外文翻译--行星齿轮固有频率-精品Word文件下载.doc
- 文档编号:4641919
- 上传时间:2023-05-03
- 格式:DOC
- 页数:32
- 大小:3.29MB
外文翻译--行星齿轮固有频率-精品Word文件下载.doc
《外文翻译--行星齿轮固有频率-精品Word文件下载.doc》由会员分享,可在线阅读,更多相关《外文翻译--行星齿轮固有频率-精品Word文件下载.doc(32页珍藏版)》请在冰点文库上搜索。
Toachievenoiseandvibrationreductioninplanetarygearapplications,keydesignparametersareoftenvariedtoavoidresonances,optimizeloaddistribution,andreduceweight.Intheplotsofnaturalfrequenciesversusplanetarygearparameters,veeringphenomenaoccurwhentwoeigenvaluelociapproacheachother,butthenabruptlyveeraway.Theimportanceofveeringismanifestedinthedramaticchangesinthevibrationmodesoftheveeringnaturalfrequenciesandtheconsequentimpactonresponse.Thisworkanalyticallycharacterizestherulesofeigenvalueveeringinplanetarygears.Thecouplingfactorsbetweentwocloseeigenvaluelociareapproximatedbyperturbationanalysis.Specialveeringpatternswereobtainedusingtheuniquepropertiesofplanetarygearvibrationmodes.Theresultsareillustratedbyanexample.Keydesignparameterswereinvestigated,andgeneralizedguidanceisprovidedfortuningplanetarygearnaturalfrequencies.
I.INTRODUCTION
Noiseandvibrationreductionarecriticalconcernsinplanetarygearapplications.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,shownatural
frequencyveeringphenomenawhentwoeigenvaluelociapproacheachotherasaparameterisvaried,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.Theassociatedeigenvalueproblemis
II.VEERING/CROSSINGCRITERION
Amethodfordetectingeigenvalueveering/crossingingeneraldynamicsystemsisintroducedhere.PointBinFig.2aisanexampleofveering.Whentwoeigenvaluelociveeraway,theirlocicurvaturesindicatetheabruptnessofcurvedirectionchanges.PerkinsandMote[7]proposedaveering/crossingcriterionbyestimatingthelocicurvatureintheveeringneighborhoodfordistincteigenvalues.LetrandλsbetwoeigenvaluesapproachingeachotheraroundpointBasaparameterrisvaried.TheunperturbedeigensolutionsatBare(λr,Φr)and(λs,Φs).ApplyingTaylorexpansionaroundB,theeigenvaluelociareapproximatedby
whereå
isasmallperturbationofthevaryingparameter;
λ¢
=¶
λ/¶
r;
¢
=.Thesecondderivativesλ¢
randλ¢
srepresentthecurvaturesoftheloci.Ifλ¢
r=λ¢
s=0,thesetwolociarelocallyindependentandfreetocrosseachother.Ifλ¢
r,λ¢
s¹
0,thelocidivergeandveeringoccurs.Largerλ¢
indicatessharperchangesofthelociandstrongerveering.Foradistincteigenvalueλr,theeigenvaluederivativesare
Intheveeringvicinitywhereλr≈λs,λ¢
saredominatedbytermswithasmalldenominator,thatis,
Thecouplingfactorscrandcsapproximatethelocalcurvatures.Theyareusedtoestimateveeringstrength.
Figure2bshowsc=c14=c18versusthevaryingparameterkspfortheveeringlociw14andw18inFig.2a.NoticethesharplychangingvibrationmodesindicatedinFig.2b.ThetwoveeringlociexchangemodeshapesfrompointsAtoC,eventhoughthelocidonotintersect.ThemodesarestronglycoupledatBanddonotlooklikeeitheroftheveeringmodesjustoutsidetheveeringzone.Whentheparameterisadjustedintheveeringzone,thedrasticchangesinthevibrationmodescanhaveagreatimpactontheoperatingconditiondynamicresponse,toothloads,loadsharing,andbearingforcesandpossiblyleadtomodelocalization.Thedegreetowhichindividualmodesareexcitedbydynamicmeshforces(i.e.,themodalforces)alsochangesdramaticallyasveeringaltersthemodes.
Thisveeringcriterioncanbegeneralizedtothecasewhenλs…λs+m-1aredegeneratewithmultiplicitym.Thesecondderivativesoftheseeigenvaluesare[14]
wherethesummationindexk=1,…L,butk¹
s,…,s+m-1.Whenanotherdistincteigenvalueλrisclosetothedegenerateλi,thedominanttermsinEqs.5and3are
Ifλr…λr+n-1isalsodegenerate,thecouplingfactorsare
Ifthecouplingfactorsareallzero,λrandλslocicross;
otherwise,veeringoccurs.Thisistheconditionweexaminebelow.
III.VEERINGPATTERNSINPLANETARYGEARS
Whenappliedtoplanetary(oranyepicyclic)gears,theaboveresultsreducetoparticularlysimpleformsbecauseoftheuniquestructureofthevibrationmodes.Allplanetarygearvibrationmodescanbeclassifiedasoneofthefollowingtypesundertheassumptionofcyclicsymmetrybetweentheplanets[10].Theanalysisisrestrictedtothiscommonlyusedclassofepicyclicgears.
1.Sixrotationalmodeswithdistinctnaturalfrequencies.Theyhavepurerotationofthecarrier,ring,andsun,thatis,xh=yh=0,h=c,r,s.Allplanetshavethesamedeflection
2.Sixpairsoftranslationalmodeswithdegeneratenaturalfrequenciesofmultiplicitytwo.Theyhavepuretranslationofthecarrier,ring,andsun,thatis,uh=0,h=c,r,s.ForapairoforthonormaltranslationalmodesΦi=and=Φj=(MΦj=0),,wherethesuperscriptsiandjindicatethemodesΦiandΦj,respectively.Theplanetdeflectionshavetherelation
whereYn=2(n-1)/Nforequallyspacedplanets.
3.ThreegroupsofplanetmodeswithdegeneratenaturalfrequenciesofmultiplicityN-3.Theyhavenomotionofthecarrier,ring,andsun,thatis,xh=yh=uh=0,h=c,r,s.Theplanetdeflectionsarerelatedby
WherewnareN-3independentsetsofscalarssatisfyingnsinYn,ncosYn,n=0
Twoapproachingplanetarygearlocicanbeassociatedwithanyoftheabovethreetypesofmodes.Fivecasesofpotentialveeringareexaminedbelow.TheuniquemodalpropertiesofEqs.10–12wereusedtoanalyzetheveeringpatterns.Forconcreteness,letthesun-planetmeshstiffnesskspbethevaryingparameter(Fig.1)andM¢
=0.
Case1.Tworotationalmodelociλrandλs.Rotationalmodeshavedistincteigenvalues,andtheirsecondderivativewithrespecttokspis[12]
wheredsn=yscos(Yn-as)-xssin(Yn-as)-hncosas-znsinas+us+unisthemodaldeflectionofthespringrepresentingthenthsun-planetmeshstiffness;
asisthepressureangleofthesun-planetmesh;
andthesuperscriptsrandkindsnindicatevibrationmodesjrandjk,respectively.Themodalstrainenergyinthenthsun-planetmeshisd¢
sn=ksp/2.TherotationalmodepropertyofEq.10resultsin,.ThecouplingfactorsofEq.4arethedominanttermsinEq.13
(14)
If,thencr,cs0,andtheapproachingeigenvaluelociveeraway.Ifor,thencr=cs=0,andthelocicrosseachother.Thisrequiresthatallstrainenergyinthesun-planetmeshesUsn=0forvibrationmoderorjs.However,allrotationalmodesexcepttherigidbodymodemusthavestrainenergyinthesun-planetmeshes.Therefore,tworotationalmode
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 外文 翻译 行星 齿轮 固有频率 精品