外文翻译--行星齿轮固有频率-精品

附录附录1英文原文NATURALFREQUENCYVEERINGINPLANETARYGEARSABSTRACTToachievenoiseandvibrationreductioninplanetarygearapplications,keydesignparametersareoftenvariedtoavoidresonances,optimizeloaddistribution,andreduceweight.Intheplotsofnaturalfrequenciesversusplanetarygearparameters,veeringphenomenaoccurwhentwoeigenvaluelociapproacheachother,butthenabruptlyveeraway.Theimportanceofveeringismanifestedinthedramaticchangesinthevibrationmodesoftheveeringnaturalfrequenciesandtheconsequentimpactonresponse.Thisworkanalyticallycharacterizestherulesofeigenvalueveeringinplanetarygears.Thecouplingfactorsbetweentwocloseeigenvaluelociareapproximatedbyperturbationanalysis.Specialveeringpatternswereobtainedusingtheuniquepropertiesofplanetarygearvibrationmodes.Theresultsareillustratedbyanexample.Keydesignparameterswereinvestigated,andgeneralizedguidanceisprovidedfortuningplanetarygearnaturalfrequencies.I.INTRODUCTIONNoiseandvibrationreductionarecriticalconcernsinplanetarygearapplications.Duringthedesignprocess,systemparametersarevariedtoevaluatealternativedesignchoices,avoidresonances,optimizeloaddistribution,andreduceweight.Itisimportanttocharacterizetheeffectsofparametervariationsonthenaturalfrequenciesandvibrationmodesforeffectivevibrationtuning.Inplanetarygeardynamicmodels(Fig.1),thekeydesignparametersincludethemeshstiffnesses,support/bearingstiffnesses,componentmasses,andmomentsofinertia.SomeplotsofnaturalfrequenciesversusplanetarygearparametersarepresentedbyCunliffeetal.[1],Botman[2],Kahraman[3,4],andSaadaandVelex[5].Naturalfrequencyplotsinthesestudies,especiallyRef.3,shownaturalfrequencyveeringphenomenawhentwoeigenvaluelociapproacheachotherasaparameterisvaried,butthenabruptlyveerawayliketwosimilarchargesrepelling(pointBinFig.2a).Thevibrationmodesoftheveeringeigenvaluesarestronglycoupledandundergodramaticchangesintheveeringneighborhood.Thephenomenonhasbeenstudiedextensively[6–9],butithasnotbeenexploredinplanetarygears.Eigenvalueveeringisalsorelatedtomodelocalizationthatcanoccurwhendisorderisintroducedintonominallysymmetricsystemsliketurbineblades,spaceantennae,multispanbeams,andotherstructures[6].Inthecaseofespeciallysharpveering,itissometimesdifficulttodistinguishbetweenintersectionandveeringjustbyobservingeigenvalueplots.Curveveering/crossingcomplexityobstructsthetracingofeigenvaluelociunderparameterchanges.Also,whenmultiplecurvesveerorintersectclosetogether(Fig.3),strongmodalcouplingandtheassociatedoperatingconditionresponsechangesthatoccurarenotidentifiablefromfrequencylociplots.Theobjectivesofthisworkweretwofold.Thefirstwastoanalyticallyderivesimplerulesthatpredicteigenvalueveeringinplanetarygears.Thesecondwastousetheveeringresults,alongwithpreviouslydevelopedmodalpropertiesandeigensensitivityanalysis,todefinemorefullytheinfluenceofmodelparametersonfreevibrationandtogiveguidancefortuningnaturalfrequencies.LinandParker[10,11]analyticallycharacterizedtheunique,highlystructuredpropertiesofplanetarygearnaturalfrequencyspectraandvibrationmodes.Theyalsoprovidedsimple,closed-formexpressionsforthesensitivitiesofnaturalfrequenciesandvibrationmodestodesignparameters[12].Theseanalyticalresultsprovidethenecessaryfoundationforthepresentstudyofveeringrulesinplanetarygears.Theveeringrulesyieldconcreteconclusionsexpressedinsimpleformsforwhentwoapproachingeigenvaluesveerorcross.Theimportanceoftheveeringrulesistoidentifythoserangesinwhichsmallchangesindesignparameterscandramaticallychangethevibrationmodesandconsequentlytheresponse.Theresultsareillustratedonabenchmarkplanetarygear(themodelparametersandnaturalfrequenciesaregiveninTables1and2,respectively)usedinahelicopterpowertrain.ThelumpedparametermodelderivedinRef.10isusedhere.ThemodelisapplicableforgeneralepicyclicgearswithNplanets.Eachcomponenthastwotranslationalandonerotationaldegreeoffreedom(DOF)inplanarmotion,sothesystemhasL=3(N-3)DOF.Numericalresultspresentedareforfixedringconfigurations,andL=3(N-2)inthiscase.TheassociatedeigenvalueproblemisII.VEERING/CROSSINGCRITERIONAmethodfordetectingeigenvalueveering/crossingingeneraldynamicsystemsisintroducedhere.PointBinFig.2aisanexampleofveering.Whentwoeigenvaluelociveeraway,theirlocicurvaturesindicatetheabruptnessofcurvedirectionchanges.PerkinsandMote[7]proposedaveering/crossingcriterionbyestimatingthelocicurvatureintheveeringneighborhoodfordistincteigenvalues.LetrandλsbetwoeigenvaluesapproachingeachotheraroundpointBasaparameterunperturbedeigensolutionsatBare(λr,Φr)and(λs,Φs).ApplyingTaylorexpansionaroundB,theeigenvaluelociareapproximatedbyofthevaryingparameter;λ=λ/;λ=.Thesecondderivativesλrandλsrepresentthecurvaturesoftheloci.Ifλr=λs=0,thesetwolociarelocallyindependentandfreetor,λs0,thelocidivergeandveeringoccurs.Largerλeigenvalueλr,theeigenvaluederivativesareIntheveeringvicinitywhereλr≈λs,λrandλsaredominatedbytermswithasmalldenominator,thatis,Thecouplingfactorsrandsapproximatethelocalcurvatures.Theyareusedtoestimateveeringstrength.Figure2bshows=14=18versusthevaryingparameterkspfortheveeringloci14and18inFig.2a.N