DYNAMICSOFHORIZONTALAXISWINDTURBINESANDSYSTEMSWITHPARAMETRICSTIFFNESSByGizemDilberAcarADISSERTATIONSubmittedtoMichiganStateUniversityinpartialful˝llmentoftherequirementsforthedegreeofMechanicalEngineeringDoctorofPhilosophy2017ABSTRACTDYNAMICSOFHORIZONTALAXISWINDTURBINESANDSYSTEMSWITHPARAMETRICSTIFFNESSByGizemDilberAcarThedynamicsofawindturbinebladeunderbend-bend-twistcoupledvibrationsisinves-tigated.Thepotentialandkineticenergyexpressionsforastraightnonuniformbladearewrittenintermsofbeamparameters.Then,theenergiesareexpressedintermsofmodalcoordinatesbyusingtheassumedmodesmethod,andtheequationsofmotionarefoundbyapplyingLagrange'sformula.Thebend-bend-twistequationsarecoupledwitheachother,andhavesti˙nessvariationsduetocentrifugale˙ectsandgravitationalparametrictermswhichvarycycliclywiththehubangle.Todeterminethenaturalfrequenciesandmodeshapesofthesystem,amodalanalysisisappliedonthelinearizedcoupledequationsofcon-stantanglesnapshotsofabladewithe˙ectsofconstantspeedrotation.Lowermodesofthecoupledbend-bend-twistmodelaredominantlyin-planeorout-of-planemodes.Toinvesti-gatetheparametrice˙ects,severalblademodelsareanalyzedatdi˙erentangularpositions.Thesti˙nesstermsinvolvingcentrifugalandgravitationale˙ectscanbesigni˝cantforlongblades.Tofurtherseethee˙ectofbladelengthonrelativeparametricsti˙nesschange,theblademodelsarescaledinsize,andanalyzedatconstantrotationalspeeds,athorizontalandverticalorientations.Blade-hubdynamicsofahorizontal-axiswindturbineisalsostudied.Bladeequationsarecoupledthroughthehubequation,andhaveparametrictermsduetocyclicaerodynamicforces,centrifugale˙ectsandgravitationalforces.Themodalinertiaofasinglebladeisde˝nedbythelinearmassdensitytimesthesquareoftransversedisplacementsfromblade'sunde˛ectedaxis.Forreasonabletransversedisplacements,themodalinertiaofabladeisusuallysmallcomparedtotherotorinertiawhichisthecombinedinertiaofthehubplusallthreebladesabouttheshaft.Thisenablesustotreatthee˙ectofblademotionasaperturbationontherotormotion.Therotorspeedisnotconstant,andthecyclicvaria-tionscannotbeexpressedasexplicitfunctionsoftime.Bycastingtherotorangleastheindependentvariable,andassumingsmallvariationsinrotorspeed,theleadingorderbladeequationsaredecoupledfromtherotorequation.Theinterdependentbladeequationscon-stituteathree-degree-of-freedomsystemwithperiodicparametricanddirectexcitation.Theresponseisanalyzedbyusingthemethodofmultiplescales.Thesystemhassuperharmonicandsubharmonicresonancesduetodirectandparametrice˙ectsintroducedbygravity.Amplitude-frequencyrelationsandstabilitiesoftheseresonancesarestudied.TheMathieuequationrepresentsthetransientdynamicsofasingle-modeblademodel.ApproximatesolutionstothelinearunforcedMathieuequation,andtheirstabilities,areinvestigated.Floquettheoryshowsthatthesolutioncanbewrittenasaproductbetweenanexponentialpartandaperiodicpartatthesamefrequencyorhalfthefrequencyofexcitation.AnapproachcombiningFloquettheorywiththeharmonicbalancemethodisinvestigated.AFloquetsolutionhavinganexponentialpartwithanunknownexponen-tialargumentandaperiodicpartconsistingofatruncatedseriesofharmonicsisassumed.Then,performingharmonicbalance,theFloquetexponentsandandharmoniccoe˚cientsarefound.Fromthisfrequenciesoftheresponseandstabilityofthesolutionaredeter-mined.Thetruncatedsolutionisconsistentwithanexistingin˝niteseriessolutionfortheundampedcase.ThetruncatedsolutionisthenappliedtothedampedMathieuequationandtoparametricexcitationwithtwoharmonics.Solutionsandstabilityofmulti-degree-of-freedomMathieu-typesystemsarealsoinvesti-gated.AproceduresimilartotheoneappliedfortheMathieuequationisusedto˝ndtheinitialconditionsresponse,frequencycontent,andstabilitycharacteristics.Theapproachisappliedtotwo-andthree-degrees-of-freedomexamples.Forafewparametersets,theresultsobtainedfromthismethodarecomparedtothenumericalsolutions.Thisstudyprovidesaframeworkforatransientanalysisofthree-bladeturbineequations.CopyrightbyGIZEMDILBERACAR2017ACKNOWLEDGMENTSFirstandforemostIwouldliketothankProfessorBrianFeeny,forhisguidanceandhelp.Healways˝ndstheperfectbalancebetweenlettinghisstudentstobeindependentresearchers,andsupportingthemwheneverneeded.Ialsoenjoyedourconversationsonanythingandeverything.Thanksforbeingthebestadvisorthereis.IwouldliketothankProfessorStevenShawforbeingagreatinstructor,andfortheintellectualconversationswehad.Heisverysupportiveandhealwaysencouragesmetofollowmyacademicdreams.Healsoisaveryfriendlyandawarmperson,andthetimeswehadwithShawfamilyalwaysmademefeelhome.ManythankstoProfessorAhmedNaguibandProfessorRanjanMukherjeeforservinginmythesiscommittee,andfortheirinvaluablefeedbackonmyresearch.Iwouldliketothankmyfriendsfortheircontinuoussupport.Mehmet,thanksforbeingmybestfriend,motivatingmeallthetime,andbeinglikeamentortomethroughtheacademiclife.Mert,reachingintoyourbrilliantmindthroughourconversationsisalwaysagreatexperience.Thanksforexpandingmyhorizonstotheskies.Asli,theonlythingIregretaboutourfriendshipisthatwemetlaterthanweshouldhave.Thanksforyourcontinuousemotionalsupport.Merve,nomatterhowearlyinlifeweparted,wehavegrownuptobegreatfriends,andIamthankfulforthetimesweshared.Yagmur,thanksforbeingmyoldestfriend,andmakingthatlittledormroomwesharedfeellikeahometome.Selcan,thanksforbeingaverygoodfriend,andyourguidanceonacademicmatters.Thankstomycolleagues,Fatemeh,Ayse,Pavel,Ori,Scott,Xing,MingandRickey,fortheexperiencewesharedtogether.Ialwaysappreciateourconversationsonscienceandresearch,aswellasondailylifeproblems.Wordscannotexpressmyappreciationofhavingyoualwaysnear,mydearMustafa.Wearenotjusthusbandandwife,butcolleagues,reallygoodfriends,andcompanionsinbothvthepursuitofthePhDandthisjourneyinlife.IwouldnotbethesamepersonifIdidnotmeetyou.NomatterhowfarfromhomeIam,tome,homeisrightwhereyouare.Iwouldliketothankmyfamily,whoalwayssupportedmeinmydecisions.SinemandEnesarenotjustmysiblings,butalsotwoofmybestfriends,andnodistancecanchangethis.Ialsothankmymom,forbeingthemostwarm-heartedandleastjudgmentalmomthereis.Iwouldnotaccomplishanyofthesewithoutherandmysiblings'support.FinallythankstoeveryonewhoIchattedwithaboutresearch,scienceandlife,whoImadepaintingswith,whoIsharedabeerwith,whotouchedmylifeonewayortheother.Really,thankyou!TheprojectsIworkedforthepastfouryearwerefundedbyNSF,undergrantno:CMMI1335177.viTABLEOFCONTENTSLISTOFTABLES...................................ixLISTOFFIGURES...................................xCHAPTER1INTRODUCTION...........................11.1MotivationandObjectives............................11.2BackgroundandLiteratureReview.......................31.2.1Bend-Bend-TwistVibrationsofaWindTurbineBlade........31.2.2In-PlaneDynamicsofaThreeBladeTurbine..............51.2.3TheMathieuEquationandMDOFSystemswithParametricSti˙ness61.2.4AerodynamicStall............................71.3ThesisOverview..................................81.4Contributions...................................10CHAPTER2EQUATIONSOFMOTIONOFABLADEUNDERBEND-BEND-TWISTVIBRATIONS.........................112.1Introduction....................................112.2Methodology...................................122.2.1EnergyFormulations...........................132.2.2AssumedModesandEquationsofMotion...............162.2.3ModalAnalysis..............................172.3Results.......................................182.3.1AHollowRectangularBeamwithaStructuralTwist.........182.3.2NREL's23mBlade...........................222.3.3NREL's5MWTurbineBlade......................232.3.4Sandia's100mBlade...........................242.3.5TheRelationBetweentheBladeSizeandtheParametricE˙ects...262.3.5.1AScalingStudy:EachDimensionScaledEqually......262.3.5.2AScalingStudy:OnlytheLengthScaled..........272.3.5.3AScalingStudy:NREL's5MWTurbineBlade.......272.4Conclusion.....................................28CHAPTER3THREE-BLADETURBINEEQUATIONSFORIN-PLANEBENDING303.1Introduction....................................303.2Three-BladeTurbineEquations.........................313.3ApplicationofMethodofMultipleScales....................363.3.1NonresonantCase.............................383.3.2SuperharmonicResonance(2r1ˇ1orˇ!n2)...........393.3.3PrimaryResonance(r1ˇ1orˇ!n2)................423.3.4SubharmonicResonance(r1ˇ2orˇ2!n2).............44vii3.3.5ExistenceofResonanceConditions...................453.4Conclusions....................................46CHAPTER4ANANALYSISOFTHEMATHIEUEQUATION...........494.1Introduction....................................494.2Floquet-basedSeriesSolution..........................514.2.1In˝niteSeriesSolution..........................524.2.2TruncatedSeriesSolution........................534.3ResponseCharacteristicsoftheUndampedMathieuEquation........574.3.1StabilityAnalysis.............................574.3.2ResponseFrequencies...........................574.4DampedMathieuEquation...........................604.5ParametricExcitationwithTwoHarmonics..................644.6Conclusions....................................66CHAPTER5APPROXIMATEGENERALRESPONSESOFMULTI-DEGREE-OF-FREEDOMSYSTEMSWITHPARAMETRICSTIFFNESS...685.1Introduction....................................685.2Analysis......................................685.2.1Two-Degree-of-FreedomExample....................695.2.2Three-Degree-of-FreedomExample...................725.3Discussion.....................................775.4Conclusion.....................................79CHAPTER6CONCLUSIONSANDFUTUREWORK...............816.1ConcludingRemarks...............................816.2FutureWork....................................83APPENDICES......................................85AppendixAEQUATIONSOFMOTIONOFABLADE...........86AppendixBIN-PLANETHREEBLADETURBINEEQUATIONS......89AppendixCAREVIEWONDYNAMICSTALLMODELS..........94BIBLIOGRAPHY....................................103viiiLISTOFTABLESTable2.1Naturalfrequencies(Hz)oftherectangularpre-twistedbeam,=0...19Table2.2Naturalfrequencies(Hz)oftherectangularpre-twistedhorizontalbeam..20Table2.3Thelowestnaturalfrequency(Hz)oftherectangularpre-twistedbeamwithE=20GPa................................21Table2.4Naturalfrequencies(Hz)forNREL's23mturbineblade..........22Table2.5Naturalfrequencies(Hz)ofNREL's5MWturbineblade..........23Table2.6Naturalfrequencies(Hz)forSNL's100mturbineblade...........24Table2.7Firstmodalfrequencies(Hz)forthescaleduphollowrectangularbeams:estimationsfromthescalingrelationsandFEAresults,for=2......27Table4.1Principalcharacteristicexponents1and2fortruncatedsolutionswithn=1;:::;4...................................55Table4.2Lowestresponsefrequenciesobtainedfromthen=2andn=1Floquetsolutions,andFFTsofnumericalsolutions.................59ixLISTOFFIGURESFigure1.1Windenergy'sshareinelectricityproductionintheU.S.[1].......1Figure1.2Bladeunderbend-bend-twistdeformation..................5Figure1.3Anairfoilcross-section.............................8Figure2.1Coordinatesystemsinthedeformedandundeformedblade........12Figure2.2Bladecross-sectionbeforedeformation....................15Figure2.3Isometricviewofthehollowrectangularpretwistedbeam.........19Figure2.4Analyticalmodeshapesforthestationary50mbeamindownwardposition.21Figure2.5SideandtopviewsoftheFEAmodeshapesforthestationary50mbeamindownwardposition..........................22Figure2.6FirsttwomodeshapesforNREL's61.5mbladewith_˚=0and˚=ˇ=2.25Figure2.7Parametricsti˙nessratiosforscaledblademodelsandactualblademodels.28Figure3.1Athree-bladeturbinewithbladesunderin-planebending.........32Figure3.2In-planemodeshapesofathree-bladeturbine................32Figure3.3Steadystatesuperharmonicresonanceresponseamplitude,frequencyplotsfor=1,=4,=0:05;0:1;0:2...................41Figure3.4Steadystatesuperharmonicresonanceresponseamplitude,frequencyplotsfor=0:05,=1,=1;2;4.....................42Figure3.5Stabilityplotsforthesubharmonicresponsefor~e=0,=0;0:25;0:5.Simulationsaredonefor=0:5,=0:1,Fj0=0:1,Fj1=0:1,=1:5,f=0:1,(+):stable,():unstable......................46Figure3.6Stabilityplotsforthesubharmonicresponsefor~e=1,=0;0:25;0:5.Simulationsaredonefor=0:5,=0:1,Fj0=0:1,Fj1=0:1,=1:5,f=0:1,(+):stable,():unstable......................47Figure3.7Campbellplotshowing!n2asafunctionof,fork0=1,k1=0:1,mb=1.....................................47xFigure4.1Transitioncurvesforthen=2Floquet-basedapproximation,in˝niteseriessolution,andthen=2Hill'sdeterminantsolution..........58Figure4.2Analyticallypredictedresponsefrequenciesfor=0:8,n=2.......59Figure4.3Numericalandtheoretical(n=2)solutionsoftheundampedMathieuequationforn=2...............................60Figure4.4FFTsofnumericalandtheoretical(n=2)solutionsoftheundampedMathieuequationforn=2..........................61Figure4.5FreeresponseandFFTplotsfor!=0:4,=0:6,withn=3harmonics.62Figure4.6TransitioncurvesforthedampedMathieuequationfor=0:005,=0:025;and=0:05,approximatedwithn=2................63Figure4.7Decayandgrowthfactorsfor=0:05and=0:8.............64Figure4.8Numericalandtheoretical(n=2)solutionsofthedampedMathieuequation....................................64Figure4.9FFTofnumericalandtheoretical(n=2)stablesolutionsofthedampedMathieuequation...............................65Figure4.10Numericalandtheoreticalsolutionsofthetwo-frequencyMathieuequation.66Figure5.1AtwoDOFspring-masschain........................69Figure5.2Stabilityregionsforthe2DOFmassspringchain..............72Figure5.3Modesexcitedbythesymmetricandtheanti-symmetricinitialcondi-tions,for!=2:3and=0:4.AmplitudeFFTplotsofdisplacementsofm1,generatedwithn=2truncatedsolution...............72Figure5.4ResponseandamplitudeFFTplotsforn=2,!=1:2,=0:6,x(0)=[10:5]Tand_x(0)=[00]T........................73Figure5.5ResponseandFFTplotsforn=2,!=2:3,=0:4,x(0)=[15]Tand_x(0)=[00]T...............................74Figure5.6A3DOFmass-springsystem.........................74Figure5.7Stabilityregionsforthe3DOFmassspringsystem,for=1and=1.75Figure5.8Modesexcitedbytheinitialconditionswhichresonatethefrequenciesthatareassociatedwith(a)1=0:975,(b)2=0:520,(c)3=0:315..76xiFigure5.9ResponseandFFTplotsforn=2,!=0:7,=0:5,=0:4,x(0)=[000]Tand_x(0)=[111]T......................77Figure5.10ResponseandFFTplotsforn=2,!=3:5,=0:3,=0:4,x(0)=[110:5]Tand_x(0)=[000]T...................78FigureC.1Arepresentativeplotshowingstallphenomenon[2].............94FigureC.2Arepresentative˝gureshowingdynamicstall[3]..............96FigureC.3AirfoildiagramusedinONERA'smodel[2].................96FigureC.4Diagramsshowingfullyattachedandseparated˛owconditions[4]....99FigureC.5Mappingfromairfoilpro˝letoaunitcircle[4]...............100xiiCHAPTER1INTRODUCTION1.1MotivationandObjectivesThroughthepastfewdecades,windpowerhasattractedconsiderableattentionasarenew-ableenergysource.In2015,windenergycomprised4:7%oftheelectricityproductionintheU.S.[5].TheDepartmentofEnergyhasstudiedscenarioswheretheaimistoincreasethewindenergy'sshareinelectricityproductionto20%by2030,andto35%by2050[1,6],asshowninFigure1.1.Astheglobalenergyindustryhasstartedtoinvestinwindpowerasacleanenergysource,researchonwindturbineshasbeenofgreatimportance.Sincetheinstallationandmaintenancecostsaresigni˝cant,e˙ortshavebeenmadeonreliabledesignsforwindturbineparts.Figure1.1Windenergy'sshareinelectricityproductionintheU.S.[1]Theamountofenergyproducedbyawindturbineisproportionaltotheareasweptby1itsblades,andhencelargerbladedesignshavebecomemorepopularrecently.TheDutchO˙shoreWindEnergyProjecthasabladedesignwith62.6m,TheNationalRenewableEnergyLaboratory(NREL)hasamodelof61.5mbladeforaturbinewitha5MWcapacity,andSandiaNationalLaboratory(SNL)hasadesignfora100mbladeforaturbinewitha13:2MWcapacityHowever,longbladedesignscomealongwithdynamicalproblems.Thegravitationalforceandcentrifugale˙ectsinducevariationsinbladesti˙ness,whichcanbesigni˝cantforlongblades.Alsothebladevibrationshavecouplingbetweenin-plane,out-of-planeandtorsionaldirections,althoughtheyhaveusuallybeenconsideredseparatelyinindustrialapplications.Theseconsiderationsarethemotivationbehindthewindturbinebladeresearch.Inhorizontal-axiswindturbinesfailureusuallytakesplaceinthehubandgearbox.Todescribetheloadingsinducedbythebladesonthehub,itisimportanttostudythebladedynamics.Then,theinteractionsbetweenthebladesandrotorneedtobeinvestigated.Bladesapplyinertialandparametricloadsontherotor,whicharetransmittedtoeachotherthroughthehub.Thesee˙ectsinduceparametricanddirectsecondaryresonancesintheblades,evenwhenthemodelislinear.So,evenforasimplescenario,thecoupleddynamicsofbladesandthehubisaninterestingresearchproblem.Becauseofthedynamicvariationsinthebladesti˙ness,thebladeequationsaresimilartoaforcedMathieuequation.SolutionstotheMathieuequationarethereforeofgreatimportancesincetheyapplytothetransientdynamicsoftheblades.Also,three-bladeandhubturbineequationscanbeconsideredasamulti-degree-of-freedom(MDOF)systemwithparametricsti˙ness.Thetransientdynamicsofthesingle-bladeequationsandthethree-bladeturbineequationsmotivatedthestudyofthegeneralresponsesoftheMathieuequationandMDOFsystemswithparametricsti˙ness,respectively.Modelingtheaerodynamicforcesonaturbinebladeisimportantforacompleteunder-standingofforcedturbinedynamics.Duetooscillationsintheblades,dynamicstallmightoccurwhenthebladesoperateathighanglesofattack.Anextensivereviewisdoneon2existingaerodynamicstallmodelsinordertounderstandthenatureoftheexternalforcesontheblades.Howeverapplicationofthesemodelsarenotincludedinthisthesis.Thesemotivatingissuesleadtotheobjectivesofthisstudy,whichare-Tomodelthebladeasabeamunderbend-bend-twistvibrations,andstudyitsmodaldynamicstoestimatethedegreeofcouplingbetweenin-planeandout-of-planevi-brations,andto˝ndtheimportanceoftheparametricgravitationalandcentrifugale˙ects,-Toinvestigatetheinteractionbetweenbladesandtheirdynamicale˙ectsonthehub,andtodeterminethesecondaryresonancesinthebladesbyusingaperturbationmethod,-TostudythegeneralresponsesoftheMathieuequationtogaininsighttothetransientdynamicsofasingleblade,-To˝ndthegeneralresponsesofMDOFsystemswithparametricsti˙nesstobuildaframeworkforthetransientdynamicsofathree-bladeturbine,-Toreviewtheaerodynamicstallmodelswhichcanrepresenttheaeroelasticloadingathighanglesofattack,andunderunsteadyoscillatoryconditions.1.2BackgroundandLiteratureReview1.2.1Bend-Bend-TwistVibrationsofaWindTurbineBladeVibrationalanalysisofawindturbinebladeplaysanimportantroleinturbinedesign.Inhorizontal-axiswindturbinesfailureoftentakesplaceinthehubandgearbox[10].Thecyclicloadsappliedbythebladesarethoughttoplayalargerole.Topreventfailureandmakeimprovementsinturbinedesign,dynamicsofthebladesmustbeinvestigated.3Bladesareundercyclicloadingduetoturbinerotation.Asteadywindspeedusuallyvarieswithaltitude,soasthebladerotateswiththehubitsaltitudevaries,andsoitisa˙ectedbyacycliclyvaryingamountofwindforce.Alsothetangentialandradialcompo-nentsofgravityforcevarycyclicly,changingthee˙ectivesti˙nessofthebladeasthehubrotates.Thesee˙ectsintroduceparametrictermsintotheequationsofmotion.Abladeunderbend-bend-twistdeformationisshowninFigure1.2.Thedeformationin^ydirectioniscalledin-plane(edgewise)bending,whereasthedeformationin^ziscalledout-of-plane(˛apwise)bending.Forpracticalreasons,thebladevibrationsareusuallyin-vestigatedin˛apwiseandedgewisedirectionsseparately[8,11,12].Thesetwodirectionsareuncoupledonlywhentheproductmomentofinertiaofabladecross-sectioniszero.Yet,forageneralairfoilcross-section,itisnotzero,whichintroducescouplingbetweentwobendingdirections.Dawson[13]studiedbend-bendcoupledvibrationsinpre-twistedbeams,andformulatednaturalfrequenciesusingenergymethods.TorsionalvibrationsarealsocoupledwithbendingvibrationsformostbladesDokumaci[14]derivedthetorsion-bendingcoupledequationsanalytically,andBishopetal.[15]improvedthetheorybyintroducingwarping.CooleyandParker[17]alsoworkedonbend-twistcoupledvibrationofspinningbeams,takingthecentrifugale˙ectsintoaccount.HodgesandDowell[18]foundtheequa-tionsofmotionforabladegoingunderbend-bend-twistmotionwithastructuraltwist.Sincetheyworkedonahelicopterblade,theydidnottakethee˙ectsofgravityandvaryingrotorspeedintoaccount.Kallesøe[19]usedHodgesandDowell'sequationsforaturbineblade,addingthee˙ectsofgravityandpitchingmotion.Heneglectedshearcenterbeingo˙setfromthemasscenter,andshafttiltandprecone.Windturbineshavebeenstudiedthroughexperimentalandnumericalmodalanalyses,and˝niteelementmethodLarsenetal.[20]explainedaprocedureforanexperimen-talmodalanalysiswhereeachcross-sectionismodeledwiththree-degrees-of-freedomandthebladeisexcitedwithanimpacthammer.Hansen[23]usedtheHAWCStabsimulationtool,wherethebladestructureismodeledwith˝niteelementmethodandtheaeroelasticforces4Figure1.2Bladeunderbend-bend-twistdeformation.aremodeledwithbladeelementmomentummethod,toconductaneigenanalysisto˝ndtheaerodynamicstabilityofaturbine.Bir[24]appliedmultibladecoordinatetransformationmethodtoNREL's5MWturbineto˝nditsfrequencies.CrespodaSilva[25]developedamodelforbeamsunderbend-bend-twistvibration.Ne-glectingtheparametrice˙ectsofgravity,hederivedtheequationsofmotionusingHamilton'sprinciple.InChapter2,asimilarapproachisgivenindetail,whereinsteadofHamilton'sprinciple,Lagrange'smethodisusedto˝ndtheequationsofmotion,andthemodaldynamicsofthebladeareinvestigatedaswellastheparametrice˙ects.1.2.2In-PlaneDynamicsofaThreeBladeTurbineInChapter3,in-planevibrationequationsarederivedforathree-bladeturbine.Thebladeequationsarecoupledthroughtherotorequation,andthemodalmassinthebladeequationissmallcomparedtotheinertiatermintherotorequation.Theseequationsaresimilartocentrifugalpendulumvibrationabsorberequations,wheretheabsorberinertiaissmallcomparedtotherotorinertia[26,27].Thebladeandrotorequationsarecoupledthroughtheinertialterms.UsingtheFouriermatrixexplainedinOlsonetal.'swork[28],thecouplingcanbetransferredtothesti˙nesstermsthroughacoordinatetransformation.Then,onecanapplythemethodofaveragingtostudythesteady-statedynamics.Inordertodecoupletheabsorberequationsfromtherotor5equation,Chaoetal.[29,30]usedrotorangleastheindependentvariable,insteadoftime,andusedascalingschemewheretheabsorbermotionisconsideredassmallperturbationstotherotorequation.Also,Theisen[31]investigatedthee˙ectsofgravityonabsorbermotion,andinternalresonancesassociatedwithit.AsimilarapproachisusedinChapter3inordertoinvestigatethethree-bladeturbinedynamics,andtheresonancesintroducedbytheparametrice˙ects.1.2.3TheMathieuEquationandMDOFSystemswithParametricSti˙nessHorizontal-axiswind-turbinebladeequationsunderconstantrotationrateshaveparametricsti˙nesstermsduetogravity[32,33].Thereforesolutionstoequationswithparametricexcitationareimportanttounderstandthetransientandsteady-statedynamicsoftheblades.Numerousmechanicalsystemshaveparametricterms,suchasswings,base-excitedpen-dulumsandshiproll[32,ParametricallyexcitedsystemssuchastheMathieuequationhavebeenstudiedusingavarietyofmethods.Floquettheoryhasbeeninvokedtoprovidestabilitycriteria,asdiscussedinmoredetailinChapter4.PerturbationmethodshavealsobeenusedtoapproximatethesolutionstotheundampedanddampedMathieuequations,aswellastheforcedandunforcedMathieuequations[32,34,Theseincludetheappli-cationofsecond-ordermultiplescales[32],andthemethodofvanderPol[34]todeterminestabilitycharacteristics,andtheuseofaveraging[37],multiplescales[38],andharmonicbalance[46]toanalyzeforcedMathieuequations.Ecker[47,48]workedonparametricallyexcitedvibrationabsorberswhicharerepresentedbyaMathieu-typeequation.Heusednumericalintegrationtosolveforthemonodromymatrixanditseigenvaluesto˝ndthestabilitycharacteristicsofthesystem.Rand[45]foundanexpressionforstabilitytransitioncurvesofaMathieu-typesystemwiththreeharmonicsbyapplyingaperturbationanalysis,whereasKlotterandKotowski[49]usedHill'smethodtoinvestigatestabilityofaparamet-ricallyexcitedsystemwithtwoharmonics.SofroniouandBishop[50]workedonamoregenericequationwithperiodicandquasi-periodicexcitation,andinvestigatedbifurcation6characteristics.Systemidenti˝cationmethodshavebeenusedtoinvestigatetheresponsecharacteristicsofthetime-periodicsystems.Allenetal.[51]usedanoutput-onlysystemidenti˝cationmethodto˝ndtheFloquetexponentsandthemodalfunctionsofthedampedMathieuequation.AccordingtoFloquettheory,solutionstotheMathieuequationarecomposedofanexponentialpartandaperiodicpart[52].AnanalysisoftheFloquet-typesolutionsuggeststhattheresponsecanbe(i)quasi-periodic,(ii)periodic,or(iii)unstable.Asthesystemparametersvary,thesolutioncantravelbetweenthequasi-periodicandunstablezones,andatthetransitions,itisperiodic[53].Thereforethestabilityboundariescanbefoundbyassumingaperiodicsolution[46,5Howeverthisapproachdoesnotprovidetheresponseitself.InChapter4,insteadofassumingaperiodicsolution,theFloquet-typesolutioniskeptasis,andharmonicbalancemethodisapplied.Thisapproachispresentedtoapproximatethegeneralresponsestotheundamped,dampedandtwo-harmonicexcitationMathieuequations,aswellasthestabilityofthesolutions.TheapproachfollowedintheMathieuproblemisextendedtomulti-degree-of-freedomsystemswithparametricsti˙nessinorderto˝ndthesolutionsandthestabilities.Thisstudycanbeappliedtothree-bladeturbineequationsinordertounderstandthetransientdynamicsoftheturbine.1.2.4AerodynamicStallTheaerodynamicloadsdependontheangleofattack,thatistheanglebetweentherelativewindspeedandtheairfoilcross-section,asshowninFigure1.3.Forsmallangles,theliftforceincreaseswiththeangleofattackinalinear-likefashion.Howeveraroundacriticalangle(i.e.stallangle),theliftforcedropssuddenly.Thisphenomenoniscalledaerodynamicstall[57,58].Whentheangleofattackvariesslowly,aquasistaticaerodynamicloadcanbemodeled,wheretheliftforceisadirectfunctionofangleofattack.However,whentheangleofattack7Figure1.3Anairfoilcross-section.oscillates,theairfoilmightgounderdynamicloading,wheretheliftforceismuchmorecomplicatedanda˙ectedbythedynamicsoftheangleofattack.Indynamicstall,thereisahysteresisinlifttheforcecurveduringincreasinganddecreasinganglesofattack[2,59].Liftforceonanairfoilhasbeenstudiedmanyresearchers[2,4,Theodorsen[60]explainedamethodusingpotential˛owandKuttaconditiontodeterminetheaerodynamicforcesandmoments,andtostudy˛utterconditions.Petersetal.[62]developedatheorywherethetimehistoryofthecirculatoryliftistreatedasafeedbackinduced˛owontheblade.Alsovortexlatticemethod,wheretheairfoilisdividedintoa˝nitenumberofelementswithavortex,canbeusedtocalculatetheliftforcenumerically[61].Theliftfunctioncanbemodeledviasemi-empiricalmethods,wherethedynamicsofthefunctionismodeledwithdi˙erentialequations,parametersofwhichareestimatedexperi-mentally[2,4,63].InAppendixC,ONERA'sandLarsenetal.'ssemi-empiricaldynamicstallmodelsarereviewedindetail.1.3ThesisOverviewThisthesiscontainsmodelingofasinglewind-turbinebladeunderbend-bend-twistvibra-tions,andderivationofequationsofmotionusingLagrange'smethodtogetherwithassumedmodes.Alinearmodalanalysisisshownto˝ndthemodalfrequenciesaswellasthemodeshapesoftheblade.Bladesti˙nessvarieswiththerotorangleduetoparametrice˙ectsofgravity,whicharemoresigni˝cantforlongblades.Ananalysisonthee˙ectoflengthon8theparametricsti˙nessisprovided.Thenathree-bladeturbinemodelwhichrepresentsthecoupledblade-hubdynamicsisstudied.Onlyin-planebendingvibrationsaretakenintoaccount,andasinglecantileverbeammodeisassumedforeachblade.Thelinearizedbladeandrotorequationsarecoupledwitheachotherinthetimedomain.Todecoupletherotorequationfromthebladeequations,theindependentvariableistransformedfromtimetorotorangle,andascalingschemeandanondimensionalizationprocedurearefollowed.Themethodofmultiplescalesisappliedtoexplorethee˙ectsoftheparametricsti˙nessonthebladeresponse.ThisthesispresentsananalyticalapproachforapproximatingthegeneralresponsesoftheMathieuequation.Ageneralnon-periodicFloquet-typesolutionisassumed,andthefrequencycontentandthestabilityofthesolutionaredeterminedviatheharmonicbalancemethod.GeneralresponsesareconstructedbyusingthecharacteristicexponentsandFouriercoe˚cientsfoundthroughtheharmonicbalance.AnextensionoftheapproachtoMDOFsystemswithparametricsti˙nessisalsoprovided.TheresultsofboththeSDOFandtheMDOFexamplesareveri˝edthroughsimulations.Thismethodcanbeusedtoapproximatethetransientresponseofparametricallyexcitedturbineequations.Lastly,anextensivereviewonaerodynamicmodelingoftheliftforceisgiven.Thevariationsinangleoftherelativewindspeedduringbladeoscillationscanintroducedy-namicstall,wheretheliftforcedropsdramatically,andcausehysteresisintheliftforce.Thisphenomenoncanbemodeledbysemi-empiricalmethods,wheretheparametersinthemathematicalmodelsaredeterminedexperimentally.Acomprehensivereviewoftwooftheexistingsemi-empiricalmethodsaregiven.Thesemethodscanbeusedtobuildaframeworkforaforcedturbinemodel.91.4ContributionsThecontributionsofthisthesistotheliteratureare:-Theenergiesofslendernonuniformpretwistedbladeareformulatedforbend-bend-twistvibration,andarepresentedintermsoffamiliarbeamparameters.-Themodalequationsofmotionarefoundforabladeunderbend-bend-twistvibration,andthefrequenciesandmodeshapesarefoundbyapplyingamodalanalysis.Itisshownthatthedegreeofcouplingbetweenin-planeandout-of-planevibrationsdependonthebeamparameters.-Theparametricandthecentrifugale˙ectsinbladesti˙nessareinvestigated.Thesti˙nessvariationsarefoundtobemoreseriousforlongerblades.-In-planeequationsofmotionforathree-bladeturbinearederived.Thesecondaryresonancesintroducedbythedirectandtheparametrice˙ectsareinvestigated.-Anapproachisdevelopedtodetermineapproximategeneralresponsestotheun-damped,dampedandtwo-harmonicexcitationMathieuequations.Theapproachal-lowsustodeterminetheresponsefrequencycontentanddampingrate,andcanbeappliedtootherlinearsecondorderdi˙erentialequationswithperiodiccoe˚cients.TheapproachisalsoextendedtoMDOFsystemswithparametricsti˙ness.10CHAPTER2EQUATIONSOFMOTIONOFABLADEUNDERBEND-BEND-TWISTVIBRATIONS2.1IntroductionAhorizontalaxiswindturbinebladeisconsideredasacantileverbeamunderbend-bend-twistcoupledvibration,rotatingataconstantspeed.ThekineticandstrainenergiesareformulatedbythehelpofCrespodaSilva'smethod[25].Giventheenergyexpressions,CrespodaSilvathenusedHamilton'sprincipleto˝ndthepartialdi˙erentialequationsofmotionofabeamwithnogravitationalloading.Instead,inthisstudy,theassumedmodesmethodisused.Firsttheenergyexpressionsarewrittenintermsofassumedmodalcoordinates,andtheequationsofmotionarederivedbyLagrange'smethodappliedtotheassumedmodalcoordinates.Understeadyrotationale˙ects,theequationshaveparametrictermsinvolvingthehubangleandtheangularspeed.Thegravityintroducesperiodicsti˙nessterms,whichleadstoaforcedMathieuequationinasinglemodemodel[32],andtheangularspeedleadstoanincreaseinsti˙ness[64,65].Inthischapterthestructuralmodaldynamicsarestudied,separatefromthedynamice˙ectsoftheparametricterms.Assuch,theequationsarelinearizedandmodalanalysisisappliedto˝ndthemodalfrequenciesandmodeshapes.Themodeshapesarecomposedofbend-bend-twistdeformations,andthedegreeofcouplingdependsonsystemparameters,suchasthedegreeofstructuraltwistandtheproductmomentofinertiaofthecross-section.Thegoalofthisstudyistogaininsightonthreeaspectsofthestructuralmodalbehavior.Theaimistoexaminemodeshapestoevaluatetheindustry'sdistinctionbetween˛apwiseandedgewisemodes,quantifythee˙ectofangularrotorspeedonthemodalfrequencies,andstudythevariationine˙ectivesti˙nesswithvaryingrotorangleposition,asitwill11contributetoparametricexcitationsunderconditionsofsteadyrotation.Todemonstratethestrengthoftheparametricterms,existingblademodelsareanalyzedatdi˙erenthubangles,rotatingwithdi˙erentspeeds.Inordertoinvestigatetherelationsbetweenthebladesizeandthesigni˝canceoftheparametrice˙ects,existingblademodelsarescaledinvolume,bykeepingtheirshapethesame.Thedownward/uprightfrequencyratiosareinvestigatedforthescaledmodels,andfoundtobeincreasingalmostlinearlywiththebladesize.Asaseparatestudytheblademodelsarescaledinlengthonly,bykeepingtheirthicknessthesame,andthefrequencyratiosareestimatedasafunctionoflength.Thesestudiesshowthattheparametrice˙ectsbecomemoreimportantasthehorizontalaxiswindturbinebladesgetlarger.2.2MethodologyThewindturbinebladeismodeledasastraightbeamwithavaryingcross-section,underbend-bend-twistcoupleddeformation,asshowninFigure2.1.Thecoordinatesystem(^x;^y;^z)is˝xedtotheundeformedbladeatthe˝xedendwithitsoriginatanarbitrarypoint,whereas(^˘;^;^)is˝xedtoacross-sectioninthedeformedblade.Thepositionvectortotheoriginof(^˘;^;^)isr0=x^xbeforedeformation.Figure2.1Coordinatesystemsinthedeformedandundeformedblade.Thecombinedmotionismodeledasthreetranslationalandthreerotationaldisplace-12ments:u,v,warethetranslational,andx,y,zaretherotationaldisplacementsin^x,^yand^zdirections,respectively.Thecross-sectionofthebladeisassumedtoremainplanar,andthedisplacementsarefunctionsofxandtonly.vandwarebendingde˛ectionsinin-planeandout-of-planedirections,andyandzarethecorrespondingslopes,sotheycanbewrittenintermsofvandw.Thepositionvectortotheoriginof(^˘;^;^)isr1=x^x+v^y+w^zafterdeformation.Thebeamisassumedtobeinextensible,whichmeanstheaxialde˛ection,u,isonlyduetotheforeshorteningofbendingmotion.Itsslopecanbeexpressedthroughderivativesofvandwasu0=1p1v02w02;(2.1)where()0denotesthepartialderivativewithrespecttox.2.2.1EnergyFormulationsThestrainenergyisformulatedintermsofthedeformationsv,wandx,andtheirspatialderivatives,byusingtherelationsinCrespodaSilva'sstudy[25].Thestrainenergydensityatanarbitrarypointisfound˝rst,andthenthetotalstrainenergyiscalculatedbyintegratingtheenergydensitythroughthevolume.To˝ndthestrainenergydensity,thestrainandstressdistributionsarefoundthroughthedisplacements.ThepositionofanarbitrarypointPbeforedeformationiswrittenasrP0=x^x+^y+^z,whereasitisgivenasrP1=(x+u)^x+v^y+w^z+^+^afterdeformation.ItisconvenienttoexpressrP1inthe(^x;^y;^z)coordinatesystem.Transformationfrom(^˘;^;^)to(^x;^y;^z)canbewrittenintermsoftherotationsas^=^y+x^zv0^xand^=^zx^yw0^x.ThereforerP1canbewrittenasrP1=(x+uv0w0)^x+(v+x)^y+(w++x)^z.UsingGreen'sformulathestrainstateatapointcanbewrittenintermsofrP0and13rP1[66,67],as2(dx)(")0BBBB@dx1CCCCA=drP1drP1drP0drP0;(2.2)where"istheLagrangianstraintensor.StraincomponentsareexpresseduptoquadratictermsinAppendixA.1.ThestressdistributionisfoundusingHooke'slaw,˙ij=Cijkl"kl,where"kl=12kl(fork6=l)withshearstrainskl.Thestrainenergyisexpressedwiththefollowingvolumeintegral[68]:U=12ZV˙xx"xx+˙"+˙"+˙+˙+˙dV:(2.3)Notethatthestraincomponents","andhaveonlyquadraticterms,asshowninAppendixA.1,andhencetheydonotappearinthestrainenergyexpressionofthelinearmodel.ThekineticenergyofthebladecanbewrittenasT=12ZV(vP1vP1)dm;(2.4)wherevP1=_rP1+_˚^zrP1.The˝rsttermcorrespondstothederivativeofrP1withrespecttotime,writtenintherotating(^x;^y;^z)coordinates,andthesecondtermstandsfortherotationofthecoordinatesystem.ThegravitationalpotentialenergycanbewrittenasVg=ZVrP1gdm:(2.5)SincerP1isgivenin(^x;^y;^z)coordinates,thegravityvectorisalsowritteninthesamecoordinates,g=g(cos˚^xsin˚^y),where˚istherotorangle,suchthat˚=0whenthebladeisindownwardposition,asshowninFigure2.1.Thereforethee˙ectofgravitychangeswiththerotoranglewhichresultsincyclicchangesinthebladesti˙ness.Allthreeenergyexpressionsarevolumeintegrals,thecalculationsofwhicharenotstraightforward.Yettheycanbereducedtoanareaintegralinsideanintegralalongthe14length.Sincethecrosssectionisassumedtobeplanar(nowarpingassumed)withshear-ingonlyduetox,thede˛ectionsarefunctionsofxonly(independentofand),andtheycanbepulledoutfromtheareaintegrals(e.g.RV_w(x;t)2dVcanbewrittenasRL0RA_w(x;t)2dAdx=RL0_w(x;t)RA2dAdx).Then,theareaintegralsareleftwithtermsinvolvingandonly,asgivenbelow:ZAdA=A;ZAdA=0A;ZAdA=0A;ZA2dA=I+20A;ZA2dA=I+20A;ZAdA=I+00A;whereI,I,Iaresecondmomentsofareaaboutthecentroid,and0and0arecomponentsoftheo˙setoftheoriginofthecoordinatesystem(O)fromthecentroid(C),asshowninFigure2.2.Fixingtheorigintothecentroid(i.e.0=0;0=0)greatlysimpli˝estheseexpressions.TheresultingpotentialandkineticenergiesaregivenforalinearmodelinAppendixA.2forthecentroid-basedcoordinatesystem.Figure2.2Bladecross-sectionbeforedeformation.Ashearforceappliedonacross-sectioncreatesatorsionalmomentiftheforceisnotappliedattheshearcenter.Formostairfoilcross-sections,theshearcenteriso˙setfromthecentroid.Therefore,onemightwishtowritetheenergyexpressionsfortheshearcenterdisplacements.Whenwrittenintermsofshearcenterde˛ections,kineticandpotentialenergieshaveafewextratermsinvolvingtheo˙setfromthecentroid,asshowninAppendixA.2.Thesetermsleadtotorsion-bendingcouplinginthekineticandgravitationalpotentialenergiesevenwhentherotorisnon-rotating(i.e._˚=0),whereascentroidbasedenergyformulationsdonothavetorsion-bendingcouplingwhen_˚=0.Theshearcenterbased15energyformulationswereconsistentwiththoseinWeaveretal.'sstudyoncoupledin-plane˛exuralandtorsionalvibrationsfornon-rotatingbeam(_˚=0)[69].2.2.2AssumedModesandEquationsofMotionInordertoderivetheequationsofmotion,energyexpressionsarewrittenintermsofmodalcoordinatesbyusingthemomethod,inwhichthedisplacementsareexpandedintermsofspatialtrialfunctions[70].Thede˛ections,v,wandx,arewrittenasv(x;t)=nXi=1vi(x)qvi(t);w(x;t)=nXi=1wi(x)qwi(t);x(x;t)=nXi=1i(x)qi(t);whereji(x)'sareassumedmodesortrialfunctions,qji(t)'saremodalcoordinatesandnisthenumberofassumedmodesineachcoordinate.Foreachdisplacement,cantileverbeammodesareassumed,suchthatvi(x)=coshkixlcoskixlrisinhkixlsinkixl;(2.6)wi(x)=coshkixlcoskixlrisinhkixlsinkixl;(2.7)i(x)=sinxˇ2l:(2.8)Thede˛ectionsarewrittenintermsofassumedmodesandinsertedintotheenergyexpres-sions.TheLagrangianisfoundintermsofthemodalcoordinates,andthen,theequationsofmotionarefoundbyapplyingLagrange'sequationstothemodalcoordinates.Forthecaseofonlyoneassumedmodeineachdirection,theequationsofmotionaregiveninAppendixA.3.Formostairfoilcross-sections,theproductmomentsofareaandinertia(I;J)arenonzero,andthisleadstoacouplingbetweentheqvandqwequa-tions.Furthermorethetorsionequationiscoupledwiththebendingequationsthroughthegyroscopicterms.162.2.3ModalAnalysisTheequationsofmotionareputintomatrixform,givenas266664MvvMvw0MvwMww000M3777750BBBB@qvqwq1CCCCA+26666400Gv00GwGvGw03777750BBBB@_qv_qw_q1CCCCA+266664KvvKvw0KvwKww000K3777750BBBB@qvqwq1CCCCA=0BBBB@QvQwQ1CCCCA;(2.9)whereqi'sarevectorsconsistingassumedmodalcoordinates,Qi'sarecorrespondinggener-alizedforcingterms,andMij,GijandKijareblockmatricesdenotingcouplingbetweencoordinatesiandj.ThemassmatrixisaconstantmatrixwhereasGij'scontain_˚andthesti˙nessmatrixhasparametrictermsinvolving˚and_˚.Forasystemwithparametricsti˙-ness,˝ndingthenaturalfrequenciesandmodeshapesisnotstraightforward.Parametricallyexcitedsystemswithdirectforcinghavebeenstudiedusingvariousperturbationmethods,includinganapplicationofsecondordermultiplescales[71],averaging[37],themethodofvanderPol[34],harmonicbalance[46]andsystemidenti˝cationmethods[51].Inthischap-ter,modalresponseofasinglebladeisstudied,andinsteadofworkingondynamice˙ectsofparametricterms,thehomogeneoussystemissolvedassumingsteadyconditions.Theaimistoseewhethertheparametrictermsmaybestrongenoughtorisktheparametricresonancesuncoveredinthepreviousworks,andalsotoevaluatethedistinctionbetween˛apwiseandedgewisemodes.Amodalanalysisisappliedtodi˙erentanglesnapshotsofabladerotatingataconstantspeedinorderto˝nditsnaturalfrequenciesandmodeshapes.ThegeneralizedforcesQiaccommodateaeroelasticloadinganddampingterms.Al-thoughtheseareimportantfortheoperationofthewindturbine,wedonottreattheirdetailsforthismodalanalysiswork.Theequationofmotionwithaeroelasticandgravita-17tionalloadsisconsideredtobeoftheformMq+G_q+Kq=Qg(˚)+Q0+Q1(q;_q),suchthattheaerodynamicsisdominatedbymeanloadQ0andhasasmallvariation,inwhichthecyclicvariationisneglected.Onecanconsideranequilibriumde˛ectionsuchthatKq0=Qg(˚)+Q0+Q1(q0).Applyingasmalldeformationq1aboutequilibrium,suchthatq=q0+q1,theequationofmotionbecomesMq1+G_q1+Kq0+Kq1=Qg(˚)+Q0+(Q1(q0)+q1@Q1@q(q0)).Then,the˝rstapproximationisMq1+G_q1+Kq1ˇ0.Assuch,amodalanalysisisperformedbasedonthestructuralmodelinEquation(2.9).2.3ResultsThemodalanalysisexplainedintheprevioussectionis˝rstappliedtoahollow,pre-twistedbeamto˝ndthenaturalfrequenciesandmodeshapes.A˝niteelementanalysisisalsoconductedonthesamebeam,toverifythemethoddeveloped.Then,theanalyticalmethodisappliedtoexistingblademodels,andtheresultsarecomparedtothosefoundintheliterature.Themodalanalysisisconductedfordi˙erenthubanglesandrotorspeeds,andthee˙ectsoftheseparametersareinvestigated.2.3.1AHollowRectangularBeamwithaStructuralTwistApre-twistedbeamwithahollowrectangularcross-section,asshowninFigure2.3,wasanalyzed.Thebeamhasdimensions3m1.5m50mwith0.2mthickness,andstructuraltwistanglechanginglinearlywithaxialposition,givenbytherelation(x)=ˇx3l.Thematerialisassumedtobelinearlyelasticandisotropic,havingYoung'smodulusE=40GPa,Poisson'sratio=0:3,andmassdensityˆ=2500kg=m3.Themodalanalysisisappliedtobeaminhorizontalandverticalpositions,byassumingstaticconditions,suchthat_˚=0and˚=˚0.The˝rsttwonaturalfrequenciesandmodeshapeswerefoundbyassuminguptofourmodesineachdirection(Table2.1).Asthenumberofassumedmodesisincreased,thenaturalfrequenciesapproachedconvergenceto18Figure2.3Isometricviewofthehollowrectangularpretwistedbeam.their˝nalvalues.Then,analyzingthesamebeamona˝niteelementanalysissoftware,naturalfrequencieswerefound,anditisshownthattheanalyticalmodelpredictedthesamefrequencieswith<1%error,asgiveninTable2.1.n=1n=2n=3n=4FEA1stmodedownward0:61570:60920:60690:60690:6046horizontal0:60930:60270:60040:60040:59822ndmodedownward0:99900:98680:98390:98370:9784horizontal0:99510:98300:97990:97970:9744Table2.1Naturalfrequencies(Hz)oftherectangularpre-twistedbeam,=0Togiveanindicationofthesigni˝canceofthegravity'sparametrice˙ect,naturalfre-quenciesarefoundatdi˙erentrotationangles.Sincethegravityhassti˙eningandsofteninge˙ects,thebeam'snaturalfrequencieschangewiththerotorangle.Whenthebeamisup-right,˚=ˇ,thegravitationalforcecompressesit,andmakesitlesssti˙inbending.Whentherotorangleis˚=0,thegravitationalforcepullsthebeamandmakesitsti˙er.Thesevariationsinsti˙nesscanbeestimatedfromthenaturalfrequencies.Forabladewithelas-ticsti˙nessk0andparametricsti˙nessk1,theratioofthefrequenciesofdownwardandhorizontalorientationsis!d!h=sk0+k1k0;19where!dand!haredownwardandhorizontalbeamfrequencies,respectively.Thereforetheratiooftheparametricsti˙nesstoelasticsti˙nesscanbefoundask1k0=!d!h21:Forthe50mrectangularbeamthesti˙nesscontributionoftheparametrice˙ectisaround2:2%.The˝rsttwomodeshapesofthebeamareplottedanalyticallyforthedownwardposition,asshowninFigure2.4.Thereisnovisibledistinctionbetweenmodeshapescalculatedatdi˙erentrotorangles.Bothmodesarecoupledbendingmodes,andtheydonothaveanycontributionfromtorsionalmodes.The˝rstmodeisdominantlyinthein-planedirection,whereasthesecondmodeisintheout-of-planedirection.Themodeshapesarealsoplottedthrougha˝niteelementanalysis,asshowninFigure2.5,andtheyareconsistentwiththeanalyticalones.Inordertoinvestigatethecentrifugale˙ects,thebeamisanalyzedwithdi˙erenthubspeeds.The˝rstnaturalfrequencyincreasedby4%for_˚=1rad/sand15%for_˚=2rad/s,withthecorrespondingchangeinsti˙ness8:2%and32%,respectively.Thecentrifugale˙ectgetslargerasthebeamspinsfaster.Althoughingeneralhorizontal-axiswind-turbinesdonotspinathighspeeds,thecentrifugale˙ectsonthesti˙nessshouldbetakenintoaccountwhiledesigningturbineblades.n=3FEA_˚=10:62430:6219_˚=20:69040:6875Table2.2Naturalfrequencies(Hz)oftherectangularpre-twistedhorizontalbeam.Thesti˙nesscontributionsofthegravityandthecentrifugalforcearecarriedthroughtheinertialterms,ascanbeseeninequationsofmotioninAppendixA.3.Inordertoshowtheparametrice˙ectsonalesssti˙beam,thesamemodelisanalyzedwithhalftheYoung'smodulus,E=20GPa.TheresultingfrequenciesaregiveninTable2.3.Theparametric20Figure2.4Analyticalmodeshapesforthestationary50mbeamindownwardposition.sti˙nesscontributiontothemodalsti˙nessofthelowestfrequencymodeisaround4:3%,whichistwiceaslargeasthatofthebeamwithE=40GPa.Besides,for_˚=1rad/s,thesti˙nessincreasedby16:1%,whichisalmosttwicethatofthebeamwithE=40GPa.Thereforeonecandeducethattheparametrice˙ectsofthegravityandthecentrifugalforcearelargerforlesssti˙blades.n=3FEA_˚=0,downward0:43370:4321_˚=0,horizontal0:42460:4229_˚=1,horizontal0:45760:4557Table2.3Thelowestnaturalfrequency(Hz)oftherectangularpre-twistedbeamwithE=20GPa.21Figure2.5SideandtopviewsoftheFEAmodeshapesforthestationary50mbeamindownwardposition.2.3.2NREL's23mBladeTheNationalRenewableEnergyLaboratory's(NREL's)23mbladeisanalyzedusingn=4assumedmodes.DistributedbladeparametersandnaturalfrequenciesobtainedfromtheexperimentsaretabulatedinBirandOyague'sstudy[72].ThefrequenciescalculatedwithourmodelandNREL'sstudyaregiveninTable2.4.Inordertoseetheparametrice˙ectofgravity,thebladeisanalyzedbothinhorizontalanddownwardpositions,andtoseethecentrifugale˙ectswecomparedthestationaryandspinningbladefrequencies.1stmode2ndmodeNREL's1:722:41_˚=0,downward1:7672:523_˚=0,horizontal1:7602:519_˚=1,horizontal1:7652:526Table2.4Naturalfrequencies(Hz)forNREL's23mturbineblade.Forhorizontalanddownwardblades,variationinthe˝rstnaturalfrequencyisabout0:4%,correspondingtoa0:8%changeinmodalsti˙ness.Furthermore,whentheturbineisrotatingwith_˚=1rad=s,the˝rstnaturalfrequencyincreasesby0:3%,correspondingtoa0:6%increaseinmodalsti˙ness.Thesevariationsarenotsigni˝cantsincethecentrifugal22andgravitationale˙ectsaresmallerforshorterblades.Thereforeitisacceptabletoneglectthesee˙ectswhiledesigningshortblades.2.3.3NREL's5MWTurbineBladeNREL's5MWturbinebladeisbasedonamodelusedintheDutchO˙shoreWindEnergyConverter(DOWEC)project[7].Thebladeis61.5mlong,havingavaryingcross-sectionandastructuraltwist.Inorderto˝ndthesystemnaturalfrequencies,Jonkmanetal.[8]conductedalineareigen-analysisonADAMSandalsoonFASTforthehorizontalblade.ThesefrequenciesaregiveninTable2.5togetherwiththefrequenciescalculatedanalyti-callybyourmethod,bothindownwardandhorizontalpositions.Forthehorizontalblade,spinningbladefrequenciesarealsoprovidedtoshowthecentrifugale˙ects.1stmode2ndmodeNREL's0:66641:0900_˚=0,downward0:67361:1088_˚=0,horizontal0:66681:1052_˚=1,horizontal0:68201:1211[73],horizontal0:6980:975[73],upright0:951Table2.5Naturalfrequencies(Hz)ofNREL's5MWturbineblade.Themethodusedherepredictedthe˝rsttwofrequenciesofthestationaryhorizontalbladein>99%agreementwith[8].The˝rstnaturalfrequencydi˙ersby1%fordownwardandhorizontalblades.Thisimpliesa2%di˙erenceinmodalsti˙nessofthe˝rstmode.Thisprobablyindicatesaparametrice˙ectthatistoosmalltocontributesigni˝cantparametricresonances.Fora61.5mbladeturbine,therotorspeedisexpectedtobearound1rad/s.Forthisrotorspeed,the˝rstfrequencyincreasesby2:3%,correspondingtoa4:6%increaseinthemodalsti˙ness.Thee˙ectof_˚onthesecondmodeislesssigni˝cant.Boththegravity'sparametrice˙ectandthecentrifugale˙ectarelargercomparedtothosewefoundforthe23mblade,whichsupportstheclaimthatthesee˙ectsaremoresigni˝cantfor23largerblades.Chauhanetal.alsocalculatedthebladefrequenciesthroughsimulatingthestationaryturbine[73].Theyfoundtheedgewisesti˙nessvaries4:9%whichishigherthanourprediction(whichmightbelargeenoughtoinduceobservableparametricresonances).Thisisprobablybecausethetowerdynamicsareinvolvedintheresponseoftheuprightbendingmodeinsimulations.ThemodeshapesforNREL's61.5mbladearegiveninFigure2.6fordownwardblade.The˝rsttwomodesarecoupledinbendingdirections.The˝rstmodeisdominantlyinthe˛apwisedirection,whereasthesecondmodeisdominantlyintheedgewisedirection.Thisstudyshowsthatinsteadofstudyingthetwobendingdirectionsseparately,takingthecouplingintoaccountgivesmoreaccurateresults.However,themodeshapespresentedaredominatedbyanin-planeoranout-of-planede˛ection,andsoitisreasonableforengineerstospeakofin-planeandout-of-planemodes,atleastforlowordermodels.2.3.4Sandia's100mBladeSandiaNationalLaboratory(SNL)hasaprototypebladedesignfora100mblade.SNL's100mbaselinebladewasdevelopedbyscalingupNREL's61.5mblade'sdimensions.Scalingupalldimensionsbyafactorofcausesthehorizontalnaturalfrequenciestodropbyafactorof,althoughthescalingoftheparametrice˙ectislessclear.Theblade'sdistributedparametersaretabulatedinGri˚thandAshwill'sstudy[9].Thefrequenciesfordi˙erentbladeorientationsanddi˙erentrotorspeedsaretabulatedinTable2.6.1stmode2ndmode_˚=0,downward0:43190:7450_˚=0,horizontal0:42550:7417_˚=1,horizontal0:44530:7646Table2.6Naturalfrequencies(Hz)forSNL's100mturbineblade.The˝rstnaturalfrequencyvaries1:5%,betweenthehorizontalanddownwardpositions,andthiscorrespondsto3%changeinmodalsti˙ness.ItishigherthanthatoftheNREL's24Figure2.6FirsttwomodeshapesforNREL's61.5mbladewith_˚=0and˚=ˇ=2.61.5mblade,becauseforlongerbladesthebendingsti˙nessislowerwhilethegravitationalloadishigher.Thereforetherelativee˙ectofgravitationalforcesishigher.Also,for_˚=1rad=s,the˝rstfrequencyincreasesby4:7%,whichmeansa9:5%changeinthemodalsti˙ness.Whilethisdi˙erenceisalotlargerthanthatofthe61.5mblade,notethatthe100mbladewilltendtooperateatlowerrotorspeedsthanthe61:5mblade.Thisstudyshowsthatboththecentrifugalandgravitationale˙ectsonsti˙nessmustbetakenintoaccountwhiledesigninglargeblades.252.3.5TheRelationBetweentheBladeSizeandtheParametricE˙ectsAshorizontalaxiswindturbinebladesgetlargertheparametrice˙ectsbecomemoresig-ni˝cant.Inordertoquantifytherelationbetweenthesizeofthebladeandtheparametrice˙ects,thee˙ectofthegeometricparametersonthe˝rstnaturalfrequencyareinvestigated.Tosimplifythemodelforthisstudy,weconsideronlyin-planebending.Forabladewithmodalmassmb,elasticsti˙nessk0andparametricsti˙nessk1,the˝rstnaturalfrequencyinupright,horizontalanddownwardpositionsare!u=sk0k1mb;!h=sk0mb;!d=sk0+k1mb;wheremb/AL,k0/EIL3andk1/ALL,ascanbeseenfromtheequationsofmotioninAppendixA.3.2.3.5.1AScalingStudy:EachDimensionScaledEquallyForascaledupmodelwitheachdimensionincreasedbywhilekeepingthematerialprop-erties˝xed,themodalmassandsti˙nessesareincreasedbytherelationsmb=3mb;k0=k0;k1=2k1:Forexample,therectangularhollowbeammodelinSection2.3.1isscaledupbydoublingeachdimension(=2),andthenaturalfrequenciesareestimatedbyusingthescalingrelations.Alsoa˝niteelementanalysisisappliedto˝ndthecorrespondingfrequencies,andresultsarecomparedasgiveninthe˝rsttwocolumnsofTable2.7.Theestimationsareabletopredictthemodalfrequencieswith99:6%accuracy.Forthescaledupbeam,thevariationinthemodalsti˙nessforthehorizontalandthedownwardpositionsis4:3%whichisabouttwiceofthatfortheoriginalbeam(2:2%).Thereforeonecandeducethatasthebladesgetbiggerinsize,theparametrice˙ectsbecomemoresigni˝cant.26BeamL=2L,A=4AL=2L,A=Aangleest.FEAest.FEAdownward0:30550:30670:16250:1627horizontal0:29910:30020:15010:1502Table2.7Firstmodalfrequencies(Hz)forthescaleduphollowrectangularbeams:estima-tionsfromthescalingrelationsandFEAresults,for=2.2.3.5.2AScalingStudy:OnlytheLengthScaledAshorizontalaxiswindturbinebladesaredesignedlonger,theirthicknessisnotnecessarilychangedinthesameproportion,andismorelikelytochangeinasmallerproportionsuchthatthebladesbecomerelativelymoreslenderwithincreasingsize.Consideranotherspecialcaseofascalingpattern,wherethelengthofthebladeisscaledwhiletheotherdimensionsarekeptthesame.Thisleadstomb=mb;k0=k03;k1=k1:Asanexample,thelengthofthebeaminSection2.3.1isdoubled(=2),whilethecross-sectionalareaiskeptthesame.Themodalfrequenciesareestimatedbyusingthesescalingrelationsandarecomparedtotheonescalculatedviathe˝niteelementanalysis,asshowninthetworight-handcolumnsofTable2.7.Thevariationinthemodalsti˙nessforthehorizontalandthedownwardpositionsis17:2%whereasitis2:2%fortheoriginalbeam.Therefore,onecanconcludethattheparametrice˙ectsbecomeamoreseriousissueifbladelengthsincreaseproportionallymorethancross-sectionaldimensions.2.3.5.3AScalingStudy:NREL's5MWTurbineBladeToshowtherelationbetweenthebladesizeandtheparametrice˙ects,theratiooftheparametricandelasticmodalsti˙nessesk1=k0=!2d=!2h1isestimatedforscaledversionsoftheNREL's61:5mblade,asshowninFigure2.7.Therearetwocasesshown:1)thewholevolumeisscaled,2)onlythebladelengthisscaled.Weexpectarealistictrendtobesomewherebetweenthetwo.SNL's100mbladeandNREL's23mbladearealsoshownin27thisplot,andtheylayonthescaledvolumecurve.Thismakessenseforthe100mblade,sinceitisdevelopedbyscalingthewholevolumeofthe61.5mblade.Figure2.7Parametricsti˙nessratiosforscaledblademodelsandactualblademodels.2.4ConclusionBend-bend-twistbladevibrationequationsforahorizontalaxiswindturbinewerefoundbyusingLagrange'sequationsandtheassumedmodesmethod.Thekineticandthepotentialenergiesofthebladeasaninextensiblenonuniformstraightrotatingslenderbeaminbend-bend-twistdeformationwereexpressedintermsoffamiliarbeamparameters.Thecoupledequationsofmotionwerelinearized,andamodalanalysiswasappliedtotheunexcitedsystemto˝ndthenaturalfrequenciesandmodeshapes.Resultssuggestthatwhilethebend-bend-twistcoordinatesarecoupled,thelower˛exuralmodesaredominatedbyeither28in-planeorout-of-planede˛ections.Thus,itisreasonableforengineerstospeakofin-planeandout-of-planemodelsinapproximation,althoughbetteraccuracyisheldinthecoupledmodel.Thelinearizedequationshavesti˙nesstermsthatvarywiththeangularspeed,andhaveadependenceontherotorangle.Theanalysiswasappliedtoexistingblademodels,andthee˙ectsofrotorangleandrotationspeedonnaturalfrequencieswereinvestigated.Resultssuggestthattherotorangledependenceonsti˙nessonexistingblademodelsmaynotbestrongenoughtoinducesigni˝cantparametrice˙ects.However,thisresultisforbladesalone,andfactoringinthetowermayincreasetheparametrice˙ect.Thecentrifugalsti˙eningandparametrice˙ectsbecomemoresigni˝cantasthebladesgetlonger.Toshowthee˙ectofbladesizeontheparametrice˙ects,atwistedbeammodelwasscaledbychangingonlythelengthandalsobychangingthewholevolume.Scalingonlythelengthresultedindramaticchangesintheparametrice˙ects,whereasscalingthewholebladeresultedinalinear-likeincreaseintheparametrice˙ectswithincreasinglength.Realbladedesignsareexpectedtoscalesomewherebetweenthesetwotrends,suggestingthatverylongbladeswillhavesigni˝cantparametricsti˙nesse˙ects.29CHAPTER3THREE-BLADETURBINEEQUATIONSFORIN-PLANEBENDING3.1IntroductionDynamicsofathree-bladehorizontal-axiswindturbineisstudied.Thebladeandthehubequationsarefound˝rst.Then,applyingaperturbationmethod,steady-statedynamicsareinvestigated.TheenergyexpressionsderivedinChapter2wereapproximatedbyassumingasingleuniformcantileverbeammodeforeachblade.Towermotionisneglected,andtheequationsforbladesandthehubwerederivedbyusingLagrange'smethod.Thetangentialandnormalcomponentsofthegravityforceactingonasinglebladechangeperiodicallywiththebladeangle.Thiscausesacyclicvariationinthee˙ectivebladesti˙ness.Astherotorspins,centrifugalforcesintroduceasti˙eninge˙ect.Furthermore,windspeedusuallyvarieswithaltitude,whichcausestheamountofwindforceappliedonabladetochangeperiodicallyasthebladerotates.Theseparametricsti˙nessanddirectforcinge˙ectsweretakenintoaccount.Theindependentvariableischangedfromtimetorotorangle,forconvenience.Then,anon-dimensionalizationandascalingprocedureareappliedtodecouplethebladeequationsfromtherotorequations.A˝rst-ordermethodofmultiplescalesisappliedtothecoupledthree-bladeequations.Theparametricanddirectexcitationofgravityintroducesasuperharmonicresonance,andtheparametricexcitationleadstoasubharmonicresonance.Thesteady-stateamplitude-frequencyrelationsarefoundforeachcase,andthestabilitiesofthesolutionsarestudied.303.2Three-BladeTurbineEquationsThetotalenergyoftheturbineiswrittenbyusingtheenergyexpressionsforasingleblade,asfoundinChapter2,andassumingasinglecantileverbeammodeforeachblade.Asimpli˝edmodelisusedwhereonlyin-planevibrationsaretakenintoaccountasshowninFigure3.1,andtowermotionisneglected.Thehubismodeledasanunrestrainedrigidbodyina˝xedaxisrotationwithdampingthatcoarselyaccomodatesenergyremoval.ThetotalkineticandpotentialenergiesofthesystemcanbewrittenintheformTT=12Jhub_˚2+3Xj=1T(qj;_qj;_˚);(3.1)UT=3Xj=1U(qj);(3.2)VgT=3Xj=1Vg(qj;˚j):(3.3)Thefour-degree-of-freedomsystemhasthestatevariablesq1,q2,q3and˚,whereqjaretheassumedin-planemodalcoordinatesofeachblade,and˚istherotorangle.˚jisthehorizontal-axisrotationangleofthejthblade,whichdi˙ersfrom˚byaconstant(i.e.˚j=˚+2ˇ3j).TheenergyexpressionsforasinglebladearegiveninAppendixB.1.ApplyingLagrange'sequations,thebladeandtherotorequationsofmotionarefound.Thebladeequationsarecoupledthroughtherotorequation.Thelinearizedequationsofmotionforthejthbladeandfortherotorare,(forj=1,2,3)mbqj+cb_qj+(k0+k1_˚2+k2cos˚j)qj+dsin˚j+e˚=Qj;(3.4)Jr˚+cr_˚+3Xk=1(dcos˚kqk+eqk)=Q˚;(3.5)wherembistheinertiaofasinglebladeabouttheaxisofitsownunde˛ectedshape,asshownwithdashedlinesinFigure3.1,Jristhetotalinertiaofthreebladesplusthehubabouttheshaftaxis,k0istheblades'elasticsti˙ness,andk1andk2aresti˙nesscontributionsof31Figure3.1Athree-bladeturbinewithbladesunderin-planebending.centrifugalandgravitationale˙ects.Thecentrifugalsti˙nesstermk1hasasti˙eningandasofteningpart[74],asshownintheAppendixB.2,butitssti˙eningpartovercomesthesofteningpartforthecantileverbeammodeshapeweused.QjandQ˚aregeneralizedforcingtermsduetoaeroelasticloading,andcbandcraredampingcoe˚cientswhichareyettobedetermined.Thesecoe˚cientsaremodalparameterswhichdependontheassumedmodeshapes,andareexpressedintheAppendixB.2.Thezero-gravitysystemhasmodalfrequencies!n1=0withmodeshapev1=(0;0;0;1)(rigidbodyrotation),!n2;3=rk0+k12mb(frequencyofasingleblade)withv2=(1;1;0;0)andv3=(1;0;1;0),!n4=rk0+k12mb3e2=Jr(withcouplinge)withv4=(Jr3e;Jr3e;Jr3e;1).Blade-rotormodeshapesareshowninFigure3.2.Figure3.2In-planemodeshapesofathree-bladeturbine.Therotorspeed_˚isnotconstant,butassumedtovaryasmallamountaroundamean32value.Weintroduce=_˚=,whereisthemeanspeed,andchangetheindependentvariableto˚,asdone,forexample,intheanalysisofcentrifugalpendulumvibrationabsorbersystems[26,29,30].Thisresultsinthederivativerelationsddt=d˚dtdd˚andd2dt2=d˚2dd˚+22d2d˚2.Re-writingEquations(3.4)and(3.5)with˚astheindependentvariable,onecanobtain2d2qjd˚2+d˚dqjd˚+~cbdqjd˚+(~k0+~k12+~k2cos˚j)qj+~dsin˚j+~d˚=~Qj;(3.6)d˚+~cr+˜3Xk=1~dcos˚kqk+~e(2d2qkd˚2+d˚dqkd˚)=~Q˚;(3.7)where~cb=cbmb;~k0=k0mb2;~k1=k1mb;~k2=k2mb2;~d=dmb2;~Qj=Qjmb2;~e=emb;~cr=crJr;˜=mbJr;~Q˚=Q˚Jr2:Thequantityd˚accountsforthevariationsintherotorspeed,anditcanbethoughtofasthedimensionlessrotoracceleration(dt=d˚d˚dt=d˚)).ThiscanbeseenbyconsideringthesummationinEquation(3.7),whichstandsfortheloadsappliedbythebladesontherotor.TherotorinertiaJrincludesboththehubinertiaandtheinertiaofthebladesaboutthehubwhenundeformed,whereasmbistheinertiaofasinglede˛ectedbladeaboutitsownneutralaxis.Wede˝neasmallparameteras=mb=Jr.Thisisalegitimateassumption,sincev(x)canbescaledsuchthatmb<<Jr.Inordertodecouplethebladeequationfromtherotorequation,thefollowingscalingrelationsareused:=1+21;~cb=^cb;~k2=^k2;~d=^d;~cr=2^cr;˜=qj=j;~Qj=^Qj~Q˚=2^Q˚:33Parameters~k1and~k2arethesti˙nesstermscontributedbycentrifugale˙ectsandgravity,andaresmallcomparedtotheconstantsti˙nessterm,~k0.ThesephysicalparameterscanbefoundintheAppendixB.2.AnalternativescalingschemeisstudiedinAppendixB.3where=1+1,anditisshownthatthescalingusedinthischapter(=1+21)isvalid.Equations(3.6)and(3.7)arerewrittenintermsofthescaledbladecoordinates,sj,andthescaledhubcoordinate,1,asd2sjd˚2+^cbdsjd˚+(~k0+~k1+^k2cos˚j)sj+^dsin˚j+~e1d˚=^Qj+H:O:T:;(3.8)1d˚+^cr+3Xk=1(^dcos˚ksk+~ed2skd˚2)=^Q˚+H:O:T:(3.9)whereH:O:T:standsforhigher-orderterms.ObtainedfromEquation(3.9),1d˚ispluggedintoEquation(3.8)toobtaind2sjd˚2+^cbdsjd˚+(~k0+~k1+^k2cos˚j)sj+^dsin˚j+~e"^Q˚^cr3Xk=1(^dcos˚ksk+~ed2skd˚2)#=^Qj+H:O:T:(3.10)Forcingterms^Qjand^Q˚accommodateaerodynamicloadings,andareassumedtohavesmallvariations,sothat^Qj=Qj0+^Qj1(˚)and^Q˚=Q˚0+˚1(˚).Forthisanalysis,weuseasimpli˝edmodelwherethe˛owisassumedtobesteady,andthewindspeedisassumedtobeslightlyincreasinglinearlywithheighth(i.e.uwind=u0+1=u0cos˚ju1).Thebladespeedisgovernedbytherotorspeed(i.e.ublade=_˚x).Thereforetherelative˛owvelocityconsistsof˚and_˚terms.Assumingconstantangleofattack(i.e.neglectingthecontributionofstatevariationsintheangleofattack),theliftforceisproportionalto34u2rel(i.e.^Qj/u2rel).Therefore,theassumedformoftheaerodynamicforceis^Qj=Qj0+j1cos˚j:(3.11)Thedetailsbehindthisassumedformin^QjaregivenintheAppendixB.2.Byneglectingthestate-variablecontributionsto^Qj,weareseekingtomodeltheleading-ordere˙ectsofresonance,andarenotseekingtodescribeotheraeroelasticinstabilitiesorself-excitatione˙ects.Recallthatthebladeanglesaresuchthat˚j=˚+2ˇ3j.ThenpluggingEquation(3.11)intoEquation(3.10)andarrangingterms,thebladeequationisobtainedasd2sjd˚2+^k0sj=Qj0^dsin˚+2ˇ3j+Qj1cos˚+2ˇ3j~eQ˚0+~e^cr^cbdsjd˚^k2cos˚+2ˇ3jsj+~e23Xk=1d2skd˚2#;(3.12)where^k0=~k0+~k1,andthetermwith^k2istheparametricexcitationassociatedwithgravity.Themodalordersoftheunexcitedangle-basedsystemequationsarep1;2=rk0=2+k1mb,p3=rk0=2+k1mb3e2=Jr.Thesearescaledversionsofthetime-basedsystemfrequencies,suchasp1;2=!n2;3=andp3=!n4=.Noticethatthecouplingbetweenthebladesisthroughtheinertialterms.Onewaytohandlethisistouseacoordinatetransformationtomaketheequationscoupledthroughsti˙nessterms,andthenapplyaveraging.ThiscanbedonebymakinguseofFouriermatrix,asexplainedin[28].Oronecanusethemethodofmultiplescalesto˝ndtheslow˛owrelations.Thelatterisusedinthisstudytoinvestigatetheinternalresonancesonebyone.353.3ApplicationofMethodofMultipleScalesWerescaletheindependentvariablefrom˚to =p1˚,wherep1=q^k0=rk0=2+k1mb.Theequationsofmotionin ares00j+sj=Fj0sinr1 +2ˇ3j+Fj1cosr1 +2ˇ3j+fs0jcosr1 +2ˇ3jsj+~e23Xk=1s00k#;(3.13)where()0standsford=d andFj0=Qj0=^k0;=^d=^k0;r1=1=p1;Fj1=Qj1=^k0;f=~e(^crQ˚0)=^k0;=^cb=p1;=^k2=^k0:Equation(3.13)isbothdirectlyexcitedwithconstantloadFj0andcyclicloadstrengthsandFj1,andparametricallyexcitedwithstrength.Thequantityr1isascaledandisgivenbyr1=!n2;where!n2=rk0+k12mbisamodalfrequencyoftheturbine.Thereforevariationsinmeanrotorspeed,,willresultinvariationsintheexcitationorderr1.Toinvestigatethesteadystatedynamicsofinterdependentbladeequations,a˝rstordermethodofmultiplescalesisapplied[38].sjisdividedintoadominantpartandacorrec-tion,witheachpartdependentonawandaassj=sj0( 0; 1)+j1( 0; 1),where 1=0.Derivativerelationsared=d =D0+1whereD0=@=@ 0andD1=@=@ 1.ThisformofsjisappliedtoEquation(3.13),andcoe˚cientsoflikepowersofarecollectedbelow.360equation:D20sj0+sj0=Fj0sinr1 0+2ˇ3j(3.14)1equation:2D0D1sj0+D20sj1+sj1=Fj1cosr1 0+2ˇ3j+fD0sj0cosr1 0+2ˇ3jsj0+~e23Xk=1D20sk0(3.15)SolvingEquation(3.14),sj0isfoundassj0=Fj02+Ajei 0+iBei(r1 0+2ˇ3j)+c:c:(3.16)whereAj=12ajeijiscomplex(suchthatAjei 0+c:c:=ajcos( 0+j))andB=2(1r21).PluggingEquation(3.16)andtherelations,D0sj0=iAjei 0Br1ei(r1 0+2ˇ3j)+c:c:;andD0D1sj0=iA0jei 0+c:c:,intoEquation(3.15),weobtainthegoverningequationforsj1:D20sj1+sj1=Fj12ei(r1 0+2ˇ3j)+f22iA0jei 0Fj02ei(r1 0+2ˇ3j)iB2e2i(r1 0+2ˇ3j)Ajiei 0r1Bei(r1 0+2ˇ3j)2Ajei(r1+1) 0+i2ˇ3j+Ajei(r11) 0+i2ˇ3j+~e23Xk=1Akei 0ir21Bei(r1 0+2ˇ3k)+c:c:(3.17)373.3.1NonresonantCaseEquatingthetermsthatareleadingtoanunboundedsolution(i.e.secularterms)tozero,thesolvabilityconditioncanbefound.Foranarbitraryr1,2iA0jiAj~e23Xk=1Ak=0:InCartesianform,Aj=Xj+iYjisinserted,andrealandimaginarypartsareseparatedasRepart:Y0j=2Yj+~e223Xk=1Xk;(3.18)Impart:X0j=2Xj~e223Xk=1Yk:(3.19)Inmatrixform0BBBBBBBBBBBBBB@X01X02X03Y01Y02Y031CCCCCCCCCCCCCCA=12266666666666666400~e2~e2~e200~e2~e2~e200~e2~e2~e2~e2~e2~e200~e2~e2~e200~e2~e2~e20037777777777777750BBBBBBBBBBBBBB@X1X2X3Y1Y2Y31CCCCCCCCCCCCCCA:(3.20)Atsteadystate,thesolutionisXj=0andYj=0.Eigenvaluesofthecoe˚cientmatrixare1;2;3;4=,5;6=i3~e2.Forall,Re()0.Thereforethenonresonantsolutionisstable.383.3.2SuperharmonicResonance(2r1ˇ1orˇ!n2)2r1=1+Superharmonicresonancesmaybeofsigni˝cancetowindturbineblades,astheturbinesaredesignedtooperatewellbelowthebladesfundamentalmodalfrequencies.Asuper-harmonicoforder1=2maystillbeoutsideoftypicaldesignranges,butitispossiblethatthesetrendsarepushedastheturbinesizescontinuetoincreaseorifoperatingregimesarepushed.Furthermore,studieshaveshownthatsuperharmonicsoforder1=3canberevealedinsingle-modebladedynamicsbysecond-orderperturbationanalyses[71].Inanycase,itisimportantfordesignerstobeawareofsuperharmonicbehavior.Furthermore,theymayalsobeexcitedbyhigherharmonicsofcyclicexcitations.Equatingthetermsthatareleadingtoanunboundedsolution(i.e.secularterms)tozero,thesolvabilityconditioncanbefoundfor2r1ˇ1as2iA0jiAjiB2ei(˙ 1+4ˇ3j)~e23Xk=1Ak=0:PuttingAj=jei(˙ 1+4ˇ3j)withj=Xj+iYj,wecanseparatetherealandimaginarypartsasbelow.Repart:Y0j=˙Xj2Yj+~e223Xk=1Xkcos4ˇ3(kj)Yksin4ˇ3(kj);(3.21)Impart:X0j=˙Yj2XjB4~e223Xk=1Xksin4ˇ3(kj)+Ykcos4ˇ3(kj):(3.22)AtsteadystatethesolutionisXj=B22+4˙2,Yj=B2+4˙2,whichmeansAj=+i2˙22+4˙2Bei(˙ 1+4ˇ3j).Eigenvaluesofthecoe˚cientmatrixare1;2=2i˙,3;4=2+i˙and5;6=2i(2˙3~e2)2.Forall,Re()0.Thereforethesolutionisstable.39TheamplitudeandthephaseofAjarejAjj=jBj2p(2+4˙2);(3.23)\Aj=arctan2˙+˙ 1+4ˇ3j;(3.24)andthehomogeneoussolutionforthejthbladeissjh=acos( 0+˙ 1++4ˇ3j);(3.25)wherea=2jAjj=jBjq(2+4˙2),and=arctan2˙.Since1+=2r1,forthespeci˝csuperharmoniccasethehomogeneoussolutioncanalsobewrittenassjh=acos(2r1 0++4ˇ3j);(3.26)andin˚domainsjh=acos(2˚++4ˇ3j):(3.27)Thesystemhasaunisonamplituderesponse.Theamplitudedoesnotdependonthecouplingterm,~e,anditisdirectlya˙ectedbythedirectandtheparametricexcitationterms,Band.ThepeakamplitudeisAp=jj=3.Amplitude-frequencyplotsaregiveninFigure3.3andFigure3.4.Decreasingsharpenstheresonance,whileincreasingjjraisesthewholecurve.Sincevariationsin1areorderofand_˚=+21),thesevariationswillap-pearin_˚withorderof3.Thereforetherotoranglecanbeapproximatedas˚=t+311cos((p1+t).Intimedomain,thebladeresponsecanbeapproximatedbyqj=(Fj02Bsint+2ˇ3j)+acost++4ˇ3j))+O(4):(3.28)Inordertoinvestigatetheresponseoftherotor,thesolutioniswrittenin˚domainassj=Fj02Bsin(˚+2ˇ3j)+acos(2˚++4ˇ3j),andinsertedintoEquation(3.9).This40Figure3.3Steadystatesuperharmonicresonanceresponseamplitude,frequencyplotsfor=1,=4,=0:05;0:1;0:2.leadsto1d˚=^Q˚^cr+^d sin(˚)3Xk=1Fk0sin(2ˇ3k)cos(˚)3Xk=1Fk0cos(2ˇ3k)32acos(3˚+)!:(3.29)TheleadingordercomponentofthenondimensionalizedaerodynamicforceFk0=Qk0=^k0isthesameforallk,ascanbeseenintheAppendixB.2.PluggingFk0=F0intotheaboveequation,weobtain1d˚=^Q˚^cr32^dacos(3˚+):(3.30)41Figure3.4Steadystatesuperharmonicresonanceresponseamplitude,frequencyplotsfor=0:05,=1,=1;2;4.3.3.3PrimaryResonance(r1ˇ1orˇ!n2)r1=1+Bydesign,turbinesareunlikelytooperatenearprimaryresonanceexceptinraresitua-tionssuchasrunawayrotorspeeds.Inordertoinvestigatetheprimaryresonanceresponse,theharmonicforcingismodeledaseak(i.e.=^).ThischangestheformsofEquations(3.13)and(3.14),andleadstoasolutionintheformsj0=Fj02+Ajei 0(3.31)andan1equationoftheform2D0D1sj0+D20sj1+sj1=Fj1cosr1 0+2ˇ3j^sinr1 0+2ˇ3j+fD0sj0cosr1 0+2ˇ3jsj0+~e23Xk=1D20sk0:(3.32)42Equatingthetermsthatareleadingtoanunboundedsolution(i.e.secularterms)tozero,thesolvabilityconditioncanbefoundforr1ˇ1as2iA0jiAj+ Fj12Fj02+i^2!ei(˙ 1+2ˇ3j)~e23Xk=1Ak=0:PuttingAj=jei(˙ 1+2ˇ3j)withj=Xj+iYj,wecanseparatetherealandimaginarypartsasbelow.Repart:Y0j=˙Xj2Yjc1+~e223Xk=1Xkcos2ˇ3(kj)Yksin2ˇ3(kj);(3.33)Impart:X0j=˙Yj2Xj^2~e223Xk=1Xksin2ˇ3(kj)+Ykcos2ˇ3(kj);(3.34)wherec1=Fj14Fj04.AtsteadystatethesolutionisXj=^+4˙c12+4˙2,Yj=2^˙2c12+4˙2,whichmeansAj=r^2+4c212+4˙2ei(˙ 1+2ˇ3j+),where=tan12c12^˙^+4˙c1.Theequationshavethesameeigenvaluesasthoseinthesuperharmoniccase.Thereforetoa˝rst-orderapproximationthesolutionisstable.Single-degree-of-freedomparametricallyexcitedsystemscanhaveslenderinstabilitywedgesnearprimaryresonance,butasecond-orderMMSwouldbeneededtouncoverit[38,71].Thuswehavenotruledoutthatsuchaphenomenonexistsinthissystemaswell.Inordertoinvestigatetheresponseoftherotor,thesolutionsj=Fj0+acos(˚++2ˇ3j)isinsertedintoEquation(3.9),wherea=2pX2+Y2and=tan1(YX).Thisresultsin1d˚=^Q˚^cr32^dacos:(3.35)Notethat1standsforthesmalloscillationsfromthemeanspeed.Thereforespeedvariationsintroducedbytheconstantterms(directforcing)canbeconsideredasavariationinthemeanspeed.So,onecanconcludethatthebladeresponsesdonothaveanye˙ectondynamics.433.3.4SubharmonicResonance(r1ˇ2orˇ2!n2)r1=2+Subharmonicresonancesarewelloutoftherangeofturbinedesigns.However,thephenomenaareofgeneraldynamicalinterest,andarethereforeincludedhere.Equatingthetermsthatareleadingtoanunboundedsolution(i.e.secularterms)tozero,thesolvabilityconditioncanbefoundforr1ˇ2as2A0jiiAjAj2ei(˙ 1+2ˇ3j)~e23Xk=1Ak=0PuttingAj=jei(˙ 12+ˇ3j)withj=Xj+iYj,wecanseparatetherealandimaginarypartsasbelow.Repart:Y0j=˙2Xj2Yj+4Xj+~e223Xk=1hXkcosˇ3(kj)Yksinˇ3(kj);(3.36)Impart:X0j=˙2Yj2Xj+4Yj~e223Xk=1hXksinˇ3(kj)+Ykcosˇ3(kj):(3.37)Equations(3.36)and(3.37)areautonomous.Atsteadystate,thesolutionisXj=0,Yj=0.Eigenvaluesofthissystemare1;2=2p24˙24,3;4;5;6=2r29~e4(2˙3~e2)26~e2q(2˙3~e2)224.ThesolutionisstableifRe()0forall,andunstableifRe()>0foratleastone.StabilitycurvesofthissystemaregiveninFigure3.5andFigure3.6.Whenthereisnocouplingbetweenblades(i.e.~e=0),thesysteme˙ectivelyisofasingledegreeoffreedom,andthereisonlyoneinstabilitywedgestartingat˙=0,boundedbythe˙=2lines(Figure3.5).ThiscorrespondstothewellknownsubharmonicinstabilitywedgeasseenintheMathieuequation[75].44Howeverwhenthereiscoupling(Figure3.6),asecondinstabilityregionemergesfromthe˙=32~e2point,boundedbylinesde˝nedaccordingto˙=32~e22:(3.38)Thereforecouplingbetweenthebladesintroducesanewinstability.Formulti-degree-of-freedomsystems,theobservedinstabilitywedgeshavebeengenerallyidenti˝edwith(!nj+!nk)=N,where!njand!nkaremodalfrequenciesandNisapositiveinteger[76](SeealsoChapter5).ThesystemgiveninEquation(3.13)hasthemodalordersrn1;2=1andrn3=1+32~e2+O(2).ThesecondinstabilitywedgeinFigure3.6isbasedat˙=32~e2.Forthesubharmonicresonanceatr1=2+,thiscorrespondstor1=2+32~e2=rn1+rn3.Inordertoverifytheanalyticalresults,Equation(3.13)wassimulatednearr1=2,andstabilityofthesolutionwasstudied.Thesystemwassimulatedfrom =0to500,overagridonthe!plane.Asastabilitycriterion,wecheckedifthenormofthesjat =500wassmallerorlargerthanthatoftheinitialconditions(i.e.stableifsj(500)<sj(0),unstableotherwise).SimulationresultsareshowninFigures3.5and3.6,anditcanbeseenthattheanalyticalresultsfromperturbationanalysiswithasmallparameterareconsistentwiththesimulations.3.3.5ExistenceofResonanceConditionsTheresonanceconditionsfoundfromEquation(3.17)are!n2ˇ=2,,,where!n2=rk0+k12mb.Existenceoftheresonanceconditionsdependsontheparametersinvolvedinthemodalfrequency.For!n2=(where=1=2forsuperharmonic,1forprimary,and2forsuperharmonicresonances),wehavek0+k12mb=22;(3.39)whichleadstotheexcitationfrequencycondition=sk0mb2k1:(3.40)45Figure3.5Stabilityplotsforthesubharmonicresponsefor~e=0,=0;0:25;0:5.Simula-tionsaredonefor=0:5,=0:1,Fj0=0:1,Fj1=0:1,=1:5,f=0:1,(+):stable,():unstable.Thereforefortheresonanceconditionassociatedwithtoexist,k1<mb2musthold.AnexampleCampbellplotisgiveninFigure3.7.Itshows!n2asafunctionof,andthefrequenciesthatareexcitingresonanceconditionscanbefoundbylookingattheintersectionsoflineswiththe!n2curve.Ifalinedoesnotcrossthe!n2curve,thentheresonanceconditionassociatedwithitdoesnotexist.3.4ConclusionsInthisstudy,in-planevibrationsofathree-bladewindturbineandhubwereinvestigated.Therotorwasmodeledasarigidinertiawithnosti˙ness,andthebladeswereeachmodeledwithasinglein-planemode.E˙ectsofgravityandcentrifugalforcesweretakenintoaccount.Thebladeandtherotorequationswereputinanangle-dependentformforconvenience.Thebladeequationswerethenseparatedfromtherotorequation.Inthebladeequations,gravityintroducesdirectandparametricexcitationterms,whichleadtoasuperharmonicandasubharmonicresonance,respectively.Theamplitudeand46Figure3.6Stabilityplotsforthesubharmonicresponsefor~e=1,=0;0:25;0:5.Simula-tionsaredonefor=0:5,=0:1,Fj0=0:1,Fj1=0:1,=1:5,f=0:1,(+):stable,():unstable.Figure3.7Campbellplotshowing!n2asafunctionof,fork0=1,k1=0:1,mb=1.phaseequationswerefoundfortheprimary,superharmonicandsubharmonicresonances.Theseequationswerethensolvedanalyticallyforeachcase.Fortheprimaryandsuperhar-monicresonances,thebladespossessaunisonamplituderesponsewhichisstableforbothcases.However,forthesubharmoniccase,thebladeshaveazeroresponse.Stabilityofthis47solution,andthee˙ectofcouplingbetweenthebladeswereinvestigated.Theseresultswerethenveri˝edwithsimulations.48CHAPTER4ANANALYSISOFTHEMATHIEUEQUATION4.1IntroductionAlinearoscillatormodelwithperiodiccoe˚cientsintheinertial,damping,andsti˙nesstermsisknownasInce'sequation.Iftheperiodiccoe˚cientisonlyinthesti˙nesstermandthereisnodamping,thesystemcanbewrittenasaHill'sequation:x+f(t)x=0;(4.1)wheref(t)isperiodic.Iff(t)=1+cos!t;Hill'sequationreducestothenondimensionalMathieuequation,where!isthenondimensionalexcitationfrequency,andistherelativestrengthofparametricexcitation.TheMathieuequationisamenabletoFloquettheory,whichcanbeappliedtosystemsoftheform_x=A(t)x;(4.2)wherexisanN1vectorandA(t)isanNNmatrixthatisperiodicwithperiodT.AccordingtoFloquettheory,Equation(4.2)hassolutionsoftheformxr=e^rtpr(t),forr=1;:::;N,wherepr(t)isanN1vectorofperiodTor2T[75].Dependingonthenatureof^,solutionsmaybestableorunstable.WhenappliedtoHill'sequation,Floquettheorysuggeststhatontheboundariesbetweenthestableandunstableregions,thesolutionisperiodicwithperiodTor2T[53].Thereforethemostcommonapproachtotheproblemistoassumeaperiodicsolution(^=0),and˝ndtheconditionsonparametersforexistenceofthesolutionbysolvingaHill'sdeterminant[46,Theseconditionsresultintransitioncurvesbetweenstableandunstableregionsontheparameterplane[38,52,49Inthisstudy,insteadofassumingaperiodicsolutionfordescribingtransitionalbehavior,thegeneralFloquetsolutioniskeptasitis,withanonzero^andaperiodicpartcomposedofaseriesofharmonics.Forasingle-degree-of-freedomsystem,suchastheMathieuequation,thedisplacementvariablexhastwoindependentsolutionsoftheformxr=e^rtpr(t)=e^rtP1j=c(r)jeij!t,forr=1;2.A˝niteseriestruncationwouldrepresentanapproximation.Indeed,thefullin˝niteseriessolutionwasusedbyMcLachlan[54]andCoïssonetal.[85],to˝ndthestabilitycurvesthroughthetransitionsinvalues.Thecharacteristicequationanditsrootsforthesolutionwithin˝niteharmonicscanbefoundbyusingMcLachlan'scalculations[54],usingaprocedurethatinvolves˝ndingthedeterminantofanin˝nitematrix.Thesolutiontothein˝nitedeterminantisabstruse,anditmaynotbeamenabletootherFloquettypeequations,suchasthedampedMathieuequation,Ince'sequationsorparametricexcitationwithmultipleharmonics.Furthermore,itisnotconvenientforobtainingtheeigenvectors,norasolutionforthetimeresponse.To˝ndthetimeresponse,andtomakethemethodapplicabletootherproblems,andalsomoreaccessibletoengineersandscientistswithbroadbackgrounds,anapproximatetruncatedFloquetsolutionisapplied.Tocheckthevalidityofthetruncatedsolution,itscharacteristicexponentsarecomparedtothoseobtainedfromthein˝niteseriessolution.After˝ndingthattheresultsareconsistent,thetruncatedsolutionisappliedtothedampedMathieuequationandtoanotherparametricexcitationproblem.PluggingtheassumedFloquetsolutionintotheequationofmotion,andapplyinghar-monicbalance,ahomogeneoussetofalgebraicequationsfortheunknowncj'sisobtained.Theseequationsarethenputinmatrixform.Toenableanonzerocjsolution,thedetermi-nantofthecoe˚cientmatrixisequatedtozero,andacharacteristicequationfor^isfound.FloquettheorysuggeststhatsolvingthisequationfortheundampedMathieuequationre-sultsintwodistinct^'s,satisfying^1=^2+i!k,wherekisaninteger,andbywhichthek=0casede˝nesprincipalexponents.ForRe(^)6=0,thesolutiongrowsexponentially,50whereasforRe(^)=0,thesolutionisbounded.Thereforetransitioncurvescanbefoundbyexaminingchangesin^intheparameterspace.Associatedwitheachof1and2aredistinctsetsofcoe˚cientsc(1)jandc(2)j,which,iffound,wouldcompletetheexpressionsofx1andx2whichcouldthenbecombinedtoexpresstheresponsetoinitialconditions.AtruncatedFloquet-typesolutionwasalsousedbyInspergerandStépánto˝ndthestabilityregionsoftheundampedanddampedMathieuequationswithatimedelay[86,87].Theypluggedtheassumedsolutionintotheequationswithdelayterms,andappliedaharmonicbalanceprocedure.Then,assumingaperiodicsolution,theydeterminedtherelationsbetweentheparametersonthestabilityboundaries.Inthisstudy,insteadofassumingaperiodicsolutionand˝ndingparametersthatde˝netransitionalbehavior,agenerallynon-periodicsolutionisretainedto˝ndtherootsforthecharacteristicexponents,^0s.4.2Floquet-basedSeriesSolutionInthissection,theseriessolution,x=e^Pnj=ncjeij!t,isappliedtotheMathieuequation.ThenondimensionalundampedMathieuequation,x+(1+cos!t)x=0;(4.3)canbewritteninstatespaceformas_x=0B@_xx1CA=26401(1+cos!t)03750B@x_x1CA=A(t)x;(4.4)forwhichA(t)hasperiodT=2ˇ=!:Inthefollowingsections,thein˝niteseriessolution(n!1)isreviewed.Then,atruncatedsolutionisapplied,anditsstabilitycharacteristicsarecomparedtothoseofthein˝niteseriessolution.Thetruncatedsolutionleadstoaneigenvalueproblem,bysolvingwhichthetwoindependentelementsofthetimeresponsecanbeconstructed,andthencombinedforresponsestospeci˝edinitialconditions.514.2.1In˝niteSeriesSolutionInthiscase,theassumedsolutionisxr(t)=ert1Xj=c(r)jeij!t;(4.5)wherer=i^r,suchthattheexponentialpartisexpressedmoreconvenientlyfortheanalysisasertinsteadofe^rt.AccordingtoFloquettheory[53],thereexisttwodistinct's,whichsatisfye(1+2)T=exp ZT0tr(A(s))ds!:(4.6)Sincetr(A)=0fortheundampedMathieuequation,therelationbetweenthe'sis(1+2)T=2iˇk(i.e.1+2=0!k).InsertingtheassumedsolutionintoEquation(4.3)andbalancingcoe˚cientsofeij!tleadstothejthequationintheform2cj1+h1(+j!)2icj+2cj+1=0;(4.7)forj=;:::;1;Inmatrixform,266666666664......0:::0a11a10...0a01a00...0a11a10:::0......3777777777750BBBBBBBBBB@...c1c0c1...1CCCCCCCCCCA=0BBBBBBBBBB@00......01CCCCCCCCCCA;(4.8)whereaj=21(+j!)2:(4.9)Notethateachcjequationisdividedby1(+j!)2tomakethediagonalelements1.Thisisrequiredtoshowthatthein˝nitedeterminantisconvergent[54].Fornon-zero52cj's,thedeterminantofthecoe˚cientmatrixhastobezero.Muchlikeaneigenvalueproblem,thisyieldsanequationinvolving.Avalueofsatisfyingthisconditioniscalledcharacteristicexponent.AccordingtoMcLachlan[54],andworkscitedtherein,isgivenas=cos111cos[2ˇ!];(4.10)whereisthedeterminantofthematrixfor=0,which,forsmall,canbeapproxi-matedbyˇ1ˇcotˇ!22!4!2:(4.11)BysolvingEquation(4.10)andEquation(4.11)together,twodistinctvalues,satisfy-ing1=2,canbefoundforany(;!)pair.Sinceappearsintheterme,itgivesinformationaboutthefrequencyoftheresponse,andstability.ForIm()6=0,thesolutionisunstable,whereasIm()=0leadstoaboundedsolution.Usingthisinformation,ap-proximatetransitioncurvesbetweenstableandunstableregionscanbeplottedonthe!plane.Theapproximationimprovesasdecreasesinmagnitude.Althoughthisanalysisprovidesaclosedformsolutionforcharacteristicexponents,McLachlan[54]didnotprovidetheassociatedeigenvectors,andtimeresponsestoinitialconditions.Thereforetoobtainthefrequencycontentandtimeresponse,atruncatedsolutionisuseful,sinceitiseasiertosolvetheeigenvalueproblem.4.2.2TruncatedSeriesSolutionWhileanin˝nitedeterminantleadstoaclosed-formexpressionapproximatinginthecaseoftheundampedMathieuequation,itmaynotbesoifadditionaltermswereincludedinEquation(4.3).Insuchcase,atruncatedsolutionmaybepractical.HereatruncatedseriessolutionisappliedtoEquation(4.3)anditiscomparedtothein˝niteseriesapproximation.Atruncatedsolutionhastheformxr(t)=ertnXj=ncjeij!t:(4.12)53InsertingthisintoEquation(4.3)leadstoKc=266666666664kn20002...20002k0...000......20002kn3777777777750BBBBBBBBBB@cn...c0...cn1CCCCCCCCCCA=0BBBBBBBBBB@0...0...01CCCCCCCCCCA;(4.13)wherekj=1(1+j!)2,forj=n;:::;n.NoticethatthisisconsistentwithEqua-tion(4.8),excepttherows(i.e.cjequations),arenotdividedby1(+j!)2.NonzerosolutionsofvectorcrequirethedeterminantofKtobezero.Thisisa˝nitedeterminant,andbyequatingitto0,thecharacteristicequationforisfound.Thechar-acteristicequationisa(2n+1)thorderpolynomialintermsof2,whichresultsin(4n+2)rootssatisfying2r1=2r,r=1;2:::;2n+1.Yet,Floquettheorystatesthatthenumberof'sshouldbeequaltothedimensionNofthestatespace,whichinthecaseoftheMathieuequationis2.Whilethetruncatedsolutionresultsin(4n+2)rootsfor,itturnsoutthatnotallofthemareindependent.Asthenumbernofassumedharmonicsisincreased,thecharacteristicexponentsconvergetoasetofelementswhichsatisfyi=rk!;(4.14)wherekisaninteger.Thedi˙erencek!doesnotleadtodistinctsolutions,sinceeik!tisofperiodTandcanbeputintop(t).Toseethis,considerep1(t)=ei(rk!t)Pj=nj=neij!t=ertPj=nj=nei(jk)!t=ertp2(t);forwhichp1(t)andp2(t)havethesameperiod.There-foretherearee˙ectivelytwodistinctprincipalroots,1=2.Forafew(!;)pairs,theprincipalrootsaregiveninTable4.1,forn=1;:::;4.Ineachcase,therootsconvergetotheir˝nalvaluesforn=2.Therefore,n=2truncatedsolutionisstudiedheretocomparewiththein˝niteseriessolution.Equation(4.13)isessentiallyaneigenvalueproblem(EVP),withprincipaleigenvalues1=2.To˝ndtherelationshipbetweentheeigenvectorssatisfyingK(r)cr=0,the54Principalcharacteristicexponents,rn=1n=2n=3n=4!=0:6;=0:40:37520:22500:21190:2118!=0:8;=0:60:14360:16710:16730:1673!=1:3;=0:80:38960:36570:36560:3656!=2:5;=0:71:06061:06111:06111:0611Table4.1Principalcharacteristicexponents1and2fortruncatedsolutionswithn=1;:::;4equationforn=2isevaluatedforbothprincipal's.For1,c1satis˝es266666666664k220002k120002k020002k120002k23777777777750BBBBBBBBBB@c21c11c01c11c211CCCCCCCCCCA=0BBBBBBBBBB@000001CCCCCCCCCCA:(4.15)Ontheotherhand,theeigenvectorassociatedwith2=1satis˝es266666666664k220002k120002k020002k120002k23777777777750BBBBBBBBBB@c22c12c02c12c221CCCCCCCCCCA=0BBBBBBBBBB@000001CCCCCCCCCCA:(4.16)Thedi˙erencebetweenEquation(4.15)andEquation(4.16)isthattheorderofthediagonalentriesisreversed,sincek(1)j=1(1+j!)2=1(2+j!)2=1(2j!)2=k(2)j.Thereforetheequationsforelementsofc2arethesamewiththoseassociatedwiththeelementsofc1,exceptthattheorderoftheequationsischanged.Speci˝cally,ifc1=[c21c11c01c11c21]T;(4.17)thenc2=[c21c11c01c11c21]T:(4.18)55Thepatterncarriesoverforothervaluesofn.Thegeneralsolutioniscomposedoftwoindependentsolutions,x(t)=A1x1(t)+A2x2(t);(4.19)wherexj(t)=ejtcj(t)T[e2i!t;ei!t;1;ei!t;e2i!t]T,andA1andA2canbedeterminedfromtheinitialconditions.Asanexample,theinitialconditionsx(0)=0and_x(0)=v0resultinA1=A2andthesolutionisgivenbyx(t)=A1c21ei(12!)t+c11ei(1!)t+c01e1t+c11ei(1+!)t+c21ei(1+2!)tc21ei(12!)t+c11ei(1!)t+c01e1t+c11ei(1+!)t+c21ei(1+2!)t;(4.20)whereA1=v0=0@2i2Xj=2(1+j!)cj11A:(4.21)Forthecaseofastablesolution(Im()=0),thisreducestox(t)=0@v0=2Xj=2(1+j!)cj11A[c21sin(12!)t+c11sin(1!)t+c01sin1t+c11sin(1+!)t+c21sin(1+2!)t]:(4.22)Thus,thistruncatedexpansionpredictsresponsefrequencies12!,1!,1,1+!,and1+2!.Dependingontheinitialconditions,Ar'smightberealorcomplex,andtogetherwithcr'stheydeterminetheamplitudesandthephasesofthesolutionharmonics.Thus,bylookingattheindividualelementsofcr's,thedominantharmonicscanbedetermined.564.3ResponseCharacteristicsoftheUndampedMathieuEquationSolvingtheEVPgiveninEquation(4.13)allowsustopredictthecharacteristicsoftheresponse,suchastheresponsefrequenciesandstabilityofthesolution.Thesecharacteristicsareinvestigatedforthen=2truncatedsolutionandthein˝niteseriessolution.4.3.1StabilityAnalysisStabilityoftheassumedsolutiondependsonthevalueof.Forahavingazeroimaginarypart,eisperiodicandthesolutionisbounded,whereasanonzeroimaginarypartleadstoanexponentiallygrowingsolution,sincetheprincipalvaluesare1=2.Thereforethetransitioncurvesbetween'shavingzeroandnonzeroimaginarypartsareessentiallytheboundariesbetweenstableandunstableregions.Theseboundariesareplottedforthein˝nitesolutionandforthen=2approximation,andthesearecomparedtothecurvesobtainedfromtheclassicalHill'sdeterminantsolution(n=2),asgiveninFigure4.1.Whilethereareanin˝nitenumberoftransitioncurves,thetruncatedsolutioniscapableofpredicting2nofthem.Asincreases,thedeviationbetweenthetransitioncurvesgetslarger,sincetheapproximationineachsolutionisvalidforsmall.4.3.2ResponseFrequenciesTheassumedsolutionisaproductofanexponentialpartandaperiodicpart.Thefrequencyoftheresponseisacombinationofthefrequenciesofthetwoparts.TheexponentialpartisoffrequencyRe(),andbyusingtrigonometricidentities,thepredictedresponsefrequenciesareRe()+j!,wherej=1;2:::;n.Foragivenvalueof,responsefrequenciescanbeplottedasafunctionof!.Asanexample,for=0:8,thepredictedlowerresponsefrequenciesareplottedversus!inFigure4.2.Figure4.2showsthatwithvariationsin!,twofrequencybranchescancollideandmergeintoonebranch.Thesefrequencycollisionscorrespondtostabilitytransitions.Thetop57Figure4.1Transitioncurvesforthen=2Floquet-basedapproximation,in˝niteseriessolution,andthen=2Hill'sdeterminantsolution.branchinFigure4.2hasnocollisions.Ifwechoosealargervalueofn,weexpecttoseemorebrancheswithwhichthistopbranchcouldcollide.Noticethatintheunstableregions,thesolutionconsistsofanexponentialgrowthandaperiodicpartwithperiodTor2T,whichisknownsincetheFloquetmultipliersarerealpositiveornegativefortheunstableHill'sequation[52].TheresponsefrequenciescomingfromtheperiodicpartcanbeclearlyseenfromtheFigure4.2.Forexamplenearthe!=2instability,theresponsefrequenciesare!=2,3!=2and5!=2,whereasnearthe!=1instability,thefrequenciesare!,2!and3!.Tovalidatethetruncatedsolution,foranumberof(;!)pairs,frequenciesofstableresponsesarecomparedtothosefoundfromthefastFouriertransforms(FFTs)ofthenu-mericalsolutions,asshowninTable4.2.BysolvingtheeigenvalueproblemgiveninEquation(4.13)forn=2,theeigenvectorsforthetruncatedsolutionarefoundfor(!;)pairsgiveninTable4.2.TheresponseisthenconstructedusingEquation(4.20)andEquation(4.22)withinitialconditionsx(0)=0,and58Figure4.2Analyticallypredictedresponsefrequenciesfor=0:8,n=2.LowestResponseFrequencies(Hz)!n=2n=1Numerical0:60:40:225;0:388;0:825;0:211;0:389;0:811;0:211;0:388;0:812;0:988;1:4250:988;1:4110:988;1:4120:80:60:167;0:655;0:966;0:173;0:627;0:973;0:171;0:637;0:969;1:455;1:7671:423;1:7731:432;1:7681:30:80:366;0:934;1:690;0:369;0:931;1:669;0:365;0:930;1:661;2:224;2:9662:231;2:9692:265;2:9662:50:71:061;1:439;3:561;1:061;1:439;3:561;1:061;1:440;3:561;3:945;6:0053:939;6:0613:939;6:060Table4.2Lowestresponsefrequenciesobtainedfromthen=2andn=1Floquetsolutions,andFFTsofnumericalsolutions_x(0)=1.Thesesolutionsarethencomparedwiththosefoundfromthenumericalsolution.ThetimeresponsesandFFTplotsaregiveninFigures4.3and4.4.Theresultsareconsistentforlarger!values,whereasthepredictedfrequencycontentandassociatedamplitudesarealittlebito˙forsmallervaluesof!.Toachievebetterresultsinthesmall!region,thenumberofharmonicsassumedinthetruncatedsolutionshouldbeincreased.Toshowthee˙ectofnumberofassumedharmonics,thefreeresponseandFFTplotsfor!=0:4,=0:6aregivenforn=3inFigure4.5.ComparingtheseplotstoFigure4.3(a)and4.4(a),itcan59beseenthatwiththeaddedharmonic,theassumedsolutionwasabletocapturethe6thpeakintheFFTplot,andthetimeresponsematchedbetterwiththenumericalsolution.Figure4.3Numericalandtheoretical(n=2)solutionsoftheundampedMathieuequationforn=2.4.4DampedMathieuEquationAtruncatedexpansionmaybedesirableforoscillatorsotherthantheundampedMathieuequation,whichmaythereforenothaveasimpleclosedformexpressionfromthein˝nitedeterminant.ForexampletheexistenceofsuchanexpressionisnotobviousforthedampedMathieuequation.Here,atruncatedsolutionwithn=2isassumedforthedampedMathieuequation,whichisgivenbyx+2_x+(1+cos!t)x=0:(4.23)60Figure4.4FFTsofnumericalandtheoretical(n=2)solutionsoftheundampedMathieuequationforn=2.Insertingthen=2expansion,Equation(4.12),intoEquation(4.23)resultsin266666666664b220002b120002b020002b120002b23777777777750BBBBBBBBBB@c2c1c0c1c21CCCCCCCCCCA=0BBBBBBBBBB@000001CCCCCCCCCCA;(4.24)wherebj=1(+j!)2+i2(+j!):(4.25)InthiscasetheconditiongiveninEquation(4.6)resultsin1+2=i2k!,sincetr(A(t))=2.Thiscauseschangesinthetransitioncurves,leadingtolargerstableregionsforlargerdampingratios.Fordampingratios=0:005,=0:025,and=0:05,transitioncurvesaregiveninFigure4.6.Thecurvesresultingfromthetruncatedsolutionareconsistentwiththosefoundintheliterature.61Figure4.5FreeresponseandFFTplotsfor!=0:4,=0:6,withn=3harmonics.Thetimeconstants,andthereforethedecayorgrowthrates,canbedeterminedfromthecharacteristicexponents.Theimaginarypartsofthe'sgoverntheexponentialpartofthesolution.Asanexample,thedecayandgrowthfactorsareplottedasafunctionof!for=0:05and=0:8,inFigure4.7.Itcanbeseenfromthe˝gurethatthetwofactorsalwaysaddupto2,whichisconsistentwiththeFloquettheory.NoticethatinsomeregionswehaveIm(1)=Im(2)=,whichmeanstheresponsehasasingledecayrateindependentof!and.Tovisualizethis,for=0:025,initialconditionresponsesareplottedinFigure4.8fortwodi˙erentcases.Itcanbeclearlyseenthat,thetworesponsesaredecayinginthesameenvelope,e0:025t.Inotherregions(i.e.Im(1)6=Im(2)),there62Figure4.6TransitioncurvesforthedampedMathieuequationfor=0:005,=0:025;and=0:05,approximatedwithn=2.aretwotimeconstants.ThismeansthatforIm(1)>Im(2)>0,thesolutionisstableandiscomposedofaslowandafastpart,whereasforIm(1)>0>Im(2),thesolutionisunstable.InthecaseofFigure4.8(a)and(b),Im(1)=Im(2)=,thereforetheresponsehasonlyonedecayrate.HoweverinFigure4.8(c)and(d),therearetwodi˙erentdecayrates,andtheresponseisdominatedbytheonehavingtheslowerdecayrate.Forparametervalues!=1;=0:3;=0:025,theimaginarypartsareIm(1)=0:019andIm(2)=0:031.Thetimeresponsehasthee˙ectsofbothdecayingpartsatthebeginning,butthenitisdominatedbytheslowlydecayingone,e0:019t.Thesamee˙ectcanbeseeninFigure4.8(d),wheretheimaginarypartsareIm(1)=0:0125andIm(2)=0:0875.ThecorrespondingFFTplotsaregiveninFigure4.9.63Figure4.7Decayandgrowthfactorsfor=0:05and=0:8.Figure4.8Numericalandtheoretical(n=2)solutionsofthedampedMathieuequation.4.5ParametricExcitationwithTwoHarmonicsThetruncatedsolutionisappliedtotheproblemx+(1+cos!t+cos2!t)x=0:(4.26)64Figure4.9FFTofnumericalandtheoretical(n=2)stablesolutionsofthedampedMathieuequation.Thissystemissuspectedtohavepossiblesimultaneousinteractionswiththee˙ectsof!and2!excitationterms.Performingharmonicbalance,thematrixequationisfoundas266666666664k222002k122022k022022k120022k23777777777750BBBBBBBBBB@c21c11c01c11c211CCCCCCCCCCA=0BBBBBBBBBB@000001CCCCCCCCCCA;(4.27)wherekj=1(1+j!)2.Thecharacteristicexponentsarefoundbyequatingthedetermi-nanttozero.AsintheundampedMathieuequation,theconditiongiveninEquation(4.6)resultsin1=2.For(!=0:8;=0:6;=0:5)andn=3,andfor(!=1:5;=0:5;=1)andn=2,theresponsesforx(0)=0,_x(0)=1initialconditionsandFFTplotsaregiveninFigure4.10.65Theresultsareconsistentwiththenumericalresults,whichmeanstheproposedmethodcanpredicttheresponseforthisproblemaswell.ComparingFigure4.10(a)withFigure4.3(c)showsthat,inthiscase,thesecondparametricexcitationharmonicresultsinaslightlysuppressedresponse.Itispossiblethatmultipleharmonicexcitationsmayhavesomeusefule˙ects.Futureresearchwillbeconductedtoexaminethebehaviorofthissystem.Figure4.10Numericalandtheoreticalsolutionsofthetwo-frequencyMathieuequation.4.6ConclusionsAtruncatedFloquetsolutionhasbeenappliedtotheanalysisoftheMathieuequation.TheformofthesolutioninvolvesanunknownFloquetexponentialtermtimesatruncatedFourierserieswithunknowncoe˚cients.TheFloquetexponentandFouriercoe˚cientsarethendeterminedbyaneigenvalueproblem.Fromthisinformation,thestabilityandtheresponsefrequenciesandtheirrelativestrengthscanbeapproximated.ThisapproachwasappliedtotheundampedMathieuequationandthefrequenciesandassociatedstrengthswereestimatedasfunctionsoftheexcitationamplitudeandfrequency.66Thetheoreticalapproximationtotheresponsefrequenciesandamplitudescomparedfavor-ablytonumericalsimulations.Informationaboutthestabilitytransitionswasalsoextractedandcomparedtoexistingapproximations.TheapproachwasthensuccessfullyappliedtothedampedMathieuequationandatwo-frequencyMathieuequation.Theapproachgivenhereisasimpli˝cationofanexistingbutuncommonsolutionin-volvinganin˝niteseries,whichresultsinanin˝nitedeterminantapproximationwhichisapplicableonlytotheundampedMathieuequation.Thisin˝nite-seriesapproachhadbeenappliedinane˙orttoobtainstabilitytransitioncurves.Thetruncatedseriesapproachofthispaperisexpectedtobeapplicabletootherparametricallyexcitedoscillators,suchasInce'sequation,orparametricexcitationwithmultipleharmonics.Italsodoesnotin-volvetheenigmaticanalysisofthein˝nitedeterminant,therebymakingtheapproachmoreaccessibletoengineersandscientistswithbroadbackgrounds.67CHAPTER5APPROXIMATEGENERALRESPONSESOFMULTI-DEGREE-OF-FREEDOMSYSTEMSWITHPARAMETRICSTIFFNESS5.1IntroductionInthisstudy,generalsolutionstoMathieu-typemulti-degree-of-freedom(MDOF)systemsoftheformMx+K(t)x=0;(5.1)areinvestigated,wherexisad1vectorofcoordinatedisplacements,wheredisthenumberofdegreesoffreedom,andMandK(t)arethemassandtime-varyingsti˙nessmatrices.Theaimistoobtainageneralinitialconditionresponseaswellasthestabilitycharacteristicsofthesystem.Tothisend,insteadofassumingaperiodicsolution,aFloquet-typesolutionisassumedasxr=eirtnXl=nc(r)leil!t;r=1;:::;2d(5.2)wheretheindexrdistinguishesbetween2dindependentFloquetsolutiontermsforand-degree-of-freedomsystem.Theassumedsolutionispluggedintotheequationsofmotion,andbyapplyingharmonicbalance,thecharacteristicexponents,r,andassociatedFouriercoe˚cients,c(r)l,aredetermined.Theresponsetoanarbitraryinitialconditioncanthenbefoundbyconsideringalinearcombinationofthexr.ThesamemethodwasappliedtotheSDOFundampedanddampedMathieuequations[88](SeealsoChapter4).5.2AnalysisWedemonstratetheresponseanalysisprocedurecanbeexplainedbygoingoverexampleMDOFsystems,namelyatwo-DOFcaseandathree-DOFsystem.685.2.1Two-Degree-of-FreedomExampleTothisend,amass-springchainasshowninFigure5.1,withperiodicsti˙nessisused.TheFigure5.1AtwoDOFspring-masschain.equationsofmotionarex1+k(2x1x2)=0;(5.3)x2+k(2x2x1)=0;(5.4)wherem=1andk=1+cos!t.Forthecaseof=0,thesystemhasconstantcoe˚cientsandthemodalfrequenciesarewellde˝nedas!1=1and!2=p3,withmodeshapesu1=(11)Tandu2=(11)T.To˝ndtheresponse,aFloquettypesolutionwith˝niteharmonics,asgiveninEqua-tion(5.2),isassumedandpluggedintotheequations.Speci˝cally,inthisexample,weseekel-ementsofx(t)=[x1(t)x2(t)]Tasx1(t)=eiPnl=nc1;leil!tandx2(t)=eiPnl=nc2;leil!t.Applyingharmonicbalance,governingequationsforcj;l'sarefound.Inmatrixform,Ac=264A11A12A21A223750B@c1c21CA=0B@001CA;(5.5)wherecj=[cj;n:::cj;1cj;0cj;1:::cj;n]T,andApq'scorrespondto(2n+1)(2n+1)blockmatrices.Sincethereisastructuralsymmetryinthisexample,thecoe˚cientmatrixissymmetricwithA11=A22andA12=A21asgivenbelow:69A11=A22=266666666664......0:::0...2(!)2......022...............2(+!)20:::0...377777777775;(5.6)A12=A21=2666666666641=20:::0=21=2......0.........0............=20:::0=21377777777775:(5.7)Tohaveanonzerocsolution,thedeterminantofthecoe˚cientmatrix,A(),mustbezero.Thisconstitutesacharacteristicequationfor,intermsoftheparametersand!.Thecharacteristicequationyields2d(2n+1)rootsfor(d=2inthisexample),wherenisthenumberofassumedharmonics.Yet,Floquettheoryimpliesthattherearee˙ectively2dprincipalroots,andtheotheronesarerelatedtotheprincipalrootsviatherelationi=jp!,wherepisaninteger.Noticethattheseextrarootsdonotcontributetoextrasolutions,sincethecorrespondingexponentialpartcanbewrittenaseijteip!tandthesecondpartcanbepluggedintotheperiodicpart.Byinsertingjsintothecoe˚cientmatrix,solutionsforc(j)=[c(j)1c(j)2]TcanbefoundasthenullspaceofA(j).TherelationsA11=A22andA12=A21leadtotherelationc(j)1=c(j)2.Furthermore,theprincipalrootscomeinpairsasj+1=j,andthisleadstothediagonalelementsA(j+1)bethesameasthoseofA(j),butinreverseorder.Thisrelationresultsinac(j+1)havingthesameelementsasc(j),inthereversedorder.Therootsofthecharacteristicequationgiveinformationaboutboththestabilityandthefrequencycontentofthesolution.Ifanyoneoftherootshasanegativeimaginarypart,theexponentialpart,eimakesthesolutiongrowunstable,whereasifallrootshave70anon-negativeimaginarypart,thesolutionisbounded.Morespeci˝cally,iftherootsarereal,thesolutioniseitherperiodicorquasi-periodic.Thestabilitytransitioncurvesforthe2DOFmass-springchainareplottedbyevaluatingtheimaginarypartsofthecharacteristicroots,asgiveninFigure5.2.Thefrequencycontentcanbedeterminedbycombiningthefrequenciesoftheexponentialpartandtheperiodicpart,asjRe()l!j.Forthis2DOFsystem,therearefourdistinct'sas2=1and4=3.Inthefrequencycontentoftheresponse,halfofthefrequenciesaredeterminedbythe˝rstpair,andtheotherhalfaredeterminedbytheotherpair.Inthiscase,symmetricinitialconditions(i.e.x1(0)=x2(0),_x1(0)=_x2(0))excitethefrequenciesassociatedwith1and2,whereasanti-symmetricinitialconditions(i.e.x1(0)=x2(0),_x1(0)=_x2(0))excitethoseassociatedwith3and4.Thesetworesponsescanbeusedtorepresenttheresponseintermsofdalcomponensuchasx(t)=a1x(1)(t)+a2x(2)(t),wherea1anda2aretobedeterminedfromtheinitialconditions.Asanexample,for!=2:3and=0:4,fromacomputationwithn=2,thechar-acteristicrootsareapproximatedas1=2=0:599and3=4=1:035.Thesymmetricinitialconditionsexcitethefrequenciesj1:0352:3lj,whereastheanti-symmetricinitialconditionsexcitethefrequenciesj0:5992:3lj,forl=2;1;0;1;2,asshownintheFFTplotsofm1inFigure5.3.ArbitraryinitialconditionsproduceresponseswithallofthefrequenciescontributingtothefastFouriertransformsplottedinFigure5.3.Theeigenvectorsarec(1)=[0:031790:689470:15270:017010:000425]Tandc(3)=[0:0038340:224520:670370:0132490:000086]T.Sincethe5thelementsinbothvectorsaresmall,the5thfrequencies(j+2!)arenotvisibleintheFFTplots.Moregenerally,˝vefrequencieswouldshowupfromann=2approximation.The2DOFspring-masschainwasanalyzedwithn=2harmonics,fordi˙erentsetsofparametersandinitialconditions.Theresultswerecomparedtothoseobtainedfromanumericalstudy,andtheresponseandFFTplotsaregiveninFigures5.4and5.5foranexample.71Figure5.2Stabilityregionsforthe2DOFmassspringchain.Figure5.3Modesexcitedbythesymmetricandtheanti-symmetricinitialconditions,for!=2:3and=0:4.AmplitudeFFTplotsofdisplacementsofm1,generatedwithn=2truncatedsolution.5.2.2Three-Degree-of-FreedomExampleAnotherMDOFsystemwithparametricsti˙nessisgiveninFigure5.6.Twomassesareconnectedtoamainmassthroughparametricsprings,whicharedrivenoutofphase,andthemainmassisgroundedwitharegularspring.Thissystemisanalogoustoahorizontal-72Figure5.4ResponseandamplitudeFFTplotsforn=2,!=1:2,=0:6,x(0)=[10:5]Tand_x(0)=[00]T.axisturbinewithtwobladesunderaconstantrotationrate,wherethemainmassrepresentsthehub,andtheothermassesrepresenttheblades.Foranormalizedsystem,theparametersareassumedasm1=m2=1,M=m1,K=,k1=1cos!tandk2=1+cos!t.Equationsofmotionrepresentingthissystemarex1+(1cos!t)(x1x3)=0x2+(1+cos!t)(x2x3)=0(5.8)x3(1cos!t)x1(1+cos!t)x2+(2+)x3=0:PluggingtheapproximatesolutiongiveninEquation(5.2)intotheequations(5.8),andapplyingthestepsexplainedintheprevioussection,characteristicexponentsandthecor-respondingeigenvectorsarefound.Byexaminingtheimaginarypartsofthecharacteristicexponents,thestabilityregionsofthe3DOFsystemareplottedfor=1,=1inFigure5.7.73Figure5.5ResponseandFFTplotsforn=2,!=2:3,=0:4,x(0)=[15]Tand_x(0)=[00]T.Figure5.6A3DOFmass-springsystem.Inorderto˝ndthedalasdoneintheprevioussection,thefollowingprocedureisapplied.Therearesixprincipalcharacteristicrootsandsothegeneralsolutioncanbewrittenasx(t)=6Xj=1ajxj(t);(5.9)wherexj(t)=nXl=ncj;leil!t:(5.10)74Figure5.7Stabilityregionsforthe3DOFmassspringsystem,for=1and=1.Arbitraryinitialconditionscanbeexpressedasasetoflinearequationsintermsoftheconstantsaj,as264x0_x0375=264x1(0):::x6(0)_x1(0):::_x6(0)375266664a1...a6377775:(5.11)NoticethattheinitialconditionhxT0_xT0iT=hxj(0)T_xj(0)TiTresultsinaj=1andal6=j=0.Thereforeascalarmultipleofeachcolumncanbeusedasaninitialcondi-75tionto˝ndtheseparatemodalfunctions.Forexamplefor=1,=1,=0:5and!=2:2,thecharacteristicexponentsare=0:975;0:520;0:315.Theinitialcondi-tionvectorx1(0)=[1:3881:3640:263000]excitesthefrequenciesj0:9752:2lj,x2(0)=[1:1951:3110:926000]excitesthefrequenciesj0:5202:2lj,andx3(0)=[1:0720:0011:513000]excitesthefrequenciesj0:3152:2lj,asshowninFig-ure5.8.Bywritinganinitialconditionasalinearcombinationofhxj(0)T_xj(0)TiT's,theresponsecanbefoundasthesamelinearcombinationofthecorrespondingmodalfunctions.Figure5.8Modesexcitedbytheinitialconditionswhichresonatethefrequenciesthatareassociatedwith(a)1=0:975,(b)2=0:520,(c)3=0:315.76TheinitialconditionresponsesandamplitudeFFTplotswereobtainedassumingn=2harmonics,forvariousparametersets,andwerecomparedtothoseobtainedfromanumericalstudy,asshowninFigures5.9and5.10.Figure5.9ResponseandFFTplotsforn=2,!=0:7,=0:5,=0:4,x(0)=[000]Tand_x(0)=[111]T.5.3DiscussionIntheMDOFexampleswepresented,closeexaminationofthestabilitywedgessuggeststhattheymaybebasedat!=(!i+!j)=N;whereNisapositiveinteger.Forthetwo-DOFcase,with!1=1and!2=p3=1:73,weseethemajorwedgesat2!1=2and2!2=3:46,representingthesubharmonicinstabilityofeachThe77Figure5.10ResponseandFFTplotsforn=2,!=3:5,=0:3,=0:4,x(0)=[110:5]Tand_x(0)=[000]T.symmetryinthetwo-DOFmodeldrivestheresonancessuchthat!i=!j(i.e.i=j)inthesuggestedinstabilitycondition.Wealsoseetheotherslenderinstabilitywedgesformodeoneat!=!1,!=2=3;andeven!=1=2;andlikewiseformodetwoat!=!2,!=2!2=3ˇ1:15;and!=!2=2ˇ0:87:Inthethree-DOFsystem,for=1and=1,themodalfrequenciesare!1=0:518,!2=1,!3=1:932.Weseetwomajorsubharmonicinstabilitywedgesoriginatingatfrequencies!ˇ!2+!3=2:93and!ˇ!2+!1=1:52.Someofthesuperharmonicwedgesarealsobasedatfrequenciesthatmatchthepattern,suchas!3=1:93;(!1+!3)=2=1:22;!1=1;(!2+!3)=3=0:98;!3=2=0:97;(!1+!3)=2=1:22;!1=0:52;and(!1+!2)=3=0:51:Simulationsweredoneatvariousparametervaluestocheckconsistency.Inthe3-DOF78example,at=0:3and!=3:1,thesimulationwasunstable,whileat!=3:2,thesimulationwasstable,consistentwiththestabilitywedgeinFigure5.7.BasedontheFloquetsolutionasalinearcombinationoftermsoftheformep(t);wherep(t)isaperiodicfunction,itisclearthatinitialconditionscanbespeci˝edsuchthatonlyoneofthesetermsisactive.Inthissense,wecancallthemdalrespEachofthesemodeshasitsownfrequencycontent,butthep(t)arenotsynchronous,andnotdescribedusingsimpleshapevectorsliketraditionalmodes.Wehavedemonstratedsuchindependentoscillationsolutionsinthetwoexamplesgivenhere.Wemightconsiderwhetheracoordinatetransformationexistssuchthatagivenmodalresponsecanbeisolatedasasingledegreeoffreedom,andthenwhethereachmodaldegreeoffreedomisparametricallyexcitedandfollowsthestabilitypatternofaHill'sequation(ormoregenerallyanInceequation,whichhasaperiodiccoordinatetransformationbetweenitandaHill'sequation[52]).Complicatingthesespeculationsistheentanglementofthereferencemodalfrequencies(thosefromthe=0system)intheexcitedsystem'sstabilitywedgepatterns.TheanalysispresentedhereinvolvesthesolutiontoanonstandardeigenvalueproblemintheformofEquation(5.5),inwhichmatrixAincludeselementswithquadraticterms.Theresultingcharacteristicequationisapolynomialofdegree2d(2n+1)in.Theremaybecomputationallimitson˝ndingsymbolicsolutionsfortheasthedegreeoffreedom,d,orthenumberofharmonics,n,increases,althoughn=2harmonicswasabletopredicttheresponseforthesystemsanalyzedabove.5.4ConclusionInthiswork,aprocedurefor˝ndinganapproximatesolutiontoaMDOFsystemwithpara-metricsti˙nesshasbeenrepresented.AFloquet-typesolutioncomposedofanexponentialandaperiodicpartwasassumed,andapplyingharmonicbalancetothesystemequations,aneigenvalueproblemresulted,witheigenvaluesthatprovidetheFloquetcharacteristic79exponentsandtheeigenvectorsthatprovidetheFouriercoe˚cients.Theinitial-conditionsresponsewasexpressedintermsofindependentmodalcomponents,whichweredemonstratedbyseparatingtheinitialconditionsexcitingthemodesassociatedwithseparatecharacteristicexponents.Thestabilitytransitioncurveswereobtainedbyexaminingtheimaginarypartsofthecharacteristicexponents.TheresponseandFFTplotsweregeneratedforvariousparametersandinitialconditions,andcomparedtonumericalresultsforvalidation.Themethodusedinthisstudywasappliedtodeterminetheinitialconditionresponseaswellasthestabilityofthesystem,whereasthecommonlyappliedanalysesseeninpreviousstudiesre˛ectinterestonlyinthelatter.Theprocedureistobeappliedtothree-bladewindturbinemodelsto˝ndtheresponsecharacteristics.80CHAPTER6CONCLUSIONSANDFUTUREWORK6.1ConcludingRemarksThisthesisaimedtostudysomeaspectsofthedynamicsofwindturbines,andfundamentaldynamicsystemsrelevanttowindturbinedynamics.Ananalysisofahorizontal-axiswind-turbinebladeunderbend-bend-twistvibrations,andinplanecoupledblade-hubvibrationsofathree-bladeturbinewerestudied.Inspiredbytheperiodicallyvaryingsti˙nesstermsinbladeequations,solutionstotheMathieuequation,andMDOFsystemswithparametricsti˙nesswereinvestigated.Modelingthebladeasastraightbeamwithavaryingcross-sectionandpretwist,theenergyexpressionswerefoundintermsofbendingandtorsionaldisplacements.Theenergieswereexpressedintermsoffamiliarbeamparameters.Thenassuminguniformcantileverbeammodesasshapefunctionsforeachdisplacement,theenergyformulationswerewrittenintermsoftheassumedmodalcoordinates.Thebend-bend-twistcoupledequationsofmotionwerefoundviaLagrange'smethod.Therotorspeedwasassumedtobeconstantforthisanalysis,soitwasnottakenasoneofthemodalcoordinates.Naturalfrequenciesandmodeshapeswerefoundforexistingblademodelsbyapplyingamodalanalysis.The˝rsttwomodeshapesaredominantlyin-planeandout-of-planebendingmodes.Bladesti˙nessvarieswithrotorangleduetoparametrice˙ectsofthegravity.Toshowthesti˙nesschanges,bladeswereanalyzedathorizontalandverticalorientations.Parametricsti˙nesstermswerefoundtobemoresigni˝cantforlongerblades,andtheyareimportantsincetheycanintroducesecondaryresonancestothesystem.Equationsofmotionforathree-bladeturbineandhubwerefoundbyapplyingLagrange'smethodtothesystem'stotalenergies.Forthisanalysis,onlyin-planevibrationsweretaken81intoaccount,andthevariationinhubspeedwasincludedasavariable.Assumingonemodeforeachblade,therotorandbladeequationswerederived.Thenthetimedomainequationsweretransformedintorotorangle-basedequations.Variationsintherotorspeedwereassumedtobesmall,andapplyinganondimensionalizationandascalingscheme,therotorequationwasdecoupledfromtheleading-orderbladeequations.Interdependentbladeequationswereanalyzedwitha˝rst-ordermethodofmultiplescales.Parametricanddirectexcitationsintroducedasuperharmonicandasubharmonicresonancetothesystem.Thesuperharmonicresonanceoccursnear=!n2,whereistherotorspeed,and!n2isthebladesti˙ness.Foracycliclysymmetricturbine(i.e.allthreebladeshavethesameinertiaandsti˙ness),thebladeshaveaunisonamplituderesponsewithaphasedi˙erence.Thesubharmonicresonanceoccursnear=2!n2,withzeroamplitudebladeresponse.Thestabilityofthebladeresponseforsubharmonicresonancewasinvestigated.TheMathieuequationissimilartoaSDOFmodelofbladeequations,anditrepresentsmanyothermechanicalsystemsaswell.Approximategeneralsolutionstotheundamped,dampedandtwo-harmonic-excitationMathieuequationswerestudied.AtruncatedFloquet-typesolutionwasassumed,andinsertedintotheequations.Then,applyingharmonicbal-ance,thecharacteristicexponentsandtheFouriercoe˚cientswereapproximated.Thegeneralresponsesandstabilityofthesolutioncanbefoundforanyparametervalues.Theinitialconditionresponseswerefoundforasetofparametervalues,andtheyagreedwellwiththesimulationsdoneviatheODEsolveronMATLAB.ForthedampedMathieuequation,thedecayrateoftheresponsewasalsoquanti˝edthroughthecharacteristicexponents.Ingeneral,thisapproximatesolutionisapplicabletoSDOFsystemswithparametricexcitation.ToshowthatthemethodusedinChapter4isalsoapplicabletoMDOFsystems,example2DOFand3DOFsystemswithparametricsti˙nesswerestudied.AMDOFFloquet-typesolutionwasappliedto˝ndthegeneralresponsesandthestabilitycharacteristics.The3DOFsystemshowedstabilitycharacteristicsdi˙erentthanexpected.Theinstabilityregionsarebasedat(!j+!k)=N(j=1:::dandk=1:::d,wheredisthedegreeoffreedom)points,82whereasfortheSDOFMathieuequationtheyarebasedat2!j=N.Aeroelasticmodelingwasreviewedasanappendixtothiswork.Aerodynamicforcesdependonthebladeangle,bladevelocity,andblademotionhistories.Thustheaeroelasticmodelwilla˙ectthepredictedvibrationresponsesandstabilities.Assuchitisimportanttostudyaeroelasticmodelsforwind-turbinebladevibrationanalysis.Whenoperatingathighanglesofattack,turbinebladesmightexperiencestall,whereatacriticalangletheliftforcedropssuddenly.Furthermorerapidvariationsintheangleofattackcanintroducedynamicstallwheretheliftforcehasahysteresis.Semi-empiricalmodelsoftheliftforceindynamicstallwerestudied.Thesemodelsusedi˙erentialequationstoexpressthedynamicsoftheliftforce.Coe˚cientsoftheseequationsandtheirdependenceonstatevariableswereexplainedfortwodi˙erentmodels(ONERA's[2]andthatofLarsenetal.[4])existingintheliterature.Insummary,thisthesisprovidedamodalanalysisofasinglebladeandaperturbationanalysisofathree-bladeturbine.Anextensivereviewontwoaerodynamicstallmodelswasgiven.Alsotoinvestigatethetransientdynamics,ananalysisoftheMathieuequationandMDOFsystemswithparametricsti˙nesswerestudied.6.2FutureWorkTheworkdoneinthisthesiscanbeextendedtothefollowing:-Theresultsfoundthroughthemodalanalysisusingabend-bend-twistbeammodelcanbeveri˝edbyexperiments.Asimplebeamcanbeanalyzedatvariousorientationstoshowtheparametrice˙ects.Alsoitcanbespunataconstantspeedtoinvestigatethesecondaryresonances,analyzedintheliteratureforsinglebeams,andforcoupledbeam-rotorsystems,asmodeledinChapter3.-Thethree-blademodelusedinChapter3hasacyclicsymmetrywhichresultsinsymmetricbladeresponses.Tounderstandthee˙ectsofbrokensymmetry,whichis83likelytobeseeninrealsystemswithimperfections,theanalysiscanbeextendedtoamodelwhereoneortwoofthebladesaremistuned.Alsotheanalysisforin-planethree-bladedynamicscanbeextendedtobend-bend-twistanalysisofthree-bladeturbines.Thetowermotionandrotorsti˙nesscanbetakenintoaccount.-Oneoftheaerodynamicstallmodelscanbeappliedtoanexistingbladeto˝ndthee˙ectontheforcedresponse,andonpossiblegenerationoflimitcycleoscillations.-TheFloquet-typesolutionassumedinChapter4canbeextendedtootherparametri-callyexcitedsystems,to˝ndtheirinitialconditionresponses,andstability.Forexam-pleverticalaxiswindturbinebladesmayshowperiodicdampingcharacteristics[89],andtheFloquet-typesolutioncanbeassumedtomodeltheirtransientdynamics.-TheanalysisonMDOFsystemswithparametricsti˙nesscanbeappliedtothree-bladeturbineequations,toexaminewhethertheremayexistparametricinstabilitiesinthetransientdynamicsofcoupledblade-hubturbinemodels,andtomodelstabletransientresponsesandpredictthefrequencycontent.84APPENDICES85APPENDIXAEQUATIONSOFMOTIONOFABLADEA.1StrainsTheLagrangianstraintensorofaninextensiblestraightbeamunderbend-bend-twistvibra-tionsisdeterminedusingGreen'sformula[66].Strainsareexpressedintermsofdisplace-mentsas"xx=v00w00+(v00+w00)22+(2+2)02x2+(w0v0)0x+O(3);(A.1)"=v022+2x2+O(3);(A.2)"=w022+2x2+O(3);(A.3)=0x+v0(v00+w00)+x(w0+0x)+O(3);(A.4)=0x+w0(v00+w00)+x(v0+0x)+O(3);(A.5)=v0w0+O(3):(A.6)A.2KineticandPotentialEnergiesForalinearstraightinextensiblenonuniformwithbend-bend-twistdeformations,rotatingaboutanaxisthroughtheattachmentpoint,perpendiculartoboththegravityvectorandthecentroidalaxis^x,theenergyexpressionsarisingfromEquations(2.3)through(2.5),afterincorporatingEquations(A.1)-(A.6)andtheEquationsforrP1andvP1fromSection2.2.1,writtenintermsofde˛ectionsofthecentroidareU=12ZL0hI(x)(Ew002+02x)+I(x)(Ev002+02x)+2EI(x)v00w00idx;(A.7)86T=12ZL0hJ(x)(_x2+(_˚+_v0)2+_˚2v02)+J(x)(_x+_˚w0)2+(x_˚_w0)22J(x)_˚2x_˚_xv0_˚2v0w0+_x_v0_˚_w0_v0_w0+m(x)_˚2v2+(_˚(x+u)+_v)2+_w2dx;(A.8)Vg=ZL0m(x)g[(x+u)cos˚vsin˚]dx;(A.9)whereIijaresecondmomentsofarea,Jijaremomentsofinertia,A(x)iscross-sectionalarea,m(x)ismassperunitlength,EiselasticmodulusandGisshearmodulus.Whenwrittenintermsofshearcenterde˛ections,theenergyexpressionsareasfollows:Us=12ZL0hIs(x)(Ew002s+02x)+Is(x)(Ev002s+02x)+2EIs(x)v00sw00sidx;(A.10)Ts=12ZL0hJs(x)(_x2+(_˚+_vs0)2+_˚2v02s)+Js(x)(_x+_˚w0s)2+(x_˚_ws0)22Js(x)_˚2x_˚_xv0s_˚2v0sw0s+_x_v0_˚_ws0_vs0_ws0+m(x)_˚2v2s+(_˚(x+us)+_vs)2+_ws2+s(x)(_ws_xx_˚2v0sv0s_vs_us+2vs+vs_vs0)+s(x)(x_x_vs_xx2w0s_vsw0s2xvs+vs_ws0)dx;(A.11)Vgs=ZL0m(x)g(x+us)cos˚vssin˚+s(x)(v0scos˚+sin˚)+s(x)(w0scos˚xsin˚)dx;(A.12)whereIijsandJijsaresecondmomentsofareaandmomentsofinertiaabouttheshearcenter,andus,vsandwsaredisplacementsoftheshearcenter.87A.3EquationsofMotionForthecaseofasingleassumedmodeforeachdeformationcoordinate,theequationsofmotionforthede˛ectionscenteredatthecentroidareqv1ZL0m(x)2v1(x)+J(x)02v1(x)dx+qw1ZL0J(x)0v1(x)0w1(x)dx+qv1ZL0EI(x)002v1(x)+_˚2xm(x)Zx002v1(˘)d˘J(x)02v1(x)m(x)2v1(x)+m(x)gcos˚Zx0v1(˘)02d˘dx+qw1ZL0EI00v1(x)00w1(x)_˚2J0v1(x)0w1(x)dx2_q1ZL0_˚J0v11dx+ZL0gm(x)sinv1(x)dx=Qv1;(A.13)qw1ZL0m(x)2w1(x)+J(x)02w1(x)dx+qv1ZL0J(x)0v1(x)0w1(x)dx+qw1ZL0EI(x)002w1(x)+_˚2xm(x)Zx002w1(˘)d˘J(x)02w1(x)+m(x)gcos˚Zx0w1(˘)02d˘dx+qv1ZL0EI00v1(x)00w1(x)_˚2J0v1(x)0w1(x)dx2_q1ZL0_˚J0w11dx=Qw1;(A.14)q1ZL0(J(x)+J(x))21(x)dx+q1ZL0hG(I(x)+I(x))021(x)_˚2J(x)21(x)idx+2_qvZl0J_0v11dx+2_qwZL0J_0w11dx+ZL0_˚2J1(x)dx=Q1:(A.15)88APPENDIXBIN-PLANETHREEBLADETURBINEEQUATIONSB.1In-PlaneEnergyExpressionsInthisanalysisonlyin-planevibrationsaretakenintoaccount.v(x;t)isapproximatedwithasinglemode.Foreachbladeitisassumedthatvj(x;t)=v(x)qj(t),wherev(x)isthe˝rstcantileverbeammode,andqj(t)istheassumedmodalcoordinateforthejthblade.In-planeenergyformulationsforasinglebladearegivenintermsoftheassumedmodalcoordinatesbelow.T(qj;_qj;_˚)=12ZL0m(x) _˚ x+q2j2Zx00v(˘)2d˘!+_qjv(x)!2+m(x)(_˚qjv(x))2+J(x)(_˚+_qj0v(x))2+(_˚qj0v(x))2dx;(B.1)U(qj)=12ZL0EI(x)q2j00v(x)2dx;(B.2)Vg(qj;˚j)=ZL0m(x)g" x+q2j2Zx00v(˘)2d˘!cos˚jqjv(x)sin˚j#dx:(B.3)89B.2ParametersusedintheEquationsofMotionExpressionsfortheparametersinEquations(3.4)and(3.5)aregivenbelow:mb=ZL0m(x)v(x)2+J(x)0v(x)2dx;k0=ZL0EI(x)00v(x)2dx;k1=ZL0xm(x)Zx00v(˘)2d˘m(x)v(x)2J(x)0v(x)2dx;k2=ZL0gm(x)Zx00v(˘)2d˘dx;d=ZL0gm(x)v(x)dx;e=ZL0(xm(x)v(x)+J(x)0v(x))dx;Jr=Jhub+3ZL0(x2m(x)+J(x))dx;Qj=ZL0fj(x)v(x)dx;Q˚=3Xj=1ZL0xfj(x)dx:wherexistheaxisalongthelengthoftheundeformedblade,m(x)ismassperunitlength,EIandJarethein-planebendingsti˙nessandmassmomentofinertiaperlengthabouttheneutralaxis,Jhubisthehubinertia,vistheassumedmodalfunction,whichisthe˝rstuniformcantileverbeammode,andfj(x)accountsforthedistributedaerodynamicloadsonthejthblade.Intheseexpressions,()0=d()=dx.Asimpli˝edmodelisusedwherethe˛owisassumedtobesteady,andthewindspeedisassumedtobeslightlyincreasinglinearlywithheighth(i.e.uwind=u0+1=u0cos˚ju1).Neglectingthecontributionofstatevariationsontheangleofattack,theliftforceisproportionaltoj~urelj2,where~urel=~uwind~ubladeand~ublade=x_˚^yj.~ureland90fj(x)arefoundas~urel=(u0cos˚ju1)^zx_˚^yj;fj(x)=cph(u0cos˚ju1)2+(x_˚)2i;wherecpisaconstantwhichiscomposedoftheairdensity,liftcoe˚cient,andothergeo-metricparameters,^xjand^yjaretheaxialandthein-planebendingdirectionsofthejthblade,andzistheout-ofplanedirection,asshowninFigure3.1.Pluggingfj(x)intotheQjandQ˚expressions,weobtainQj=ZL0cpu2020u1xcos˚j+_˚2x2+O(2)v(x)dx;Q˚=3Xj=1cpu20L22+_˚2L4420u1cos˚jL33+O(2):NotethatP3j=1cos˚j=0,thereforeQ˚canbewrittenasQ˚=^Q˚0+^Q˚1_˚2.Forsmall,onecanassumethatQjhastheformQj=^Qj0+^Qj1cos˚j+^Qj2_˚2.Plugging_˚==+21),weobtainQ˚=^Q˚0+^Q˚12+O(2)andQj=^Qj0+^Qj22+^Qj1cos˚j+O(2):Sinceisconstant,onecansimplywriteQ˚=Q˚0+O(2)andQj=Qj0+j1cos˚j+O(2).B.3AnAlternativeScalingSchemeAnalternativescalingschemeisinvestigatedwhere=1+1;~cb=^cb;~k2=^k2;~d=^d;~cr=^cr;˜=qj=j;~Qj=^Qj~Q˚=^Q˚:Theserelationsleadtod2sjd˚2+^k0sj+^dsin˚j+~e1d˚+"21d2sjd˚2+1d˚dsjd˚+^cbdsjd˚+^k2cos˚jsj+~11d˚=^Qj;(B.4)911d˚+^cr+"11d˚+^cr1+3Xk=1~ed2skd˚2#=^Q˚:(B.5)Notethat1standsforthesmalloscillationsfromthemeanspeed.Thereforespeedvariationsintroducedbytheexternalforces(^Q˚^cr)canbeconsideredasavariationinthemeanspeed,andcanbeomittedfromthe1equation.Thebladeandrotorequationscannotbedecoupledatthisstage.Insteada˝rstordermultiplescalesanalysiscanbeapplieddirectly.Writingtheequationsin domain,onecan˝nds00j+sj+~ep101=Fjsin˚j+"21s00j+01s0j+^cbp1s0j+^k2^k0cos˚jsj+~ep1101#;(B.6)01="101+^crp11+3Xk=1~ep1s00k#;(B.7)whereFj=^Qj^k0and=^d^k0.Toapplythemethodofmultiplescales,onecanwritesj=sj0+j1and1=10+11.The0and1equationsare:0:D20sj0+sj0+~ep1D010=Fj0sin(r1 0+2ˇ3);(B.8)D010=0;(B.9)921:D20sj1+2D0D1sj0+sj1+~ep1(D011+D110)+D010D0sj0=210D20sj0^cbp1D0sj0^k2^k0cos(r1 0+2ˇ3)sj0~ep110D010(B.10)D011=D11010D010^crp110~ep13Xk=1D20sk0:(B.11)Equation(B.9)impliesthat10isnotafunctionof 0,itisonlyafunctionof 1(10=10( 1)andD010=0).Thisleadsto~ep1D010termtodropo˙fromtheEquation(B.8).Thereforetheleadingordersjequation,aswellasthesolution,arethesameasthoseinChapter3.AlsotheEquation(B.11)reducestoD011=D110^crp110~ep13Xk=1D20sk0:(B.12)Integratingbothsides,onecanobtain11=(D110^crp110) 0~ep13Xk=1ZD20sk0d 0:(B.13)Thetermsthataremultipliedby 0canleadtounbounded10solution.Equatingtheseterms(i.e.secularterms)to0,thefollowingequationfor10isfound.D110=^crp110;(B.14)whichhasasolutionoftheform:10=he^crp1 1.Therefore10decayswithtime(orrotorangle),andinthesteadystateitgoesto0.So,theassumptionmadeinChapter3islegitimate.93APPENDIXCAREVIEWONDYNAMICSTALLMODELSTheaerodynamicloads(lift,dragandmoment)arefunctionsoftherelativewindspeedandtheangleofattack,theanglebetweenthechordofthebladecross-sectionandtherelativewindvelocity.Intermsofthebladeparameters,theliftforcecanbewrittenasFL=12cLˆV2cL;(C.1)whereˆisairdensity,Vistherelativewindspeed,cisthechordlength,ListhelengthofthebladeandcLisliftcoe˚cient,whichisafunctionofangleofattack,airfoilshape,aircompressibilityetc.Forsmallanglesofattack,theliftforceusuallyincreaseswiththeangle.However,atacriticalangleofattack(i.e.stallangle),itdropsdramatically.Thisphenomenoniscalledaerodynamicstall,showninFigureC.1.FigureC.1Arepresentativeplotshowingstallphenomenon[2].94C.1StaticStallWhentheangleofattackchangesslowly,andtherearenooscillations,staticstalloccurs.Inthiscase,theliftcoe˚cientisadirectfunctionoftheangleofattackasshowninFigureC.1.Therelationcanbedeterminedby˝ttingamathematicalmodeltotheliftdataobtainedfromstatictests.Staticstallmodelscanbeusedforsystemsinwhichtheangleofattackvariesquasi-statically.C.2DynamicStallInawindturbineblade,theangleofattackchangesrapidlyduetovariationsinangularspeed,pitchingmotionandthevibrationsoftheblade.Rapidvariationsinangleofattackmakeliftforcehistorydependent,meaningthatahysteresisoccursinliftforcecurvebetweenincreasinganddecreasingangleofattackcases,asshowninFigureC.2.Tomodeltheliftforceindynamicstallconditions,semi-empiricalmethodsarewidelyused[2,4,63].Thesemethodsrequireexperimentsindi˙erentrangesofangleofattackandamathematicalmodeltode˝nethebehavioroftheempiricalliftcurves.C.3ONERA'sDynamicStallModelThismodelisdevelopedbyFrenchAerospaceLabONERA[2],andhereisasummarygiven.Themodelisusedforhelicopterbladesinforward˛ightwheretherelativevelocitycanbeseparatedintotwopartsv=v0+v1;averagerotorspeed(constant)andvariationsduetooscillations,blade˛apandchangesintherotorspeed[2].Themotionoftheairfoilisde˝nedbythreefunctions;theangleofattack(˝),thepitchrate_(˝)andthevelocityratio˙(˝)=v=v0,where˝isreducedtime,˝=2jv0jt=c.AdiagramshowingtheseparametersisgiveninFigureC.3.ThevelocityratioisdirectlyproportionaltoMachnumber,andaccountsforcompressibilitye˙ects.Thelift,dragand95FigureC.2Arepresentative˝gureshowingdynamicstall[3].momentcoe˚cientsaregivenasFigureC.3AirfoildiagramusedinONERA'smodel[2].Normalliftcot:cL=N12ˆjvj2cL(C.2)Thecoe˚cientdependsonthetimehistoryof˙,,_andtheirsuccessivederivatives.Tointroducethetimehistorye˙ects,thebehaviorofcoe˚cientsareexpressedwithadi˙erentialequation,asAL(cL;˙;;_;_cL;_˙;_;;:::)=0:(C.3)96AccordingtotheexperienceacquiredatONERAUnsteadymovementsofanairfoilarelimitedeitherinfrequencyorinamplitude.Itisassumed_˙;_;_;_cL,andtheirhigherderivativesareInmostcases,cLdeviatesonlyslightlyfromitsstaticcharacteristicscL0.AdmittingALisdi˙erentiable,Eq.(C.3)canbewrittenasAL(P0)+XkhALcL(cLcL0)+AL_cL_cL+:::i+AL__+AL__+AL_˙_˙+:::=0;(C.4)whereP0(cL0;˙;;0;:::)isthesetofparametersinthestaticcase,cL0isthestatic(mean)valueofcLand_˙0=_0=_0=:::=0.ALxarepartialderivativesasALcL=@AL=@cL,AL_cL=@AL=@_cL.AL(P0)=0equationsde˝necL0=CL(;˙),foundfromstaticwindtunneltests.ThestationarylimitofALcanbewrittenasAL(P0)=cL0CL(;˙)=0.RewritingEq.(C.4),weget3Xk=1(ALcLcL+AL_cL_cL+:::)=3Xk=1ALcLCLAL__AL__:::(C.5)Notethatthecoe˚cientsvarywithand˙.Ifthemotionofanairfoilisimposed,_(˝),(˝)and˙(˝)areknown,andEq.(C.5)canbewrittenas3Xk=1(ALcLcL+AL_cL_cL+:::)=Sr(˝):(C.6)Presumablythecoe˚cientsareknownfromexperiments,andthusSr(˝)canbeknownexplicitly.IfALissuchthatALcL,AL_cLareconstants,thenwehavealinearforcedODEwithconstantcoe˚cients.TheexperienceacquiredatONERAjusti˝esThederivativesof,˙andoforderhigherthantwocanbeneglected.Ingeneral,onesinglerealpoleandtwocomplexconjugatepolesaresu˚cienttoeval-uatecorrectlytheevolutionofharmonicresponses.97Then,Eq.(C.5)canberewrittenasALcLcL+AL_cL_cL+ALcLcL+AL...cL...cL=ALcLCL(˙;)AL__AL__AL_˙_˙ALALAL˙˙:(C.7)CL(;˙)canbemeasuredforvariousincidences.Aleastsquarecurve˝ttingcanbeappliedto˝ndapproximateanalyticalformula.Thisformulahastwodomains:linearandstall.Betweenthetwo,thecontinuityofCLmustbeinsured.Identi˝cationoftheothercoe˚cientsrequiresunsteady˛owtesting.C.3.1ONERA'sModelAppliedonaHelicopterAirfoilCross-sectionThisisanapplicationofONERA'smodeltoahelicopterblade.AlltheworkwasconductedbyNASA[3].Ahelicopterbladewastestedundervariousmeanincidences[3],oscillatingwithvariousamplitudes.Themovementisde˝nedbyonlythepitchangle(=)andtheconstantspeed(v=v0).Theliftcoe˚cientiscomposedoftwopartscL=cL1+cL2.Inthelinearregion,changesintheliftcoe˚cientaresmoothandcanberepresentedbycL1,dynamicsofwhichisgovernedbyasinglenegativerealpole,whereasinstallregion,rapidvariationsoccurandcL2isintroducedwhichhastwocomplexconjugatepolestorepresentthebehavior.TheequationsgoverningthecL1andcL2are_cL1+L1=L0L+(+)_+˙;(C.8)cL2+2_cL2+2(1+2)cL2=2(1+2)cL0+cdcL0d˝:(C.9)wherecL0ListhestaticliftinthelinearregionandcL0=cL0LcL0isthedeviationofrealstaticliftcurvefromthelinearlift.Bydoingtestsatsmallincidences,therealnegativepolecanbefoundfromthe˝rstequation.However,to˝ndthepolesofthesecondequation,i,dynamictestinginhigherincidencesisrequired.98C.4ASemi-empiricalModelbyLarsenetal.ThemodelusedbyLarsenetal.wasdevelopedforfullyattachedandseparated˛owcon-ditionsseparately,asshowninFigureC.4[4].Theymodeledthedelaye˙ectsinseparationandre-attachmentwithdi˙erentialequations.Thecontributionofleadingedgevortexanditsmovementswerealsotakenintoaccount.FigureC.4Diagramsshowingfullyattachedandseparated˛owconditions[4].C.4.1StationaryLiftandSeparationUnderfullyattached˛ow,theliftcoe˚cient,cL0,islinearizedforsmall,ascL0=@cL@0(0):(C.10)Forseparated˛ow,however,theliftcoe˚cientdeviatesfromcL0,viatherelationcL'1+pf22cL0;(C.11)wherefisthedegreeofattachment.Forfullyattached˛ow,f=1andcL=cL0,whereasforfullyseparated˛ow,f=0andcL=14cL0.ChangesindegreeofattachmentresultsinchangesincLviatherelationdcL=141+1pfcL0df:(C.12)99Noticethatforfullyseparated˛owconditions(f=0),averysmallchangeinfresultsinaverylargevariationinliftcoe˚cient.Togetridofthissingularity,thephysicalpro˝lewasmappedonaunitcircle,asshowninFigureC.5.FigureC.5Mappingfromairfoilpro˝letoaunitcircle[4].Insteadoff,isusedtode˝nethedegreeofattachment.Theconversiontakesplaceas2f=1+cos.InsertingthisintoEq.(C.11),weobtaincL'cos44cL0:(C.13)C.4.2DynamicLiftForfullyattached˛ow,forasmallchangeinangleofattack,,asmallseparationoccursbeforetheattachingisre-established.Thisconstitutesachangeinliftforce.Hence,theincrementdcL0duetoisnotachievedinstantaneously,Thisdelaycanbemodeledviaadelayfunctiont),dcL0;d(t)=t˝)dcL0(˝):(C.14)Forincompressible˛ow,forathinpro˝le,halftheincrementisfeltinstantaneously.So,=12and1)=1.So,thelineardynamicliftcoe˚cientforattached˛ow,canbewrittenascL0;d(t)=Ztt˝)_cL0(˝)d˝;(C.15)wherethedelayfunctioncanbeexpressedast)=1A1e!1tA2e!2t.Thecoe˚cientsandexponents,A1;A2;!1and!2arepro˝ledependentvariablesdescribingthetimedelay.100Forathinpro˝leA1+A2=12,and!1and!2representtimescaleforlowandhighfrequencycontributions,respectively._cj(t)+!jcj(t)=Aj_cL0(t)j=1;2andcL0;d(t)=cL0()c1(t)c2(t):(C.16)Forseparated˛ow,astepchanceinresultsinachangein.Yet,atimeintervalisobservedduringwhichtheseparationanglemovestoitsnewstationaryvalue.Thedynamicattachmentangleisgovernedbytheequation_d(t)=!3(d(t)())Then,thedynamicliftcoe˚cientiscL;d'cos4d4cL0;d(t):(C.17)Theexperimentaldataindicatesthatleadingedgeseparationgeneratesalinearlyin-creasingliftcurveevenatfullseparation.Tomodelthislinearliftcurve,acorrectiontermisaddedtocL;d(t).cL(t)=cL0;d(t)cL;d(t)Atacertainanglev,theleadingedgevortexdetachesfromtheleadingedge,andtravelsdownstreamoverthepro˝le.Thetravelingvortexbuildsupstrength,andasitreachesthetrailingedge,itstops,correspondingto_cL(t)=0andavortexwithoppositecirculationstartstodevelopatthetrailingedge.Thetrailingedgevortexcounteractstheleadingedgevortexandtheliftstartsdiminishing._cL;v(t)+!4cL;v(t)=8>><>>:_cL(t)for>vand_>00otherwise(C.18)andcL(t)=cL;d(t)+cL;v(t):(C.19)101Theequationsgoverningtheliftcoe˚cientscanbewritteninstatevariableform,as_z(t)=Az(t)+b0()+b1_cL0(t);(C.20)wherez(t)=0BBBBBBB@c1(t)c2(t)d(t)cL;v(t)1CCCCCCCA;A=266666664!10000!20000!30000!4377777775;b0()=0BBBBBBB@00!3()_cLH(v)H(_)1CCCCCCCA;b1=0BBBBBBB@A1A2001CCCCCCCA:C.5ConclusionsExamplesofsemiempiricalaerodynamicstallmodelsarereviewed.ONERA'smodelwasusedonhelicopterbladeswheretherelativewindvelocityisassumedtooscillatearoundameanspeed.Thebehaviorofliftforceisexpressedviaathirdorderdi˙erentialequation,takingpitchingmotionandcompressibilitye˙ectsintoaccount.To˝ndthecoe˚cientsoftheequation,theairfoilmustbetestedundersteadyandunsteady˛owconditions.ThemodelusedbyLarsenetal.,ontheotherhand,wasappliedtowindturbineblades.Sinceturbinebladesoperateatsmallspeeds,compressibilitye˙ectsareomitted.Themathematicalmodeltakes˛owseparation,leadingedgevortexanddelaye˙ectsintoaccount.Larsen'smodelissimplerthanONERA's,andmoresuitableforwindturbines.Oneoftheabovemethodscanbeusedonanexistingblademodelto˝ndtheliftcoe˚cientasafunctionofangleofattack.102BIBLIOGRAPHY103BIBLIOGRAPHY[1]R.Wiser,L.Eric,T.Mai,J.Zayas,E.DeMeo,E.Eugeni,J.Lin-Powers,andR.Tusing.Windvision:ANewEraforWindPowerintheUnitedStates.TheElectricityJournal,2015.[2]C.T.TranandD.Petot.Semi-empiricalmodelforthedynamicstallofairfoilsinviewoftheapplicationtothecalculationofresponsesofahelicopterbladeinforward˛ight.Vertica,1980.[3]K.W.McAlister,O.Lambert,andD.Petot.ApplicationoftheONERAmodelofdynamicstall.Technicalreport,DTICDocument,1984.[4]J.W.Larsen,S.R.K.Nielsen,andS.Krenk.Dynamicstallmodelforwindturbineairfoils.JournalofFluidsandStructures,2007.[5]Monthlyenergyreview.U.S.EnergyInformationAdministration(EIA).Publishedonline,2015.http://www.eia.gov/totalenergy/data/monthly/.[6]DOE/GO-102008-2578.20%WindEnergyBy2030:Increasingwindenergy'scontri-butiontoU.S.electricitysupply.U.S.DepartmentofEnergy,2008.[7]C.Lindenburg.AeroelasticmodellingoftheLMH64-5blade.EnergieonderzoekCentrumNederland,02-KL-083,Petten,December,2002.[8]J.M.Jonkman,S.Butter˝eld,W.Musial,andG.Scott.De˝nitionofa5-MW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