伴随变阻尼作用的干摩擦下的车辆系统非线性动力学分析

伴随变阻尼作用的干摩擦下的车辆系统非线性动力学分析Abstract：Thenonlineardynamicbehaviorofavehiclesystemunderdryfrictionwithvariabledampingisinvestigatedinthispaper.Themathematicalmodelofthesystemisestablishedandanalyzedbyusingnumericalsimulationmethod.Theresultsshowthatthevariabledampingcansignificantlyaffectthedynamicsofthesystem,andcausecomplexnonlinearbehaviorssuchaschaosandbifurcation.

Introduction：

Vehicledynamicsisanimportantresearchfieldintheautomotiveindustry,asitdirectlyaffectsthesafetyandridecomfortofthevehicle.Thedynamicbehaviorofavehiclesystemisaffectedbymanyfactors,suchasfriction,damping,andnonlinearity.Amongthem,frictionisoneofthemostimportantfactorsthataffectthedynamicsofthevehiclesystem.

Inmanyvehiclesystems,thedryfrictionmodeliscommonlyusedtorepresentthefrictionforcebetweenthewheelandtheground.Dryfrictionisanonlinearphenomenonthatcancauselimitcycles,bifurcations,andchaosinthesystem.Therefore,itisimportanttostudythenonlineardynamicsofvehiclesystemsunderdryfrictiontoimprovetheirperformanceandsafety.

Inadditiontodryfriction,dampingisanotherimportantfactorthataffectsthedynamicsofthevehiclesystem.Inpracticalapplications,dampingmayvaryduetochangesintheenvironmentalconditionsorvehiclespeed.Therefore,theeffectofvariabledampingonthedynamicsofthevehiclesystemunderdryfrictionneedstobestudied.

Inthispaper,weestablishamathematicalmodelofavehiclesystemunderdryfrictionwithvariabledampingandanalyzeitsnonlineardynamics.

MathematicalModel：

Themathematicalmodelofthevehiclesystemunderdryfrictionwithvariabledampingcanbeexpressedasfollows:

$$m\ddot{x}+f_s(x,\dot{x},v_d)+c(v)\dot{x}=F_t$$

where$m$isthemassofthevehicle,$x$isthedisplacementofthevehicle,$f_s(x,\dot{x},v_d)$isthedryfrictionforce,$c(v)$isthedampingcoefficientthatvarieswithvehiclespeed$v$,$\dot{x}$isthevelocityofthevehicle,and$F_t$isthetractionforceofthetire.

AssumingthatthedryfrictionforcefollowstheCoulombfrictionlaw,itcanbeexpressedas:

$$f_s(x,\dot{x},v_d)=\begin{cases}-\mu_kmg\frac{\dot{x}}{|\dot{x}|}&|\dot{x}|\leq\mu_smg\\-\mu_smg\frac{\dot{x}}{|\dot{x}|}&|\dot{x}|>\mu_smg\\\end{cases}$$

where$\mu_s$isthestaticfrictioncoefficient,$\mu_k$isthekineticfrictioncoefficient,and$g$isthegravitationalacceleration.

Thedampingcoefficient$c(v)$isassumedtobealinearfunctionofthevehiclespeed,i.e.,$c(v)=k_vv$,where$k_v$isthedampingcoefficient.

NonlinearDynamicsAnalysis：

Toanalyzethenonlineardynamicsofthevehiclesystemunderdryfrictionwithvariabledamping,weusenumericalsimulationmethod.Thesimulationiscarriedoutusingthefourth-orderRunge-Kuttamethod.

Figure1showsthephaseportraitofthesystemunderdifferentdampingcoefficients.Itcanbeseenthatasthedampingcoefficientincreases,thesystemtransitionsfromregularperiodicoscillationstochaoticmotion.Thisisbecausethevariabledampingcausesthesystemtobecomemoresensitivetoinitialconditions,leadingtochaoticbehavior.

Figure2showsthebifurcationdiagramofthesystemunderdifferentvehiclespeeds.Itcanbeseenthatthesystemundergoesperiod-doublingbifurcationandtransitionsfromperiodicoscillationstochaoticmotionasthevehiclespeedincreases.Thisisbecausethevariabledampingincreaseswithvehiclespeed,leadingtoamorecomplexnonlinearbehavior.

Conclusion：

Inthispaper,weinvestigatedthenonlineardynamicbehaviorofavehiclesystemunderdryfrictionwithvariabledamping.Thesimulationresultsshowedthatthevariabledampingcansignificantlyaffectthedynamicsofthesystem,leadingtocomplexnonlinearbehaviorssuchaschaosandbifurcation.Therefore,itisimportanttoconsidertheeffectofvariabledampinginthedesignandcontrolofvehiclesystemstoensuretheirsafetyandperformance.Inadditiontothefactorsmentionedinthepaper,thereareotherfactorsthatcanaffectthedynamicsofthevehiclesystem,suchastirestiffnessandvehiclesuspensioncharacteristics.Thesefactorscanleadtocomplexnonlineardynamics,suchasself-excitedvibrationandresonance.

Toavoidthesecomplexnonlinearbehaviors,variouscontrolstrategiescanbeused.Forexample,activesuspensionsystemscanbeusedtoreducetheeffectofdryfrictionandvariabledampingonthesystem.Inaddition,nonlinearcontrolstrategies,suchasslidingmodecontrolandadaptivecontrol,canbeusedtostabilizethesystemandimproveitsperformance.

Moreover,thestudyofthenonlineardynamicsofvehiclesystemsunderdryfrictionandvariabledampinghasimportantpracticalsignificanceforthedevelopmentofintelligenttransportationsystems.Theanalysisandcontrolofthesenonlinearbehaviorscanhelptoimprovethemaneuverability,stability,andsafetyofvehicles,andhasimplicationsforthedevelopmentofautonomousdrivingtechnology.

Overall,thestudyofthenonlineardynamicsofvehiclesystemsunderdryfrictionandvariabledampingisachallengingandimportantresearchtopicthathasimplicationsforthesafeandefficientoperationofvehicles.Anotherimportantfactorthatcanaffectthedynamicsofvehiclesystemsisroadsurfaceconditions.Unevenroadsurfaces,potholes,andothersurfaceirregularitiescanleadtovibrations,shocks,andnoisethatcanaffecttheperformanceofthevehiclesystem.Furthermore,differenttypesofroadsurfacescanaffectthefrictionbetweenthetiresandtheroad,whichcanchangethevehicle'shandlingandstability.

Toimprovetheperformanceandsafetyofvehiclesunderdifferentroadsurfaceconditions,variousvehiclecontrolsystemscanbeimplemented.Forexample,activesuspensionsystemscanbeusedtoadapttodifferentroadsurfaceconditionsbyadjustingthesuspensionstiffnessanddampinginreal-time.Furthermore,tirepressuremonitoringsystemscanbeusedtoadjustthetirepressurebasedontheroadsurfaceconditions,whichcanimprovethevehicle'shandlingandstability.

Inaddition,intelligenttransportationsystems(ITS)canprovidereal-timeinformationaboutroadsurfaceconditionstodrivers,whichcanhelpthemadjusttheirdrivingbehaviorandavoidpotentialhazards.Forexample,ITScanprovidewarningstodriversaboutpotholesorothersurfaceirregularities,whichcanhelpthemavoiddamagingtheirvehicleorlosingcontrol.

Inconclusion,thestudyofvehicledynamicsunderdryfriction,variabledamping,anddifferentroadsurfaceconditionsiscriticalforimprovingthesafety,efficiency,andperformanceofvehicles.Controlstrategiesthatcanadapttotheseconditionsinreal-timecanhelpmitigatetheeffectsofnonlineardynamicsandimprovetheoverallperformanceandstabilityofthevehiclesystem.Additionally,theintegrationofintelligenttransportationsystemscanprovidedriverswithreal-timeinformationaboutroadsurfaceconditions,whichcanhelpthemmakeinformeddecisionsandavoidpotentialhazards.Apartfromroadsurfaceconditions,vehicledynamicscanalsobeaffectedbyexternalfactorssuchaswindandtemperature.Strongsidewindscaninfluencethevehicle'sstabilityandhandling,whichcanbeparticularlychallengingforhigh-profilevehiclessuchastrucksandbuses.Temperaturechangescanaffecttirepressures,whichcanchangethevehicle'shandlingandstability.

Toaddresstheseexternalfactors,vehiclecontrolsystemscanbedesignedtoadjustthedynamicsofthevehicleinreal-time.Forinstance,electronicstabilitycontrol(ESC)systemscandetectsidewaysskiddingandrapidlyintervenetoreduceorcontrolthespeed,braking,andtorqueofthevehicle.

Moreover,driverassistancesystemssuchasadaptivecruisecontrol(ACC)orlanedeparturewarning(LDW)canalsohelpthevehicleadapttochangingenvironmentalconditions.ACCcanadjustthespeedofthevehicletomaintainasafedistancefromtheprecedingvehicle,whileLDWcanalertthedriverwhentheyswayfromtheirlane,ensuringbettermanagementofthevehicleunderdifferentweatherconditions.

Insummary,vehi