9781447142393-c2

IntroductiontoRotorDynamics

Rotordynamicsisthebranchofengineeringthatstudiesthelateralandtorsionalvibrationsofrotatingshafts,withtheobjectiveofpredictingtherotorvibrationsandcontainingthevibrationlevelunderanacceptablelimit.Theprincipalcomponentsofarotor-dynamicsystemaretheshaftorrotorwithdisk,thebearings,andtheseals.Theshaftorrotoristherotatingcomponentofthesystem.Manyindustrialapplicationshaveflexiblerotors,wheretheshaftisdesignedinarelativelylongandthingeometrytomaximizethespaceavailableforcomponentssuchasimpellersandseals.Additionally,machinesareoperatedathighrotorspeedsinordertomax-imizethepoweroutput.Thefirstrecordedsupercriticalmachine(operatingabovefirstcriticalspeedorresonancemode)wasasteamturbinemanufacturedbyGus-tavDelavalin1883.Modernhighperformancemachinesnormallyoperatesabovethefirstcriticalspeed,generallyconsideredtobethemostimportantmodeinthesystem,althoughtheystillavoidcontinuousoperatingatornearthecriticalspeeds.Maintainingacriticalspeedmarginof15%betweentheoperatingspeedandthenearestcriticalspeedisacommonpracticeinindustrialapplications.

Theothertwoofthemaincomponentsofrotor-dynamicsystemsarethebear-ingsandtheseals.Thebearingssupporttherotatingcomponentsofthesystemandprovidetheadditionaldampingneededtostabilizethesystemandcontainthero-torvibration.Seals,ontheotherhand,preventundesiredleakageflowsinsidethemachinesoftheprocessingorlubricatingfluids,howevertheyhaverotor-dynamicpropertiesthatcancauselargerotorvibrationswheninteractingwiththerotor.Gen-erally,thevibrationinrotor-dynamicsystemscanbecategorizedintosynchronousorsubsynchronousvibrationsdependingonthedominantfrequencyandsourceofthedisturbanceforces.Synchronousvibrationshaveadominantfrequencycom-ponentthatmatchestherotatingspeedoftheshaftandisusuallycausedbytheunbalanceorothersynchronousforcesinthesystem.Thesecondtypeisthesub-synchronousvibrationorwhirling,whichhasadominantfrequencybelowtheop-eratingspeedanditismainlycausedbyfluidexcitationfromthecross-couplingstiffness.

Inthischapterwepresentashortintroductiontorotordynamics,withthein-tentiontofamiliarizethereaderwithbasicconceptsandterminologiesthatareof-S.Y.Yoonetal.,ControlofSurgeinCentrifugalCompressorsbyActiveMagneticBearings,AdvancesinIndustrialControl,

DOI10.1007/978-1-4471-4240-9_2,©Springer-VerlagLondon2013

17

182IntroductiontoRotorDynamics

tenusedindescribingAMBsystems.Thematerialpresentedhereisbasedontherotor-dynamicscoursenotespreparedbyAllaire[5],andthemanybooksavail-ableinrotordynamicsbyauthorssuchasChilds[30],Genta[49],Kramer[78],Vance[115],andYamamotoandIshida[119].First,themathematicsbehindthebasicrotor-dynamicprinciplesareintroducedthroughtheexampleofasimplero-tor/bearingsystemmodel.Theprimaryconcernsinrotor-dynamicsystems,includ-ingthecriticalspeed,unbalanceresponse,gyroscopiceffectsandinstabilityexci-tation,arediscussedinthesectionsthroughoutthischapter.Finally,thestandardspublishedbytheAmericanPetroleumInstituteforauditingtherotorresponseincompressorsarepresentedindetail.MostofthesestandardsaredirectlyapplicabletocompressorswithAMBs,andtheywillplayanimportantroleinthedesignoftheAMBlevitationcontrollerforthecompressortestriginChap.7.

2.1Föppl/JeffcottSingleMassRotor

Rotor-dynamicsystemshavecomplexdynamicsforwhichanalyticalsolutionsareonlypossibletoobtaininthemostsimplecases.Withthecomputationalpowerthatiseasilyavailableinmoderndays,numericalsolutionsfor2Dandeven3Drotor-dynamicanalysishavebecomethestandard.However,thesenumericalanalysesdonotprovidethedeepinsightthatcanbeobtainedfromastep-by-stepderivationofananalyticalsolution,suchashowthedifferentsystemresponsecharacteristicsareinterconnectedinthefinalsolution.Forexample,numericalanalysiscanaccuratelyestimatethelocationoftheresonancemodeofthesystem,butitcannotgiveananalyticalrelationshipbetweenthatmodefrequencyandtheamountofdampingandstiffnessontherotor.

Thevibrationtheoryforrotor-dynamicsystemswasfirstdevelopedbyAugustFöppl(Germany)in15andHenryHomanJeffcott(England)in1919[5].Em-ployingasimplifiedrotor/bearingsystem,theydevelopedthebasictheoryonpre-dictionandattenuationofrotorvibration.Thissimplifiedrotor/bearingsystemthatiscommonlyknownastheFöppl/Jeffcottrotor,orsimplytheJeffcottrotor,isof-tenemployedtoevaluatemorecomplexrotor-dynamicsystemsintherealworld.Inthissectionweoverviewtheanalyticalderivationoftheundampedanddampedre-sponsesoftheFöppl/Jeffcottrotor.Wewillusetheseresultsthroughoutthischaptertocharacterizethedynamicsofcomplexrotor-dynamicsystemsthatcanbefoundinactualindustrialapplications.

Figure2.1illustratesthesinglemassJeffcottrotorwithrigidbearings.Therotordiskwithmassmislocatedattheaxialcenteroftheshaft.ThemassoftheshaftintheJeffcottrotorisassumedtobenegligiblecomparedtothatofthedisk,andthusisconsideredtobemasslessduringtheanalysis.ThegeometriccenterofthediskCislocatedatthepoint(uxC,uyC)alongcoordinateaxisdefinedaboutthebearingcenterline,andthediskcenterofmassGislocatedat(uxG,uyG).TheunbalanceeccentricityeuisthevectorconnectingthepointsCandG,anditrepresentstheunbalanceintherotordisk.Therotatingspeedofthedisk/shaftisgivenbyω,and

2.1Föppl/JeffcottSingleMassRotorFig.2.1SinglemassJeffcottrotoronrigidbearings

19

withoutlossofgeneralityweassumethateuisparallelwiththex-axisattheinitialtimet=0.Lastly,uCisthedisplacementvectorwithphaseangleθthatconnectstheoriginandthepointC,andφisdefinedtobetheanglebetweenthevectorsuCandeu.

Undertheassumptionthattherotordiskdoesnotaffectthestiffnessofthemass-lessshaft,thelateralbendingstiffnessattheaxialcenterofasimplysupporteduniformbeamisgivenby

48EI

,(2.1)L3

whereEistheelasticmodulusofthebeam,Listhelengthbetweenthebearings,andIistheshaftareamomentofinertia.Forauniformcylindricalshaftwithdiam-eterD,theequationfortheareamomentofinertiais

ks=πD4I=.

(2.2)

Additionally,weassumethatthereisarelativelysmalleffectivedampingactingonthelateralmotionofthediskattherotormidspan,andthecorrespondingdampingconstantisgivenbycs.Thisviscousdampingisacombinationoftheshaftstructural

202IntroductiontoRotorDynamics

damping,fluiddampingduetotheflowinturbomachines,andtheeffectivedampingaddedbythebearings.

ThedynamicequationsfortheFöppl/JeffcottrotorarederivedbyapplyingNew-ton’slawofmotiontotherotordisk.Withtheassumptionthattheshaftismassless,theforcesactingonthediskaretheinertialforceandthestiffness/dampingforcesgeneratedbythelateraldeformationoftheshaft.Thelateralequationsofmotioninthex-andy-axesasshowninFig.2.1arefoundtobe

˙xC,mu¨xG=−ksuxC−csumu¨yG=−ksuyC−csu˙yC,

(2.3a)(2.3b)

where(uxG,uyG)and(uxC,uyC)arethecoordinatesofthemasscenterandgeomet-riccenter,respectively.Thecoordinatesofthediskcenterofmasscanberewritten

intermsofitsgeometriccenterCandtherotorangleofrotationωtattimet,

uxG=uxC+eucos(ωt),uyG=uyC+eusin(ωt).

(2.4a)(2.4b)

SubstitutingthesecondtimederivativeofEqs.(2.4a),(2.4b)intoEqs.(2.3a),(2.3b),weobtaintheequationsofmotionfortheFöppl/Jeffcottrotorintermsofthediskgeometriccenteras

mu¨xC+ksuxC+csu˙xC=meuω2cos(ωt),˙yC=meuω2sin(ωt).mu¨yC+ksuyC+csu

(2.5a)(2.5b)

Wenoteherethat,asthebearingsareconsideredtobeinfinitelystiffandthe

rotordiskdoesnottilt,thismodeldoesnotincludethegyroscopiceffectsactingontherotor.Theshaftisfixedatthebearinglocations,thusitisalwaysalignedtothebearingcenterline.Theeffectofthegyroscopicforcesinrotor-dynamicsystemswillbediscussedinSect.2.2.Additionally,noaerodynamicsorfluid-filmcross-couplingforcesareincludedinthissimplifiedanalysis.Thesedisturbanceforcesaremostlygeneratedatthesealsandimpellersoftherotorduetothecircumferentialdifferenceintheflow,andtheyarenotmodeledinthissection.Aerodynamiccross-couplingforceswillbediscussedinSect.2.3.Asaresultofallthis,theequationsofmotioninEqs.(2.5a),(2.5b)aredecoupledinthex-andy-axes.

2.1.1UndampedFreeVibration

Theundampedfreevibrationanalysisdealswiththerotorvibrationinthecaseofnegligibleunbalanceeccentricity(eu=0)anddamping(cs=0).TheequationsofmotioninEqs.(2.5a),(2.5b)aresimplifiedto

mu¨xC+ksuxC=0,mu¨yC+ksuyC=0.

(2.6a)(2.6b)

2.1Föppl/JeffcottSingleMassRotor21

Thesolutiontothissecondorderhomogeneoussystemtakestheformof

uxC=Axest,uyC=Ayest,

(2.7a)(2.7b)

forsomecomplexconstants.ThevaluesoftheconstantsAxandAyareobtainedfromtheinitialconditionsoftherotordisk.SubstitutingthesolutioninEqs.(2.7a),(2.7b)intoEqs.(2.6a),(2.6b)weobtain

󰁢󰁡

(2.8a)ms2Axest+ksAxest=ms2+ksAxest=0,

󰁢󰁡

(2.8b)ms2Axest+ksAxest=ms2+ksAyest=0.TheaboveequationsholdtrueforanyvalueofAxandAyiftheundampedcharac-teristicequationholds,

ms2+ks=0.

(2.9)

Solvingtheaboveequalityforthecomplexconstants,weobtainthefollowing

solution:

s1,2=±jωn,

whereωnistheundampednaturalfrequencyoftheshaftdefinedas

󰁒󰁒ks48EI=.ωn=mL3m

(2.10)

(2.11)

Thus,thesolutionstotheequationofmotioninEqs.(2.6a),(2.6b),areundamped

oscillatoryfunctionswithfrequency±ωn.Theundampedcriticalspeedofthesys-temisdefinedas

ωcr=±ωn,

(2.12)

correspondingtothepositiveforward+ωnandthenegativebackward−ωncompo-nents.Theforwardcomponentindicatesthelateralvibrationthatfollowsthedirec-tionoftheshaftrotation,andthebackwardcomponentrepresentsthevibrationthatmovesintheoppositedirection.ThefinalsolutionstotheundampedfreevibrationaregivenbythelinearcombinationofthetwosolutionsfoundinEqs.(2.7a),(2.7b)andEq.(2.10),

uxC=Ax1ejωnt+Ax2e−jωnt

=Bx1cos(ωnt)+Bx2sin(ωnt),

and

uyC=Ay1ejωnt+Ay2e−jωnt

=By1cos(ωnt)+By2sin(ωnt),

(2.14)(2.13)

forsomevaluesofAxiandAyi,orBxiandByi,whichcanbefoundfromtheinitialconditionsoftherotor.

222IntroductiontoRotorDynamics

2.1.2DampedFreeVibration

NowconsiderthefreevibrationoftheFöppl/Jeffcottrotorwithanon-zeroeffectiveshaftdampingactingonthesystem.Newton’sequationofmotioninEqs.(2.5a),(2.5b)becomes

mu¨xC+ksuxC+csu˙xC=0,˙yC=0.mu¨yC+ksuyC+csu

(2.15a)(2.15b)

Thesolutionstotheabovesystemofhomogeneoussecondorderdifferentialequa-tionstakethesameformasinEqs.(2.7a),(2.7b).Substitutingthesesolutionsinto

Eqs.(2.15a),(2.15b),weobtain

󰁢󰁡2

ms+ks+csAxest=0,(2.16a)

󰁢󰁡2

(2.16b)ms+ks+csAyest=0.Theseequationsholdforanyinitialconditionifthedampedcharacteristicequation

holds:

ms2+ks+cs=0.

(2.17)

Thezerosofthecharacteristicequation,alsoknowasthedampedeigenvaluesofthesystem,arefoundtobe

󰁓󰁳󰁴cskscs

±j−.(2.18)s1,2=−2mm2mGenerally,therotor/bearingsystemisunderdamped,whichmeansthat

csks

<,2mm

andswillhaveanimaginarycomponent.Definethedampingratioas

ζ=

cs

.2mωn

(2.19)

Thisvaluecorrespondstotheratiooftheeffectivedampingcstothecriticalvalueinthedampingconstantwhenthesystembecomesoverdamped,ortheimaginarypartofthesolutioninEq.(2.18)vanishes.Withthisnewlydefinedratio,thesolutionstoEqs.(2.16a),(2.16b)canberewrittenas

󰁑

s1,2=−ζωn±jωn1−ζ2.(2.20)Theimaginarycomponentofs1,2isknownasthedampednaturalfrequency,

󰁑

ωd=ωn1−ζ2.(2.21)

2.1Föppl/JeffcottSingleMassRotor23

Fortraditionalpassivebearings,thevalueofthedampingcoefficientcanvarybe-tween0.3>ζ>0.03,althoughaminimumofζ=0.1isnormallyconsideredasneededforthesafeoperationofthemachine.ThefinalsolutionstotheundampedfreevibrationarefoundtobethelinearcombinationofthesolutionsfoundinEqs.(2.7a),(2.7b)andEq.(2.18),thatis,

󰁡󰁢

uxC=e−ζωntAx1ejωdt+Ax2e−jωdt

󰁡󰁢

=e−ζωntBx1cos(ωnt)+Bx2sin(ωnt),(2.22)and

󰁡󰁢uyC=e−ζωntAy1ejωdt+Ay2e−jωdt

󰁢󰁡

=e−ζωntBy1cos(ωnt)+By2sin(ωnt),

(2.23)

forsomevaluesofAxiandAyi,orBxiandByi,dependentontheinitialconditionoftherotor.

AtypicalresponseforanunderdampedsysteminfreevibrationisshowninFig.2.2.Weobservethattheresponseisoscillatory,wherethefrequencyisgivenbythedampednaturalfrequencyωd.Becauseofthedamping,themagnitudeoftheoscillationisreducedovertime,andtherateofdecayisafunctionofthedampingratioζandtheundampednaturalfrequencyωn.Formostrotor-dynamicsystems,thedampingratioissmallerthan0.3andthefreevibrationresponseissimilartotheunderdampedresponseinFig.2.2.

2.1.3ForcedSteadyStateResponse

Finally,weconsidertheforcedresponseoftheJeffcottrotorwithanon-zeromasseccentricity.Usingthedefinitionofωnandζasgivenabove,theequationsofmotionfortherotorarerewrittenintotheform

2

u¨xC+2ζωnu˙xC+ωnuxC=euω2cos(ωt),2u¨yC+2ζωnu˙yC+ωnuyC=euω2sin(ωt).

(2.24a)(2.24b)

Inordertosimplifytheequationsofmotion,wewillcombinethexandydisplace-mentsoftherotorintothecomplexcoordinatesas

uC=uxC+juyC,

(2.25)

whereuCisthedisplacementofthediskgeometriccenteronthecomplexcoordinate

axis.

Weassumethatthesteadystatesolutionsofthesystemofthedifferentialequa-tionsinEqs.(2.24a),(2.24b)areincomplexexponentialform,

uxC=Uxejωt,uyC=Uyejωt.

(2.26a)(2.26b)

242IntroductiontoRotorDynamics

Fig.2.2Typicalresponseofanunderdampedsysteminfreevibration

Itisobservedherethat,sinceEqs.(2.24a),(2.24b)isalinearsystemwithasinu-soidalinputoffrequencyω,thesteadystateoutputsolutionswillalsobesinusoidalsignalsofthesamefrequency.Then,thesolutionofthediskdisplacementinthecomplexformis

uC=Uxejωt+jUyejωt.

(2.27)

Combiningtheexponentialtermsintheexpressionfortheabovecomplexrotordisplacement,weobtainthesolutionintheform

uC=Uejωt,

where

U=Ux+jUy.

(2.29)

Next,thesetofsolutionsinEqs.(2.26a),(2.26b)aresubstitutedintoEqs.(2.24a),(2.24b),andtheresultingsystemofequationsis

󰁡2󰁢2−ω+2jωζωn+ωnUxejωt=euω2cos(ωt),(2.30a)

󰁢󰁡2

2

(2.30b)Uyejωt=euω2sin(ωt).−ω+2jωζωn+ωnTheequationsforthex-axisandy-axisdisplacementsarecombinedintothecom-plexformasdoneinEq.(2.25)bymultiplyingEq.(2.30b)bythecomplexoperator

(2.28)

2.1Föppl/JeffcottSingleMassRotor25

1j,andaddingittotheexpressioninEq.(2.30a).Theresultingcomplexequationofmotionis

󰁡2󰁢jωt

2

−ω+2jωζωn+ωnUe=euω2eωt,(2.31)or

󰁡2󰁢2

−ω+2jωζωn+ωnuC=euω2,

(2.32)

whereeuistheunbalanceeccentricityinthecomplexcoordinatesasillustratedin

Fig.2.1(b).

ConsideringthatthevaluesofboththerotordiskdisplacementuCandtheun-balanceeccentricityeuarejustcomplexnumbers,wecancomputefromEq.(2.32)theratiobetweenthesetwocomplexvaluesas

uCfr2

,=

eu[1−fr2+2jfrζ]

where

fr=

ω

ωn

(2.34)(2.33)

isknownasthefrequencyratio.WenoticethatrighthandsideofEq.(2.33)isnotafunctionoftime,anditonlydependsonthefrequencyratio.ThecomplexsolutioninEq.(2.33)canberewrittenastheproductofamagnitudeandaphaseshiftintheformof

|U|−jφuC=eeueu

=󰁐fr2e−jφ(1−fr2)2+(2ζfr)2.

(2.35)

Theratio|U|/euisknownasthedimensionlessamplituderatiooftheforcedre-sponseandisgivenby

|U||Uy||Ux|fr2===󰁐.eueueu(1−fr2)2+(2ζfr)2(2.36)

Theaboveequationgivestheexpectedamplitudeoftherotorvibrationasafunction

ofthefrequencyratio.Additionally,theangleφisthephasedifferencebetweentheuCandeuandisfoundfromEq.(2.32)tobe

󰁳󰁴2ζfr

φ=tan−1.(2.37)

1−fr2Thedimensionlessamplituderatio|U|/euisplottedinFig.2.3overthefre-quencyratiofrfordifferentvaluesofdampingratio.Forverylowfrequencies,theamplituderatioisnearlyzerosincetheunbalanceforcesaresmall.Astheshaft

262IntroductiontoRotorDynamics

Fig.2.3DimensionlessamplitudeoftheforcedresponsefortheJeffcottrotorvs.frequencyratio

speedincreases,theamplitudeshowsalargepeaknearfr=1whenωisneartheresonancefrequencyofthesystem.Theamplituderatioatthecriticalspeedfr=1canbefoundfromEq.(2.36)tobe

|U|1

.=

eu2ζ

(2.38)

Whenthedampingratioissmall,theamplituderatioincreasesrapidlynearfr=1

astheunbalanceforcesexcitetherotorresonancemode.Forlargervaluesofζ,thesystemisnearlycriticallydamped,andonlyalittleoftheresonanceisseenintheamplituderatioplot.Finally,forfr󰁽1theamplitudeofvibrationapproaches1.ThephaseangleφcorrespondingtodifferentvaluesofthedampingratioisalsopresentedhereoverarangeoffrequencyratiosinFig.2.4.Atlowfrequencies,thephaseangleisnearzero,andthecenterofgravityGisalignedwiththegeometriccenterofthediskduringtherotationoftheshaft.Whenthefrequencyratioisnear1andtheshaftspeedisclosetothenaturalfrequency,weseeinFig.2.4thatthephaseangleisabout90degreesforallvaluesofdampingratios.Thischaracteristiccanbehelpfulinidentifyingexperimentallythecriticalspeedofactualmachines.Lastly,athighfrequencieswherefr󰁽1,thephaseangleapproaches180degrees.Inthiscase,thecenterofgravityofthediskisinsidetherotororbitdrawnbytherotatingpathofC,andtheunbalanceforcesworkintheoppositedirectiontotheinertialforcesoftherotor.

2.2RotorGyroscopicEffects27

Fig.2.4PhaseangleφoftheforcedresponsefortheJeffcottrotorvs.frequencyratio

2.2RotorGyroscopicEffects

Sofar,wehavefoundthattherotorlateraldynamicsaredecoupledinthehorizon-talandtheverticaldirectionsofmotionwhenrigidbearingsareassumed.IntheFöppl/JeffcottrotorconsideredinSect.2.1,theshaftaxisofrotationwasalwaysalignedwiththebearingcenterline,andthustheinertiainducedmomentsactingonthediskwereneglected.Inthissectionweinvestigatehowthegyroscopicmomentsaffectthedynamicsofthesystem,astheadditionofflexiblebearingsallowstheshaftrotationalaxistodivergefromthebearingcenterline.Throughanexampleofasimplecylindricalrotorsupportedonflexiblebearings,theundampedfreevibra-tionoftherotorisanalyzed,andthenaturalfrequencyoftherotorispredictedasafunctionoftheshaftspeed.Theresultswilldemonstratethesensitivityoftheactualcriticalspeedofrotor-dynamicsystemstothegeometryandrotatingspeedoftherotor.

Thetiltofarotatingshaftrelativetotheaxisofrotationgeneratesgyroscopicdis-turbanceforces.Aswewillfindlaterinthissection,themagnitudeofthegeneratedforceisproportionaltotheangleoftilt,angularmomentofinertiaoftherotor,andtheshaftrotationalspeed.Inthemodelingandanalysisofrotor-dynamicsystems,therearetwomainphenomenathatareattributedtothegyroscopiceffects.First,thegyroscopicmomentstendtocouplethedynamicsinthetworadialdirectionofmotions.Achangeintheverticalstateoftherotoraffectsthehorizontaldynamics,

282IntroductiontoRotorDynamics

Fig.2.5Cylindricalrotorwithisotropicsymmetricflexiblebearings[115]

andviceversa.Second,gyroscopicmomentscausethecriticalspeedsofthesystemtodriftfromtheiroriginalpredictionsatzerospeed.Aswewillseelaterinthissec-tion,thegyroscopicmomentactingonarotorcanincreaseordecreasethecriticalspeedsrelatedtosomesystemmodesasafunctionoftherotationalspeed.

2.2.1RigidCircularRotoronFlexibleUndampedBearings

ConsidertherigidrotorasshowninFig.2.5withalongcylindricaldiskofmassm,lengthL,androtatingspeedω.Thesupportbearingsareconsideredtobeflexiblewithstiffnesscoefficientsofk1andk2inthelateraldirectionsasshowninFig.2.5.TheaxialdistancebetweenthebearinglocationandtherotorcenterofgravityGisafortheleftbearingandbfortherightbearing.ThetotaldistancebetweenthebearingsisLb.

Undertheassumptionthattheshafthasnegligiblemass,thepolarmomentofinertiaoftheuniformrigidcylindricalrotorisgivenby

mR2

,Jp=2

(2.39)

whereRistheradiusoftherotor.Thisrepresentstherotationalinertiaofthecylin-deraboutitsmainaxisofrotation.Thetransversemomentofinertiaforthesamerotoris

󰁳󰁴m122

R+L,(2.40)Jt=43whichrepresentstherotationalinertiaabouttheaxisperpendiculartothemainaxis

ofrotation.AcharacteristicoftherotorthatwillbeimportantinthederivationstofollowthroughoutthissectionistheratioPofthepolartothetransversemoment

2.2RotorGyroscopicEffects29

ofinertia,whichisgivenby

P==JpJt

2

L2

1+13(R)

.(2.41)

Wenoticethatthevalueofthisratioisaffectedbythegeometryoftherotor.For

cylindricalrotorswheretheradiusismuchlargerthanthelength,orR󰁽L,thevalueofthemomentofinertiaratioapproachesP≈2.Ontheotherhand,forthecaseofalongthinrotorwithR󰁼L,thedenominatorofEq.(2.41)approachesinfinityandthevalueofthemomentofinertiaratioisapproximatelyP≈0.Finally,theratio√inEq.(2.41)isequaltooneiftheratioofthelengthLtotheradiusRisequalto3.2.2.2ModelofRigidCircularRotorwithGyroscopicMoments

ConsidertherigidcylindricalrotorpresentedinFig.2.5.ThelateraldisplacementsoftherotorcenterofmassaregivenbyxGinthex-direction,andyGinthey-direction.Additionally,therotationoftherotoratthecenterofmassGaboutthex-axisisdenotedasθxG,andtheequivalentrotationaboutthey-axisisθyG,asFig.2.5illustrates.Thedisplacementsandrotationsabouttherotorcenterofmasscanbecomputedas

xG=yG=θxG≈θyG≈

1

(bx1+ax2),Lb

1

(by1+ay2),Lb

1

(y2−y1),Lb

1

(x2−x1),Lb

(2.42a)(2.42b)(2.42c)(2.42d)

wherex1andy1arethelateraldisplacementsoftheshaftatthefirstbearingloca-tion,asshowninFig.2.5.ThecorrespondingdisplacementsatthesecondbearinglocationinFig.2.5aregivenbyx2andy2.Forcomputingtherotortiltangle,theapproximationsin(θ)≈θforθ󰁼1wasused.

TheequationsofmotionforthetranslationandrotationoftherotoraboutitscenterofmasscanbefoundonceagainasinSect.2.1throughtheuseofNewton’slawofmotion.Theresultingequationsare

mx¨G+αxG−γθyG=0,my¨G+αyG−γθxG=0,

(2.43a)(2.43b)

302IntroductiontoRotorDynamics

˙yG+γxg+δθxG=0,¨xG+JpωθJtθ

˙xG+γyg+δθyG=0.¨yG−JpωθJtθ

Thedefinedstiffnessparametersintheaboveequationsare

α=k1+k2,γ=−k1a+k2b,δ=k1a2+k2b2.

(2.43c)(2.43d)

(2.44a)(2.44b)(2.44c)

ThefirsttwoequationsinEqs.(2.43a)–(2.43d)describethelateraltranslationof

therotor,andthelasttwoequationsdescribestheangulardynamics.Thesecondtermintheleft-handsideofEq.(2.43c)andEq.(2.43d)isthelinearizedgyroscopicmomentaboutthex-andthey-axes,respectively,forsmallamplitudemotionsasdiscussedin[119].Animportantcharacteristicoftheabovedynamicequationsisthatthetwoequationsoftranslationalmotionaredecoupledfromtheequationsofangularmotionwhenγis0,inwhichcasetheycanbesolvedseparately.

ThedifferentialequationsofEqs.(2.43a)–(2.43d)aresometimeswritteninthevectorform

¨+ωGX˙+KX=0,MX(2.45)wherethegeneralizedstatevectorisgivenby

⎡⎤xG⎢yG⎥

⎥X=⎢⎣θxG⎦,θyG

(2.46)

andthemassmatrixM,gyroscopicmatrixG,andstiffnessmatrixKaregivenby

⎡⎤m000⎢0m00⎥

⎥(2.47)M=⎢⎣00Jt0⎦,

000Jt⎡⎤0000⎢0000⎥⎢⎥,(2.48)G=⎣

000Jp⎦00−Jp0and

⎡

⎤γ0⎥⎥,0⎦δ

α⎢0K=⎢⎣0

γ

respectively.

0αγ00γδ0

(2.49)

2.2RotorGyroscopicEffects31

Wenoticeherethatthemassmatrixisalwaysdiagonal,andthestiffnessmatrixisdiagonalwhenγiszero.Ontheotherhand,thegyroscopicmatrixisskewsym-metric,anditrepresentsthecouplingbetweenthemotionsinthex-andthey-axes.Thisisoneofthemaincharacteristicsofthegyroscopiceffectsasmentionedatthebeginningofthissection.Fortheremainderofthissection,wewillmakethesimplifyingassumptionthatthestiffnessesofallsupportbearingsarethesame,

k=k1=k2,

andthattherotorisaxiallysymmetricaboutitscenterofmass,

Lb

=a=b.2

Thisprovidesthedecouplingconditionofγ=0fortherotorequationsofmotioninthetranslationalandtheangulardirectioninEqs.(2.43a)–(2.43d).Inthiscase,thesystemstiffnessmatrixbecomes

⎡⎤α000⎢0α00⎥

⎥K=⎢(2.50)⎣00δ0⎦.

000δ

2.2.3UndampedNaturalFrequenciesoftheCylindricalMode

HerewearetosolvetherotorequationsgiveninEq.(2.43a)andEq.(2.43b)cor-respondingtotherotortranslationalorparallelmotion.UsingthemethodsasinSect.2.1,weassumethatthesystemofhomogeneouslineardifferentialequationshassolutionsinthecomplexexponentialform

xG=UxGest,yG=UyGest,

(2.51a)(2.51b)

forsomeconstantvaluesofUxGandUyG.SubstitutingthesesolutionsintoEq.(2.43a)andEq.(2.43b),werewritetheequationsofmotionas

󰁡2󰁢

ms+αUxG=0,(2.52a)

󰁢󰁡2

(2.52b)ms+αUyG=0.Theexpressionwithintheparenthesesontheleft-handsidesoftheabovetwoequa-tionsisknownasthecharacteristicpolynomial.WeknowfromSect.2.1thatthe

zerosofthecharacteristicequation,

ms2+α=0,

(2.53)

322IntroductiontoRotorDynamics

aretheeigenvaluesofthesystemcorrespondingtothecylindricalmode.Thechar-acteristicequationsforthehorizontalx-andtheverticaly-axesofmotiongivenaboveareidenticalanddecoupled.Thisisexpectedsincethelateraltranslationdoesnotcauserotortilt,andthecorrespondinggyroscopicmomentiszero.

ThenaturalfrequencyωncorrespondingtotherotorparallelvibrationisfoundfromthezerosofthecharacteristicequationinEq.(2.53).Moreprecisely,theimag-inarycomponentsofthezerosgivethenaturalfrequency

s=±jωn.

(2.54)

Inthecaseofthecylindricalmode,thehorizontalundampednaturalfrequencyhastheforwardmodeωn1andthebackwardmodeωn2.Theundampednaturalfrequencyintheverticaldirectionhastheforwardmodeωn3andthebackwardmodeωn2.Thesenaturalfrequenciesarefoundtobe

󰁐

ωn1=ωn3=2k/m,(2.55a)

󰁐

ωn2=ωn4=−2k/m.(2.55b)

2.2.4UndampedNaturalFrequenciesoftheConicalMode

Wenowconsidertheangulardynamicsoftherotor,giveninEq.(2.43d)and

Eq.(2.43c).Wewillassumeonceagainthatthesolutionstothehomogeneoussys-temofdifferentialequationstaketheform

θxG=ΘxGest,θyG=ΘyGest,

(2.56a)(2.56b)

forsomeconstantvaluesofΘxGandΘyG.SubstitutingthesesolutionsintoEq.(2.43c)andEq.(2.43d),weobtainthefollowingsystemofhomogeneousequa-tions:

󰁡2󰁢

Jts+δΘxG+JpωsΘyG=0,(2.57a)

󰁢󰁡2

(2.57b)Jts+δΘyG−JpωsΘxG=0.Thecharacteristicequationfortheabovesystemis

󰀂2󰀃Jts+δJpωsdet=0.

−JpωsJts2+δ

(2.58)

Theangulardynamicsaboutthedifferentlateralaxesofmotionarecoupledthroughthetermscorrespondingtothegyroscopicmomentintheabovecharacteristicequa-tion.Intheremainderofthissection,wewilldiscusshowtherotatingspeedoftheshaft,andthusthegyroscopicmomentactingontherotor,affectsthenaturalfrequenciesoftheconicalmode.

2.2RotorGyroscopicEffects33

2.2.4.1ConicalModeatZeroRotatingSpeed

Forthespecialcasewheretherotationalspeediszero(ω=0),thecharacteristicequationinEq.(2.58)becomesdecoupledinthex-andthey-axes.Theconicalnaturalfrequenciesforthenon-rotatingrotorcanbefoundbysolvingforthezerosoftheundampedcharacteristicequationinEq.(2.58),

󰁓kL2b

s=±j.(2.59)

2JtTheresultingnon-rotatingconicalnaturalfrequencyis

󰁓kL2b

ωnC0=.

2Jt

(2.60)

Thenon-rotatingconicalnaturalfrequencyωnC0willappearagaininthecalculationoftherotorconicalmodewithnon-zerorotatingspeed.

2.2.4.2ConicalModeatZeroRotatingSpeed

Inthegeneralcasewithnon-zerorotatingspeed(ω=0),thecharacteristicequation,afterexpandingthedeterminantofthematrixinEq.(2.58),becomes

󰁡2󰁢2

Jts+δ+(Jpωs)2=0.

(2.61)

InthesamewayasinSect.2.1,theundampedconicalnaturalfrequencyωnCisfoundfromthecomplexzerosofthecharacteristicequationinEq.(2.61),

s=±jωnC.

Thisisanexpressionequivalentto

2

s2=−ωnC.

ReplacingtheaboveexpressionsforsinthecharacteristicequationinEq.(2.61),

weobtain

󰁡󰁢2

2−JtωnC+δ−(JpωωnC)2=0.(2.62)Factoringtheaboveexpressionintotwotermsgives

󰁡2󰁢󰁡2󰁢JtωnC−δ+JpωωnCJtωnC−δ−JpωωnC=0.

(2.63)

ThisequationisfurthersimplifiedbydividingbothsidesoftheaboveequalitybyJt,

andsubstitutinginthederivedexpressionforthemomentofinertiaratioPandthe

342IntroductiontoRotorDynamics

non-rotatingconicalnaturalfrequencyωnC0.Theresultingcharacteristicequationis

󰁢󰁡2󰁢󰁡2

22

(2.)+PωωnCωnC−ωnC0−PωωnC=0.ωnC−ωnC0Next,wedefinethedimensionlessconicalmodenaturalfrequencyratioω¯nCand

thedimensionlessconicalmodefrequencyratiofrC0as

ω¯nC=

and

frC0=

ω

,ωnC0

(2.66)

ωnC

,ωnC0

(2.65)

respectively.Then,bydividingbothsidesofEq.(2.)bythesquareofωnC0,andsubstitutinginthenon-dimensionalparametersdefinedinEqs.(2.65)and(2.66),weobtain

󰁡2󰁢󰁡2󰁢ω¯nC+PfrC0ω¯nC−1ω¯nC−PfrC0ω¯nC−1=0.(2.67)ThenaturalfrequenciesoftheconicalmodesarethefourzerosofEq.(2.67).

Hereweorganizethesemodesasthelowermodesandthehighermodes.ThezerosofthefirstterminEq.(2.67)providefrequenciescorrespondingtotheforwardcomponentofthenon-dimensionallowermodeω¯n3,andthebackwardcomponentofthenon-dimensionalhighermodeω¯n8as

󰁑

ω¯n5=−PfrC0/2+(PfrC0/2)2+1>0,(2.68a)

󰁑

(2.68b)ω¯n8=−PfrC0/2−(PfrC0/2)2+1<0.Ontheotherhand,thezerosofthesecondterminEq.(2.67)providefrequencies

correspondingtothebackwardcomponentofthenon-dimensionallowermodeω¯n6,andtheforwardcomponentofthenon-dimensionalhighermodeω¯n7as

󰁑

(2.69a)ω¯n6=PfrC0/2−(PfrC0/2)2+1<0,

󰁑

(2.69b)ω¯n7=PfrC0/2+(PfrC0/2)2+1>0.TheforwardandbackwardconicalmodesareplottedinFig.2.6overthefrequency

ratiofrC0andfordifferentvaluesofP.Thedashedlineinthefiguresconnectsthepointswheretherotorspeedmatchesthefrequencyofthemodeatthecorrespondingfrequencyratio,andthesystemisintheconditionofresonance.

Figure2.6showshowthegyroscopicseffectsactingontherotorcausesthenat-uralfrequencyofthesystemtodrift.ForlongrotorswhereP≈0,thegyroscopicmomentissmall,andthefrequencyoftheconicalmoderemainsunaffectedtotherotationalspeedandfrC0.AsthevalueofPincreasesfordifferentgeometriesoftherotor,wecanobserveamoresignificantdriftinthemodefrequency.Forexample,

2.2RotorGyroscopicEffects35

Fig.2.6Dimensionlessconicalnaturalfrequencyratioversustheconicalmodefrequencyratio

362IntroductiontoRotorDynamics

fortheextremecaseofP≥1,weobserveinFig.2.6thattheshaftrotationwouldneverexciteoneoftheforwardconicalmodesasthegyroscopiceffectskeepthemodefrequencyalwaysabovetherotoroperatingspeed.

2.3InstabilityduetoAerodynamicCrossCoupling

Cross-couplingforcesareinmanycasesthemaincauseofinstabilityinrotor-dynamicsystems.Theseforcesaregeneratedincomponentssuchasfluid-filmbear-ings,impellersandseals,whichareessentialfortheoperationoftheturbomachines.Theaerodynamiccross-couplingforcesaregeneratedbytheflowdifferenceintheunevenclearancesaroundimpellersandsealscausedbytherotorlateralmotion.Machineswithtraditionalfluid-filmbearingsaresometimesmorevulnerabletotheseeffects,astherotorisnotcenteredintheclearanceanditissusceptibletogointothewhirlingmotion.Itiscommonforcross-couplingdisturbanceforcestogeneratelargerotorvibration,andeventuallydrivethemachinetoinstability.Inthissectionwefocusontheaerodynamiccross-couplestiffnessgeneratedbytheflowofgasthroughtheimpellerandsealclearances.

Acommonlyobservedeffectofthecross-couplingforcesistherapidlossofdampingintherotor/bearingsystemmodes,particularlytheforwardmodecorre-spondingtothefirstcriticalspeed.Thisresultsinlargesubsynchronousrotorvi-brations,asthecross-couplingforcesincreasetogetherwiththepressurebuild-upinthecompressororpump.Eventually,thesystemmodelosesallitsdampingforlargeenoughmagnitudesofthecross-couplingforces,andtherotor-dynamicsys-tembecomesunstable.Thedestabilizingeffectsoftheaerodynamiccross-couplingforcesareamplifiedwhentheyaregeneratedneartherotormidspan,farfromthesupportingbearings,wheretheeffectivenessoftheaddeddampingbythebearingsissignificantlyreduced.

2.3.1AerodynamicCrossCouplinginTurbines

J.S.Alfordin1965studiedtheforcesfoundintheclearancesaroundtheaircraftgasturbineenginerotors,whichtendtodrivetheturbinewheelunstable[3,30].Theseforces,affectingbothturbinesandcompressors,cametobeknownasAlfordforcesoraerodynamiccross-couplingforces.Theaerodynamiccross-couplingforcesarenormallyexpressedintermsofstiffnessvalues,connectingthetwoaxesoftherotorlateralmotion.DefinetherotorlateralaxesofmotionasshowninFig.2.1.Giventhattherotorxandydisplacementsatthelocationofaturbinestagealongtherotorlengtharedenotedbyxdandyd,thecross-couplingforcesactingontheturbinerotortaketheform

󰁶󰁵󰁶󰁵󰁶󰁵

qsxxqsxyxdFdx

=,(2.70)Fdyqsyxqsyyyd

2.3InstabilityduetoAerodynamicCrossCoupling37

whereFdxandFdyarethex-axisandy-axiscomponentsoftheresultingcross-couplingforces,respectively.Thecoefficientsqsxxandqsyyarerelatedtotheprinci-pal(direct)aerodynamicstiffness,andqsxyandqsyxareknownasthecross-couplingaerodynamicstiffnesscoefficients.

Itisnormallythecaseinactualmachinesthattheprincipalaerodynamicstiffnesscoefficientsarenegligiblewhencomparedtothecross-couplingcoefficients,and−qsxy=qsyx.Then,theexpressionforthecross-couplingforcescanbesimplifiedtotheform

󰁶󰁵󰁶󰁵󰁶󰁵

xdFsx0−qa

=,(2.71)Fsyqa0ydforsomecross-couplingstiffnesscoefficientqa.Asimpleestimateofthecross-couplingaerodynamicstiffnesscoefficientforoneturbinestagewasintroducedby

Alfordinhisderivationas

Tβqa=,(2.72)

DmLtwhereTisthetorqueontheturbinestage,βisacorrectionconstant,Dmisthemeanbladediameter,andLtistheturbinebladeradiallength.Baseduponhisexperiencewithaircraftgasturbines,Alfordsuggestedthevalueofthisconstanttobe1.0<β<1.5.

2.3.2AerodynamicCrossCouplinginCompressors

Inthecaseofcompressors,theimpellersaresubjecttothesamecross-couplingstiffnessaspresentedinEq.(2.71)forasingleturbinestage.Inindustrialcom-pressorapplications,acommonrangeforthevalueoftheimpelleraerodynamiccross-couplingcoefficientpereachstageorimpelleris

175,000N/m≥qa≥525,000N/m.

(2.73)

Intherotor-dynamicanalysisofcompressors,therotorvibrationlevelandstabil-ityareoftenevaluatedattheaveragecross-couplingstiffnesscoefficientvalueofqa=350,000N/mperimpellerstage[5].Moreover,acommonruleforcompres-sorsthatisalsobasedonexperienceisthatthecross-couplingstiffnesscontributionoftheendimpellersinmulti-stagemachinesisnegligibleandnotcountedwhencomputingthetotalcross-couplingstiffnessofcompressors.

Sealsareemployedincompressorsandotherturbomachinestopreventthegasleakagebetweenthedifferentmachinestages.Thecompressibleflowinthesesealsgeneratelateralforcesthatactontherotorintheformofstiffnessanddamping,

󰁵󰁶󰁵󰁶󰁵󰁶󰁵󰁶󰁵󰁶Fsxksxxksxyxdcsxxcsxyx˙d

=+,(2.74)Fsyksyxksyyydcsyxcsyyy˙d

382IntroductiontoRotorDynamics

whereFsxandFsxarethexandycomponentsofthecross-couplingforcesgener-atedbytheseals,respectively.Onceagain,theprincipalstiffnesscoefficientsandthedampingtermsarerelativelysmallwhencomparedtothecross-couplingstiff-nesscoefficients,andareusuallytakentobeequaltozero.Thus,theequationforthesealcross-couplingforcesisoftensimplifiedto

󰁶󰁵󰁶󰁶󰁵󰁵

xd0ksxyFsx

=,(2.75)Fsyksyx0ydwhereksxy<0andksyx>0areknownasthesealcross-couplingstiffnesscoeffi-cients.

Finally,thetotalaerodynamiccrosscouplingforcompressorsissometimesesti-matedbasedonthehorsepowerofthemachine.Thisapproximationisgivenas

Qa=

63,000(HP)β

.

DhN

(2.76)

TheparametersoftheaboveexpressionarethecompressorhorsepowerHP,theimpellerdiameterD(in),thedimensionofthemostrestrictiveflowpathh(in)andtheshaftrotatingspeedN(rpm).AcommonvalueofthecorrectionconstantintroducedbyAlfordisβ=1.0baseduponexperience[5].Thetotalcross-couplingstiffnessisgivenintheEnglishunitoflbf/inandcanbeconvertedintotheequivalentSIunitN/mbyafactorof175.AnexpressionsimilartoEq.(2.76)isemployedbytheAPItopredicttheappliedaerodynamiccross-couplingstiffnessinthestabilityanalysisforcompressors.ThisexpressionwillbediscussedbelowinSect.2.4.

2.4Rotor-DynamicSpecificationsforCompressors

Turbomachinessuchascompressorsplayanintegralroleinthemanufacturingpro-cessesofthechemicalandpetrochemicalindustries.Therefore,eachmachineiscarefullyauditedbeforebeingcommissionedinordertoguaranteethatitmeetstheperformanceandreliabilitystandardsagreedtobeneededforcontinuousoperation.BoththeInternationalOrganizationforStandardization(ISO)andtheAmericanPetroleumInstitute(API)publishedsetsofspecificationsdevelopedfordifferenttypesofturbomachineusedinindustrialapplications,althoughtheAPIstandardsarelargelypreferredinthechemicalandpetrochemicalindustries.AlistofthoseAPIstandardsrelevanttodifferenttypesofturbomachinearepresentedinTable2.1.Inthissectionwepresentabriefsummaryofthedifferentlateralrotor-dynamicanalysesthatarerequiredbytheAPIspecificationsforcompressors.Theseanalysesguidecompressorend-users,originalequipmentmanufacturers(OEM),componentmanufacturers,servicecompaniesandeducationalinstitutionsonproperdesign,manufacturingandon-siteinstallationofmachines.Foramoredetaileddescriptionoftherequiredanalysesforcompressors,pleaserefertotheoriginalAPIStandard617[6].

2.4Rotor-DynamicSpecificationsforCompressorsTable2.1APISpecificationforCompressors,FansandPumps[111]

APIstandardnumber610612617673

MachinetypeCentrifugalpumpsSteamturbines

39

AxialandcentrifugalcompressorsCentrifugalfans

2.4.1LateralVibrationAnalysis

TheAPIdefinesthecriticalspeedtobetherotationalspeedoftheshaftthatcausestherotor/bearing/supportsystemtooperateinastateofresonance.Inotherwords,thefrequencyoftheperiodicexcitationforcesgeneratedbytherotoroperatingatthecriticalspeedcoincideswiththenaturalfrequencyoftherotor/bearing/supportsystem.Generally,thelateralcriticalspeedisthemostrelevant,anditisgivenbythenaturalfrequencyofrotorlateralvibrationinteractingwiththestiffnessanddampingofthebearings.Inthepresentday,ithasbecomecommonforhighperformancemachinestooperateabovethefirstcriticalspeed,butthecontinuousoperationatornearthenaturalfrequenciesisgenerallynotrecommended.

Figure2.7illustratesthelateralvibrationamplitudeversustherotatingspeedforatypicalrotor-dynamicsystem.ThebasiccharacteristicsofthevibrationresponsethatAPIemploystoevaluatethemachineareidentifiedinthefigure.TheithcriticalspeedisdenotedasNci,whichislocatedattheithpeakinthevibrationresponseplotwithamplitudeofAci.Theamplificationfactorofacriticalspeedisdefinedastheratioofthecriticalspeedtothedifferencebetweentheinitialandfinalspeedabovethehalf-powerofthepeakamplitudeN1−N2,asshowninFig.2.7.Lastly,themaximumcontinuousoperatingspeed(MCOS)ofthesystemcorrespondstothe105%ofthehighestratedspeedofthemachineinconsideration,andthespeedsbetweentheMCOSandtheminimumoperatingspeedofthemachineisknownastheoperatingspeedrange.

Theeffectivedampingataparticularcriticalspeedinarotor-dynamicsystemismeasuredthroughtheamplificationfactor,

AF=

Nc1

.

N2−N1

(2.77)

ThemeasurementoftheamplificationfactorisillustratedinFig.2.7forthefirstcriticalspeed.Alargeamplificationfactorcorrespondstoasteepresonancepeakwithlowdamping.Therefore,asmallvalueofAFisdesiredformodeswithinorneartheoperatingspeedrangeofthemachine.Formodeswithlargeamplificationfactors,aminimumseparationmarginSMisrequiredbetweenthecorrespondingcriticalspeedandtheoperatingspeedrangeofthemachine.

Thecriticalspeedsoftherotor/supportsystemcanbeexcitedbyperiodicdis-turbanceforcesthatneedtobeconsideredinthedesignofthemachine.TheAPIidentifiessomeofthesourcesfortheseperiodicdisturbancestobe[6]:

402IntroductiontoRotorDynamics

NciNmcN1,N2AFSMAci

=======Rotorithcriticalspeed(rpm).

MaximumcontinuousoperatingspeedMCOS(105%ofhighestratespeed).

Initialandfinalspeedat0.707×peakamplitude.Amplificationfactor.Nc1N2−N1.

Separationmargin.AmplitudeatNci.

Fig.2.7Exampleofarotorforcedresponse[6]

•••••••••••••rotorunbalance,oilfilminstabilities,internalrub,

blade,vane,nozzle,anddiffuserpassingfrequencies,geartoothmeshingandsidebands,couplingmisalignment,looserotorcomponents,hystereticandfrictionwhirl,boundarylayerflowseparation,

acousticandaerodynamiccross-couplingforces,asynchronouswhirl,

ballandracefrequencyofrolling-elementbearings,andelectricallinefrequency.

2.4Rotor-DynamicSpecificationsforCompressors41

Fig.2.8Undampedcriticalspeedvs.stiffnessmap[6]

Manyofthesedisturbancesarerelatedtothemechanicalandelectricalcharacteris-ticsofthemachinehardwareandtheycanbecorrectedatthedesignstageorthroughpropermaintenance.Forthelateralvibrationanalysis,wefocusontheforcedre-sponseduetotherotorunbalance.Thecross-couplingforceswillbediscussedintherotorstabilityanalysislaterinthissection.

2.4.1.1UndampedCriticalSpeedAnalysis

Estimatingthecriticalspeedsandthemodeshapesoftherotor-dynamicsystembetweenzeroand125%oftheMCOSisgenerallythefirststepinthelateralanalysis.Thecriticalspeedsoftherotor/supportsystemareestimatedfromtheun-dampedcriticalspeedmap,superimposedbythecalculatedsystemsupportstiffnessinthehorizontaldirection(kxx)andtheverticaldirection(kyy)asshowninFig.2.8.Aquickestimateofaparticularcriticalspeedcanbefoundfromthefigureattheintersectionofthecorrespondingcurveinthecriticalspeedmapandthebearingstiffnesscurve.TheactuallocationsofthecriticalspeedsofthesystembelowtheMCOSshouldbevalidatedinateststandasrequiredbytheAPIstandard[6].Modeshapeplotsfortherelevantcriticalspeedsshouldalsobeincludedinthisinitialanalysis.

422IntroductiontoRotorDynamics

2.4.1.2DampedUnbalanceResponseAnalysis

Adampedunbalanceorforcedresponseanalysisincludingallthemajorcompo-nentsoftherotor/bearing/supportsystemisrequiredbytheAPIstandardtobein-cludedinthemachineaudit.Thecriticalspeedandthecorrespondingamplificationfactorareidentifiedhereforallmodesbelow125%oftheMCOS.

TheproperlevelofunbalanceincompressorrotorsfortheforcedresponsetestisspecifiedintheSIunitstobe4×Ub,where

Ub=6350

W.N

(2.78)

ThetwoparametersintheabovedefinitionisthejournalstaticloadW(kg),andthemaximumcontinuousoperatingspeedN(rpm).ThejournalloadvalueusedforWandtheplacementoftheunbalancealongtherotoraredeterminedbythemodetobeexcitedasillustratedinFig.2.9.Forexample,toexcitethefirstbendingmode,theunbalanceisplacedatthelocationofthemaximumdeflectionneartherotormidspan,andWisthesumofthestaticloadsatbothsupportbearings.Figure2.9appliestomachineswithbetween-bearingandoverhungrotorsasgivenin[6].Basedontheresultsfromtheforcedresponse,aseparationmarginisrequiredforeachmodebelowtheMCOSthatpresentsanamplificationfactorequaltoorgreaterthan2.5.Therequiredminimumseparationmarginbetweenthemodecriticalspeedandtheoperatingspeedrangeisgivenas

󰁻󰁳󰁴󰁼

1

SM=min171−,16.(2.79)

AF−1.5Ontheotherhand,ifthemodewithanamplificationfactorequaltoorgreaterthan2.5isabovetheMCOS,thentherequirementfortheminimumseparationmarginbetweenthemodecriticalspeedandthemachineMCOSisspecifiedtobe

󰁻󰁳󰁴󰁼

1

SM=min10+171−,26.(2.80)

AF−1.5Therequirementsontheseparationmarginisemployedtodetermineanoperatingspeedofthemachinethatavoidsanycriticalspeedwiththepotentialtodamagethesystem.

Finally,fortraditionalfluid-filmandrolling-elementbearings,thepeak-to-peakamplitudelimitoftherotorvibrationisgivenby

󰁒

12,000

A1=25,(2.81)

NwhereNisthemaximumcontinuousoperatingspeedinrpm.Atthesametime,thepeakamplitudeoftherotorvibrationatanyspeedbetweenzeroandNmcshouldnotexceed75%oftheminimummachineclearance.WewilllaterseeinChap.7thatthisparticularspecificationgenerallydoesnotapplytosystemswithactivemagneticbearings.

2.4Rotor-DynamicSpecificationsforCompressors43

Fig.2.9UnbalancevaluesandplacementsasspecifiedbyAPI[6]

2.4.2RotorStabilityAnalysis

Asthenameindicates,thisanalysisinvestigatesthestabilityoftherotor-dynamicsysteminthepresenceofcommondestabilizingforcesthatcompressorsandtur-binesaresubjectedtoduringnormaloperation.Thedominantforcesinthisgroupareoftentheaerodynamiccross-couplingforces,whichwereintroducedinSect.2.3.ThestabilityanalysisisrequiredbytheAPIforcompressorsandradialflowrotorswiththefirstrotorbendingmodebelowtheMCOS[6].

Thestabilityoftherotor-dynamicsystemintheAPIstandardisnormallyeval-uatedbytheamountofdampingonthefirstforwardmode.ThestandardmeasureofmechanicaldampingemployedintheAPIstandardisthelogarithmicdecrement,whichiscomputedasthenaturallogarithmoftheratiobetweentheamplitudesoftwosuccessivepeaks.Therelationbetweenthemodelogarithmicdecrementδand

442IntroductiontoRotorDynamics

thecorrespondingdampingratioζcanbefoundtobe

δ=󰁐2πζ1−ζ2.

(2.82)

2.4.2.1LevelIStabilityAnalysis

TheLevelIstabilityanalysisisthefirststepofthestabilityanalysis.Itisintendedtobeaninitialscreeningtoidentifythemachinesthatcanbeconsideredsafeforoper-ation.Theinletanddischargeconditionsforthestabilityanalysisareselectedtobeattheratedconditionofthemachine,althoughitisallowedforthevendorandthepurchasertoagreeonadifferentoperatingconditiontoperformthetest.Thepre-dictedcross-couplingstiffnessinkN/mmateachstageofacentrifugalcompressorisgivenby

qA=HP

BcCρd

,

DcHcNρsBtC

.

DtHtN

(2.83)

andthatofanaxialcompressorisgivenby

qA=HP

(2.84)

TheparametersintheaboveequationsareHPBcBtCDc,Dt

HcHtNρdρs

==========

ratedcompressorhorsepower,3,1.5,9.55,

impellerdiameter(mm),

minimumofdiffuserorimpellerdischargewidth(mm),effectivebladeheight(mm),operatingspeed(rpm),

dischargegasdensityperimpeller/stage(kg/m3),andsuctiongasdensityperimpeller/stage(kg/m3).

Thepredictedtotalcross-couplingstiffnessQAisthesumoftheqAforalltheimpellers/stagesinthecompressor.

IntheLevelIanalysis,thestabilityoftherotor-dynamicsystemistestedforavaryingamountofthetotalcross-couplingstiffness.Theappliedcross-couplingstiffnessvaluerangesfromzerotothesmallestbetween10QAandthemaximumcross-couplingstiffnessbeforethesystembecomesunstable.Thispointofinstabil-ityisidentifiedbytheAPItocorrespondtothecross-couplingstiffnessvalueQ0wherethedamping,orlogarithmicdecrementofthesystemfirstforwardmodebe-comeszero.FortheLevelIanalysis,thecross-couplingstiffnessisassumedtobeconcentratedattherotormid-spanforbetween-bearingmachines,oratthecenterofmassofeachimpeller/stageforcantileveredsystems.

2.4Rotor-DynamicSpecificationsforCompressors45

Fig.2.10Typicalplotoflogarithmicdecrementcorrespondingtothefirstforwardmodevs.ap-pliedcross-couplingstiffnessforLevelIstabilityanalysis

AnimportantgraphthatisrequiredbytheAPItobeincludedintheLevelIanalysisistheplotofthelogarithmicdecrementδforthefirstforwardmodeversustheappliedcross-couplingstiffnessQ,aspresentedinFig.2.10.Thepredictedtotalcross-couplestiffnessQAandthecorrespondinglogarithmicdecrementofthefirstforwardmodeδAaremarkedinthefigure.Additionally,Q0correspondstothecross-couplingstiffnesswhenthelogarithmicdecrementofthefirstforwardmodebecomeszero.Theboundaryatδ=0.1correspondstothepass/failconditionofthestabilityanalysis,whichwillbediscussedlaterinthissection.

Wenoteherethat,althoughwithtraditionalpassivebearingsthefirstforwardmodeisgenerallythefirstonetobedriventoinstabilitybythecross-couplingstiff-ness,thesituationisnotasstraightforwardwithAMBs.Astheactivecontrollerinthesemagneticbearingsnormallyhasadirectinfluenceonanysystemmodewithinthecontrollerbandwidth,theinteractionbetweenthecontrollerandthecross-couplingeffectshasthepotentialtodestabilizeagroupofmodeswithinandabovethecompressoroperatingspeedrange.Therefore,thelogarithmicdecrementofallmodeswithinthelevitationcontrollerbandwidthissometimesinspectedduringtheLevelIstabilityanalysisformachineswithmagneticbearings.

BasedontheresultsfromtheLevelIstabilityanalysis,machinesthatdonotmeetcertainstabilitycriteriaarerequiredtoundergoamoreadvancedLevelIIsta-bilityanalysis.Forcentrifugalcompressors,aLevelIIstabilityanalysisisrequired

462IntroductiontoRotorDynamics

ifeitherof

Q0/QA<2,

δA<0.1,

(2.85a)(2.85b)

isfoundtobetrue.Inthecaseofaxialcompressors,aLevelIIanalysisisrequiredonlyif

δA<0.1.

(2.86)

2.4.2.2LevelIIStabilityAnalysis

TheLevelIIstabilityanalysisisacompleteevaluationoftherotor/bearingsystemwiththedynamicsofallthecompressorcomponentsgeneratingtheaerodynamiccross-couplingstiffnessoraffectingthestabilityoftheoverallmachine.Someofthesecomponentsare[6]•••••

seals,

balancepiston,

impeller/bladeflow,shrinkfit,and

shaftmaterialhysteresis.

Detailsonthemethodologyoftheanalysisislefttoagreatextenttobedecidedbasedonthelatestcapabilitiesofthevendor.APIdoesnotspecifyhoweachdy-namiccomponentishandledintheanalysis.TheoperatingconditionofthemachineusedintheanalysisisthesameasintheLevelIanalysis.

DuringtheLevelIIanalysis,theAPIrequiresthevendortoinitiallyidentifythefrequencyandlogarithmicdecrementofthefirstforwarddampedmodeforthebarerotor/supportsystem.Then,theanalysisisrepeatedafteraddingthedynamicsofeachcomponentpreviouslyidentifiedtoaffectthestabilityoftherotor-dynamicsystem.Finally,thefrequencyandlogarithmicdecrementδfofthefirstdampedforwardmodeiscomputedforthetotalassembledsystem.

Thepass/failconditionoftheLevelIIstabilityanalysisstatedbyAPI617is

δf>0.1.

(2.87)

Ifthisissatisfied,thenthemachineisconsideredtohaveguaranteedstabilityintheratedoperatingcondition.Ontheotherhand,ifthepass/failconditioncannotbesatisfied,APIallowsthevendorandpurchasertomutuallyagreeonanacceptablelevelofδfconsideredtobesufficientforthesafeoperationofthemachine.Finally,itisrecognizedintheAPI617thatotheranalysismethodsexistforevaluatingthestabilityofrotor-dynamicsystems,andthesemethodsareconstantlybeingupdated.Therefore,itisrecommendedtofollowthevendor’sstabilityanalysismethodsifthevendorcandemonstratethatthesemethodscansuccessfullypredictastablerotor.

2.5RotorFiniteElementModeling47

2.5RotorFiniteElementModeling

Thefirstpriorityofthebearingsinarotor-dynamicsystemiscommonlythesupportoftherotorlateraldynamics.Althoughtherotoraxialvibrationsalsoneedtobecarefullyanalyzedforpossiblesignsoftrouble,themainsourceofrotorinstabilityinmostrotatingmachinescomesfromthelateralorradialvibrations.Forthisrea-son,anaccuratemodelofthesystemlateraldynamicsisessentialfortheanalysisandsimulationtestingthatarerequiredduringthedesignandcommissioningphasesofthesemachines.InthecaseofsystemswithAMBs,theneedforanaccuratemodelisevenhigherastheunstablebearingsystemrequiresreliablemodel-basedrotorlevitationcontrollersfornormaloperation.

Thelateraldynamicsofflexiblerotorsaredescribedbypartialdifferentialequa-tions.Thesearecomplexequationswithspatiallydistributedparameters,anditusu-allyisnotpossibletoderiveanalyticsolutionsforrotorswithcomplexgeometries.Inrealworldapplications,alinearizedapproximationmodeloftherotorlateraldy-namicsisnormallysufficientforanalyzingrotor-dynamicsystemsanddesigningrotorlevitationcontrollersforAMBs.Suchamodelcanbeobtainedbymeansofthefiniteelementmethod(FEM),wherethedescriptionofthespatiallycontinu-ousrotorissimplifiedtothedegreesoffreedomcorrespondingtoafinitenumberofshaftelements,effectivelyeliminatingthespatialvariableintheoriginalbeamequation[119].

Inthissectionwepresentabriefsummaryoftheprocessforobtainingthetwo-dimensionalfiniteelementmodelofarotor-dynamicsystem.Detailedstep-by-stepdescriptionofthefiniteelementmethodcanbefoundinthemanyavailablefiniteelementtextbookssuchas[4],andtheapplicationofthismethodformodelingtherotor-dynamicsystemisthoroughlydiscussedin[5]and[119].Inthissection,weonlypresentaconcisedescriptionoftheprocessforderivingthefiniteelementmodel,asanintroductiontowhatwilllaterbeusedinChap.7forthesynthesisoftheAMBlaterallevitationcontroller.

2.5.1DiscretizingRotorintoFiniteElements

Asthefirststepofderivingafiniteelementmodel,therotorisaxiallydividedintosimpleuniformbeamelementsconnectingtwoadjacentnodepoints.AtypicalmeshofasimplerotorisillustratedinFig.2.11,wherethenodepointsareshownasdarkdots.Theselectionofanadequaterotormeshmustfollowsomerulesthatarebasedontherotorgeometry,aswellasthelocationsoftherotordisks,bearings,andotherrotor-dynamiccomponents.First,anodalpointmustbeplacedateachloca-tionalongtherotorwithastepchangeinthediameter,sothatallshaftelementshaveauniformradius.Thiswilllatersimplifythemodelingofthedynamicsforeachindividualshaftelement.Second,anodepointisdefinedateachlocationwithamass/inertiadisk,bearing,seal,andanyothersourceofexternaldisturbanceforce.Bythesametoken,allsensorlocationsandothermeasurementpointsarealsocollo-catedwiththeshaftnodepoints.Thisrulesimplifiesthedefinitionoftheinputand

48

Fig.2.11Rotormeshexample

2IntroductiontoRotorDynamics

Fig.2.12Beamelementandgeneralizeddisplacementsofnodesiandi+1

outputvariablesinthefinalexpressionofthefiniteelementmodel.Finally,theratiooftheelement’slengthtodiametermustbeaboutoneorlessinordertoguaranteetheaccuracyofthefiniteelementformulation.

TherotorshowninFig.2.11hasatotalof17elementsand18nodepoints.Itiscommonfortheelementsandnodestobenumberedfromlefttoright,asdemon-stratedinthefigure.Thesupportbearings,withgivenstiffnessanddampingcoeffi-cients,arelocatedinthisexampleatthenodes4and15.Fortheremainingofthissection,wewillassumethatthegeneralrotormeshconsideredhereiscomposedofnbeamelements,correspondingtoatotalofn+1nodepoints

2.5.2ApproximatingElementDisplacementFunctionsandNodal

Displacement

Oncetheshaftissectionedintosmallerelements,thedynamicsofeachshaftsectionisstudiedindependently.Thegeneralizeddisplacementsandrotationsoftheshaftelementaredescribedthroughthedegreesoffreedomthataredefinedateachnodepoint.ThedegreesoffreedomforatypicalbeamelementareshowninFig.2.12.Consideringonlythelateraldynamicsoftherotorforsimplicity,eachshaftsec-tionhaseightdegreesoffreedom,correspondingtothetwodisplacementsandtworotationsaboutthelateralaxesateachnodepoint.

AsshowninFig.2.12,thelateraldisplacementsoftheithnodearegivenasuxiinthehorizontalx-axis,anduyiintheverticaly-axis.Theangulardisplacementsatthesamenodeaboutthey-andx-axesaredefined,respectively,as

θy=∂ux

,∂z∂uy

.θx=∂z

(2.88a)(2.88b)

2.5RotorFiniteElementModeling49

Thedegreesoffreedomoftheithnodepointarecollectedinthegeneralizeddis-placementvectorqi,whichdescribesthepositionandrotationofthenodeatagiventime.Thedisplacementandrotationvariablesaresortedinthementionedvectoras

⎡⎤uxi⎢uyi⎥

⎥(2.)qi=⎢⎣θyi⎦.θxiLastly,thegeneralizeddisplacementoftheithshaftelementillustratedinFig.2.12combinesthegeneralizeddisplacementvectorsattheendnodesqiandqi+1.ThegeneralizeddisplacementvectorfortheithelementinFig.2.12isdefinedas

󰁵󰁶qi

Qi=.(2.90)

qi+1Thegeneralizeddisplacementvectordefinedaboveisusedinthederivationofthedynamicmodeltoestimatethestateoftheentireshaftsection.Thus,theeightvari-ablesinQiuniquelydescribetheshapeoftheithbeamelementinthefiniteelementformulation.

Basedonthedegreesoffreedomdefinedatashaftelementoftherotormesh,thelateraltranslationandrotationisinterpolatedatanyarbitrarypointalongtheshaftelement.TheshapeoftheentireshaftelementisestimatedintermsofthegeneralizeddisplacementvectorQiandtheshapefunctionsNi.Theshapefunctionsthatformathirdorderpolynomialbasisoftheshaftelementaregivenas[4]

N1=

󰁢1󰁡323

L,−3zL+2zL3

󰁢1󰁡

N2=2zL2−2z2L+z3,

L

󰁢1󰁡23

N3=33zL−2z,

L

󰁢1󰁡

N4=2−z2L+z3.

L

(2.91a)(2.91b)(2.91c)(2.91d)

TheparameterListhelengthoftheshaftelement,andthevariablezistheaxialpositionalongtheelement’slength.TheaboveshapefunctionsareillustratedinFig.2.13.

ForthegivenbasisofshapefunctionsinEqs.(2.91a)–(2.91d),thegeneralizedlateraltranslationoftheithshaftelementatanarbitraryaxialpositionzisexpressedasafunctionofthetimetandtheaxialoffsetfromtheleftmostnodeas

󰁵󰁶󰁵󰁶uxi(z,t)N10N20N30N40

=(2.92)Qi.

uxi(z,t)0N10−N20N30−N4Inthesameway,thelateralrotationsθyi(z,t)andθxi(z,t)atanarbitraryaxialpositionzcanbefoundbycomputingthepartialderivativeofEq.(2.92)withrespect

502IntroductiontoRotorDynamics

Fig.2.13ElementHermiteshapefunction

totheaxialoffsetzasshowninEqs.(2.88a),(2.88b).AnimportantobservationfromtheexpressionsinEq.(2.92)isthatthespatialvariableziscontainedinthematrixofbasisfunctionsinEq.(2.92),whileonlythegeneralizeddisplacementvectorQiisafunctionoftime.Thus,thedescriptionofthedynamicsoftheoriginalcontinuousshaftelementissimplifiedinthefiniteelementformulationintoafinitenumberofdegreesoffreedomcorrespondingtoadiscreteshaft[119].

2.5.3FormulatingEquationsofMotionforEachElement

TheequationofmotionfortheithshaftelementisdeterminedfollowingtheLa-grangeformulation:

󰁳󰁴

∂Rid∂Li∂Li+=0.(2.93)−

dt∂q˙i∂qi∂q˙i

2.5RotorFiniteElementModeling51

TheLagrangianoftheithelementLiisdefinedasthedifferencebetweentheele-ment’skineticenergyTiandpotentialenergyUi,

Li=Ti−Ui.

(2.94)

Additionally,Ricapturestheenergydissipationinthesystemduetotheinternalfrictionordamping,anditisknownasthedissipationfunction.Giventhatthegen-eralizeddisplacementofashaftelementisapproximatedasshowninEq.(2.92),thetermsforboththekineticandpotentialenergiescanbeeasilyfoundbasedoneithertheBernoulli–EulerortheTimoshenkobeamtheories[4].Foreachofthebeamele-ments,thepotentialenergycomesmainlyfromthebeambendingandsheareffects.Ontheotherhand,thelevelofthekineticenergyisdeterminedbyboththelateralandtherotatoryinertialeffectsintheshaftelement.

ByexpandingtheLagrangeequationinEq.(2.93)withtheenergyformulationfortheindividualshaftsection,anexpressiondescribingthelateraldynamicsoftheithelementoftherotormeshisobtainedintheformofthevectordifferentialequation,

˙i+GiQ¨i+CQ˙i+Kiqi=Fi.MiQ

(2.95)

ThesystemmatricesarethemassmatrixMi,gyroscopicsmatrixGi,stiffnessmatrix

Ki,andthedampingmatrixCi.ThegeneralizedexternalforcevectorFiisaddedtotheLagrangeequationtoaccountfortheexternalforcesandtorquesperturbingthesystem.Theobjectiveofthefiniteelementformulationistofindtheexpressionsforthesystemmatrices,basedonEq.(2.93)andthegeneralizeddisplacementsinEq.(2.92).

2.5.4ElementMassandGyroscopicMatrices

Thekineticenergyofameshelementcomesfromthetranslationalandangularmomentumoftheshaft.Forauniformithbeamelementwiththegeneralizeddis-placementasdefinedbyEq.(2.92),theresultingexpressionofthekineticenergytakestheform

1˙T1˙T

˙Ti=QM+(2.96)ωQiWiQi.Qii

2i2

ThematrixMicorrespondstothemassmatrixoftheshaftelement,andthematrixWiisrelatedtothepolarmomentofinertiaoftheelementwitharotationalspeedofω.Adetailedstep-by-stepdescriptionofhowtodeterminetheexpressionsforthesematricescanbefoundin[5]and[119].

ThecontributionofthekineticenergyintheLagrangeequationappearsinthefirstandsecondtermsofEq.(2.93).ThefirsttermoftheLagrangeequationinEq.(2.93)withtheaboveformofthekineticenergyisgivenby

󰁳󰁴d∂T¨i+1ωWIQ˙i.(2.97)=MiQ

˙idt∂Q2

522IntroductiontoRotorDynamics

ThecorrespondingsecondtermoftheLagrangeequationis

−

1∂T˙i.=−ωWiTQ

∂Qi2

(2.98)

CombiningthetwotermsofthekineticenergyintheLagrangeequation,weobtain

theequation

󰁳󰁴

󰁢󰁡∂Td∂T˙i,¨i+1ωWi−WiTQ=MiQ−

˙idt∂Q∂Qi2

¨i+GiQ˙i,=MiQ

(2.99)

wherethegyroscopicmatrixGiisdefinedaboveintermsofthematrixWiandthe

rotorspeedω.

ThefinalexpressionsofthemassmatrixMiandthegyroscopicmatrixGiforauniformshaftelementcanbefoundin[5]and[119].Thesematricesareexpressedintermsoftheelement’slength,crosssectionalarea,andmaterialdensity.Therefore,astheexpressionsareidenticalforallelementsinthemesh,itisrelativelysimpletoautomatetheprocessoffindingthesematricesforallshaftsections,giventhattherotormeshhasbeenselectedaccordingtotherulesdescribedatthebeginningofthissection.

2.5.5ElementStiffnessMatrix

BasedontheBernoulli–Eulerbeamtheory,thepotentialenergyofauniformshaftelementcomesfromtheinternalstrainenergyduetothelateralbending.Fortheithuniformbeamelementwiththegeneralizeddisplacementsuxi(z,t)anduxi(z,t)definedasinEq.(2.92),theresultingexpressionofthepotentialenergytakesthequadraticform

1

KiQi.(2.100)Ui=QT2i

ThematrixKiisthestiffnessmatrix.Itdescribestheaxialstrain/stressduetothelateralbendingofthebeamelement.AdetailedderivationofthepotentialenergytermUiandthestiffnessmatrixcanbefoundin[5]and[119].SubstitutingtheaboveexpressionofthepotentialenergytothesecondtermoftheLagrangeequationinEq.(2.93),weobtain

∂Ui

=KiQi.∂Qi

(2.101)

ThecoefficientsofthestiffnessmatrixKiarefoundin[5,119],andtheyaregivenintermsoftheelement’slength,crosssectionalareamomentofinertiaaboutthelat-eralaxes,andmodulusofelasticity.Therefore,sameasinthemassandgyroscopicmatrices,theprocessofcomputingthestiffnessmatrixforallshaftelementscanbeeasilyautomated,providedtheinformationabouttherotormesh.

2.5RotorFiniteElementModeling53

2.5.6ElementDampingMatrix

Thedissipationoftheenergyintheshaftduetotheinternalfrictionisgenerallysmall,andthusthedissipationfunctionisnormallyneglectedinthefiniteelementformulation.Forspecialcaseswherethedissipationfunctionisnotnegligible,theexpressionforRitakestheform

1˙T˙

CiQi,Ri=Q2i

(2.102)

wherethematrixCiisthedampingmatrixoftheshaftelement.Withtheaboveformofthedissipationfunction,thethirdtermoftheLagrangeequationinEq.(2.93)becomes

∂Ri

=CiQi.(2.103)˙i∂QFinally,combiningthetermsintheLagrangeequationcorrespondingtothekinetic

energyinEq.(2.99),potentialenergyinEq.(2.101)anddissipationfunctioninEq.(2.103),weobtainthevectordifferentialequationfortheshaftelementasshowninEq.(2.95)

2.5.7AddingLumpedMass,StiffnessandDampingComponents

Complexrotordesignscanincludeimpellers,motorcore,andothermassdisksthatcontributetothedynamicsoftherotor/supportsystem.Thesecomponentsaretreatedinthetwo-dimensionalfiniteelementformulationasrigiddiskslocatedatthedifferentshaftnodepoints,andthecorrespondingmassandmomentofinertiaareaddedtotheshaftmodel.Asdiscussedatthebeginningofthissection,thecentersofmassofthedisksareassumedinthefiniteelementformulationtobecollocatedwithsomenodalpointsintherotormesh.Undertheassumptionthatthegeneralizeddisplacementvectorcorrespondingtothenodeatthelocationofthediskisgivenas

⎡⎤uxd⎢uyd⎥

⎥(2.104)qd=⎢⎣θyd⎦,θxdthevectordifferentialequationofthedisktakestheform[119]

Mdq¨d+Gdq˙d=0,

(2.105)

whereMdisthediagonalmassmatrixofthedisk,andGdistheskew-symmetric

gyroscopicmatrix.TheexpressionsforthemassandthegyroscopicmatricesareasdescribedinSect.2.2

542IntroductiontoRotorDynamics

Sealsandbearingsarealsoimportantcomponentsinrotor-dynamicsystems,addingstiffnessanddampingtotherotoratparticularnodelocations.Giventhatqbisthegeneralizeddisplacementvectoratthenodepointcorrespondingtothebearing/seallocation,thevectordifferentialequationforthestiffnessanddampingcontributionis

Cbq˙b+Kbqb=0.

(2.106)

ThematrixCbisthedampingmatrix,andKbisthestiffnessmatrixofthebearing

orseal.Thesematricesaredesignparametersthatarecommonlyprovidedbythemanufacturer,andinmanycasestheyarefunctionsoftheshaftspeed.

2.5.8AssemblingtheGlobalMass,Gyroscopic,Stiffness,Damping

Matrices,andForceTerms

Finally,thesystemmatricesfortheshaft,disksandothercomponentsareassem-bledtoformthecorrespondingglobalmatrices.Giventhattheglobalgeneralizeddisplacementvectorisdefinedas

Q=[q1q2q3···qn+1]T,

(2.107)

thevectordifferentialequationforthecompleterotor-dynamicsystemhasthefinalformof

¨+GQ˙+CQ˙+KQ=F.MQ

(2.108)

ThesystemmatricesoftheequationofmotioninEq.(2.108)aretheglobalmassma-trixM,theglobalgyroscopicmatrixG,theglobaldampingmatrixCandtheglobal

stiffnessmatrixK.ThegeneralizedexternalforcevectorfortheglobalsystemisgivenbyF,whichincludesalltheexternaldisturbanceforces/torquesperturbingthedynamicsoftheglobalsystem.AllsystemmatricesandvectorsaredefinedinthesameorderasthenodaldisplacementsinthevectorQ.

TheglobalmatricesinEq.(2.108)areassembledbycombiningateachshaftnodepointthecontributionofallthecomponentsinthefiniteelementmodel.Herewedescribetheprocessfortheassemblyoftheglobalmassmatrix.Thesamestepscanbefollowedforformingtheremainingsystemmatrices.TheassemblyoftheglobalmassmatrixfromtheindividualmassmatricesoftheshaftelementsisshowninFig.2.14,whereMiisthemassmatrixfortheithshaftelement.Theoverlap-pingregionsbetweentheblockscorrespondingtoadjacentelementsinFig.2.14aresummedintheglobalmatrix.Next,themassmatricesfortherotordiskandanyothercontributingcomponentsareaddedintotheglobalsystembysummingthematrixentriestotheappropriateblocksinM.Foracomponentlocatedattheithnodeoftheshaft,themassmatrixofthecomponentisaddedtothesquareblockofMbetweenthecolumnandrownumbers4i−3and4i.Thefinalglobalmassmatrixisa4(n+1)×4(n+1)squaresymmetricmatrix,whichisconsistentwiththelengthofthedisplacementvectorQ.

2.6ConclusionsFig.2.14Globalmassmatrixassembly

55

2.6Conclusions

Abriefintroductiontorotor-dynamicswaspresentedinthischapter,withthein-tentionoffamiliarizingthereaderwiththeconceptsthatwillbeexpandedinthelatterchaptersofthisbook.ThediscussioninrotordynamicswasinitiatedherebystudyingtheequationsofmotionfortheJöppl/Jeffcottrotor.Basedonthissim-plifiedrotor-dynamicsystem,differentcharacteristicsthatareusedfordescribingthedynamicsofcomplexrotatingmachineswereidentified.Next,thegyroscopicmomentandthecross-couplingstiffnessweredefined,andtheireffectsonrotatingshaftswerediscussedinsomedetail.ThesearegenerallyknowntobethetwomainsourcesofinstabilityinAMBsupportedsystems,aswewilllaterobserveduringthedesignoftheAMBlevitationcontrollerinChap.7.Finally,theAPIstandardthatiswidelyusedforauditingtherotorresponseincompressorswerereviewed.Al-thoughmostofthesestandardsweredevelopedbasedontheresponseoftraditionalpassivebearings,manymanufacturersandend-usersrelyontheAPIspecificationsforauditingAMBsystems.

Aspreviouslymentioned,manyoftheconceptsintroducedherewillberevisitedduringthecharacterizationofthecompressortestriginChap.4andthedesignoftheAMBlevitationcontrollerinChap7.Rotordynamicsisaveryrichfieldofstudy.Itisnotpossibletopresentallthematerialwiththesamelevelofdetailasfoundinspecializedbooksonthetopic.SomeconceptswillplayamoreimportantrolethanothersinthedevelopmentofthestabilizingAMBcontrollersfortherotorvibrationandthecompressorsurge.Inthischapterwefocusedonaselectednumberoftopicsthatarerelevanttotheobjectivesofthisbook.Forfurtherreadingonthetheoryofrotordynamics,werecommendtheliteraturethatwasreferencedthroughoutthischapter.

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