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附錄附錄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,Botman2,Kahraman3,4,andSaadaandVelex5.Naturalfrequencyplotsinthesestudies,especiallyRef.3,shownaturalfrequencyveeringphenomenawhentwoeigenvaluelociapproacheachotherasaparameterisvaried,butthenabruptlyveerawayliketwosimilarchargesrepelling(pointBinFig.2a).Thevibrationmodesoftheveeringeigenvaluesarestronglycoupledandundergodramaticchangesintheveeringneighborhood.Thephenomenonhasbeenstudiedextensively69,butithasnotbeenexploredinplanetarygears.Eigenvalueveeringisalsorelatedtomodelocalizationthatcanoccurwhendisorderisintroducedintonominallysymmetricsystemsliketurbineblades,spaceantennae,multispanbeams,andotherstructures6.Inthecaseofespeciallysharpveering,itissometimesdifficulttodistinguishbetweenintersectionandveeringjustbyobservingeigenvalueplots.Curveveering/crossingcomplexityobstructsthetracingofeigenvaluelociunderparameterchanges.Also,whenmultiplecurvesveerorintersectclosetogether(Fig.3),strongmodalcouplingandtheassociatedoperatingconditionresponsechangesthatoccurarenotidentifiablefromfrequencylociplots.Theobjectivesofthisworkweretwofold.Thefirstwastoanalyticallyderivesimplerulesthatpredicteigenvalueveeringinplanetarygears.Thesecondwastousetheveeringresults,alongwithpreviouslydevelopedmodalpropertiesandeigensensitivityanalysis,todefinemorefullytheinfluenceofmodelparametersonfreevibrationandtogiveguidancefortuningnaturalfrequencies.LinandParker10,11analyticallycharacterizedtheunique,highlystructuredpropertiesofplanetarygearnaturalfrequencyspectraandvibrationmodes.Theyalsoprovidedsimple,closed-formexpressionsforthesensitivitiesofnaturalfrequenciesandvibrationmodestodesignparameters12.Theseanalyticalresultsprovidethenecessaryfoundationforthepresentstudyofveeringrulesinplanetarygears.Theveeringrulesyieldconcreteconclusionsexpressedinsimpleformsforwhentwoapproachingeigenvaluesveerorcross.Theimportanceoftheveeringrulesistoidentifythoserangesinwhichsmallchangesindesignparameterscandramaticallychangethevibrationmodesandconsequentlytheresponse.Theresultsareillustratedonabenchmarkplanetarygear(themodelparametersandnaturalfrequenciesaregiveninTables1and2,respectively)usedinahelicopterpowertrain.ThelumpedparametermodelderivedinRef.10isusedhere.ThemodelisapplicableforgeneralepicyclicgearswithNplanets.Eachcomponenthastwotran