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基于Poincare變換的滑動軸承非線性油膜力數(shù)據(jù)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:
$$
\fracarthvb2{dt}\begin{pmatrix}x\\y\\z\end{pmatrix}=
\begin{pmatrix}-\alpha&\omega&0\\-\omega&-\alpha&\omega^2\\0&-\omega^2&-\alpha\end{pmatrix}
\begin{pmatrix}x\\y\\z\end{pmatrix}
$$
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
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