithasprocketkeyedinagenericposition.

Infact,thechainrotatesthroughanangleΔηwhichissmallerthantherotationangleΔϑimposedontheswingingarm.Furthermore,thedistancebetweenthepointsoftangencyofthechainwiththerearsprocketandthedrivingsprocketvarieswiththevariationoftheswingingarmrotationangle.

Thelengthoftheupperbranchofthechainisatamaximumwhentheaxisofthedrivingsprocketisalignedwiththeswingingarmaxis(Fig.5-29).

Fig.5-29Maximumlengthoftheupperbranchofthechain.

ThewheelrotatesinacounterclockwisedirectionequaltoΔϑandasimultaneousclockwiserotationequaltoΔβ:

ΔLcrepresentsthevariationinlengthoftheupperbranchofthechain(areductionhasanegativesign).ThepointPofthetirethereforerotatesthroughtheangle:

Δα=Δϑ−Δβ

SincethetermΔLcisnegligiblewithrespecttotheproductΔηrp,theslidingvelocitydependssubstantiallyonthetransmissionratiobetweenrearsprocketanddrivingsprocketandontheangleΔϑwhichdependsonthelengthoftheswingingarm.

5.6Models withone degree offreedomThemotorcycle,instraightrunning,ischaracterizedbyfivedegreesoffreedom(oneofwhichis

associatedwiththevehicleforwardmotion),andismadeupofthreerigidbodies:thesprungmass(chassis,engineandrider);therearunsprungmass(wheel,brakeandpartoftheswingingarm);thefrontunsprungmass(wheel,brakeandpartofthefork).

Ingeneralthein-planemotionofthemotorcyclecanbeconsideredasthecombinationofaverticalmotion(bounce)andarotatingmotion(pitch).Thesetwomotionscorrespondtothevibrationmodesofthemotorcycleintheplane.

Onlybyignoringtheunsprungmasses,andconsideringthesystemoftwodegreesoffreedomasuncoupled,wecantreatthemotorcycleintheplanewithtwomodelseachhavingonedegreeoffreedom:onefortheverticalbounceandtheotherforthepitchingmotion(Fig.5-30).Consideringthemotorcycleasanuncoupledsystemisthesameasassumingthat,imposingaverticaldisplacementofthechassis,theconsequentmovementiscomposedonlyofverticaloscillationsorthat,imposingarotationofthechassisaroundanaxispassingthroughthecenterofgravity,theresultingmotionisoneofpurepitch.

We,therefore,considerthatthemotorcycleintheplaneiscomposedofonlyonesprungmass,sustainedbytwosprings,whichrepresenttheactionofthesuspensionandthetires.Thereducedstiffnessofthesuspensionisconnectedinserieswiththestiffnessofthetires,sothattheequivalentelasticconstantsKfandKrforthefrontandrearsectionsrespectively,are:

where:

kfisthereducedstiffnessofthefrontsuspension;kristhereducedstiffnessoftherearsuspension;istheradialstiffnessofthefronttire;istheradialstiffnessofthereartire.

Fig.5-30Principalvibrationmodesintheplane.

Example9

Consideramotorcyclewithareducedrearsuspensionstiffnessofkr=20N/mm.

Comparetheoverallstiffnessofthesystemwitharatherstiff( )andaratherdeformabletire( )reartire.

WitharadiallystiffreartiretheoverallstiffnessisKr=18.5N/mm(-7.4%),whilewitharatherdeformablereartiretheresultisKr=17.1N/mm(-14.3%).

5.6.1Bounce andpitchmotion

Theequilibriumequationsofverticalforcesandthemomentaboutthehorizontalaxis:

providetheexpressionsfornaturalfrequenciesoftheverticalbouncemode,andforthepitchingmode,where isthepolarmomentofinertiaaroundtheyaxis.

Thenaturalfrequenciesνb,νprespectively,fortheverticalbouncemotionandthepitchmotionare:

Ingeneral,theradiusofinertiaρislessthanthedistancesband(p−b)sothatthefrequencyofthepitchingmotionνpisgreaterthanthatoftheverticalbouncemotionνb.

Inthissection,wehaveassumedthatthetwomainvibrationmodesareuncoupled.Thisisanidealconditionthatisnotmetinrealitybecausethesetwomodesaregenerallycoupledtoeachother.Thecouplingofthemodescanbeexperiencedbyuniformlyloweringtheentiremotorcycle,withoutallowingittorotateandthenleavingitfreetovibrate.Itiseasytoverifythatboththeverticalandthepitchmotionsareexcited.Inthesameway,byimposingarotationofthemotorcyclearoundthecenterofgravityboththepitchandtheverticalmotionsareexcited.

Fig.5-31Modesofvibrationintheplane.

Inconclusion,becauseoftheroadirregularities,themotorcycleoscillateswithamotionthatisacombinationoftwovibrationmodes;i.e.,themotorcycleoscillatesverticallyandpitchesatthesametime.

Example10

Consideramotorcyclewiththefollowingcharacteristics:

•wheelbase: p=1.4m;

•distancefromthecenterofgravitytotherearwheel: b=0.7m;

•sprungmass m=200kg;

•pitchmomentofinertia:

•reducedstiffnessofthefrontsuspension: kf=15kN/m;

•reducedstiffnessoftherearsuspension: kr=24kN/m;

•radialstiffnessofthetires:

Determinetheequivalentstiffnessofthefrontandrearsections.Thendeterminethebounceandpitchfrequencies.

Theequivalentstiffnessesare:

•frontstiffness: Kf=13.85kN/m;

•rearstiffness: Kr=21.18kN/m.

Thenaturalfrequenciesofthetwomodesofvibrationare:

•bouncemotion: νb=2.11Hz;

•pitchmotion: νp=3.38Hz.

Thefrequencyoftheverticalbouncemotionislessthanthatofthepitchmotion(νb<νp).Resonanceoccursincorrespondencewiththefollowingvaluesofthevehiclevelocityinthepresenceofanirregularityontheroadplanewithawavelengthof6m,

•bouncemotion: criticalvelocity=12.64m/s;

•pitchmotion: criticalvelocity=20.29m/s.

5.6.2Wheel hopresonance

Thesuspensionstiffnessvaluesaresignificantlylowerthanthosefortirestiffness(tirestiffnessisapproximately6-12timessuspensionstiffness).Theunsprungmassmis,therefore,connectedtothegroundwithahardspringandtothesprungmasswithasoftspring.Inafirstapproximation,theinfluenceoftheconnectiontothesuspendedmasscanbeignored.Inthisway,theunsprungmass,elasticallysupportedbytheverticalstiffnessofthetirealone,canberepresentedbyasimplesystemwithonedegreeoffreedom.Thereforethenaturalfrequencyoftheverticalmotionoftheunsprungmassis(Fig.5-32):

Fig.5-32Modelwithonedegreeoffreedomoftheverticalmovementofthewheel.

Example11

Considerthemotorcycleofthepreviousexample.

Withtheunsprungmasses,15kgonthefrontand18kgontherear.Determinethenaturalfrequencies.

Thenaturalfrequenciesare:

•verticalmotionofthefronttire:

•verticalmotionofthereartire:

Withanirregularityintheroadplanewithawavelengthof3m,theconditionofresonanceoccursin

correspondencewithavehicleforwardvelocityof52.3and47.5m/s,respectively,forthefrontandrearwheels.

5.7Twodegree offreedommodelIfweignoretheunsprungmasses,thesystemhastwodegreesoffreedom.Theycanbeassociated

withtheverticaldisplacementofthemotorcyclecenterofgravity(sprungmass),andwithitspitchrotationaboutahorizontalaxis(Fig.5-33).

Fig.5-33Modelofthemotorcycleintheplanewithtwodegreesoffreedom.

Whenstudyingthemodesofvibration,havingignoredtheunsprungmasses,doesnotleadtosignificantinaccuraciessincethestiffnessofthesuspensionis6to12timessmallerthantheverticalstiffnessofthetires.Theinfluenceoftheunsprungmassesbecomesimportantatmediumandhighfrequenciesoftheirregularitiesintheroadplane,i.e.,athighvelocitiesandshortwavelengths.

Thefreeoscillations,ignoringthedampingeffect,aredescribedbythefollowingequations:

Thefrequencyequationisthen:

Thetworootsoftheequationaretheundampedsystem’stwonaturalfrequencies.Theratiobetweentheamplitudesoftheverticaloscillationandthepitchoscillationisgivenbytheequation

ωiindicatesthenaturalfrequencyofthevibrationmodeconsidered.

Theseexpressionsrepresentthedistancefromthecenterofrotationtothecenterofgravity.

Example12

Considerthemotorcycleinexample10.

Thenaturalfrequenciescalculatedwiththetwodegreesoffreedommodel,are:

•verticalbouncemotion: νb=2.02Hz;

•pitchingmotion: νp=3.42Hz.

Thesevaluesdifferatmostby4%withrespecttothecorrespondingvaluescalculatedwhenconsideringtwouncoupledsystemswithonedegreeoffreedomeach.

Thefirstmodeofvibration,representedinFig.5-34,isbasicallyaverticalbouncemotionofthemotorcycle.Theinstantaneousrotationcenterislocatedbehindthemotorcycleatadistances1=2.08mfromthecenterofgravity.

Fig.5-34Firstmodeofmotorcyclevibrationintheplane(verticalbouncemode,i=1).

Thesecondmode,representedinFig.5-35,isessentiallyapitchmode,itsrotationcenterislocatedimmediatelyinfrontofthecenterofgravity,atadistanceofonlys2=0.09m.

Fig.5-35Secondmodeofmotorcyclevibrationintheplane(pitchingmodei=2).

Onlyifthefollowingconditionbetweenstiffnessesanddistancesisfulfilledcanthetwomotionequations,forzandμrespectively(p.174),beuncoupled:

−Krb+Kf(p−b)=0

Therefore,thenaturalfrequencieswillbeequaltothosepreviouslydeterminedinthemodelswithonedegreeoffreedom.

Inthiscasethefirstmode,calledthebouncemode,nowbecomesapureverticaltranslationwhilethesecondmode,calledthepitchmode,becomesapurerotationaroundthecenterofgravity.Inthiscasewherethemodesarenotcoupled,thedistances1fromtherotationcenterofthefirstmodetendstowardsinfinity,whilethedistances2fromthecenterofrotationofthesecondmodeiszero.

Example13

Considerthemotorcycleofthepreviousexample.Determinethevalueofbneededtouncouplethebounceandpitchmodes.

Theuncouplingofthepitchmotionfromthebouncemotioncanbeaccomplishedbyreducingthedistancebtoavalueof0.55m.Consequentlythedistributionoftheweightsbecomes40%onthefrontsectionand60%ontherear.

Theuncouplingcanalsobeobtainedbymodifyingthestiffnessofthesuspensions.Sincethedistances(p−b)andbareequal,thestiffnessofthefrontsuspensionneedstobeequaltothatoftherearsuspension,i.e.,equalto24kN/m.

5.8Fourdegrees offreedommodelAsmentionedearlier,themotorcycleinitsplaneofsymmetrycanberepresentedasthreerigid

bodieswhosevibratingmotionisdescribedbyfourindependentcoordinates,asillustratedinFig.5-

36:theverticaldisplacementofthesprungmasscenter;thepitchingrotationofthesprungmass;theverticaldisplacementsofthetwounsprungmasses(theequivalentmassesatthecenterofthewheels).

Fig.5-36Diagramofthemotorcyclewithfourdegreesoffreedom.

Theequationsoffree,undampedmotionaresummedupinthefollowingmatrixequation:

Thenaturalfrequenciesarecalculatedbysolvingtherelatedeigenvalueproblemnumerically.Rewritingtheequationsoffreemotion,usingcoordinateszf,Zf,zr,Zr,bringssomeinterestingpointstolight.

Observingthemassmatrix,wenotethatthefirsttwoequationsareuncoupledfromtheothertwoifthefollowingtermiszero:

Thatis,theproductofthedistances(p−b)andbequalsthesquareoftheinertiaradius .

Inthiscase,thefourequationsrepresenttwomono-suspensionswithtwodegreesoffreedomthatareindependentofeachother,asshowninFig.5-37.Thefirsttwoexpressionsdescribethebehaviorofthefrontmono-suspension,whilethelasttwodescribetherearone.

Fig.5-37Mono-suspensionsofthefrontandrearsections.

Fromaphysicspointofview,thismeansthatthesuspendedmasscanberepresentedbyanequivalentdynamicsystemcomposedoftwomassesplacedattheextremitiesincorrespondencewiththetwowheels:

Thesuspendedmassmisdistributedbetweenthefrontsection(Mf)andtherearsection(Mr)inthesameproportioninwhichthestaticloadsonthewheelsaredistributed.

Itisvitaltospecifythatthisconditionisdifficulttoaccomplishunderrealconditions(withreferencetothesuggestedexample,theproductofthedistances(p−b)bisequalto0.49,whilethesquareoftheinertiaradiusis0.44),neverthelessitisimportantfromaphysicspointofview.Therefore,inassumingthatthisconditiondoesapply,itispossibletoconcludethattheverticaldisplacementsofthefrontandrearsectionswilloccurindependentlyofeachother,andthat,therefore,thepitchingmotiondependsonthevalueofthephasebetweenthetwoverticalmotions.

Ignoringtheunsprungmasses,thenaturalfrequenciesofthetwosystemscanbeeasilycalculated.Forthefrontandrearsections,wehave:

ThesefrequenciesareequalifthenecessaryconditionfortheuncouplingoftheverticalmotionfromthepitchingKrb=Kf(p−b)issatisfied.

Inthiscase,thebouncehasthesamefrequencyasthepitchmotion,sothateachfreemotionismadeupofacombinationofverticalandrotationaloscillationswiththesamefrequency.

Generally,thefrontsuspensionhasarelativelylowerstiffnessthantherearsuspensionand,therefore,thefrequencyofthefrontsectionislowerthanthefrequencyoftherearsection.Inpercentagetermsνfisequalto70to80%ofνr.

Thesuspensionstiffnessrequirementcanbeevaluatedonthebasisofconsiderationsonthefrequenciesofthefrontandrearsections.Forgoodridingcomfort,thetwofrequenciesshouldhavevaluesaround1.5Hz,andthepitchrotationcentershouldbelocatedintheareaoftherider ’sseat.

Inracingvehicles,knownfortheirratherrigidsuspensions,thenaturalfrequenciesvaryfrom2Hzto2.6Hz.

Usuallyinstreetmotorcyclesandinscooters,thedistance(p−b)isgreaterthanb,andthereforeinordertoobtainthesamenaturalfrequencyatboththefrontandrearsections,thestiffnessatthefrontshouldbeassumedtobelessthantherear.

Toevaluatetheapproximatevaluesofthefrequenciesofthefrontandrearsectionsexperimentally,thestaticdeflectionsΔf,Δr,duetotheweightpressingonthetwowheelsneedtobemeasured.Assuchthecorrespondingfrequenciesaregivenbytheequations:

Hence,forafrequencyof2Hzastaticdeflectionof60mmisrequired.

Example14

Considerthevehicledescribedpreviouslyandsupposethattheunsprungmasseshavethefollowingvalues:

•frontmass: mf=15kg;

•rearmass: mr=18kg.

Determinethefournaturalfrequenciesofthesystem.

Thefournaturalfrequenciesare:2.03,3.42,16.98,18.18Hz;itisusefultorecallthatthenaturalfrequenciesofthesprungmass,calculatedwhiletheunsprungmassesareignored,areequalto2.11,3.38Hz,whilethoseoftheunsprungmassesareequalto15.91and17.43Hz.

Example15

Thesprungmassesonthefrontandrearsectionsare,inthecaseofloaddistribution50%-50%,equaltoMf=Mr=100kg.Determinethenaturalfrequenciesofthefrontandrearsections.

Ignoringtheunsprungmasses,thenaturalfrequenciesare:

•frontsectionfrequency: νf=1.87Hz;

•rearsectionfrequency: νr=2.32Hz.

Inpercentageterms,νfisequalto81%ofνr.

Example16

Αmotorcyclewithaweight(includingtherider)of1970Νhasafrontunsprungmassof14kgandarearoneof16kg.Witha50%-50%distributionoftheloads,calculatethefrontandrearreducedstiffnessesinsuchawaythatthefrontfrequencyisequalto1.9Hzandthatoftherear2.3Hz.

Themassespressingdownonthesuspensionsare:

Theoverallfrontandrearstiffnessesare:

Thereducedstiffnessesofthesuspensions,takingintoaccountthetirestiffnesses,havethevalues:

InFig.5-38thein-planemodesofvibrationofthemotorcyclearegiven,calculatedwithamathematicalmodelthattakesintoaccountthemotorcyclegeometry.Themeaningofthedampingratiowillbeexplainedinthenextsection.

Fig.5-38Modesofvibrationintheplane.

5.9Onedegree offreedommono-suspens ion

5.9.1Osci l latorymotionimposedbyroadirregulari ties

ConsideramotorcyclerunningonasinusoidalprofileroadatconstantvelocityVandsupposethatthemotorcyclecanberepresentedbytwoseparatemono-suspensions.Supposefurthermorethattheunsprungmassesarenegligible.Themodelofthemono-suspensionwithonedegreeoffreedomcan

representeitherthefrontorrearsuspensions.

Fig.5-39Suspensionexcitedbytheroadplane.

Considerthefrontsuspensionaffectedbythemotioncausedbyroadirregularities,representedinFig.5-39.Thecontactpointofthewheelwiththeroadprofilemovesinharmonicmotionaccordingtothelaw:

y=yosin2πνt

ν=V/Lrepresentsthefrequencyofthemotionimposedonthesystembytheroadirregularities:

Itcaneasilybedemonstratedthatinsteadystate(periodicresponse)theratiobetweentheamplitudeZooftheverticalmotionofthesprungmassandtheamplitudeyooftheimposedmotionis:

νnrepresentsthefrequencyofthemono-suspension.TheratioTiscalledtransmissibility.InFig.5-40thetransmissibilitycurvesTforvariousvaluesofthedampingratioζaregiven.Itisusefultorecallthatthedampingratioisgivenbytheequation:

Fromthegraphitisclearthatthetransmissibility,foranyvalueofthedamping,isalwaysequaltooneatthevalue ofthefrequencyratio.Thisvalueappearswhentheforwardvelocitysatisfiestherelationship:

Ifthewavelengthisequalto6mandthenaturalfrequencyisequalto2Hz,thetransmissibilityhasavalueofunitywhentheforwardvelocityisequaltoapproximately17m/s.

Theplotclearlyhighlightsthat:forvaluesofthefrequencyratiolessthan (velocityV<17m/s),theintroductionofsuspensionincreasestheoscillationamplitude(T>1).Therefore,theapplicationofsuspensionisuseful(T<1)onlyforvaluesofthefrequencyratiogreaterthan (velocityV>17m/s);highvaluesofthedampingratioattenuatetheincreaseintransmissibilityforratiosofthefrequencieslessthan (velocityV<17m/s),buttheyworsentheresponsivenessofthesystemathighvelocities(velocityV>17m/s).

Forthestudyofridingcomfort,theverticalaccelerationgraphofthemotorcycleismoreinteresting.Therider(whichinthismodelisassumedtobefixedtothemotorcycleandforcedtomovealongtheverticalaxis),perceivesasensationofcomfort,whichisrelatedtotheaccelerationstowhichhisbodyissubjected.

Fig.5-40Displacementtransmissibilityversusthefrequencyratio.

Figure5-41representsthetransmissibilityofverticalacceleration,asafunctionoftheratioofthe

frequencyforvariousvaluesofthesuspensiondampingratio.

Fig.5-41Transmissibilityofaccelerationsversusthefrequencyratio.

Allthecurvesassumethevalueof2,whenthefrequencyratioisequalto ,butwithdifferentslopes.Theslopeiszerowhenthevalueofthedampingratioisequalto:

Thecurvecharacterizedbysuchaslopeensuresminimumaccelerationsaroundthepointν/νn=2.Theratioζ=0.354representstheoptimalvaluearoundthepointconsidered.Thegraphallowsustodrawthefollowingimportantconclusions:

thesuspensionbehaveslikeafilterthatcutsthehighfrequenciesandamplifiesthosefoundinanarrowbandaroundtheconditionofresonance.asignificantincreaseinridingcomfortcanbeobtainedbyreducingthemotorcycle’snaturalfrequencyvalues.Thisreductioncanbeobtainedbydiminishingsuspensionstiffness,thatisbyusingsoftersprings.Itmustbenoted,however,thatexcessivelysoftspringscancompromisethevehicletrimespeciallyinphasesofrapidaccelerationorsuddenstops.

5.9.2Optimal value ofthe dampingratio

Considerthemono-suspensionillustratedinFig.5-42,andsupposethatthemassisoscillatingfreelyandthat,attheinitialinstant,itpassesthroughthepositionofstaticequilibriumatvelocityZo.

Thelawofharmonicmotion,assumingzerodamping,is:

Themaximumvalueofaccelerationis:

Fig.5-42Mono-suspension.

Inordertoreducemaximumacceleration,itisnecessarytoattachaviscousshockabsorbertothespring;hence,themassoscillatesfreelyafteradisturbanceaccordingtothefollowinglaw:

whereqnindicatesthenaturalfrequencyofthedampedmono-suspension.

Theevaluationofridingcomfortcanbeassociatedwiththemaximumpeakofverticalaccelerationofthesprungmass.Thebestcomfortoccurswhenpeakaccelerationisataminimum.

ThegraphofFig.5-43showsthecourseoftheverticalaccelerationofthesprungmassforseveralvaluesofthedampingratio.Theidealconditiontoprovideacomfortablerideiswithadampingratiovalueof0.35,atwhichtheaccelerationbecomesminimal.

Itisimportanttonotethattheoptimalvalueofthedampingratiocoincideswiththatderivedpreviouslyonthebasisoftheperiodicresponseinsteadystate.

Fig.5-43Accelerationafteraroadbumpforvariousvaluesofthedampingratio.

5.9.3Cons iderations ons ingle anddouble e ffectshockabsorbers

Supposethatthemono-suspensionmovesalongacosinusoidalshapedbumpandassumethatthetimenecessaryforthevehicletotransittheirregularityissmallcomparedtotheinverseofthenaturalfrequencyofthesystem(Fig.5-44).

Theviscousshockabsorberaffectsthemotionofthesprungmassduringthepassageovertheroadirregularity.Weevaluateitsinfluenceinthreecases:

ashockabsorberwithconstantcactinginboththecompressionandextensionphases(doubleeffect);ashockabsorberwithconstant2cactingonlyinthecompressionphase(singleeffect);ashockabsorberwithconstant2cactingonlyintheextensionphase(singleeffect).

Fig.5-44Mono-suspensiontravelingoverabumpandoverastep.

Example17

Consideramono-suspensionwiththefollowingproperties:

•sprungmass: M=140kg;

•stiffness: K=20kN/m;

•dampinginextensionandcompression C=1171Ns/m,ζ=0.35;

•dampingonlyinextension: C=2342Ns/m;

•dampingonlyincompression: C=2342Ns/m.

Determinethenaturalfrequency.

Thenaturalfrequencyνnisequalto1.9Hz.

Passingoverabump

Thebumphasaheightof0.008mandalengthof0.6m.Withaforwardvelocityof15m/s,thebumpispassedoverin0.04s;itisassumedthatthewheelneverdepartsfromtheroadprofile.

Figure5.45showsthecourseofthedisplacementofthesprungmassversusthetime.Itcanbenotedthatadoubleeffectdamperbehavesoptimally,sinceithaslessdisplacementfromthepositionofstaticequilibrium.

Fig.5-45Travelingoverabump:evolutionofthedisplacementofthesprungmass.

Thebehaviorofthedoubleeffectshockabsorbercanbeeasilyexplainedifthetimeemployedindrivingoverthebumpiscomparedwiththenaturalperiodofthesuspension.Sincethenaturalperiod

isofalargerorderofmagnitude,analmostimpulsiveforceonthesprungmassisexertedontheshockabsorber,firstwithapositivesignandthenwithanegativesign,whichproducesoverallequaleffectsofoppositesign.

However,inthecaseofashockabsorberactingonlyinthecompressionorextensionphase,thesprungmassundergoes,respectively,apositiveornegativeimpulse.

Fig.5-46Travelingoverabump:evolutionoftheaccelerationofthesprungmass.

Theaccelerationgraph(Fig.5-46)highlightsanimportantaspect.Theshockabsorberwithdampingonlyinextensionhasthecharacteristicofdrasticallyreducingthepositiveaccelerationofthesprungmass,sothattheriderisnotsubjectedtoannoyingaccelerationsupwardthatcouldthrowhimoffthesaddle.

Passageoverastep

Nowconsiderthemono-suspensionpassingoverastep.Figure5-47highlightsthedifferentbehaviorofthemono-suspensioningoingoverthestepcomparedtothepassageoverabump.Thesuspensionwiththeshockabsorber,whichoperatesonlyinextension,enablesapassageoverthestepwithoutharmfuloscillationsofthesprungmass.

Fig.5-47Passageoverastep:evolutionofthedisplacementofthesprungmass.

Theshockabsorberthatisactiveonlyincompressionisstressed(theforceisproportionaltovelocity)atalevelsuchastocompromiseitsintegrity.

Fig.5-48Passageoverastep:evolutionofaccelerationofthesuspendedmass.

Forthepurposeofridingcomfort,thegraphrepresentingtheaccelerationofthesprungmassagainsttime(Fig.5-48)clearlyshowsthattheshockabsorbermustactprimarilyinextension.

5.10Characteris tics ofshockabsorbersWehaveseenthatthedampingcoefficientincompressionshouldbelowerthanthatinextension,

becausewhenawheelencountersasteporabump,itmustfollowtheprofileoftheobstaclewithoutgeneratingtoomuchopposingforce.Whileifitencountersarutorapothole,itcanjumpoveritwithonlyatemporarylossofwheelcontactwiththeroadplane.

Theshockabsorbercharacteristicsarerepresentedinatypicalgraphthatshowstheforceontheordinateandthedisplacementoftheimposedharmonicmotionontheabscissa,asillustratedinFig.5-49.

Fig.5-49Diagramoftheforcegeneratedbyashockabsorber.

Asthefrequencyoftheimposedmotionincreases,wideclosedcurvesarise,theareaofwhichrepresentstheenergydissipatedbythedamping.Theenergydissipatedbythedouble-effectshockabsorberisproportionaltothefrequencyandtothesquareoftheamplitudeΔoftheimposedharmonicmotion:

ceandccindicatetheconstantdampingcoefficientsrespectivelyinextensionandcompression.

Inadditiontoitsnon-linearity,duetotheasymmetryofthedampingcoefficientinrebound(deformationvelocity>0)andincompression(deformationvelocity<0),theshockabsorberwillhave

adampingcoefficientthatvarieswiththevelocity.Dependingonhowmuchthedampingcoefficientdependsonthevelocity,wecanhaveprogressiveordegressivebehavior.

linearshockabsorber

progressiveshockabsorber

degressiveshockabsorber

Theexponentngivesthedegreeofdependenceofthedampingcoefficientonthevelocity.

Theareaoftheforce-displacementgraphrepresentstheenergydissipated.Itcanbeobservedthatwithequalmaximumforce,thedegressiveshockabsorberdissipatesmoreenergythanthelinearandprogressiveones.Fig.5-50givesacomparisonofthecharacteristicgraphsforthreecases:

lineardamping;progressivedamping;degressivedamping.

Thechoiceoftheshockabsorberanditscalibrationaremadewiththefollowingfeaturesinmind:theoverallenergytobedissipatedinacycle;thedistributionoftheenergytobedissipatedinthetwophasesofextensionandcompression;thevalueoftheprogressiveordegressivepropertyoftheshockabsorber.Asfarasthequantityofenergytobedissipatedisconcerned,wehaveseenthat,forthesakeofacomfortableride,theaverageofthecompressionandextensioncoefficientsmustgenerallyhaveavalueofaround30-35%withrespecttothecriticaldamping.

Ingeneral,thedampinginthecompressionphaseislessthanhalfofthatinextension.Thedistributiondependsonthemotorcycletypeaswellassuspensionstiffness.Someridersprefersomewhatrigidsuspensionwithlittledampingincompression;otherspreferadditionaldampingincompression,andsoftersuspension.

Degressivedampinghastheadvantagethatitdissipatesagreaterquantityofenergyatanequalmaximumforcelevel.ItcanbenotedinFig.5-50thattheareaoftheforce-strokegraphisgreaterinthecaseofdegressivedamping.

Wehaveseenthatthereduceddampingisequaltotheproductoftheactualdampingcoefficientandthesquareofthevelocityratio.Therefore,ifweusealinearshockabsorber(constantcoefficient)andaprogressivesuspension(τm,ycincreasingwiththeamplitudeofthewheel),thereduceddampingisprogressivebecauseofthegeneralprogressivebehaviorofthesuspensionmechanism.

Thechoiceoftheshockabsorber ’sdegreeofdegressiveratemust,therefore,bemadebytakingintoaccountthecontraryeffectgeneratedbyanyprogressiverateofthesuspension.

Fig.5-50Shockabsorberwithdoublelinear,progressiveanddegressivecharacteristics.

5.11The influence ofthe unsprungmassTheonedegreefreedom,mono-suspensionmodelhasallowedustobringtolightsomeinteresting

considerationsregardingthemostappropriatevalueofthedampingratio.However,itmustbehighlightedthatthemodelwithonedegreeoffreedomdisregardstheinfluenceoftheunsprungmass.Wehaveseenthatthehopfrequencyofthesystemcomposedofonlytheunsprungmassandthetireradialspringisintherange12-18Hz.

Nowforexample,letusconsideramono-suspensionwithtwodegreesoffreedom,representingtherearsection,andobservetheinfluenceofthevalueoftheunsprungmassoncomfortandroadadherence.

Supposeamotorcycleproceedsatconstantvelocityalongaroadwithaprofilethatimposesaharmonicmotiononthewheel,asshowninFig.5-51.

Theequationsofthemotionofthesprungmassandunsprungmassare:

whereMrepresentsthesprungmass,mtheunsprungmass,ctheshockabsorberdamping,cpthetiredamping,kthesuspensionspringstiffness,andkpthetireradialstiffness.

Fig.5-51Mono-suspensionwithtwodegreesoffreedom.

Theimposedmotioncanbedescribedas:

whereν=V/Lrepresentsthefrequencyoftheimposedmotion,dependingbothontheforwardvelocityandtheroadprofilewavelength.

Inasteadystateperiodicmotion,themassesoscillatewiththesamefrequencyastheimposedmotion.Thecomplexamplitudesofthesprungandunsprungmassesnormalized,inrelationtotheamplitudeoftheimposedmotion,aregivenbythefollowingequation:

Thedynamicloadonthewheeliscomposedofaconstantcomponent,equaltothestaticload,andofafluctuatingcomponent.Theminimumdynamicloadonthewheel,normalizedinrelationtothestaticload,isgivenby:

Themaximumnormalizeddynamicloadisgivenby:

Example18

Considerasuspensionwiththefollowingcharacteristics:

•sprungmass: M=110kg;

•tirestiffness: kp=130kN/m;

•reducedstiffnessofthesuspension: k=30kN/m;

•unsprungmass: m=15kg.

Determinethenaturalfrequenciesofthesuspension,consideredasasystemwithtwodegreesoffreedom:

1stmode(displacementofthesprungandunsprungmasses,inphase):ν1=2.29Hz;2ndmode(displacementofthesprungandunsprungmasses,inoppositephase):ν2=14.88Hz.

Bywayofexample,comparethesetwovalueswiththefrequenciescalculatedwiththeonedegreefreedommodels:

asystemcomposedofonlysprungmassMandsuspensionstiffnessk:νs=2.63Hz(seeexample15);asystemcomposedofunsprungmassmandtirestiffnesskp:νt=15.92Hz(seeexample11).

Itcanbeobservedthat:thefirstfrequency(ν1=2.29Hz),isnearthenaturalfrequencyofthesprungmass(νs=2.63Hz);thesecondfrequency(ν2=14.88Hz),isnearthevalueofthenaturalfrequencyoftheunsprungmass(νt=15.92Hz).

Nowletusintroducethenon-dimensionalratioofthefrequencies:

where:

νrepresentsthefrequencyoftheimposedmotion;

νsrepresentsthefrequencyofthesystemcomposedofonlysprungmassMandsuspensionstiffnessk.

Fig.5-52Periodicresponseamplitudeofthesprung(solidline)andunsprungmasses(dottedline)versusthefrequencyratio.

Fig.5-53Accelerationofthesprung(solidline)andunsprungmasses(dottedline)versusthefrequencyratio.

Fig.5-52showsthesystem’sresponseasafunctionofthefrequencyratioΩ,inthethreefollowingcases:

rigidsuspension;suspensionwithoutashockabsorber;suspensionwithdampingratioof0.3.

Thecalculatednaturalfrequenciesofthetwovibrationmodesofthesuspension(ν1=2.29Hz,ν2=14.88Hz)correspond,respectively,toΩ=0.87andΩ=5.66.Withoutsuspension,i.e.,withinfinitesuspensionstiffness,thenaturalfrequencydependsonlyonthetirestiffnessandthesumofthemasses:

Itsvalue,withthedataoftheprecedingexercise,isequalto4.5Hz(Ω=1.71).

Thedampingratioiscalculatedinrelationtothecriticaldamping ofthesystemwithonedegreeoffreedom,whichisobtainedbyassumingthattheunsprungmassiszeroandthetirestiffnessinfinite.

Forthepurposesofridingcomfort,theaccelerationgraphscanbeobtainedfromthepreviousonesbymultiplyingthedisplacementbythesquareofthefrequencyratio.Thefollowinginterestingconclusionscanbedrawn:

allthecurvesforthesprungmassdisplacementamplitudepassthroughthepointsA,B,CandD;inthefrequencyrangesA−BandC−Dthemaximumaccelerationofthesprungmassdiminisheswithanincreaseindamping,whileforvaluesofthefrequencyratio,betweenBandC,orhigherthanD,theincreaseindampingcausesanincreaseinaccelerationofthesprungmass;intherangeoflowfrequencies,theoptimalcurveistheonethatgivestheminimumvalueofaccelerationincorrespondencewiththefirstresonance.SincethecurvemustpassthroughpointB,theoptimalcurveistheonethathasitsmaximumatB.Itcanbeshownthatthiscurveisobtainedwithavalueofthedampingratioequaltoaboutζ=0.35.ThisvalueofζalsomakestheaccelerationapproximatelyminimalintherangeC−D;intheintermediatefieldofthefrequencyratioA−BandforvaluesbeyondpointD,forcomfortpurposesitwouldbeappropriatetoadoptlowerdampingvalues,butthatwouldmeananincreaseinthemaximumvaluesofaccelerationsunderresonanceconditions;theaccelerationtowhichtheunsprungmassissubjectedisnotsignificantlyinfluencedbythevalueofthedampingratio.

Fig.5-54Sprungmassaccelerationforvariousvaluesoftheunsprungmassversusthefrequencyratio.

Figure5-54showstheinfluenceoftheunsprungmassontheamplitudeofaccelerationofthesprungmass.Dividingtheunsprungmassinhalf(m=7.5kg),theaccelerationofthesuspendedmassdiminishesatlowfrequencies,butincreasesathighones.Alternatively,indoublingtheunsprungmass(m=30kg),thereisanetaccelerationdecreaseathighfrequenciesandanincreaseatthelowones.

Nowletusanalyzethefluctuationsoftheloadsonthewheels.Asmentionedearlier,theloadiscomposedofaconstanttermtowhichafluctuatingharmoniccomponent,withfrequencyequaltothatoftheimposedmotion,isadded.Itsamplitudenormalizedwithrespecttothestaticloadisgivenby:

Itisclearthatthefluctuatingcomponentworksagainstroad-holding.Furthermore,ifthefluctuatingtermexceedsthestaticload,thetirewilllosecontactwiththeroadplane.

InFig.5-55,thefluctuatingcomponentisplottedagainstthefrequencyratio.Thefluctuatingcomponentisnormalizedinrelationtothestaticloadanddividedbytheamplitudeofthesinusoidalprofileoftheroad.Itcanbeobservedthatthevalueofthedampingratioequalto0.3ensuresgoodgripatlowfrequencies;athighfrequencies,itwouldbeappropriatetohaveagreatervalueofζwhich,however,couldreduceridingcomfortandadherenceintheB-Cfieldofthelowfrequencies.ThelossofadherencehappenswhenΔNaisequaltoone.Forexample,atafrequencyof5Hzandwithadampingratioζ=0.1separationoccursforvaluesoftheprofileheightgreaterthanyo=1/142m.

Thereductionoftheunsprungmassesbenefitsroad-holdingespeciallyatlowfrequencies;moreoverthevalueofthesecondfrequencyofthemono-suspensionincreasesasthevalueoftheunsprungmassdecreasesasshowninFig.5-56.

Fig.5-55Amplitudeofthefluctuatingpartoftheverticalloadonthewheelforvariousvaluesofthedampingratio.

Fig.5-56Amplitudeofthefluctuatingpartoftheverticalloadonthewheelforvariousvaluesoftheunsprungmass.

5.12The rearsuspens ionofthe s cooterThescootercanberegardedasaspecialcaseofthemotorcycleinwhichtheengineisanintegral

partoftheswing-arm.Thisconstructionsolutionfacilitatesthetransmissionofmotionfromtheenginetotherearwheel,butinterfereswithitsdynamicbehaviorintheverticalplaneforthefollowingreasons:

theunsprungrearmassofthevehicle(massofthewheelandpartofthemassoftheengine),withrespecttothesprungmass(chassis+rider)hasasignificantlyhighervaluethanthatofaconventionalmotorcycle;theunbalancedalternatingforces(theenginesofscootersarenotusuallyprovidedwithequilibratingcountershafts)generatedbytheenginearetransmittedtothechassisandarethe

causeofunwantedvibrations,whicharefeltbytheriderinthehandlebars,saddleandfootrest.

5.12.1Cons iderations onthe pos i tionofthe attachmentpointoftheengine

Considerascooterwiththeengineconnecteddirectlytothechassiswithapivot,asshowninFig.5-57.

WesetagoalofseekingthebestpositionfortheattachmentpointPoftheenginewiththechassis,inordertominimizetheforcestransmittedtothechassis,generatedbyroadunevenness.Forthispurpose,wesubstitutethe“engine”withanequivalentsystemfromadynamicpointofview,composedofamomentofinertiaandtwomassesplacedincorrespondence:

totheintersectionpointOofthehorizontallinepassingthroughthecenterofgravitywiththeverticallinepassingthroughthewheelcontactpoint;tothepointPwheretheengineisattachedtothechassis,whosepositionisunknown.ThereducedmassesatthepointsOandPandthemomentofinertiaare:

where:IGrepresentsthemomentofinertiaaroundthecenterofgravity;mindicatesthemassoftheengine;a,barethedistancesindicatedinFig.5-57.

ThemomentofinertiaIoiszerowhenthedistancebsatisfiestheequation:

ThedistancebetweenthetwopointsOandPisthereforeequalto:

InthiscasethepointOisthecenterofpercussioninrelationtopointP.TheverticalforcesactingonOdonotgeneratereactionforcesatP.

Itisinterestingtonotethattheenginesuspendedatthepercussionpointpresentsinterestingbehaviorfromthedynamicpointofview.

Theswingingarm-enginesystem,attachedatPandelasticallysupportedatthepointO,constitutesavibratingsystemwithonedegreeoffreedom.Givenkr,whichisthereducedverticalsuspensionstiffnessatthepointO,thenaturalfrequencyofthesystemis:

Fig.5-57Classicsuspensionofthescooter.

Ifthedistancecisvariedwithaconstant,thenaturalfrequencychanges.ItcanbeshownthatitreachesitsmaximumvaluewhenthedistancecsatisfiesthenecessaryconditionformakingthepointOthecenterofpercussionwithrespecttothepointofattachmentoftheengineP.

Thefrequencyinthiscaseis:

Fig.5-58.Reducedsystem.

Atthemaximumvalueofthefrequencywiththesamereducedstiffness,theequivalentunsprungmasscorrespondstothatoftheminimumfrequency;thusprovidingbenefitsforbothridingcomfortandroad-holding.

Thedirectattachmentoftheenginetothechassishassomedisadvantagesfortheisolationofvibrations.Infact,theenginegeneratesalternatingforcesofimbalance,withfrequencyequaltoanddoublethatoftheengine’srotation,whichvaryinthe50to400Hzrange.Theunbalancingforcesaretransmittedtothechassisthroughtheattachmentpointoftheengineandalsothroughthespring-shockabsorbergroup.Theseforcesgeneratevibrationsthatarefeltbytherideronthehandlebars,footrestandsaddle.Toreducethevibrationstransmittedtothechassisthroughthemounts,thejunctionwiththechassiscanbeaccomplishedelasticallywithasystemconsistingofasimplerockerarmorarockerarmwithalinkrod.

5.12.2Attachmentofthe engine witharockerarm(twodegrees offreedom)

Theengineconnectedtothechassis(assumedlocked),withasimplerockerarmconstitutesasystemwithtwodegreesoffreedom,asshowninFig.5-59.

Thetwovibrationmodesofthesystemhavetheircentersofrotationalignedalongtheaxisoftherockerarm.Theirpositiondependsontheinertialcharacteristicsoftheengine,thetireradialstiffness,thespringstiffnessandthemountstiffnessoftherockerarm.

Forthepurposesofoperatingasasuspension,pointΡ,wheretherockerarmisattached,shouldbethecenterofpercussionwithrespecttothepointO,sothattheverticalforcesgeneratedbytheirregularitiesintheroaddonotstressthemountsoftherockerarm,butareopposedbythespring-shockabsorbergroup.

Sincetheshockabsorberbehavesathighfrequenciesalmostlikeastrut,theenginewiththespring-shockabsorbergroupandwiththerockerarmmakesupafour-barlinkage.Thecenterofrotationoftheengineinrelationtothechassiscaneasilybefound.Itistheintersectionpointoftheaxesoftherockerarmandtheshockabsorber.

Thelargestcomponentoftheunbalancingforceshouldbenormaltotheaxisoftherockerarm.

Thecomponentoftheunbalancingforceparalleltotheaxis,however,iscompletelytransmittedtothechassisandthisisthedisadvantageofsuchasuspension.Forthisreason,therockerarmisjoinedelasticallytothechassisbyanelastomerplug,toreducethetransmissionofvibrations.

Ifthebalancingoftheengineisaccomplishedinsuchawayastohaveanormalcomponentequalto100%ofthealternateunbalancingforce,thistypeofsuspensionassuresanexcellentisolationofthevibrations.Suchadistributionofthealternateforcecangenerateexcessivelyhighloadsonthedriveshaftbearings.

Fig.5-59Rearsuspensionofthescooterwitharockerarm.

Fig.5-60Centersofinstantaneousrotationofthescooterwitharockerarm.

Fromatheoreticalpointofview,foranoptimaldynamicperformance,thissuspensionarrangementshouldhavethefollowingproperties:

thedistributionoftheunbalancingforceequalto:100%normalcomponentinrelationtotherockerarmand0%inthetangentialdirection;thepointofattachmentPoftheenginecoincidingwiththecenterofpercussionwithrespecttoO;thecenterofrotationincorrespondencewiththecontactpoint;thecenterofrotationofonevibrationmodeincorrespondencewithpointP(modeexcitedbyroadirregularity);thecenterofthesecondvibrationmodeincorrespondencewiththecontactpoint(modeexcitedbyunbalancedforce).

5.12.3Rockerarmandl inkrodattachmentofthe engine (threedegrees offreedom)

Theswingingarm-engineassemblyattachedtothechassiswitharockerarmandwithalinkrod,representedinFig.5-61,hasthreedegreesoffreedomandthereforethreemodesofvibration.Theintroductionofanadditionaldegreeoffreedomrespondstotheneedtoisolatebothcomponentsofthealternateunbalancingforce.

Fig.5-61Suspensionofthescooterwithrockerarmandlinkrod.

Thepositionoftherotationcentersofthethreevibratingmodesdependsontheinertialcharacteristicsoftheengine,thestiffnesscharacteristicsofthemounts,thetireandthesuspensionspring.

Theenginebehaviorcanbestudiedbysubstitutingthesystem,madeupoftherockerarm,linkrodandtherelativeelasticplugs,withanappropriatestiffnessmatrixasshowninFig.5-61.Thismatrixisgenerallynotdiagonal:aforceappliedtotheswingingarm-enginesystemalongthexdirectionbringsonadisplacementalongtheydirectionandviceversa.

Inprinciple,thelocationofthecentersofrotationofthethreemodesissetupintheidealdiagram,illustratedinFig.5-62.

Fig.5-62Suspensionofthescooterwithrockerarmandlinkrod.

5.13Roadexcitation

Themotorcycle’sin-planeresponse,excitedbyanunevenroad,isimportantforhumanperception(ridercomfort)andalsofortireadherencewiththeroad.Suchexcitationisrandomandisdescribedbytheroadprofile’sstatisticalproperties.

Fig.5-63Thein-planemotorcycledynamicsystem.

Consideramotorcyclewithaforwardvelocityequalto20m/s.Forlongundulationsexceeding40mtheequilibriumofthemotorcyclemaybeconsideredwithastaticanalysis.Infactthefrequencyoftheexcitations(lessthan0.5Hz)arelowenoughinrelationtothepitchandbouncefrequenciestoconsiderthemotorcycleaquasi-staticsystem.Theexcitationfrequencyrange0.5-25Hzmaybedefinedastheriderange.Inthisrangethedynamicresponsedependsprimarilyonthesuspension.Withaforwardvelocityof20m/sthisfrequencyrangecorrespondstowavelengthsfrom40mto0.8m.Theexcitationsabove25Hzareinthenoisefield.

Toevaluatetherider ’sindexofcomfortitisimportanttoconcentrateonaccelerationsratherthan

displacements:infactthehumanbodyisespeciallysensitivetotheRMSvalueofaccelerations.Discomfortisfeltmoreinarangeoffrequenciesthatliebetween4Hzand8HzasshowninFig.5-64.ThisfigureshowsthelinesofmaximumtolerablelevelsfordifferentdurationsofthevibrationexposureestablishedbytheISO2631Standards[Mechanicalvibrationandshock,Evaluationofhumanexposuretowhole-bodyvibration,InternationalOrganizationforStandardization,1997].

Fig.5-64Discomfortthresholdfordifferentexposuretimes.

Ofcourseaccelerationlimitsarerelatedtothetimeofexposure:highaccelerationsaretoleratedforashortertimewhilelowaccelerationsaretoleratedforalongertime.

5.13.1Powerspectral dens i tyofthe road

TheelevationprofilemeasuredoveralengthofroadcanbedecomposedintoaseriesofsinewavesvaryinginamplitudeandphaserelationshipbymeansofthePowerSpectralDensityfunction.Thisfunctionrepresentstheamplitudedensityversuspathfrequency.Pathfrequencyistheinverseofthewavelengthandisexpressedincycles/mandthusthePSDfunctionoftheroadelevationprofileisexpressedins.

FromexperimentalmeasurementsoftheroadprofilesomelawsregardingthePSDfunctionhavebeenproposed.

AccordingtoISOstandard[ISO2631,ISO5349,ISO,DraftStandardISO/TC108/WG9]thepowerspectraldensityofaroadprofileisdescribedbythefollowingequation:

where:n1=2,n2=1.5areexponents;PSD0istheroughnessmagnitudeparameterthatdependsonthequalityoftheroad(m2/(cycle/m));υ=1/Lw=ν/Visthepathfrequency(cycle/m);υ0isthecutoffpathfrequency(cycle/m).

PSD0valuesliebetween4*10−6and1024*10−6(m2/(cycle/m)).SomeexamplesofPSDfunctionsareplottedasafunctionofpathfrequencyυinFig.5-65.

Fig.5-65Powerspectraldensityofsomeroad[ISO,DraftStandardISO/TC108/WG9].

5.13.2In-plane frequencyresponse function

Thein-planefrontandreartransferfunctionofthemotorcycleistheratiobetweenthebounceorpitchaccelerationamplitudeofthemotorcyclecenterofmassandthedisplacementofthecontactpointofthewheelsversusfrequency.

Figure5.66showsanexampleoftransferfunctions.Itcanbeseenthatthebounceresonanceismorevisibleinrelationtothepitchresonance,whichismoredamped.Alsothewheelhopresonancesareclearlyvisible;itshouldbehighlightedthattheradialdampingofthetiresarenegligibleinrelationtothedampingoftheshockabsorbers.

Fig.5-66Exampleoffrontandreartransferfunction.

5.13.3Motorcycle response

Oncethepowerspectraldensityoftheroadprofileisknownitispossibletocalculatethesprungmassaccelerationspectrumbymultiplyingtheroadspectrumbythesquareofthemotorcycletransferfunction.Thepowerspectraldensityissimply:

where:

PSDsprungmassisthePSDoftheaccelerationofthesprungmass;

PSDroadisthePSDoftheroadinput;

|H(ω)|sprungmassisthemagnitudeofthein-planecomplextransferfunctionofthemotorcycle.

Theisolationpropertiesofthesuspensiongenerateanaccelerationspectrumofthesprungmasswithhighamplitudesatthesprungmassresonances,withamoderateattenuationintherangeofthewheels’resonancesandarapidattenuationthereafter.

ThemotorcycleresponsestotheroadexcitationwithtwodifferentforwardvelocitiesarerepresentedinFig.5-67foranaverageroadprofile.Inthiscasethebounceandpitchmodesofvibrationarecoupled.Thebounceresponseismorepronouncedatlowfrequencieswhilethepitchaccelerationofthesprungmassismorepronouncedatmiddlefrequencies.Thefiguresalsohighlight

thewheelbasefilteringphenomenon(dipsinthecurves).

Wheelbasefilteringiscloselyrelatedtotheratiop/Vbetweenthevehiclewheelbaseandforwardvelocity.Infacttherearwheelseesthesameroadprofileasthefrontwheel,onlywithatimedelaywhichisequaltotheratiobetweenthewheelbaseandtheforwardvelocity.Wheelbasefilteringcausescharacteristiclobes(inbetweenthedips)intheseplots:whenthespeedincreases(astheratiop/Vdecreases)thenumberoflobesdiminishesandonlythepeakcharacteristicsofthesystemresponsetotheroadsurfacePSDremainintheplots.Thefirstfrequencyoftheminimumbounceaccelerationresponseisequaltothevelocitydividedbytwicethewheelbasewhilethepitchaccelerationpresentsthefirstminimumwhentheratiobetweenthevelocityandthewheelbaseisequaltoone.Thecouplingofthebounceandpitchmodemakesthefilteringphenomenonlessclear.Themotorcyclehasa1.5mwheelbase,withaspeedof20m/snullresponse(dip)occursatapproximately6.5Hzforthebounceand11Hzforthepitch.

Fig.5-67Exampleofsprungmassresponse.

Fig.5-68Exampleofverticalloadresponse.

Thesuspensionsarealsoimportantfortireadherence,whichdiminisheswiththeincreaseoftheverticalloadfluctuations.ThePSDoftheverticalloadsinducedonthewheelareshowninFig.5-68.Atlowspeedsthefrontwheelissubjectedtohigherloadvariationscomparedwiththerearwheel,whileasthespeedincreasestherearwheelloadfluctuationsbecomemoreimportantandaredistributedoveralargefrequencyrange.

Theloadfluctuationriseswiththeincreaseoftheunsprungmassinrelationtothesprungmassandwiththeincreaseofthetires’radialstiffness.

MVGrandPrix125ccof1953

6MotorcycleTrim

Inthepreviouschapters,theforcesactingonthemotorcyclewerecalculated:resistanceforces,drivingforceanddynamicloadsonthewheels,indifferentconditionsofbothstationaryandnon-stationarymotion,inaccelerationandinbraking.Inthischapter,variationsinthetrimexhibitedbythemotorcycleundervariousdrivingconditionswillbestudied,andtheimportanceofthechainforcewillbehighlighted.

Thetermvehicletrimimpliesthegeometricconfigurationthatthemotorcycleacquiresindifferentconditionsduringtransientandsteadymotion,inaccelerationandinbraking.

Asshownbelow,themotorcycletrimdependsonthestiffnesscharacteristicsofthefrontandrearsuspensions,ontheforcesoperatingonthemotorcycle,andontheinclinationangleofthechainandtheswingingarm.

6.1Motorcycle trimins teadys tate motionFigure6-1illustratesthesystemofforcesoperatingonthemotorcycleinsteadystateconditions.In

thiscase,thestudyofthetrimisspecificallydesignedtodeterminetheattitudeandconfigurationofthemotorcycle,andinparticular,thatoftherearsuspension.

Fig.6-1Motorcycletriminthethrustphase.

6.1.1Rearsuspens ionequi l ibrium

NowletusconsidertherearswingingarmwithitswheelrepresentedinFig.6-2,assumingtheforwardmotionofthevehicletobeatconstantspeed.Thefollowingforcesareappliedtotherearswingingarmandwheelsystem:

thethrustforceS;theverticaldynamicloadNr;thechainforceT;theelastictorqueM.

Fig.6-2Rearsuspensionbalancewithchaintransmission.

Thebalanceofthemomentsontheswingingarmpivotgivesthefollowingexpression:

Mv=NtrLcosφ−S(Rr+Lsinφ)+T[rc−Lsin(φ−η)]

TheequilibriumequationdoesnotreportthestaticelasticmomentMsexertedbythesuspensionspringandthemomentgeneratedbythestaticverticalloadNsrwhichisequalto−Ms.

Thefourmomentsactingontheswingingarmare:themomentgeneratedbytheloadtransferNtrthatcompressesthesuspension;themomentgeneratedbythedrivingforceSthattendstoextendthesuspension;themomentgeneratedbythechainforceTthatcompressesthesuspension.theadditionalelasticmomentgeneratedbythesuspensionmovementMvthatcanbepositiveornegative.

Thedrivingforceisassumedtobeconstantandisrelatedtothechainforce,exertedbythechainonthewheel,throughtheequation:

Fig.6-3Rearsuspensionbalancewithtransmissionshaft.

Ifthereisnothrust,thechainforceandtheloadtransferarenull,sothemomentexertedthestaticloadbalancestheelasticmoment:

Conversely,ifthereisathrustforce,thetrimoftherearsuspension(armpositionwithrespecttotheframe),dependsonthevaluesofthethreeaforementionedcomponents.Expressingthedrivingforceasafunctionofthechainforce,theequilibriumequation,withrespecttotheswingingarmpivot,can

berearrangedasfollows:

whereMvisthepartoftheelasticmoment,necessarytobalancethemomentsgeneratedbytheloadtransfer,chainforceanddrivingforce.

Ifthetermduetotheloadtransferislargerthanthatduetothechainforceanddrivingforce,thesuspensionisfurthercompressed,withrespecttothedeflectioncausedbyonlythestaticload(Mv>0).Viceversa,ifthecomponentduetothechainforceanddrivingforceprevailsovertheloadtransfercomponent,thenthesuspensionisextended(Mv<0).

Itisinterestingtonotethatthechainforcedisappearsforboththeshaft-drivetransmission,asshowninFig.6-3,andinthecaseofscooters(withengineintegratedontheswingingarm).Therefore,onlytheloadtransferforceandthethrustforceexertamomentontheswingingarm.

Inthiscase,thebalanceofthemomentsresultsin:

Mv=NtrLcosφ−S(Rr+Lsinφ)

6.1.2Incl inationangle ofchain

Inordertostudythebalanceofthemoments,itisnecessarytoexpresstheinclinationangleofchainη,asafunctionoftheangularinclinationφoftheswingingarm.

Fig.6-4Geometryoftherearswingingarmwithchaintransmission.

WithreferencetoFig.6-4,wemaywritethefollowingequation:

b−(rc−rp)cosη=Lcsinη

whererpandrcrepresenttheradiiofthedrivesprocketandtherearsprocket,brepresentstheverticaldistancebetweentheaxesofthesprockets,Lcisthelengthofthestraightlinesectionofthechain.Theequationgivestheinclinationangleasafunctionoftheswingingarmangle.

Forpracticalpurposes,thefollowingapproximateexpression(assumingcosη≅1)issufficient:

whereyPistheverticalcoordinateofthedrivesprocketshaft.

6.1.3Squatratioandsquatangle

Chaintransmission

ConsidertheintersectionpointA,betweentheaxisoftheupperchainbranchandthestraightlinepassingthroughthecenterofthewheelandthroughtheswingingarmpivot,showninFig.6-5.ThestraightlinethatconnectsthepointofcontactbetweentherearwheelPrandpointAiscalledthesquatline.Itsinclinationtothehorizontalplaneiscalledthesquatangleσ.

Fig.6-5Squatandloadtransferlines.

TheresultantFr,fromthesumoftheloadtransferforceandthedrivingforce,appliedtothepointofcontactoftherearwheel,isinclinedwithrespecttohorizontalbyananglenamedtheangleofloadtransferτ.ThelineofactionoftheresultantFriscalledthestraightlineoftheloadtransfer.

Wedefinethesquatratioℜastheratiobetweenthemomentgeneratedbytheloadtransferandthe

momentgeneratedbythesumofthechainforceandthedrivingforce:

Expressingtheloadtransferasafunctionofthedrivingforce,theratioisafunctionofonlythegeometriccharacteristics,andinparticular,itisequaltotheratiobetweenthetangentoftheloadtransferangleandthetangentofthesquatangle:

Theratiovariesaccordingtothevariationoftheswingingarminclinationangleanddependsonthedifferencebetweenthearminclinationangleandthechaininclinationangle.Suchadifferenceissensitivetothepositionoftheaxisofthedrivesprocketinrelationtothepositionoftheswingingarmpivot.

Threecasescanoccur:PointAliesonthestraightlineoftheloadtransfer,thatisσ=τ;inthiscaseℜ=1.Duringthethrustphasetherearenoadditionalmomentsoperatingontheswingingarm,sothesuspensionspringisnolongerstressedcomparedwiththestaticcondition;PointAliesunderthestraightlineoftheloadtransfer,thatisσ<τ;inthiscaseℜ>1:themomentgeneratedbytheresultantFrcausesacompressionofthespringinadditiontotheonecreatedbythestaticload;PointAliesabovethestraightlineoftheloadtransfer,thatisσ>τ;inthiscaseℜ<1.ThemomentgeneratedbytheresultantFrcausestheextensionofthespring.

Transmissionshaftwithuniversal joints

Inthecaseoffinaltransmissionshaftandalsoinscooters,thesquatratiois:

Fig.6-6Swingingarmbalancewithtransmissionshaft.

Theratiocanbeexpressedasafunctionofthegeometriccharacteristicsofthesuspensionandalsoasaratiobetweenthetangentoftheloadtransferangleandthetangentofthesquatangle.InthiscasethesquatlinepassesthroughthepointofcontactandtheswingingarmpivotascanbeseeninFig.6-6:

Ingeneraltheloadtransferangleτwiththetransmissionshaft,issmallerthantheangleσ,thustheratioℜislessthanone,i.e.,thesuspensionisalwaysextendedinthethrustphase.Inordertoobtainratiosclosetounitaryvaluesswingingarmsofgreatlengthshouldbeused.

Four-barsuspensionwithtransmissionshaft

AsshowninFig.6-7,inthiscasetheratioℜcanbeexpressedasafunctionofboththeloadtransferangleτandthesquatangle.Inthiscasethesquatlineisthestraightlinepassingthroughthecenterofrotationwithrespecttotheframe(pointA)ofthesuspensionconnectinglinksandthepointofcontactoftherearwheel.

Fig.6-7Motorcyclebalancewithfour-barrearsuspension.

Example1

Wewillnowmakesomeremarksregardingamotorcyclewiththefollowingproperties:

•motorcyclewheelbase: p=1370mm;

•heightofgravitycenter: h=600mm;

•swingingarmlength: L=590mm;

•rearwheelradius: Rr=317mm;

•rearsprocketradius: rc=111.3mm;

•drivesprocketradiusof: rp=43.2mm;

•sprockethorizontalpositioncomparedwithswingingarmpivot: xP=75mm;

•sprocketverticalpositioncomparedwithswingingarmpivot: yP=0mm.

Letusexaminethemotionoftheswingingarminrelationtotheframe(assumedtobefixed).

Figure6-8depictsthedeviationofsquatratioversusthevariationoftheverticalpositionycoftherearaxlewithrespecttotheframe.Itisworthpointingoutthatinthisexampletheloadtransfertendstocompressthesuspensionspring,whereasthechainforceandthedrivingforcetendtoextendit.

Itispossibletoobservethatinthereferencecase,forthosevaluesregardingthewheelverticalpositionsthatarelowerthanapproximately-65mm,theeffectofthechainforceandthrustprevailsasthevalueofthesquatratioislessthanone(ℜ<l),whereasforthehigherycvalues,theeffectoftheloadtransferforceprevailssincethevalueoftheratioisgreaterthanone(ℜ>1).

Fig.6-8Typicaldeviationofthesquatratioℜasycvaries.

Thefigurealsoillustratesacomparisonamongthevaluesoftheratioℜ,obtainablewhenonlyonesinglegeometricparametervariesatatime:itisinterestingtoobservethatasthegeometryvaries(radiiofthedrivesprocketandoftherearsprocket,positionofthedrivesprocketaxiswithrespecttotheswingingarmpivot)thevaluesalsovary,however,thecurvesstillmaintainanincreasingdeviationastheverticalpositionoftherearwheelaxlevaries.

6.1.4Motorcycle trimas the squatratiovaries

Letusnowcontinuewithafewobservationsonthetrimofthevehicleinmotionataconstantspeedandinthepresenceofathrust(balancedbytheaerodynamicresistantforce).

Withtheincreaseofthechainforceand,thus,ofthedrivingforce,thefrontaxleliftsupbecauseofthefrontwheelloaddecrease,whilethebackoftherearframeliftsuporlowersasafunctionofthesquatratio.

Theverticalextensionofthefrontsuspensionisequaltotheratiobetweentheloadtransferandthereducedverticalstiffness:

AscanbeseeninFig.6-9,inthecaseofaunitaryratio(ℜ=1),asthechainforcevariestheforceoperatingontherearsuspensionspringdoesnotundergoanyvariationastheactionoftheloadtransferisperfectlybalancedbythechainforce.Underthesespecificconditionsvariationsinthe

thrustaffectonlythefrontsuspension.

Fig.6-9Motorcycletrimwithunitarysquatratio.

Iftheratioisgreaterthanone(ℜ>1),therearsuspensionspringwillbecompressedcomparedtotheconditionofbalancewithanullchainforce.Withtheincreaseofthechainforcevalue,thefrontaxleliftsupwhilethebackoftherearframelowersproportionaltothevalueoftheratio.Furthermore,iftheratioincreaseswiththeincreasingyc(suchasthecaseinexample1),withtheincreaseofthechainforceand,therefore,ofthedrivingforce,themovementbecomesmoreappreciableasrepresentedinFig.6-10.

Iftheratioislessthanone(ℜ<1)therearsuspensionspringwillbeextendedcomparedwiththeconditionofbalancewithanullchainforce.Withtheincreaseofthechainforce,boththefrontaxleandtherearaxleareextended,causingthemotorcyclecenterofgravitytoraise,alsoillustratedinFig.6-11.

ThedeformationofthespringΔLr,reducedtotherearwheelaxle(whichbecomesnegativeduringcompression),isprovidedbythefollowing(approximate)expression:

Thepreviousobservationsallowustoconcludethatthetrimofthevehicledependsonthevalueofthesquatratio.

Thevariationinthetrim,causedbythedrivingforce,canalsobeexpressedbythevariationoftheframepitchangle:

Fig.6-10Motorcycletrimwithsquatratiogreaterthanone.

Fig.6-11Motorcycletrimwithsquatratiolessthanone.

Example2

Consideramotorcyclewiththefollowingcharacteristics:

•motorcyclewheelbase: p=1370mm;

•heightofcenterofgravity: h=600mm;

•reducedstiffnessoftherearsuspension: kr=20kN/m;

•reducedstiffnessofthefrontsuspension: kf=13kN/m.

TheaimistodeterminethevariationsinthemotorcycletrimafterapplyingachainforceTequalto4000N,asthesquatratiovaluevaries:

ℜ=0.7;ℜ=1.0;ℜ=1.3.

Theloadtransferisequalto615Νand,therefore,thevariationΔμofthepitchangleis:1.42°,foravalueoftheratioℜ=0.7;1.98°,foravalueoftheratioℜ=1.0;2.27°,foravalueoftheratioℜ=1.3.

Wecanseethat,asthesquatratioℜincreases,thevariationinthetrimalsoincreases.Thisisrepresentedbythevariationoftheframepitchangle.Itisworthpointingoutthatthepositivedirectionofthepitchiscounter-clockwise.

6.2Motorcycle triminacurveAsrepresentedinFig.6-12,themotorcyclethatrunsataconstantspeedfromastraightlinetoa

corneringmotionlowersandalsoslightlypitchesforward.

Fig.6-12Loweringofthemotorcyclecenterofgravityinacurve.

Theloweringiscausedbytheincreaseinloadthatactsinthemotorcycleplanethatincreasesininverseproportiontothecosineoftherollangleϕ.Thepitchforwardisduetothefactthatthereducedstiffnessofthefrontsuspensionislessthanthatoftherearsuspension.

TheloweringofthecenterofgravityΔh,andthevariationofthepitchangleΔμ,inthechangefromastraightlinetoacurve,isprovidedbytheexpressions(Fig.6-13):

anditispossibletonotethat,ifbkr>(p−b)kf,thepitchanglewillbenegative,and,thus,forward.

Fig.6-13Variationsofthemotorcycletriminacurve.

6.2.1Squatratioinacurve

Thesquatratio,inthechangefromastraightlinetoacurve,decreasesbecauseoftheloweringofthecenterofgravity:

Actually,inthechangefromstraightrunningtocornering,theratiodecreaseslessthanthechainforceangleσbecausepointAisalsolowered.

6.2.2Triminenteringacurve

Inordertoevaluatethevariationsofamotorcycletriminthechangefromastraightlinetoacurveweshouldconsidertheexampleshownbelow.

Example3

Consideramotorcyclewiththefollowingcharacteristics:

•motorcyclemass: m=200kg;

•wheelbase: p=1.4m;

•longitudinaldistanceofthecenterofgravity: b=0.6m;

•reducedstiffnessoftherearsuspension: kr=25kN/m;

•reducedstiffnessofthefrontsuspension: kf=13kN/m;

•tirestiffness:

•rollangle: ϕ=45°.

Fig.6-14Motorcycletriminastraightlineandinacurve.

Case1:Straightlinemotion.

Theloweringofthevehiclecenterofgravity,duetoitsweight,andinthehypothesisofnospringpreload,isequalto60mm(loweringof52mmwithrespecttotherearwheeland70mmtothefrontwheel),whiletheclockwisepitchrotationisequalto-0.73°.Obviouslyifthespringsarepreloadedtheloweringwillbeless.Forexampleiftherearsuspensionispreloadedat28mmtheloweringoftherearframeshouldbe24mminsteadof52mm.

Case2:Changefromstraightlinemotiontocorneringmotion.

Inthechangefromastraightlinetoacurvetheloadonthewheelsincreasesgraduallyaccordingtotheincreaseinthemotorcycleangleofinclination.Atarollangleϕthatisequalto45°theloweringΔhofthegravitycenter,causedbytheincreaseofthewheelload,isequalto24mm.

Thevehiclefrontframelowerstoagreaterextentcomparedwiththerearframe(29and22mmrespectively)becausethefrontsuspensionspringissofterthantherearone,andtheresultingforwardpitchrotationΔμisequaltoanadditional-0.3°.

Theaboveexamplesuggestssomeinterestingconsiderationsregardingthemotorcyclepitchmotion,inthechangefromastraightlinetoacurve:

themotorcyclepitchisfurtherincreasedinthebrakingphasewhileenteringthecurve,theforwardpitchrotationcausesadecreaseinthefronttrail,whichpotentiallyhelpsenteringacurve.Infact,thelateralforcemomentthattendstoalignthefrontframedecreases.

6.2.3Triminexi tingacurve

Ifathrustforceisappliedinacurve,andintheexitingphase,anothervariationofthetrimappearsduetoseveralfactorsthatcomeintoplay:

theloadtransferfromthefrontwheeltotherearone,duetothethrust,determinesadecreaseoftheloadoperatingonthefrontwheelandhence,causesapositiverearwardpitchrotation;thechangefromthecorneringmotiontothestraightlinemotiondeterminesadecreaseintheloadoperatingonthewheelsand,therefore,themotorcycle,withthetypicalvaluesofsuspensionstiffness,undergoesapositivecounter-clockwisepitchrotation.Letusnowillustrate,withanexample,anothereventrelatedtothesquatratio.

Example4

Considerthemotorcyclefromthepreviousexample.

Inthecurvebalanceconfiguration,theloweringofthegravitycenterisequaltoΔh=24mm.

Let’snowtakealookattheeffectsofapplyingthethrustforce.Fig.6-15depictsthedeviationofthesquatratio,bothinastraightlineandinacurve(angleofrollϕ=45°)versusvariationintheverticalpositionofthewheelcomparedwiththeframe.

Fig.6-15Squatratioinastraightlineandinacurve.

Itispossibletoobservethatinthebalancepositions,bothinastraightrunningandinacurve,thesquatratiowillbedifferent.

Applyingathrustinacurvecausesavariationinthetrim,inthesamewaynotedforthestraightline:

withℜ=1therearsuspensiondoesnotchangetrim;withℜ1therearsuspensionisextended;withℜ>1therearsuspensioniscompressed.Inthisexample,thesquatratiointhechangefromacurvetoastraightlinedecreasesfrom1.02to0.95.Thisdecrease,therefore,causestherearsuspensionspringtoextend.

Letusnowexaminetheparametersthatcanbemodifiedinordertolimitthevariationsofthemotorcycletriminthethrustphase:

thevariationofthetrimduetothedifferenceinstiffnessbetweenthefrontandrearsuspensionisunavoidablebecauseofthelowerstiffnessofthefrontsuspension;thevariationofthetrimcausedbytheloadtransferisalsoinevitableinthethrustphase;conversely,sinceinthethrustphasethetrimvariesdependingonthevalueofthesquatratio,itispossibletomodifythisparameterbychangingthesuspensiongeometry.

6.3Motorcycle triminacce leratedmotionOneofthemostimportantcharacteristicsofmotorcycledynamicbehaviorintheacceleration

phase,isits,moreorless,easetomountuporwheeliewhenitissubjectedtoahighdrivingforce.

Asidefromtheamountoftorqueprovidedbytheenginethemotorcyclewheeliedependsontherearsuspensioncharacteristicsandthetransmissionsystemthatlinkstheenginetotherearwheel.

Fig.6-16Motorcycleswithdifferenttypologiesofrearsuspensions.

Considerthreemotorcycleswithequalinertialandgeometriccharacteristics,subjecttothesamedrivingforce,butfeaturingdifferentrearsuspensionsandfinaldrivesystems:

motorcyclewithaclassicswingingarmwithchain;motorcyclewithaclassicswingingarmandtransmissionshaft;motorcyclewithafour-barlinkagerearsuspensionandtransmissionshaft.

Ifintheinitialinstant,motorcycleshaveaconstantspeedequalto100km/h,andsuddenlytheenginegeneratesahightorquetransmittedtotherearwheel,thewheelacceleratesandtransfersthethrustforcetothegroundbymeansoflongitudinalslipbetweenthewheelandtheground.

Firstofall,considerthreemotorcycleswithaclassicswingingarmrearsuspensionandwithchaintransmission,butwithdifferentsquatratiovalues:

ℜ=1referenceconfiguration;ℜ=0.7obtainedbymovingthedrivesprocketdown;ℜ=1.3obtainedbymovingthedrivesprocketup.

Highratiovaluescauseacompressionoftherearsuspensionduringthethrustphaseandthusreducesthetendencyforthevehicletomountup,asshowninFig.6-17.Asquatratiolessthanone,whilecausingtherearsuspensiontoextend,facilitatesmountup.Thefiguredemonstratesthat,whereℜ=1andℜ=1.3,thefrontwheelliftsupandreturnstothegroundafteratimeintervalof0.8-1seconds,whereasinthecaseoftheratiobeingℜ=0.7,thevehicletravelswithitsfrontwheelskyward.

Fig.6-17Motorcyclemountupasthesquatratiovaries.

Fig.6-18Motorcyclemountupwithfinalshaftdrive.

Let’snowseehowmotorcyclesfeaturingdifferentrearsuspensionsystemsbehave.Figure6-18showsthatthemotorcyclewithshafttransmissionmountsupeasily,andinthenumericalsimulationrepresentedhere,iftheappliedtorqueisnotcontrolleditreachesthepointwherethevehicleoverturns.Theeasetomountupisproportionaltotheswingingarmlength.Thefour-barlinkagesuspension,inthecaseofshafttransmission,behavesmuchbetter.Thefigureillustratesthattheconfigurationtakenintoconsideration,thathasthecenterofrotationoftheconnectingrodapproximatelyabovethefrontwheel,showsabehaviorwhichissimilartothatofthemotorcyclewithaclassicswingingarmandwithahighsquatratio.TheBMW“Paralever”suspensionisbasedonthiskinematicscheme.

6.4Influence ofrearwhee l s l ippage onthe trimConsideramotorcycleinstraightlinemotion,withaconstantspeed,thatsuddenlylosesitsgripon

anoilyoricyspot.

Instraightrunningathrustforce,equalandoppositetotheresultantoftheresistantforces,isappliedtotherearwheel.Thelossandthesuddenpickupofgripcreateaviolentvariationofthedrivingforcethatimpulsivelyexcitesboththesuspensionandtherearpartofthemotorcycle.The

behaviorofthemotorcyclecanbeunderstoodbyconsideringthiseventasbeingconstitutedoffivedifferentsequentialphases,representedinFig.6-19:

steadystatewiththemotorcycletravelingataconstantspeed,suddenslippageoftherearwheel,rearwheelaccelerationduetopoorgrip,suddenpickupofgrip,dampedtransientoscillation.

Steadystatewiththemotorcycletravelingataconstantspeed.

TheelasticmomentMvdependsontheverticaltransferloadNtr,drivingforceSandchainforceT:

Mv=NtrLcosφ−S(Rr+Lsinφ)+T[rc−Lsin(φ−η)]

Itisimportanttorememberthatifℜisequalto1,themomentexertedbythechainforceT,thedrivingforceSandtheloadtransferNtrbalancethemselvesout,therefore,thevariablecomponentMvisequaltozero(thestaticelasticmoment

MsbalancesthestaticloadNsr).Ifℜ<1,thespringisslightlyextended,converselyifℜ>1,itwillbeslightlycompressed.

Suddenslippageofthedrivingwheel.

Thelossofgripsuddenlynullifiesthedrivingforce,sothattheswingingarmandwheelsystemarenolongerbalanced.

Theswingingarmissubjecttoasuddenangularaccelerationinaclockwisedirection(negativevalue),thatistransmittedthroughthespringandshockabsorberassemblyandeventheframe.Theimpulsive,upwardsaccelerationoftherearpartofthemotorcycletendstothrowthedriverforward.

Rearwheelaccelerationduetopoorgrip.

Therearwheel,subjecttothetorquegeneratedbythedrivingforce,acceleratesand,furthermore,themotorcyclepitchesforwarduntilitnullifiestheloadtransfer.Thisforce,insteadystate,isproportionaltothedrivingforce.Ontheotherhand,intransientconditionsitoccurswithadelaycomparedwiththedrivingforce,asitislinkedtothevehiclepitch.Onlyintheabsenceofsuspension,theloadtransferisperfectlyinphasewiththethrust.

Inthisphase,therearsuspensionisextendedbecauseoftheslowdecreaseintheloadtransferforceand,therefore,therearframetendstoliftupwhilethefrontframelowers.

Suddenpickupofgrip.

Thehighvalueofthelongitudinalslippagecreatesalargeandsuddendrivingforce,theswingingarmissubjecttoasuddenmomentthatcausesanacceleration(whichispositive)inacounter-clockwisedirection(towardsextension)thattransmitsthroughthespringandshockabsorber

assemblytotheframe.

Fig.6-19Rearsuspensiontrimduringrearwheelslippage.

Dampedtransientoscillation.

Αdampedtransientoscillationoftheframepitchoccursandafteracertaintimeintervaltheinitialsteadyconditionisattained.

Figure6-20showsthedeviationofthedrivingforceandtheverticalloadontherearwheel.Aswecansee,theloadtransferforceinthelowgripzoneslowlydecreasesandthedrivingforcebecomesveryhighwhenthepickupofthegripoccurs.

Fig.6-20Deviationoftheverticalloadanddrivingforce.

Figure6-21illustratesthedeviationoftheverticalaccelerationabouttherearaxleandtheframepointtowhichthespringandshockabsorberassemblyisattached.Theupwardaccelerationpeakatthebeginningofthelowgripsection,andthedownwardaccelerationpeakatthegrippickuppointarequitenoticeable.

Fig.6-21Verticalaccelerationoftherearpartofmotorcycle.

Themomentbalancefortheswingingarminsteadyconditions,asrepresentedinFig.6-22:

0=Mv−NtrLcosφ+S(Rr+Lsinφ)−T[rc−Lsin(φ−η)]

allowsustoexpressafewobservationsregardingthechainforceandthedrivingforce.Aswecansee,theelasticmomentbecomesindependentofthechainforceandthedrivingforceifthefollowing

fourconditionsareverified:thechainangleofinclinationηisequaltotheswingingarmangleofinclinationφ;theinclinationangleoftheswingingarmφisequaltotheloadtransferangleτ=h/p;theloadtransferisproportionaltothedrivingforce,asitwasforthestationaryconditionNtr=Sh/p=Stanτ;thewheelangularspeedisconstantinordertohaveadirectproportionalitybetweenthedrivingforceandthechainforceT=SRr/rc.

Fig.6-22Geometricconfigurationoftherearsuspensionthatshouldensure,insteadyconditions,atrimindependentoftheforcesapplied.

Infact,themotionequationcanbesimplifiedinthefollowingway:

Mv=NtrLcosφ−SLsinφ

ThevariableelasticmomentumMvisnullandtherefore,asthechainforceandthenthedrivingforcevarytheswingingarmremainsinequlibrium.

Twonewarchitectureshavebeenproposed[Romevaetal.,1993]thatmeettheconditionsnecessarytokeepthesquatratioconstantandunitaryundersteadyconditions.Thefunctionalschemesofthetwosolutions,respectivelynamedthe“Bilever”and“Tracklever”systems,areshowninFig.6-23.

Fig.6-23Rearsuspensions“Βileνer”and“Tracklever”.

Unfortunately,aswehavealreadyseen,intransientconditions,theloadtransfer(functionofthedrivingforce)occurswithacertaindelaycomparedwiththedrivingforceand,therefore,theseinnovativesuspensionschemesdonotpresentanyadvantagesovertheclassicswingingarmsuspension.Infact,Fig.6-24showsthateveniftheη=φconditionismetinthetransienteventbothinthechangefromdrytowetgroundandviceversa,thedrivingforce,theloadtransferandthechainforcemomentsontheswingingarmarenotinequilibrium.Thisbehavioristhuslikethatofatraditionalswingingarmsuspension.

Fig.6-24Forcesoperatingonthesuspensionduringslippageandgrippickup.

6.4.1Rearsuspens ionwiththe pinionattachedtothe swingingarm

Anothervalidsolutiondesignedtoimprovethesuspensiondynamicbehaviorinthepresenceofthechainforceisrepresentedbytheintroductionofasecondsprocketontheswingingarm(ATKSystem:AntiTensionChainSystem).

Intheoriginalversion,thechaininclinationangleisequaltotheangleoftheswingingarm.Thesecondsprocket,integraltotheswingitself,ispositionedsothatthelineofactionofthereactionforceexertedbythepinionontheswingingarmwouldpassthroughtheswingingarmpivotwithoutcausinganyadditionalmoment.

Letusconsiderthemoregeneralcasewhichpresentsaconstantvaluebetweenthechaininclinationangleandtheswingingarminclinationangle(Fig.6-25):

ν=φ−η

Ifthereactionforcegeneratedbythesecondsprocketontheswingingarmdoesnotgenerateamoment,duetothefactthatthelineofactionpassesthroughtheswingingarmpivot,theequilibriumbetweenthemomentsactingonthewheel-swingingarmsysteminrelationtotheswingingarmaxle,willbe:

Letusanalyzetheaboveequilibriumequation:themomentcausedbytheloadtransfercompressesthesuspension;thedrivingforcecompressesthesuspensionifφ<0,orextendsitifφ>0;themomentgeneratedbythechainforceextendsthesuspensionifν>0,orcompressesitifν<0.

Inordertoobtainvaluesofthechainratio,veryclosetotheunit,thechainmustbemoreinclinedinrelationtotheswingingarm,thatis:ν=(φ−η)>0.

ThecurvesinFig.6-26showthegrowingtrendofthechainforceratioastheverticalwheeltravelycbecomeshigher.

Fig.6-25Suspensionwithasecondsprocketattachedtotheswingingarm.

Fig.6-26Chainforceratiovaryingwiththeycdisplacementforseveralvaluesoftheνangle.

Thecurveshapecanbemodifiedbyvaryingthesprocketpositioninrelationtotheswingingarmpivot.Todothis,placingthesecondpinioninvariouspositionsontheswingingarm(seeFig.6-27),itispossibletotakeadvantageofthereactionforcethusgeneratingamomentwhichwillextendorcompressthesuspension.

Fig.6-27Influenceofthesprocketpositiononthechainforceratio.

6.5The brakingactionTheloadtransferduringthebrakingactionisdirectlyproportionaltothetotalforceofthebraking

action,totheheightofthemasscenter,andinverselyproportionaltothewheelbase(seeFig.6-28):

whereFindicatesthesumofthefrontandrearbrakingaction.

Fig.6-28Motorcycleequilibriumduringbraking.

Thebrakingactiongeneratesapitchingmotioninthemotorcycle,especiallyatthebeginning,whenthebrakingforceissuddenlyapplied.Duringtheresidualtimeofthebrakingaction,assumedtobeuniform,thefrontandrearsuspensionstakeadifferenttrimdependingonthetypeofsuspension.

Inthefollowingparagraphtheeffectsofthebrakingforceonthefrontandrearsuspensionsareexamined.

6.5.1The frontsuspens ion

Duringbrakingthefrontsuspensionissubject,inadditiontothestaticverticalload,totwoadditionalforces:

thefrontbrakingforceFf,theloadtransferNtrgeneratedbythetotalbrakingforceF.

Thesetwoforcesdefinetheloadtransferangleofthefrontwheelτf,i.e.,theinclinationangleofthelineofactionoftheresultingforceactingonthefrontwheelinrelationtotheroad(Fig.6-29).

Fig.6-29Forcesactingonthefrontsuspension.

Thete lescopicforks

Inthecaseoftelescopicforks,thecontactpointtrajectoryofthefrontwheelinrelationtotheframecanbepresumedtobestraightandparalleltothesteeringaxis.Duringthebrakingaction,theforkiscompressed,duetotheloadtransfercomponentNtrandtheeffectgeneratedbythebrakingforcecomponentFf,asshowninFig.6-30.Themagnitudeofthecompressiondependsprimarilyontheforkinclinationangleγ=π/2−ε(trajectoryangle).

Fig.6-30Forcescompressingthesuspension.

Letussupposethatthebrakingforceappliedtotherearwheeliszero.Inthiscase,thenormalizedcompressionforceonthefork,expressedbytheratiobetweenthesumofthecompressing

componentsonthesuspensionandthefrontbrakingforce,dependsonlyonthesteeringinclinationangleandontheloadtransferangle:

Thedivebehaviorofthesuspensionismaximumwhentheloadtransferangleτfcorrespondstotheforkinclinationangleγ.

Fig.6-31Normalizedcompressionforceonthetelescopicfork[h/p=0.5].

Figure6-31showsthatthedivedisplacement,proportionaltothecompressionforce,isatitsmaximumwhentheforkhasaninclinationangleofabout63°.Wecanobservethatundernormalusage,thedivebehaviorismorepronouncedastheinclinationangleoftheforkbecomeshigher.

Anothertypicaleffectofthetelescopicforkisthedecreaseinthetrailasforkcompressionincreases.

Fig.6-32Trailvarianceduringbrakingaction(telescopicfork).

Let’ssupposethatthewheelshavethesameradiusandthatwecanignoretherearsuspensiondeformation.Itiseasytodetermineaformula,whichwouldindicatethetrailvariation,asaconsequenceofthefrontsuspensioncompression.Thesteeringinclinationangleduringbrakingis,therefore:

whereΔs=s−sorepresentsthefrontsuspensioncompression.

Thetrailduringbrakingisexpressedbythefollowingformula:

Fig.6-33showsanexampleofthevariationoftrailandsteeringinclinationdirectlyrelatedtothevariationofthefrontsuspensioncompression.

Fig.6-33Trailandcasterangleversusthefrontsuspensioncompression.[p=1.4m,a=0.116m,ε=27°,Rr=Rf=0.36m]

Neutral suspens ionandanti -dive suspens ion

Ifthebrakingbehaviorofthefrontsuspensionhastobeneutral,thatis,withoutanyeffectfromthebrakingforce,thetrajectorydescribedbythecontactpointshouldbeaverticalline(seeFig.6-34).Inthiscasecompressionofthesuspensioniscausedonlybytheloadtransfer.

Fig.6-34Neutralfrontsuspensionandanti-divesuspension.

Verticalaxlepathsduringbrakingareeasilyobtainedbyapplyingfour-barlinkagesuspensionswiththewheelattachedtotheconnectinglinkofthefour-baritselforwithasimpleleadingortrailingarm.

Ifthetrajectoryofthewheelisnormalinrelationtotheforcegeneratedbythesumoftheloadtransferandthebrakingforce(Fig.6-34),thefrontsuspensionspringwillnotbestressedduring

braking.Thisfunctionaldiagramisalsoobtainedbyapplyingafour-barlinkagesuspension,orwiththesuspensionpush-arm.

Suspensionsystems,whichdiminishthesuspensioncompressionduetotheloadtransfer,arecalledanti-divesuspensions.

Four-barl inkage suspens ion

The“Telelever ’suspension(seeFig.6-35),appliedasweallknowbyBMW,isgeneratedbythefour-barlinkage.Themechanismisinfactaspatialmechanismwithtwodegreesoffreedom:

the1stdegreeoffreedom,givenbytherotationofthetwocomponentsconnectedtotheprismaticcouplearoundtheaxisthatpassesthroughthecenterofthetwosphericalcouples.Itcorrespondstothesteeringaction.the2nddegreeoffreedom,whichimpliesthemotionofallthecomponentsofthemechanism,correspondstothefrontsuspensionmovement.

Fromakinematicpointofview,thebehaviorshowninthisdiagramisquitedifferentfromthatofatraditionalsuspensionwithtelescopicfork.Oneoftheadvantagesofthisschemeisthatitallowsacertaindegreeofanti-divesuspension.

Fig.6-35Teleleverfrontsuspensionbasedonaspatialmechanism.

Fig.6-36Varianceofwheelbase,trailandinclinationangleversusthesuspensionmovement.

ThediagramsofFig.6-36showthevarianceofthewheelbase,thetrailandthesteeringheadangle,fortwomotorcycles,geometricallysimilarinstaticequilibriumthoughequippedwithdifferentfrontsuspensions.Forthetwosuspensionswecanobserveadifferent,almostoppositebehavior:withthetraditionalsuspensiontheincreaseofthesuspensioncompressioncausesadecreaseinthewheelbase,thetrailandthecasterangle,whereaswiththeTeleleversuspension,thewheelbasestaysalmostconstantandtheothertwoparametersaugment.

Push-armsuspens ion

Ifthepivottrajectoryofthefrontwheeliscircular(asforexamplewiththe‘Earles’fork)theloadtransfercompressesthesuspension,whilethebrakingforceextendsit,asshowninFig.6-37.

Fig.6-37Frontpush-armsuspensionandfour-barsuspension.

Ifweapplyonlythefrontbrake(Fr=0),theequationofthemomentsaroundtherotationcenterchangesto:

Thesuspensionisthereforeeithercompressedorextended,dependingonitsgeometricalcharacteristics,andinparticulardependingonthedistancescandb.Ifthemomentispositive,thesuspensioniscompressed.Ifthemomentisnegative,thesuspensionisextended.

Ifthebrakecaliperisnotapartofthearm,butconnectedtotheframewitharod,thepreviousformulaisstilltrue.Inthiscasethevaluescandbwillrepresentthedistancesmeasuredalongthexandyaxesbetweenthewheelaxleandtheintersectionpointofthestraightlines(seeFig.6-37).

6.5.2The rearsuspens ion

Letusconsidertherearsuspensionandsupposethatthechaindoesnottransferanyforce(duringbrakingthelowersectionofthechainisslack).

ThedirectionofthesumoftherearbrakingforceFr,withtheloadtransferNtr,maybemoreorlessinclinedinrelationtothestraightlinewhichconnectsthecontactpointwiththeswingingarmpivot,dependingonthevaluesofthetwocorrespondingforces.

Fig.6-38Swingingarmsuspensionduringbraking.

Theangleoftheloadtransferduringrearbrakingτrcorrespondstotheinclinationofthestraightlineoftheforceactingontherearwheelinrelationtotheroadplane(seeFig.6-38).

IftheloadtransferNtrishigh,duetothebrakingactionofboththefrontandrearbrakes(Ff≠0)andiftherearbrakingforceFrislow,theresultingforcewillgenerateamomentthatwilltendtoextendtherearsuspension(Fig.6-38a).Inthiscase,theloadtransferangleislargerthantheγangle.Thisangleisformedbythestraightlinewhichconnectsthecontactpointwiththeswingingarmpivotandthexaxis.

Onthecontrary,ifweapplyonlytherearbrake,andthereforetherearbrakingforceFrhasahighvalue,theangleγcouldbelargerthantheloadtransferangle.Inthiscase,therearsuspensionwillbecompressed(Fig.6-38b).

Intheabovediagram,wehavesupposedthatthebrakecaliperisconnectedtotheswingingarm.Thismeansthattheforcesexchangedbythebrakingelementswillreactwithintherearswingingarm-wheelassembly.

Fig.6-39Four-barlinkagerearsuspensionduringbraking.

Thebrakinganglewillchangeifthesupportofthebrakecaliperisfreetorotatearoundthewheelaxleandisfixedtotheframebymeansofarod.Theswingingarm,theconnectingrod,thesupportandtheframeformafour-barlinkage.Theintersectingpointoftheswingingarmaxisandtherodcorrespondstotheinstantaneouscenterofrotationoftheconnectingrodinrelationtotheframe.Thewheel-swingingarmequilibrium,inrelationtothispoint,showsthatthesuspensioniscompressedorextendeddependingonitsposition,asshowninFig.6-39.

MotorcycleDucati125cc.of1956

7MotorcycleVibration

ModesandStabili tyAseveryoneknows,thefrontand/orrearendofamotorcycleinmotioncanstarttooscillate

aroundthesteeringaxis,evenifthewheelsarewellbalanced.Thisphenomenoniseasytoobserveexperimentally,forexamplebygraduallyslowingdownthemotorcyclefromafairlyhighspeed.Oscillationscanbeobservedatcertainspeedsespeciallyifthefrontwheelisoutofbalance.Theyreachtheirmaximumamplitudeandthendecreaseasspeeddecreasesuntiltheydisappearcompletely.Rear-endoscillationscanbeobservedwhentravelingoveratransversebumporbyexcitingtherearframewithanimpulsivemovementoftherider ’strunk.Atalowspeeditcanalsobeeasilyobservedthatthemotorcycletendstofalloversideways,regardlessofwhattheriderdoes.Theseexperimentalobservationsofmotorcycledynamicsshowthattherearethreemajormodes:

capsize,anon-oscillatingmodeusedandcontrolledbytherider;weave,anoscillationoftheentiremotorcycle,butmainlytherearend;wobble,anoscillationofthefrontendaroundthesteeringaxiswhichdoesnotinvolvetherearendinanysignificantway.

Therider ’scontroltaskcanbeconsideredtoinvolveeitherfixedcontrolorfreecontrol,i.e.withorwithouttheirhandsgraspingthehandlebar,respectively.Withthesteeringrotationfixedthemotorcycle-ridersystemisunstableinrollatallspeeds,likeacapsizingship,whereasintheunconstrainedconditionthesteeringsystemisfreetosteeritself,potentiallyrelievingtherideroftheneedtoapplysteeringcontrolactionforstabilization.Aviablemotorcycleneedstoself-steereffectively,thuscontributingtoautomaticstabilization,withoutbecomingtoooscillatoryunder

certainrunningconditions.

Atverylowspeedsamotorcycleisunstablebecauseofcapsize,twonon-oscillatingmodesinvolvingrespectivelyrollandsteermotionofthevehicle.Around1m/stheserealpolesmeet,coalesceandbecomethecomplexpolepairassociatedwiththeweavevibrationmode.Weavemodeisusuallyunstableupto7-8m/s.Overthisspeedmotorcyclesusuallyenterintoastablezone,suchthattheridermayremovehishandsfromthehandlebarwithoutfalling.Asspeedincreases,theweave,wobble,orcapsizemodesmaybecomeunstable,dependingonthemotorcyclecharacteristics,andtheriderhastocounteractthesemodeswithatorqueappliedatthehandlebar.Weavemodeisusuallypoorlydampedorunstableathighspeeds,whereaswobbleispoorlydampedorunstableinthemid-rangeofspeeds.Capsizeinstabilityatmidtohighspeedsistypicallynotverysignificant.

Inthischapter,wewillfirststudythesemodesusingsimplifiedmodelsandlaterthein-planeandout-of-planemodeswillbestudiedbymeansofanelevendegreeoffreedommodel.Finally,theeffectofframecomplianceandridermobilityonmotorcyclestabilitywillbepresented.

7.1Simpli fiedmodel

7.1.1Caps ize

Thismodeisdeeplyinfluencedbyrideractiononthehandlebar,i.e.bythemechanicalimpedance(inertia,stiffness,damping)whichtheriderprovides.Thereforethemodeeasilyshiftsfromtheunstablezonetothestablezone.

Capsizeisamodeactuallyusedbytheridertorollthemotorcycle.Thisrollingactionisachievedthroughtherider ’sefforttoholdormovethesteeringheadrotationtosomenon-equilibriumposition(fixed-control).Asmentioned,thismodeisalwaysunstable,sinceinessencemotorcycleexistsasaninvertedpendulum.Thismodeisalsopresentinfree-controlcondition(i.e.withouttherider ’shandsonthehandlebar)ashighlightedbyeigenvalueanalysis.Inthelattercasethemodeisusuallysomewhatstableatlowspeedsbecauseofthesteeringmodalcomponentofitseigenvector;whereasathigherspeedsitmaybecomeslightlyunstable.

Thecapsizemodeconsistsmainlyofarollmotioncombinedwithalateraldisplacementplussomelessimportantsteeringandyawmovements(Fig.7.1).Itdependsonanumberoffactors:

speedofthemotorcycle;wheelinertia(gyroscopiceffect);positionofthecenterofgravity;motorcyclemass;motorcyclerollinertia;casterangle;mechanicaltrail;propertiesofthetires,primarilycrosssectionalsizeofthetires,twistingtorqueandpneumatictrailofthefronttire.

Fig.7-1Capsize.

Tohighlighttheinfluenceofsomegeometricalandinertialpropertiesofthemotorcycleonthecapsizemodeitisusefultoanalyzethefallmotionofamotorcyclewiththesteeringheadlocked.

Inthishypotheticalcase,withinthelimitsoflinearapproximation,capsizecanbeexpressedasanexponentiallaw:

whereτisapositivetimeconstant,therefore,thecapsizeisalwaysunstable.

Basicallythetimeconstantisameasureofhoweasilythemotorcyclewillleanover.Forexample,racingmotorcyclesneedasmalltimeconstantsotheycancornerandchangetrajectoryquickly.Touringmotorcyclesneedtorollmoreslowly,makingthemeasierfortheridertocontrol.

Capsizeinstabilityshouldnotbeviewedasadrawback,however,sinceitispreciselythisphenomenonthat,givenpropercontrolaction,enablesthemotorcycletoleanintoandexecutecurvescorrectly.Thesmallerthetimeconstant(i.e.,thegreaterthecapsizeinstability),thelessleadtimeisneededtostartleaningthemotorcycleintoacurve.

Thesimplifiedmodels,withthesteeringheadlockedandnegligiblegyroscopiceffects,yieldsmallertimeconstantvaluesthantheoneswhichcanbeobtainedbystudyingthecompletemodelofthemotorcycle.Thesimplifiedmodelsclearlyshowhowgeometricandinertialpropertiesaffectthe

capsizetimeconstant.

Modelusingthindiskwheels

Mathematicalmodelingofmotorcyclecapsizeiscomplicatedbythepresenceofthesteeringhead,gyroscopiceffects,andtirecontactforcesarisingfromtheslipandcamberangles.Herewewillconcentrateonaverysimplemodeltounderstandaspecificaspectofcapsize,thefallingtime.

Fig.7-2Capsizeforamotorcyclewiththindiskwheels.

Thissimplifiedmodelmakesthefollowingassumptions:themotorcycleismovingindirectionxatspeedV;thethicknessofthecrosssectionofthetiresisnull;thereisnoslippagebetweenthetiresandtheroad;thesteeringheadislockedinplace;gyroscopiceffectsarenegligible.

Basedontheseassumptions,capsizeisasimplerotationofthemotorcyclearoundtheaxisdefinedbythepointsinwhichthetirescomeintocontactwiththeroadway(Fig.7-2).

Theequilibriumofmomentswithrespecttothecontactpointgivesthefollowingequation:

Linearizingtheequationaroundtheverticalequilibriumposition:

andintroducingthesolution:

ϕ=ϕoest

yieldsthefollowingfrequencyequation:

Itssolutionisarealnumberandthereforecorrespondstoanon-oscillatingmotion:

Thetimeconstantofinterestτisgivenbytheinverseofthepositiverealeigenvalue:

Notethatthetimeconstantisdeterminedbytheheightofthecenterofgravity,themassofthemotorcycle,andthemotorcycle’smomentofinertiaaboutthex-axisthroughitscenterofmass.

Usingtheradiusofgyrationρ(IxG=mρ2)toexpressthemotorcycle’smomentofinertia,thetimeconstantτtakesonthefollowingform:

Nowletusassumethattheheightofthecenterofgravityandthemassofthemotorcycleareconstant,butthatthemasscanbedistributeddifferentlytovarythemomentofinertia ,i.e.theradiusofgyrationρ.

Fig.7-3Normalizedtimeconstantforcapsizeasafunctionoftheratioofgyrationradiustocenterofgravityheight.

Fig.7-4Timeconstantforcapsizeasafunctionofradiusofgyrationandheightofcenterofgravity.

Thevalueofthetimeconstantτincreasesastheradiusofgyrationρincreases(masslocatedfurtherawayfromthecenterofgravity).Figure7-3showsthenormalizedtimeconstantcurvewithrespecttotheminimumvaluefortheidealcaseofallthemassbeingconcentratedatthecenterofgravity

Figure7-4isacontourplotthatshowshowthetimeconstantvariesasafunctionoftheheightofthecenterofgravityandtheradiusofgyration.

Notethat,foragivenradiusofgyrationvalue,thetimeconstantdecreasesastheheightofthecenterofgravityincreasesuntilitreachesaminimumvalue,andthenincreases.Thismeansthatoncetheradiusofgyrationρisset,thetimeconstantisatitslowestvaluewhentheheightofthecenterofgravityisequaltotheradiusofgyration,ascanreadilybeshownanalytically.

Example1

Basedonthefollowingdata,determinethecapsizetimeconstant.

•totalmassofmotorcycle: m=248kg

•heightofmasscenter: h=0.648m

•momentofinertiaaboutthex-axisthroughitscenterofmass:

(radiusofgyrationρ=0.284meters)

Theresultingtimeconstantis:τ=0.281s.Intheidealcaseofallthemassbeingconcentratedatthecenterofgravity( )thetimeconstantdecreasesto0.257s.

Modelusingtireswithcircularcrosssection

Αsecondsimplifiedmodelcanbebuiltfromthefirst,byremovingtheassumptionthatthewheelsarethindisks,andinsteadassumingamotorcyclewithcirculartirecrosssectionswhichdonotsliplaterallyontheroadwayduringcapsize(Fig.7-5).Onceagain,thesystemhasonlyonedegreeoffreedom.

Theresultingequationsareasfollows:

wherethethirdequationistheequilibriumofmomentswithrespecttothemasscenter.

Underpureforwardrollingconditionsthesystemstillhasjustonedegreeoffreedom,andthereforey,yGandzGcanbeexpressedasafunctionofϕ:

y=ϕt

yG=ϕt+hosinϕ

zG=−t−hocosϕ

Simplemathematicalsubstitutionyieldsthefollowingfrequencyequation:

[IxG+m(ho+t)2]s2−mgho=0]

Theresultingtimeconstantisgivenby:

Thisequationshowsthatthetimeconstantincreaseswiththeradiusofthetirecrosssection.Therefore,enteringacurve,amotorcyclewithlargetirestakeslongertoleanthanonewithsmalltires.

Fig.7-5Capsizewithpurelateralrollingonthetire.

Fig.7-6Capsizetimeconstantasafunctionoftheratiobetweenthetirecrosssectionradiusandtheheightofcenterofgravity.

Figure7-6showstheratiobetweenthetimeconstantforamotorcyclewithtiresofnon-zerothicknessandthetimeconstantforthesamemotorcyclewiththindisktiresasafunctionoftheratiobetweenthetirecrosssectionradiusandtheheightofthecenterofgravity.

Example2

UsingthemotorcycleinExample1butreplacingeachthindisktire(t=0)withatirehavingcrosssectionradius(t=0.10m).Determinethetimechangeinthecapsizetimeconstant.

Thetimeconstantincreasesfromτ=0.281stoτ=0.305s.

Modelusingtireswithlateralsideslip

Thismodelisthesameasthepreviousone,exceptthatthepurelateralrollingconditiononthetire-to-roadhasbeenremoved.Inotherwords,thetirescanslipsidewaysontheroadwaywhenthemotorcycleleansover.Thissimplifiedmodelhastwodegreesoffreedom:

rotationϕaroundthex-axis(leaningthemotorcycle);displacementindirectiony(thecontactpointsideslips).

Themotorcycleequilibriumequationsthatassumepureforwardrollingmotionarevalidevenwhenthetireshowssideslip.

Fig.7-7Capsizewithtireslippage.

Inthiscase,however,thelateralforceexertedonthetiresbytheroadwayisdefinedbyboththesideslipangleandthecamberangle.Thislateralforcecanbedescribedbylinearlawasafunctionofthetiressideslipandcamberangles:

F=(kλλ+kϕϕ)mg

wherethesideslipangleisgivenbyλ=−ẏ/V

Thelinearizationaroundtheverticalequilibriumpositionyields:

Manipulationoftheequationstoeliminatetimedependenceyieldsthefollowingcharacteristicequation:

Itcanbeshownthatjustoneofthefourroots(eigenvalues)ofthecharacteristicpolynomialhasapositiverealpart.Αsecondrootiszero,andtheothertwohavenegativerealparts.

Thepositiverootrelatestocapsizeinstability,whereasthetwonegativeonesrepresenttwostablemotions.Rememberthatthetimeconstantforcapsizeisgivenbytheinverseoftherealeigenvalue.InthismodelthetimeconstantdependsonthespeedofforwardmotionV.

Example3

AssumethatthemotorcycleinExample2hastireswiththefollowingstiffnessvalues:

•corneringstiffnesscoefficient: kλ=11.0rad−1;

•camberstiffnesscoefficient: kϕ=0.93rad−1;

Determinetheeffectonthecapsizetimeconstantasafunctionofspeed.

Figure7-8showsthetimeconstantforcapsizeasafunctionofspeed.Thetimeconstantincreaseswithspeed.Thecamberstiffnesskϕhasatowingeffectoncapsize(increasingthetimeconstant)bygeneratingasortofelasticresistingtorquewhichactsagainstrollkϕϕNt.Withoutthecambercomponenttherewouldbemoresideslipandthecapsizetimewouldbeshorterasaresult.

Fig.7-8Timeconstantforcapsizeasafunctionofspeed.

Figure7-8alsoshowstheborderlinecaseofzerocamberstiffnessinwhichthetimeconstantdecreasesasspeedincreases.Thisisduetothefactthattomaintainthesideslipangleorratherthesideslipforceconstant,thelateralvelocityincreasesasspeedVincreases.Thus,higherspeedcontributestothelateraldisplacementofthecontactpoint,therebydecreasingthetimeconstant.

Example4

Intheborderlinecaseofzerostiffnessvaluesforbothkλandkϕ(thatis,asiftheroadsurfacewereasheetofice)determinethetimeconstantforthemotorcycledescribedinExample3.

Inthiscasethereisnoforcecounteractingtiresideslipandthetimeconstantdecreasestoitslimitvalueof:

whichisobtainedbyputtingkλ=0andkϕ=0intothecharacteristicequation.

Substitutingnumericalvaluesyieldsτ=0.122s,whichistheminimumvaluethatcanbeobtainedwiththesteeringheadlockedinplacewithouttakinggyroscopiceffectsintoconsideration.

7.1.2Wobble

Wobbleisanoscillationofthefrontassemblyaroundthesteeringaxisthatcanbecomeunstableatfairlylowtomiddlespeeds(Fig.7-9).

Fig.7-9Wobblemode.

Wobbleoscillationsresembleshimmyofacar ’sfrontwheelsorairplanelandinggear.Typicalfrequencyvaluesrangefrom4Hzforheavymotorcyclesto10Hzforlightweightmotorcycles.

Wobblefrequencygoesupastrailincreasesandfront-frameinertiadecreases,andisdeterminedmainlybythestiffnessanddampingofthefronttire,althoughthelateralflexibilityofthefrontforkalsoplaysapart.

Intheforwardspeedrangefrom10to20m/s(40to80km/h),wobbleisonlyslightlydampedandcanthereforebecomeunstable.Addingasteeringdamperincreasesthedampingeffectand,consequently,thestability.

Modelofwobblewithonedegreeoffreedom

Wobblecanfirstbethoughtofincompleteisolationfromrear-assemblymotionandroll.Thus,the

front-endisarigidbodythatcanrotatearoundthesteeringaxiswhiletherearframeisfixed(Fig.7-10).

Fig.7-10Frontassemblygeometry.

Theequilibriumequationaroundthesteeringaxisleadstothefollowingrelationship:

where:isthefrontassemblymomentofinertia(includingthefrontwheel)aroundthe

steeringaxis;cisthedampingcoefficientofthesteeringdamper;isthelateralforceactingonthetire.

Thislasttermisassumedproportionaltothesideslipangleλaccordingtothefollowingequation:

withnullvalueoftherelaxationlength.Forsmalldisplacementsthefollowingequationcanbeused

tocalculatethesideslipangle:

Thesideslipangleisthereforethesumoftwocomponents:thefirstdependsonthelateralspeedofthecontactpointduetosteeringvelocity ;thesecondonthesteeringanglemeasuredattheroadsurface.

Theeffectsduetothefronttirenormalloadandfrontframeweightforcearesignificantlysmallerthanthatduetotirelateralforce.So,makingthepropersubstitutionsthemotionequationforsmalloscillationsbecomes:

Introducinganoscillatingsolutionintotheequationandeliminatingtime-dependencegivesthefollowingcharacteristicequation:

whichyieldsthefollowingroots:

Thesystemisoscillatingwhenthediscriminantisnegative,i.e.,neglectingthedampingc,forforwardspeedsgreaterthan:

Thefrequencyforthedampedsystemνis:

Thedampingratioζisgivenby:

Notethatζdecreasesastheforwardspeedofthemotorcycleincreases.

Figures7-11and7-12showhowthenaturalfrequencyanddampingratio,respectively,ofthewobblemodevarywithspeed.Specifically,Fig.7-12showshowasteeringdamperaffectsthedampingratioofthewobblemode,especiallyathighspeeds.

Fig.7-11Naturalfrequencyofwobbleasafunctionofspeed.

Fig.7-12Dampingratioforwobbleasafunctionofspeed.

7.1.3Weave

Weaveisanoscillationoftheentiremotorcycle,butmainlytherearend,asshowninFig.7-13.

Fig.7-13Weave.

Thenaturalfrequencyofthisside-to-sidemotioniszerowhentheforwardspeedisalsozeroandrangesfrom0to4Hzathighspeed.Weaveisdeterminedbymanyfactors:

positionofthecenterofgravityoftherearassembly(andsecondarilythatofthefrontassembly);wheelinertia;casterangle;trail;corneringstiffnessofthereartire.

Thedampingofweavecanbeshowntodecreaseasspeedincreases.

Weaveisusuallyunstableatalowspeed(upto7-8m/s).Itisgenerallystableinthemiddlespeedrange,butitmaybeuncontrollablefromthepracticalstandpointathighspeedsinceitsdampingmaydecreasesubstantiallyanditsnaturalfrequencymaybetoohighfortheridertocontrol..

Theweavemodeisgeneratedbythecoalescence,atverylowspeeds,oftwounstablenon-oscillatingmodes:body-capsizeandsteering-capsize(Fig.7-14).

Bodycapsize

Body-capsizeindicates–capsizeoftheentiremotorcycle,andcanbeestimatedwithoneofthethreesimplemodelspresentedinsection7.1.1.Thetimeconstantwiththesteeringfreedecreasesslightly.Infact,supposethemachinebeginstofalltotherider ’sright.Inbodycapsizemode,steering

geometrycausesthemachinetosteerlefttherebymovingthefrontwheelcontactpointtowardstherider ’sright.Consequently,thegroundcontactlinethatjoinsthefrontandrearwheelgroundcontactpointsrotatestotherider ’sright.Thismeansthegravitationaltorqueincreasesandsothevehiclecapsizeslessquickly.

Fig.7-14Weavecoalescence.

Steeringcapsize

Steering-capsizeisacapsizeofthesteeringhead,duetothedisaligningeffectofboththefronttirenormalloadandfrontframeweightforce.Onceagainconsiderthesimplifiedsituationinwhichtherearframeisfixedwhereasthefrontframeisfreetosteer.Theequilibriumequation,neglectingthesteeringdamperandthetirelateralforce,whichdoesnotactatthebeginningofmotion,becauseofitslag,yieldsthefollowingrelationship:

Therootsare:

Thetimeconstantofthesteeringcapsizehasvaluesintherange0.1-0.2sforspeedlessthen1m/s.

Modelofweavewithonedegreeoffreedom

Weavecanfirstbeseenasanoscillationoftherearendaroundthesteeringheadaxisalmostindependentoffront-assemblymotionandrollmotion.Thismodelofweavewithonedegreeoffreedomisbasedonthefollowingassumptions:

therollvalueforthemotorcycleisnull;thesteeringheadislockedinplace.

Thisassumptionarisesfromtheobservationofrealweaveoscillations,inwhichlateraldisplacementofthesteeringheadaxisissubstantiallylowerthanthelateraldisplacementofthereartire.

Fig.7-15Weave.

Themotionequationisobtainedbyimposingequilibriumonrotationaroundthesteeringheadaxis.Therelationshipobtainedisonceagainthesamerelationshippresentedintheprevioussteering-capsizesection,wherenormalreartraillsubstitutesnormaltrailan.Theeffectsduetothefronttirenormalloadandfrontframeweightforcearesignificantlysmallerthanthatduetotirelateralforce,soweassumethefollowingsimplifiedexpressionofsteeringequilibrium:

where(Fig.7-15):istherearassemblymomentofinertia(includingtherearwheel)aroundthe

steeringaxis;cisthedampingconstantforthesteeringheadrotation;isthelateralforceactingonthetire.

Thislasttermisassumedproportionaltotherearsideslipangleλaccordingtothefollowingequation:

withnullvalueoftherelaxationlength.Forsmalldisplacementsthefollowingequationcanbeusedtocalculatethesideslipangle:

Thesideslipangleisthereforethesumoftwocomponents:thefirstresultsfromthelateralspeedofthecontactpointduetoyawvelocity ;thesecondfromtherearframeyawanglemeasuredattheroadsurface.

Makingthepropersubstitutionsthemotionequationforsmalloscillationsbecomes:

Introducinganoscillatingsolutionintotheequationandeliminatingtime-dependencegivesthefollowingcharacteristicequation:

whichyieldsthefollowingroots:

Thesystemisoscillatingwhenthediscriminantisnegative,i.e.,forforwardspeedsgreaterthan:

Inthiscase,thefrequencyforthedampedsystemνandthedampingratioζaregivenby:

Thus,wecandrawthefollowingconclusions:thedampingratioζfallsrapidlyandasspeedincreasesitapproachesalimitvalue:

thedampednaturalfrequencyνdecreasesasinertiaincreasesandincreaseswithspeedandtirestiffness;withnegligibledampingc,increasingthespeeditapproachesthevalue:

thislastequationconfirmsthattheweavemodefrequencyincreaseswithtirestiffnessandthelengthofthewheelbase,butdecreasesasthecasterangleandinertiaofthemotorcycleincrease.

Example5

Letusconsideramotorcyclewiththefollowingdata:

•casterangle ε=27°

•rearframeinertiaaboutsteeringaxis

•distancebetweenrearcontactpointandsteeringaxis l=1.38m

•distancebetweenmasscenterandsteeringaxis l1=0.67m

•steeringdamper c=6.8Nm/rad/s

•tirecorneringstiffnes

Wewillinvestigatehowthecharacteristicsoftheweavemodechangewithspeed.Figures7-16and7-17showhowthenaturalfrequencyanddampingratio,respectively,oftheweavemodevarywithspeed.

Theweavefrequencytendstotheundampednaturalfrequencyvalueasthevelocityincreases.Thesimplifiedmodelshowsthatthesteeringdamperincreasestheweavedampingratio.Inthiscasethesimplifiedmathematicalresultsarenottruebecauseinrealitythesteeringdampinghasanegativeeffectonweavestability.

Fig.7-16Naturalfrequencyofweaveasafunctionofforwardspeed.

Fig.7-17Dampingratioforweaveasafunctionofforwardspeed.

7.1.4Combinedmodel forweave andwobble

Uptothispoint,wehavelookedatweaveandwobbleinisolationassumingthatthesteeringheaddoesnotdisplacelaterallyineithervibrationmode.Nowwewilluseasystemwiththreedegreesoffreedomtorepresentthemotorcycle.Lookingatthemotorcycleinthedirectionofthesteeringaxis,itiseasytoseethatthethreedegreesoffreedomaretheabsoluterotationsofthefrontassemblyθf,

andrearassemblyθraroundthesteeringaxis,andthelateraldisplacementyofthesteeringaxis(Fig.7-18).

Theresultingequationsofmotionareasfollows:

where[M]ismassand[Κ]isstiffnessmatrix:

[C]isthedampingmatrix:

Fig.7-18Front-endandrear-endgeometry.

Figures7-19and7-20confirmthat:thetwomodesofvibrationaresubstantiallyindependentofeachother;therearenosignificantdifferencesbetweenthemodelwithonedegreeoffreedomandtheonewiththreedegreesoffreedomwithrespecttothefrequencyvaluesordampingratios.

Fig.7-19Naturalfrequenciesofweaveandwobbleasafunctionofspeed.

Fig.7-20Dampingratiosforweaveandwobbleasafunctionofspeed.

7.2Multi -bodyModel

7.2.1Introduction

Themotorcyclehastwokindsofmodes:thein-planemodes,whichinvolveframe,suspensionandwheelmotionintheverticalplane,theout-of-planemodes,whichinvolveroll,yaw,steeringangleandsteeringheadlateraldisplacement.

Thein-planemodesarerelatedtoridecomfortandroad-holding,whereastheout-of-planemodesarerelatedtovehiclestabilityandhandling.Instraightrunning,in-planeandout-of-planemodesareuncoupledandtheycanbeexaminedseparately.

Theeigenvaluesofthein-planeandout-of-planemodesarecomplex:

s=sr+isi

Thenaturalfrequencyoftheoscillatingmodescorrespondstotheimaginarypartoftheeigenvalue:

Therealpartoftheeigenvaluesgivesinformationonthedampingofthemodes.Inthecaseoftheout-of-planemodes,therealpartoftheeigenvaluesprovidesinformationonthestabilityofthemotorcycle.Themotionisunstableiftherealpartispositive,anddampediftherealpartisnegative.

Thedampingratioisgivenby:

Iftheimaginarypartiszerothemodeisanon-oscillatingoneandthetimeevolutionofthegenericmodalcomponentcanberepresentedbyadecreasingorincreasingexponentiallaw:

whereτisthetimeconstant,positiveforunstablemodesandnegativeforthestableones.

Thepropertiesoftheeigenvaluesareclearlyrepresentedintherootlocusgraph,asshowninFig.7-21.Thepositionoftheeigenvaluesinthecomplexplanerepresentsdifferentcases.Theeigenvalueslocatedontheright,inrelationtotheverticalaxis,representunstablemodeswhiletheeigenvalueslocatedontheleftarestablemodes.Toclarifythemeaningoftheeigenvalues,thetimeevolutionlawscorrespondingtodifferenteigenvaluesareshowninthesamefigure.

Fig.7-21Rootlocusandexamplesoftimeevolutionlawsfordifferenteigenvalues.

Thefirstout-of-planemotorcycleequations,linearizedaroundtheverticalequilibriumposition,weredevelopedbyR.S.Sharpinaninfluentialarticle[Sharp,1971].

Sharp’smodelofthemotorcyclehasfourdegreesoffreedom(Fig.7-22):rear-frameroll;rear-frameyaw;rotationofthefrontframearoundthesteeringaxis;lateraldisplacementoftherearframe.

Thetire-to-roadcontactforcesaretakenaslinearfunctionsofthesideslipandcamberangles.Thelagintheforceswithregardstothesideslipanglesisintroducedinthemodelusingtwofirst-orderdifferentialequations.

Fig.7-22Motorcyclemodelwithfourdegreesoffreedom.

Table7-1.

Fig.7.23Motorcyclediagram.

Figure7-24showsthefrequenciesandrealpartsofvibrationmodesasafunctionoftheforwardvelocity,derivedwithSharp’smodel.

Fig.7-24Frequenciesandrealpartsofvibrationmodesasafunctionofspeed.

Thewobblecoversthefrequencyrange8.5to9.6Hz;thespeedofthemotorcyclehaslittleeffectonwobblefrequency.

Themaximumrearwobblefrequencyisapproximately6.5Hzattheminimumvelocity;therearwobblefrequencydropsoffmarkedlyatspeedsof18m/sandaboveandceasestovibrate.

Themaximumweavefrequencyisapproximately3.6Hzatthemaximumvelocity.Infacttheweavemodefrequencyincreaseswithspeed.

TheFig.7-24alsoshowshowtherealpartoftheeigenvaluesvarieswithspeed.Notethat:weavemodeisunstableatlowspeed,dampedinthemediumspeedrange,andweaklydampedathighspeed;rearwobblemodebycontrastisstronglydampedandbecomesovercriticalaboveacertainspeed;frontwobbleisdampedinthemediumandlowspeedranges,butbecomesalittleunstableathighspeed.Infact,theoscillationsassociatedwithwobblearethemostdangeroussincetheirhighfrequencymakesthemhardfortheridertocontrol.

Aboveapproximately18m/stherearwobblebecomessupercriticalandsplitsintotwomodes.Thefirstoneisverystableandcharacterizedbynearlyconstantdamping,whilethesecondoneischaracterizedbyadecreasingrealpartwhichgraduallybecomesmorestable.

Figure7-24showsthattherealpartofthecapsizeeigenvalueisalwaysnegative,itisverystableatlowvelocityandtendstobecomeborderlineunstableathighvelocity.Stabilityatlowvelocityisin

contrastwiththeexperimentalevidence.However,itisworthrememberingthatthefirstSharpmodelisbasedonwheelswithzerothickness;sincethecapsizemodedependsonthelocationofthefrontcontactpoint,theevaluationofthecapsizemodeneedsamoreaccuratetiremodel.

Thefrequenciesanddampingsoftheout-of-planemodesinstraightrunningarequitewellvalidatedbyexperimentresults.Incornering,thefrequenciesandthedampingratios,ofbothin-planeandout-of-planemodes,varywithrespecttothestraightrunningcondition.Furthermore,thetwotypesofmodesarecoupledtoeachother.

7.2.2Motorcycle multi -bodymodel

Amorecomplexmathematicalmodelisneededtogiveamoreaccuratedescriptionofmotorcyclebehaviorthanthecurrentone.ForthisreasonthefirstSharpmodelcanbedevelopedfurther,byaddingthesuspension’sdegreesoffreedomanddescribingthetirepropertiesaccurately.

Themulti-bodymodelofthemotorcyclefromwhichtheremarksincludedinthischapterariseiscomposedofsixrigidbodies,respectively:(formoreinformationsee[CossalterandLot,2002]):

therearframe(whichincludeschassis,engine,tank,rider,partoftherearsuspensionandpartofswingingarm),thefrontframe(whichincludeshandlebars,sprungforkcomponentsandthesteeringhead),therearunsprungmass(whichincludespartoftheswingingarmandtherearbrakecaliper),thefrontunsprungmass(whichmainlyincludespartoftheforksandthefrontbrakecalipers),therearwheelandthefrontwheel.

Thissetofbodieshaselevendegreesoffreedominall,ascanbeseeninFig.7-25,theyare:thepositionoftherearframecenterofmass(threecoordinates);theorientationoftherearframegivenbypitch,rollandyawangles;thetravelofbothsuspensions(thefrontoneistelescopic,therearoneisaswinging-type);thespinanglesofthewheels;thesteeringangle.

Thedynamicsofthismulti-bodysystemisverycomplexbecauseofthelargenumberofbodiesandforces.Theaerodynamiceffectsarereducedtoonlythreeforces(drag,lift,andlateral)whichactonthecenterofpressureoftherearframe.Thebraketorquesareappliedontherespectivewheelaxes,whereasthepropulsiveforceistransmittedfromtheenginetotherearwheelbyachaintransmission.Thesteeringtorqueactsbetweenthefrontandrearframealongthesteeringaxis.

Fig.7-25Motorcyclemodelwithelevendegreesoffreedom.

Thetiremodelaccuratelydescribesthegeometryofthetreadwhichisfundamentalinevaluatingthetirebehavioratlargecamberangles.Thecarcassisthoughtofaselasticallydeformablealongradial,lateralandtangentialdirections.Thecontactforces(seealsochapter2)areappliedonapointwhosepositionisdefinedbytheequilibriumbetweentheexternalforces(duetothecamberangle,thelongitudinalslipandthesideslip)andtheinternalelasticreactions.Thispointrepresentsthecenterofthecontactpatch.Thecontactyawtorqueisappliedonthesamepointandactsaroundtheverticalaxis,whereastherollingresistanceactsinthewheelplaneandtendstoslowthewheel’srotation.

7.2.3Modes ofvibrationins traightrunning

Fig.7-26showstherootlocusplotofbothin-planeandout-of-planemodesforasportmotorcycleinstraightrunning.Thedirectionofthearrowsshowstheincreaseofthevelocityfrom3to60m/s.Thegreylinesstartingfromtheoriginareconstantdampingratioloci.

Fig.7-26Root-locusplotinstraightrunningatdifferentspeeds,(speedfrom3to60m/s).

Thein-planemodes,pitchandbounce,arealsodiscussedinchapter5.Inthecaseplottedherethepitchandbouncearecoupledtogether,thepitchmodeinvolvesanotnegligibleverticaldisplacementofthemotorcyclemasscenterwhilethebouncemodeinvolvesanotnegligiblepitchrotation.Generallythepitchmodeismoredampedthanthebouncemode.Infact,duetothedampingselectedfortheshockabsorbers,thebouncemodehasadampingratioofabout0.3-0.5whereasthepitchmodegenerallyhasadampingratioofabout0.9andcanalsobeoverdamped.Thefrequencyofthebouncemodeisintherange1.4-2.0Hz;thepitchfrequencyincreasesfrom2Hzto2.9Hzat60m/s.

Thefronthopmode(around10-11Hz)andtherearhopmode(around13-14Hz),respectively,ofthefrontandrearunsprungmassarecharacterizedbynegligiblemotionofthesprungmass.

Moreover,Fig.7-26highlightsthevaluesandthedependenceoftherealandimaginarypartsofthemainoutofplainmodesonthevelocity:

capsize;wobble;weave;rearwobble.

Thewobbleandweavemodesarevibratingmodesintheentirerangeofvelocityconsideredherewhiletherearwobblebecomesovercriticalwhenvelocityincreases.Thecapsizemodeisanon-

vibratingmodethatisquiteunstableintheexampleconsidered.Thewobblemodebecomesunstablewhenincreasingthevelocityduetothefactthatinthismodelthesteeringdamperwassettozero.Figure7-27showsthefrequenciesandthedampingpropertiesofthemainout-of-planemodesasafunctionoftheforwardvelocity.

Fig.7-27Frequenciesandrealpartsofvibrationmodesasafunctionofspeed.

Thespeedofthemotorcyclehaslittleeffectonwobblefrequency;themaximumwobblefrequencyisapproximately11Hzattheminimumvelocityandafter15m/sremainsconstantandequalto7.8Hz.Thewobblemodeisdampedinthelowandmediumspeedrangesbutbecomesunstableathighspeed.

Therearwobbleceasestovibrateatavelocityofabout14m/s;itisastronglydampedmodethatbecomesovercritical.

Theweavemodefrequencyincreaseswithspeed;itsmaximumvalueisapproximately3.0Hzatthemaximumvelocity.Theweavemodeiswelldampedinthemediumspeedrange,andweaklydampedathighspeed.

Thecapsizeeigenvalueisalwayspositive.Itisunstableatlowvelocityandtendstobecomelessunstableathighvelocityduetothegyroscopiceffects.

Inthefollowingsectionsdetailsoftheout-of-planemodesinstraightrunningarepresented.

Capsize

Simplifiedmodelswithlockedsteeringshowthatcapsizeisalwaysunstable.Actuallythecapsizemodemaybeunstableormoderatelystabledependingonvelocityandonmotorcycleandtireproperties.

Adetailedanalysisshowsthatcapsizebecomesmoreunstablebyreducingthemechanicaltrailofthemotorcycleandincreasingthecasterangle.Asfarastiresareconcerned,wecansummarizethatonlythefrontonehasappreciableinfluenceduetothepresenceofthesteeringmechanism.Anincreaseofthecrosssectionsize(sectionradius)reducesthestability,butdoesnotmakethemodeunstable.Onthecontrary,theyawtorqueparameterscanalterthesignoftherealpartoftheeigenvalue:twistingtorquetendstostabilizewhereastheself-aligningtorquetendstodestabilizethecapsize.Figure7-28showsthetimeevolutionofastablecapsizeatlowvelocity,whereasFig.7-29showsthecaseofanunstablecapsizeatthesamevelocity.Wecanobservethatinthestablecase,rollandsteeringmotionhaveoppositephases.

Fig.7-28Exampleofamotorcyclewithstablecapsize(speed4m/s).

Figure7-30highlightsthetimeevolutionofastablecapsizeathighvelocity.Fromthesethreefiguresonecanobservethatatthelowervelocityof4m/sthesteering,roll,andyawvariationsaremuchmorepronouncedthananyvariationinlateraldisplacement.Converselyat30m/sthelateraldisplacementisthedominatemodalcontentwithmuchsmallervariationsintheothersignals.Thisisduetothewheelgyroscopiceffectsthatbecomeimportantathighvelocity.Therefore,thecorneringmaneuverathighspeedinvolvesotherout-of-planemodestoagreaterextentthanthecapsizemode.

Fig.7-29Exampleofamotorcyclewithunstablecapsize(speed4m/s).

Fig.7-30Capsizeataspeedof30m/s.

Figure7-31showshowtherealpartoftheeigenvalueofthecapsizemodechangeswhenthevaluesofseveralgeometricandinertialparametersofthemotorcycleandseveraltirepropertiesarechanged.Ineachcaseonlyoneparameterwaschangedby10%andalltheotherparametersweremaintainedconstant.Inrealityitwouldbedifficult,inseveralcases,tovaryjustoneparameteratatime.Forexample,theincreaseintheradiusofthefrontwheel,causesvariationsinthetrail,thecasterangle,andotherparametersatthesametime.

Thefigureshowsthatthestabilityofthecapsizemodecanbeimprovedinthefollowingways:-decreasing:

thecasterangle;thecrosssectionradiusofthefronttire;theheightofthemotorcyclemasscenter;thefrontwheelspininertia;thetrailofthefronttire;thecamberstiffnessofthefronttire.

-increasing:thetwistingtorqueofthefronttire;themechanicaltrail;thedistancebetweenthemotorcyclemasscenterandtherearwheelaxis;thecrosssectionradiusofthereartire;themotorcyclerollinertia;thefrontwheelradius;thecamberstiffnessofthereartire.

Thefigurealsoshowsthattheinfluenceofalltheparametersdecreaseswhenthespeedincreases;infactasthespeedincreasesthegyroscopiceffectsbecomedominant.

Fig.7-31Predictedchangeincapsizemodeat10m/s(lightgray)at30m/ss(gray),at60m/s(darkgray)forparameterincreased10%withrespecttothereferencecase.

Wobble

AsshowninFig.7-32,wobbleischaracterizedbyrotationsofthefront-framewhiletherearframeisonlyslightlyaffected.Thelateraldisplacementofthemotorcycleandtheyawandrolloscillationsaresubstantiallysmallerthanthesteeringoscillations.

Figure7-33showssensitivitiesofthewobblemodedampingtothevariationinthemotorcycleandtireparameters.Thefigureshowsthatthestabilityofthewobblemodecanbeimprovedinthefollowingways:-increasing:

thelateralstiffnessofthecarcassofthefronttire;

thevalueofthesteeringdamper;thefrontwheelradius;

-decreasing:thedistancebetweenthemotorcyclemasscenterandtherearwheelaxis;thecorneringstiffnessofthefronttire;thefrontwheelspininertia.

Fig.7-32Wobbleataspeedof30m/s.

Theneedtoincreasethelateralstiffnessand,atthesametime,todecreasethecorneringstiffnessofthefronttiremeansthattherelaxationlengthofthefronttiresideslipforceshouldbeshortertoobtainstrongimprovementsinstability.

Notethatanincreasein:theheightofthemotorcyclemasscenter;thecasterangle;therearwheelradius;

givesadvantagesatlowvelocityanddisadvantagesathighvelocity.

Theincreaseoftherollinertiaofthemotorcycleandofthemechanicaltrailgivesanoppositebehavior,i.e.,givesadvantagesathighvelocityanddisadvantagesatlowvelocity.

Notethatincreasingthedampingconstantofthesteeringdamperisaveryeasywaytoincreasethedampingratioforthewobblemode,althoughithastheundesiredeffectofslightlyreducingtheweavemodedamping.

Fig.7-33Predictedchangeinwobbledampingat10m/s(lightgray)at30m/ss(gray),at60m/s(darkgray)forparameterincreased10%withrespecttothereferencecase.

Weave

Figure7-34showstheweavemodewiththemotorcyclemovingataforwardspeedof10m/s.

Notethatthismodeischaracterizedbysizableoscillationsinroll,yawandsteeringangle.Therotationofthesteeringheadisoppositeinphasewithrespecttotheyawoscillation,whichisinturn90°outofphasewiththerolloscillation.Thelateraldisplacementofthemotorcyclelagsbehindroll.

Asspeedincreases,thelateraldisplacementbecomesmorepronounced,asdoestherotationofthesteeringheadwithrespecttotherollandyawoscillations(seeFig.7-35).

Fig.7-34Weavemode(V=10m/s).

Fig.7-35Weavemode(V=30m/s).

Figure7-36showshowthedampingratiofortheweavemodevarieswhenthevaluesofseveralparametersofthemotorcyclearechanged.Notethattheincreaseofthedampingconstantofthesteeringdamperhastheundesiredeffectofslightlyreducingdampingoftheweavemode.

Thefigureshowsthatforthemotorcycleusedinthisexampletheweavemodestabilityathighvelocitycanbeimprovedinthefollowingways:-increasing:

thedistancebetweenthemotorcyclemasscenterandtherearwheelaxis;thecasterangle;thefrontwheelspininertia;thelateralstiffnessofthecarcassofthereartire;

-decreasing:

thefrontwheelradius;themotorcycleyawinertia.

Fig.7-36Predictedchangeinweavedampingat10m/s(lightgray)at30m/ss(gray),at60m/s(darkgray)forparameterincreased10%withrespecttothereferencecase.

Theincreasesintheheightofthemotorcyclemasscenterandoftherearwheelradiusgiveadisadvantageatlowvelocityandanadvantageathighvelocity.Onthecontrary,theincreaseofrollinertiaisadvantageousatlowvelocityandprovesadisadvantageathighvelocity.

Theincreaseofthemechanicaltrailgivesaslightadvantageatmediumvelocitiesbutadisadvantageathighvelocity.Anincreaseinthecorneringstiffnessofthereartireproducessimilarbehavior.

Itisworthpointingoutthattheincreaseinfrontwheelspininertiaisstabilizingduetotheincrease

ofthegyroscopicphenomenon.Onthecontrary,theincreaseinthefrontwheelradius(keepingfrontwheelspininertiaconstant)isdestabilizingbecauseitcausesareductioninfrontwheelspinangularvelocityand,hence,areductioningyroscopiceffects.

Inthesimulationtheriderisassumedtoberigidandattachedtothemainframe.Effectivelytherideractslikeadamperthatimprovesbothwobbleandweavestability.Forexample,apillionpassengerimprovesstabilitywhereasthesamerigidmassattachedinthesamepositioncausesadecreaseofthestability.

Rearwobble

Rearwobbleischaracterizedbyrollandyawfluctuationsthatarealmostinphasewitheachother,andinphaseoppositiontothelateraldisplacementofthemotorcycleandthesteeringangle(Fig.7-37).

Fig.7-37Rearwobble(V=10m/s).

7.2.4Modes ofvibrationincornering

Instraightrunningthein-planeandtheout-of-planemodesareuncoupled.Sincetherollangleisnull,thetireverticalloadslieinthemotorcyclesymmetryplanetowhichthetirelateralforcesareorthogonal.Forthisreasonthenormalloadsonlyexcitethein-planemodes,whereasthelateralforcesonlyactontheout-of-planevibration.

Whenthemotorcyclecorners,itisinclinedbysomerollangleandeachofthepreviousforceshascomponentsinbothdirections.Itshouldnowbeclearthatinsomemodesbothin-planeandout-of-planedegreesoffreedomareinvolved.Thisphenomenoniscalled“modalcoupling”.

InFig.7-38therootlocusplotincorneringatconstantspeedispresented.Inthesamefigure,forcomparison,therootlocusinstraightrunningisplottedingrey.

Forthemotorcycleconsideredherethemaindifferencesbetweenthelociare:therearhopmovesalittletowardsamorestableregion;thefronthopdoes-notpresentsignificantmodifications;

thereisaninteractionbetweenthebounceandtheweavemodesatmediumspeed;therearetwopitchmodesthatdifferinphasebetweenthepitchmotionandtheyawandsteeringmotions;thecapsizemodebecomesmoreunstable.thefrequencyofthewobblemodeincreasesslightly.

Inthefollowingsectionsthetimeevolutionofsomemodesincorneringarepresented.

Fig.7-38Root-locusplotincorneringatdifferentspeeds,(speedfrom3to60m/s,centripetalacceleration=0.5g).

Capsize

Whenthemotorcycleisrolledthecapsizemodeinvolvesbothin-planeandout-of-planedegreesoffreedom,asFig7-39shows.

Inthisconfiguration(thatiswithaspeedof4m/sandacamberangleof30degrees)themodeshowsapronouncedinstability;infact,themotioncomponentsdepartfromthesteadystatevalueasthetimeincreases.

Wobble

Theplotofthewobble(Fig7-40)showsonlyslightcoupling,becauseonlytheout-of-planecomponentsareinvolvedintheoscillation,whereasthein-planequantitieskeeptheirconstanttrimvalues.

Weave

Theweaveisthemodeshowingthemaincouplingeffect:allcomponentsoftheeigenvectorareinvolved.Figure7-41showsthatthemodeisstableandtheoscillationvanishesafter1second.

Fig.7-39Unstablecapsizeincorneringataspeedof4m/s.

Fig.7-40Wobbleincorneringataspeedof25m/s;rollangle=30°.

Fig.7-41Weaveincorneringataspeedof25m/s;rollangle=30°.

Bounce

Thetimeevolutionofthemotioncomponentsforthebouncemodeshowsthestronginteractionbetweenin-planeandout-of-planeoscillations.

Fig.7-42Bounceincorneringataspeedof25m/s;rollangle=30°.

7.2.5Effectofframeflexibi l i ty onmodes ofvibration

Theassumptionsofrigidbodiesandafixedriderinmotorcyclemodelingarenotstrictlytrue,particularlyforalessrigidchassisasinthecaseofascooter.Therefore,inordertohighlighttheeffectofflexibility,ascooterwasconsidered.Asiscommoninscooterslow-pricefrontforkshaveanon-negligiblebendingandtorsioncompliance;theengineiselasticallyconnectedtothemainchassisandthelow-slung,cradleshapeoftheframedoesnotmakeitpossibletodesignahigh,stiffchassis.Inadequatestructuralstiffnessmaynotablyreducestabilityandhandlingofthesevehicles.Inordertoinvestigatethesephenomena,amathematicalmodelofthescooterwhichalsoincludes

vehiclecompliancesandridermobilityisused(Fig.7-43).

Thevehiclecompliancesaretakenintoaccountbymeansofaforkbendingstiffness(25-75kN-m/rad),aforktorsionstiffness(4-10kN-m/rad),aswingarmbendingstiffness(30-70kN-m/rad)andaswingarmtorsionstiffness(10-20kN-m/rad).Rider-vehiclemobilityistakenintoaccountaccordingtothesuggestionofthereference,[Katayamaetal.,1997].

Simulationresultsarepresentedintermsoftheroot-lociinFig.7-44.Thewell-knownweave,wobbleandcapsizemodesareclearlyvisible,anddifferencesbetweentherigidandlumpedstiffnessmodelestimationarealsoevident.Moreoverthelumpedstiffnessmodelshowstwoadditionalmodesthatcorrespondtotheriderleanandshake.

Fig.7-43Flexiblebodiesmodel.

Fig.7-44Root-locusplotinstraightrunningatdifferentspeeds:rigidmodelvs.flexiblemodel(speedfrom1to40m/s).

Inmoredetail,therigidmodelestimatesawobblefrequencythatdecreasesfrom8to7.5Hzasthespeedincreases;onthecontrarythelumpedstiffnessmodelpredictsafrequencywhichrisesfrom6to8Hz.Moreover,atlowspeedsthewobblestabilityoftherigidmodelisgreaterthanthatofthelumpedstiffnessmodel,whereasathighspeedsthewobbleinstabilityoftherigidmodelisgreaterthanthatofthelumpedstiffnessmodel.

Themaindifferencesintheweavemodeareduetothemodebranchinginthelumpedstiffnessmodel,whichgivesrisetotwomodescoupledwiththeriderleanmotion.Bothmodelsshowthattheweavemodehasaverylowfrequencyatlowspeeds;thisfrequencyclimbsto3Hzasthespeedincreases.Weavemodeisunstableatverylowspeeds,becomesverystableinthemediumspeedrangeanditsdampingratiodecreasesathighspeeds.

Thecapsizemodeisnonoscillatoryinthewholespeedrangeandtherearenodifferencesbetweenthetwomodels.Theridershakemodeisverystableanddoesnotappeartoinfluenceothermodes.

Figure7-45presentsthesimulationresults,obtainedtakingintoaccountonlythefrontforkbendingcompliance.Thewobblemodeisdeeplyinfluencedbyforkcompliance.Inparticularaflexibleforkdecreaseslowspeedstability,butincreasesstabilityintheupperspeedrange.Thisbehaviourcanbeattributedtoapairofopposingeffects:theincrementofforkflexibilitytendstoreducestability,atthesametimethecombinationofwheelspinandforkbendinggeneratea

gyroscopictorquearoundthesteeringaxiswhichtendstostabilizethewobble.Atalowspeedthefirstnegativeeffectispredominant.Athighspeedthesecondpositiveeffectdominates.

Fig.7-45Frequencyandstabilityofthewobblemodetakingintoaccountonlytheforkbendingcompliance.

Theweavemodeisonlyslightlyaffectedbyforkcompliance.Figure7-46showstheeffectoftheswingarmbendingcomplianceontheweavemode:theflexibilitymayincreaseweavestabilityslightlyathighspeeds.

Finally,Fig.7-47showsthatthetorsionalcomplianceoftheswingarmalwaysworsenstheweavestabilityatmedium-highspeeds.

Fig.7-46Frequencyandstabilityofweavemodetakingintoaccounttheswingarmbendingcomplianceonly.

Fig.7-47Frequencyandstabilityoftheweavemodetakingintoaccountonlytheswingarmtorsionalcompliance.

MotoGuzziFalconetouringmotorcycle,1957(ownedbyVittoreCossalter)

8MotorcycleManeuverabil i ty andHandling

Amotorcycle’sdynamicpropertiesaredescribedusingtermslikemaneuverability,handlingandstability.Maneuverabilityandhandlingdescribethemotorcycle’sabilitytoexecutecomplicatedmaneuvers,andhowdifficultitisfortheridertoperformthem.Stability,ontheotherhand,meansamotorcycle’sabilitytomaintainequilibriuminresponsetooutsidedisturbanceslikeanunevenroadsurfaceorgustsofwind.

8.1Directional s tabi l i ty ofmotorcyclesMotorcyclesinmotionneedtobecontrolledbytherideratalltimes.Riderinputaffectsthe

motorcycle’sequilibriumanddirectionofforwardmotion.

Inrectilinearmotion,amotorcycleiscalled“directionallystable”ifitiseasytocontrolornaturallytendstomaintainitsequilibriumandfollowarectilinearpath.

Itiseasytosee,however,thatalargetendencytowardsdirectionalstabilitymakesamotorcyclehardtohandle,i.e.,cumbersometoturnandcontrolthroughtwistsandturns.

Thissectiondiscussesthedirectionalstabilityofmotorcycles,whichisdeterminedbyanumberoffactors:

inertialpropertiesofthemotorcycle;forwardspeed;geometricpropertiesofthesteeringhead(whichcollectivelydeterminethealigningeffectofthetrail);gyroscopiceffects;

tireproperties.

Obviously,thegreaterthemotorcycle’squantityofmotion(mV),thelessitwilldeviatefromitsrectilineartrajectoryasaresultofoutsidedisturbances.

Takingagustofwindasanexample(Fig.8-1).SupposetheaerodynamicpressuregeneratedbythegustactsonthemotorcycleforshorttimeintervalΔtthattendstozero.Thedisturbancecausesanangulardeviationofthemotorcyclefromtherectilineartrajectoryequalto:

Theangleofdeviationisinverselyproportionaltothemassofthemotorcycleanditsforwardspeed,anddirectlyproportionaltothelateralaerodynamicforce.Thelengthofthemotorcycle’swheelbasealsoplaysaratherimportantroleindeterminingdirectionalstability.Figure8-2showshowamotorcyclewithashortwheelbasebehavesdifferentlyfromonewithalongwheelbase.Ifadisturbancecausesadisplacementofthefrontwheel,theangleofdeviationfromtherectilineartrajectoryisinverselyproportionaltothelengthofthewheelbase.

Fig.8-1Directionalbehaviorofamotorcyclestruckbyagustofwind.

Fig.8-2Directionalbehaviorofamotorcycleasafunctionoflengthofwheelbase.

Intermsoftheeffectofmotorcyclegeometry,wehavealreadyseenthatthemomentexertedbyresistanceonthefronttirehasanaligningeffectthatincreaseswithforwardspeedandthelengthof

thetrail.

Wecancalculatethisaligningeffectusingasimplifiedmodel:amotorcycleinrectilinearmotiontravelingatconstantspeed.Supposethatanoutsidedisturbancecausesthefrontendtorotatetotheright,andthereforethemotorcyclebeginstofollowacurvedtrajectorytotherightwithalargeradius.Letusalsoassumethattherollangleisnegligible.Basedonthesesimplifyingassumptions,wecancalculatethemomentexertedaroundthesteeringheadaxis.

Thefollowingforcesareactingonthevehicleasawhole:thrustS,exertedatthecontactpointoftherearwheel;dragFD,whichisassumedtobeexertedatthecenterofgravity;rollingresistance exertedatthecontactpointofthefrontwheel;lateralforces ,exertedatthecontactpointsofthewheels;verticalloadsNf,Nsexertedatthecontactpointsofthewheels.

Fig.8-3Corneringmotorcyclewithnegligiblerollangle.

Theequilibriumequationsforforcesandmoments(Fig.8-3):

givethedynamicverticalloads:

thefrontlateralforce:

andthethrustneededtomakethemotorcycletravelthroughtheturnatconstantspeed:

Amomentisexertedaroundthesteeringheadaxis(Fig.8-4)through:thecomponentoftheverticalloadNfperpendiculartothesteeringheadaxiswhichhastheeffectofincreasingthesteeringangle:Nfsinβ,thecomponentoflateralforce perpendiculartothesteeringheadaxis,whichhastheeffectofaligningthewheel: cosβ.

Theresultingmomentexertedbythetwoforces,neglectingtherollingresistance,isgivenby:

Sincethepositivetermisproportionaltothesquareofthespeed,thealigningmomentincreaseswithspeed.Thelongerthenormaltrailis,themoremarkedtheeffectis.Asexpected,motorcyclestabilityisstronglyinfluencedbythelengthofthetrail.

Fig.8-4Forcesactingonthefrontwheel.

Wehavesuggestedthatgyroscopiceffectsplayanespeciallyimportantroleindirectionalstabilityandmaneuverability.Manygyroscopiceffectsareexperiencedwhilecornering,andenteringorexitingturns.Therotationofthesteeringhead,wheelsandrotaryenginepartsgenerategyroscopicmomentsasaresultofmotorcyclerolland/oryawmotions.

8.2Gyroscopice ffects onthe motorcycleAgyroscopiceffectisgeneratedbyarigidbodyrotatingaroundanaxisa−a,whichinturnis

rotatingaroundasecondaxisb−baskew(notparallel)tothefirstaxisa−a.Thegyroscopiceffecttakestheformofacoupleexertedaroundanaxisperpendiculartobotha−aandb−b.ThevalueofthegyroscopicmomentisequaltothevectorproductoftheangularmomentumIωofthebodyaroundaxisa−aandthespeedofrotationΩaroundthesecondaxisb−b.AngularmomentumisequaltothepolarmomentofinertiaofthebodyIaroundaxisa−amultipliedwiththespeedofrotationωaroundthesameaxis.

Motorcycledynamicsincorporateavarietyofgyroscopiceffects,whichmaybebrokendown

accordingtothesecondaxisofrotationb−b:yawgyroscopiceffects:whereaxisb−bpassesthroughtheturncenterofthepathandisperpendiculartotheroadway;rollgyroscopiceffects:whereaxisb−bisthestraightlinelyingintheplaneoftheroadwaywhichpassesthroughthetirecontactpoints;steeringgyroscopiceffects:whereaxisb−bisthesteeringheadaxis.

8.2.1Gyroscopice ffects generatedbyyawmotion

Gyroscopiceffectgeneratedbythewheelsduringcornering(wheelrotation- yawmotion)

Letusconsiderthefrontwheelalone,rotatingataconstantspeedωfasthemotorcycletravelsthroughaturnofradiusRcataconstantyawvelocityΩ(Fig.8-5).

Fig.8-5Gyroscopiceffectgeneratedbythefrontwheelduringcornering(thecoordinatesystemwithsubscriptmisattachedtotheforkofthemotorcycle).

Themotionofthewheelasitcornersgeneratesagyroscopicmomentaroundthehorizontalaxis,whichhastheeffectofstraighteningthewheel:

ThesecondapproximateexpressionisvalidiftheyawvelocityΩcanbeconsideredsmallwith

respecttothespeedofrotationωf.Thisassumptionisverifiedinpracticebecausetheturningradiusismuchgreaterthanthewheelradius.Axisxmisfixedtothefork,andtherefore,itisamobileaxis.

Lookingnowattheeffectofbothwheelsandsettingasidethefactthatthewheelshaveslightlydifferentrollanglesanddirectionsduringcornering,theirgyroscopiceffectscanbeaddedtogether:

Motorcycleequilibriumoccurswhentheresultantoftheweightforceandthecentrifugalforceintersectsthelinejoiningthecontactpointsofthetwowheels.Disregardingthegyroscopiceffectandassumingzerothicknesswheels,theidealrollangleforamotorcycleinsteadystatecorneringisgivenbythefollowingsimpleequation:

Aswehaveseen,thegyroscopiceffectofthewheelsduringcorneringismanifestedbyarightingmoment.Tocounteractthegyroscopiceffectofthetwowheelsandtherebymaintainequilibrium,theridercanleanintotheturninsuchawaythattheresultantoftheweightforceandthecentrifugalforcegeneratesamomentequalandoppositetothegyroscopicmomentofthetwowheels,asshowninFig.8-6.

Fig.8-6Influenceonequilibriumofgyroscopiceffectgeneratedbywheelsduringcornering.

Ofcourse,theridercanachieveequilibriumwithoutdisplacinghistrunkinordertoproduceadisplacementofthemasscentertowardsinsideofthecurve,buttheleanangleofthemotorcyclewillbegreaterthantheidealrollanglecalculatedontheassumptionthatthegyroscopiceffectiszero(Fig.8-7).

Inthiscase,therightingmomentgeneratedbythecentrifugalforceandthemomentgeneratedbythegyroscopiceffect(whichalsohasarightingeffect)arebothoffsetbytheoverturningmomentoftheweightforce.Thegyroscopiceffectmakestheactualrollanglegreaterthantheidealrollanglethatwouldbeachievedifthegyroscopiceffectwereabsent.

TheincreaseintherollangleΔϕneededtocounterbalancethegyroscopiceffectisgivenby:

SinceΔϕissmallwithrespecttoϕ,itcanbedisregardedinthenumeratorontherighthandside,thefollowingsimplerequationholds:

Here,thenumeratorrepresentsthegyroscopicmomentgeneratedbythetwowheelsofthemotorcycle.ThemomentMwhichcounterbalancesthegyroscopicmomentisgeneratedbytheresultantoftheweightforceandthecentrifugalforce.TheincreaseΔϕmakesthemotorcyclelessmaneuverable,sincethemotorcycletakesmoretimetoreachtheincrementallylargerequilibriumrollangle(whichisgreater).

Fig.8-7Increaseinrollanglecausedbyyawgyroscopiceffect.

Example1

Assumeamotorcycleinstationarymotionduringcornering:

•turningradius: Rc=200m;

•forwardspeed: V=40m/s;

Thepropertiesofthemotorcycleareasfollows:

•mass: m=200kg;

•heightofcenterofgravity: h=0.6m;

•wheelradius: Rf=Rr=0.32m;

•spininertiaoffrontandrearwheels:

Nowdeterminethegyroscopicmomentgeneratedbythemotionofthetwowheels,andtheresultingincreaseintherollangle.

Sinceweknowtheturningradiusandforwardspeed,wecancalculatetheangularvelocityvalues:

•yawvelocityofthemotorcycle: Ω=0.2rad/s;

•angularvelocityofthewheel: ω=125rad/s;

andtherefore:

•idealrollangle: ϕideal=39.20°;

•gyroscopicmomentgeneratedbythemotionofthetwowheels: Mg=23.25Nm.

Theridercanachieveequilibriuminoneoftwoways:eitherdisplacingthemasscentertowardtheinsideoftheturnby:d=9.2mm;orincreasingtherollangleby:Δϕ=0.88°.

Gyroscopiceffectgeneratedbytransversallymountedengine(enginerotation- yawmotion)

Thegyroscopiceffectgeneratedbytheengineisdeterminedbytheengine’sspeedofrotation,whichdependsonwhatgearthemotorcycleisin.

Letusassumeamotorcycleinsteady-statecorneringmotionanddisregardtheinertiaofthewheels.Inotherwords,wearegoingtolookatthegyroscopiceffectgeneratedbytherotationoftheengineonly.Themainshaftoftheenginegenerallyrotatesinthesamedirectionasthewheels,asshowninFig.8-8.

Fig.8-8Gyroscopiceffectgeneratedbyatransverseengineduringcornering.

Asbefore,thegyroscopiceffectgeneratedbytheenginecausesequilibriumtobeachievedbyleaningthemotorcycleoveratanactualrollanglegreaterthantheidealanglethatwouldbenecessaryifthegyroscopiceffectwereabsent.

Theresultingincreaseintherollangleisequalto:

Thesignispositivewhentheenginerotatesinthesamedirectionasthetiresandnegativeforacounter-rotatingengine.

Theterm expressestheengine’stotalangularmomentum,incorporatingtheangularmomentumofthedriveshaft,transmissionshaftsandanyotherrotatingshaftparalleltorearwheelaxisandrotatingwithsamesense:

Incalculatingtheresultingangularmomentum,thesignoftheangularvelocityisassumedtobepositiveifthedirectionofshaftrotationagreeswiththedirectionofrotationofthewheels.Otherwise,itisassumedtobenegative:

Forexample,iftheengine’srotationisinthesamesenseasconcordantwiththewheelspinthemaingearshaftrotateswithanopposingvelocity,anditscontributionisthereforenegative.

Ofcourse,theengine’scontributionmustbeaddedto,orsubtractedfrom,thecontributionofthewheels,dependingonthedirectionofrotationfollowingtheconventionestablishedabove(addedifthedirectionofrotationisconcordantwiththewheelspin,andsubtractedifnot).

Toreducethegyroscopiceffect,themomentumoftherotatingbodiesmustbereduced.Sincelightweightmaterialscanonlybeusedtoreducethemomentofinertia,anattractiveoptionistoreducetheangularmomentumoftheengine,oreventogiveitanegativesign,bychoosingarotationinthedirectionoppositetothewheelspin.

Forexample,insometwo-cylinderracingenginesthetwodriveshaftsrotateinoppositedirections.Thus,thegyroscopiceffectsofthetwocrankshaftscanceleachotherout,leavingonlytheeffectofthetransmissionshafts.

However,themaingearshaftandtransmissioncontributelesstothegyroscopiceffectthanthedriveshaftdoes,sincetheyhavelessinertiaandrotateatslowerangularvelocities;thevelocityratiobetweenthedriveshaftandthemaintransmissionshaftisoftheorderof2to2.5.

Example2

UsingthemotorcycleinExample1insteady-statecorneringmotion,determinethegyroscopiceffectofanenginewiththefollowingproperties:

•momentofinertiaofcrankshaft:

•primaryshaftinertiamoment(includingclutch):

•secondaryshaftinertiamoment:

•engine-primaryshafttransmissionratio: τm,p=3;

•primary-secondarygearshafttransmissionratio: τp,s=2;

•enginerpm: n=12,000rpm.

Addingupthevariouscomponents,theengine’stotalangularmomentumisequalto:

Thegyroscopicmomentis:Mg=2.66Nm;theincreaseinrollanglesolelyduetothegyroscopiceffectgeneratedbytheengineisΔϕ=0.1°.

Notethatthegyroscopiceffectgeneratedonlybytheengineislessthanthatgeneratedbythewheels(Mengineisabout8%ofMwheels);theengine’scontributiongenerallyfallswithintherangeof5%to15%ofthegyroscopiceffectgeneratedbythewheels.Ifthecontributionbythewheelsistakenintoaccountaswell,theincreaseintherollanglegoesuptoΔϕ=0.97°.

Gyroscopiceffectgeneratedbylongitudinallymountedengine(enginerotation- yawmotion)

Nowconsideramotorcycleequippedwithalongitudinaldriveshaftcorneringatconstantspeed,asshowninFig.8-9.

Fig.8-9Gyroscopiceffectgeneratedbyanenginewithalongitudinalaxis.

Iftheturnistotheleftwithrespecttothedirectionofforwardmotionthemotorcycleleansovertotheleft.Assumingthatthelongitudinaldriveshaftisrotatingtowardtheoutsideoftheturn,thegyroscopicmomentactingaroundtheymaxisisequalto:

Thegyroscopicmomenthastheeffectofextendingthefrontsuspensionandcompressingtherearsuspensiontoagreaterdegreemakingthemotorcyclepitchbackwards.

Thegyroscopicmomenthastheoppositeeffectswhencorneringtotheright,i.e.,thefrontsuspensioncompressesandtherearsuspensionextends.

Nowconsiderthemotorcyclecorneringtotheleftagain,butthistimeassumethatthedriveshaftrotatestowardtheinsideoftheturn.

Thegyroscopicmomentactingaroundtheymaxisreversessign:

Inthisinstance,thegyroscopicmomenthastheeffectofreducingtheloadontherearsuspensionandincreasingtheloadonthefrontsuspension.Themomentgeneratedbythesuspensions’forcesbalancesoutthegyroscopicmoment,withtheendresultofthemotorcyclecharacteristicallypitching

forwardslightly.

Example3

Nowassumethattheengineinexample2ismountedlongitudinally.

Thestiffnessvaluesforthefrontandrearsuspensionsare:kf=9kN/mandkr=2kN/m,respectively.Wewanttocalculatethechangeintrimgeneratedbythegyroscopiceffectoftheengine.

Thegyroscopicmomentoftheengineisequalto:Mg=2.43Nm.

Fig.8-10Motorcycleridingtrim.

Tocalculatethepitchangle,weneedtotakeintoaccounttheequilibriumofthemotorcycleonwhichthegyroscopicmomentisacting(Fig.8-10).Thestaticequilibriumequationseasilyyieldtheverticaldisplacementofthecenterofgravityandthepitchangle:

Boththepitchangleof0.01°andtheincreaseinheightofthecenterofgravityareentirelynegligible.

8.2.2Gyroscopice ffects generatedbyrol l motion

Gyroscopiceffectgeneratedbythefrontwheel(frontwheelrotation–rollmotion)

Fig.8-11Frontwheelrotation-rollmotioninduceagyroscopicmomentactingonthefrontend.

Nowwewilllookatthefrontwheelwhilethemotorcycleisrollingtotheright.Thefront-wheelspin,coupledwiththerolltotheright,generatesagyroscopicmomentMgthatactsonthefrontframearoundanaxislyingintheplaneofthemotorcycleandperpendiculartothelongitudinalrollaxis,asshowninFig.8-11:

Theprojectionalongthesteeringaxisprovidesthebeneficialmomentaroundthesteeringaxis:

Thus,thegyroscopicmomenthastheeffectofturningthesteeringheadtotheright,therebyhelpingthemotorcycleentertheturn(increasingthesteeringanglereducestheturningradius).Analogously,whentherollvelocitychangessignasthemotorcyclereturnstotheverticalpositionthegyroscopicmomenthastheeffectofreducingthesteeringangle,therebyhelpingthemotorcycleexittheturnandreturntorectilinearmotion.

Example4

Nowconsideramotorcyclerollingfromlefttoright.Tocalculatethegyroscopicmomentactingon

thesteeringhead,assumethatthemotorcycleisrollingfromthelefttotherightatavelocityof0.5rad/s.

Thepropertiesofthemotorcycleareasfollows:

•momentofpolarinertiaoffrontwheel:

•motorcyclerollvelocity:

•spinvelocityofwheel: ω=100rad/s;

•rakeangle: ε=25°;

ThegyroscopicmomentaroundthesteeringaxisisequaltoMg=27Nm.

Itcanbedemonstratedthatintransientmaneuverssuchasthelanechangemostofthesteeringtorqueappliedbytheriderisusedtoovercomethisgyroscopicmoment.

Gyroscopiceffectgeneratedbywheels(wheelsrotation- rollmotion)

Ifthemotorcycleisassumedtobearigidbody(i.e.,withthesteeringheadlockedinplace),thegyroscopiceffectofthewheelspinduringrollcaneasilybeshowntogenerateayawingmoment,asshowninFig.8-12.

Fig.8-12Gyroscopiceffectgeneratedbymotorcyclerollmotion-wheelsrotation.

Again,consideramotorcyclerollingfromlefttoright.Thegyroscopicmomentactingonthemotorcycleisequalto:

Thegyroscopicmomenttendstomakethemotorcycleyawtotheright,andisbalancedbythelateralresistanceexertedonthewheelsbytheground.Thus,thefrontlateralforceincreasesslightlyΔF,whiletherearlateralforcedecreasesbythesameamount:

Whenexitingtheturnthemotorcyclerollsfromrighttoleft.Thegyroscopicmomentreversessign,and,hence,alsothevariationintirelateralforceschangessign.

Example5Nowconsideramotorcyclerollingfromlefttorightatarollvelocityof0.5rad/s.Thepropertiesofthemotorcycleareasfollows:

•momentofpolarinertiaofwheels:

•spinvelocityofwheels: ω=100rad/s;

•lengthofwheelbase: p=1.37m;

•motorcyclerollvelocity:

Determinethegyroscopiceffectgeneratedbythewheelsandthechangeinlateralforce.

Thegyroscopicmomentactingonthemotorcycleisequalto60Nm.

ThechangeΔFinlateralforceneededtocounterbalancethegyroscopicmomentis44Nwhenthemotorcycleisintheverticalposition.

Thisisafairlyhighvaluecomparedtothevaluesforthelateralforceneededtomaintainequilibriumundersteady-statecornering.ForexamplewithspeedV=30m/s,turningradiusRc=200m,andmassm=180kgthesumofthetwolateralforcesmustbeequalto872Ν.

Ifthetotallateralforceisdistributedevenlybetweenthetwowheels,therebyexertingatransverseforceof436Νoneachwheel,thevariationduetothegyroscopiceffectisontheorderof10%.

8.2.3Gyroscopice ffects generatedbys teering

Sincethewheel’sdirectionofspinisperpendiculartothesteeringheadaxis,turningthehandlebarsfromrighttoleftgeneratesagyroscopicmomentaroundanaxisperpendiculartoboththesteering

headaxisandtheaxisofthefrontwheel,asshowninFig.8-13:

Thishastheeffectofleaningthemotorcycleovertowardstheright.Theprojectionofthegyroscopicmomentontherollaxis(thelineconnectingthecontactpointsofthetwowheels)isasfollows:

Fig.8-13Gyroscopiceffectgeneratedbythefrontwheelandsteeringheadrotations.

Basedonthesegyroscopiceffects,onemightconcludethatamotorcyclewithzerowheelinertiaisideal.Itisimportanttopointout,however,thatthegyroscopiceffectgeneratedbythefrontwheelandsteeringmotionplaysanimportantpartinmotorcyclestabilityduringrectilinearmotion.

8.3Motorcycle equi l ibriuminrecti l inearmotionatlowspeedSincethesameprinciplesgoverncontrolofanytwo-wheeledvehicleatlowspeed,amotorcyclist

alsoknowshowtorideabicycle.

Achildlearningtorideabicyclebeginsbyrollingdownagentlehill,andquicklylearnsthat,ifthebicyclebeginstoleantotherightandheturnsthehandlebarsinthesamedirection,thebicycleeasilyreturnstotheverticalpositionafterturningright.Ifthebicycleleanstotheleft,theequilibriumisachievedbyasimilarmaneuver.

Thepaththebicyclefollowsisinfluencedbythecontrolactionscontinuallymadebythecyclisttokeepthebicyclevertical.Therefore,asaresultthebicyclefollowsaweavingpath,andhowmuchitweavesisdeterminedbytheskilloftherider.

Thesameprinciplesareusedtocontrolthemotorcycle’sverticalequilibriumatlowspeed.

Asamotorcycleleanstotherightitsfalliscounteractedbyturningthehandlebarstotheright.Themotorcyclebeginstoturnright,creatingacentrifugalforcethatstraightensthemotorcyclebackup,asshowninFig.8-14.

Fig.8-14Motorcyclebalanceinrectilinearmotionatlowspeed.

8.4Motorcycle equi l ibriuminrecti l inearmotionathighspeedLetussupposeanexternaldisturbancethatcausesthemotorcycletorolltotheright,asshownin

phase1ofFig.8-15andthattheriderdoesnotapplyanytorquetothehandlebars;inotherwords,he/sheremainspassive.Letωbetheangularvelocityofthefrontwheel, therollvelocity,and thesteeringvelocity.

Duetothefrontwheel-spinandrollvelocitiesagyroscopicmomentisgenerated(phases2)thathastheeffectofturningthehandlebarstotheright(phases3).

Thehandlebarssteertotherighttherebyreducingtheturningradius(phase3).

Astheturningradiusdecreases,thecentrifugalforceincreases,therebystraighteningupthemotorcycle(phases3-4).

Atthesametime,duetothefrontwheelspinandsteeringandyawvelocitiesanoverturninggyroscopicmomentisgeneratedthatcounteractstherollmotiontowardstheright(phase3-4).

Themotorcyclestopsrollingtotheright(phase4).ThecentrifugalforcehastheeffectofstraighteningthemotorcycleupandreversingtheroIlmotion(phase5-6).

Theroll-inducedgyroscopicmomenthastheeffectofturningthehandlebarstotheleft,therebysteeringthemotorcycletotheleft.

Finally,themotorcyclereturnstotheverticalpositionshowninphase6,butitcontinuestorolltotheleft,andtheresultinggyroscopicmomentturnsthesteeringheadtotheleft.Whenrollingtotheleftthemotorcyclegoesthroughasimilarsequence.

Fig.8-15Motorcyclebalanceinrectilinearmotionathighspeed.

8.5SlowenteringinaturnAfterlearninghowtokeepatwo-wheeledvehicleinverticalequilibriuminrectilinearmotionone

mustlearnhowtoturn.

Thebeginnerriderofabicyclefindsagentlehillandsetsoff,worryingonlyaboutkeepingthebicycleupright.Atacertainpointhespiesanobstaclee.g.,apotholetohisleft,anddecidestochangedirection,soheturnsthehandlebarstotheright.Thebicyclebeginstofollowacurvedtrajectoryto

theright.Thecentrifugalforce,generatedasthebicycleroundsthecurve,rapidlyleansthebicycleandridertowardtheleft,makingafallinevitable.

Afterseveralunsuccessfulattempts,theridercomesupwithastrategythatisneithersimplenorintuitive.Ifhewantstochangehisorherdirectiontowardtherighthemustfirstapplyleftwardtorquetothehandlebars(“youturnlefttogoright”).Thatmakesthesteeringheadturntotheleft,andthebicyclebeginstoturnleft,creatingacentrifugalforcethatleansthebicycletotheright.Oncethebicyclehasbegunrollingtotherighttheridercanturnthehandlebarstotherighttocontinuehisentranceintoarightwardturn.

Letusnowconsideramotorcycletravelingat20m/senteringaturnwithaslowmaneuver.Theriderbeginsthemaneuverwellbefore(approx.14m)theturninordertoreachthesteadystateconditionslowly(Fig.8-16).

Fig.8-16Trajectorywhenenteringslowlyinaturn.

Initiallytheriderappliesatorquetowardsleftcausingthefrontwheeltosteertowardsthesameside(Fig.8-17).Thelateralforcegeneratedatthefronttirecontactpoint,causesayawmotiontotheleftandthebeginningofarollmotiontotheright.

Fig.8-17Steeringangle,steeringtorque,lateralforce,rollandyawwhenenteringslowlyinaturn(computedwithFastBikecode).

Thevehiclefollowsaheadingtowardtheleft,oppositethedirectionofthedesiredtrajectory.

Afterabout5mfromtheinitiationofthemaneuver(atapprox.19m),thelateralforcechangesdirectionfromtheoutsidedirectiontowardtheinsideofthecurve.Afteranadditional4m(atapprox.23m),withthemotorcyclealreadyrolledtowardtheright,theleftwardsyawmotionendsandthemotorcyclestartstoyawinthedesireddirection.Nowthepathistowardstheright.

Nowtheriderfollowsthedesiredpathcontrollingthevehiclewiththesteeringtorque.Afterabout

13mfromtheinitiationofthemaneuver(atapprox.27m)thesteeringanglechangesfromlefttoright.Themotorcyclereachesthesteady-stateturningconditionabout70mafterthebeginningofthemaneuver.Inthesteadystateconditionthecentripetalaccelerationisequalto0.9g.Itisworthnotingthatthesteeringtorquevalueinthesteadyconditionissmallwithrespecttothemaximumvalueofthetorqueappliedduringtheentrancephase.

8.6Fas tenteringinaturnToenterfastinaright-handturn,theriderappliesaquicktorqueonthehandlebarstotheleft.The

movementofthefrontwheelaroundthesteeringheadgeneratesafronttirelateralforcethathastheeffectofleaningthemotorcycletotheright.Thegyroscopiceffectgeneratedbythefrontwheelandthesteeringrotationalsohasanimportanteffectonfastenteringinaturn.Thisgyroscopicmomenthastheeffectofleaningthemotorcycletotheright.Oncethemotorcyclehasbeguntorolltotherighttheridercanslowlyturnthehandlebarstotheright,andthemotorcycleenterstheturn.

Fig.8-18Trajectorywhenenteringfastinaturn.

Figure8-18showsthemotorcycletravelingat22m/sandexecutingarelativelyquickenteringinaturnmaneuver.Theaverageturningradiusoftheroadis50mandtheroadwidthis16m.

Therider(atapprox.14m)quicklysteerstowardstheleftandthemotorcycleimmediatelyyawstothesamesidefollowingapathtowardstheleft.Theminimumturningradiusinthisphaseisabout100m.Atthesametimethemotorcyclestartstorollrightwards.Theleftwardlateralforcereachesamaximumvalueof50Nafterabout2-3m.Itstiltingmomentwithrespecttothemasscenter(height=

0.6m)isequaltoabout30Nm.

Fig.8-19Steeringangle,steeringtorque,lateralforce,rollandyawwhenenteringfastinaturn(computedwithFastBikecode).

Letuscomparethismomentwiththegyroscopicmomentduetofrontwheelspin-steeringmotion.Thesteeringratereachesitsmaximumvalueafterabout1mbeforethelateralforcereachesitsmaximum.Ifthefrontwheelspininertiaisequalto0.6kgm2thegyroscopicrollmomentisabout3.5Nm.

Thismomentcontributestothegenerationoftherollmotionduetothefactthatitispresentsince

theinitiationofthemaneuverwhile,dependingontherelaxationlengthofthetire,thelateralforcerequiresmoretimetoreachitsmaximumvalue.

Obviously,thegyroscopiccontributionbecomesmoreimportantasthefrontwheelspeedandthesteeringrateincrease.

Thisexamplehighlightsthe“out-tracking”techniquesthatconsistofenteringarightturnsteeringforashorttimetowardstheleft.Theenteringphasecanbeimprovedbythelateraldisplacementsoftherider ’sbodyintotheturnneglectedinthepreviousexample.Riderlateraldisplacementcausesthemotorcycletoleanandcanbeusedtoreducetheinitialcounter-steer.

8.7TheoptimalmaneuvermethodforevaluatingmaneuverabilityandhandlingMotorcycledynamicsisanotoriouslythornytopictodealwith,mainlyduetothefollowing

problems:firstofalltheprecisedescriptionofvehiclekinematicsiscomplexbecauseofthepresenceofthesteeringhead;secondly,dependingontheridingconditions,suchasspeedforexample,motorcyclesarenaturallyunstablevehicles(aswehaveseeninpreviouschapters)andthedriver ’sactionsarealwaysneededtoprovidecontrol;moreoverthepersonaldrivingstyle,thedriver ’sskillandtheirexperiencealsoaffectthevehicle’sperformance.

Thus,itisclearthatthewholedriver-motorcyclesystemmustbestudiedifwewishtounderstandwhatdesignparametersinfluencethevehicle’sperformanceandhandling.

Aclearwayofdealingwiththemotorcycledynamicsproblemistoanalyzeitusingasystematicapproach.Thismeanssplittingtherider-motorcyclesystemintosubsystems,asshowninFig.8-20.Itispossibletoconsiderthreemainsubsystems.

Thedriver,whocontrolsthemotorcycleviathesteeringtorque,thebrakelever,thethrottle,andthemovementofhisbody.Theseareinputstothemotorcyclesystem.Itisobviousthatthereexistsomelimitsontheseactions.Forexamplethedrivercannotexertaninfinitevalueforthesteeringtorque,normayheexertitwithoutsomedelay.

Themotorcycleitselfcanbeconsideredasbeingmadeupofmanyothersubsystems,dependingonthecase.Forvehiclelateraldirectionalcontrol(usuallyreferredtoas“lateraldynamics”)twosubsystemsmaybeidentified:

thesteeringsubsystem,whichisamechanismthattransformssteeringtorqueintoalateralforcethatactsonthefrontwheel;thevehiclesubsystem,whichistreatedasarigidbodyandincludesthegyroscopiceffectofthewheels,onwhichwheel-to-groundcontactforcesareexerted.

LetussupposethatwewouldliketoaccomplishagivenmaneuversuchasaU-shapedcurveorS-shapedchicane,forexample,intheleasttimepossible(minimumtimeistheobjective).Theinitialandfinalpositionsonthetrackareknown,buttherearealotofpossibletrajectoriesthatcanleadthevehiclefromthestartingpointtothefinalone.Wearelookingforthefastestone,whichbestexploitstheintrinsicmotorcyclecharacteristics.

Fig.8-20Functionaldiagramofamotorcycle.

Furthermore,letussupposethatweobserveonlythemotorcyclesystemwithoutconsideringthedriver ’sphysicalandpsychologicallimitssuchasthemaximumtorqueheisabletoapply,orthemaximumsteeringrateheisabletoachieve.Thissituationcorrespondstohavingamotorcycledrivenbyanidealperfectdriver.Thebestperformancethatwegetfromthemotorcyclequantifiesitsmaneuverability.Inthissensemaneuverabilityisrelatedtotheabilityofthemotorcycletodocomplexmaneuversintheshortesttimepossible.Ifwealsoconsiderthedriver ’sperformancelimits(wehavearealdriverridingthemotorcycle),thebestperformancewegetfromthemotorcyclequantifiesitshandling.Thus,handlingmeanstheabilityofthemotorcycletodocomplexmaneuverstakingintoconsiderationthedriver ’slimits.Inotherwords,amotorcycle,whichhasbetterhandlingthananother,isfasterandatthesametimethedrivercanrideitwithlessphysicalandphysiologicaleffort.

Itispossibletolookatthesameproblemfromthesafetypointofview.Infact,amotorcycle,whichismoremaneuverablethananotherone,isabletoaccomplishthesamemaneuverwithoutreachingitslimits,forexamplethetireadherencelimits.Thismeansthatsomemarginsremain,forexampleadditionallateralforceisavailable,thatcanbeusedshouldadangeroussituationarise.

However,theextraeffortthatthedriverhastoexertinordertousetheseleftovermargins(forexampletheadditionaltireforces)tellsushoweasyorhowdifficultthemotorcycleistodrive.

Thediscussionsofarwouldseemtoindicatethatwithappropriateconstraintsplacedonthesystem,maneuverabilityandhandlingareintrinsicvehiclecharacteristics,thatqualifynotonlyitsbestperformancebutalsohoweasilythedriverrealizesthisperformance.

Anotherquestionthatmayarisewhentalkingaboutmaneuverabilityandhandlingisthatthespecifictrajectoryfollowed,andthereforetheforcesneededtoproducethattrajectory,dependonchoicesmadebytherider.Fromthisstandpoint,itwouldnotbeentirelycorrecttotalkaboutmaneuverabilityasanintrinsicpropertyofthemotorcycle.Instead,weshouldtalkabouttheoverallperformanceofthemotorcycle/ridersystem.Theperformanceofthesystemcouldonlybeassessedifweassumeaspecificmodelfortherider.

However,ontheotherhand,itseemsequallyclearthatsomemotorcyclesareintrinsicallybetter

thanothers,independentoftherider ’sdrivingskills.Sohowcantheintrinsicmaneuverabilityofmotorcyclesbedefined?Theansweristoassumeaperfect,oridealrider,capableofchoosingthebestpossibletrajectoryforagivenvehicle.Thebesttrajectoryandthenecessarydriveractiontogetherarecalledtheoptimalmaneuver.

8.7.1Optimal maneuver

Letusseehowitispossibletodefinetheconceptoftheoptimalmaneuverwithanexample.Letussupposethatthedesiredmaneuverisenteringaturnwhere:

thestartingstateissteady,rectilinearmotion;thedesiredendstateissteady,circularmotioncharacterizedbyagiventurningradius.

Tomovefromthestartingstatetotheendstate,thelateralforceappliedtothefrontwheel(whichisassumedtobecontrolledbytheriderthroughthesteering)mustvaryinspecificwaysovertime.Ofcourse,therearemanypossiblewaysofgettingthemotorcycletothedesiredendstate,buteachsolutionischaracterizedbyadifferentseriesofinputsovertime,whichaffectthesystemindifferentwaysintransit.Ofallpossiblesolutions,theonly“optimal”solutionisthemost“efficient”one,i.e.,theonethatminimizesagivenperformanceobjective.

Ifthetotaltimerequiredtocompletethemaneuverisusedastheindexofefficiency,the“optimal”maneuverwillbetheonethatminimizesthatobjective.Intheexample,thesolutiontotheproblem(enterastateofsteadycircularmotionasquicklyaspossible)isgivenbythetimetakentoexecutethatparticularmaneuver.

Obviously,the“optimal”solutionwillbedifferentfordifferentmotorcycles,aswilltheminimumtimeittakesthebestpossibleridertoenteraturn.Inshort,theperformanceindexassociatedwiththeobjectivetobeoptimizedduringthemaneuver(time,inthisexample)canbeusedtomeasuretheperformanceofthemotorcycle,whichquantifiesthemotorcycle’smaneuverability.Ifwealsoaddsomeconstraintsonthedriver ’sphysicalandpsychologicalefforttotheobjective,theperformanceindexquantifiesmotorcyclehandling.

8.7.2Anexample ofanoptimal maneuverforan“S”trajectory(chicane)

Forourfirstexampleofapplyingtheoptimalmaneuverprinciple,wewilllookatamotorcyclefollowingan“S”trajectory(chicane),asshowninFig.8-21.

Thefollowingconstraintsareplacedontheproblem:thevehiclemustcoverthesectionoftrackfromthecenterlineatthestartingpointtothecenterlineattheendpoint;thestartingspeedisknown,butnottheendingspeed,sinceitisdeterminedbythesolutionforthemaneuverwhichoptimizesthespecifiedperformanceindices.

Thefollowingcriteriamustallbeoptimizedatthesametime,intheorderofimportancegiven:minimizethetimetocompletethemaneuver;preventthetirereactionsfromsurpassingtheedgesofthefrictionellipse;preventthemotorcyclefromgoingovertheedgesofthetrack.

Figure8.21showsthe“optimal”trajectorysolutiontotheproblemasasolidlineplottedagainsta

dottedlineindicatingthecenterlineofthesectionoftrack.Notehowthesolutiontrajectory“shaves”theturnswithoutovercomingthem.

Fig.8-21“S”trajectory(chicane).

Fig.8-22“S”trajectory:motorcyclespeed,normalizedlongitudinalforce(thrustorbraking)andverticalforceswithrespecttotheweightforce.

InFig.8.22,notehowthespeedofthemotorcycledecreasesbeforethechicane,andthenincreasesagainattheexit.Inotherwords,the“optimal”solutioncallsforbrakingwhenenteringthefirstturnandaccelerationwhenexitingthesecond.

Theverticalreactionsexertedonthewheels(Fig.8-23)illustratethephenomenonofloadtransfer,

movingfirstfromtherearendtothefrontend(brakingphase),andthenintheoppositedirection(accelerationphase).Itisnecessarytohighlightthatinthemiddleofthechicane,quicklytiltingfromonesidetotheother,thewheelloadsdiminishforashorttimebecauseofcentrifugalaccelerationduetorollmotion.

Fig.8-23“S”trajectory:steeringangle,steeringtorque,lateralforces,rollangleandrollvelocity,yawangleandyawvelocity.

Theevolutionsofthesteeringtorqueandofthesteeringangleillustratethemaneuvercarriedoutbytherider(Fig.8-23).Initiallytheridersteersquicklytowardtherightinordertomovefromthecenteroftheroadtotheleftsideoftheroad.Havingreachedtheleftsideinabout40m,theridersteersleftwardstoinsertthemotorcycleintothefirstturnofthechicane.Thesteeringangleneeded

forequilibriumduringthefirstturnisverysmallduetothehighvalueoftherollangle.Inthemiddleofthechicanethequasi-impulsivemovementofthehandlebarsinducesthemotorcycletorollquicklyfromonesidetotheotherandyawintheoppositedirection.

Theexitfromthesecondturnisperformedmoresmoothlycomparedtotheenteringphase.Increasingthedrivingforcecausesagradualincrementintheforwardvelocityandconsequentlytheincreasedcentrifugalforcetiltsthemotorcycletotheverticalposition.Inthisphasetheridercontrolsthepathtobefollowedbymeansofthesteeringangle.

Thefigureshowswhathappenstothelateralforcesactingonthetires,whichhavebeennormalizedwithrespecttocurrentverticalload;inotherwords,itshowstheinclinationoftheresultingreactionforcesonbothwheels.Thisisoneoftheparameterstheoptimalmaneuverisdesignedtocontrol.Notethattheratiooflateraltoverticalforcereachesmaximumvaluesofabout1.5forbothfrontandrearwheels,andisslightlylesspronouncedforthefrontwheel.Inthisspecificexample,therefore,themotorcycleiswellbalancedbecausethefrontwheelandtherearwheelreachcriticalgripforceatthesametime.

Intermsofrollangleandrollvelocity,Fig.8-23clearlyshowsthemotorcyclefirstleaningovertotheleft(ϕ<0)andthentotheright(ϕ>0),totravelthroughthefirstturn,andthenbacktothelefttocompletethesecondturn.Notethattherollmaneuverbeginsrightawaywhenthemotorcycleisstillfarfromthefirstturn.Anticipatingthemaneuverreducesthelateralforcesonthetirestosomeextentandslightlydecreasestheturningradiusofthefirstturn.

Thechangeinyawangleandyawvelocityovertimeshows,especiallyearlyon,howcomplexamaneuverisrequiredtobetostartfromaninitialstateofperfect,steady,rectilinearmotionandgetreadyforthenextmaneuver.Thistransitionoccursinanextremelyshortspaceoftime,inordertoleaveasmuchtimeaspossiblefortherestofthemaneuver(whichisthemostimportantsegment).

8.7.3Anexample ofanoptimal maneuverfora“U”trajectory

Asasecondexampleofapplyingtheoptimalmaneuverprinciple,wewilllookata“U”trajectorywiththesameconstraintsandoptimizationcriteriaasabove(Fig.8-24).

The“optimal”trajectorysolutiongivesrelativelylittleweighttothedistancetobekeptfromtheedgesofthetrack.Asaresult,themotorcycletendstotravelonatrajectorywhichminimizesthecurvature(and/orkeepsthecurvatureconstantforaslongaspossible)usingtheentireroadwidth.Infact,themotorcyclemovesawayfromthecenterlinetomakeasmoothertransitionwherethetrackturnssharplyatthebeginningandendofthearc.

Thespeedandlongitudinalthrustgraph(Fig.8-25)showsthemotorcyclebrakinghardbeforetheturnandreacceleratingoutofit.Asinthefirstexample,theverticalforcecurvesillustratetheloadtransferphenomenoncausedbyinitialbrakingandsubsequentacceleration.

Fig.8-24“U”trajectory.

Fig.8-25“U”trajectory:motorcyclespeed,normalizedlongitudinalforce(thrustorbraking)andverticalforceswithrespecttotheweightforce.

Theinclinationoftheresultingreactionforces(Fig.8-26)againshowsthemotorcycletravelingthroughtheU-bendinthetrackwithanearlysteady,circularmotion.Hereaswell,thefirstpartofthegraphshowsaninitialmaneuvertowardstheoutsideoftheturn.

TherollanglegraphinFig.8-26showsthatthe“optimal”solutioncallsforthemotorcycletoleanawayfromdirectionoftheturninthefirstpartofthemaneuver(“entryphase”).Themotorcyclethenleansoverinthedirectionoftheturnandholdssteadyforalmosttheentire“U”.

Thegraphofyawangleandyawvelocityshowshowthemotorcycletravelsthroughtheturngradually,withalinearincreaseinyawangle.Italsoshowsaninitialmaneuvertowardstheoutsideoftheturnaspartofthe“optimal”solution.

Fig.8-26“U”trajectory:steeringangle,steeringtorque,lateralforces,rollangleandrollvelocity,

yawangleandyawvelocity.

8.7.4Influence ofthe adherence onthe trajectory

InthissectiontheeffectofdifferenttireadherenceisstudiedconsideringaU-turnmaneuver.Theothermaneuversarenotreportedbecausetheyshowsimilarconclusions.

Figure8-27isveryinterestingasitshowsthedifferentpathsfollowedbyavehicleasthelimitofadherencedecreases.Asisshown,withreducedadherencethevelocityatwhichthecurveistakendecreases.Inthiscase,itmakesmoresensetodrivestraightduringbrakingandaccelerationphasesyieldingapathwithasharpcurve.

Iftheadherencebecomessmallerthan0.5,thetrajectorycannolongertouchtheinnerborder.Infact,whenadherenceis0.31themotorcyclegoesstraightonandtouchestheexternallaneborderatverylowspeed.

Figure8-28showsthatduringtheapproachwithhighadherencethemotorcycleaccelerateswhilewithlowadherencethemotorcyclebeginstobrakeimmediately.Figure8-29showsthatdecreasingtheadherencedecreasesthetimeduringwhichthemotorcycleistilted.

Fig.8-27Trajectorycomparisoncarriedoutwithdifferenttireadherence.

Fig.8-28Velocitycomparisoncarriedoutwithdifferenttireadherence.

Fig.8-29Rollanglecomparisoncarriedoutwithdifferenttireadherence.

8.8Handl ingtes tsInpracticedefiningthemotorcycle’shandlingqualityisnotaneasytaskbecauseitconstitutesan

overallcharacteristicdeterminedbydifferentcomponentsofthevehicle(engine,brakes,aerodynamics,frame,tires).Moreover,thereisastrongsubjectiveinvolvementintheuseandratingofthemotorcycleonbehalfofthedriver,accordingtothedrivingstyleandsensitivity.

Handlingisusuallyassociatedwiththevehicle’sresponsetothecontrolaction.First,apromptresponsetothecontrolactionisrequiredintermsoflateralaccelerationandyawrate.Thispropertyhowevermustnotdecreasethestabilityandthecapabilityofdampingtheoscillationsthatmightariseduringcertainmaneuvers.Alowsideslipofthemotorcycleisalsorequired.Alowsensitivitytoexternaldisturbancesisneeded,aswellasauniformresponsetothecontrolactionatdifferentspeeds,differenttiresandroadsurfaces.Finally,constantfeedbackbetweenthevehicleandthedriverisrequired,sothatthedriveriscontinuallyawareofthevehicle’sdynamicstate.

Experimentalandsimulationtestssupplyusefuldataforthecomprehensionoftheactualdynamicbehaviorofthevehicle.Themotorcycleisassumedtobeasystemwithsomecontrolinputs(steeringangleortorque,forwardvelocity)andsomekinematicanddynamicoutputs.Thebehaviorofthemotorcycleisthusdescribedbythefunctionthatlinksinputstooutputsinperformingtypicalmaneuvers,suchasslalom,transientandsteadyturning,lanechange,obstacleavoidanceandsoon.

Theneedfordifferentkindsoftestsonthemotorcycleisaconsequenceofthefactthatitisnotpossibletodivideagenerictrajectoryintoasimplesequenceofcurvesandstraights,becausethecommandsgivenbythedrivertoperformamaneuveralsodependonthepreviousones.Forexample,thecommandsgiven(throughthesteeringandthethrottle)duringacurveofaslalomtestarecompletelydifferentfromthosegivenduringasteadyturningmaneuverperformedwiththesamespeedandturningradius.Inaddition,thewaythemotorcycleisdrivenalongaknownpathisdifferentfromthewayitisdrivenperformingthesamepathforthefirsttime,additionallyapremeditatedmaneuverisdifferentfromanemergencyoneeventhoughtheresultisthesame(forexample,theavoidanceofaknownvs.anunexpectedobstacle).Asaconsequence,differenttestshavetobeplannedtostudymotorcyclebehaviorintransientandsteadystateconditions.

8.8.1Steadyturningtes t

Thesteadystateturningtesthasproventobeanefficientandquantitativewaytoassesslow

frequencyandnon-transienthandlingpropertiesofmotorcyclesandothertwo-wheelers.Inputandoutputquantitiesareindeedconstantandthesteadystatevehicleresponseratiosandgainscanbemeasuredinarepeatableandusefulway.

Thequantitiesdescribingthedriver ’scontrolactionarethesteeringtorqueandthedriverleanangle,sincebothtrajectory(i.e.radiusofcurvature)andforwardvelocityaregiven.Inmostcasesdrivercontrolmainlyconsistsofthesteeringtorque,whereasleanangleandbodylateraldisplacementcanbeconsideredassecondarycontrolinputs(Fig.8-30).

Fig.8-30Steadyturningtest.

Vehiclemaneuverabilityandsteeringbehaviorcanbequantitativelyinvestigatedinsteadyturningtests.

Ingeneral,driverhandlingfeelingisrelatedtothesteeringeffortneededtoperformacertainmaneuver.Insteadyturningsuchfeelingisrelatedtothesteeringtorquenecessarytofollowtherequiredpathwiththegivenforwardvelocity.Foragoodfeeling,littletorqueshouldbeappliedtothehandlebarandpreferablyitshouldbenegative(i.e.awayfromthecurve).

Therelationbetweendriveractionandvehicleresponsecanbequantifiedbytheratiobetweenthesteeringtorqueandrollangle:

rollindex=τ/ϕ

orbetweenthesteeringtorqueandlateralacceleration:

Figure8-31showsexperimentalresultsintermsoftheaccelerationindexasafunctionofforwardspeedforasportmotorcycle.Theaccelerationindexismainlynegative(i.e.negativesteeringtorque,awayfromthecurve).Characteristicallyforagivenradiusittransitionsfromnegativetopositive(i.e.positivesteeringtorque,towardsthecurve)asspeedincreases.

Fig.8-31Accelerationindexversusvelocityforseveralturningradii.

Negativeappliedsteeringtorqueispreferablebecauseinthissituationthemotorcycle’sturningbehaviortendstobestable(Fig.8-32).Infact,atasufficientvelocity,ifthecontrolofthedriverissuddenlyremoved,themotorcycleaftersomelateraloscillationstendstofollowastraightpathwithoutcapsizing(i.e.thecapsizemodeisstable).Onthecontrary,withpositivetorque,ifthecontrolsuddenlystoppedapplyingtorquethesteeringanglewoulddecreaseandthemotorcyclecapsizes(i.e.thecapsizemodeisunstable).

Fig.8-32Positiveandnegativesteeringtorque.

Withregardtomotorcyclesteeringbehavior,theratiobetweentheactualturningradiusRcandtheidealturningradiusRc0(i.e.associatedwiththeidealtirebehavior)isconsidered:

Inparticular:

ξ<1

under-steering:theactualcorneringradiusisgreaterthantheidealoneandthemotorcycletendstorunonalargertrajectory(i.e.thefrontsideslipisgreaterthantherear);

ξ=1

neutralsteering:theactualcorneringradiusisequaltotheidealoneandthemotorcyclefollowsthekinematictrajectory(i.e.therearsideslipisalmostthesameasthefront);

ξ>1

over-steering:theactualcorneringradiusissmallerthantheidealoneandthemotorcycletendstorunonasmallertrajectory(i.e.therearsideslipisgreaterthanfront);

ξ=∞

criticalcondition:thevehicleturnsevenifthesteerangleisnull(i.e.Δ=0):thiscorrespondstocriticalspeed;

ξ<0

counter-steering:inthisconditionthehandlebarmustbesteeredawayfromthecurve(i.e.therearsideslipismuchgreaterthanfrontandnegativesteeringanglemustbeadoptedtocompensate).

Correlationswithexperttestriders’subjectiveopinionshaveshownthatthebestratingsoccurforvehicleswithneutralormodestover-steeringproperties(thistrendisinsharpcontrasttotypicalresultsforautomobiles,wheresmallamountsofunder-steerareuniversallypreferred).

Letusconsideranunder-steeringmotorcycle:sincethevehicletendstoexpandthecurve,therider,tocorrectthetrajectory,isobligedtoincreasetherollanglewhichinturnincreasesthesteeringangle(inordertoincreasethelateralreactionforceofthefrontwheel).Whentherotationofthehandlebarbecomesconsiderable,thereactionforceneededcanexceedthefrictionlimitbetweenthefronttireandtheroadsurface,withtheresultthatthewheelslidesandtheriderfalls.Amotorcyclethatisunder-steeringisthereforedangerous,sincevehiclecontrolisverydifficultafterthefrontwheelhaslostadherence.Ontheotherhand,withanover-steeringmotorcycle,incaseswheretheneededreactionforceovercomesthemaximumfrictionforcebetweenthereartireandtheroadplane,therearwheelslips,butanexpertrider,throughacounter-steeringaction,hasabetterchanceofcontrollingthevehicle’sequilibriumandavoidingafall.

Figure8-33showsthesteeringratioasafunctionofvelocityfordifferentturningradiifrom10to50m.Itcanbeobservedthatthesportmotorcyclehasawell-definedover-steeringbehaviorwhichbecomesmoreandmoremarkedasspeedincreases.Anywaycriticalspeedisnotreachedandnocounter-steeringisevident.Steeringratiofittinghasbeenperformedbythesimplifiedexpression

wheretheconstantγdependsonthecorneringandcamberstiffnessofthetires.Criticalspeedisnotreachedexperimentallybutisextrapolatedbylinearfitting(≃20m/s).

Fig.8-33Steeringratioversusvelocityforseveralturningradii.

Figure8.34showsexperimentaltestsofthesportmotorcycleconsideredaboveinaspeed–lateralaccelerationdiagram.Zerosteeringtorque(τ=0),zerosteeringtorquegradient(∂τ/∂A=0)andzerosteeringangle(ξ=∞)linesareplottedasforwardspeedandlateralaccelerationvaryanddifferentover-steeringandcounter-steeringzonescanbeidentified.

Fig.8-34Drivingzones.

Withregardtoover-steering,theO1zoneischaracterizedbynegativesteeringtorque,positivesteeringtorquegradientandpositivesteeringratio.Basedontheconsiderationsoftheprevioussections,theseconditionsarecorrelatedtogoodhandling:thecapsizemodeisinfactstable,thesteeringtorquedecreasesapproachingzeroaslateralaccelerationincreases(thusrequiringlighterridereffortasrollangleincreases),andfavorableover-steeringbehaviorisachieved.Itfollowsthatthesecombinationsofspeedandlateralaccelerationcanbeconsidereda“preferabledrivingzone”.

TheO2zoneissimilartothepreviouswiththeexceptionthatthesteeringtorquegradientisnegative.Thismeansthatsteeringtorquebecomesmorenegative(approachingitsrelativeminima)aslateralaccelerationincreases,thusrequiringgreaterridereffortasrollangleincreases.

TheO3zoneisalsosimilartotheO1butthesteeringtorqueispositive.Thismeansthecapsizemodeisunstable,thusrequiringevenmorerollstabilizationtobeachievedbytherider.Evenifthisconstitutesaninappreciablefractionoftheglobalcontrolbeingexerted,iteffectivelymakesdrivingevenmoredifficultandunsafe.Furthermoreinthiszonethepositivesteeringtorquebecomesunfavorablylargerasthelateralaccelerationincreases,thusrequiringgreaterriderefforttoperformvehiclecontrolathighrollangles.

Thesethreeover-steeringzonesarefullyaccomplishedexperimentally.

TheC1,C2,C3zonesaresimilarrespectivelytotheO1,O2andO3over-steeringzones,withtheexceptionthatsteeringratioisnegative.Thatiscounter-steeringanglebehaviorisachieved(i.e.steeringanglenotinaccordwithturningdirection),whichmayrequireacertainamountofexperienceandskilltobepracticedsafelyandmayormaynotbeperceptibletotherider.

Thesethreecounter-steeringzonesarenotaccomplishedexperimentally.

8.8.2“U”turntes t

Thecharacteristicsofamotorcycle’shandlingarenotdefinedsolelyonthebasisofthevalueof

thetorquetobeappliedunderconditionsofmovementonastationaryturn,butalsoonthebasisofotherparameters,suchasthetorqueneededtoleanthevehicletoasetanglefromthevertical,andthetimeusedtoreachthedesiredangle.

Fig.8-35“U”turntest.

J.Koch(1978),afterexperimentaltestsin“U”turns,proposedthefollowingindextoevaluatethevehicle’scapacitytoenteraturn:

Whereτpeakisthepeakvalueofsteeringtorque,ϕpeakisthepeakvalueofrollvelocity,andVistheforwardvelocity.

Fig.8-36Kochindexfordifferentmotorcyclesversusvelocity.

AstheforwardvelocityincreasestheKochindextendstowardsalimitvaluewhichdependsonthekindofmotorcycleandontheturnradius.Itisworthnotingthatallthepeakvaluesarereachedintransientphasesandnotinsteady-stateconditions.

AlowvalueofKochindexhighlightsthatwithahighforwardvelocitythereisahighrollspeedwithalowpeakinsteeringtorque;thesearethecharacteristicsofmotorcycleswithgoodhandling.

Thetransientbehaviorwhenenteringaturnismainlyinfluencedbycenterofmassheight,frontwheelinertia,frontframeinertiawithrespecttosteeringaxis,frameinertiawithrespecttorollingaxisandyawaxis.Figure8-36showsKochindexwhenenteringa“U”turnhavingaradiusequalto100mfordifferenttypesofmotorcycles.ThefigureshowsthatwhileincreasingthevelocitytheKochtendstoalimit.Theindexhighlightsandquantifiesthefactthatitiseasiertomaneuverwithalightscooterinsteadofaheavytouringmotorcycle.

8.8.3Slalomtes t

Insteady-slalomingconditionstheridercontrolsthemotorcyclethroughaperiodicactiononthesteeringsystemandthevehiclereactswithperiodicroll,yawandlateralmotion.Theslalomfrequencyis:

wherePisthespacingofthecones.

Thereforetheslalomtestconstitutesastudyoftheforcedresponseofthesystem,wheretheforcingexcitationisrepresentedbythesteeringtorqueand/orthesteeringangle.Themotorcycle’sresponsechangesbothinamplitudeandphaseasafunctionofspeedandslalomingfrequency.

Fig.8-37Slalomtest.

Figure8-38showsasampleoftheacquiredsignals’timehistoriesfora14mspacingslalomtestperformedatlowspeed.Thesteeringtorqueistendentiallyoppositetothepathcurvature.

Fig.8-38Slalomtestatlowvelocity:V=4.8m/s.

Figure8-39showsthesamesignals’timehistoriesforaslalomtestperformedatmediumspeed.Atthisparticularfrequencythesteeringtorqueis180°outofphasewithrespecttotherollangle.Thesteeringtorqueamplitudeisalmostunchangedifcomparedtothepreviousplot.

Figure8-40showstheslalomingbehaviorathighspeed.Itcanbeobservedthatthephasebetweensteeringtorqueandrollangleshiftsto90°.Thesteeringtorqueamplitudeismuchincreased.

Fig.8-39Slalomtestatmiddlevelocity:V=7.2m/s.

Fig8-40Slalomtestathighvelocity:V=15.2m/s.

Themostsuitablemathematicaltooltointerprettheresultsofaslalomtestisthetransferfunction,whichmakesitpossibletodescribethesystemresponse(rollangle)inrelationtotheinput

characteristic(steeringtorque)asafunctionofthefrequency:

Boththeamplitudeandthephaseofrolltransferfunctionareveryinteresting.Highratiobetweenrollangle|ϕ|andsteeringtorque|τ|meansthatalargemotorcyclerollmotionisobtainedwithlittlesteeringeffort,whilealargephasemeansthattherollanglefollowsthesteeringtorquewithatimelag.

Increasingthefrequency,thatisincreasingthespeediftheconespacingisfixed,themotorcyclesneedanincreasedsteeringefforttofollowtheslalompath.However,thedriverfeelingisdeterminedbythephaselagbetweenrollangleandsteeringtorqueratherthanthemaximumsteeringtorque:betterhandlingisassociatedwithmotorcycleshavingaquickresponsetothesteeringinput.Fig.8-41showsanexperimentaltransferfunction.Themagnitudeexhibitsamaximumvalueatabout8m/sthenincreaseswithfrequency.Themaximuminthemagnitudeoftherolltransferfunctioncorrespondstotheminimumeffortonthehandlebar;atthisvelocitytheweavemodeswitchesfromtheinstabilitytothestabilityzone.

Fig.8-41Frequencyresponsefunction.

8.8.4Lane change tes t

Thelanechangemaneuverrepresentsatypicaltransientmaneuver,andstronglydependsondriverskillandridingstyle.Variousdrivingstrategiesamongridersdifferfromeachotherdependingontheinitialcounter-steercarriedoutandonthemovementoftherider ’sbodywithrespectthemotorcycle.

Expertriderscarryoutthismaneuverwithahighinitialout-trackingandusetheirbodyinclination

toremainverticaloreventogenerateanadditionalinputwithrespecttosteeringtorque.

Inveryfastmaneuvers,inrealityriderstendtoapplynotonlysteeringtorques(i.e.torquesparalleltothesteeringaxis),butalsorollingtorques(i.e.perpendiculartosteeringaxis)tothesteeringsystem.

Fig.8-42Lanechangetest.

Thephasebetweenyawvelocityandsteeringtorqueseemstobethequantitymorehighlyperceivedbytheriderwhencarryingoutsuchamaneuver.

Basicallytheriderimpartssomecontrolaction,torque,causingthevehicletorollandyaw.Theratioofthepeak-to-peakmagnitudeofsteeringtorquetothepeak-to-peakrollrateisagoodindicatorofamotorcycle’smaneuverability.NormalizingthisquantitybyvelocityweobtaintheLaneChangeRollIndex:

wherethesubscriptp-pindicatespeak-to-peakvalues.

Fig.8-43ParametereffectsonLaneChangeRollIndexfordifferentgeometriesandspeed.

Thisindexrepresentstheeffortrequiredoftheriderintheformofsteeringtorquetoobtainadesiredvehicleresponseinrollrate.Figure8-43showstheresultsofnumericalsimulationvaryingcriticaldesignparameters.

Wecanseethatbyorientingtheengineinacounter-rotatingdirectionorbyreducingthefrontwheelspininertiathevalueofLCRollindexisdecreased.Thelowervaluesmeanthatlesseffortisneededtoperformthelanechange,inthiscaseduetothelessergyroscopiceffects.

AlsoobservetheasymptoticnatureoftheLCRollindexwithspeed.Atlowspeedsallfourvehicleconfigurationsaresimilarbutasspeedincreasesweapproacharegionwherethegyroscopiceffectsbecomedominant.Inthisregionthedifferencesbetweentheconfigurationsareobvious.InasimilarmannertheLCRollindexcanbeusedtocontrastthebehaviorofdifferentclassesofmotorcycles:touring,sport,cruiser,etc.Typicallyascooterwillexistatorbelow1whileatouringmotorcyclecanreachorexceed2.5N/(rad/s2).

8.8.5Obs tacle avoidance tes t

Atypicalmaneuverwhichcauseshighrollandyawspeedsistheobstacleavoidancetest.Insuchamaneuverthegyroscopiceffectofthefrontwheelcombinedwiththerollmotionhasafundamentalroleindeterminingthesteeringtorquethathastobeappliedbythedriver.

Thegyroscopiceffectcausesatorquearoundsteeringaxiswhosevalueis:

Fig.8-44Obstacleavoidancetest.

Intheprevioussectionswehavehighlightedthatthetotalsteeringtorqueisdeterminedbyalotoffactorsactingonthefrontframe,particularlybyverticalandlateralforces.Neverthelessinrapidmaneuversalmostthewholeeffortappliedbythedriveropposesthegyroscopicmomentcausedbyfrontwheelandrollmotion.

ThiscanbeeasilyseeninFig.8-45,wherethesteeringtorqueappliedbythedriveriscomparedwiththegyroscopiceffectcalculatedbymeansofthepreviousformula.

Firstofallweobservetheout-trackingtechnique;thedriverinitiallyappliesahighsteeringefforttotherighttomakethevehiclerollrapidlytowardstheleft.Afterthis,thereisasteeringtorquepeaktotheleft,correspondingtothenecessaryrotationofthesteeringsystem,andfinallyapositivepeakcorrespondingtothefinallineupofthesteeringsystemitself.

Themotorcyclehandlingsensationinsuchamaneuverisrepresentedbythetimelagbetweenthesteeringtorqueandtheyawvelocity;theshorterthistimelagis,thebettervehiclehandlingis.

Fig.8-45Steeringtorqueappliedinaobstacleavoidancetest.

8.9Dangerous dynamicphenomena

8.9.1Highs ide

Thisdangerousphenomenonisduetotheinteractionbetweenthesideslipforcewiththelongitudinalforceappliedontherearwheel.Itcanhappenduringabrakingmaneuverwhileenteringacurveorduringathrustingmaneuverwhileexitingfromacurve.Thehighsideduetothebrakinghasalreadybeenexplainedinthe2ndchapterwithreferencetothefrictionellipse.

Figure8-46showshowthe“high-side”fallduetothedrivingforcecomesabout.Toexitfromthecurvetheriderstartstothrusttherearwheel,thereforethelongitudinaldrivingforceincreasesasdoesthetotalfrictionforce(phase1).Thetotalfrictionforcereachesthelimitvalue,therearwheellosesgripandthereforetherearofthemotorcyclemovesoutwards(phase2),Theriderstopsaccelerating,reducingthethrustingforcesuddenly,andtherearwheeltakesgripagain(phase4),Thelargesideslip,whichisstillpresent,generatesalateralforceimpulsethatisnotbalanced.Theresultisthatthemotorcycleisviolentlytwistedandthrownupwards(phase5).

Tirebehaviorduringa“high-side”maybebetterunderstoodbylookingatFig.8-47,whichshowstheavailablelateraltireforcewhenalongitudinaldrivingtireforceispresent.Inbothdiagrams,theenvelopeofthefamiliesofcurvesisthefrictionellipse.TheinitialconditionisrepresentedbypointAinwhichalateralforceispresentandthesideslipangleisabout1.5°.Whenthedriverstartstoacceleratethemotorcycle,thepointmovesinthehorizontaldirectionandthesideslipangleincreasesinordertokeepthelateralforceconstantinthepresenceofanincreasinglongitudinalforce.

Fig.8-46Exampleofphenomenonknownas“highside”duetothrustingmaneuver.

Fig.8-47Lateralandlongitudinalforcesforvariousvaluesoflongitudinalslipκandsideslipλ.

ThelossofgriptakesplaceatpointBwhentheboundaryofthefrictionellipseisreached;alargesideslipangle(ofabout5°)ispresent.Whentheriderreleasestheacceleratortherearwheeltakesgripagain.

ThenewconditionisrepresentedbypointC,wherethereisstillalargesideslipanglebutwherethelongitudinalslipisnegligible.

Therefore,thelateralforceimpulsetakesplacebecausethelateralforceincreasessuddenlyfromthevalueofpointBtothevalueofpointC.Theimpulsetorque,producedbythelateralforce,isnotbalanced,consequentlythemotorcyclefalls.

Fig.8-48Exampleofsimulationofthephenomenonknownas“highside”.

8.9.2Kickback

Thesocalled“kickback”effectisaphenomenonthatconcernsmotorcyclestability.

Fig.8-49Exampleof“kickback”phenomena.

Roadundulationsortransversejointsontheroadsurfaceduringhighspeedriding(150-200km/h)canunloadthefrontwheelcausingittoliftfromtheroadsurface(phase1).Whenthefrontwheelisunloadedthefrontassemblyisnotinequilibriumaroundthesteeringaxis.Theridermovesthehandlebarinstinctivelyandthefrontwheelplanemovesoutoflinewithrespecttotheforwarddirectionofthemotorcycle(phase2).

Whenthefrontwheelmakescontactwiththeroadsurfaceagain(phase3)thefrontframeisnotinequilibriumwithrespecttothesteeringaxis.Duetothesteeringanglealargelateralforceisgenerated.Thissideforcekicksbackthehandlebarintheoppositedirectionwithrespecttothesteeringangle(phase4).Consequentlytheridercanlosecontrolofthemotorcyclewithdramaticeffects.

Thekickbackeffectdecreaseswhenusingfrontandreartireswithlowercorneringstiffness.Frameswithhighstructuralstiffnessmakethemotorcycle’sbehaviorworse.

8.9.3Chattering

Thechatterofmotorcyclesappearsduringbrakingandconsistsofavibrationoftherearandfrontunsprungmassesatafrequencyintherange17-22Hzdependingonthemotorcycles.Itappearsnearlyexclusivelyintheracingmotorcyclesandonlyinsometracksandinsomekindofmaneuvers.Thisvibrationcanbeverystrongandtheunsprungmassesaccelerationcanreach5g.Itisoftenobservedthatthewheelrotationfrequencyisclosetothechatterfrequency,whichsuggestsatirenonuniformity;forcevariationorrun-out,ormaybeimbalance.Thiswouldnormallyoccurinthemid-corner.

Fig.8-50Exampleof“chattering”phenomenon.

Thechatterisanauto-excitedvibrationandthisfactexplainswhyitappearssuddenlywhenthemechanismofauto-excitationisgenerated.

Thesuspensionsshockabsorbersarenotabletodampthesevibrations.Inthepresenceofchatteringmotorcycleguidanceinlimitconditionsbecomesverydifficult.

Themechanismofself-excitationofthesevibrationsisduetothecouplingoftherearwheelunsprungmassresonanceoscillationswiththefluctuationofthelongitudinalfrictionalforceinthecontactpatchofthetire.Thechattervibrationsbeginontherearandappearalmostinstantaneouslyonthefrontduetoenergytransferfromthereartothefrontwhichoccurswhentherearandfronthopresonancefrequenciesarecloseeachother.

Figure8-50showsabrakingmaneuverofaracingmotorcycle.Thefigurehighlightsthatthespeedofthefrontwheelislessoftherearwheelspeed,duringthebraking.Duringthebrakingtherearandfrontunsprungmassesbegintooscillateatafrequencyofabout20Hz.Theoscillationsdecreasedecreasingthebrakingrateanddisappearcompletelyduringthefollowingaccelerationphase.

8.9.4Bounce andweave coupl ingincornering

Thecouplingbetweentheout-of-planemode“weave”andthein-planemode“bounce”hasbeenpreviouslyanalyzed.Aswehaveseenthetwomodeshavesimilarvaluesfortheirfrequenciesandsimilarmodalshapes.

Fig.8-51Exampleof“bounce-weavecoupling”phenomena.

Thisphenomenonisvisibleinsomeracingbikeswhenexitingturnsduringtheaccelerationphase.Wideoscillationsoftherearframeandmovementsoftheswingingarmarevisible.Theriderisinclinedtoslowdowntodampoutthisphenomenon.Fluctuationsofthelongitudinalslip,dependingontheangularpositionoftheswingingarm,canincreasethebounceandweavecoupling.

8.10Structural s ti ffnes sStructuralstiffnessofthemotorcycleasawholeandofeverysinglecomponent(inessencefront

forks,chassisandswingarm)isakeyfactorindefiningtheperformancewithregardtohandlingandmaneuverabilityofthemotorcycle.

Modernmotorcycleshaveframes,swingingarmsandforksstifferthanoldervehicles.Beyondcertainvaluesoflateralandtorsionalstiffnessoftheframe,themotorcyclestabilitypropertiesnolongerdependinasignificantwayonthestructuralcharacteristics.Highvalueofstiffnessguaranteesprecisioninthetrajectoryandquickresponsetotheinputoftheriderbutalsopresentssomedisadvantages.

Forexample,vehicleswithgreatframestiffnessaresometimesfelttobenervousbytherider,especiallywhenpassingonatransversebumpsandalsoonwetroads.

Simulationresultsshowthatthelateralflexibilityofthefrontfork(orthetorsionalflexibilityoftheupperpartoftheframenearthesteeringhead)stabilizesthewobblemodeathighspeedandhasanoppositeeffectatlowspeed,whereasthetorsionalflexibilityoftheforkdoesnotappeartohavearemarkableinfluence.

Thelateralflexibilityoftheswinging-armoroftherearframeslightlystabilizestheweavemodeatveryhighspeedwhereasthetorsionalflexibilityoftheswinging-armoroftherearframehasacontraryeffect.

Themotorcycleinsteadyconditions,bothinlinearmotionandwhencornering,issubjectedtoforcesactinginitsplaneofsymmetrywhereasintransientconditionsitissubjectedtolateralforces

appliedonthewheelcontactpointwiththeroadandtoinertialforcesduetolateralaccelerations.Asanexample,whenthevehicleisinlinearmotionandencountersabumpinclinedwithrespecttothesurfaceroad,impulsivelateralforcesactingonthecontactpointoccur.Consequentiallyitseemstobeappropriate,whenmeasuringthestructuralstiffness,toapplyforcesonthecontactpointbetweenthetireandthesurfaceroad.

Lateralforcescausedeformationsofthevehiclethatgeneratelateraldisplacementsofthewheelsandalsorotationsofthewheelplane.Thelateraldisplacementsandthecomponentoftherotationofthewheelaroundtheverticalaxis,causeanincreaseofthetiresideslipandthisincreasesthedampingofthestructuralvibrationexcited.

If,instead,thedeformationofthewheelisatorsion,whichcouldbethoughtasatorsionalrotationabouttheaxisconnectingthetwocontactpointswiththeroad,thestructuralvibrationwillbelessdampedsincetherotationofthewheelplanedoesnotincreasethedampingofthevibrationexcited.

Asaresult,itisdesirabletohavemoderatelateralflexibilityandahighdegreeoftorsionalstiffness.

Measurementofthestiffnesspropertiesmaybecarriedoutusingmanydifferentapproaches.

8.10.1Structural s ti ffnes s ofthe whole motorcycle

Thestructureofthemotorcycleisexcitedmainlybytheforcesgeneratedinthecontactpatchofthetires.Theseforcesinclude:thelongitudinalforce,theverticalload,andthelateralforce.Inordertohighlighttheactualbehaviourthemotorcycle,stiffnessshouldbemeasuredbyapplyingtheforceinthisregion,asshowninFig.8.52.

Lateralandtorsionalstiffness,respectively,areexpressedbytheratios:

Therotationaxisistheintersectionofthesymmetryplaneoftherearassemblywiththerearwheelplaneinthedeformedposition.Iftherotationaxisisvertical(angleβ=0)thedeformationispredominantlytheflexuraltypewhileiftherotationaxisisratherhorizontal(angle≈90)thedeformationismainlytorsion.

Therotationalaxisclosetotherearcontactpatch(smallvalueofthearmb)meansthatthereismoreplanerotationwithrespecttolateraldeformation.Onthecontrarytherotationalaxisfarfromtherearwheel(largevalueofthearmb)meansmorelateraldeformationwithrespecttorotation.

Thevaluesformodernmotorcycles,withoutthecomplianceofthetire,varyintherangeof:lateralstiffness:Krear=0.1-0.2kN/mmtorsionalstiffness:Ktrear=1.5-3.0kN/°

Withtheenginelockedinsteadofthesteeringheadtheconceptsremainthesame.Inthiscasethevaluesofthestiffnessarelargerwithrespecttotheprevious.

Fig.8-52Loadingconditionforevaluatingtorsionalandlateralstiffnessoftherearassemblyofthemotorcycle.

ThestiffnessofthefrontframecanbemeasuredasshowninFig8-51.Alsointhiscasetherearetwopossibilities:

lockthesteeringhead,locktheengine.

Inthelattercasetheupperpartoftheframecontributestothedeformationofthefrontassembly.

Theinclinationoftherotationalaxisistypicallytiltedbyaboutβ≈2εtowardsfrontofthemotorcyclewithrespecttoaplaneperpendiculartothesteeringaxis.Alsointhiscaselateralandtorsionalstiffnessareexpressedbytheratios:

Thevaluesofmodernmotorcyclevaryintherange:lateralstiffness:Kfront=0.08-0.16kN/mm;torsionalstiffness:Ktfront=0.7-1.4kN/°.

Fig.8-53Loadingconditionforevaluatingtorsionalandlateralstiffnessofthefrontassemblyofthemotorcycle.

8.10.2Structural s ti ffnes s ofthe frame

Thetorsionalstiffnessoftherearframememberisgenerallymeasuredwiththeenginefitted.Itiscalculatedaboutanaxisatarightangletothesteeringheadandpassingthroughtheswingingarmpivotaxisandapplyingacouple(torque)aroundthisaxis.

Thelateralstiffnesscanalsoberepresentedbytheratiobetweentheforceappliedalongtheswingingarmpivotaxisandthelateraldeformationmeasuredinthatdirection.Theforcecanbeappliedwithanoffsetinordertoavoidtorsionaldeformation.

Lateralstiffnesstypicallyvariesdependingonthetypeofframeandonthemethodofengineattachment.

Insomecasesthemomentisappliedonthesteeringheadandthepivotaxisoftheswingingarmislocked.Therearesomesmalldifferencesinthetwodifferentmeasurementproceduresduetotheasymmetryoftheframe.

Thevaluesofmodernmotorcycle(sport1000cc.)varyintherange:lateralframestiffness:Kf=1–3kN/mm;torsionalframestiffness:Ktf=3-7kNm/°.verticalframestiffness:Kzf=5-10kNmm.

Fig.8-54Loadingconditionforevaluatingtorsionalstiffnessoftheframe.

Fig.8-55Loadingconditionforevaluatinglateralstiffnessoftheframe.

Fig.8-56Loadingconditionforevaluatinglongitudinalstiffnessoftheframe.

8.10.3Structural s ti ffnes s ofthe swingarm

Thevaluesofswingarmstiffnessareintherange:swingingarmlateralstiffnessKs=0.8-1.6kN/mm.swingingarmtorsionalstiffnessKts=1-2kNm/°;

Themono-shockswingingarmischaracterizedbyagreaterlateralandasmallertorsionalstiffnesscomparedwiththeclassicswingingarm.

Fig.8-57Loadingconditionforevaluatingtorsionalandlateralstiffnessoftheswingarm.

8.10.4Structural s ti ffnes s ofthe frontfork

Thefrontforkisthemostflexiblepartofthestructuralmotorcycle.

Fig.8-58Loadingconditionforevaluatingtorsional,lateralandlongitudinalstiffnessofthefork.

Thestiffnessesareintheranges:forklateralstiffnessKff=0.07-0.18kN/mm.forktorsionalstiffnessKtff=0.1-0.3kNm/°;

Example6

Calculatethelateralandtorsionalstiffnessofamotorcycle(withoutthetirecompliance)whichcomponentshavethefollowingvalues:

•lateralframestiffness: Kf=2.2kN/mm;

•torsionalframestiffness: Ktf=6.0kNm/°;

•swingarmlateralstiffness: Ks=1.4kN/mm.;

•swingarmtorsionalstiffness: Kts=1.0kNm/°;

•wheellateralstiffness: Ks=0.8kN/mm.

Thegeometryofrearassemblyisthefollowing:

•framelength: Lf=0.85m,ΔLf=Lf/3;

•swingarmlength: Ls=0.6m,ΔLs=Ls/3;

•wheelradius: Lw=0.3m,ΔLw=Lw/3;

•swingarminclination: αs=8°;

•frameinclination: αf=30°;

•torsionalaxisinclination: α=-12°.

Fig.8-59Compositionofthestructuralstiffnessofthemotorcyclecomponents.

TheequivalentlateralstiffnessatthecontactpointisequaltoKrear=0.16kN/mmTherotatingaxisisinclined,respecttotheground,ofanangleequalto45°andthedistanceoftherotatingaxiswithrespecttothecontactpatchisequalto0.76m.Thetorsionalstiffnesssplitalongthexaxisandzaxisare: and respectively.Intheexperimentaltestthevalueshouldbelessbecausetheconstraintscompliancesarenotinfinitive.

8.11Experimental modal analys isExperimentalmodalanalysiswasdevelopedintheaeronauticalfieldduringtheseventiesand

nowadaysithasfoundanapplicationintwo-wheeledvehicles.

Fig.8-60Structuralmodesofamodernsportingmotorcycle(massequalto190kg).

Themodalpropertiesofastructureareindependentfromtheexcitationandtheacquisitionpoint,soaseriesofexperimentalFRFs(FrequencyResponseFunctions)canbeacquiredinlaboratory,usinganexcitationsystemlikeanimpacthammerwithaloadcellorashakerabletogenerateafrequencysweep,andanacquisitionsystemthatusuallyincludesthree-axisaccelerometers.

Theacquisitionpointsaredisplacedalongthewholevehicle,creatingameshthatcanbeanimatedafterthatthemodalidentification(obtainedusingspecificalgorithmsabletoanalyzealltheacquiredFRFsatthesametime)hasbeenperformed.

Experimentalmodalanalysisisanimportanttechniquethatmakesitpossibletoobtainthemodalmodelofcomplexstructuresformedbydifferentparts,likeamotorcycle(frontandrearframe,wheels).Thismodelisgivenasresonancefrequencies,dampingandeigenvectors,startingfromexperimentalacquisitionsandnotfromavirtualmodellikeFEMcodes.

Figure8-60showsthefirstfourstructuralmodesforamodernsportingmotorcycle.Thefirstmodeshowslateralbendingofbothfrontforkandswingarminthesamedirection.Thesecondmodeshowsabendingdeformationoftherearpartofthechassisandoftheforkinthesamedirection;thereisalsoaremarkableinplaneandlateraldeformationoftheswingarm.Thethirdandthefourthmodesaremorecomplicatedandinvolvebendingandtorsionoftheswingarm,oftheforkandoftherearpartofthechassiswhilethemainpartofthechassishassmallerdeformations.

8.12Rigidbodyproperties andMozzi axisIfweconsiderthemotorcycleasasinglerigidbodymovinginspace,subjecttotranslationinthe

groundplaneandrotationsaboutitsrollandyawaxes,anumberofinterestingobservationscanbemade.

Motorcyclemotiondependsonthekindofmaneuverandontheridingstyle.Everymaneuverstartswithavariationofthefrontlateralforcegeneratedbythesteeringmotion.Ifarightwardimpulsiveforceisappliedtothefronttirecontactpoint,themotorcyclemoveswiththefollowingleaningandyawvelocity:

Thefrontwheelcontactpointlateralvelocityis:

Therollvelocityisnegative(leftwardmotion)whiletheyawingvelocityispositive(rightwardmotion).Therollvelocityincreasesiftherollmomentofinertia isdecreased,whiletheyawingvelocityincreases,iftheyawmomentofinertia isdecreased.Bothvelocitiesincreaseifthecorrespondingproductsofinertia(intheinertiatensor)becomemorenegative.

Fig.8-61Instantaneousrotationaxis.

Knowingtheforwardvelocity,theyawvelocity,andtherollvelocityofmotorcycleonecandefineavectorquantityknowastheMozziaxis,ortheinstantaneousaxisofrotationfortherigidbody.TheMozziaxiscanbeusedtodescribedifferenttransientmaneuversmadebythemotorcycle.

TheMozziaxis(Fig.8-61)isdescribedintheSAEcoordinatesystemby:

andxiscoincidentwiththebody’scenterofmasslocation.Theanglethattheaxismakeswiththegroundplaneis:

Intermsoftherigidbodypropertiestheslopeoftheinstantaneousaxisofrotationwithrespecttotheroadplanexyis:

Theangleincreases,increasingtherollinertiaanddecreasingtheyawone(i.e.lessrollmotionandmoreyaw).

Theverticaldistanceoftheaxisfromthemotorcyclecenterofmassincreases,astherollinertiaincreasesandproductofinertiatermsdecrease.

Peopleinvolvedinmotorcycledesignandracingoftenaskthemselvesthefollowingquestions:Whereistheinstantaneousaxisofthevehicleinmotion?Howdoyawandrollratecombine?

AnswerstothesequestionsmaybefoundinamathematicalwaybymakinguseoftheconceptoftheMozziaxis.TheplotsderivedfromtheMozziaxistheoryareusefulinhighlightingtheeffectsofvariationsinpath,vehicleproperties,andridingstyle.

ThegeometriclocioftheMozzitrace(thepointwheretheaxisintersectsthegroundplane)andthatoftheturncenterduringaslalommaneuverarepresentedinFig.8-62.

Fig.8-62Mozzitraceinaslalommaneuver.

Fig.8-63Mozzitraceinaenteringturnmaneuver.

Thelocusoftheturncenterisclosetoapiecewiselinear.Itisworthhighlightingthattheturncentercoordinatestendtoinfinityiftheyawvelocitytendstozero.Inthesinusoidalslalomthisconditionhappenswhenthetrajectorycrossesthexaxis.

ThelocusoftheMozzitraceisacurvewithperiodiccusps,whichalwayslieoninnersideofthepath.ThelocioftheMozzitraceandtheturncentershowperiodicintersectionpointsthattakeplacewhentheradiusofcurvatureisclosetotheminimumvalue.

Figure8.63dealswithamotorcyclethatisenteringarightcurvewithacounter-steertechnique.InthismaneuvertheMozziaxismovesfromtheoutsideofthepathtothecenterofthecurve.

Theyawrateisnegative(counter-yaw)atthebeginningofthemaneuverandpositiveduringtherestofthemaneuver.Therollratereachesamaximumafterthebeginningoftherotationtowardstherightandthentendstozero.TheMozzitracecomesfrom-∞,becausethepathisalmoststraightandtheyawrateisnegative,thenitcrossesthepathwhentheyawrateiszero.Finally,inthesteadyturningportionofthepath,theMozzitracetendstocoincidewiththeturncenter,becausetherollrateisclosetozero.

8.13Dynamicanalys is withmulti -bodycodesNowadays“multi-body”codesmakepossiblethepreciseandcompletedynamicanalysisofa

vehicle’soperationontheroadthroughtheuseofcomputersimulation.MSA(Multi-bodySystemAnalysis)representsthecomputerstudyofmovementsinmechanicalsystemsasaresultoftheapplicationofexternalforcesorstressesthatactonthesystem.Thespatialsystemsformingthesubjectofthestudyaresimulatedwithrigidbodiesand/orflexibleelementsconnectedtoeachotherwithvarioustypesofkinematicanddynamicconnections.Theexternalforcesandresultingreactionsleadtomovementsofthesystem’scomponentsthatsatisfyconstraintconditions.TheMSAcodesplayarolewhoseimportanceisonlydestinedtoincreasethroughtheuseofmodernintegratedcomputer-aideddesign.Theymakeitpossibletoevaluateandoptimizethecharacteristicsandperformanceofa

productevenbeforetheprototypephase,therebyassuringthereductionofdevelopmentcostsandthesystematicevaluationofalternativedesigns,andespeciallyreducingthetimetomarketforanewproduct.

Figure8-64illustratesbyexampleamodelofascooterthatrunsinrectilinearmotionalongaroadwayandencountersastep.Thechassisofthescooterisfurtheraffectedbytheunbalancingforceofthemotor.

Thecomputersimulationenablesustorepresentthecharacteristicsofthespringsandtheshockabsorbersevenwithnon-linearlaws.Inthesameway,thetirescanbemodeledemployingvariouslevelsofsophistication.

Figure.8-65showsaracingvehicleperformingawheelieduringacceleration,generatedbyahighthrustforce.Thein-planemodelingcanbeveryaccurateandcantakeaccountof,forexample,thecharacteristicsoftheenginetorque,theelasticcontributionofthegaspresentintheshockabsorbers,theslipcharacteristicsofthetiresintermsoftheload,etc.

Three-dimensionalmultibodycodeshavetobeusedtosimulatetheoperationsofthevehicleout-of-plane.Sincethevehicleisunstable,themodelrequiresacontrolsystembothfortheequilibriumofthevehicleandfortheexecutionofthedesiredmaneuver(Fig.8-66).

Recentadvancesinbothmulti-bodysoftwareandincontrolstrategiesandimplementationhaveyieldedmodelscapableofin-depthstudiesofdesignchangesandparametervariations,whichalsoprovidesignificantinsightintodrivertechniqueandskill-level.Inthenearfuturethevalidationofaparticularvehicledesignscanbesubstantiatedbeforeanymetalhasbeencutorwheelshavebeenlaced.

Fig.8-64Exampleofmodelingascooterwithamulti-bodycode.

Fig.8-65Exampleofmodelingofaracingvehiclewithamultibodycode.

Fig.8-66Exampleof3Dsimulationofthedynamicbehaviorofamotorcycle.

List of symbols

Coordinatesystems

(Pr,x,y,z) mobiletriadwiththeoriginintherearcontactpointPr,accordingtoSAEJ670

x forwardandparalleltothelongitudinalplaneofsymmetry

y lateralontherightsideofthevehicle

z downwardwithrespecttothehorizontalplane

(C,X,Y,Z) mobiletriadwiththeoriginintheturncenterpointC

X paralleltoxaxis

Y paralleltoyaxis

Z paralleltozaxis

(Ar,Xr,Yr,Zr) triadattachedtorearframe

(Af,Xf,Yf,Zf) triadattachedtofrontframe

r suffixforparametersofrearframe

f suffixforparametersoffrontframe

Kinematicsanddynamicsparameters

C turningcenterpoint

C pathcurvature

pathradiusofrearwheel

pathradiusoffrontwheel

V forwardvelocity

ξ steeringratio

δ steeringangle

Δ kinematicsteeringangle

Δ* effectivesteeringangle

μ pitchangleofthemainframe

ψ yawangleofthemainframe

σ thrustchainanglerespecttothegroundplane

τ transferloadanglerespecttothegroundplane

τi,j velocityratio

Ω yawangularvelocityaboutthez-axis

ν frequency

νp naturalfrequencyofpitchvibration

νb naturalfrequencyofbouncevibration

Motorcycleparameters

A frontalareaofthemotorcycle

a mechanicaltrail(trail)

an normaltrail

b longitudinaldistancefromrearaxistothemotorcyclemasscenter

c viscousdampingcoefficientofsuspension

c viscousdampingcoefficientofsteeringdamper

CD aerodynamicdragcoefficient

CL aerodynamicliftcoefficient

d forkoffset:distancefromthecenterofthefrontwheeltothesteeringaxis

h heightofmotorcyclemasscenter

momentofinertiaofmotorcycleaboutthex-axisthroughitscenterofmass(rollinertia)

productofinertiaofmotorcycleaboutthex-z-axesthroughitscenterofmass

momentofinertiaofmotorcycleaboutthey-axisthroughitscenterofmass(pitchinertia)

momentofinertiaofmotorcycleaboutthez-axisthroughitscenterofmass(yawinertia)

k suspensionstiffness

L swingingarmlength

m motorcyclemass

p wheelbase

P tirecontactpointwiththeground

G motorcyclecenterofmass

rp radiusofdrivesprocket

rc radiusofrearsprocket

Δh loweringofthesteeringhead

ε casterangle

φ swingingarmanglerespecttothegroundplane

ϕ rollangleoftherearframe(camberangleoftherearwheel)

β rollangleofthefrontframe(camberangleoftherearwheel)

η chainanglerespecttothegroundplane

Frontframeparameters

bf distanceofthemasscenteroffrontframefromsteeringaxis

hfdistanceofthemasscenteroffrontframefromthelinepassingthroughtherearwheelcenterandperpendiculartothesteeringaxis

Gf masscenteroffrontframe

kf effective(reduced)frontsuspensionstiffness

mf frontunsprungmass

Mf frontframemass

momentofinertiaofthefrontwheel

momentofinertiaoffrontframeaboutthexf-axisthroughitscenterofmass

productofinertiaoffrontframeaboutthexf-zf-axesthroughitscenterofmass

momentofinertiaoffrontframeabouttheyf-axisthroughitscenterofmass

momentofinertiaoffrontframeaboutthezf-axisthroughitscenterofmass

Rearframeparameters

br longitudinaldistanceofthemasscenterofrearframefromrearwheelaxis

hr heightofthemasscenterofrearframe

Gr masscenterofrearframe

kr effective(reduced)rearsuspensionstiffness

mr rearunsprungmass

Mr rearframemass

momentofinertiaoftherearwheel

momentofinertiaofrearframeaboutthexr-axisthroughitscenterofmass

productofinertiaofrearframeaboutthexr-zr-axesthroughitscenterofmass

momentofinertiaofrearframeabouttheyr-axisthroughitscenterofmass

momentofinertiaofrearframeaboutthezr-axisthroughitscenterofmass

ForcesandMoments

Ns staticloadonthewheel

N dynamicloadonthewheel

Na normalizeddynamicloadonthewheel:ratioofthedynamicloadtothemotorcycleweight

Ntr dynamicloadtransfer

FD aerodynamicdragforce

Fw rollingresistanceforce

FP resistantforceduetothesloperoad

F brakingforce

Fs lateralforceonthetire

M elasticmomentappliedtotheswingingarm

MX overturningmoment

MY rollingresistancemoment

MZ yawingmoment

Mt twistingmoment

P power

S drivingforce

Sa normalizeddrivingforce:ratioofthedrivingforcetothemotorcycleweight

T chainthrustforce

τ steeringtorqueappliedbytherider

ℜ squatratio:

μ braking/drivingforcecoefficient(normalizedlongitudinalforce):ratioofthebraking/drivingforcetotheverticalload

braking/drivingtractioncoefficient:themaximumvalueofthebraking/drivingforcecoefficient

lateraltractioncoefficient:themaximumvalueofthelateralforcecoefficient

Tiresandwheels

at tiretrail

d rollingfrictionparameter

fw rollingresistancecoefficient

R outsideradiusofthetire

ρ radiusoftorusrevolutionofthetire

t radiusofcrosssectionofthetire

L relaxationlengthofthetire

kp radialstiffnessofthetire

ks lateralstiffnessofthetire

kκ longitudinalslipstiffnesscoefficient:ratiooflongitudinalslipstiffnesstotheverticalload

kλ cornering(sideslip)stiffnesscoefficient:ratioofcorneringstiffnesstotheverticalload

kϕ camberstiffnesscoefficientofthetire:ratioofcamberstiffnesstotheverticalload

Kκ longitudinalslipstiffnessofthetire

Kλ cornering(sideslip)stiffnessofthetire

Kϕ camber(roll)stiffnessofthetire

Ro effectiverollingradius

λ sideslipangleofthetire

ω spinvelocityofthewheelaboutitsaxis

κ longitudinalslipofthetire

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Index

Acceleration-inrectilinearmotion-index-traction-limited-wheelinglimited

Aerodynamics-dragarea-dragforce-liftforce

BouncevibrationmodeBraking

-forwardflipover-optimalbraking

Camberangle (seealsorollangle)CapsizeCasterangle

-definition-variation

CenterofgravityChaintransmission

-inclinationangle-squatratio,angle

ChatteringComfortDamping

-optimalratio-reduced-ratio

Directionalbehavior-criticalvelocity-neutral-oversteering-understeering

DirectionalstabilityEnteringinaturn

-fast-slow-Uturn-chicane

Equilibrium-onacurve-rearsuspension-rectilinearmotion

Force-aerodynamic-braking-camber-cornering

-contact-drag-driving-lateralintransientstate-lateral-lift-longitudinal-rollingresistance-sideslip-transferload

ForwardflipoverFrictionellipseGyroscopiceffect

-generatedbyyawmotion-generatedbyrollmotion-generatedbysteering

Handling-optimalmaneuvermethod-handlingtest-lanechangetest-Kochindex-obstacleavoidancetest-slalomtest-steadystatetest-Uturntest

Highside

Kickback

InertiatensorIn-planedynamics

MagicformulaManeuverability(seealsohandling)MasscenterMotion

-rectilinear-steadyturning

Multi-bodycodesModel

-tire-ofmotorcycleonaturn-inplane(1d.off.)-inplane(2d.off.)-inplane(4d.off.)-mono-suspension-kinematic-weave(1d.off.)-wobble(1d.off.)-multi-body

ObstacleavoidancetestOffsetOptimalManeuvermethodOversteerPath

-curvature-radius

Pitch-angle-vibrationmode

Preload

Radius-tirerolling-path

Road-powerspectraldensity-roadexcitation-roadirregularities-roadslope

Roll-effectiveangle-frontwheelangle-idealangle-index-angle-motion

Rolling-rollingradius-rollingresistance

ScootersuspensionSelf-alignmentmomentShafttransmission

-squatratioShockabsorbers

-characteristics-single,doubleeffect

SideslipangleSlalomtestSlip

-longitudinal-side(seealsosideslip)

StabilitySquatratioSquatangleSteadyturningtestSteering

-criticalvelocity-effectiveangle-kinematicangle-angle-ratio-neutral-over-under-headheight-torque

Stiffness-progressive-reducedstiffness-degressive-structural

SwingingarmSuspension

-anti-divesuspension-four-barlinkagesuspension-frontsuspension-linkagesuspension-mono-suspension-preload-rearsuspension-swingingarm-Telelever

Telescopicfork

Tire-camberforce-contactpoint-frictionellipse-lateralforce-lateralstiffness-longitudinalforce-magicformula-overturningmoment-relaxationlength-rollingradius-self-aligningmoment-twistingmoment-vibrationalmodes-yawmoment-sideslipforce

Torque-steeringcomponent-steeringtorque

Trail-effectivetrail-mechanicaltrail-normaltrail-trailofthetire-variation

Transferload-inbraking-inrectilinearmotion

TransmissibilityTrim

-inacceleration-inbraking-incornering-squatratio-squatangle-rectilinearmotion

Twistingmoment

UndersteerUnsprungmass

Vibration-inplane(seebounce,pitch,wheelhop)-incornering-instraightrunning-outofplane(seecapsize,weave,wobble)-structuralmode-tiremode

WeaveWheel

-velocityincurve-climbingastep-contactpointsliding-hop

Wheelbase-definition

Wobble

Yaw-tiresizeeffectonyaw-.tiremoment