GRAVITATIONEFFECTSONCENTRIFUGALPENDULUMVIBRATIONABSORBERS LINEARANALYSIS By MingMu ATHESIS Submitted toMichiganStateUniversity inpartialoftherequirements forthedegreeof MechanicalEngineeringŒMasterofScience 2015 ABSTRACT GRAVITATIONEFFECTSONCENTRIFUGALPENDULUMVIBRATIONABSORBERS LINEARANALYSIS By MingMu Thisworkinvestigatestheeffectsofgravityonthedynamicresponseofcentrifugalpendulum vibrationabsorbers(CPVAs).Thisanalyticalstudyconsiderssmallamplitudemotionsoftheab- sorberssothatlinearvibrationtoolscanbeapplied.Themotivationofthestudyistodeterminethe behaviorofCPVAsatlowrotorspeeds,wheregravityeffectscanbecomparabletothoseofrota- tion.Themaingoalofthepresentstudyistopredictpatternsthatwereobservedintheresponseof systemswithseveralsymmetricallyplacedabsorbers[10],andtousemoresophisticatedanalysis toolsforsymmetricsystems,namelycirculantmatrices,toinvestigatethelinearizedversionofthe model.AmathematicalmodelisdevelopedusingLagrange'sequationsforadiskrotatingabout aedhorizontalaxisand N pointmassescyclicallyarrangedontherotorthatcanmovealong pathsrelativetotherotor.Gravityprovidesbothdirectandparametricexcitationtothesependu- lummassesatorderone,whereasthetorqueappliedtotherotorisatorder n .Theequationsare linearizedandnon-dimensionalizedforanalysis.Thenumberofdistinctgroupsofabsorberswith identicalbutphase-shiftedwaveformsisconsidered,anditisshownthatthisgroupingbehavior dependsontheengineorder n andtheratio N n .Modelswithandwithouttheeffectsofparametric excitationareconsidered,anditisshownthatparametricexcitationleadstoresonanteffectswhen n = 1and n = 2.Itisshownthattherotorisaffectedonlybytheorder n componentoftheabsorber responses,becauseofthesymmetriesoftheresponseatorderonefromgravity.Theseresultspro- videusefulinformationaboutabsorberbehaviorandcanbeusedtoassesspotentialproblemsthat mayarisefromgravitationaleffects. Copyrightby MINGMU 2015 Tomywonderfulfamilyandallmyfriends. iv ACKNOWLEDGEMENTS Firstofall,Iwouldliketoexpressmyappreciationstoallthepeoplewhohaveprovidedme invaluablesupportsduringmyworkonthisthesis,especiallymyadvisor,Dr.StevenW.Shaw,and Dr.BrianFeeny,forgivingmetheopportunitytoworkonthisproject.Iamverygratefulforthe continuoussupportandguidancefromDr.Shawtoallowmecontinuingmyeducationandwork overthelasttwoandahalfyears.Inaddition,IwouldliketothankBruceGeist. SpecialthankstoFiatChryslerAutomobiles,theNationalScienceFoundation,andMichigan StateUniversityfortheirsupportofmyresearch. Iamalsogratefultomylabmates,MustafaAliAcar,MichaelThelenandmanyothersforall theendlessdiscussionsandlabworkstogether. Lastbutnottheleast,Iwouldliketothankallmyfamilymembersfortheirsupportsthroughout writingmythesisandmylifeingeneral. v TABLEOFCONTENTS LISTOFTABLES ....................................... vii LISTOFFIGURES ....................................... viii CHAPTER1INTRODUCTION ............................... 1 1.1BackgroundandMotivation..............................1 1.2ThesisOutline.....................................3 CHAPTER2MATHEMATICALMODELING ....................... 5 2.1EquationsofMotion..................................5 2.2NondimensionalizationandLinearization......................7 2.3GroupingBehaviorAnalysis.............................11 CHAPTER3LINEARMODELWITHOUTPARAMETRICEXCITATION ....... 15 3.1Steady-stateDampedResponse............................15 3.2DiagonalizationandSteady-StateSolution......................17 3.3Steady-stateUndampedResponse..........................20 CHAPTER4ACCOUNTINGFORGRAVITATIONALPARAMETRICEXCITATION -PERTURBATIONANALYSIS ........................ 22 4.1Scaling.........................................22 4.2SingleAbsorberCase.................................22 4.3MultipleAbsorbersCase...............................25 4.4SlowFlowinCartesianCoordinates.........................27 4.5AnalysiswithDampingEffects............................31 4.6RotorBehaviorAnalysis...............................34 4.7ResultsandDiscussions................................35 CHAPTER5CONCLUSIONSANDFUTUREWORK ................... 46 5.1SummaryandConclusions..............................46 5.2RecommendationsforFutureWork..........................48 BIBLIOGRAPHY ........................................ 50 vi LISTOFTABLES Table2.1:SymbolsanddescriptionsfromFigure2.1.....................6 Table2.2:PathfunctionsfromDenman[2]...........................9 Table2.3:Pathvariablesandrequiredexpansions.......................9 Table2.4:Examplesof l , k and q withdifferentvaluesof N and n ...............13 Table2.5:Numberofgroupswithdifferentvaluesof N and n .................14 Table2.6:Summaryofgroupingwithdifferent N and n ...................14 vii LISTOFFIGURES Figure1.1:designofCPVA...............................2 Figure2.1:Modelingdiagram.................................5 Figure2.2:Layoutof i thand j thabsorber...........................12 Figure3.1:Steady-stateresponsepeakamplitudesofasystemof N CPVAswithincreas- ingtorqueamplitude G q ;for g = 0 : 05,withdamping...............20 Figure4.1:Steady-stateresponseamplitudesofasingle( N = 1)CPVAwithincreasing torque, G q ;for g = 0 : 05, n = 2, m a = 0 : 04,and s = 0...............36 Figure4.2:Steady-stateresponseamplitudesofasingle( N = 1)CPVAtime-traceplot atvarioustorquelevel,for g = 0 : 05, n = 2, m a = 0 : 04,and s = 0........37 Figure4.3:Steady-stateresponseamplitudesofasingle( N = 1)CPVAwithincreasing torque, G q ;for g = 0 : 05, n = 1 : 5, m a = 0 : 03,and s = 0..............38 Figure4.4:Damped, m a = 0 : 04,versusundampedsteady-stateresponseamplitudeof1 CPVAwithincreasing G q , g = 0 : 05, n = 2with s = 0..............39 Figure4.5:Damped, m a = 0 : 03,versusundampedsteady-stateresponseamplitudeof1 CPVAwithincreasing G q , g = 0 : 05, n = 1 : 5with s = 0.............40 Figure4.6:Steady-statetimetracesof4CPVAs, G q = 0 : 01, g = 0 : 05, n = 2,with m a = 0 : 04and s = 0...................................40 Figure4.7:Steady-stateresponseamplitudesof4CPVAswithincreasing G q , g = 0 : 05, n = 2,with m a = 0 : 04and s = 0..........................41 Figure4.8:Steady-statetimetracesof4CPVAs, G q = 0 : 01, g = 0 : 05, n = 1 : 5,with m a = 0 : 03and s = 0................................41 Figure4.9:Steady-stateresponseamplitudesof4CPVAswithincreasing G q , g = 0 : 05, n = 1 : 5,with m a = 0 : 03and s = 0.........................42 Figure4.10:Steady-stateresponsepeakamplitudesplotsof4CPVAsbetweennon-linear simulation,linearsimulationandanalysis,withincreasing G q , g = 0 : 05, n = 1 : 5,with m a = 0 : 03and s = 0...........................42 Figure4.11:Dampedvsundampedsteady-stateresponseamplitudesof4CPVAswith increasing G q , g = 0 : 05, n = 2with s = 0 : 02...................43 viii Figure4.12:Rotorpeakamplitudeplotwith4CPVAswithincreasing G q , g = 0 : 05, n = 1 : 5with s = 0,analysisversusnonlinearsimulation...............44 Figure4.13:Rotorpeakamplitudeplotwith4CPVAswithincreasing G q , g = 0 : 05, n = 2 with s = 0,analysisversuslinearsimulation...................45 Figure5.1:Nonlinearsimulrotorpeakamplitudeplotwith4CPVAswithincreasing G q , g = 0 : 05, n = 2with0detuning..........................49 ix CHAPTER1 INTRODUCTION 1.1BackgroundandMotivation Torsionalvibrationisamajorconcerninpowertransmissionsystemsthathaverotatingcompo- nentssuchasshaftsorcouplings.Notonlycantorsionalvibrationcompromisetheintegrityofthe structureofthosecomponents,itcanalsoaffecttheperformanceandrobustnessofothermechani- calpartseitherdirectlyorindirectly,producingnoise,orintheworstcase,systemfailure.Ideally, thetorquewillbegeneratedandtransmitted"smoothly"throughoutthewholesystem,thusensur- ingthattherotationalspeedisconstant.However,inreality,thegeneratedtorqueisusuallynot smooth,buthasandoftentheseareorderbased,thatis,theirfrequencyisproportional totherotationrate.Acommonexampleofthisisininternalcombustionengines,wherethein- cylindergaspressurevariessubstantiallyovereachcycle[9].Inafour-strokeenginetheexcitation orderisthehalfofthenumberofcylinders,sinceeachcylinderonespertworevolutionsof thecrank.Additionally,theconnectedpartssuchasreductiongears,driveshafts,couplings,etc., canincreasetorsionalvibration.Thesevibrationsusuallyappearthroughoutthewholeoperating rangeatalloperatingspeeds,andcauseparticularproblemsinresonanceconditions. Traditionalmethodsthathavebeenusedtomitigatetorsionalvibrationininternalcombustion enginesincludetheuseoftorsionalfrictiondampers,largeinertiaandso-calledhar- monicbalancers,whicharesimplyfrequencytunedtorsionalvibrationabsorbers.However,those methodshavetrade-offssuchasengineperformance,engineefy,andlimitedspeedrange. Anothersolutiontoreducetorsionalvibrationsisbyaddingorder-tunedvibrationabsorbers, namely,centrifugalpendulumvibrationabsorbers(CPVAs).TheCPVAwasutilizedingeared radialaircraft-engine-propellersystemsduringWorldWarII[12].EarlydesignsofCPVAsuseda circleforthepathfollowedbythecenterofmassoftheabsorbers,duetotheeaseofmanufacture 1 andtheuseof(twopoint)suspensionsfortheabsorbermass,whichisconvenientdesignbe- causeofitscompactness(seeFigure1.1).However,Newland[6]showedaninstabilitycouldarise fromnonlineareffects,resultinginajumpinphasethatmadetheCPVAintoavibration. Thetraditionalremedyforthiswastoovertunetheabsorbers,whichenhancedrobustnessatthe expenseofperformance.Later,thisledtothedevelopmentofalternativepathsthatcircumvent, toalargeextent,thisnonlinearbehavior.Alongtheselines,Madden[3]patentedthesus- pensionabsorberdesignwithacycloidalpath,similartothatshowninFigure1.1,forapplication tohelicopterrotors.Later,Denman[2]introducedthetautochronicpath,whichmakesthetuning orderconstantatallamplitudes.Healsobuiltamodelthatincludedthefromtherollers usedinthesuspension. Figure1.1:designofCPVA. ThemainadvantageofaCPVAisthatitisdrivenbythecentrifugalforcesandtorsionalvibra- tions,soitdoesnotrequireextraenergyinputandactsasapassivedevice.TheCPVA'snatural frequencyisproportionaltothemeanrotatingspeedofthecrankshaft,whichplaysanessential roleintheirtuning.Theconstantratioiscalledtheabsorberorder,whichissetbyitsdesignpa- rameters,,theradiusoftheholesandrollersinthedesign,andcanbetunedto matchtheengineexcitationorder.Automotivecompanieshaverecentlybeendoingresearchonthe implementationofCPVAsincarenginesandotherpowertraincomponentsbecausetheybelieve CPVAswillhelpdecreasefuelconsumptionbyreducingtorsionalvibrations,therebyallowingen- 2 ginestorunatlowerspeeds,wherepumpinglossesarereduced[5,1].Infact,CPVAsarealready inproductionforuseintorqueconvertersanddualmass[8]. However,aconcernwhenimplementingCPVAsinenginesisthattherotationalaxisishori- zontaland,thus,therotatingplaneisvertical,therebycausinggravitationalforcestoactonthe rotor-absorbersysteminadditiontotheenginetorque.Infact,theseeffectscomeintoplayatlow enginespeeds,inparticular,atidle.Theinterplayofgravity,whichhasorderone(onecycleof forcingperrevolution)andsubjectstheabsorberstobothdirectandparametricexcitation,andthe order n enginetorqueactingontherotor,canhaveinterestingconsequences,asdescribedbelow. AninvestigationofthegeneraleffectsofgravityonthedynamicsofCPVAsystemswithmul- tipleabsorbersplacedcyclicallyaroundarotorwasdonebyT.Theisen[10],whousedthemethod ofmultiplescalestoinvestigatethenonlinearresponseofCPVAsystemswithgravity.Oneofthe resultsfromTheisen'sworkshowsthatwithdifferentengineordersandnumberofabsorbers,in somecasestheabsorbersallbehaveidentically,withaphaseshiftduetotheirplacementaround therotor,butinothercasestheybehavedifferently.Thisso-called groupingbehavior wasobserved andanalyzedintermsofpossibleresonanceconditions,butnotexaminedfromageneralpointof view.ThepresentworkwasmotivatedbyTheisen'sresults,withagoalofunderstandingtherole ofthegravitationalparametricexcitation,totakeadvantageofsomeofthespecialpropertiesof cyclicsystems,andtouncoverthefundamentalreasonsbehindtheobservedgroupingbehavior. Thesegoalscanbemetbyconsideringthelinearizedequationsofmotionwhicharevalidforsmall absorberamplitudes,andthatisthemodelusedinthepresentstudy. 1.2ThesisOutline Theremainderofthisthesisisarrangedasfollows:InChapter2,themathematicalmodelingfor arotor/absorbersystemisconducted,andthenonlinearequationsofmotion(EOM)withgravity termsareobtainedbyusingLagrange'sequation.Then,alinearizationversionoftheEOMsis obtainedbyassumingsmallabsorbermotionsandsmallrotorspeedInprevious worksseveraldifferenttypesofabsorberpathsareconsidered,butinthelineartheoryonlythe 3 smallamplitudeabsorbertuningaffectstheequations,asthemovementofabsorbersisassumedto besmall.AttheendofChapter2weanticipatewhattheformoftheabsorberresponse,intermsof itsharmoniccontent,andusethatformtoderiveageneraltheoryforthegroupingresultsobserved byTheisen,whichareinthiswork.InChapter3,apreliminarystudyisconsidered inwhichtheparametricterm,thatis,thetime-varyinggravitystiffnessterm,isomitted,giving asimplemodelthatprovidesinsightfortherestoftheinvestigation.Inthisanalysis,therotor equationisexpandedandsubstitutedintotheabsorberequations,andthelatteraretransformed intomatrixform.Sincetheparametrictermisomitted,itiseasytoperformdiagnolizationofthe systembyintroducingFouriermatrices,asthecoefmatricesarecirculant[7],andanexact formforthesteady-statesolutionsisfound.InChapter4,aperturbationanalysisisperformed tothelinearequationsinwhichtheparametrictermfromgravityisincluded.Theequationsare scaledsothattheMethodofMultipleScalescanbeused.Fromthescaledlinearequations,there isonlyonespecialcase(engineorder n = 2)inwhichtheabsorbersresonantlyinteractwiththe parametricexcitationfromgravityandtheappliedtorque.Theanalysisinthischapterisdone withoutdampingforsimplicity,afterwhichtheeffectsofdampingareconsidered.Inthelastpart ofthischapter,therotorbehaviorisrecoveredandanalyzedusingthesolutionsfromtheanalysis oftheabsorberresponse.Theresultsareshownintheformofplotsforsomecasesthat maybeofinterestforindustrialapplications.Theanalyticalresultsarecomparedwithnumerical simulationsofthelinearandnonlinearequationsofmotion,andgoodagreementisfound.Chapter 5providestheconclusionsofthisthesisandoutlinesfutureworkonthistopic. 4 CHAPTER2 MATHEMATICALMODELING 2.1EquationsofMotion Inthisthesis,aversionofamodelforarotorwithmultipleCPVAsisused,since thiswillsuffordeterminingtheessentialsystemdynamics.ThemodelisshowninFigure2.1, whichconsistsofarotatingrigiddisk(therotor),withitsaxisofrotationpointingoutofthepage, andgravityactingnormaltothataxis,asshown.Althoughthereisonlyoneabsorbershown, theequationsofmotionareestablishedwiththeassumptionofseveralabsorbersbeingequally spacedaroundtherotor.Thegravitationalisuniforminthedownwardverticaldirection.The Figure2.1:Modelingdiagram. symbolsusedinFigure2.1arelistedinTable2.1,alongwiththeirphysicaldescriptions. Therotorangleisgivenby q andisrotatingwithameanspeed < q > = W ,withsmall tionsgivenby q W .Thearisefromatorque,whichcanbemodeledasa 5 Symbols Description O Centerofrotor C Centeroftheabsorberpath;itsvertex R 0 Distancebetween O and C R ( S ) Distancebetween O andtheabsorberCOMatposition S S Arclengthpositionofabsorbermass M Absorberpointofmass W Meanrotationalspeedofrotor q rotorcrankangle g Gravity x horizontalcoordinateedinspace y verticalcoordinatededinspace Table2.1:SymbolsanddescriptionsfromFigure2.1. functionof q ,as T q sin ( n q + t ) where T q istheamplitudeofthetorque, n istheengineorder,and t isaphaseneededtoorientthetorquerelativethegraviationaldirection.Itisassumedthatthe absorbersdonotrotaterelativetotherotor,thus,eachofthemcanbetreatedasapointmassand theirmomentsofinertiaabouttheircentersofmasscanbeincorporatedintothatofthetotalrotor inertia.Thedisplacementofanabsorberisthusdeterminedbythepositionofitscenterofmass (COM),whoselocationsisdenotedby S ,whichisthegeneralizedcoordinatefortheabsorber, measuredasthearclengthalongtheabsorberpathawayfromthevertex. Theequationsofmotion(EOMs)ofthesystemareobtainedbyusingLagrange'sequations.A step-by-stepcomputationisverysimilartotheoneprovidedinpreviousworks,forexample[10], soitisnotnecessarytorepeatithere.TherotorEOMisfoundtobe, J rot ¨ q + N å j = 1 m p j [ R 2 ( S j ) ¨ q + dR 2 ( S j ) dS S j q + G ( S j ) ¨ S j + dG ( S j ) dS S j 2 g ( X p ( S j ) cos ( q j ) + Y p ( S j ) sin ( q j ))]= c 0 q + T 0 + T q sin ( n q + t ) (2.1) 6 andtheEOMforthe j th absorberisexpressedas, m p j [ ¨ S j + G ( S j ) ¨ q 1 2 dR 2 ( S j ) dS q 2 + g ( dX p ( S j ) dS sin ( q j )+ dY p ( S j ) dS cos ( q j ))] = c a j S j (2.2) where m p j ismassofthe j thabsorber, J rot istherotormomentofinertia, X p and Y p arethe x and y componentsofthelocationof R ( S j ) , c a j isthedampingcoefentforthe j thabsorber,and G ( S j ) isafunctionofthepath,givenby G ( S j )= s R 2 ( S j ) 1 4 ( dR 2 ( S j ) dS ) 2 ; whichisapathfunction,representedasapartofkineticenergy.Theseequationsarethebasisfor theanalysisandsimulationofCPVAsystems,however,theyarefullynonlinear.Astepin understandingthesystemdynamicsistoconsiderthedynamicsofthelinearizedmodelinwhich theabsorbersandrotorundergosmallamplitudeoscillations. 2.2NondimensionalizationandLinearization TheEOMsareformulatedintermsoftimedependentgeneralizedcoordinates q andthe S j s. However,thetorqueisafunctionoftherotorangle, q ,whichisamonotonicfunction oftime,andsoitisconvenienttoconverttheindependentvariablefromtimeto q ,whichrenders thetorqueasaperiodicexcitation.Thisisdonebyintroducingnon-dimensionalvariables n and w as, n = q W = 1 + w (2.3) where n isthenon-dimensionalrotorspeedand w describesnormalizedspeedabout W ,whicharegenerallysmall,thatis, j w j << 1.Wewillexpressboth n and w asfunctionsof q , ratherthantime. 7 Thetransformationfromtimedependentderivativeto q dependentderivativeisdonebyusing thechainruleandusingtheof n .Thisformulationforderivativesisgivenby ( )= d ( ) dt = d ( ) d q d q dt = d ( ) d q n W = W n ( ) 0 (2.4) ¨ ( )= d 2 ( ) d q 2 ( d q dt ) 2 + d ( ) d q ( d 2 q dt 2 )= n 2 W 2 ( ) 00 + nn 0 W 2 ( ) 0 ; (2.5) sothattheprimesaredimensionlessderivatives. ByusingEquation2.4above,Equation2.1andEquation2.2aretransformedas J rot W 2 nn 0 + N å j = 1 m p j [ R 2 ( S j ) W 2 nn 0 + dR 2 ( S j ) dS n 2 W 2 S j 0 + G ( S j )[ n 2 W 2 S j 00 + W 2 nn 0 S j 0 ] + dG ( S j ) dS n 2 W 2 S j 0 2 g ( X p ( S j ) cos ( q j ) + Y p ( S j ) sin ( q j ))]= c 0 n W + T 0 + T q sin ( n q + t ) (2.6) m p j [ n 2 W 2 S j 00 + W 2 nn 0 S j 0 + G ( S j ) W 2 nn 2 1 2 dR 2 ( S j ) dS W 2 n 2 + g ( dX p ( S j ) dS sin ( q j )+ dY p ( S j ) dS cos ( q j ))]= c a j n W S j 0 (2.7) whichrepresentthefullynonlinearequationsexpressedwith q astheindependentvariable. ToformulatetheEOMonemustspecifythepathoftheabsorber,whichiscapturedinthe function R ( S ) ,whichthendictates X ( S ) ; Y ( S ) ; G ( S ) .Thedetailsofdifferentpathformulationscan befoundinDenman'swork[2],buthereonlythesmallamplitudenatureofthepathisimportant, namelythecurvatureatthevertex.TheparametersandtheirexpressionsaregiveninTable2.2, asfunctionsofthearc-lengthvariable S ,where r 0 ispathradiusofcurvatureatthevertex(that is,at S = 0), l 2 [ 0 ; 1 ] isacharacteristicparameterdictatingthenonlinearnatureofthepath, andtheangle F j isaneffectiveangularpositionofabsorber j fromitsvertex,givenby F j = 1 l arcsin l S j r 0 [2,10].Notethattheradiusofcurvature r 0 dictatesthesmallamplitude(linear) absorbertuningorder,Ÿ n ,whichispurelygeometry-dependentandgivenbytherelation r 0 = R 0 Ÿ n 2 + 1 8 Term NonlinearExpressionfromDenman X p ( S j ) r 0 1 l 2 ( sin ( F j ) cos ( l F j ) l 2 S j r 0 cos ( F j )) Y p ( S j ) R 0 + r 0 1 l 2 ( cos ( F j ) cos ( l F j )+ l 2 S j r 0 sin ( F j ) 1 ) R 2 ( S j ) X p 2 + Y p 2 G ( S j ) r R 2 ( S j ) 1 4 ( DR 2 ( S j ) DS ) ) 2 Table2.2:PathfunctionsfromDenman[2]. ToproceedwithlinearizationoftheEOMwenondimensionalizetheabsorbervariableby s j = S j R 0 ,andnotethat j s j j << 1forrealisticmotions.Therefore,termsthatdependson s j arelinearizedbykeepingonlytheconstantandlineartermsfromtheirTaylorseriesexpansions about s j = 0.Also,since nn 0 =( 1 + w ) w 0 where w isassumedsmall,itfollowsthat nn 0 Ÿ = w 0 and n 2 Ÿ = 1 + 2 w .Termsinvolving nn 0 s 0 , s 0 2 ,andotherproductsofsmalltermsareignored.The linearizedandnon-dimensionalizedparametersarelistedinTable2.3.Notethatderivativesof sometermsareneededintheEOM,sothatquadratictermsinsomeexpansionsarereserved. Non-dim.Term RequiredExpansion x p ( s j ) X p ( S j ) R 0 s y p ( s j ) Y p ( S j ) R 0 1 1 2 ( 1 + Ÿ n 2 ) s j 2 r 2 ( s j ) x p 2 + x p 2 1 Ÿ n 2 s j 2 Ÿ g ( s j ) G ( S j ) R 0 1 1 2 ( Ÿ n 2 + Ÿ n 4 ) s j 2 Table2.3:Pathvariablesandrequiredexpansions. Withtheseexpressions,thenon-dimensionalversionofthederivativesneededintheEOMcan 9 beexpressedas dx p ( s j ) ds Ÿ = 1 dy p ( s j ) ds Ÿ = ( 1 + Ÿ n 2 ) s j dr 2 ( Ss j ) ds Ÿ = 2Ÿ n 2 s j dg ( s j ) ds Ÿ = ( Ÿ n 2 + Ÿ n 4 ) s j Theresultingequationsofmotion,linearizedforsmallabsorbermotionsandrotorspeed tuations,arethengivenby ( 1 + b 0 ) w 0 + b 0 N N å j = 1 s j 00 = G q sin ( n q + t ) (2.8) s 00 j + w 0 +( Ÿ n 2 g ( 1 + Ÿ n 2 ) cos ( q j )) s j = g sin ( q j ) m a s j (2.9) where b 0 = NmR 2 0 J rot istheratiooftotalabsorberinertiatotherotorinertia, g = g R 0 W 2 isthenon- dimensionalgravitycoef G q = T q J rot W 2 isthenon-dimensionaltorqueamplitude, m a = c a m W isthenon-dimensionaldampingcoefand q j = q + 2 p ( j 1 ) N istheangleofthe vertexofthepathofabsorber j ontherotor.Itisalsoassumedthatallabsorbersaregeometrically andmateriallyidentical,thatis, m p j = m andallpathparametersarethesame.Howeverthe absorbershavedistinctdisplacementsduringthesystemresponse,sowedonotassumethatthe s j sareequal. InordertosolveEquation2.8and2.9,itisconvenienttouncoupletherotordynamicsfromthe absorberdynamics.ThisisaccomplishedbysolvingEquation2.8for w 0 ,whichisgivenby w 0 = 1 1 + b 0 ( G q sin ( n q + t ) b 0 N N å j = 1 s j 00 ) (2.10) andtheninsertthisexpressionintoEquation2.9,resultinginanequationforthedynamicsofthe absorbersthatisuncoupledfromtherotor,givenby ( 1 + b 0 ) s 00 j b 0 N N å k = 1 s 00 k +( 1 + b 0 ) m a s j 0 +( 1 + b 0 )( Ÿ n 2 g ( 1 + Ÿ n 2 ) cos ( q + 2 p ( j 1 ) N )) s j =( 1 + b 0 ) g sin ( q + 2 p ( j 1 ) N ) G q sin ( n q + t ) : (2.11) 10 Thisreducedmodelisthebasisforourinvestigationofabsorberbehaviors. Aftersolvingforthesteady-stateabsorberresponsesusingEquation2.11,therotorresponse canbeobtainedusingEquation2.10.Notethattherotorangularacceleration, ¨ q ,isagoodmeasure fortherotortorsionalvibration,since,whenitiszerotherotorrunsataconstantspeed,andthisis givenby ¨ q = W 2 nn 0 Ÿ = W 2 w 0 Notethattheabsorbersaresubjectedtoorder n directforcing,causedbytherotorangular acceleration,aswellasbyorder1directandparametricexcitationfromgravity.Theorder n excitationisequalforallabsorbers,sincetheyareconnectedidenticallytotherotor.However, theorder1excitationiscyclicinnature,sincetheforcesonabsorber j dependonitspositionon therotor,andtheseareassumedtobeplacedsymmetricallyaroundtherotorcenter.Therefore, theexpectedresponseoftheabsorberswillhaveorder1,cyclicallyshiftedbyindex j around therotor,and n ,whichisidenticalforallabsorbers,pluspossiblelinearcombinationsofthese fromtheparametricexcitation.Thisleadstosomeinterestingconsequences,theofwhich isanobservationabouttherotorresponse,whichdependsontheabsorberresponses.Whenthe absorberresponsesareidenticalandorder n andphaseshiftedby 2 p ( j 1 ) N atorder1,theeffectsof theabsorbersontherotoratorder n adddirectly,whilethoseoforder1sumtozeroduetotheir cyclicnature.Thesecondconsequenceisthegroupingbehaviordescribedinthenextsection. 2.3GroupingBehaviorAnalysis Asseeninthesubsequentanalysisandsimulations,insomecasesalltheabsorbershavethesame waveformswithasimplecyclicphaseshift,whileinothercasesallabsorberwaveformsaredis- tinct,andinothercasestherearesubgroupsofabsorberswithmutuallyidenticalbutphaseshifted waveforms.Werefertothisbehaviorasabsorbergrouping,andthiscanbeanalyzedwithout solvingtheequationsofmotion,asdescribedinthissection. 11 Inordertopredictwhichabsorberswillgrouptogether,itisconvenienttoexpressthegeneral formoftheresponseofthe i th absorber,withacyclicorder1componentofamplitude A andan order n componentofamplitude B ,as s i ( q )= A sin ( n q + t )+ B sin ( q + 2 p ( i 1 ) N ) (2.12) where t accountsforthephaseshiftbetweentheorders.Wenowconsideranotherabsorber, s j ( q ) , where j = i + l ,thatis,theabsorberthatis l sectionsawayfromthe i th absorber,andexpressits responseas s j ( q )= A sin ( n q + t )+ B sin ( q + 2 p ( i + l 1 ) N ) (2.13) notingthattheorder n responseisthesameforallabsorbersandthattheorderonecomponentis phaseshiftedbythesectoranglebetweentheabsorbers.Notethatformholdsfor l = 1 ; 2 ;:::; ( N 1 ) . Figure2.2:Layoutof i thand j thabsorber. Wenextconsidertheconditionsunderwhichthewaveformofabsorber j willbeidenticalto thatofthe i absorber,butwithadifferentphase.Tothisend,weaddadummyphase y toboth componentsofabsorber i andexaminetheconditionsforwhich s j ( q )= s i ( q + y ) .Thefollowing expansionof s i ( q + y ) isusedtocompareitwith s j : s i ( q + y )= A sin ( n q + n y + t )+ B sin ( q + 2 p ( i 1 ) N + y ) : (2.14) 12 Sincetherearetwosinefunctionswithdifferentordersinthesteady-stateresponseofeachab- sorber,inordertohave s j and s i equal,itisnecessarytohave n y = 2 p k , k = 1 ; 2 ; 3 ;::: and y 2 p l N = 2 p q , q = 1 ; 2 ; 3 ;::: .Thus,theconditionforidenticalabsorberwaveformscanbeex- pressedbyeliminating y fromthesetwoconditions,resultinginthefollowingconditionon l asa functionofindices k and q , l = N n ( k nq ) : (2.15) Forgivennumberofabsorbers, N ,andengineorder n ,thewaveformsof s i and s j absorbers willbeidenticalifonecanintegers l if k and q thatsatisfythiscondition.Someexamplesare showninTable2.4,where"-"means"nosolution",whichmeanstheselectedwaveformswillbe different, (a)N=3,n=2 l k q 1 - - 2 - - (b)N=4,n=2 l k q 1 - - 2 3 1 3 - - (c)N=5,n=2.5 l k q 1 3 1 2 6 2 3 4 1 4 7 2 (d)N=6,n=2 l k q 1 - - 2 - - 3 3 1 4 - - 5 - - Table2.4:Examplesof l , k and q withdifferentvaluesof N and n . InTable2.4b,with N = 4and n = 2,theonlysolutionfor l is l = 2,andthereforeabsorber 3willhavethesamewaveformasabsorber1,andfromsymmetryconsiderationsabsorbers2and 4mustalsomatch,andsotherearetwogroupsofabsorbers,whichisseeninFigure3.1binthe nextchapter;moreaboutthisfollowsbelow.InTable2.4a,with N = 3and n = 2,thereisno solutionforanyvalueof l ,sothatallabsorbershavedistinctresponsesandthereare3( N )groups. ThiscorrespondstothecaseinFigure3.1ainthenextchapter.InTable2.4c,with N = 5and n = 2 : 5,thereisasolutiontoEquation2.15foreveryvalueof l ,sothatallabsorbershavethesame waveformandthereisonlyoneabsorbergroup.ThiscorrespondstothecaseinFigure3.1cinthe nextchapter.InTable2.4d,with N = 6and n = 2,theonlysolutionisfor l = 3,sothatabsorbers1 13 and4match,asdoabsorbers2and5andabsorbers3and6,thatis,therearethreeabsorbergroups. ThisisthecaseshowninFigure3.1dinthenextchapter. Table2.5showsmoreexampleswithamatrixofthenumberofgroupsformoredifferentvalues of N and n . n/N 2 3 4 5 6 1.5 2G 1G 4G 5G 2G 2 1G 3G 2G 5G 3G 2.5 2G 3G 4G 1G 6G 3 2G 1G 4G 5G 2G Table2.5:Numberofgroupswithdifferentvaluesof N and n . FromTable2.5,itisseenthatthenumberofgroupscanbedeterminedbywhether N n and/or n areintegers.ThesummaryresultisgiveninTable2.6,whichallowsonetopredictthe numberofgroupsbasedontheengineorder n andthenumberofcyclicallyplacedabsorbers N . N n n numberofgroups integer integer N n integer notinteger N 2 n notinteger either N Table2.6:Summaryofgroupingwithdifferent N and n Theanalysisaboveisgeneralforall N ,sincethesteady-stateformsassumedarevalidinthese cases. 14 CHAPTER3 LINEARMODELWITHOUTPARAMETRICEXCITATION InEquation2.11,theparametricexcitationterm, g ( 1 + Ÿ n 2 ) cos q j ,makesthestiffnesstimedepen- dent.Thatcreatesaproblemwhendecouplingtheoriginalequations.However,for linearanalysis,theparametricterminnotconsideredtohavehugeimpactsonthegeneralbehavior ofabsorbers,whichiswhyinthischapteritisomitted. 3.1Steady-stateDampedResponse Withouttheparametricexcitationterm,Equation2.11canreducedto ( 1 + b 0 ) s 00 j b 0 N N å k = 1 s 00 k +( 1 + b 0 ) m a s j 0 +( 1 + b 0 ) Ÿ n 2 s j =( 1 + b 0 ) g sin ( q j ) G q sin ( n q + t ) (3.1) WerewriteEquation3.1inmatrixform: M s 00 + C s 0 + K s = F (3.2) wherethemassmatrixis M = 0 B B B B B B B B B B B B B B @ ( 1 + b 0 b 0 N ) b 0 N ::: b 0 N b 0 N ( 1 + b 0 b 0 N ) ::: b 0 N . . . . . . . . . . . . b 0 N b 0 N ::: ( 1 + b 0 b 0 N ) 1 C C C C C C C C C C C C C C A 15 thestiffnessmatrixisdiagonalandgivenby K = 0 B B B B B B B B B B B B B B @ ( 1 + b 0 ) Ÿ n 2 0 ::: 0 0 ( 1 + b 0 ) Ÿ n 2 ::: 0 . . . . . . . . . . . . 00 ::: ( 1 + b 0 ) Ÿ n 2 1 C C C C C C C C C C C C C C A andthedampingmatrixisintheformof C = 0 B B B B B B B B B B B B B B @ ( 1 + b 0 ) m a 0 ::: 0 0 ( 1 + b 0 ) m a ::: 0 . . . . . . . . . . . . 00 ::: ( 1 + b 0 ) m a 1 C C C C C C C C C C C C C C A : Thisisalineartime-invariantsystemthatcanbesolvedinmanyways.Hereweuseacom- plexvariableapproachandtakeadvantageofthespecialcyclicnatureofthecoupledsystemof absorbersandthegravityforcesactingontheabsorbers.Byassuming s j = Im ( r j ) where r j is complex,Equation3.2canbetransformedinto M r 00 + C r 0 + K r = F gravity + F torque (3.3) Eachcomponentoftheexcitationforcesisexpressedastheimaginarypartofitsexponential formsas F gravity = Im [ e i q e i 2 p ( j 1 ) N ] F torque = Im [ e i ( n q + t ) ] Thenthesteady-statesolutionswillbeintheformof s j ss = Im ( z j ss ) 16 3.2DiagonalizationandSteady-StateSolution Matrices M , C and K aresymmetricandcirculant,soinordertodecoupletheequations,the( N N ) Fouriermatrixisintroducedasmentionedin[7],intheformof E N = 1 p N 0 B B B B B B B B B B B B B B @ 111 ::: 1 1 W 1 N W 2 N ::: W ( N 1 ) N . . . . . . . . . . . . . . . 1 W ( N 1 ) N W 2 ( N 1 ) N ::: W ( N 1 ) 2 N 1 C C C C C C C C C C C C C C A where W N = e i 2 p N andtheelementsofFouriermatrixcanbewrittenas ( E N ) jk = 1 p N e i 2 p N ( j 1 )( k 1 ) ,where j ; k = 1 ; 2 ;:::; N . TheHermitianoftheFouriermatrixis E ƒ N = 1 p N 0 B B B B B B B B B B B B B B @ 111 ::: 1 1 W 1 N W 2 N ::: W ( N 1 ) N . . . . . . . . . . . . . . . 1 W ( N 1 ) N W 2 ( N 1 ) N ::: W ( N 1 ) 2 N 1 C C C C C C C C C C C C C C A wherethe" () ƒ "istheHermitianoperation.Itcanbeshownthat E N E ƒ N = I [7],thatis, E N is unitary. 17 Notethat W N providesaconvenientwaytoexpressthecomplexexcitationforce., F =( 1 + b 0 ) g e i q 0 B B B B B B B B B B B B B B @ 1 W 1 N . . . W ( N 1 ) N 1 C C C C C C C C C C C C C C A G q e i ( n q + t ) 0 B B B B B B B B B B B B B B @ 1 1 . . . 1 1 C C C C C C C C C C C C C C A Wecomplexmodalcoordinates q forthissystemusing E N as r = E N q (3.4) SubstitutingEquation3.4intoEquation3.3andmultiplingtheequationby E ƒ N ,wehave, E ƒ N M E N q 00 + E ƒ N C E N q 0 + E ƒ N K E N q = E ƒ N F (3.5) whichareuncoupled. Thediagonalstiffnessanddampingmatricesareunchangedbythistransformation, Ÿ K = E ƒ N K E N = K and Ÿ C = E ƒ N C E N = C .Thediagonalizedmassmatrixisobtainedas Ÿ M = E ƒ N M E N = 0 B B B B B B B B B B B B B B @ 10 ::: 0 0 ( 1 + b 0 ) ::: 0 . . . . . . . . . . . . 00 ::: ( 1 + b 0 ) 1 C C C C C C C C C C C C C C A . 18 Themodalexcitationtermisgivenby E ƒ N F as E ƒ N F =( 1 + b 0 ) g e i q p N 0 B B B B B B B B B B B B B B @ 0 1 . . . 0 1 C C C C C C C C C C C C C C A G q e i ( n q + t ) p N 0 B B B B B B B B B B B B B B @ 1 0 . . . 0 1 C C C C C C C C C C C C C C A whichshowsthat,forthesemodalcoordinates,gravityexcitesonlythesecondmodeandthe torqueexcitesonlythemode. Thus,forsteady-statestudy,thereareonlytwomodalequationsthatneedtobesolved,specif- ically, q 00 1 +( 1 + b 0 ) m a q 1 0 +( 1 + b 0 ) Ÿ n 2 q 1 = G q e i ( n q + t ) p N (3.6) q 00 2 +( 1 + b 0 ) m a q 2 0 + Ÿ n 2 q 2 = g e i q p N (3.7) Thesteady-statesolutionsinmodelcoordinatesare q 1 s = p Ne i ( n q + t ) n 2 ( 1 + b 0 ) Ÿ n 2 in m a ( 1 + b 0 ) G q (3.8) q 2 s = p Ne i q Ÿ n 2 1 + i m a g (3.9) Then,thesteady-statesolutioninoriginalcoordinates, s s ,isobtainedbymultiplying q s bythe Fouriermatrix, E N ,andtakingtheimaginarypartsoftheresult. Figure3.1showstheabsorbersteady-stateresponsepeakamplitudeswithincreasingtorquefor severalsamplecases. AscanbeseeninFigure3.1,forsomegiven N and n ,someabsorbers,orinsomecaseallof them,havethesameamplitude.Inordertoexplainsuchinterestingbehavior,itisconvenientto ageneralresponseexpressionforeachabsorber.However,inexpression3.8,since q 1 s and q 2 s bothhavecomplexdenominator,theexpressionsfor s s becomeverycomplicatedtoobtainby hand.Thus,asimplercase,wherethedampingisomitted,isconsiderednext. 19 (a)Steady-statepeakamplitudesof3CPVAs, n = 2 (b)Steady-statepeakamplitudesof4CPVAs, n = 2 (c)Steady-statepeakamplitudesof5CPVAs, n = 2 : 5 (d)Steady-statepeakamplitudesof6CPVAs, n = 2 Figure3.1:Steady-stateresponsepeakamplitudesofasystemof N CPVAswithincreasingtorque amplitude G q ;for g = 0 : 05,withdamping. 3.3Steady-stateUndampedResponse Sincethedampingisomittedhere,Equation3.6canbereducedto q 00 1 +( 1 + b 0 ) Ÿ n 2 q 1 = G q e i ( n q + t ) p N (3.10) q 00 2 + Ÿ n 2 q 2 = g e i q p N (3.11) andthesteady-statesolutionsare q 1 s = p Ne i ( n q + t ) n 2 ( 1 + b 0 ) Ÿ n 2 G q (3.12) q 2 s = p Ne i q Ÿ n 2 1 g (3.13) Byusingthesamecoordinatetransformationproceduredescribedbefore,thatis s ss = Im ( r ss )= Im ( r ss )= Im ( E N q ) (3.14) 20 thesteady-stateresponseoftheabsorberscanbeobtainedas s s = 0 B B B B B B B B B B B B B B B B B B B @ p N n 2 ( 1 + b 0 ) Ÿ n 2 G q sin ( n q + t )+ p N ] Ÿ n 2 1 g sin ( q ) p N n 2 ( 1 + b 0 ) Ÿ n 2 G q sin ( n q + t )+ p N Ÿ n 2 1 g sin ( q + 2 p N ) . . . . . . p N n 2 ( 1 + b 0 ) Ÿ n 2 G q sin ( n q + t )+ p N Ÿ n 2 1 g sin ( q + 2 p N ( N 1 )) 1 C C C C C C C C C C C C C C C C C C C A (3.15) Equation3.15showsthateachabsorberresponsehasanorder n responsewithacommon amplitude,andanorder1responsewithacommonamplitude.Thisisakeytotheassumptionof thegeneralCPVAresponseformulationinthegroupinganalysisinSection2.3. 21 CHAPTER4 ACCOUNTINGFORGRAVITATIONALPARAMETRICEXCITATION- PERTURBATIONANALYSIS Whentheparametricexcitationfromgravityiskeptinthemodel,theequationsarelinearwith time-periodiccoefWhilethesteady-stateresponsescanbeexpressedintermsofintegrals, aconvenientwaytoobtainapproximationsoftheseistoemployperturbationmethods;herewe usethemethodofmultiplescales(MMS).Tothisendweneedtointroduceasmallparameter ‹ e tobeusedintheexpansions.Forsimplicityindevelopment,webeginwithananalysisofthe undampedsystemwithoneandthen N absorbers,andthenconsidertheeffectsofdampingatthe endofthechapter. 4.1Scaling ThereareseveraltermsinEquation2.11thatarescaledby ‹ e ,namely b 0 = ‹ e B d ; ‹ e P p ; s 00 = ‹ e P p 00 ; w 0 = ‹ e W x 0 ; g = ‹ e G Ÿ g ; G q = ‹ e G Ÿ G q ; Ÿ n = n ( 1 + ‹ e Q s ) : (4.1) Theseareconsistentwithapplications,astheparametersandvariablesaboveareassumedsmall. Notethat d isavariablethatisusedtotracetheinertiaratio b 0 inthescaledequations,anditcan betakentobeunitysothat ‹ e B becomestheinertiaratio. 4.2SingleAbsorberCase Theperturbationanalysisisbasedonthelinearized,non-dimensionalequationwiththeparametric excitationtermpresent.Thesimplestsystemtoanalyzeisasingleabsorberattachedtotherotor, asthecyclicphasehasnoeffectinthiscase.Theequationisformulatedas s 00 +( 1 + b 0 )( Ÿ n 2 ( 1 + Ÿ n 2 ) g cos ( q )) s = G q sin ( n q + t )+( 1 + b 0 ) g sin ( q ) (4.2) 22 ByusingthescalinginEquation4.1in4.2,theequationofmotionbecomes ‹ e P p 00 + n 2 ( ‹ e P + B d + ‹ e P ) p + 2 n 2 s ( ‹ e Q + P + B d + ‹ e Q + P ) p ( 1 + n 2 ) Ÿ g cos ( q )( ‹ e G + B + P d + ‹ e G + P ) p 2 n 2 s Ÿ g cos ( q )( ‹ e G + B + P + Q d + ‹ e G + P + Q ) p = ‹ e G Ÿ G q sin ( n q + t )+( ‹ e G + ‹ e G + B d ) Ÿ g sin ( q ) (4.3) InEquation4.3,inordertokeeptheparametrictermatleadingorder,alongwiththeothereffects ofinterest,wechoose G = 1 2 , B = 1 2 , Q = 1 2 , P = 1 2 ,and G = 1.Thus,Equation4.3,whenexpanded toleadingorderin ‹ e ,becomes p 00 + n 2 p + ‹ e 1 2 [ n 2 d + 2 n 2 s ( 1 + n 2 ) Ÿ g cos ( q )] p = ‹ e 1 2 ( Ÿ G q sin ( n q + t )+ d Ÿ g sin ( q ))+ Ÿ g sin ( q )+ HOT (4.4) whereHOTreferstohigherorderterms. Itisconvenientto ‹ e 1 2 = e ,sothatEquation4.4,withremovaloftheHOT,isgivenby p 00 + n 2 p + e [ n 2 d + 2 n 2 s ( 1 + n 2 ) Ÿ g cos ( q )] p = e ( Ÿ G q sin ( n q + t )+ d Ÿ g sin ( q ))+ Ÿ g sin ( q ) (4.5) Notethat,accordingtotheterminologyin[4],thisisacaseofhardnon-resonantandweak resonantexcitation(assuming n 6 = 1,whichisconsistentwithcasesofpracticalinterest). FollowingthestandardprocedurefortheMMS,theabsorberresponsecanbeexpressedasan expansionof p : p = p 0 + e p 1 + ::: whereboth p 0 and p 1 aredependonscales q 0 = q and q 1 = eq . Gathering e 0 and e 1 termsseparatelyprovidesthefollowingtwoequations e 0 : D 2 0 p 0 + n 2 p 0 = Ÿ g sin ( q 0 ) (4.6) e 1 : D 2 0 p 1 + n 2 p 1 = 2 D 0 D 1 p 0 [ n 2 d + 2 n 2 s ( 1 + n 2 ) Ÿ g cos ( q 0 )] p 0 + d Ÿ g sin ( q 0 ) Ÿ G q sin ( n q 0 + t ) (4.7) where D 0 isthepartialderivativerespecttotherotoranglescale q 0 and D 1 isthepartialderivative respecttothescale q 1 . 23 Fromthe e 0 equation,thesolutionfor p 0 isgivenby p 0 = Ae in q 0 + L e i q 0 + c : c : (4.8) where L = Ÿ g 2 i ( n 2 1 ) anditisconvenienttoexpress A = 1 2 ae i b ,whichisafunctionof q 1 . InsertingEquation4.8intothe e 1 equationinEquation4.6andexpanding,wethefollow- ingequationfor p 1 : e 1 : D 0 2 p 1 + n 2 p 1 = 2 D 0 D 1 ( Ae in q 0 + L e i q 0 + c : c : ) (4.9) n 2 d ( Ae in q 0 + L e i q 0 + c : c : ) 2 n 2 s ( Ae in q 0 + L e i q 0 + c : c : ) (4.10) + 1 2 Ÿ g ( 1 + n 2 )( e i q 0 + e i q 0 )( Ae in q 0 + L e i q 0 + c : c : )+ 1 2 i d Ÿ g ( e i q 0 e i q 0 ) (4.11) 1 2 i Ÿ G q ( e i ( n q 0 + t ) e i ( n q 0 + t ) ) (4.12) ItcanbeseenthatinsomecasestherearesometermsinEquation4.9leadstounboundedas q 0 evolves.Thesetermsarecalledsecularterms,whichcanvarydependingonthevalueof n . AccordingEquation4.9,thepossiblecasesare: n = 1, n = 2,and n 6 = 1 ; 2.Inthischapter,cases n = 2and n 6 = 1 ; 2areconsideredas,when n = 2,bothtorquegravityparameterscontributeto resonatingorder n responseofabosorber. n = 1caseisrecommendedforfuturework. Forthecase n = 2,theslowwequationisobtainedbyequatingtheseculartermstozero.In otherwords, ( 2 inA 0 ( n 2 d + 2 n 2 s ) A + 1 2 Ÿ g ( 1 + n 2 ) L 1 2 i Ÿ G q e i t ) e in q 0 = 0(4.13) ThefollowingequationsbelowareobtainedbytakingtherealandimaginarypartsofEquation 4.13: Re : a b 0 ( 1 2 d na + n s a ) 1 4 n ( 1 + n 2 ) Ÿ g 2 ( n 2 1 ) sin ( b )+ 1 2 n Ÿ G q sin ( b t )= 0 Im : a 0 + 1 4 n ( 1 + n 2 ) Ÿ g 2 ( n 2 1 ) cos ( b ) 1 2 n Ÿ G q cos ( b t )= 0(4.14) Itisconvenienttouse D = n 2 + 1 4 n ( n 2 1 ) Ÿ g 2 ,whichcontainstheparametriceffect,foralloftherestof analysis.(Note: n iskeptherejusttoshowthegeneralformulation). 24 InordertosolveEquationset4.14forthesteady-statesolution,theterms a 0 and b 0 aresetto zero,whichindicatesaconstantphaseandamplitudefor A .FromtheimaginarypartofEquation 4.14,ifandonlyif t = 0,theonlyvaluefor b tomakeitvalidis b = p 2 .Thus, a issolvedafter substituting b valueintotherealpartofEquation4.14 a = 1 n 2 ( d + 2 s ) Ÿ G q 2 D n ( d + 2 s ) (4.15) For n 6 = 1 ; 2,theonlydifferencefromEquation4.13isthattheterm 1 2 Ÿ g ( 1 + n 2 ) L isexcluded.So theslowequationintheformofimaginaryandrealpartsare Re : a b 0 ( 1 2 d na + n s a )+ 1 2 n Ÿ G q sin ( b t )= 0 Im : a 0 1 2 n Ÿ G q cos ( b t )= 0(4.16) andthesteady-statesolutionforEquation4.16is b = p 2 a = 1 n 2 ( d + 2 s ) Ÿ G q (4.17) 4.3MultipleAbsorbersCase UsingthesamescalingmethodfromSection4.1onthelinearizedequationsformultipleCPVAs, namelyEquation2.11,withanadditional s j 00 = ‹ e 1 2 p j 00 ,thefollowingequationisobtained ( p 00 j + n 2 p j Ÿ g sin ( q j ))+ ‹ e 1 2 [( d p 00 j 1 N N å k = 1 p 00 k ) d +( n 2 d + 2 n 2 s Ÿ g ( 1 + n 2 ) cos ( q j )) p j d Ÿ g sin ( q j )+ Ÿ G q sin ( n q + t )]+ HOT = 0(4.18) where q j = q 0 + 2 p ( j 1 ) N and j = 1 ; 2 ;:::; N .Fromtheleadingorderpart, p 00 j + n 2 p j Ÿ g sin ( q j )= 0,itcanbeseenthattheterm p 00 j isreplaceablewith n 2 p j + Ÿ g sin ( q j ) inthesummation.In addition,theterms n 2 d p j and d Ÿ g sin ( q j ) canalsobecanceledbytheterm d p 00 j .Thosearethemain differencesfromtheequationforthesingleabsorbercase.Itisalsoconvenienttouse e = ‹ e 1 2 .In 25 thiscasetheequationsatorders e 0 and e 1 aregivenby e 0 : D 2 0 p 0 j + n 2 p 0 j = Ÿ g sin ( q j ) e 1 : D 2 0 p 1 j + n 2 p 1 j = 2 D 0 D 1 p 0 j 1 N d n 2 N å k = 1 p 0 k (4.19) [ 2 n 2 s ( 1 + n 2 ) Ÿ g cos ( q j )] p 0 j Ÿ G q sin ( n q 0 + t ) : Thesolutionofthe e 0 equationsare p 0 j = A j e in q 0 + L e i ( q j ) + c : c : (4.20) where L = Ÿ g 2 i ( n 2 1 ) andthe A j aretobedetermined. Byreplacing p 0 j inthe e 1 equationwiththeexpressioninEquation4.20,thefollowingequa- tionisobtainedfor p 1 j : e 1 : D 0 2 p 1 j + n 2 p 1 j = 2 D 0 D 1 ( A j e in q 0 + L e i ( q 0 + f j ) + c : c : ) 1 N d n 2 N å k = 1 ( A k e in q 0 + L e i ( q 0 + f k ) + c : c : ) 2 n 2 s ( A j e in q 0 + L e i ( q 0 + f j ) + c : c : ) + 1 2 Ÿ g ( 1 + n 2 )( e i ( q 0 + f j ) + e i ( q 0 + f j ) )( A j e in q 0 + L e i ( q 0 + f j ) + c : c : ) 1 2 i Ÿ G q ( e i ( n q 0 + t ) e i ( n q 0 + t ) ) JustasmentionedinSection4.2forsingleabsorber, n = 2and n 6 = 1 ; 2casesareconsidered, becausebothgravityandtorqueareinvolvedinorder n responsewhen n = 2,whileforothervalue (not1)of n thereisonlytorque. Forthecasewhen n = 2,bygatheringtheseculartermsandsettingthemequaltozero,we thefollowingslowwequationforthecomplexamplitudes A j : 2 inA 0 j + 2 n 2 s A j + 1 2 i Ÿ G q e i t 1 2 Ÿ g ( 1 + n 2 ) L e i 2 f j + d n 2 N N å k = 1 A k = 0(4.21) Forothercaseswhere n 6 = 2,byequatingalltheseculartermstozero,thefollowingsloww equationisobtained: 2 inA 0 j + 2 n 2 s A j + 1 2 i Ÿ G q e i t + d n 2 N N å k = 1 A k = 0(4.22) 26 where A j isafunctionof q 1 ,and L = Ÿ g 2 i ( n 2 1 ) . AconvenientformforEquation4.21andEquation4.22isobtainedbyintroducing A j = 1 2 a j e i b j andseparatingtherealandimaginaryparts,resultinginthefollowingslowwequa- tionsfortheamplitudeandphase n = 2: Re : a j b 0 j + n s a j Ÿ G q 2 n sin ( b j t ) D sin ( 2 f j b j )+ d n 2 N N å k = 1 [ a k cos ( b k b j )]= 0 Im : a 0 j Ÿ G q 2 n cos ( b j t )+ D cos ( 2 f j b j )+ d n 2 N N å k = 1 [ a k sin ( b k b j )]= 0(4.23) (Note: n iskeptheresothatageneralformulationcanbepresented) n 6 = 1 ; 2: Re : a j b 0 j + n s a j Ÿ G q 2 n sin ( b j t )+ d n 2 N N å k = 1 [ a k cos ( b k b j )]= 0 Im : a 0 j Ÿ G q 2 n cos ( b j t )+ d n 2 N N å k = 1 [ a k sin ( b k b j )]= 0(4.24) FromEquation4.23and4.24,itisseenthatthephasesoftheabsorbersarecoupled,which makesithardtosolvefor a j and b j .However,sinceitislinear,itisconvenienttoexpressthe A j intermsofCartesiancoordinates. 4.4SlowFlowinCartesianCoordinates Theform A j = 1 2 a j e i b j isacommonexpressioninpolarcoordinateinreal-imaginarycoordinates. ItcanalsobeexpressedinCartesiancoordinates,asintheformof A j = 1 2 ( u j + iv j ) (4.25) where u j = a j cos ( b j ) ,and v j = a j sin ( b j ) .Thus,theamplitude a j canbeexpressedby q u j 2 + v j 2 ,andthephase b = arctan ( v j u j ) . 27 Bysubstitutingtheformof4.25intoEquation4.21for n = 2caseandEquation4.22, n = 2: Im : u j 0 = d n 2 N N å k = 1 v k n s v j D cos ( 2 f j )+ 1 2 n Ÿ G q cos ( t ) Re : v j 0 = d n 2 N N å k = 1 u k + n s u j D sin ( 2 f j )+ 1 2 n Ÿ G q sin ( t ) (4.26) n 6 = 1 ; 2: Im : u j 0 = d n 2 N N å k = 1 v k n s v j + 1 2 n Ÿ G q cos ( t ) Re : v j 0 = d n 2 N N å k = 1 u k + n s u j + 1 2 n Ÿ G q sin ( t ) (4.27) Itcanbeseenthatequations4.26and4.27arebothlinearin v j and u j andthereforesolvable inaquitestraightforwardmanner. Thesteady-statesolutionsarefoundbysetting u j 0 and v j 0 equaltozero.Theresultingversions ofEquations4.26and4.27canbeexpressedinamatrixformas A z = F (4.28) where A isthe(2 N 2 N )coefmatrix,whichisalsoablock-circulant[7]of N 2 2matrices, intheformof A = 0 B B B B B B B B B B B B B B @ J L ::: L L J ::: L . . . . . . . . . . . . L L ::: J 1 C C C C C C C C C C C C C C A where J and L are J = 0 B B B B @ n s + n 2 N d 0 0 n s + n 2 N d 1 C C C C A ; L = 0 B B B B @ n 2 N d 0 0 n 2 N d 1 C C C C A 28 and z =[ u 1 ; v 1 ; u 2 ; v 2 ;:::; u N ; v N ] T .The A matrixisthesameforall n . Forother n 6 = 1 ; 2,theforcevectorcontainsonlytheappliedtorqueandisgivenby F = F T = 1 2 n Ÿ G q 0 B B B B B B B B B B B B B B B B B B B @ sin ( t ) cos ( t ) . . . sin ( t ) cos ( t ) 1 C C C C C C C C C C C C C C C C C C C A For n = 2theforcevectorisacombinationoftheappliedtorqueandthegravitationalparamet- ricexcitationforce,intheform F = F T + F G = 1 2 n Ÿ G q 0 B B B B B B B B B B B B B B B B B B B @ sin ( t ) cos ( t ) . . . sin ( t ) cos ( t ) 1 C C C C C C C C C C C C C C C C C C C A + D 0 B B B B B B B B B B B B B B B B B B B @ sin ( 2 f 1 ) cos ( 2 f 1 ) . . . sin ( 2 f N ) cos ( 2 f N ) 1 C C C C C C C C C C C C C C C C C C C A whichshowswhythe n = 2casestandsout. Itiscleartoseethatmatrix A isacirculantmatrix,which,byintroducingFouriermatrix,can bediagonalized,asseeninSection3.2.Inthisway,thesolutionsfor z canbeeasilysolvedin closedform,asfollows.Thediagnoalizedversionof A isrepresentedby Ÿ A = E ƒ 2 N A E 2 N ,which isablockmatrixwithtwo D ( N N )arrangeddiagonally,givenby Ÿ A = 0 B B B B @ D 0 N N 0 N N D 1 C C C C A 29 where D = 0 B B B B B B B B B B B B B B B B B B B @ n s + n 2 d 00 ::: 0 0 n s 0 ::: 0 00 n s ::: 0 000 . . . 0 000 ::: n s 1 C C C C C C C C C C C C C C C C C C C A Sincethetorqueexcitationpartintheforcevector F isthesameforall n 6 = 1,thenthetrans- formedformisalsothesame,whichcanbetreatedastwo N 1, F T 1 and F T 2 ,vectorsjoined together, Ÿ F T = E ƒ N F T = p N 2 n Ÿ G q 0 B B B B @ F T 1 F T 2 1 C C C C A where F T 1 = 0 B B B B B B B B B B B B B B @ sin ( t )+ cos ( t ) 0 . . . 0 1 C C C C C C C C C C C C C C A ; F T 2 = 0 B B B B B B B B B B B B B B @ sin ( t ) cos ( t ) 0 . . . 0 1 C C C C C C C C C C C C C C A When n = 2,thephasesinthegravitypartintheforcevector, F G makesthetransformationpro- ceduremorecomplicatedandthetransformedexpression, Ÿ F G ,varieswiththenumberofabsorber, N .Thus,theform Ÿ F G willnotbepresented. Theproblemwiththeseundampedcasesisthat,whentheabsorbersareperfecttuned, s = 0, thecoefmatrix,eithertheoriginalordiagonalized,becomessingular.Thismakesthesystem unsolvable.Thus,fortheperfectlytunedabsorbercases,itisnecessarytoadddamping. 30 4.5AnalysiswithDampingEffects Equation2.11isusedinthissection.Fortheperturbationanalysis,thescalingfactorsarethesame withanadditionalterm Ÿ m a = e L m a ,and L = 1 2 .AlongwithotherscalingfactorsfromSection 4.1,afteromittinghigherorderterms,the e 0 equationis,ofcourse,unchanged,andthenew e 1 equationisexpressedas e 1 : D 0 2 p 1 j + n 2 p 1 j = 2 D 0 D 1 ( A j e in q 0 + L e i ( q 0 + f j ) + c : c : ) 1 N d n 2 N å k = 1 ( A k e in q 0 + L e i ( q 0 + f k ) + c : c : ) 2 n 2 s ( A j e in q 0 + L e i ( q 0 + f j ) + c : c : )+ 1 2 Ÿ g ( 1 + n 2 )( e i ( q 0 + f j ) + e i ( q 0 + f j ) )( A j e in q 0 + L e i ( q 0 + f j ) + c : c : ) 1 2 i Ÿ G q ( e i ( n q 0 + t ) e i ( n q 0 + t ) ) Ÿ m a ( inA j e in q 0 + i L e i ( q 0 + f j ) + c : c : ) (4.29) TheCartesianslowwisagainapplied.Theslowwequationshavethesameforcing vectorsinthiscase,givenbyEquation4.26and4.27.Dampingaltersthe A matrixandthethe EOMcanbeexpressedas A D z = F (4.30) where A D isablockcirculantmatrixwhereeachrowandcolumnareformedby N 2 2matrices arrangedas, A D = 0 B B B B B B B B B B B B B B @ B L ::: L L B ::: L . . . . . . . . . . . . L L ::: B 1 C C C C C C C C C C C C C C A where B and L (sameasintheundampedcase)are B = 0 B B B B @ n s + n 2 N d Ÿ m a 2 Ÿ m a 2 n s + n 2 N d 1 C C C C A ; L = 0 B B B B @ n 2 N d 0 0 n 2 N d 1 C C C C A 31 Thediagonalizationmethodforblock-circulantmatricesisdescribedin[7].Thetransformation ofcoordinate z iscarriedoutusingthe??Kroneckerproduct E N N I 2 where E N isthe N N Fouriermatrixand I 2 isthe2 2identitymatrix.Thisismathematicallyrepresentedby z = E N O I 2 k (4.31) where k arethemodalcoordinateswhichwillblockdecouplethesystem.Thesystem A D z = F is thustransformedas ( E ƒ N O I 2 ) A D ( E N O I 2 ) k =( E ƒ N O I 2 ) F or,incompactform Ÿ A D k = Ÿ F D : (4.32) wherematrix Ÿ A D isblockdiagonalizedwith N 2 2matricesalongitsdiagonal,whichoccurs since B isnotsymmetric.Thesematricesaregivenby Ÿ A D = 0 B B B B B B B B B B B B B B @ X 0 ::: 0 0 Y ::: 0 . . . . . . . . . . . . 0 0 ::: Y 1 C C C C C C C C C C C C C C A where X = 0 B B B B @ n 2 d + n s Ÿ m a 2 Ÿ m a 2 n 2 d + n s 1 C C C C A ; Y = 0 B B B B @ n sd Ÿ m a 2 Ÿ m a 2 n sd 1 C C C C A Fromthispointofview,Equation4.32canbetreatedas N separatecoupledequations,eachwith 2unknowns,whicharesolvablebyusingasymbolicmathematicssoftware. FromSection4.4,thediagonalizationtransformationonthegravitypart( F G )ofvector F when n = 2iscomplicatedsincethephasesin F G dependonthenumberofabsorbers.Thisalsooccurs 32 hereinthissection.However,assimilartothetorquepart, F T ,in F ,thediagonalizedtorqueforce vector, Ÿ F T D ,hasacleanexpression,as Ÿ F T D =( E ƒ N O I 2 ) F T = p N 2 n Ÿ G q 0 B B B B B B B B B B B B B B B B B B B @ sin ( t ) cos ( t ) 0 . . . 0 1 C C C C C C C C C C C C C C C C C C C A Forthesingleabsorbercase, N = 1,thelinearequationisexpressedas s 00 +( 1 + b 0 ) m a s 0 +( 1 + b 0 )( Ÿ n 2 ( 1 + Ÿ n 2 ) g cos ( q )) s = G q sin ( n q + t )+( 1 + b 0 ) g sin ( q ) (4.33) Withthesameprocessofgatheringtheseculartermsfordifferentcasesof n ,theslow equationsareputintoCartesiancoordinates.Forthespecialcaseofinterest, n = 2,thesloww equationsbecome Im : u 0 = ( n s + n d 2 ) v Ÿ m a 2 u D + 1 2 n Ÿ G q cos ( t ) Re : v 0 =( n s + n d 2 ) u Ÿ m a 2 v + 1 2 n Ÿ G q sin ( t ) (4.34) For n 6 = 1 ; 2,theslowequationisexpressedas Im : u 0 = ( n s + n d 2 ) v Ÿ m a 2 u + 1 2 n Ÿ G q cos ( t ) Re : v 0 =( n s + n d 2 ) u Ÿ m a 2 v + 1 2 n Ÿ G q sin ( t ) (4.35) ThesolutionstoEquation4.34andEquation4.35atsteadystatecanbeobtainedbyequating u 0 and v 0 tozero.Theresultin u and v for n = 2case( n iskeptasasymbol)is u = Ÿ G q Ÿ m a cos ( t ) 2 n D Ÿ m a n ( d + 2 s ) Ÿ G q sin ( t ) n Ÿ m 2 a + n 3 ( d + 2 s ) 2 v = Ÿ G q Ÿ m a sin ( t )+ n ( d + 2 s ) Ÿ G q cos ( t ) 2 n 2 D ( d + 2 s ) n Ÿ m 2 a + n 3 ( d + 2 s ) 2 (4.36) 33 Thenorder n responseamplitudeisobtainedas a = s Ÿ G 2 q + 4 n 2 D 2 4 n Ÿ G q D cos ( t ) n 2 ( Ÿ m 2 a + n 2 ( d + 2 s ) 2 ) (4.37) andwhen t = 0, a canberewrittenas a = s ( Ÿ G q 2 n D ) 2 n 2 ( Ÿ m 2 a + n 2 ( d + 2 s ) 2 ) (4.38) Itcanbeseenthat,when t = 0,ifthenon-dimensionalgravityterm, Ÿ g ,isconstant,thentherewill beacriticalvaluefor Ÿ G q thatwillkillorder n (inthiscase, n = 2)response. For n 6 = 1 ; 2,thesolutionsfor u and v havethesamedenominator,andthereisno D ,suchthat u = Ÿ G q Ÿ m a cos ( t ) n ( d + 2 s ) Ÿ G q sin ( t ) n Ÿ m 2 a + n 3 ( d + 2 s ) 2 v = Ÿ G q Ÿ m a sin ( t )+ n ( d + 2 s ) Ÿ G q cos ( t ) n Ÿ m 2 a + n 3 ( d + 2 s ) 2 (4.39) andtheorder n amplitudeisexpressedas a = Ÿ G q n p ( Ÿ m 2 a + n 2 ( d + 2 s ) 2 ) (4.40) Forthemultipleabsorbercases,thetransformationfromcoordinates k backto z isverymessy. Thus,theexpressionsarenotpresentedinthispaper. Wepresentresultsfromthisanalysisafterconsideringhowtheseabsorberresponsesaffectthe dynamicsoftherotor,whichistheultimatemotivationforusingCPVAs. 4.6RotorBehaviorAnalysis TheoverallgoalforCPVAsistoreducetorsionalvibrationsontherotor,sointhissectionthe rotorperturbationequationisformulated.Therotoranalysisisbasedonthesolutionsfromthe absorbers'perturbationanalysis.BysubstitutingscalingfactorsandtheirvaluesfromEquation 4.1intoEquation2.8,therotorequationscanbeexpressedas ‹ e W x 0 + 1 N ‹ e 1 å j = 1 Np j 00 + HOT = ‹ e 1 Ÿ G q sin ( n q + t ) 34 where W = 1inordertocancel ‹ e frombothsidesoftheequation.Thus,thescaledversionofthe rotorequationisintheformof x 0 = Ÿ G q sin ( n q + t )+ 1 N N å j = 1 ( n 2 p j Ÿ g sin q j )= Ÿ G q sin ( n q + t )+ n 2 N N å j = 1 p j Then,thenon-dimensionalrotorangularacceleration, w 0 fromtheEquation4.1,isformulated as w 0 = ‹ e ( Ÿ G q sin ( n q + t )+ n 2 N N å j = 1 p j ) (4.41) and p j Ÿ = p 0 j as p 1 j isconsideredtobeverysmall.Thus, p j isintheformof p j Ÿ = 1 2 ( u j + iv j ) e in q 0 + L e i ( q 0 + f j ) + c : c : (4.42) where L = Ÿ g 2 i ( n 2 1 ) . Becauseofthesummation,theorder1responsefromgravityoftheabsorbersiscanceledin therotorresponse.Theonlyeffectsfromgravitycomefromtheparametricexcitationwhen n = 2, whichcanbeseenintheresultsinthenextsection. 4.7ResultsandDiscussions Previouslyinthischapter,theanalysisisbasedonlineartheory,anditshowsthatfortheabsorbers therearetwo-orderresponses:order1andorder n .Thus,evenwhenthereisnotorque, thereshouldstillberesponsefromthegravity,fromtheassumedforminEquation4.42.Fromthe singleabsorbercase,itcanbeseenthatfororder n responses,theamplitudeisproportionaltothe torque Ÿ G q forallof n ,yetinthe n = 2resonantcase,thereisanadditionaltermthatisproportional to Ÿ g 2 . Therearetwocaseswithdifferentengineordersthatareusedfornumericalresultsinthis section.Thecaseisfor n = 2,whichcorrespondstoafour-strokefour-cylinderengine.The othercaseis n = 1 : 5,relevanttofour-strokethree-cylinderengines.Inbothcasesdifferentnumbers ofabsorbers N areconsidered.Thedifferenceofthesetwocases,asmentionedinSection4.4,is 35 theappearanceoftheparametrictermwhen n = 2.Infact,theseparametricresonancetermsappear onlywhen n = 1 ; 2,and n = 1isnotofmuchpracticalinterest,soweselect n = 2todemonstrate theeffectsofparametricresonance. Thesimplestsystemthatdemonstratestheeffectsoftheparametriceffectsontheabsorbers fromgravityisthatwithasingleabsorber, N = 1.ThenumericalsimulationofEquation2.11with asingleCPVAisconductedinMATLABwithparameters d = 1, e = 0 : 0355, g = 0 : 05, t = 0 and G q intherangefrom0to0.01,atameanspeedof400rpm,thedampingratioissettobe z = 0 : 01,sothat m a varieswithdifferentvaluesof n .For n = 2thedampingcoef m a = 0 : 04 andfor n = 1 : 5, m a = 0 : 03.Inordertoseetherelationshipbetweentheabsorberresponseand theamplitudeofthetorque,thepeakamplitudeoftheabsorberresponseateachtorque levelisplotted. Figure4.1:Steady-stateresponseamplitudesofasingle( N = 1)CPVAwithincreasingtorque, G q ;for g = 0 : 05, n = 2, m a = 0 : 04,and s = 0. Theresultsfromanalysisandsimulationmatchwell,ascanbeseenintheplotsshownbelow. InFigure4.1,withtheamplitudeofthetorqueincreasing,theabsorberamplitudegoes downatandthenincreases,whichiscausedbyorder n responseinthiscase,supportedby 36 Equation4.38.Thisbehaviorcomesfromtheparametricexcitationeffectsfromgravity.Atime- traceplotofthiscase n = 2isprovidedinFigure4.2batthecritictorque( G q 2 fromFigure4.1). Itclearlyshowsthattheabsorbershaveonlyoneharmonicresponse.Figure4.2alsoshowsthe (a)At G q 1 (b)At G q 2 (c)At G q 3 Figure4.2:Steady-stateresponseamplitudesofasingle( N = 1)CPVAtime-traceplotatvarious torquelevel,for g = 0 : 05, n = 2, m a = 0 : 04,and s = 0. transitionofabsorberbehaviorsfromlesstorquelevels( G q 1 fromFigure4.1)tothehighertorque levels( G q 3 fromFigure4.1). Figure4.3showstheabsorberresponsefor n = 1 : 5,inwhichcaseparametrictermisnotreso- nantandtheabsorberamplitudegrowslinearlyatthetorqueincreases.Aninterestingobservation fromthesetwoisthatwhenthereisnotorque, G q = 0,theabsorberhasa non-zeroresponseduetothedirectexcitationfromgravity,withamplitudegivenbyEquation4.8. Fromthesolutionexpressionsoforder n amplitudesfortheundampedcase,Equation4.15and Equation4.17,itiscleartoseethedifferencefromtheonesofthedampedcases,in Equation4.37andEquation4.40.Thenitismorevisualtoseetheeffectsofthedampingingraphic forms.Theresultsoftheundampedresponseandthosewithdampingof z = 0 : 01areplottedin Figure4.4andFigure4.5for n = 2and n = 1 : 5,respectively. InFigure4.4,itisclearlyshownthatthe"dip"occurswithorwithoutdamping.Also,whenthe torqueamplitudeissmall,thedampingdoesnothavemuchontheabsorberamplitude. However,asthetorqueincreases,thedampedandundampedresponseamplitudesstarttodiffer. Thisseparationalsooccursinthe n = 1 : 5case,asshowninFigure4.5.Inaddition,asexpected, theresponseamplitudesatmoderateandlargetorquesaresmallerinthedampedcases. 37 Figure4.3:Steady-stateresponseamplitudesofasingle( N = 1)CPVAwithincreasingtorque, G q ;for g = 0 : 05, n = 1 : 5, m a = 0 : 03,and s = 0. Thecasesshownnextareformultipleabsorbers.Fourabsorberswithdampingareinvestigated. Heretheinertiaratioistakentobe e = 0 : 1419andtheotherparametersarethesameasforthe singleabsorberexampleabove.For n = 2,accordingtothegroupinganalysisofSection2.3,there shouldbetwoabsorbergroupssince N n = 4 2 = 2;seeTable2.6.ThiscanbeseeninbothFigures4.6 andFigure4.7.Figure4.6showsthewaveformsofthefourabsorbers,aspredictedfromanalysis, usingEquation4.42.ItiscleartoseethatAbsorber1andAbsorber3sharethesame waveformwitha p phasedifference,andsimilarlyforAbsorbers2and4.Thephasedifferences comefromthecyclicpositionsoftheabsorbers,andthereareindeedtwogroups. Figure4.7containsadetailedcomparisonbetweentheanalyticresultsandlinearsimulation results.Becausetheequationislinear,thereareonlytwoorderswithclearpeaksintheFast Fouriertransform(FFT)ofthesimulatedresponse:order1andorder n = 2.Theorder1amplitude comesfromgravity,intheformof L = Ÿ g 2 i ( n 2 1 ) ,andordertheorder n amplitudeisobtainedfrom solvingEquation4.30orEquation4.32. FromeachorderplotinFigure4.7,itiscleartoseethefromtheparametricterm. 38 Figure4.4:Damped, m a = 0 : 04,versusundampedsteady-stateresponseamplitudeof1CPVA withincreasing G q , g = 0 : 05, n = 2with s = 0. Theamplitudeoforder1responseisconstant,because L isnotafunctionof Ÿ G q .However,order n responseamplitudeisincreasingwhileseparatingintotwogroups,whichthenresultingrouping behaviorinthetotalpeakamplitudeplots.Byutilizingasymboliccomputingsoftware,itcanbe shownthatorder n amplitudeexpressionshaveasimilarformtothesingleabsorbercase,which meansitisnotlinearlydependenton G q .Italsocanbeseenthatthetotalpeakamplitudesarethe sumoforder1andorder n amplitudes. Figure4.8andFigure4.9showthesteady-stateresponseoffourabsorbersandtheirpeakvs torqueplots,respectively,for n = 1 : 5case.Fromthetime-traces,thepeakamplitudesofeach absorberaredistinct,whichisexpectedaccordingtothegroupingresultsinTable2.6.However, aninterestingobservationshowsthattheabsorberswith p differenceinpositionhavesimilarwave formsbutareaboutthe q -axisrelativetoeachother,indicatingthatadditionalsymmetry considerationsshouldbeinvestigated. Figure4.9showsthattheorder n = 1 : 5amplitudesincreaselinearlywithtorqueinasamerate 39 Figure4.5:Damped, m a = 0 : 03,versusundampedsteady-stateresponseamplitudeof1CPVA withincreasing G q , g = 0 : 05, n = 1 : 5with s = 0. Figure4.6:Steady-statetimetracesof4CPVAs, G q = 0 : 01, g = 0 : 05, n = 2,with m a = 0 : 04and s = 0. andorder1amplitudesarethesame.Yet,thetotalpeakamplitudeofeachabsorberisdifferent fromtheothers.Onereasonforthisscenarioisthattheamplitudesoftwoharmonicresponses cannotbesimplyaddedtogetherduetophasedifferences. Alloftheresultsabovearecomparisonsbetweenanalysisandlinearsimulations.Itisalso 40 Figure4.7:Steady-stateresponseamplitudesof4CPVAswithincreasing G q , g = 0 : 05, n = 2, with m a = 0 : 04and s = 0. Figure4.8:Steady-statetimetracesof4CPVAs, G q = 0 : 01, g = 0 : 05, n = 1 : 5,with m a = 0 : 03and s = 0. importanttoensuretheyalsomatchwithnonlinearequations.Onecasechosenforsuchacom- parisonisfourabsorberswith n = 1 : 5becauseofthedistinctresponseofeachabsorber.Thepeak amplitudesversustorqueareshowninFigure4.10,showingthatthelinearanalysis,thelinear simulations,andthenonlinearsimulationsallmatchquitewelloverthetorquerangeshown.As expected,theresultsbegintodeviateasthetorqueamplitudeincreases. Inordertogetsolutionsfortheresonantcase n = 2withoutdamping,asseenfromEquation 4.28,itisnecessarytodetunetheabsorbers.Figure4.11showsanalyticalresultsofcomparison 41 Figure4.9:Steady-stateresponseamplitudesof4CPVAswithincreasing G q , g = 0 : 05, n = 1 : 5, with m a = 0 : 03and s = 0. Figure4.10:Steady-stateresponsepeakamplitudesplotsof4CPVAsbetweennon-linearsimu- lation,linearsimulationandanalysis,withincreasing G q , g = 0 : 05, n = 1 : 5,with m a = 0 : 03and s = 0. for n = 2withdampingandwithoutdamping,for0 : 02detuning.Ingeneral,dampingmakesthe absorbersteady-stateresponseamplitudechangemoreslowlyasthetorqueincreases.Thisistrue 42 Figure4.11:Dampedvsundampedsteady-stateresponseamplitudesof4CPVAswithincreasing G q , g = 0 : 05, n = 2with s = 0 : 02. forasingleabsorber(seeFigure4.4),aswellasthepresentcasewith N = 4showninFigure4.11 aswell,whichshowsthattheseparationofthetwogroupsisalsodecreasedwithdamping. Equation4.41isusedtopredicttherotorresponseforthelinearabsorbermodel.Duetothe summation,theorder1effectsfromtheabsorberscanceleachother,aresultthatfollowsfrom theircyclicplacementaroundtherotor.Thus,inlinearanalysis,therotorresponsehasonlyan order n component.InFigure4.12,therotorpeakamplitudeversustorqueamplitudeisplotted alongwithnonlinearsimulationresultsfor n = 1 : 5.Thereasonforcomparinglinearanalysiswith nonlinearsimulationsisthat,sincetherotorresponseisrelativelysmall,higherordercomponents oftheresponsescan,infact,haveontherotorresponse.Figure4.12shows thattheanalyticalpredictionsofthepeakrotoramplitudealignswiththeorder n componentfrom simulations,computedbyanFFT,verywell.Italsoshowsthatthereisnoorder1response. However,thetotalamplituderesultshowsagrowingseparationasthetorquegetslarger,showing thefromthehigherorderresponsesgeneratedbynonlinearity.Asthetorque increases,thesehigherorderresponsecomponentsbegintodominatetheorder n response. 43 Figure4.12:Rotorpeakamplitudeplotwith4CPVAswithincreasing G q , g = 0 : 05, n = 1 : 5with s = 0,analysisversusnonlinearsimulation. Thiscanalsobeseenfor n = 2case.Figure4.13showstherotorresultforthecase n = 2.In Figure4.13,theanalyticalresultiscomparedwithlinearsimulationresultsfor n = 2.Itcanbe seenthattheamplitudesgrowlinearlywithincreasingtorque.At G q = 0,boththeanalyticaland simulationrotoramplitudesare0. 44 Figure4.13:Rotorpeakamplitudeplotwith4CPVAswithincreasing G q , g = 0 : 05, n = 2with s = 0,analysisversuslinearsimulation. 45 CHAPTER5 CONCLUSIONSANDFUTUREWORK 5.1SummaryandConclusions Thepurposeofthisworkwastocontinuetheinvestigationoftheeffectsofgravityonrotorsys- temswithcentrifugalpendulumvibrationabsorbers(CPVAs),asinitiatedbyTheisen[10]. Themaingoalsoftheworkwere:(i)toinvestigatethespeeffectsofgraviationalparametric excitationonthesystem,(ii)toexploitthecyclicnatureofthegravitationalexcitation,and(iii)to analyzethegroupingbehaviorofabsorbersforagivennumberofCPVAs, N andengineorder, n . Theseweredoneusingalinearizedmathematicalmodelandnumericalsimulations. AbriefbackgroundoftorsionalvibrationsandCPVAswasprovidedTheequationsof motionforanidealizedmodelwerederived,linearized,andnon-dimensionalized,whichformsthe basisfortheanalysis.Ageneralformoftheabsorberresponseatorders1and n wasusedtoanalyze thegroupingbehavioroftheabsorbers.Thegroupinganalysiswasconductedbyaddingadummy phasevariableintoageneralformofthe j thabsorber'ssteady-stateresponseandcomparingthat withthe i thabsorber'ssteady-stateequation,todeterminetheconditionsunderwhichthesetwo absorberswouldhaveidenticalwaveforms.Theanswertowhetherabsorberswouldbehavein groupswasdeterminedbywhether N n and n areintegers,assummarizedinTable2.6.Theresults fromsimulationsshowedthatdampingmadethegroupingmoredistinct,atleastforsomecases. Intheanalysis,bothresonant( n = 2)andnon-resonant( n = 1 : 5)casesofpracticalinterest werestudied.Modelswithandwithoutparametricexcitation,andwithandwithoutdamping, wereconsidered.Expressionsforthesteady-statesolutionsfortheabsorberswereobtainedandthe resultsshowedthat,withouttheparametricexcitationterm,solutionofthesystemmodelcouldbe reducedtothediagonalizationofacirculantmatrix.ThiswasachievedusingtheFouriermatrix[7] totransformthesystem,andthetransformedversionoftheexcitationforcesshowedthatgravity 46 excitedonlyasinglemode(anorderonetravelingwavemode)andthetorqueexcitedonlyanother singlemode(theunisonmode).ThisapproachallowsforanexactsolutionoftheEOMinthis case. Forthelinearsteady-stateresponsewithparametricexcitation,theequationswererescaled sothatamultipletimescalesperturbationmethodwasabletobeconducted.AsetofCartesian coordinates( u and v )wasusedfortheorder n amplitudesandphasesinsteadoftraditionalpolar coordinates,sinceittransformedtheslowequationsintolinearfunctionsof u and v .Itwas shownthattheorder1amplitudes(fromdirectexcitationfromgravity)forallcaseswereconstant foragivenmeanrotorspeed,asexpected.Forthesteady-statesolutionsoftheorder n amplitudes, itshowedthatthecoefmatriceswereblock-circulant,andtheparametrictermfromgravity onlyappearedinresonant n = 2case.Onenotefromtheanalysiswasthatwhendampingwas considered,thecoefmatrixcouldnotbefullydiagonalized,however,therewere N sets oftwo-by-twolinearequationsthatcouldbesolvedinclosedform.Theanalyticresultswere comparedwithnumericalsimulationsofboththelinearandnonlinearmodelequations,showing thattheanalyticalpredictionsareaccurateforsmallabsorberamplitudes.Asingle-absorbercase wasstudiedinordertounderstandtheeffectsofgravityforresonant, n = 2,andnon-resonant, n = 1 : 5,cases.Inthenon-resonantcase( n = 1 : 5)itwasseenthattheabsorberamplitudegrows linearlywithincreasingtorque,astheorder n responsewasnotaffectedbythegravity.Forthe resonantcase( n = 2)theresultsshowthat,foragivenvalueof g ,therewasacriticaltorque levelwheretheparametricexcitationfromthegravitycancelsthetorque,whichleft onlytheorder1responsefromthegravityatthatpoint.Forthecaseofmultipleabsorbers,the resultsshowthatforbothresonantandnon-resonantcases,theorder1componentoftheresponse hadacommonamplitudeforallabsorbers.For n = 2,theorder n responseamplitudegrowsnon- linearlywiththetorque,duetotheresonantinteraction,andtherewasnoscenariowhenthegravity canceledthetorque,aswasthecasewithasingleabsorber.For n = 1 : 5,theresultsshowthatthe order n componentoftheabsorbers'responseshavethesameamplitudewhichgrowslinearly withtorque,similartothesingleabsorbercase.However,becauseofthephasedifferences,the 47 peakamplitudesofthecombinedharmonicsforeachabsorbercanbedifferent.Fromcomparisons betweenthedampedandundampedcases,itwasobservedthatdampingdoesnotchangethe groupingbehavior,butdoesaffecttheresponseoftheabsorbersintheexpectedmanner,reducing responseamplitudes. TheresponseoftherotorwasreconstructedusingEquation4.41andthesteady-stateabsorber response.Itwasshownthat,formultipleabsorbersystems,theeffectsoftheorder1responseson therotorcanceleachotheroutinasummation,duetothecyclicnatureoftheorder1response. Thus,therotorresponseincludesonlytheorder n absorberresponses.Itwasalsoshown,using simulationsofthenonlinearequations,thatnonlineareffectscomeintoplayatmoderateampli- tudes,resultinginhigherorderharmonicsintherotorresponse,whichareimportantsincethe order n componentislargelyeliminatedbytheabsorbers;thisisconsistentwithpreviousobserva- tions[11]. 5.2RecommendationsforFutureWork Informationfromtheresultsofthisstudyofferopportunitiestoexamineothertopicsbeyondthe conclusionsofthestudy.Theseinclude: Theeffectsofgravityonmultipleabsorbersystemswithgeneralordertorques,includingthe resonantcase n = 1. Analysisofnonlinearsystemmodelswithgravityandnear-tautochronicabsorberpaths. Analysisofmodelsinwhichtheinertiaratioisnotsmall. Re-examinationofthegroupingbehaviorandanalysisofhigherorderharmonicsinthere- sponseinlightofnonlineareffects.Asanexample,itisnotedfromFigure5.1thattherotor amplitudepeak,fromsimulationsofthenonlinearEOM,hasanon-zerovaluewhentheor- der n torqueiszero.Itisclearfromtheplotthatthisnon-zeroresponsecomesfromanorder 48 4gravitationalcomponent,whichisignoredinlinearanalysis.Infact,Theisennotedthat thisbehaviorstemsfromnonlineareffects[10],whichareclearlyimportantforthiscase. Figure5.1:Nonlinearsimulrotorpeakamplitudeplotwith4CPVAswithincreasing G q , g = 0 : 05, n = 2with0detuning. ExperimentalinvestigationsofCPVAsystemswithhorizontalaxes,forwhichgravitycomes intoplay. 49 BIBLIOGRAPHY 50 BIBLIOGRAPHY [1] LindsayBrooke.fiChryslerseestheICEfuturefl.In: AutomotiveEngineeringInternational 20.S2(2013),pp.16Œ19. [2] HHDenman.fiTautochronicpendulumtorsionabsorbersforreciprocatingenginesfl. In: JournalofSoundandVibration 159.2(1992),pp.251Œ277. [3] J.F.Madden. 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