Stress in Rotating Disks and Cylinders

8/13/2019 Stress in Rotating Disks and Cylinders

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ARMYR E S E A R C HL A B O R A T O R Y

Stressin RotatingDisksandCylinders

ThomasB.BahderARL-TR-2576 October2002

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Thefindingsin thisreportar eno tto be construedasanofficialDepartmento fth eArmypositionunlesssodesignatedby otherauthorized documents.Citationof manufacturer'sortrade namesdoesnotconstitute an officialendorsementorapprovalo fth eusethereof.Destroy thisreportwhenitisnolongerneeded.D onotreturnittoth eoriginator.


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ArmyResearchLaboratoryAdelphi,MD 20783-1197ARL-TR-2576 ctober2002

Stressin RotatingDisksandCylindersThomasB.Bahder

SensorsandElectronDevicesDirectorate

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Stress inRotatingDisksandCylindersThomasB.Bahder

U . .ArmyResearchLaboratory2800PowderMillRoad

Adelphi,Maryland,SA20783-1197(Augus t24 ,2001)

AbstractT hesolutionoftheclassicproblem of stressin a rotatingelasticdiskorcylin-der,assolvedinstandardtextson elasticitytheory,hastw ofeatures:dynam-icalequations are usedthatare valid only in an inertialf rameofreference,andquadratict e rmsare d r op p e din displacementgradientinth edefinitionofthestrain. showthat, in aninertialframeofreferencewhereth edynamicalequations are valid,it is incorrect to dropth equadrat ict e rmsbecausetheyare aslargeas thel ineart e rmsthatare kept. provideanalternateformula-tionoftheproblem by t ransformingthedynamicalequat ionsto acorotatingframeofreferenceofthedisk/cylinder, wheredropping th equadrat ic termsin displacementgradient sjustified.T heanalysisshowsthatth eclassictext-bookderivationof stressand strainm u s tbeinterpretedasbeingcarried ou tin thecorotatingf rameofthem ed i u m .


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CKGROUNDTheU.S.Armyis deve lop ingan electromagneticgu n (EMG) fo rbattlefieldapplications.

Duringhepastewyears ,onarecurringbasis,D r.ohnLyons former RLDirector)andD r..C .McCork leDirectoro fU.S.A r m yAviationandMissi leCommand)haverequested that I lookat s o m e ofthephysicsoftheE M G .narecentrequest, w as askedtoookatstressesnheotoroftheEM Gcompulsator.hesimplestphysicalm o d e lfo rarotors rotatingcylinder.herefore, pen t om eimeooking atthephysicsofstressesnelasticrotatingcylinders.hecaseof an lasticcylinders classicproblemthat is solved in manytextson linearelasticity1-6].owever ,w h e nthesederivationsareexaminedclosely,Ifoundcertainshortcomingsin thetreatments. nparticular,w h e nabody(cylinder)deformsduring increasingangularvelocity ofrotationaboutitssymmetryaxis,theundeformedstateat eroangularveloci ty) ndhedeformedstate at initeangularveloci ty)arerelatedby alargeangleofrotation.Wh e nlargeanglesofrotationcoexistwithdeformations, t is wellk n ow nhat thequadraticerms se eEq.20 ) )n thedefin i t ionofthestrainensorcannotbedropped,andhiseadsocomplicatednonlineardifferentialequations.owever ,heproblemofstressesnherotatingcyl inderca nbeanalyzednaframeo f reference coordinate y s t e m)hatsrotatingwithhecylinder.nhis pecialnon-inertialframe,thequadratictermso fthestraint ensorca n be dropped,andthes tressesfoundby solvinglineardifferentialequations.hephysicsdescribed bove isnot exposedin thestandardtreatmentsandthisreport is thesubjectofthisexplanation.

II .INTRODUCTIONTheproblemofs tresses in rotatingdisksandcylinders isimportantin practicalappl ica-

tionstorotatingmachinery, such asturbinesandgenerators, and w h er ev e r largerotationalspeedsareused.he textbookproblemofstressesn lasticrotatingdisksandcylinders,using theassumptionofplanestrainorplanestress,spublishednclassic texts, u ch s


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Love1] ,LandauandLifshitz2],Nadai3] ,Sechler4],TimoshenkoandGoodier5],andVolterraandGaines [6].Thestandardapproachpresented in thesetextshas tw ocharacter-isticfeatures:

1.ewton's econdaw ofmotion isappliednan inertial f rameof referenceoderivedynamicalequationsfo rthecontinuum seeEq .1)below),and

2.uadratictermsndisplacementgradientaredropped in thedefin i t ionofthestraintensor (seeEq.(20)below) .

Inthispaper, Ishow that, fo rarotatingelasticbody, the secondfeatureofthesolutionisinconsistentwiththefirst:droppingthequadratict e r msin thedi sp lacemen tgradient is anunjustifiedapproximation in n inertialframeofreference.nwhat follows, refertothemethodthat is e mp l oy e d in Ref.1-6]as 'thestandardmethod',andfo rbrevity, Iwil lrefertoacylinderasageneralizationofbothadiskandacyl inder .

Theclassicproblemofs tress in an elasticrotatingcylinder iscomplexbecause theun-deformedreference stateofthebodyisthenon-rotatingstate.Thedeformed stateis one ofsteady-staterotation.Theanalysis oftheproblemm u s tconnectthenon-rotatingreferencestateoherotatingstressed/strainedstate.hese tw ostatesare typicallyconnectedbylargeanglesofrotation.henarge ngleso frotation represent,hequadraticermsin thedisplacementgradientcannotbedropped in an inertial f r a meofreference)nthedefinitionofthestrain7-9].Theproblemofstressanalysis w h e n la rge-anglerotationsarepresent is wel lk n o w nandhasbeen discussed by an u mb e rofauthors in genera lcontexts,see forexample7-9].owever ,arge-anglerotations inheproblemofarotating elasticcylinderhavenotbeendealtwithin atechnicallycorrectmanner,becausequadraticstraingradienttermsareincorrectlydropped in the 'standardmethod'1-6].

Inthisw or k ,Iformulatetheelasticproblemo farotatingcyl inderin af r a meofreferencethatscorotatingwith thematerial.n thiscorotating f rame ,hequadratictermsn thedisplacementgradientcanbedropped,andtheresultingdifferentialequations arelinearandcanbesolved .


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Insection I, ev iewhestandardmethod'of solutionusednRef.1-6]and howthat fo rarotatingcylinderthedisplacementgradient in aninertialf rame ofreference is o forderuni ty ,andhereforequadraticterms in strainensordefin i t ion)cannotbe droppedwhencomparedtothelinearterms.SectionIIIconta insthebulk oftheanalys is . describethecorotatingsystemsofcoordinatesandheransformationofthevelocity ieldohecorotatingframe. u se thevelocitytransformationrulestotransformthedynamicalEq.1)from theinertialframetothecorotating frame(seeEq . (61) or(62)) ,whereextratermsarisek n o w nasthecentrifugalaccelerationandthecoriolisacceleration.nsectionIV ,Iwritetheexplicitcomponentequations fo rstress incylindricalcoordinates)ortherotatingelasticcylinderntscorotating f r ame.odisplay theesultingsolutionconcretely, derive thewel l -knownormula fo rhestressnherotatingcyl inderorhecaseofplane t ress ,scomputed in thecorotatingf rame.tresss anobjectivetensor,.e., tress isindependentofobservermotion10,11], so thephysicalmeaningofstress in thecorotatingf rame isthesameas in theinertialframe.There fore ,thestress fieldcomponentsin thecorotatingf r a meareequaltothestress field components in theinertial f r a me ,seeEq.36) .

III.STANDARDSOLUTIONMETHODInthe standardmethod' [1-6],thestressanalysisofelasticrotatingcylindersstartswith

thedynamicalequations,which , in generalizedcurvilinearcoordinatesare givenby [2,10,11]*% +/>/*=pa* 1)

w h e r e akjarethecontravariantcomponentsofthes tresstensor,fkisthevectorbodyforce,andakistheaccelerationvector.nEq. 1 ) ,repeatedindicesares u m m e dand thesemicolonindicatescovariantdifferentiationwithrespecttothecoordinates.Expressed in termsofthevelocityf ieldinspatialcoordinates,theacceleration isgiven by10,11]

k dyka=Tvvi 2)


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wherevashevelocity ield,ndhe emicolonndicatescovariantdifferentiationwithrespectohecoordinates,andvjvk.jscalled theconvectiveerm.nEq.1 ) ,he s tressakj,ccelerationak,ndbodyorcefk,regenerallyimedependent.quation1)sder ivedby applyingNewton'ssecondla w ofmotionto ane l e me n tofthem e d i u m .Newton'ssecondaw is validonly in an inertial f r a meofreference,andconsequently thevalidityo fEq.1)slimitedtoinertialf rames ofreference.

In the 'standardmethod'ofsolut ion,Eq.1)sapplied by invoking an effectivebodyforce",ofmagnitudeequalohecentr i fugalorce in therotating f rame.nhenertialf r a me ,there is actuallyno effectiveforce (suchasCoriolisorcetrifugalforce).orthecaseofabodyrotatingaboutitsprincipleaxis,a m o r ecarefuldeterminationofthetermsfk-akin Eq.1)comes fromsettingthebodyforcetozero (o rsettingequaltosome appliedforce)andcomputingthematerialaccelerationakfo ragivenbodymotion.orarigidbody,orauniformdensityelasticcyl inderthat is rotatingaboutitsaxisofsymmetryataconstantangularvelocity cj0,heCartesianvelocity fieldcomponen ts re :1=-w0yjv2=w0x,andv3= ,w h e r esuperscripts ,2,3ndicatecomponen tsonheCartesianbasisvectorsassociatedwiththex,y,z-axes in theinertialf r ame) .orrespondingtothisvelocity field,thecylindricalcomponentso ftheacceleration fieldar egiven by

f c= + =(- > > )3 ) w h e r e havechosenhe -axis shesymmetry xisandhebaroverhecomponentsindicatesthattheyare intheinertialf r a meo freference in cylindricalcoordinates.orthecasew h e r e therearenobody forces,withtheacceleration in Eq.3 ) ,Eq.1)n cylindricalcoordinates leadstothethreeequations

+ + +-ri -pruit4)O

r13 -23 -33 a,1 + O+ O+ = 6

w h er ethesuperscripts1,2,3enumeratetensorcomponen tsonther,< j ,zcoordinatebasisvec-


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tors respectively ,n cylindricalcoordinatesandthec o m m a sindicatepartialdifferentiationwithrespecttothesecoordinates.

Forsteadyrotationata uniformangularvelocityw0,andassumingtheabsenceo felasticwa ves ,thereisrotationalsymmetryaboutthez-axissothes tresscomponentsdonotdependonazimuthalangle j > .here fore , ll deriva t ives withespect to f are zero,eadingtotheequations:

+ +-r -prul7 )12 , ,+233+V=0 8 )131+333+= 0 9)

Iintroducephysicalcomponentsofs tress ,arr,a^,ozz,a ,a TZ ,anda**,withunitsofforceperunitareaandwhicharerelatedto the tensorcomponents11,22,33,12,13,and23,by10,11]

=rr_.11 (10)^ H)

zz=33 12)T*=rn 13)rz=13 14 )+z=r23 15)

ExpressingEq.7)-(9)n termsofthephysicalcomponen ts , Iobtainthewel l -knownequa-tionsvalid in aninertialframeofreference [1-6],

drr drz rT 2 .R,or oz r

9(V)+1*+3,0 1 7 )or\r ) r oz rlddrz dzz rz n 1fix --+-5+=0 18)or oz r


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Note thatEqs.16)-(18)havebeenderivedusingNewtons 's second law, nd oheyarevalidonly in an inertialf rameofreference.n particular,Eqs.16)-(18) ar eno tvalid in arotatingf rameofreference.

When rotat ingdiskorcylinder is analyzed,he ssumptionofplanestressorplanestrainisoftenmade.nbothcases,stresses mu stberelatedtostrainsbyconst i tut iveequa-t ions.Fo rthe simplestcaseofa homogeneous, isotropic,perfectly elasticbody, th e consti-tut ive equat ions incurvilinearcoordinates in an inertialframeca nbe writ tenas [2,10,11]

Tik\ Ak r,.ika lK=Xeglk+2elk 19)where nd reheL a m emater ia lconstants, ik rehecontravariant t rainensorcomponents , = e aaisthecontractionofthe strain tensor, ndgik re thecontravariantmetrictensorcomponents.

T he Eulerian strain tensoreikisrelated tothedisplacementfieldu * by [10,11]eik=2 (i;*+uk- i+um;ium; J f c) 20 )

Inthe standardm ethod 'ofsolution [1-6], th e quadratictermsum . i U m .kar edropped,whichleadsoinearequations thatca nbe olved fo r example,by using theAiry tressunc-t ion [12]).

However ,droppingthequadrat ic t e rm s in Eq.20 )snotjustified fo rarotatingbodybecause thesedimensionless)e r msu m;i reoforderunity.oprove thisassertion,t issufficient to consider th e limitingcaseofa rigidbody in steady-state rotationatconstantangularspeed o 0 .Thedeformationm app ingfunctiongivesthecoordinates zk here takentobeCartesian)ofaparticleat t imetin t e rmsofth eparticle's coordinatesZkn somereferencestate (configuration)at t ime t=t0:

zk=zk(Zm,t 21 )so thatzk(Zm,t)=Zk.T hedeformationmappingfunction has an inverse,whichIquoteherefo rlaterreference


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Zm=Zm{z*,t) (22)BothzkandZmreferto th esameCartesiancoordinatesystem.T hecoordinatesof aparticleinitiallyatZkatt=t0=0 rotat ingaboutthez-axisar egivenbythedeformationma ppingfunction

zk=RkmZm (23)wheretheorthogonalmatrixRkmisgivenby

*=cosw 0i sinw 0t

sinw 0 osw 0t01 (24)T hedisplacementvectorfield fo rthisdeformationmappingfunctionisgivenby 1 3 ]u=umlm= (zm-Zm)l=( 5 J T -K\)z*l (25)

whereImareth eunitCartesianbasisvectorsandRmkisthe t ransposematr ixthatsatisfiesRmkRk,= ; m (26)

whereS f1=+1 ifm= and 0 if mITherefore, fromEq .25), itisclearthatgradientsofdisplacementum]k ppearing in Eq20 ) reoforderunity nd therefore th equadrat ict e rm sum .^ um .kcannotbedroppedbecausetheyarenotsmall.Morespecifically,th e droppedt e rm s in Eq .20 )vary int imebetween-2 and 0 (inCartesiancomponents):

um;iu ;k~ l COSLJ0

01+coso;0t0(27)T heabovecalculation w as done fo ra rigidbody,bu tclearly, similarerror is int roducedfo relasticbodies.Therefore, in general,fo rarotating elasticbody, th equadrat ict e rms indisplacementgradients in Eq.20 )cannotbe dropped 1 4 ] .8


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Inylindricalomponents,heelatione tweenhehysicalomp on e n t sseeRef.10,11])fstrain,rrnde ,ndphysicalcomponen tsof thedisplacement ield,(u r,u ,ttz), is given by

dur 1crr dr 2 ' h ( h m (28)H*= ~ (t*r)a+[u4Y] 29 )

(Iuseabarovererrande toindicatethatthesear ecylindricalcomponents,and indicesrrandj x f ) asdistictfrom 1and22)o indicatethat thesearephysicalcomponen ts andnottensorcomponents.eetheAppendixand TableIandIIfo rnotationconventions.)nEq.28 )and 29) , h a v eassumedhat there isnodependenceonj andz, o Ihavesetderiva t ives withrespecttothesevariablesto zero.

Inhestandardethod'f olvingorhe tressnarotatingcyl inder1-6],hequadraticterms in Eq.28 )and29 ) are incorrectlydropped.

Thestraightorwardapproachocorrectlystudyingthe s tressesn rotatingdiskorcylinder,i nvolveskeepingthequadratictermsin di sp lacemen tgradientinEq. (20) .However,thisapproachd oe snotappearpromisingbecauseit leads to insolublenonlineardifferentialequations.nhenextsection,pproachheproblembyusingaransformationoacorotatingframeof reference,nwhichdropping thequadratic termsca nbe justifiedormoderateangularvelocityofrotationw0.

IV.TRANSFORMATIONTOTHEROTATINGFRAMEA sdiscussednhentroduction,heproblemofan lasticrotatingcylinders om -

plicatedbecauseheunstressed/unstrained eference state is thenon-rotatingstate,whilethestressed strained)stateis rotating,andthesetw ostatesaretypicallyrelatedby ala rge(time-dependent)angle.Theanalysisoftherotatingdiskorcyl indermustrelatethes tressesin therotating statetothereferenceconfigura t ion,whichItaketobe thenon-rotatingstate.Idefineatransformationfrom aninertialf r ameofreferencetothecorotatingframeofrefer-


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enceofthecylinder.histransformationprov idesarelationbe tween therotatings tressedstateandthenon-rotatingreferenceconfiguration.

A .CoordinateSystemsStarting romannertial rameof reference,,definedbyheCartesiancoordinates

zk=x,y,z), m a k ea transformationoarotatingframeof reference5" .herotatingframewillbecorotatingwithhecylinder sohatnhis frameS'heazimuthalvelocityfieldwillbe e roat ll times.he transformation from nnertial systemofcoordinatestoarotatingsystemofcoordinates ismost s implydoneusingCartesiancoordinates.ntheotherhand,theassumedcylindricalsymmetryoftheproblembegs fo ruseofcylindricalcoordinates.ence Iwillm a k e u seoffoursystemsofcoordinates.n theinertialframeS,Ihavew osystemsofcoordinates:Cartesian sys temo fcoordinateszk Z k=(x,y,z),andacylindricalsystemofcoordinatesxl= r,f > ,z) .nthecorotating f r ameofreference,", haveaCartesiansystemofcoordinatesz'k= x 1,y', z')andacylindricalcoordinatesystemx'1= r',(/)',z').Thesecoordinatesaresummarized in Table IandtheAppendix.also introducenotationfor tensorcomponentsneacho fthe fou rcoordinatesystems, eeTable I heCartesiancomponentsofthe t ress tensornhenertial f r ameSwillbedenotedby alk.nhesameinertialframeS,hecylindricalcomponentsofstresswillbelk.TheCartesiancomponentsofstress in thecorotatingframeS'willhaveaprime,a 'xk .Inhissamecorotatingframe,",hecylindricalcomponentsofstresswillbe denotedbyusing atilde,dxk.

Promthevantagepointofan inertialframeofreference,S,withCartesiancoordinateszh,consideracylinderw h o s esymmetryaxisis aligned andcolocatedwiththecoordinatez-axis.Attimet=-co,takethecylindertobenon-rotating.N ow a s s u methatin thedistantpast,aroundthetimet~-T,thecylinderbeginsaslow angularaccelerationlastingalongtime,onheorderof1/e,w h e r e1/eT.n xampleofsuchanangularaccelerationfunction is

10


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w(t)= iw0[l+ taah(c(t+ r))] (30)whereIassumethat r1/eTandristhelonges ttimeconstantin theproblem.Thisinequalitystates thattheaccelerationoccurs slowly,r1/e, slowerthananytimescalein theproblem,andthatthisaccelerationoccurs inthedistantpast,1/eT, so thatatt=0,1havea steady-statesituationofacylinderrotatingatconstantangularspeedw0.Byslowlyacceleratingthecylinder,Iavoid introducingm o d e sofvibration.shecylinder'sangularvelocityincreases from t=-oo, eachparticle compr i s ingthecylinderm o v e salongaspiral trajectorywith increasingadius).rom thepointofview ofthenertialframeS,he s tressesonagiven l emen toftheme d i u m particle) re such thatheycauseheparticletoexperienceanacceleration,mov i n galongthespiralpath.A tt=0, thecylinderhasachieved it smaximumangularvelocityu0.D u e to theassumptionofaperfectlyelasticmedium,att=0 thevelocity fieldhaszeroradialcomponen t ; ll particlesofthecylinderaremov i n gazimuthally(inaplaneperpendiculartothez-axiswithzeroradialcomponent).Thevelocity fieldis thatofarigidbodyandtheacceleration field is givenby Eq. 3 ) .

N ow ntroducehecorotatingframeofreference,5",withtheCartesiancoordinates4,w h o s eangularvelocityofrotations qualohatofthecylinderat llimes.hecoordinateszk(=zk)andzk(= )arerelatedby

z-=Aik(t)zk (31)w h e r ethetimedependentmatrixAik(t)is givenby

cos(0-0o) sin(0-0o)Aik(t)= -sin(0-0o)os(0-0o)0where 0 is afunctionoftimegivenby theintegralofu(t):n (t)= [e(t+T)+ lo g cosh(et+eT )+lo g 2 ]Z6

and0(0)=0O .

(32)

(33)

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Byconstruct ion, inth ecorotat ingframeS 'th eparticlescomprisingthematerialarenotrotat ing aboutthez'-axis;thereiszeroazimutha l componentofthe velocityfieldatallt imes.As th e angularvelocity u(t)ncreases from t=-co,eachparticlecomprising th ecylinderexperiences nncreasingeffectivecentrifugal forcethatdisplacesheparticle to largerradius.n this rotat ingf rame5", there will in general) lsobe aCoriolis force.owever ,in S ',fo rm odera t eangularspeedw 0,thestrain eikwillbe small ,and thegradientsofthedisplacement fieldwillalsobe small.Consequently,droppingthequadrat ict e rmsum]j U m .kin Eq .20 )willprovide agood approximationtoe^.

Notethatth et ransformationthatrelatescylindricalcomponents ininertialf rameSandrotat ingf rameS'isgivenbytheidenti tymatr ixdx{_x ^ _z^ _dz^__{dx'k ~dzadz'm x'k kFur thermore ,heim edependentransformat ionbetweennertialcylindricalcoordinatesx{=(r ,< f > ,z)in Sandcorotat ingcylindricalcoordinatesx'{=(r',< f > ,z' )in S , isgivenby

r'=T

z'=zwhere9(t)sgivenby Eq.33).he relat ionbetweencylindricalstresscomponents tknth e inertialf rameSandcylindricalstresscomponentslkinth e corotat ingf rameS'is

Ofcoursethet ransformationin Eq .36) m u s tbeusedsothatth e componentsarereferringto th es a mephysicalpoint in spacehaving coordinatesxn= r,


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where aparticleatt im et=t0wth ereferenceconfigurationhas(curvilinear)coordinatesXk,and in thedeformed stateatt imetth eparticle has coordinates * inthesamecurvilinearcoordinate ystem) .heEulerian t rainensoreij(X,x) epends'onw opoints:nthe eferenceconfiguration ndxnhedeformed tate.onsequently,under generalcoordinateransformat ion to moving f rame,xl ) xn=i{xk,t,heEulerian trainis two-point tensor ,which t ransforms s second rank tensorunder t ransformationofdeformed coordinates

si-x'i=h , * 0 Jr|=


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Eq.21),whereheparticle in th e referenceconfiguration t t ime t= 0 ascoordinatesZm.T hecoordinates Zmlabel theparticle in th eLagrangeandesription.singthe t rans-formation to therotatingf ramein Eq.31), th emotionoftheparticlewith labelZmwithrespect to therotatingS'f rameCartesian coordinates isgivenby

z'k(Zm,t=Akj(t)zi(Zm,t 41)Indiscussingth et ransformat iontoth erotating system ofcoordinates,Imus tdistinguish

betweenhevelocityofagivenparticle in th emedium th eLagrangean picture) nd th evelocityfield (theEulerianpicture) .TheLagrangeanvelocityv ^ S ;Zm ;t)ofagivenparticle(whosecoordinates in thereferenceconfigurationareZm)withrespect toth einertialf rameS isdefinedasthepartialt imederivativeof thatparticle's -coordinates,whenholdingZmconstant

Vi(S;Z-i)=M|V) 42)T heEulerianvelocityfield,Vi(zk,t),withrespectto th ef rameS isafunctionofcoordinateszkand t ime tand is relatedto th e Lagrangean (particle) velocityby

Vi{S-Zm -t=ViOSS'V;*),*)=Vi{z\t)43)where IusedEq .22)oexpress th e particlecoordinate Zmin t e rmsofit spositionzk tt imet.

Ica ndescribehe am eparticle'svelocitywhosecoordinatesnhe inertial rame)referenceconfiguration reZm)withespectohe rotat ing frameof reference5'. Thevelocityofthisparticle withrespectto th erotat ingframe 5" isdz'(Zm) dv Zt)= iKm=-A ik(t)zk(Z ,t}=A ik(t)zk(Zm,t+Aik(t)vk(S-Zm;t)

(44)whereIusedEq .31) to express theparticle'scoordinates in t e rmsofcoordinatesin th eSframeandthe do tonAikindicates differentiationwith respect to t ime.

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T hecomponents Vi(S;Zm ;t ndv-(5';Zm;i)epresentphysically distinctvectorsge-ometricobjects).ach ofthesevectorsca nbe expressedon th e otherbasis.nparticular,accordingto th e standard t ransformat ionrulesfo rvectorcomponents, Ihave

v'(S,;Zm]t=Aimvm(S'-Zm;t 45)v,m(S;Zm-t=Amivi{S;Zm;t 46)

Equations (45)and (46)areth es tandardtensort ransformationrulesfo rvectorcomponentsunderhecoordinateransformat iongivennEq .31).n u mmary , mu stdistinguishbetweenfour (Cartesiancomponent) velocities 1 8 ] :

Vj(5;Zm\t=Zi(Zm,t=componentsonz-axesofparticlevelocity withrespecttoSVi(S'\Zm\t)=componentsonz-axesofpa rticlevelocitywithrespect to S

Vi(S";Zm;t)=a[Zm,i=components onz'-axesofparticlevelocity withrespect, to 5"v'm{S ;Zm ;t)=componentsonz'-axesofparticlevelocitywith respecttoS

Thesefourvelocitiesarerelated.MultiplyingEq . (44)byA im(t),summingoverindexiandusingtheorthogonalmatrixpropert ies

A in(t)Aim(t)=5 nm 47)Ani{t)Ami{t)=8 nm 48)

leads to 1 8 ]vn(S";Zm ;t)=vn(S;ZT-1)+unk(S',S )zk(Zm, t)49)

wherewnk(S',S)istheCartes ianangularvelocitytensorofframeS 'withrespect to frameS :

1 5


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Unk(S',S)=Ain{t)ik{ t) 50)TheensorwnJt(S ",5)describesheime-dependentrotationofframeS'wi thespectoframeS.imilarly,usingEq .21 ) ,substitutingtheinverserelation

zi(Zm,t)=Ani(t)z (Zm,t 51)carryingoutheimedifferentiation,multiplyingbyAmi, ummingover ndu seoftheorthogonalityrelations in Eq.4 8 )eadsto

v'n(S; Zm; t)=Vn(S';Zm; t)+u'kn(S',S)z'(Zm, t)52)w h e r eheangularvelocitytensorcomponents re expressedwithrespect to the5' r a meCartesianbasis:

u'nmi iS)Amini =AniAmkL ik(S',S)53)Equation 4 9 )and52) rehewel l -known ulesorransformatingpart iclevelocityoarotatingframeofreference [18].

Next, deriveheequationhatrelatestheCartesiancomponentsofthevelocity fieldin S,Vi(zk,t,tothevelocityfield in5",v'(z'k,t.Equation (43) relatesthevelocityfield intheSframetotheLagrangeanparticle)velocity.imilarly,theveloci tyfieldwithrespecttothe5"frame is givenbyvKz'^t)=vJ(S*i2it)=9z-(Zm,t)=y{sl{z* .t=Viis>-Z (Ankz\t);t) (54)

w h e r eIused the inverserelationzk=Ankz'n.singEq .4 3 )and (54) in theleftand rightmostterms in Eq. 4 4 ) , Iobtainarelationbetween th eveloci tyfields in f rames Sand5"

uKA* )=Aik{ t)zk+Aik(t)vk{zn,t) 55)InEq.55) ,hecoordinatesz'kandznarerelatedbyEq.3 1 ) . MultiplyingEq.55)byAim,summingover index, i,using the inversetransformation in Eq.51)and

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o jjk(S')S)=AmjAnkum n(S',S) 56 )leadsto

vj i zm,t)=Mt)v[{z t)-Akj(t)u'kn(3',S )z' n57)Eq . (57) isthedesiredrulefo rt ransformationofthe Eulerianvelocityfieldfromth einertialf rameSto the rotat ingf rame 5".Notethat th e rightsideofEq .57 )dependsonlyon5'f ramecoordinatesz' nandtheleftsidedependsonSframecoordinateszm-Furthermore,th evelocity fieldcomponentson th e leftsideofEq.57) re takenon th eCartesian)nertialf rameSbasisvectors,andonth erightsideallcomponentsar eexpressedonthe(Cartesian)rotat ingf rameS'basisvectors.

C .Dynamica lEquationintheRotatingFrameInw h a tfollows,It ransformthemo men t um balanceEq.1) to th ecorotatingsystem of

coordinatesS'.Fo rsimplicity,Idothist ransformat ionusingCartesiancoordinatesfo rboththe inert ialf rameSandcorotatingf rameS".use th et ransformationofthe velocityfieldgiven in Eq .57 ) to computeth e termsthatappearin Eq .1).Takingth e gradientofthevelocity in Eq.57 )

^ ^=4* 4* P 4* U^.S )'5 8 ) ozknwhereIusedthechainrulefo rdifferentiation =Amfcsfrsincezkandz'mare relatedbyEq .41 ) .ext, Icompute th e t imederivat iveofthevelocitythatoccurs in Eq .1)and Iexpresstherightsidein t e rmsofS 'f rame componentsand coordinates:

(59)whereIhaveomit tedthef ramelabelsof theangularvelocityandthecoordinateargumentsinth evelocity.T hedivergenceofth estresst ransformsasavectorunderthet ransformationto th erotatingf rame

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dzkz'SubstitutingEqs.57)-(60) intoth e inertial-framemomentu m balanceEq .1) nd sim-

plifying,eadsohedynamicalequation fo r th eCartesian) tressensorn th eotatingf rameS'lda'ik dv[ v[ ( du'i n\, . ,.

wherea ' ikare th e stress components in Cartesian coordinates 4,v' nisth eEuler ianvelocityfieldthatdependsonz' kandt,andth e (Cartesian)componentsof th eangular velocitytensorin5" regivenbyEq.53).nEq .61), ll repeated ubscripts re su mmed.otehatal lvelocities thatappearnEq.61 )efer tohecorotating frameS' nd that llensorcomponentsare takenon th e S'f rameCartesianbasisvectors.he first tw o t e rm s in Eq .(61) re th eacceleration includingtheconvective term) s seenn th e corotat ingsystemofcoordinates.Thethird term u'im u'm nz^ is th ecentrifugalacceleration.Thefourth term,C j' in z'nis th e angularacceleration.T he lastt e rm,2u' i nv'nistheCoriolisacceleration.

T he dynamica lEq.61 ) isatensorequation; th equantities a'kn,w'm n,andv' n ,areCarte-sianensors.nderorthogonalransformations ro moneCartesian ystemo nother ,Eq.61 )scovariant:thashe same form.hegroupofsy mmet ryoperat ionsm aybeextendedto t ransformat ionsbetweencurvilinearcoordinates bywritingEq .61 ) in aman-ifestlycovariantform as :

J **= +finSii n+2iB+m


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Thequant i t ies(n rehecontravariantcomponentsof th epositionvectorncurvilinearcoordinates x'm,whichare related to the Cartesianpositionvectorcomponents z'kby

r)r m< ~ = ?*z 63 )

Therotat ingf ramecurvilinear components ofstress,velocityandangularvelocity, ab ,va,and D a6,are related to theirrotat ingf rameCartesiancomponents,a'^,v'k,andw ( 5 - ,by:

b=^i* 64)/a

=5?* 65 > 5 = r fw 66)

where " ndx'a re theCartesianand curvilinearcoordinates in th e rotat ing f rame '.Notethat thepartialderivativewithrespectto coordinatesz'kinEq . (61) hasbeenreplacedbyacovariantderivativewith especto thecurvilinearcoordinates x'knEq.62).eeTables Iand II fo r s umma r y ofth enotat ion.dentificationofth e meaningofth evarioustermsinEq. (62),suchasth eCoriolisacceleration an dcentrifugalaccelerationisclearfromcomparisonwithEq.61).V .ROTATINGCYLINDEREQUATIONSINCOROTATINGCOORDINATES

In order to obtainthe explicitequat ions fo rarotating cylinder from Eq.62), needtocomputen,D mn, nd"sdescribedpreviously, ake th e referenceconfigurationofthecylinder to be the s ta t ionarycylinderat t=-co in th einertialf rameS .assumethat the cylinderexperiences slow angularaccelerationsuchas givenby Eq.30))hatlastsapproximatelyt imeAt= 1/e,and haspeakmagnitude at t= T,with 1/e


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inert ialframe ,th econtravariantcomponentsofth evelocityfieldincylindricalcoordinatesare

t Ji= (S1,w(t),e8) 67 )where v1andv3are theradialand z-components ofvelocityand wheretheazimutha lcom-ponen tu)(t)isgivenbyEq. (30). obtainthevelocitycomponentsincylindricalcoordinatesin thecorotating f rameS" sollows:irst,ransform v o inert ial frameCartesian com-ponentsvlusingth e standardvector t ransformation rulebetweenCartesian and cylindricalcoordinates.ext,use Eq.57 ) to t ransform th e inertialf rameCartesiancomponets v{totherotatingS frameCar tes ianvelocityfieldv'\Finally,use thestandard tensort ransfor-mat ion rules,betweenCartesian ndcylindricalcoordinatesboth in th e rotat ing f rame) ,to t ransform th evelocityfieldfromCar tes iancomponentsvato cylindrical(rotat ingframe)componentsv'\where Im a d euse ofth e angularvelocitycomponents

u )ik{S',S)=AliAk=u t)o-10 +10 0

y 0 Oy u'ik {S',S )=AimAknumn{S',S )' (68)Followingthisprocedure, Iobtainthevelocity componentsin cylindricalcoordinatesinth ecorotatingframe5"

vi={v\Q,v3) 69 )whereth e azimutha lco mpo nen tofvelocity iszerofo rallt ime,by constructionofthecoro-tatingf rameS ,as expected.

In th e corotatingf rameS',there isparticle motionaround th e t ime t-T.However,ati=0 th e particleshave reached theirnewdeformed)steady-state positions andmotionhasceased; th evelocityfield isgivenby

*= 0,0,0) 7 0)whichisth evelocityfieldofarigidbody.Sinceth evelocityfieldiszeroatt=0, th eCoriolisacceleration term in Eq.62 )doesnotcontributein steady-staterotat ion.

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UsingEq.(66),he ngularvelocity tensorncylindricalcoordinates in th e corotatingf rameS 'is

w \

V (71)

0 u)(t)/r'-r'u)(t) 0 0

00 A tt= ,he t ime derivative ofth e ngularvelocity tensor iszero. Using this fact, ndEq .70 )and71),thestressEqs.62 )nthecorotatingframear egivenby

-12 -23 -12_na i+o QH-o u C T 13^-13 -33 u _na i+a 3 Hu (72)(73)(74)

Equat ions(72)-(74) areth eequationssatisfiedbyth estresstensorincylindricalcomponentsatt=0,ins teady-s ta terotationin thecorotating frameS '. n termsofphysicalcomponents(seeEq. 10)- (15) ) ,Eq .72)-(74)become:

3dTr darz dTT-a * dz''darz dzz ~ TZ =-pr'u% dr dr ' dz' +=0 (75)(76)( 7 7 )NotethatEq .75)-(77) inth ecorotat ingframe5 'haveth esameform as Eq.16)-(18)

in th einert ialf rameS .owever,he keypoint isthatthecorotating frameEq.75)-(77)haveadistinctadvantage:hequadraticstraingradientt e rmsinth edefinitionof th estrainin Eq .20 )ca nbe droppedbecause theyare small inthe corotatingf rameS '.Thesameisnot t rue in theinert ialframe S .

Toproceedwiththesolutioninth ecorotat ingframe,theconstitutive Eq.(19)(ininertialf rameS )m u s tbet r ansformedto th e corotat ingframeS 'by takingcylindricalcomponentsin th eSf rameand usingEq .34 )toobtainanexpressionofth esameform as in Eq .19 )but in th ecorotating f rameS '.nthis t ransformation,theLameconstantsare t reated as

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invariants .oheconstitutiverelations in thecorotatingframeS' re the s a me s in theinertialframeS.

Inhecorotating rameS',roppinghequadraticermsndisplacementgradients,whichrelatethedisplacementfieldtostrain, isjustifiedsince in therotatingframeS'theset e rms canbeconsideredsmallformoderateangularvelocity.Therefore,thesolution in thecorotatingframeS'canproceed in anana logousw aytothatofthe standardmethod',butIhavenotmadethe(incorrect)approximationofdroppingquadraticdisplacementgradientt e rms in theinertialframe.

Thestresstensorisobjective,sothestressintherotatingframehasthes a m emeaningasin the inertialframe,see Eq.36) .Theboundaryconditionsonthestresstensorcomponentsinthecorotatingframearethes a me asintheinertialf rame ,duetotheobjectivityofstresstensor .lternatively,neca nverifyhatheboundaryconditionsonhestressnherotatingframearethesameas in theinertialframe [20].

VI.PLANESTRESSSOLUTIONThesolutionoftheproblemofs tressnarotatingcylinder in thecorotatingframeofreferencenowollows.hesolutionnhecorotating rameparallelshesolutionnhe

'standardmethod'1-6],exceptthattheincorrectapproximationofdroppingthequadraticstraingradientterms in theinertialf r a meisavoided.

Iassumethat in theinertialf rame S,thecylinderis rotatingatangularvelocity u0andhasradiusb.Undertheassumptionofplane s tress5],where

cr = *z=orz=0 78)with boundaryconditionofzero s tressonthe long peripheralsurface:

Asmentioned bove ,stresss nobjectiveensor10,11], ohephysicalmeaningoftheboundaryconditionsnEq.78)and79 )nhecorotating frameS' re thesame she

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physicalmeaningoftheanalogousconditionsinth einertialframe(asused, fo rexample byTimoshenko [5]).Also,asshownabove,Im aytaketh e(transformed) linearelasticrelationsin th e corotatingf rameto beoft hesame form as nth einertialframe:

g r r=i (yr_vdM \ 80) Eg*=Io**-v3 8 1 ) EwhereYoung'smo d ulusEandPoisson'sratio var erelated to th eLameconstantsby

E=^(A+2/x) 82)A v= ( 8 3 )2(A +/ x )

wheretheLameconstants aret reated asinvariant scalarsinth et ransformation.T heelasticrelationsin Eq .80) nd81) re in thecorotatingf rameS '.heyca nbeobtained fromth elinearelasticrelat ionsintheinertialf rame5,Eq.19) ,byusingth et ransformationstothecorotatingf rame in Eq.36 ) nd40) , nd th ecoordinate t ransformation in Eq.35).In.doingth et ransformat ionofth eelasticrelations to th e corotatingframe, Iam assumingthatthetensorthatenters inth e inertialf rameelasticrelationsin Eq .19 ) istheEulerianstraintensorgivenin Eq.(20) andth equadrat ictermshavenotbeendropped.Asdiscussedearlier,droppingthequadrat ict e rmsinthedisplacementgradientsisdoneinthecorotat ingframe.)

Forsmalldisplacementsincorotatingf rame",th egradientsofthedisplacementvectorinf rameS 'ca nbe assumedsmall form odera t eangularvelocityw 0 ,sothequadratict e rmsin th egradientofthedisplacementca nbe neglected.herefore,n th ecorotat ingf rameS",Itaketherelat ionbetweenth e radialcomponentof th edisplacementvector,,andth ephysicalcomponents ofstrain,errand e , to be (comparewithEq .28 )and29))

*-= 84)eH=l 8 5 )

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where th e t ilde onindicatesthatthe radialcomponentofdisplacement field is taken inth e corotat ingframeS'and th e pr imeonr'indicatesthat th ecylindricalradialcoordinateis in th ecorotatingframeS",se ethet ransformat ionin Eq .35).

Substitut ingEq.84) nd85)ntoEq .80) nd81),eads to elat ionsbetweenhephysicalcomponents ofstressand the radialdisplacementfield

E fdu ua = 1 v2dr ' rE ( du

+-. 8 6)a* \ vlr orJ

Substitut ing theserelationsintoEq. (75)leadstoadifferentialequationfo rthedisplacementin therotatingframe5"/2< 9 2 ,d _ 1 v2 r~+r -=-pu2T,z88)dr'2 dr'T hegeneralsolutionis 5 ]. 1U=E (l_I)C1r-(l+i/)C2 -pJr3r(89)Substi tut ion ofthissolutionintoEq. 86)and87) Iobtain

rr=C+C2J 1+Zpur2 90)r d**=Cl2- pulr * 91)

T hestressesatr'= mus tremainfinite,soItakeC 2= 0 .Applying th eboundaryconditionon th e longperipheralsurface,Eq.79 ) leadsto C\-( 3 +v)pu2 b 2/and th e stresses

TT~ 2 2 -2P(2-r2) 92)=^[3+ -(l+3^ ) r '2] 93)

T hephysicalstresscomponentsinEq. (92)and (93)are inthecorotating f rameS '.However,dueto th e t ransformationbetweenthecorotatingframeandth einertialf rame in Eq .36),and th ecoordinatet ransformation inEq.35),thecorotatingframecomponents inEq.92)and93 ) re equalto th e inert ialf r am ecomponentsof stress.Usingth e expressions in th erotating rame, uch sEq .7 5 ) (77 ) , xpressionsorplane train ndotherboundaryconditionsca nbe derivedfo rrotatingcylinders,disksandannular rings,se eRef.1-6].

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VII.SUMMARYTheclassicproblemofs tressin rotatingdisksorcylindersisimportantinapplicationsto

turbines,generators,andwhenever largerotationalspeedsexist.hetextbookproblemofstress in perfectlyelasticdisksorcyl inders is solvedin standardtexts1-6].The 'standardmethod'ofsolutionbeginswithEq. (1 )anddropst e r msthatare quadraticin straingradien tin thedefinitionofthestrain,seeEq.20 ) .Equation(1 ) isvalidonly inaninertialframeofreference, inceit is derivedfrom Newton's second law ofmotion,which itselfis only validin aninertial referenceframe.

Inthisw o r k ,I haves h ow nthatdroppingthet e r msquadraticin thedisplacementgradient(in Eq.20 ) )sincorrectintheinertialframe in whichEq.1)sapplied in thestandardmethod'ofsolution[1-6]. provideanalternativeformula t ionoftherotatingelasticcylinderproblemin aframeof referencethat iscorotatingwi ththecylinder. nthiscorotating f rame,Iderivethedynamicalequation fo rthestressse eEq .61)or (62) )andIshow thattermsquadraticin thedisplacementgradientcanbedroppedbecausetheyaresma l l(formoderateangularspeedofrotation).hisanalysis in thecorotatingf rame showsthatthestandardmethod'ofsolution [1-6]shouldbeinterpretedasbeingcarried out in thecorotatingf r ameofreferenceofthecylinder.

Furthermore,w h e n stressesarecomputedin rotatingdisksorcylindersc omp os e dofm a-terialsthathave morecomplex constitutiveequations,such aselastic-plasticorviscoelast icbehavior,onemustcarefullyjustifydroppingthequadratictermsindisplacementgradients.Ifdroppingthese termscannotbejustified,hentheproblem canbe na lyzed inarotat-ingframe,using thederivedEq.61 )or62) .notherpracticalapplicationofthe s tressEq.62)nherotating frame.is tostudyelasticwavesnbodiesduringrotation,wherecorioliseffectsm ayplayarole.

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ACKNOWLEDGMENTS

TheauthorthanksDr..C.McCorkle ,U.S.A r m y AviationandMissi leCommand,fo rsuggestingthisproblemand providingn u me r ou sdiscuss ions .TheauthorthanksH o w a r dBrandt fo rdiscussionsandpointingoutRef.2].

APPENDIXA :CONVENTIONSIspecifytensorcomponentsoncoordinate(non-holonomic)basisvectorsusingnumerical

indices,1,2,3, such sa12.orphysicalcomponen ts , whichhavethedimens ionsassociatedwiththatquantity,Iuseletteredindices,suchasa^.naddition,Imustdistinguishbe tweenfourcoordinatesystems:artesianandcylindricalcoordinates in theinertialframeSandCartesianandcylindrical in thecorotatingframe".u sezk= x,y ,z)andxk= r,f > ,z)fo rCartesianandcylindricalcoordinatesnnertial rameS, espect ively.ncorotatingframe", u sez'k=x',y',z')ndxlk r',',z') orCartesian ndcylindricalcoordi-nates, espectively.ordistinguishingcomponents in theseou rcoordinatesys tems , u seanadditionalmark sollows:bsenceofmark nd bar, orCartesianandcylindricalcomponents in inertialframeS,respectively.o rcomponents in thecorotatingframeS", Iu se aprimeanda tilde,forCartesianandcylindricalcomponents, respectively.eeTableIand II .

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R F R N S[I]A ...ove ,.4 8 , reatisenheMathema t i ca lheo ryofElastic i ty ,over

Publications,4thEdition,N ewYork ,1 9 4 4 ) .[ 2 ]L.D .LandauandE.M .Lifshi tz ,p.22,heoryofElasticity,PergamonPress ,N ew

Y o r k1 9 70 ) .[ 3 ]A .Nadai,TheoryofFlowandFractureofSolids",p. 87,McGraw-Hil l ,N ew York

(1950) .[ 4 ]E.E.Sechler ,p. 64 ,ElasticitynEngineering, ohnWiley&Sons,nc.,N ewYork

(1952) .[ 5 ]S.P.Timoshenko.andJ.N .Goodie r ,p.81,TheoryofElasticity,3 rd Edition,M c G r a w -

HillBook Company,N ew Y o r k ,1970) .[ 6 ]E.VolterraandJ.H .a m e s ,p.56,AdvancedStrengthofMaterials ,Prentice-Hall,

Inc.,EnglewoodCliffs,N.J.,USA,1971) .[ 7 ] .K.Dienes ,ActaMechan ica ,Adiscuss ionofmaterialrotationands tressrate ,65,1-111 9 79 ) .[ 8 ]J.K.Dienes,ActaMechan ica ,A discussionofmaterialrotationands tressrate ,32,

217-232 1 9 8 6 ) .[ 9 ]P.M .NaghdiandL Vongsarnpigoon, Smal lStrainAccompan ied byModerateRota-

tion ,TechnicalReporUCB/AM-81-4,1981) .[ 1 0 ] A.C .Eringen,NonlinearTheo ryofCont inuousMedia,McGraw-Hi l lBookC o m p a n y ,

N ew Y o r k ,1962) .[II]M .N .L.Narasimhan,PrinciplesofContinuumMechanics ,JohnWiley&Sons,nc.,

N ewY or k1 9 9 2 ) .[ 1 2 ] Forexample, eeRef.6].

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[13]A . .M .Spencer,p .63 ,ontinuumMechanics ,LongmanMathematicalTexts ,L o n g -man,N ew Y or k 1 9 8 0 ) .

[14]Forheimitingcaseofarotating igidbody,hestrain ik= becauseheineardisplacementgradienttermsw i;Jt+uhiicancel th e quadratictermsumiUm.k,so thatthefullstraintensor ik 0.

[15]F.D .Murnaghan,A m .J.Math., FiniteDeforma t ionsofanElasticSolid",59 ,235-260(1937) .

[16]J.M .BGambi,A .SanMiguel ,andF.Vicente ,Gen.Rel .Grav., TheRelativeDefor -mationTensor fo rSmallDisplacementsinGenera lRelativity ,21,279-2861 9 8 9 ) .

[17]T.B.Bahder,Noteon DerivationofLagrangianandEulerianStrainTensors in FiniteDeformationTheory ,ArmyResearchLaboratoryTechnicalReport in press).

[18] .L.SyngeandA .child,ensorCalculus,DoverPublications,ncorporated,ewYork ,1 9 8 1 ) .

[19] .L.Synge,Relativity:heGeneralTheory ,(North-HollandPublishingCo. ,N ew York ,1960) .

[20]T.B.Bahder,Boundaryconditions in arotatingf r a meofreference",unpublished.

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T A B L ES

CartesianCylindrical

T A B L E .oordinatesInertialFrameS Rotat ingFrameSzk=zk

x

JkJk

CartesianCylindrical

TABLEII .TensorComponentsInertialFrameS

a


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R E P O R TDOCUMENTATIONPAGE Form Approved OM BN o.0704-0188 Publicreportingburden fo r thiscollectionofinformationis estimated to average hourpe rresponse,inc ludingth em e for reviewing structlons-.searching exlsng sourcesttherlnclandIraftttnlngth edata needed,an d completing an dreviewing th e collectionof Information.Sendcommentsregarding thisburdenesmateor an yotheraspeOid[this ffiSnrtKX?fnclu su5SSbnsfo rreducingthisburden,to WashingtonHeadquartersServices,Directorate* SS?mxS5^^DavisH ighwaySue1204 ,Arlington,A2202 -4302 ,an d to th eOfficeof Managementan dBudget ,Paperwork Reduc t ionProject 0704-018B) ,Washing ton ,DC 20503.

1. AG ENCY US E ONLYLeaveblank) 2.REPORT DATEOctober 2002

3 REPORT TYPE ND D TES OVEREDFinal,FY1998-FY2000

4. TITLE AN D SUBTITL EStress inRotat ingDisks andCylinders6. A U T H O R S )T h o m a sB. Bahder7. PERFORMING ORGANIZATION N A M E ( S )AN DADDRESS(ES) U.S.A r m yResearch LaboratoryAtta:A M S R L -SE-EE2 800P o w d e rMillRoad

Adelphi,MD07 83- 1197

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13 .ABSTRACTMaxi m um 20 0 words)Thesolutionofth eclassicproblemofst ressinarota t ingelasticdiskorcyl inder,assolvedin s tandardtextson elasticity theory,hastw ofeatures:dynamicalequat ionsare us edthatarevalidonlyin aninertialf rameofreference,andquadratic termsar edroppedindisplacement gradientin thedefinitionofth estrain.T heauthorshows tha t,in an inertialf ram eofreferencew here th edynamicalequat ionsarevalid,itisincorrecttodrop th equadratic termsbecausetheyare aslargeas th elinear terms thatar ekept .T heauthor providesanalternateformulat ionofth eproblem byt ransforming th edynamicalequat ionsto acorotating f r ameofreferenceofth edisk/cylinder,wheredroppingth equadrat ict e rm sind isp lacementgradientis justified.T heanalysisshowsthatth eclassic textbook derivationofstressandstrain m u s tb einterpretedas beingcarried ou tin thecorotatingf rameofth em e d i u m .

14 . SUBJ ECTTERMSElasticity,cylinder,disk,rotation,rotatingcoordinates ,coriolis,rotatingf r ame,strain,stress,dy namica lequat ions,deformation,nonl inearstrain1 7. SECURITY CLASSIFICATION 1 8. SECURITY CL ASSIFICATIONOF THIS PAG E 19 .SECURITY CLASSIFICATION OF ABSTRACT

18 .N U M B E ROF P A GE S3316 .PRICEC O D E 20 .LIMITATIONOF ABSTRACT

Stress in Rotating Disks and Cylinders

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