质量不平衡转子系统发生碰摩后振动特性分析.docx

质量不平衡转子系统发生碰摩后振动特性分析.docx

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质量不平衡转子系统发生碰摩后振动特性分析

VIBRATIONCHARACTERISTICSANALYSISOFTHE

RUB-IMPACTROTORSYSTEMWITHMASS

UNBALANCE

5JIAJiuhong,HANGTianqi

10

15

20（KeyLaboratoryofPressureSystemsandSafety,MinistryofEducation,EastChinaUniversityofScienceandTechnology,ShangHai200237）Abstract:

Inthispaper,vibrationcharacteristicsofarub-impactJeffcottrotorsystemexcitedbymassunbalanceincludingtheeccentricmassandtheinitialpermanentdeflectionareinvestigated.Throughthenumericalcalculation,rotatingspeeds,masseccentricities,initialpermanentdeflectionsandphaseanglesbetweentheeccentricmassdirectionandtherotorinitialpermanentdeflectiondirectionareusedascontrolparameterstoinvestigatetheireffectontherub-impactrotorsystemwiththehelpofbifurcationdiagrams,Poincaremaps,frequencyspectrumsandorbitmaps.Resultshowsthatthesetwokindsofmassunbalancehavegreatbutdifferenteffectonthedynamiccharacteristicoftherubbingrotorsystem.Differentmotioncharacteristicsappearwiththevaryingofthesecontrolparameters,andcomplicatedmotionssuchasperiodicandquasi-periodicvibrationsareobserved.Correspondingresultscanbeusedtodiagnosetherub-impactfaultanddecreasetheeffectofrubimpactintherotorsystemwithmassunbalance.Keywords:

Rub-impactrotorsystem;Massunbalance;Nonlinearmotion

0Introduction

Massunbalanceoftherotoristhemainfactortoincreasevibrationandbringunstable

characteristicstothewholerotorsystem[1-2].Sincetherotor-statorrubisoneofthemainfaultsforlargerotarymachines,ithasattractedgreatconcernandlotsofresearchworkhasbeendonebymanyresearchers.JunyiCao[3]investigatednonlineardynamiccharacteristicsofrub-impactrotorsystemwithfractionalorderdamping.Fromthestudy,variouscomplicateddynamicbehaviorsandtypesofroutestochaoswerefound,includingperioddoublingbifurcation,suddentransitionandquasi-periodicfromperiodicmotiontochaos.Jawaidi[4]investigateddynamicsofarigidrotorsupportedbyload-sharingbetweenmagneticandauxiliarybearingsforarangeofrealisticdesignandoperatingparameters.Manyphenomenawerefoundandstudied.WenmingZhang[5]carriedoutananalyticalinvestigationonthestabilityoftherubsolutionsoftherotorsysteminMEMS.Numericalcalculationdemonstratedthecomplexnonlinearmotionformsinthissystem.

Althoughmuchworkhasbeendonerespectivelyontheresearchofrotormassunbalanceand

rotor-statorrubbingimpact,thevibrationcharacteristicsofthiskindofrub-impactrotorswithmassunbalancehaverarelybeenstudiedspeciallyintheliterature.Duetomassunbalance’subiquity,greateffectonthevibrationoftherub-impactrotorsystemandrub’slargethreattothesafeoperationoftherotorsystem,Therefore,inthispaperattentionispaidtotheresearchofvibrationcharacteristicsofrub-impactrotorswithmassunbalance.First-orderheadline253035

1Physicalmodelandequations

40Rub-impactforcesIntheO-xycoordinatesystemare[6]:

Fx=−FNcosγ+FTsinγ=−（1−hr）Kr（x−μy）

（1）Fy=−FNsinγ−FTcosγ=−（1−hr）Kr（μx+y）

whenr

Foundations:

NewTeacherFundforDoctorStation,theMinistryofEducation（20090074120005）.

Briefauthorintroduction:

JIAJiuhong,（1979-）,Female,Associateprofessor,Mainresearch:

Integrityofstructure.E-mail:

jhjia@

-1-

nondimensionalprocedure,thegoverningmotionequationsareasfollows:

ˆ=rh<1.0,therubdoesn’thappen,andthenondimensionalmotionequationsare:

WhenR

45

ˆ''+2ξxˆ'+xˆ=εω2cosωT+rˆ0cos（ωT+α0）x

ˆ''+2ξyˆ'+yˆ=εω2sinωT+rˆ0sin（ωT+α0）y

（2）

ˆ=rh≥1.0,therubhappens,andthenondimensionalmotionequationsare:

WhenR

ˆ（1−Rˆ）（xˆ''+2ξxˆ'+xˆ+Kˆ−μyˆ）=εω2cosωT+rˆ0cos（ωT+α0）x

（3）

2ˆˆˆ''+2ξyˆ'+yˆ+K（1−R）（μxˆ+yˆ）=εωsinωT+rˆ0sin（ωT+α0）y

ˆ=KK,ξ=

C2ωM,ω=thenaturalfrequencyofˆ=xh,yˆ=yh,KWhere:

x

r

00

50

ˆ0=r0h,T=ω0t,'=ddT.theshaft,ω=Ω0thefrequencyratio,ε=emh,r

Sincethenondimensionalgoverningequationsofmotionhavebeengotabove,theyare

&=f（u）.Thenthefourth-ordertransferredintoasetoffirstorderdifferentialequationsu

55

Runge-Kuttamethodisusedtointegratethissetofequations.Accordingtotheanalysisneed,

someparameterscanbeusedasthecontrolparameterssuchastherotorrotatingspeed,theinitialpermanentdeflectionandsoon,whileotherparameterskeepfixedduringeverytimeofcalculation.Togetthestableresult,asmallintegrationstephastobechosentoavoidthenumericaldivergenceatthepointwherederivativesofFxandFyarediscontinuous.Inthispaper,theintegrationstepischosentobe2π/500,i.e.,withinoneperiod,thereare500timesofintegralcalculation.Generally,longtimemarchingcomputationisrequiredtoobtainaconvergentorbit.Inthispaper,duringeverycalculation,resultsofthefirst500periodsareabandoned,andthenresultsofthenext100periodsaregottocarryoutvariouskindsofanalysis.Tostudythevibrationcharacteristicsoftherubbingrotorsystemwithmassunbalance,bifurcationdiagrams,Poincaremaps,frequencyspectrumsandorbitmapsareemployed.Theyareallusefulandeffectivewaystoillustratethemotionbehavioroftherotorsystem.

60

2Numericalsimulationofmotionsoftherotorsystem

65

2.1Effectofrotorrotatingspeeds

Althoughmanyparameterssuchasthesystemdampingandthefrictionalcoefficientbetweentherotorandthestatorcanbeusedtoinvestigatethevibrationcharacteristicsoftherub-impactsystem,themostcommonandusefulparameterusedistherotorrotatingspeed.Thewholestartingprocesscanbeobservedbyusingtherotatingspeedtosimulatetherub-impactrotor

ˆ

0070

-2-

00=/475

Fig.1Bifurcationdiagram

80

85

90

95

Fig.1.（a）isthebifurcationdiagramoftherub-impactrotor’svibrationwhichisexcitedonlybytheeccentricmass.Fig.1.（b）isthebifurcationdiagramoftherub-impactrotor’svibrationwhichisexcitedonlybytheinitialpermanentdeflection.Fig.1.（c）isthebifurcationdiagramofthevibrationexcitedbythesetwokindsofmassunbalancejointly.VeryclearlyfromFig.1,differentshapesofthreediagramsdemonstratethattheeccentricmassandtheinitialpermanentdeflectionhavedifferentinfluencetothevibrationoftherubbingrotorsystemandbothcontributealottothevibrationcharacteristicsoftherub-impactrotorsystem.AsshowninFig.1.（a）,therub-impactmotionexcitedonlybytheeccentricmasskeepssynchronouswithperiod-oneforthewholecalculatingrotatingspeedrange.Fig.2showsthatonlyonepointiscorrespondinglyshowninthePoincaremap,onlyonepeakappearsinthefrequencyspectrum,andtheorbitoftherotorcenterisanellipse.Allthesefurtherdemonstratethemotionisperiod-onemotion.FromFig.1.（b）,itcanbeseenthatwhentherotatingspeedratioisfrom0.0to1.1,themotionisperiod-one.From1.1to1.6,themotionisquasi-periodic,thenthemotiontransitstoperiod-onemotionagainforthehigherspeedratios.ThiscanbeseenfromFig.3.Itillustratestherubbingmotionexcitedonlybytheinitialpermanentdeflectionatω=1.2.ThePoincaremapisaclosedcircle,severalpeaksappearinthefrequencyspectrum,andtherotorcenterorbitisirregular.Theseprovethatthemotionisquasi-periodic.

FromFig.1.（c）,itcanbeseenthatwhentherotatingspeedratioisfrom0.0to1.1,themotionisperiod-one,from1.1to2.4,themotionisquasi-periodic,thenthemotiontransitstoperiod-onemotionagainforthehigherspeedratios.ThismotiontypeisclearlydifferentfromtheoneexcitedonlybytheeccentricmassinFig.1.（a）andtheoneexcitedonlybytheinitialpermanentdeflectioninFig.1.（b）.Fig.4demonstratesthatthemotioninthiscaseatω=1.5isquasi-periodicasanalyzedabove.

100

Fig.2Motionexcitedbyε=0.2atω=1.5

-3-

105

ˆ0=0.8,α0=π/4atω=1.2

Fig.3Motionexcitedbyr

110

ˆ0atω=1.5Fig.4Motionexcitedjointlybyεandr

2.2Effectoftheeccentricmass

Eccentricmassisoneofthemassunbalancetoexcitethemotionoftherotorsystem,soit’sessentialtostudyitseffectondynamiccharacteristicoftherub-impactrotorsystem.Fig.3arebifurcationdiagramsofrotor’svibrationatdifferentrotatingspeedsusingthemasseccentricityratioεasthecontrolparameter.Theinitialpermanentdeflectionisnotincludedinthisanalysisandissetas0.0.

115

120

Fig.5Bifurcationdiagramusingεasthecontrolparameter

125

Fig.6Motionwithε=0.3at

ω=1.0

-4-

130

Fig.7Motionwithε=0.3at

ω=1.5

135

140

AsshowninFig.5.（a）,whentherotatingspeedratioisnothighlike1.0,themotionoftherub-impactrotorsystemkeepssynchronouswithperiod-oneforthewholecalculatingrange.FurtherdemonstrationisshowninFig.6withε=0.3atω=1.0thatonlyonepointiscorrespondinglyshowninthePoincaremap,onlyonepeakappearsinthefrequencyspectrum,andtheorbitoftherotorcenterisanellipse.Allthesefurtherdemonstratethemotionisperiod-onemotion.

AsshowninFig.5.（b）,atω=1.5whenthemasseccentricityratioεisfrom0.0to0.26,themotionisperiod-one.From0.26tohighervaluesofthemasseccentricityratio,themotionkeepsquasi-periodic.InFig.7withε=0.3atω=1.5,thePoincaremapisaclosedcircle,severalpeaksappearinthefrequencyspectrum,andtherotorcenterorbitisirregular.Theseprovethatthemotionisquasi-periodic.

2.3Effectoftheinitialpermanentdeflection

Theinitialpermanentdeflectionisanotheroneofthemassunbalancetoexcitethemotionoftherotorsystem,it’salsoimportanttostudyitseffectondynamiccharacteri