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4BENTLYROTOR DYNAMICS
ORBIT December1989
__ ~) ShaftCenterLINESRESEARCH CORPORATION
FigureIRotatingshaftcircularsynchronous(IX)orbitingarounda
neutralaxisataconstantspeed.Constantstressinshaftfibers.
By Dr.AgnesMuszynskaSeniorResearchScientist
BentlyRotorDynamicsResearchCorporation
This articlediscussesrotorlat-eral vibrations and radial
preload-related rotor dis-placementsas a source oflow
andhighcyclefatigueof
rotormaterial.
The availability of vibration
trans-ducersandvibrationmDnitoringsystemscreatedundisputableawarenessofvibra-tion
levelsin machinery.Vibrations ofrotorsandof other
machineparts,suchas blades, couplings, pedestals
andcasings,occurasasideeffectof themaindynamicmode, namelyshaft
rotation.Due to the existenceof severalphysicalmechanisms,part of
the rotativeenergybecomestransferredinto the energyofvibration at
various modes.The rotor
itself,thecarrierof
therotativeenergy,ismostpronetovibrationalmotion.Vibra-tionsof
othermachineelementsareusu-
ally secondaryto the rotor. They aretransmittedfrom
therotor.Shaftlateral
vibrationsareamongthemostdamagingandoftendestructivevibrationalmodes.
Vibration monitoringsystems,whicharenowwidelyusedon
machinery,havecreatedanoverallimpressionthatvibra-tions are
unwelcome and that the
vibrationlevelon machineryshouldbekept aslow aspossible.This
philosophygenerallyholds true.There is, however,onecatch.
A horrorstory"Our machineshaveverylowvibration
levels;' the maintenance technician
__ NEUTRAL AXIS
SHAFT
PRECESSION (ORBITING)
WlTH FREQUENCY w
of a chemical plant told us proudly."Since we installedthe
vibration mon-
itoringsystem,wecanmonitorandcontrolvibrations.If the vibration
level is too
high,wesimplypreloadtheshaftbymis-aligning it within the machine
train,until vibration sufficiently decreases.
No vibration,no problem!"
COUNTERCLOCKWlSE ROTATION
'MTH FREQUENCY W
HIGH SPOT
1 ROTAllON I TIME
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ONE ORBITING PERIOD
December1989
Twomonthslater,thistechnicianwasverysurprisedwhen,subsequently,theshaftsin
twomachinesbroke,creatingamajordisasterfortheplant.Why?Therewasnobadvibrationandyettheshaftsbroke?
ShaftstressThis unfortunatetechnicianover-
looked one important
aspect.Themachineshaftnotonlyvibrates;theshaftalsorotates.Thus,theshaft'sdynamicmotionhastwoimportantcomponentswhichcontributetoshaftmaterialstress.Shaftbreakageis
duetoeitherstressinthematerialexceedingtheultimatelimitorduetolowcycleorhighcyclefatigue.In
thenextsectionswewillexplainhowto correlateshaftvibrationand
shaftmaterialstress.
Theleastdamagingmode:Shaftlateralsynchronousvibrationaroundaneutralaxis
Synchronous(IX) lateralvibrationsof rotatingshaftsare mainlydue
tounbalance.Evenwell-balancedshaftsexperiencesomeresidualsynchronousvibration
in two orthogonallateraldirections(e.g,horizontalandvertical),
ONE ROTATION
ORBIT
whichare perpendicularto the shaftaxis.The resultingshaftmodeis
repre-sentedbythefamiliarorbiting.FigureIillustratestheinstancewheretherotat-ingshaft'sIX
orbitiscircular.Thecon-ventionofthedrawingisasfollows:Theorbitis,in
fact,muchsmallerthantheshaftradius.Imagine,however,thatthecirclesdonotrepresenttheentireshaftcrosssections,but
just tiny portionsaroundtheshaftcenter.The
stressattheshaftsurfacewillberoughlypropor-tional to the stressat
the
surfaceofthesetinycircles.Sincetheconsidera-tionsherearequalitative,thisconven-tionprovidessatisfactoryresults.
When the
shaftvibrationmodeispurelysynchronousandtheorbitiscir-culararoundaneutral(nostress)axis,then,ata
constantrotativespeed,partof the shaft fibers are
constantlystretched.Another part is
constantlycompressed.Themoststretchedsectionoccursattheshafthighspot.If
theIXvibrationamplitudeis not large,themaximumstressremainswithin
anacceptableconstantlevel.Althoughshaftvibrationis
alwaysunwelcome,this synchronouscircular vibrationmodearoundthe
neutralaxisis lessdamagingtotheshaft.
COUNTERCLOCKWISE ROTATION
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ElUPTICAL 1X ORBIT
TIME
5
Preloadeffect:Periodicallyvariablestress
Assumethat the unbalancedshaft,rotatingat a constantspeed,is
pre-loadedbyaconstantradialforce.Suchforcecanbegenerated,for
instance,byamisalignment,bygravityonhorizontalrotorsorbyaradialpumpingsideload,suchasin
asinglevolutepump.In thiscase, the shaft
centerlinebecomesdeformedand thusput
understress,roughlyproportionalto the
preload-resultingdisplacement(Figure2).Dueto thecombinationof
rotationandIXorbiting,theshaftfibersareperiodicallystretchedandcompressed.Thus,
thedisplacementof theshaftneutralaxisresultsin
muchmoresevereconditionsontheshaft.EvenasmallIX
vibrationamplitude(asmeasuredbyaproximityprobe)andsomeshaftpreload-relateddisplacement(also
measuredby theproximityprobe as the dc gap) mayresult in
significantlyhigh reversal,cyclicstressin theshaft.
Thisperiodicvariablestress(withIXfrequency)leadstomaterialfatigue,tolowcyclefatiquewhenstressamplitudesare
largeand to highcyclefatique~
~I
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TIME
Figure2Preloadedrotatingshaft:displacedIX
orbitexhibitsamoreellipticalshape.Shaftfibersareunderreversalcyclicstress.
Figure3Shaftstressduringwhipconditions.Whipfrequency=1/4rotativefrequency.Shaftstressfrequency=
3/4rotativespeedfrequency.
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6cyclefatigueresultswhenthe
ampli-tudesaresmaller.Thelowcyclefatigueoccursrelativelyquicklydue
to non-linearplasticdeformations.The
highcyclefatiguerequiresmanymorecyclesof reversalstress(of108to
109range)todamagetheshaftmaterial.Thenumber108seemstobeaverylarge.Calculate,however,howmanycyclesa
machinemakeswhilecontinuouslyoperatingataconstantspeedof3600rpm.In
only500hoursof operation,thenumberof
IXfrequencyreversalstresscyclesreaches1.08X 108.Therefore,in
abouttwentydaysthe highcyclefatigueconditionsmayeasilyoccur.This
mayresult inshaftcrackingandbreakage.
Othercasesof shaftstress
Highor lowcyclefatigueconditionsmayresultfromeither(1) a
largepre-load andsmallvibrationas discussedabove,(2)a
largevibrationof thenon-synchronoustypeor (3) a
largevibra-tiontogetherwithasmallpreload.
In thecaseof highpreload-resultingdisplacementof the
shaftcenterline,thefrequencyof shaftreversalstressisalwaysIX for
anyvibrationpatternoftheshaft,if thedisplacementis larger
ORBIT
thanthevibrationamplitude(peak).
In thecaseof largenonsynchronousvibrationsof the shaftwith no
initialdisplacement,the
reversalstressfre-quencycanbeestimatedastheabsolutevalueof
thedifferencebetweenvibra-tion androtativefrequencieswith
therelativedirectionsofrotationandorbit-ingtakenintoaccount.Thus,if
therotoris in forwardwhipconditionwithsub-synchronousfrequencyl/4X,
theshaftwillbecyclicallystressedwithfrequency3/4X(Figure3).If
theshaftoperatesatits halt-balanceresonancespeed,andtheorbithasa IX
andasignificant2Xforwardcomponent,then, the shaftstresswillhaveIX
frequency(Figure4).The
rub-relatedself-excitedbackwardvibrations("drywhip")withfrequency,for
instance,twicehigherthanrotativespeed,result in 3X
frequencyshaftreversalstress(Figure5). In
thiscase,however,therubbingdamagewillmostprobablyoccurmuchearlierthananyfatiguedamage.
Theunbalance-relatedIX shaftvibra-tionsseldomresultin
circularorbits.Asymmetryofstiffnessandotherparam-etersin
twoorthogonaldirectionsof theshaftitself,as well as the
supporting
December 1989
structure,usuallycausestheIX
orbitstobeelliptical.FortheellipticalIX
orbits,theshaftstresshasapulsatingcharacterwithfrequency2X
(Figure6).Pulsatingstressoccursalsointheshaft,vibratinginthecircularorbitmodewithashaftcen-teronlyslightlydisplaced(Figure7).
Shaftmode
The shaft-observingproximitytrans-ducersareusuallymountedat,or
nextto,bearings.At themachineoperatingspeed,theselocationsmightbe
verycloseto nodalpointsat whichlateralvibrationsdo notoccur.The
anticipa-tionof theactualshaftvibrations,andlocationof
theanti-nodal,highampli-tudevibrationsectionsshouldbebasedonmodalconsiderations.Twomils(5011m)of
shaftvibrationat the
bearingmaytranslateinto50mils(127011m)atmid-spanof the rotor.On the
otherhand,thesetwomilsatthebearingmaybe
accompaniedbythemisalignment-relatedpreloadandhighreversalstressin
theshaft.The axiallocationof thepreloadforcein considerationof
thevibration mode shapessignificantlyaffectsthe shaft
stressdistribution.Additionalstressconcentrationfactorsalongthe
shaft,suchas press-fitted
TIME
SHAFT
ORBIT
SHAFTORBIT
Figure4Orbit, consistingof IX and a
significant2Xcomponent,resultsin shaftstressfrequencyIX.
Figure5Rotor/sealbackward"drywhip"orbitwith2Xfrequencyresultsin
3Xfrequencyreversalstressontheshaft.
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December 1989
partsor diameterdiscontinuities,mayjeopardizethe shaft
conditionevenmore.
Diagnosisusingvibrationmonitoringwarnsaboutshafthigh stress
The
casesdiscussedaboveindicate.thattheshaftstressconditionscanbeeasily
predicted if the machine isequippedwith a
vibrationmonitoringsystembasedon
shaftobservingdis-placementproximitytransducers.Theadvantageof
theuseof displacementtransducers,as opposedto
VelocitySeismoprobesand accelerometers,consistsof
theabilitytomeasureshaftstatic(zerofrequency)displacements,i.e,theshaftcenterlineposition,aroundwhichvibrationtakesplace.Changesinshaftstressconditionsdependnotablynot
onlyon thevibrationlevel,tradi-tionallyconsideredas the main
andoftenonlycauseof
machineproblems,butalsoontheshaftcenterlineposition.The latter
should be
continuouslyobservedandcorrelatedwiththeshaftvibrationlevelin
ordertoavoidprema-turestress-relateddamageofrotors.
ORBIT
ShaftcrackingShaftcrackinitiationmayhavevar-
iousorigins,suchascorrosion,materialirregularity,and/orinclusions,etc.Interms
of
mechanicalperformance,thesematerialdefectsactasstresscon-centratingfactors.Theshaftstressisthemajor
reasonthat
crackspropagate;thus,highstresspreventionbecomesanimportanttask to
maintainmachineintegrity.The earlydetectionof
shaftcrackingbyusingvibrationmonitoringequipmenthasbeendescribedin
sev-eralpublications[1-4].
Closing remarksNo easilyapplicableandwidely-used
directstressmeasuringinstrumentationisavailableasitexistsforvibrationmon-itoring.Shaft
stressduringmachineoperationhasbecomesomehowhiddenand forgotten;it
represents,l).owever,the major sourceof potential
cata-strophicfailuresof machines.Thepur-pose of this article is to
swaythevibration-orientedto a shaft stress-orientedphilosophy,by
showingthecorrelationbetweenshaftvibrationandstress.
7
References
1:BentlyNevadaPublicationsonShaftCrackDetection.
2.Bently,D.E.,
Muszynska,A.,DetectionofRotorCracks.Pro-ceedingsofTexasA&M
University15thTurbomachinerySymposiumandShortCourses.CorpusChristi,Texas,November10to13,1986,pp.129-139.
3.Bently,D.E., Muszynska,A.
EarlyDetectionofShaftCracksonFluid-HandlingMachines.ProceedingsofASME
InternationalSymposiumonFluidMachineryTroubleShoot-ing.1986WinterAnnualMeeting,Anaheim,California,December7to12,1986,.pp.53-58.
4.Bently,D.E.,Muszynska,A.,Thomson,A.
S.,VibrationMon-itoringTechniquesandShaftCrackDetectiononReactorCoolantPumpsandRecirculationPumps.EPRI
WorkshoponReactorCool-antPumpRecirculation.PumpMonitoringSymposium,Toronto,Ontario,Canada,March29to31,1988.
SHAFT
ORBIT
TIME 1 ROTATION TIME
Figure6IX elliptical orbit causes shaft pulsating stress
withfrequency2X.
Figure7IX circular orbit with
slightlydisplacedcentercausesshaftpulsatingstresswithfrequencyIX.
