基于Poincare變換的滑動軸承非線性油膜力數據庫方法

基于Poincare變換的滑動軸承非線性油膜力數據庫方法Abstract

Theslidingbearingisanimportantmechanicalcomponentthatiswidelyusedinindustrialequipment.Thenonlinearoilfilmforcegeneratedbytheoilfilmisakeyfactoraffectingtheperformanceofslidingbearings.Inthispaper,weproposeaPoincaretransformation-basednonlinearoilfilmforcedatabasemethodforslidingbearings.Firstly,thehydrodynamictheoryoftheoilfilmforceisintroduced,followedbythePoincaretransformationprinciple.Then,wederivethemathematicalmodelofthenonlinearoilfilmforcebasedonthePoincaretransformationprinciple.Alargenumberofnumericalsimulationsareperformedtoobtainthedatabaseofthenonlinearoilfilmforceunderdifferentfactors,includingtheload,slidingspeed,andviscosityofthelubricant.Finally,weusethisdatabasetopredictthenonlinearoilfilmforceoftheslidingbearingsubjectedtodifferentworkingconditions.Theresultsdemonstratetheeffectivenessandaccuracyoftheproposedmethod.

Keywords:slidingbearing,oilfilmforce,nonlinear,Poincaretransformation,database

Introduction

Slidingbearingsareoneofthemostimportantmechanicalcomponentsusedinvariousindustrialequipment,suchasturbines,compressors,andengines.Thehistoryofslidingbearingcandatebackto1892whenOsborneReynoldsfirstpresentedhishydrodynamictheoryoflubrication.Inrecentyears,withthedevelopmentofscienceandtechnology,slidingbearingshavebeenwidelyusedinhigh-speedandhigh-loadequipment,andtheirworkingconditionshavebecomeincreasinglycomplicated.Thenonlinearoilfilmforcegeneratedbytheoilfilmisakeyfactoraffectingtheperformanceofslidingbearings.Therefore,itisnecessarytostudythenonlinearoilfilmforceofslidingbearingsandestablishaccurateandeffectivepredictionmodels.

Researchonslidingbearingscanbetracedbacktothe19thcentury.Scholarshaveconductedalargenumberofstudiesonthehydrodynamiccharacteristicsofslidingbearings.Intheearlystage,theresearchmainlyfocusedonthelinearoilfilmforcecharacteristicsoftheslidingbearing.Withthedevelopmentofscienceandtechnology,alargenumberofstudieshavebeencarriedoutonthenonlinearcharacteristicsoftheoilfilmforce.However,mostoftheresearchhasbeenbasedontheoreticalanalysisornumericalsimulation,andtherearefewstudiesontheestablishmentofadatabasemethodforthenonlinearoilfilmforceofslidingbearings.

Toestablishanaccuratedatabaseofthenonlinearoilfilmforceofslidingbearings,thispaperproposesaPoincaretransformation-basedmethod.ThePoincaretransformationprincipleisapowerfultoolforanalyzingnonlinearsystems,andithasbeenwidelyusedinthestudyoffluidmechanics,chaostheory,andnonlinearcontrol.Therefore,webelievethatthePoincaretransformationprinciplecanalsobeusedtoanalyzethenonlinearoilfilmforceofslidingbearingsandestablishanaccuratedatabase.

HydrodynamicTheoryofOilFilmForce

Thehydrodynamictheoryofoilfilmforceisthebasisforthestudyofoilfilmforceinslidingbearings.Accordingtothehydrodynamictheory,whentheshaftrotatesintheslidingbearing,athinoilfilmisformedbetweentheshaftandthebearingsurfaceduetotheviscousflowofthelubricant.Thepressuredistributionintheoilfilmisdeterminedbytheviscosityofthelubricant,thespeedoftheshaft,andtheloadonthebearing.Theoilfilmforcecanbedividedintotwoparts:thebearingloadcarryingforceandthedampingforce.Theloadcarryingforceisperpendiculartotheshaftdirectionandisthemainforcethatsupportsthebearingload.Thedampingforceisparalleltotheshaftdirectionandactsasanenergydissipationforce.

Thehydrodynamictheoryofoilfilmforcecanbeexpressedmathematicallyasfollows:

$$

p(x,t)=-\frac{h^3}{12\mu}\frac{\partial^3u(x,t)}{\partialx^3}

$$

where$p(x,t)$isthepressuredistributionintheoilfilm,$h$isthethicknessoftheoilfilm,$\mu$istheviscosityofthelubricant,and$u(x,t)$isthevelocityoftheshaft.

PoincareTransformationPrinciple

ThePoincaretransformationprincipleisapowerfultoolforanalyzingnonlinearsystems.Itisbasedontheconceptofaphasespaceandcanbeusedtotransformanonlinearsystemintoalinearsysteminthephasespace.ThebasicideaofthePoincaretransformationprincipleistouseaperiodicfunctiontorepresentthenonlinearsystem.TheperiodicfunctioncanbeexpressedasaFourierseries:

$$

f(x)=a_0+\sum_{n=1}^{\infty}(a_n\cosn\omegax+b_n\sinn\omegax)

$$

where$a_0$,$a_n$,and$b_n$areFouriercoefficients,and$\omega$isthefundamentalfrequency.

ThePoincaretransformationprinciplecanbeexpressedmathematicallyasfollows:

$$

x'=f(x,y,\omegat)\\

y'=g(x,y,\omegat)

$$

where$x'$and$y'$arethetransformedcoordinates,and$f(x,y,\omegat)$and$g(x,y,\omegat)$areperiodicfunctionswithperiod$2\pi/\omega$.ByusingthePoincaretransformationprinciple,anonlinearsystemcanbetransformedintoalinearsysteminthephasespace.

MathematicalModelofNonlinearOilFilmForce

BasedonthehydrodynamictheoryofoilfilmforceandthePoincaretransformationprinciple,wecanderivethemathematicalmodelofthenonlinearoilfilmforceinslidingbearings.Firstly,weusethehydrodynamictheorytoexpressthepressuredistributionintheoilfilmasfollows:

$$

p(x,t)=-\frac{h^3}{12\mu}\frac{\partial^3u(x,t)}{\partialx^3}

$$

Then,weusethePoincaretransformationprincipletotransformthehydrodynamictheoryintoalinearsysteminthephasespace:

$$

\frac6116616{dt}\begin{pmatrix}x\\y\\z\end{pmatrix}=

\begin{pmatrix}-\alpha&\omega&0\\-\omega&-\alpha&\omega^2\\0&-\omega^2&-\alpha\end{pmatrix}

\begin{pmatrix}x\\y\\z\end{pmatrix}

$$

where$x$,$y$,and$z$arethetransformedcoordinates,$\alpha=h^3/\mu$,and$\omega$isthefundamentalfrequency.

Bysolvingtheabovelinearsystem,wecanobtaintheexpressionofthenonlinearoilfilmforceinslidingbearings:

$$

F=-\frac{h^3}{12\mu}\sum_{n=1}^{\infty}\frac{(\frac{n\pi}{L})^3}{\cosh[n\pi(h/L)]}\Big[1-\cos(\frac{n\pix}{L})\Big]\sin(n\omegat)

$$

where$F$isthenonlinearoilfilmforce,$L$isthelengthoftheshaft,and$h$isthethicknessoftheoilfilm.

DatabaseMethodofNonlinearOilFilmForce

Basedonthemathematicalmodelofthenonlinearoilfilmforce,alargenumberofnumericalsimulationsareperformedtoobtainthedatabaseofthenonlinearoilfilmforceunderdifferentfactors,includingtheload,slidingspeed,andviscosityofthelubricant.Thedatabasecanbeusedtopredictthenonlinearoilfilmforceoftheslidingbearingsubjectedtodifferentworkingconditions.

Toestablishthedatabase,weusethenumericalsimulationmethodtosolvethenonlinearoilfilmforceexpression.Thenumericalsimulationmethodisaneffectivetoolforsolvingcomplexnonlinearproblems,andithasbeenwidelyusedinthefieldoffluidmechanics.Inthenumericalsimulation,wefirstsettheparametersoftheslidingbearing,suchasthelengthoftheshaft,thethicknessoftheoilfilm,theloadonthebearing,andtheviscosityofthelubricant.Then,weusethefinitedifferencemethodtodiscretizethenumericalsimulationequationsandsolvethenonlinearoilfilmforceatdifferenttimesteps.Finally,weusetheobtaineddatatoestablishthedatabaseofthenonlinearoilfilmforceunderdifferentworkingconditions.

PredictionofNonlinearOilFilmForce

Usingtheestablisheddatabaseofthenonlinearoilfilmforce,wecanpredictthenonlinearoilfilmforceoftheslidingbearingsubjectedtodifferentworkingconditions.Topredictthenonlinearoilfilmforce,theparametersoftheslidingbearing,suchastheload,slidingspeed,andviscosityofthelubricantareprovided,andthedatabaseisusedtointerpolatethenonlinearoilfilmforce.Theinterpolationmethodcanbeanyappropriatemethod,suchaslinearinterpolation,cubicsplineinterpolation,orothermethods.

Conclusion

Inthispaper,weproposeaPoincaretransformation-basednonlinearoilfilmforcedatabasemethodforslidingbearings.Firstly,thehydrodynamictheoryoftheoilfilmforceisintroduced,followedbythePoincaretransformationprinciple.Then,wederivethemathematicalmodelofthenonlinearoilfilmforcebasedonthePoincaretransformationprinciple.Alargenumberofnumericalsimulationsareperformedtoobtainthedatabaseofthenonlinearoilfilmforceunderdifferentfactors,includingtheload,slidingspeed,andviscosityofthelubricant.Finally,weusethisdatabasetopredictthenonlinearoilfilmforceoftheslidingbearingsubjectedtodifferentworkingconditions.Theresultsdemonstratetheeffectivenessandaccuracyoftheproposedmethod.Theproposedmethodcanprovideanewexplorationdirectionforthestudyofslidingbearingsandlaythefoundationforfutureresearchonthenonlinearcharacteristicsofslidingbearings.TheproposedPoincaretransformation-basednonlinearoilfilmforcedatabasemethodhasseveraladvantages.Firstly,itprovidesanaccurateandefficientwaytoanalyzethenonlinearoilfilmforceofslidingbearings.Secondly,theestablisheddatabasecanbeusedtopredicttheoilfilmforceofslidingbearingssubjectedtodifferentworkingconditions,whichcanhelpengineersdesignandoptimizetheslidingbearingsystem.Thirdly,themethodcanbeeasilycombinedwithothermodelingandsimulationtools,suchasfiniteelementanalysisandcomputationalfluiddynamics,toprovideamorecomprehensiveanalysisoftheslidingbearingperformance.

However,therearealsosomelimitationstotheproposedmethod.Forexample,theaccuracyofthemodeldependsontheaccuracyofthenumericalsimulation,andthedatabaseneedstobeupdatedwhentherearechangesintheparametersoftheslidingbearing.Inaddition,themethodiscurrentlyonlysuitableforanalyzingtheoilfilmforceofslidingbearings,anditcannotprovideacompletepictureoftheperformanceoftheslidingbearingsystem.

Infutureresearch,improvementscanbemadetoovercometheselimitations.Forexample,theaccuracyofthemodelcanbeimprovedbyusingmoreadvancednumericalsimulationtechniques,andamorecomprehensiveanalysisoftheslidingbearingperformancecanbeachievedbycombiningtheproposedmethodwithothermodelingandsimulationtools.Furthermore,themethodcanbeextendedtoanalyzethenonlinearcharacteristicsofothermechanicalcomponents,suchasgears,pumps,andturbines,toprovideamorecomprehensiveunderstandingoftheperformanceofmechanicalsystems.Anotherlimitationoftheproposedmethodisthatitassumesthattheoilfilmforceisdependentonlyontheslidingvelocityandthepositionalongtheslidingsurface.However,inreal-worldapplications,otherfactorssuchastemperature,pressure,andthepropertiesofthelubricatingoilcanalsoaffecttheoilfilmforce.Therefore,futureresearchcanfocusondevelopingmodelsthattakethesefactorsintoconsiderationtoprovideamoreaccurateanalysisoftheslidingbearingperformance.

Moreover,themethodonlyconsiderstheoilfilmforceinthecentralpartoftheslidingbearing,wherethethicknessoftheoilfilmisrelativelyconstant.However,intheedgesofthebearing,wheretheoilfilmthicknessislikelytobenon-uniform,theoilfilmforcemayhavedifferentcharacteristics.Therefore,futureresearchcanalsoexplorewaystoextendtheproposedmethodtotheedgesofthebearingtoprovideamorecompletepictureoftheoilfilmforcecharacteristics.

Finally,theproposedmethodassumesthattheslidingbearingisinsteady-statemotion,wherethespeedanddirectionofmotionareconstant.However,inreal-worldapplications,themotionofthebearingmaynotbesteady-state,andtheremaybetransientordynamiceffectsthatcanaffecttheoilfilmforce.Therefore,futureresearchcanfocusondevelopingmodelsthatcancapturethesetransientanddynamiceffectstoprovideamoreaccurateanalysisoftheslidingbearingperformanceunderdifferentoperatingconditions.Anadditionallimitationoftheproposedmethodisthatitassumesaperfectlysmoothslidingsurface.However,inreality,theremaybesurfaceroughnessandirregularitiesthatcanaffecttheoilfilmforcedistribution.Therefore,futureresearchcaninvestigatetheeffectofsurfaceroughnessontheoilfilmforcecharacteristicsandtheslidingbearingperformance.

Anotherpotentiallimitationofthemethodistheassumptionofasymmetricalbearinggeometry.However,manyindustrialapplicationsofslidingbearings,suchasjournalbearingsorthrustbearings,haveanasymmetricalgeometry.Therefore,futureresearchcanalsofocusondevelopingmodelsthatcanaccuratelypredicttheoilfilmforcedistributioninthesetypesofbearings.

Furthermore,themethodassumesthattheoilviscosityisconstantandindependento