IN-PLANEBLADE-HUBDYNAMICSOFHORIZONTAL-AXISWINDTURBINESWITH TUNEDANDMISTUNEDBLADES By AyseSapmaz ADISSERTATION Submittedto MichiganStateUniversity inpartialful˝llmentoftherequirements forthedegreeof MechanicalEngineeringDoctorofPhilosophy 2020 ABSTRACT IN-PLANEBLADE-HUBDYNAMICSOFHORIZONTAL-AXISWINDTURBINESWITH TUNEDANDMISTUNEDBLADES By AyseSapmaz Understandingvibrationofwindturbinebladesisoffundamentalimportance.Thisstudyfocuses onthee˙ectofblademistuningonthecoupledblade-hubdynamics.Unavoidably,thesetofblades arenotpreciselyidenticalduetoinhomogeneousmaterials,manufacturertolerances,etc. Thisworkisfocusedontheblade-hubdynamicsofhorizontal-axiswindturbineswithmistuned blades.Thereduced-orderequationsofmotionarederivedforthewindturbinebladesandhub exposedtocentrifugale˙ectsandgravitationalandcyclicaerodynamicforces.Althoughtheblades andhubequationsarecoupled,theycanbedecoupledfromthehubbychangingtheindependent variablefromtimetorotorangleandbyusingasmallparameterapproximation.Theresulting bladeequationsincludeparametricanddirectexcitationterms.Themethodofmultiplescalesis appliedtoexamineresponseofthelinearizedsystem.Thisanalysisshowsthatsuperharmonicand primaryresonancesexistandarein˛uencedbythemistuning.Resonancecasesandtherelations betweenresponseamplitudeandfrequencyarestudied.Besidesillustratingthee˙ectsofdamping andforcinglevel,the˝rst-orderperturbationsolutionsareveri˝edwithcomparisonstonumerical simulationsatsuperharmonicresonanceofordertwo.Thesimulationpointtospeed-locking phenomenon,inwhichthesuperharmonicspeedislockedinforanintervalofappliedmeanloads. Additionally,thee˙ectofrotorloadingontherotorspeedandbladeamplitudesisinvestigatedfor di˙erentinitialconditionsandmistuningcases.Lastly,weaimtoanalyticallycon˝rmtheblade responseamplitudesatvariousrotorspeedsnearresonanceandverifyspeedlockingphenomenon byapplyingmethodofharmonicbalance.Thestudyshowsthatthespeed-lockingisduetothe averageinteractionbetweenthebladevibrationandrotormotionintherotorequation,andits balanceagainstthemeanrotormoment.Thephenomenonisexaminedforane˙ective(balanced) singleblade-rotorsystem. Next,asecond-ordermethodofmultiplescalesisappliedintherotor-angledomaintoanalyze in-planeblade-hubdynamics.Asuperharmonicresonancecaseatonethirdthenaturalfrequency isrevealed.Thisresonancecaseisnotcapturedbya˝rst-orderperturbationexpansion.The relationshipbetweenresponseamplitudeandfrequencyisstudied.Resonancesunderconstant loadingarealsoanalyzed.Thee˙ectofblademistuningonthecoupledblade-hubdynamicsis takenintoaccount. Tobetterunderstandparametricallyexcitedmulti-degree-of-freedombehavior,approximate solutionstotunedandmistunedfour-degree-of-freedomsystemswithparametricsti˙nessare studied.Thesolutionandstabilityofafour-degree-of-freedomMathieu-typesystemisinvestigated withandwithoutbrokensymmetry.TheanalysisisdoneusingFloquettheorywithharmonic balance.AFloquet-typesolutioniscomposedofaperiodicandanexponentialpart.Theharmonic balanceisappliedwhentheFloquetsolutionisinsertedintotheoriginaldi˙erentialequationof motion.Theanalysisbringsaboutaneigenvalueproblem.Bysolvingthis,theFloquetcharacteristic exponentsandthecorrespondingeigenvectorsthatgivetheFouriercoe˚cientsarefoundintermsof thesystemparameters.Thestabilitytransitioncurvecanbefoundbyanalyzingtherealpartsofthe characteristicexponents.Thefrequencycontentcanbedeterminedbyanalyzingimaginarypartsat theexponents.AresponsethatinvolvesasingleFloquetexponent(anditscomplexconjugate)can begeneratedwithaspeci˝csetofinitialconditions,andcanberegardedasaresponse Themethodisappliedtobothtunedanddetunedfour-degree-of-freedomexamples. Copyrightby AYSESAPMAZ 2020 Thisthesisisdedicatedtomyson,BurakAral.Youhavemademestronger,betterandmore ful˝lledthanIcouldhaveeverimagined.Iloveyoutothemoonandback. v ACKNOWLEDGEMENTS ThejourneyIhavetakenduringmyPhDhasbeenthemostimpactfulofmylife.Ilearnednotonly professionalandpracticalskills,butalsolearnedalotaboutmyselfintheprocess. ThisachievementdidnotcomewithouthelpfrommanypeoplethatIamveryluckytobe surroundedby.Firstandforemost,Iwouldliketoexpressmyheartfeltgratitudeandappreciation tomyacademicadvisorProfessorBrianF.Feenyforhisuniquecontributionstothisstudyandmy developmentasresearcher,scholar,andeducator.Yourunyieldinglypositiveattitude,boundless energy,andenthusiasmsustainedme,whileyourexcellenceanddedicationinspiredme.Thank you,Dr.Feeny,forsogenerouslyprovidingmentorshipandguidance,whilealsovaluingmeasa colleague. IwouldliketothankProfessorStevenShawforbeingagreatinstructor,andfortheintellectual conversationswehad,bothonscienceandlifeingeneral,whichhavebroadenedmyhorizonsin di˙erentdirections.IhavebeenextremelyfortunatetotakesomeclasseswithDr.Shaw,fromwhom Ilearnedmanyinterestingtheoriesandmethodsthathelpedmeinmyworkonthisdissertation.I amtrulyhonoredtohavehadanopportunitytoworkwithProfessorShawoncentrifugalpendulum vibrationabsorber(CPVA)experimentandlearnfromhim.Iwasalsofortunatetolearnfrom ProfessorThomasPenceandIamverygratefulforhispreciousinputstomyresearchtoo.His supportivecommentsandquestionsthathelpedimprovemyworkbythinkingoutofthebox. MysincerelythankstoProfessorKeithPromislowforacceptingbeingpartofmydissertation committee,andhisinvaluablefeedbackanddelightfulsuggestionsonmyresearch. Inadditiontothecommittee,Io˙ersincerethankstomyformerandfellowlab-mates:Scott Strachan,VenkatRamakrishnan,PavelPolunin,SmrutiPanigrahi,RickeyCaldwell,GizemDilber Acar,MustafaAcar,XingXing,MingMu,FatemehAfzali,MahdiehTanha,GauravChauda, ZhaobinZhan,JunGuo,AakashGupta,JamalArdister,JackMetzger.Youhavebeenconsistently supportive,collaborative,insightful,andatruejoytolaboralongside.Youhavemadethelaba funplacetowork.Thedebates,lunches,dinners,camping,jokesanddrivingtotheconferences vi aswellaseditingadvice,havingco˙eeandicecream,generalhelpandfriendshipwereallgreatly appreciated.Morespeci˝cally,IwassoluckytohaveFatemehandEhsanwhohavebeenalways besidesme,lendmetheirhandswhenIaminneed.Wordscannotexpressmyappreciationof havingyoualwaysnear,beingthemarvelousauntanduncletomysonandbeingafamilytousin theUnitedStates.AhugethanksgoestomywonderfulfriendsYen,HayriyeandFiratforlistening, motivatingandsupportingmeemotionallyallthetime.Ifeelveryluckytohaveyouallinmylife. Iwouldalsoliketoexpressmygratitudetomanyotherdoctoralstudents,facultiesandsta˙ofthe MechanicalEngineeringDepartment.Thankyouforcaringandsupportingmewholeheartedly. IgratefullyacknowledgethefundingreceivedtowardsmyPhDfromtheRepublicofTurkey MinistryofNationalEducation,NationalScienceFoundation(NSF)undergrantnumbersCMMI- 1335177andCMMI-1435126,theMSUCollegeofEngineering,DepartmentofMechanicalEn- gineering,GraduateSchool,O˚ceofInternationalStudentService(OISS),StudentParentsonthe Mission(SPOM)andTheCouncilofGraduateStudents(COGS).Theymadethisworkmaterial- ized. ImostofallwanttothankmyfamilybackinTurkeyfortheirongoingsupportsandunconditional lovethattheyhavegiventome.WordscannotexpresshowgratefulIamtomymother,Nazmiye andmyfather,Recepforallofthesacri˝cesthatthey'vemadeonmybehalf.Also,tomybrother, HasanAlithanksforbeingthewonderfulyoungerbrotherwhoisalwaysprotectiveandtakescare ofme.YoualwaysbelievedmethatIwouldachievethingsIplannedevenwhenIdoubtedmyself. Yourprayerformewaswhatsustainedmethusfar.Thisthesiswouldnothavebeencompleted withoutyourunconditionalsupport.Iknowhowmuchyouwouldproudofme,andjustimagining yourproudfacesgivemethestrengthIneedforthenextstepsinmycareer.Itrulyrespectandlove myparents! AveryspecialwordofthanksgoesformydearhusbandBilalforallhehasdonetosupport me,upliftme,comfortme,andbringjoytomysoul.Wearenotjusthusbandandwife,but bestfriendsandcompanionsinbothinthepursuitofthePhDandthisjourneyinlife.Wehave learnedandexperiencedallthestepstogetherhowtobecomemomanddadadequatelytoourlittle vii sunshine.Also,Iamthankfulyouforbearingwithmyabsencefrommyparentingdutiesespecially thelastfewweeksofmyextensivedissertationstudy.Youhavebeeninvaluablethroughoutthis dissertation.Thankyouforbeingpartofmyjourney. Lastbutnotleast,Iwouldliketoexpressmythankstomybelovedson,BurakAralwhohas beenthelightofmylifeforthelastthreeandhalfyearsandwhohasgivenmetheextrastrength andmotivationtogetthingsdonewithalltheinnocence.Thankyouforbeingsuchagoodboy whoalwayscheermeupandforkeepingmyspiritup.Ihavetakenalotfromourmomandson timeandthankyouforyourunderstanding.Youaremyinspirationtoachievegreatness.Without you,IwouldnotbewhereIamtoday.Also,Iwouldliketothankmyunbornbaby,whoiscoming inaroundAugust,forbringingthejoyandhappinessofourlives.Thankstomanyotherpeople whosenamesarenotlistedhere. viii TABLEOFCONTENTS LISTOFTABLES ....................................... xi LISTOFFIGURES ....................................... xii CHAPTER1INTRODUCTION ............................... 1 1.1Objectives.......................................1 1.2Motivation.......................................2 1.3BackgroundandLiteratureReview..........................4 1.3.1In-planeBlade-HubDynamicsofWindTurbineswithMistunedBlades..4 1.3.2ApproximateGeneralReponseofFour-Degree-of-FreedomSystemswith ParametricExcitation.............................7 1.4ThesisOverview....................................7 1.5Contributions.....................................8 CHAPTER2IN-PLANEBLADE-HUBDYNAMICSOFHORIZONTAL-AXISWIND TURBINESWITHMISTUNEDBLADES .................. 10 2.1Introduction......................................10 2.2MistunedThree-BladeTurbineEquations......................12 2.3Multiple-ScalesAnalysis...............................17 2.3.1NonresonantCase...............................19 2.3.2SuperharmonicResonance( 2 ! 1 ˇ 1 or 2 ˇ ! n 2 ), 2 ! 1 = 1 + ˙ ....20 2.3.3PrimaryResonance( ! 1 ˇ 1 or ˇ ! n 2 )..................23 2.3.4SubharmonicResonance( ! 1 ˇ 2 or ˇ 2 ! n 2 )...............28 2.3.5ExistenceofResonanceConditions.....................30 2.4NumericalSimulation.................................32 2.5SpeedLockingAnalysisforHAWTswithTunedBlades...............35 2.5.1PrimaryResonanceCase...........................36 2.5.2SuperharmonicResonanceCaseat 2 ˇ ! n 2 ................41 2.6Conclusions......................................44 CHAPTER3SECOND-ORDERPERTURBATIONANALYSISOFIN-PLANEBLADE- HUBDYNAMICSOFHORIZONTAL-AXISWINDTURBINES ...... 49 3.1Introduction......................................49 3.2ThreeBladeTurbineEquations............................49 3.3Second-OrderMethodofMultipleScales......................52 3.3.1NonresonantCaseat O ¹ º ..........................54 3.3.2SuperharmonicCaseatOrder 3 at O ¹ 2 º ..................56 3.4Results.........................................58 3.4.1TunedBladeCase( v = 0 )..........................58 3.4.2MistunedBladeCase( v , 0 )........................59 3.5RemainingTask....................................60 ix 3.5.1SuperharmonicCaseatOrder 2 at O ¹ º ...................60 3.5.2SuperharmonicCaseatOrder 2 at O ¹ 2 º ...................64 3.6Conclusions......................................68 CHAPTER4APPROXIMATEGENERALRESPONSESOFTUNEDANDMISTUNED 4-DEGREE-OF-FREEDOMSYSTEMSWITHPARAMETRICSTIFFNESS70 4.1Introduction......................................70 4.2Analysis........................................71 4.2.1TunedFour-Degree-of-FreedomExample..................73 4.2.2MistunedFour-Degree-of-FreedomExample................76 4.3Discussion.......................................77 4.4Conclusions......................................80 CHAPTER5ONGOINGWORK ............................... 86 5.1ParametricIdenti˝cationofTheMathieuEquationwithaConstantLoad......86 5.1.1IntroductionandObjective..........................86 5.1.2Background..................................86 5.1.3ParameterEstimationProcedure.......................88 5.1.4ResultsandDiscussion............................90 5.1.5ProposedWork................................91 5.2Second-OrderPerturbationAnalysisofForcedNonlinearMathieuEquation....92 5.2.1Introduction..................................92 5.2.2ANonlinearMathieuEquationwithHardExcitation............93 5.2.3Case1:NoResonanceat O ¹ º ........................95 5.2.4Case2:PrimaryResonanceat O ¹ º .....................95 5.2.5Case3:SubharmonicResonanceofOrder 1 š 2 at O ¹ º ...........96 5.2.6Case4:SuperharmonicResonanceofOrder 2 at O ¹ º ............96 5.2.7Case5:SuperharmonicResonanceofOrder 3 at O ¹ º ............97 5.2.8Conclusion..................................101 5.2.9ProposedWork................................102 CHAPTER6CONCLUSIONANDFUTUREWORK ................... 103 6.1Conclusion......................................103 6.2FutureWork......................................106 APPENDICES ......................................... 107 APPENDIXAIN-PLANETHREE-BLADEMISTUNEDTURBINEEQUATIONS . 108 APPENDIXBWINDTURBINESPARAMETERS .................. 112 APPENDIXCSECOND-ORDERPERTURBATIONANALYSISOFIN-PLANE THREE-BLADETUNEDANDMISTUNEDTURBINES ....... 122 APPENDIXDSECOND-ORDERPERTURBATIONANALYSISOFNONLIN- EARMATHIEUEQUATIONWITHHARDEXCITATION ...... 125 BIBLIOGRAPHY ........................................ 127 x LISTOFTABLES Table3.1:ResonanceChart. R 1 :Resonanceidenti˝edat˝rst-orderofMMSexpansion. R 2 :Resonanceidenti˝edatsecond-orderofMMSexpansion. :Known resonancecase/Instabilitynotuncovereduptotwoordersofexpansion.....58 Table4.1:Primary,superharmonicandsubharmonicinstabilitywedgesbasedat ! i + ! j N fortunedandmistunedcases............................85 TableB.1:StructuralPropertiesfortheBaselineWindTurbineModels...........113 TableB.2:Distributedbladestructuralpropertiesforthe5MWmodeltakenfromNREL..118 TableB.3:Blademasscomparisonforbaselinemodelandmid-pointnumericalintegra- tionmethod(Mid-pointI.M.)............................119 TableB.4:Blademodalparametersforbaselinewindturbinemodels............119 TableB.5:Blademodalfrequencies(Hz)comparisonwiththedatafromNRELandSANDIA120 TableB.6:Parametrice˙ectforscaledbladesize.......................120 TableB.7:Modalfrequencies(Hz)forthebaselineblademodels...............120 TableB.8:Modalfrequencies(Hz)forthescaledblademodelsofNREL60.8m......120 TableB.9:Parametrice˙ect ¹ k 2 š k 0 º forthescaledbladeofNRELbaselinemodels.....121 xi LISTOFFIGURES Figure1.1:Globalcumulativeinstalledwindcapacitybetween2001-2017.........2 Figure1.2:MonthlynetelectricitygenerationfromselectedfuelsbetweenJan2007-Mar 2017-shareoftotalelectricitygeneration.....................3 Figure1.3:TheWindVisionstudyscenarioaboutshareinelectricityproductionintheU.S.4 Figure1.4:Averageturbinenameplatecapacity,rotordiameter,andhubheightinstalled duringperiod....................................5 Figure1.5:AnnualFailureFrequencyperTurbineSubsystem2012..............6 Figure2.1:Amistuningthree-bladeturbinewithbladesunderin-planebending.......12 Figure2.2:Eigenvaluesversuselasticsti˙nessmistuningparameter( k v )plotfor k 0 = 5 , k 1 = 0 : 5 , Û = 0 , m = 1 , J = 1 , e = 0 : 2 , = 0 : 1 (e1:˝rstblade,e2:second blade,e3:thirdblade)...............................14 Figure2.3:In-planemodeshapesofasymmetricthree-bladeturbine.............15 Figure2.4:Steady-statesuperharmonicresonancebladeresponseamplitudesversuselas- ticsti˙nessmistuningparameter v for ~ e = 0 : 2 , ˙ = 0 : 0064 , = 0 : 1 , j B j = 1 , ^ = 0 : 0445 .Thethirdbladeisthemistunedblade............21 Figure2.5:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuning parameterfor = 0 : 1 , ~ e = 0 : 2 , v = 0 , j B j = 1 , ^ = 0 : 005 (a1:˝rstblade, a2:secondblade,a3:3rdblade).........................23 Figure2.6:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuning parameterfor = 0 : 1 , ~ e = 0 : 2 , v = 0 : 006 , j B j = 1 , ^ = 0 : 005 (a1:˝rst blade,a2:secondblade,a3:3rdblade)......................24 Figure2.7:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuning parameterfor = 0 : 1 , ~ e = 0 : 2 , v = 0 : 1 , j B j = 1 , ^ = 0 : 005 (a1:˝rstblade, a2:secondblade,a3:3rdblade).........................24 Figure2.8:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuning parameterfor = 0 : 1 , ~ e = 0 : 2 , v = 1 , j B j = 1 , ^ = 0 : 005 (a1:˝rstblade, a2:secondblade,a3:3rdblade).........................25 xii Figure2.9:Steady-statesuperharmonicresonancebladeresponseamplitudesversusfre- quencyfor = 0 : 1 , ~ e = 0 : 2 , v = 0 : 1 , j B j = 1 , ^ = 0 : 005 ; 0 : 008 ; 0 : 01 ; 0 : 015 ..25 Figure2.10:Steady-statesuperharmonicresonancebladeresponseamplitudesversusfre- quencyfor = 0 : 1 , ~ e = 0 : 2 , v = 0 : 1 , j B j = 1 ; 2 ; 3 ; 4 , ^ = 0 : 05 .........26 Figure2.11:Steadystateprimaryresonanceresponseamplitudesversuselasticsti˙ness mistuningparameter( v )for = 0 : 1 , ˙ = 0 , ~ e = 0 : 2 , F j 0 = 0 , F j 1 = 0 : 001 , ^ = 0 : 005 , t = 1 , j B j = 1 (a1:˝rstblade,a2:secondblade,a3:3rdblade)...27 Figure2.12:Steadystateprimaryresonanceresponseamplitudesversusdetuningparam- eterfor = 0 : 1 , v = 0 : 005 , ~ e = 0 : 2 , F j = 0 , F j 1 = 0 : 001 , ^ = 0 : 005 , t = 1 , j B j = 1 ,(a1:˝rstblade,a2:secondblade,a3:3rdblade)............28 Figure2.13:Steadystateprimaryresonanceresponseamplitudesversusdetuningparam- eterfor = 0 : 1 , v = 0 : 1 , ~ e = 0 : 2 , F j = 0 , F j 1 = 0 : 001 , ^ = 0 : 005 , t = 1 , j B j = 1 ,(a1:˝rstblade,a2:secondblade,a3:3rdblade)............29 Figure2.14:Steadystateprimaryresonanceresponseamplitudesversusdetuningparam- eterfor = 0 : 1 , v = 1 , ~ e = 0 : 2 , F j = 0 , F j 1 = 0 : 001 , ^ = 0 : 005 , t = 1 , j B j = 1 ,(a1:˝rstblade,a2:secondblade,a3:3rdblade)............30 Figure2.15:Campbelldiagramshowing ! n 2 asafunctionof ,for k 0 = 1 , k 1 = 0 : 5 , m b = 1 .Intersectionwiththe 2 , and 2 linesindicatepossibleresonances..31 Figure2.16:Campbelldiagramshowing ! n 2 asafunctionof ,for k 0 = 1 , k 1 = 0 : 1 , m b = 1 .Intersectionwiththe 2 , and 2 linesindicatepossibleresonances..32 Figure2.17:RealpartofeigenvaluesofXandYequationsystemversuselasticsti˙ness mistuningforsubharmoniccasefor = 0 : 1 , ~ e = 0 : 2 , ^ = 0 : 005 , ˙ = 0 : 0005 ,..33 Figure2.18:RealpartofeigenvaluesofXandYequationsystemversusdetuningparam- eter(rotorfrequency)forsubharmoniccasefor = 0 : 1 , ~ e = 0 : 2 , ^ = 0 : 005 , v = 0 : 1 ,......................................34 Figure2.19:RealpartofeigenvaluesofXandYequationsystemversusdetuningparam- eter(rotorfrequency)forsubharmoniccasefor = 0 : 1 , ~ e = 0 : 2 , ^ = 0 : 005 , v = 0 : 35 ,.....................................35 Figure2.20:CampbelldiagramforNREL5MW64meterbladeshowing ! n 2 asafunction of ,withparameters k 0 = 603670 N : m , k 1 = 4479 : 3 k g : m 2 , mb = 14441 k g : m 2 .Intersectionwiththe 2 , 3 , and 2 linesindicatepossiblereso- nancesatthecorresponding ...........................36 xiii Figure2.21: ! n 2 versus actual graphwith IC = 0 : 5 supH , m b = 1 , e = 0 : 2 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , d = 0 : 2 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 .....................................37 Figure2.22:FFTplotof q 1 fortheanalyticalsolutionandsimulationfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 , Q ˚ 0 ˇ 0 : 0198 , = 1 : 225 , ! n 2 = 2 : 2451 ......38 Figure2.23:FFTplotof q 2 fortheanalyticalsolutionandsimulationfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 , Q ˚ 0 ˇ 0 : 0198 , = 1 : 225 , ! n 2 = 2 : 2451 ......39 Figure2.24:FFTplotof q 3 fortheanalyticalsolutionandsimulationfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 , Q ˚ 0 ˇ 0 : 0198 , = 1 : 225 , ! n 2 = 2 : 2451 ......40 Figure2.25:FFTplotof Û ˚ fortheanalyticalsolutionandsimulationfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 , Q ˚ 0 ˇ 0 : 0198 , = 1 : 225 , ! n 2 = 2 : 2451 ......41 Figure2.26: actual versus Q ˚ Graphwithnegativeandpositivemistuningcasesandtuned casefor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 .................42 Figure2.27: actual versus Q ˚ graphfordi˙erentinitialconditionfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 ............................43 Figure2.28: ˙ desired versus ˙ actual withnegativeandpositivemistuningcasesandtuned casefor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 .................44 Figure2.29: ˙ desired versus ˙ actual graphfordi˙erentinitialconditionfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 ............................45 Figure2.30: Q ˚ versusperturbationsolutionamplitudegraphfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 ............................46 Figure2.31:Speed-lockinggraph versus Q ˚ forprimaryresonancefor = 0 : 1 ; 2 = 0 : 5 ;! n 2 = 2 : 23 ; d = 1 : 871 .............................47 xiv Figure2.32:Speed-lockinggraph versus Q ˚ forsuperharmonicresonanceordertwofor g = 1 ; = 0 : 01 ; 2 = 0 : 0135 ;! n 2 = 2 : 23 ; d = 1 : 871 ; = 0 : 5 ..........48 Figure3.1:Eigenvaluesversuselasticsti˙nessmistuningparameter( k v )plotfor k 0 = 5 , k 1 = 0 : 5 , Û = 0 , m = 1 , J = 1 , e = 0 : 2 , = 0 : 1 (e1:˝rstblade,e2:second blade,e3:thirdblade)...............................51 Figure3.2:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuning parameterfor ~ e = 0 : 2 , = 0 : 1 , j j = 10 , ˚ = 0 , = 0 : 1 , = 0 : 005 (a1:˝rst blade,a2:secondblade,a3:thirdblade).....................60 Figure3.3:Steadystatesuperharmonicresonanceresponseamplitudesversusfrequency for ~ e = 0 : 2 , = 0 : 1 , j j = 10 , ˚ = 0 , = 0 : 1 , = 0 : 005 ; 0 : 01 ; 0 : 03 .......61 Figure3.4:Steadystatesuperharmonicresonanceresponseamplitudesversusfrequency for ~ e = 0 : 2 , ˚ = 0 , = 0 : 1 , = 0 : 005 , = 1 ; 2 ; 3 ................61 Figure3.5:Steadystatesuperharmonicresonanceresponseamplitudesversuselastic sti˙nessmistuningparameter v for ~ e = 0 : 2 , = 0 : 1 , j j = 1 , ˚ = 0 , = 0 : 1 , = 0 : 005 , ˙ = 0 (a1:˝rstblade,a2:secondblade,a3:thirdblade).......62 Figure3.6:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuning parameterfor ~ e = 0 : 2 , = 0 : 1 , j j = 10 , ˚ = 0 , = 0 : 1 , = 0 : 005 (a1:˝rst blade,a2:secondblade,a3:thirdblade).....................63 Figure3.7:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuning parameterfor ~ e = 0 : 2 , = 0 : 1 , j j = 10 , ˚ = 0 , = 0 : 1 , = 0 : 005 (a1:˝rst blade,a2:secondblade,a3:thirdblade).....................64 Figure3.8:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuning parameterfor ~ e = 0 : 2 , = 0 : 1 , j j = 10 , ˚ = 0 , = 0 : 1 , = 0 : 005 (a1:˝rst blade,a2:secondblade,a3:thirdblade).....................65 Figure3.9:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuning parameterfor ~ e = 0 : 2 , = 0 : 1 , j j = 10 , ˚ = 0 , = 0 : 1 , = 0 : 005 (a1:˝rst blade,a2:secondblade,a3:thirdblade).....................66 Figure4.1:AfourDOFspring-masschain...........................71 Figure4.2:Eigenvaluesversussti˙nessmistuningparameter for = 0 , ! = 1 : 6 , m 2 = m 3 = m 4 = = 1 , = 0 : 4 .........................72 Figure4.3:Modeshapesofthetuned( = 0 )andmistuned( = 1 )systems.........72 xv Figure4.4:Stabilityregionsforthetuned4DOFmass-springchainfor n = 2 , = 1 and = 0 : 4 .......................................75 Figure4.5:Responsefrequenciesplotasafunctionofexcitationfrequencywithparam- eters n = 1 , = 0 , = 0 : 4 , = 1 and = 0 : 6 ...................76 Figure4.6:Responseplotsfor n = 2 , ! = 0 : 75 , = 0 : 2 , = 0 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T ....................77 Figure4.7:FFTplotsfor n = 2 , ! = 0 : 75 , = 0 : 2 , = 0 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T ....................78 Figure4.8:Responseplotsfor n = 2 , ! = 0 : 75 , = 0 : 6 , = 0 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T ....................79 Figure4.9:FFTplotsfor n = 2 , ! = 0 : 75 , = 0 : 6 , = 0 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T ....................80 Figure4.10:Stabilityplotofthemistuned4DOFsystemfor n = 1 , = 0 : 2 , = 1 and = 0 : 4 81 Figure4.11:Stabilityplotforthemistuned4DOFsystemfor n = 2 , = 1 , = 1 and = 0 : 4 81 Figure4.12:Responsefrequencyplotasafunctionofexcitationfrequencyformistuned casewithparameters n = 1 , = 0 : 2 , = 0 : 4 , = 1 and = 0 : 6 .........82 Figure4.13:Responseplotsofdetunedsystemfor n = 2 , = 0 : 2 , ! = 0 : 8 , = 0 : 2 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T ............82 Figure4.14:FFTplotsofdetunedsystemfor n = 2 , = 0 : 2 , ! = 0 : 8 , = 0 : 2 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T ................83 Figure4.15:Responseplotsofdetunedsystemfor n = 2 , = 0 : 2 , ! = 0 : 8 , = 0 : 6 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T ............83 Figure4.16:FFTplotsofdetunedsystemfor n = 2 , = 0 : 2 , ! = 0 : 8 , = 0 : 6 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T ................84 Figure5.1:AmplitudesofsimulatedresponsesofEquation(5.1)showingprimaryandsu- perharmonicresonancesandanunstableresponseatsubharmonicresonance duetoincreaseoftheparametricforcingamplitude; = 0 : 1 ; = 0 : 25 ; F 0 = 2 . Di˙erentcurvesdepict = 0 : 5 and 1 .(Figuretakenfrom[1])..........87 Figure5.2:Examplecalculationwiththesuperharmonicresonance.............89 Figure5.3:Examplecalculationwiththeprimaryresonance.................90 xvi Figure5.4:Estimatedforce F .................................91 Figure5.5:Estimated 'sand 's.(Thevaluesof 8 and 8 est areplotted.).........92 FigureB.1:Parametricsti˙nessratiosforscaledblademodelsandactualblademodels...115 FigureB.2:Superharmonicresonanceordere˙ectonactualandscaledbladetipdisplacement117 FigureB.3:Parametrice˙ectonactualandscaledbladetipdisplacementforsuperhar- manicresonanceatorder2............................117 xvii CHAPTER1 INTRODUCTION 1.1Objectives Thepurposeofthisstudyistoadvancetheunderstandingofthewindturbinebladevibration andthedynamicrelationshipbetweenthebladesandhub.Thebladesareunderthee˙ectsof gravitationalandcyclicaerodynamicsforcesandcentrifugalforces.Thetangentialandradial componentsofthegravityforcecreatecyclicchanges,causingthee˙ectivesti˙nessoftheblade tovarythewithrotationalangle.Additionally,centrifugalforcesa˙ectthesti˙nessterms,andas theturbinerotates,thebladesareexposedtocyclicallyvaryingwindforces.Therefore,parametric sti˙nessanddirectforcinge˙ectsaretakenintoaccountintheequationofmotion.Understanding thesecyclicgravitationalandaerodynamicloadingshavefundamentalimportanceforimproving theturbinelife-spananddesigningmorereliablewindturbinesstructures. Further,Mathieu-typemultidegreeoffreedomsystemswithparametricexcitationarestudiedto ˝ndthegeneralresponses.Indeed,4DOFsystemswithparameticexcitationmatchesthemotivation ofthree-bladewindturbineandrotor. Particularly,thisworkfocusesonthein-planeblade-hubdynamicsofathree-bladehorizontal- axiswindturbine,involvingcyclicallychanginggravitationalandaerodynamicloading,andaspires tothefollowing 1. Analyzetheblade-hubdynamicsofanon-identicalthree-bladehorizontal-axiswindturbine. 2. Obtainthesteady-stateamplitude-frequencyrelationsandthestabilitiesofthesolutionsfor coupledmistunedthree-bladeequationsbyapplyinga˝rst-orderperturbationmethod. 3. Analyzethespeed-lockingphenomenabothnumericallyandanalyticallybyapplyinghar- monicbalancemethod,andinterpretedforthevariousblades. 1 4. Determinethesteady-statedynamicsforin-plane,tunedbladesofhorizantal-axiswind turbinebyapplyingasecond-ordermethodofmultiplescalestotheequationsofmotion. 5. Analyzethesteady-statedynamicsforin-plane,mistunedbladesofwindturbinebyapplying asecond-orderperturbationanalysis. 6. Obtainthegeneralresponseofbothtunedandmistunedfour-degree-of-freedomsystems withparametricexcitation,andestablishabasisforthetransientdynamicsofathree-blade turbine. 1.2Motivation Renewablepowergenerationcanhelpcountriestoaccessclean,secure,reliableanda˙ordable energy.Theworldmarketforwindenergyhasbeenexperiencingsolidgrowththroughtheyear 2017.Totalinstalledcapacitywordwidereachedabout540GWbytheendof2017asseenin Fig.1.1.AccordingtotheGlobalWindEnergyCouncil(GWEC)[2],therecordsin2017point outanincrementofinstalledcapacityofabout52GW,takingthetotalinstalledwindenergylevel toabout540GW. Electricitygenerationfromwindenergysourceshasgrownconsistently[3].Forthe˝rsttime, montlyelectricitygenerationfromthewindexceeded8%oftotalelectricitygenerationinthe UnitedStatesin2017asshowninFig.1.2. Figure1.1:Globalcumulativeinstalledwindcapacitybetween2001-2017 2 Figure1.2:MonthlynetelectricitygenerationfromselectedfuelsbetweenJan2007-Mar2017- shareoftotalelectricitygeneration. TheWindVisionReportbyDepartmentofEnergy[4]projectsascenariowithwindenergy supplying10%oftheU.S.electricityin2020,20%in2030and35%in2050totalfromboth land-basedando˙shorewindenergiestypesasseeninFig.1.3.Sincethewindenergyindustryhas growingexportvolumecontinuously,researchonwindturbineshasbecomemoreimportantforthe globalrenewableenergymarket.Thisrapidgrowthinthewindindustryhasattractedattentionfor researchanddevelopmenttomodifythefundamentaldesignofwindturbines,inordertoenhance gearboxandbearinglifeofconventionalhorizontal-axiswindturbines(HAWTs)andvertical-axis windturbines(VAWT).Alsoessentialforimprovingwindturbinecapacityandoutputisblade designandtechnologydevelopments[5].Figure1.4showsthatincreasingtowerheightandrotor diameterallowsturbinestocapturemorewindenergyandthereforeproducemoreelectricaloutput, becausepoweroutputofawindturbineisproportionaltotheareasweptbytheblades[6]. SandiaNationalLaboratory(SNL)WindEnergyTechnologiesDepartmentdevelopeda100 meterall-glassbaselinewindturbineblademodelwitha13.2MWcapacity[7].TheNational RenewableEnergyLaboratory(NREL)hasadesignofano˙shore5-MWbaselinewindturbine with61.5mblades[8]andTheDutchO˙shoreWindEnergyProject(DOWEC)modeledawind turbinewith62.6mblades[9].GERenewableEnergyintroducedawindturbineHaliade-X-12, 3 themostpowerfulo˙shorewindturbineintheworld,witha12MWcapacity.Ithasa220meter rotordiameter,107mblades,anda260mheight[10].Someo˙shorewinddevelopersareworking togetheron13to15MWturbinestobeinthemarketby2024(DONGEnergy,2017)[11].However, theincreaseinsizeofturbinescreatessigni˝cantloadingonturbinecomponents.Therefore,there isafocusonmaterialsfatigueandstructureandequipmentsloadingsinordertoreducegearbox failures.ThisisimportantfordecreasinginstallationandmaintanencecostsasseeninFigure1.5 [12]. Studyingbladedynamicsisimportanttounderstandhowthebladesinduceloadinginthe hubsincefailuresoccurgenerallyinthehubandgearboxinthehorizontal-axiswindturbines. Bladesarecoupledthroughthehubsodynamicalloadingsofbladesonrotoraretransmittedto theeachotherbyhub.Avarietyofdynamicloadingscaninducevibrationsandinstabilities.So, understandingthecoupledbladeandhubdynamicresponsesisinspiringasaresearchproblem. 1.3BackgroundandLiteratureReview 1.3.1In-planeBlade-HubDynamicsofWindTurbineswithMistunedBlades Theenergyproducedbyawindturbineisproportionaltoitsrotorarea,whichmakeslargerwind turbinedesignsmorefavorable[7,8].However,asthebladesgetlargerinsize,theybecomemore susceptibletofailureduetovariationsindynamicloadings.Therefore,understandingtheblade Figure1.3:TheWindVisionstudyscenarioaboutshareinelectricityproductionintheU.S. 4 Figure1.4:Averageturbinenameplatecapacity,rotordiameter,andhubheightinstalledduring period. dynamicsandblade-hubinteractionsisimportantformakingpredictionsaboutturbinedurability aswellasbuildingaframeworkforreliabledesigns. Inthepast,researchersworkedonsinglebladedynamicsRamakrishnanandFeeny [16]foundanonlinearequationofmotionthatgovernsthein-planedynamicsforasingleblade.In theirstudy,parametricanddirectexcitationtermsduetogravityweretakenintoaccount.Through singlemodereduction,theequationofmotioncanberepresentedwithaforcednonlinearMathieu equation,whichwasthenanalyzedto˝ndthesteadystatedynamicsvia˝rst-ordermethodof multiplescales[16,18].Bychangingrelatedparameters,theyexaminede˙ectsofparametric excitation,directforcingandnonlinearity.AcarandFeenyderivedtheequationsofmotionfora bladeunderbend-bend-twistvibrations,wheretheyaccountedforsti˙nesschangesduetogravity andcentrifugale˙ects[17]. Reliabilityisoneoftheproblemsforlargewindturbinedesigns.Whenhorizontal-axiswind- turbinebladesincreaseinsize,variationsindynamicloadingbecomemorelikelytothedurability oftheturbine.Understandingvibrationofthebladesandrelationshipbetweenbladesandhubhave fundamentalimportanceforpredictingtheturbinelife-spananddevelopingmorereliabledesigns. 5 Figure1.5:AnnualFailureFrequencyperTurbineSubsystem2012. Silva[13],BirandOyague[14],andAcarandFeeny[19]dealtwithbladeloadingandvibration ofasingleblade.Experimentalstudieshavebeendonetoestimatestructuralandmodalproperties ofthebladeandtowerofathree-bladedupwindturbine[14].E˙ectsofgravity,pitchactionand varyingrotorspeedwereincludedinthepartialdi˙erentialequationsofblademotionbyKallesøe [15].Adynamicsmodelforarotor-bladesysteminhorizontalaxiswindturbinesisdevelopedand modelaccuracyisimprovedbyincludingadditionalcouplingterms[20].RamakrishnanandFeeny [16]focusedonin-planedynamicsofasinglebladeusingalinearandnonlinearsingle-modemodel. A˝rst-orderperturbationanalysisshowedthatsuperharmonicresonancesofordertwoexistedin thelinearmodel.Theyalsoappliedasecond-orderperturbationmethodtoablade-motivatedlinear andnonlinearforcedMathieuequationtodescribesuperharmonicresonancesoforderthree[21]. Asecond-ordermethodofmultiplescalesisappliedtotheequationsofmotionforin-planetuned 6 andweaklymistunedbladesofhorizontalaxiswindturbinetodeterminethesteady-statedynamics, withfocusonthesuperharmonicresonanceoforderthreeforthelinearsystemwithhardforcing [22].InonueandIshida[23,24]performedtheout-ofphasenonlinearvibrationanalysisofwind turbinebladetoinvestigatethesuperharmonicresonancecaseandtheyalsoshowedtheexistence ofsuperharmonicresonanceatorderstwoandthree.Higher-orderperturbationexpansionshave beenappliedtostudydynamicsofsystemsNayfehandMook[29]usedhigher-order perturbationmethodto˝ndthestabilitywedgesoftheMathieuequation. 1.3.2ApproximateGeneralReponseofFour-Degree-of-FreedomSystemswithParametric Excitation Manymechanicalsystemshaveparametricexcitationcharacteristics[18,23,30,31].Anumberof di˙erenttypesofmethodshavebeenusedtostudytheMathieuequation.Themethodofmultiple scaleshasbeenusedtoexamineaforcedMathieuequationforresonances[18].Likewise,stability characteristicsarefoundbyusingthemethodofvanderPol[23].Anotherwaytoapproach theMathieuequationistouseFloquettheory.AcarandFeeny[32]usedamethodcombining Floquettheorywithharmonicbalanceto˝ndthetuned2-DOFand3-DOFsystemsresponses.An assumedFloquet-typesolutioniscomposedofaperiodic p ¹ t º andanexponentialpart e ^ t suchas x ¹ t º = e ^ t p ¹ t º .ThetheoryindicatesthatthefundamentalsolutiontoaMathieuequationonstability boundariesispurelyperiodic[33].Inconsequenceofthat,stabilityregionscanbeprocuredby assumingaperiodicsolutionwithoutsolvingforthegeneralresponseitself[34],[24,The responsecharacteristicsoftime-periodicsystemshavebeenstudiedbyusingsystemidenti˝cation methods.Allen etal. [38]presentedanoutput-onlysystemidenti˝cationmethodologytoidentify themodalfunctionsoftheMathieuequationandtheFloquetexponents. 1.4ThesisOverview Thisthesisincludestheanalysisofin-planeblade-hubdynamicsofhorizontal-axiswindturbines withmistunedbladesbyapplyingperturbationanalysis.InChapter2,low-orderin-planevibration 7 equationofmotionarederivedforamistunedthree-bladewindturbine.Thebladeequationsand therotorequationarecoupledthroughtheinertialterms.Pendulumvibrationabsorberequations aresimilartotheseequationswhereabsorberinertiaissmallcomparedtorotorinertia[39,40]. Todecoupletheabsorberequationsfromtherotorequation,Chao etal. [41,42]changedthe independentvariablefromtimetorotorangle.Followingtheanalysisofthesevibration-absorber systems,theindependentvariableistransformedfromtimetorotorangleinthiswork,thenanon- dimensionalizationprocedureandascalingprocessarefollowedtoseparatethebladeequations fromthehubequation.Next,themethodofmultiplescalesisappliedtoequationsofthemistuned bladestoanalyzethesteady-stateamplitude-frequencyrelationsandthestabilitiesofsolutions. Moreover,inChapter3,second-ordermethodofmultiplescalesisappliedtotheequationsof motionforin-planetunedandweaklymistunedbladesofhorizontalaxiswindturbinetoobtain thesteadystatedynamics,withfocusonthesuperharmonicresonanceoforderthreeforthelinear systemwithhardforcing,andalsosuperharmonicresonancesunderaconstantload. Additionally,inChapter4,generalsolutionsofMathieu-typefour-degree-of-freedommass- springsystemwithparametricexcitationarestudied.AssumingaFloquet-typesolution,andusing theharmonicbalancemethod,thefrequencycontentandstabilityofthesolutionareobtained. Finally,theanalysisisextendedtoasystemwithmistunedparameters,andthee˙ectofsymmetry breakingonsystemresponseisanalyzed. Lastly,Chapter5discussesstudieswhichareunderwaybuthavenotbeencompleted.These includeanonlinearanalysisofasinglebladeandanapproachforparametricidenti˝cationofa systemwithcyclicsti˙ness. 1.5Contributions Literaturecontibutionsofthisthesisare: Equationsofmotionsarederivedforin-planevibrationsofamistunedthree-bladewind turbine.Thisstudyallowsustodeterminethatsuperharmonicandprimaryresonances, whichwereobservedinthepreviousstudyofsymmetriccase,canbesplitintomultiple 8 resonancepeaks,andthatthebladestakeondi˙erentsteadystateamplitudes. Thee˙ectsofparameters,suchasdamping,forcinglevel,positiveandnegativemistuningon thesuperharmonicresonantresponseswereanalyzed.Also,theanalyticalsolutionapproxi- mationsofthebladevibrationandrotordynamicswithnumericalsolutionsareveri˝ed.In doingso,thesimulationsexposearotor-speedlockingphenomenonatthesuperharmonic resonances.Additionally,thespeed-lockingphenomenaisveri˝edanalyticallybyapplying harmonicbalancamethod. Asecond-orderpertubationanalysisisappliedonbothtunedandmistunedthree-bladewind turbines.Theanalysisrevealsthesuperharmanicresonancesatone-thirdnaturalfrequency. Thisresonancecaseisnotabletobecapturedwitha˝rst-ordermultiplescalesanalysis.The superharmonicresonancesplitsfromasingleresonancepeakinthetunedcaseintomultiple resonancepeakswithmistuning.Theamplitudeinceaseswhilethemodaldampingfactor decreasesforsteady-statesuperharmonicresonanceresponse. Generalresponsesofa4degree-of-freedommass-springsystemwithparametricexcitation areinvestigated.Thefrequencycontentandstabilityofthesolutionareobtainedbyassuming aFloquet-typesolution,andusingtheharmonicbalancemethod.Further,theanalysisis broadentoasystemwithmistunedparameters,andthee˙ectofsymmetrybreakingonsystem responseisanalyzed. 9 CHAPTER2 IN-PLANEBLADE-HUBDYNAMICSOFHORIZONTAL-AXISWINDTURBINES WITHMISTUNEDBLADES 2.1Introduction Inthischapter,wind-turbineblade-hubinteractionsareconsideredasawhole.Dynamicsof alinearizedsymmetricthree-bladehorizontal-axiswindturbinewasstudiedpreviouslyin[43], whereallthreebladeswereassumedtohaveidenticalinertialandsti˙nessproperties.Inthat work,theparametricallyexcitedbladeequationsexhibitedsuperharmonicandprimaryresonances. Sincethesystemwaslinearandperfectlysymmetric,eachbladehadthesamevibrationamplitude. Ikeda etal. investigatedunstablevibrationsofatwo-bladewindturbinetowertheoretically[44]. Nonlinearitycouldcausethebladestodeviatefromthissymmetricresponse(e.g.inGri˚n et al. ,[45],Dick etal. [46]).Alternatively,inthischapter,oneofthebladesismistunedtoshow thee˙ectsofbreakingthecyclicsymmetry.Manyresearchersworkedonmistunedrotational systemsWhitehead[51]analyzedthee˙ectofbroken-symmetryonforcedvibrationof turbinebladeswithmechanicalcoupling.MistuninginbladeddiskshasbeenstudiedbyEwins [52]andChaandSinha[53].Localizationphenomenoninthree-bladehorizontal-axiswindturbine vibrationswereanalyzedbyIkeda etal. [54].Approximategeneralresponsesoftunedandmistuned 4-degree-of-freedomsystemswithparametricsti˙nesswerefoundbyapplyingFloquettheorywith harmonicbalance[55]. Followinguptheworkonasymmetricblade-hubsystem,themethodofmultiplescalesis appliedtoequationsofthemistunedbladesandhubtoexaminethesteadystatedynamics.By usingaprocedurethatissimilartotheoneusedin[43],linearizedequationsofmotionfornon- identicalbladesandhubwerefound.Byassumingasingleuniformcantileverbeammodefor eachblade,energyequationsareapproximated.Then,bladeandhubequationsareobtainedby applyingLagrange'sequations[43].Thetangentialandradialcomponentsofthegravityforce 10 createcyclicchanges,varyingthee˙ectivesti˙nessofthebladewithrotationalangle.Also, centrifugalforcescontributetothesti˙nessastherotorspins.Althoughhorizontal-axiswind- turbinesdonotspinathighspeedsgenerally,thesti˙nesscontributionofcentrifugale˙ectsshould betakenintoaccountwhiledesigningturbineblades.Moreover,windvelocitygenerallyvaries withaltitude,thebladesareexposedtocyclicallyvaryingwindforcesasthebladerotates.Inthe equationsofmotion,parametricsti˙nessanddirectforcinge˙ectsweretakenintoconsideration. Derivationofequationscanbefoundfrom[17]. Inplaceofassumingasymmetricmodel,inthisstudy,wemistuneoneofthebladesinorder tounderstandhowbroken-symmetrya˙ectstheturbinedynamics.Ascanbeseeninthefollowing sections,evenwhenthemistuningissmall,itcanintroducelargerforcedresponseswhencompared totheperfectlytunedsystem.Theblade-hubdynamicsofanon-identicalthree-bladehorizontalaxis windturbine,involvingcyclicallychangingaerodynamicloadings,directandparametricexcitation isanalyzed.Primaryresonanceandsuperharmonicresonanceatorder2wereunfoldedduetothe parametricanddirectexcitationofgravity.Thebladeequationsandtherotorequationarecoupled throughtheinertialterms. Pendulumvibrationabsorberequationsaresimilartotheseequationswhereabsorberinertia issmallcomparedtorotorinertia[39,40].Todecoupletheabsorberequationsfromtherotor equation,Chao etal. [41,42]changedtheindependentvariablefromtimetorotorangle.Then theyappliedmethodofaveragingto˝ndsteadystatedynamics.Gravitationale˙ectsonabsorbers andtheinternalresonancesintroducedbytheparametrice˙ectwerestudiedbyTheisen[56]. Likewiththeanalysisofthesevibration-absorbersystems,theindependentvariableistrans- formedfromtimetorotorangleinthiswork,andthen,anon-dimensionalizationprocedureand ascalingprocessarefollowed.Toanalyzethesteady-stateamplitude-frequencyrelationsandthe stabilitiesofsolutionsforcoupledmistunedthree-bladeequations,a˝rst-ordermethodofmultiple scalesisapplied.Thisanalysisisfocusedonsuperharmonicresonanceandaprimaryresonance. Thischapteralsoincludesthee˙ectsofparameters,notablynegativemistuning,damping,and forcinglevel,onthesuperharmonicresonantresponses.Ontheotherhand,theanalyticalsolution 11 Figure2.1:Amistuningthree-bladeturbinewithbladesunderin-planebending. approximationsofthebladevibrationandrotordynamicswithnumericalsolutionsisveri˝ed. Indoingso,thesimulationsexposearotor-speedlockingphenomenonatthesuperharmonic resonances.Last,thespeed-lockingphenomenonandthebladeresponseamplitudesatvarious rotorspeedsnearresonanceareveri˝edanalyticallybyapplyingharmonicbalancamethod. 2.2MistunedThree-BladeTurbineEquations Followingreference[43],thehubismodeledasarigidbodyina˝xed-axisrotationwith damping.AsshowninFigure2.1,onlyin-planevibrationistakenintoconsiderationbyusinga simpli˝edmodel,andtowermotionisneglected.Thebladesandrotorequationsarecoupled.The bladesaremodeledasnonuniformslenderbeamsoflength L within-planetransversedisplacement y ¹ x ; t º ˙ v ¹ x º q j ¹ t º ,where v ¹ x º istheassumedmodaldisplacementfunctionofposition x ,and q j ¹ t º isthemodalcoordinateofthe j th blade.Oneoftheblades'elasticmodalsti˙nesstermsis assumedtohaveasmallmistuning. Theequationsofmotionforthe j th bladeandtherotorare,for j = 1 , 2 and 3 , 12 m b Ü q j + c b Û q j + ¹ k 0 j + k 1 Û ˚ 2 + k 2 cos ˚ j º q j + d sin ˚ j + e Ü ˚ = Q j ; (2.1) J r Ü ˚ + c r Û ˚ + 3 Õ k = 1 ¹ d cos ˚ k q k + e Ü q k º = Q ˚ ; (2.2) where k 0 1 = k 0 2 = k 0 , k 0 3 = k 0 + k v and k 0 isasingleblade'selasticsti˙ness, k v istheelastic sti˙nessvariationofthemistunedblade, m b isthemodalmassofasinglebladeaccordingtothe equationgivenintheAppendixA.1, J r isthetotalinertiaofthreebladesplusthehubaboutthe shaftaxis, e isthecouplingterm, q j istheassumedmodalcoordinateforthe j th blade, ˚ isthe rotorangle,and ˚ j = ˚ + 2 ˇ 3 j istheblade-rootangle,whichdi˙ersfrom ˚ byaconstant(i.e. ˚ 1 = ˚ + 2 ˇ š 3 , ˚ 2 = ˚ + 4 ˇ š 3 , ˚ 3 = ˚ ), k 1 Û ˚ 2 isthecentrifugalsti˙ness, k 2 isthesti˙ness contributionofthegravitationale˙ect, Q j and Q ˚ aregeneralizedforcingtermsduetoaeroelastic loading,and c b and c r aregenericdampingcoe˚cients.Theseparametersarede˝nedinthe AppendixA.1. Forasystemunderzerogravity,the˝rsttwomodalfrequenciesare: ! n 1 = 0 withmode shape # 1 = ¹ q 1 ; q 2 ; q 3 ;˚ º = ¹ 0 ; 0 ; 0 ; 1 º (rigidbodyrotation), ! n 2 = s k 0 + k 1 2 m b (frequencyof asingleblade)with # 2 = ¹ 0 : 707 ; 0 : 707 ; 0 ; 0 º .Thethirdandthefourthmodalfrequenciesare rathercomplicated,andtheyaregivenintheAppendixA.2.Forspeci˝cparametervalues,the naturalfrequenciesareplottedasfunctionsofthesti˙nessvariationterm( k v ),asdemonstratedin Figure2.2.When k v = 0 ,thesymmetriccase, ! n 3 = ! n 2 ,whichisconsistentwith[43].Schematic blade-rotormodeshapesaregiveninFigure2.3forthesymmetriccase.Forthemistunedblades case,˝rstandsecondmodeshapesarethesameasshowninFigure2.3,butthethirdandforthmode shapesareslightlyperturbed.Inthecaseofmode2,themistunedbladeandhubaremotionless whilethetunedbladesvibrateequallyandoppositely. Similartoreferences[39,theindependentvariableischangedfromtimetorotorangle ˚ .Therotorspeed Û ˚ isnotconstant.By = Û ˚ š ,where isthemeanspeed,onecan˝ndthe derivativeexpressions: 13 Figure2.2:Eigenvaluesversuselasticsti˙nessmistuningparameter( k v )plotfor k 0 = 5 , k 1 = 0 : 5 , Û = 0 , m = 1 , J = 1 , e = 0 : 2 , = 0 : 1 (e1:˝rstblade,e2:secondblade,e3:thirdblade) d d t = d ˚ d t d d ˚ and d 2 d t 2 = 0 2 d d ˚ + 2 2 d 2 d ˚ 2 . Equations(2.1)and(2.2)aremodi˝edwithrotorangleasthenewindependentvariable.The equationsbecome 2 q 00 j + 0 q 0 j + ~ c b q 0 j + ¹ ~ k 0 j + ~ k 1 2 + ~ k 2 cos ˚ j º q j + ~ d sin ˚ j + ~ e 0 = ~ Q j ; (2.3) 0 + ~ c r + ˜ 3 Õ k = 1 h ~ d cos ˚ k q k + ~ e ¹ 2 q 00 k + 0 q 0 k º i = ~ Q ˚ ; (2.4) where ¹º 0 = d ¹ºš d ˚ ,and ~ k 0 1 = ~ k 0 2 = ~ k 0 and ~ k 0 3 = ~ k 0 + ~ k v andwhere ~ c b = c b m b ; ~ e = e m b ; ~ k 0 = k 0 m b 2 ; ~ k v = k v m b 2 ; ~ k 1 = k 1 m b ; ~ k 2 = k 2 m b 2 ; ~ d = d m b 2 ; ~ Q j = Q j m b 2 ;˜ = m b J r ; ~ c r = c r J r ; ~ Q ˚ = Q ˚ J r 2 : 14 Figure2.3:In-planemodeshapesofasymmetricthree-bladeturbine. Theterm 0 referstothevariationsintherotorspeed,andcanbecalledthedimensionless rotoracceleration( d dt = d d ˚ d ˚ dt = 0 ¹ º ).ThiscanbeseenfromEquation(2.4),wherethe summationrepresentstheloadsappliedbythebladesontherotor. Theparameter J r containsinertiaofthethreeundeformedbladesandthehubinertiaabout theshaftaxis. m b isthecumulativeinertiaofelementsofasinglemodallydisplacedbladeabout thetransverseaxesoftheirownunde˛ectedpositions.Since m b issmallcomparedto J r ,asmall parameterisde˝nedas = m b š J r .Theexpressionsfortheseparameterscanbefoundinthe AppendixA.1.Forthepurposeofthedecouplingoftheblade-hubequations,thefollowingscaling isappliedtoEquation(2.3)andEquation(2.4): = 1 + 2 1 ; ~ c b = ^ c b ; ~ k 2 = ^ k 2 ; ~ d = ^ d ; ~ c r = 2 ^ c r ;˜ = ; q j = s j ; ~ Q j = ^ Q j ; ~ Q ˚ = 2 ^ Q ˚ : Theequationsarerevisedwithrespecttoscaledbladecoordinates s j andhubcoordinate v 1 as s 00 j + ^ c b s 0 j + ¹ ~ k 0 j + ~ k 1 + ^ k 2 cos ˚ j º s j + ^ d sin ˚ j + ~ e 0 1 = ^ Q j + H : O : T :; (2.5) 0 1 + ^ c r + 3 Õ k = 1 ¹ ^ d cos ˚ k s k + ~ es 00 k º = ^ Q ˚ + H : O : T : (2.6) whereH.O.Tstandsforhigher-orderterms. 15 Theconstantelasticsti˙nessterm ~ k 0 j islargerrelativetothe ~ k 1 and ~ k 2 terms.Thesecanbe evaluatedaccordingtotheequationswhichcanbefoundintheAppendixA.1. 0 1 whichisobtainedfromEquation(2.6)isinsertedintoEquation(2.5)toget s 00 j + ^ c b s 0 j + ¹ ~ k 0 j + ~ k 1 + ^ k 2 cos ˚ j º s j + ^ d sin ˚ j + ~ e h ^ Q ˚ ^ c r Í 3 k = 1 ¹ ^ d cos ˚ k s k + ~ es 00 k º i = ^ Q j + H : O : T : (2.7) ^ Q j and ^ Q ˚ aregeneralizedforcingtermsduetoaeroelasticloadingandaresimpli˝edasa meanplussmallvariation,as ^ Q j = Q j 0 + ^ Q j 1 ¹ ˚ º and ^ Q ˚ = Q ˚ 0 + ^ Q ˚ 1 ¹ ˚ º .Bladespeedis approximatedbytherotorspeed(i.e. u blade = Û ˚ x ).Itisassumedthatthe˛owissteadyandthe windspeedincreaseslinearlywithheight h (i.e. u w ind = u 0 + hu 1 = u 0 cos ˚ j u 1 ).This indicatesthattherelativespeedcontains ˚ and Û ˚ terms.Thedetailedassumptionandformofthe aerodynamicsforcecanbefoundintheAppendixA.1.Inserting ~ k 0 j = ~ k 0 + ~ k v j , ^ k 0 = ~ k 0 + ~ k 1 , and ˚ j = ˚ + 2 ˇ 3 j intoEquation(2.7)andreorganizingterms,wegetthedecoupledbladeequations as s 00 j + ^ k 0 s j = Q j 0 ^ d sin ˚ + 2 ˇ 3 j + h Q j 1 cos ˚ + 2 ˇ 3 j ~ eQ ˚ 0 + ~ e ^ c r ^ c b s 0 j ~ k v j s j ^ k 2 cos ˚ + 2 ˇ 3 j s j + ~ e 2 Í 3 k = 1 s 00 k i ; (2.8) for j = 1 ; 2 ; and 3 ,where ~ k v 1 = ~ k v 2 = 0 and ~ k v 3 = ~ k v ,andthetermwith ^ k 2 isthegravitational parametricexcitation. p 1 = s k 0 š 2 + k 1 m b isthemodalorderoftheunexcitedangle-basedsystem equation.Thetime-basedsystemnaturalfrequencyisscaledby ,suchas p 1 = ! n 2 š . Aswenotedbefore,bladesarecoupledthroughtheinertialterms.Onewaytohandlethisis tomakeacoordinatetransformationtogettheequationscoupledthroughthesti˙nessterms,and 16 thenapplyanaveragingmethodtostudythesteadystatedynamics.Alternatively,thesystemwith inertialcouplingcanbestudiedbyusingaFouriermatrix,asexpressedin[57].Themethodof multiplescalescanalsobeusedtogettherelationsoftheslow˛ow.Inthispaper,weusedthe methodofmultiplescalestoanalyzetheinternalresonances. 2.3Multiple-ScalesAnalysis Werewritetheequationsofmotionintermsofnewindependentvariable = p 1 ˚ ,where p 1 = q ^ k 0 = s k 0 š 2 + k 1 m b . Thescaledequationsofmotioninthe domainare s 00 j + s j = F j sin ! 1 + 2 ˇ 3 j + h F j 1 cos ! 1 + 2 ˇ 3 j + f ^ s 0 j v j s j + ~ e 2 Í 3 k = 1 s 00 k cos ! 1 + 2 ˇ 3 j s j i ; (2.9) wherenow ¹º 0 = d d and F j = Q j 0 ^ k 0 ; = F j 0 = ^ d ^ k 0 ;! 1 = 1 p 1 ; F j 1 = Q j 1 ^ k 0 ; ^ = ^ c b p 1 ; f = ~ e ¹ ^ c r Q ˚ 0 º ^ k 0 ; = ^ k 2 ^ k 0 ; v j = ~ k v j ^ k 0 Theparameter ! 1 isascaledxcitationorderandisgivenby ! 1 = ! n 2 ; where ! n 2 = s k 0 + k 1 2 m b isamodalfrequencyoftheturbine.Asaresult,whenthemeanrotor speed, ,varies,theexcitationorder ! 1 willvaryaswell. Thesteadystatedynamicsofthedecoupledbladeequationsisanalyzedbyapplyinga˝rst-order methodofmultiplescales[29]. s j hasslowandfastscales ¹ 0 ; 1 º andissplitintodominantsolu- 17 tion s j 0 andavariationofthatsolution s j 1 ,i.e. s j = s j 0 ¹ 0 ; 1 º + s j 1 ¹ 0 ; 1 º ,where i = i 0 . Then d š d = D 0 + D 1 ,where D i = @ š @ i .TheseformulationsareinsertedintoEquation(2.9) andthenweseparateoutthecoe˚cientsof 0 and 1 . Theequationforthecoe˚cientof 0 : D 2 0 s j 0 + s j 0 = F j sin ! 1 0 + 2 ˇ 3 j (2.10) Theequationforthecoe˚cientof 1 : 2 D 0 D 1 s j 0 + D 2 0 s j 1 + s j 1 = F j 1 cos ! 1 0 + 2 ˇ 3 j + f ^ D 0 s j 0 + v j s j 0 cos ! 1 0 + 2 ˇ 3 j s j 0 + ~ e 2 Í 3 k = 1 D 2 0 s k 0 (2.11) s j 0 isfoundasthesolutionofEquation(2.10): s j 0 = F j 2 + A j e i 0 + i Be i ¹ ! 1 0 + 2 ˇ 3 j º + c : c : (2.12) where A j iscomplexand B = 2 ¹ 1 ! 2 1 º . Insertingthesolutionfor s j 0 , D 0 s j 0 = i A j e i 0 B ! 1 e i ¹ ! 1 0 + 2 ˇ 3 j º + c : c :; and D 0 D 1 s j 0 = i A 0 j e i 0 + c : c : intoEquation(2.11),we˝ndthe s j 1 equation: 18 D 2 0 s j 1 + s j 1 = F j 1 2 e i ¹ ! 1 0 + 2 ˇ 3 j º + f 2 2i A 0 j e i 0 F j 2 e i ¹ ! 1 0 + 2 ˇ 3 j º i B 2 e 2 i ¹ ! 1 0 + 2 ˇ 3 j º + v j ¹ F j 2 + A j e i 0 + Bie i ¹ ! 1 0 + 2 ˇ 3 j º º + ~ e 2 Í 3 k = 1 A k e i 0 i ! 2 1 Be i ¹ ! 1 0 + 2 ˇ 3 k º 2 A j e i ¹ ! 1 + 1 º 0 + i 2 ˇ 3 j + A j e i ¹ ! 1 1 º 0 + i 2 ˇ 3 j ^ A j ie i 0 ! 1 Be i ¹ ! 1 0 + 2 ˇ 3 j º + c : c : (2.13) ThesolvabilityconditionforEquation(2.13)isobtainedbyeliminatingsecularterms.The solvabilityconditiondependsontheresonancecase. 2.3.1NonresonantCase Weeliminatecoe˚cientsof e i 0 whichconstitutetheseculartermsandthesolvabilitycondition isfound 2i A 0 j i ^ A j v j A j ~ e 2 3 Õ k = 1 A k = 0 : Writing A j = X j + i Y j ,andsplittingtheaboveequationintorealandimaginaryparts,weobtain Realpart: Y 0 j = ^ 2 Y j + v j 2 X j + ~ e 2 2 3 Õ k = 1 X k ; (2.14) Imaginarypart: X 0 j = ^ 2 X j v j 2 Y j ~ e 2 2 3 Õ k = 1 Y k : (2.15) RepresentinginEquation(2.14)andEquation(2.15)matrixform 19 © « X 0 1 X 0 2 X 0 3 Y 0 1 Y 0 2 Y 0 3 ª ® ® ® ® ® ® ® ® ® ® ® ® ® ® ® ¬ = 1 2 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 ^ 00 ~ e 2 ~ e 2 ~ e 2 0 ^ 0 ~ e 2 ~ e 2 ~ e 2 00 ^ ~ e 2 ~ e 2 ~ e 2 + v j º ~ e 2 ~ e 2 ~ e 2 ^ 00 ~ e 2 ~ e 2 ~ e 2 0 ^ 0 ~ e 2 ~ e 2 ¹ ~ e 2 + v j º 00 ^ 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 © « X 1 X 2 X 3 Y 1 Y 2 Y 3 ª ® ® ® ® ® ® ® ® ® ® ® ® ® ® ® ¬ : (2.16) Atsteadystate,thesolutionis X j = 0 and Y j = 0 .Eigenvaluesofthesystemde˝nethestability. Assuch, 1 ; 2 = ^ ,and 3 ; 4 ; 5 ; 6 canbefoundintheAppendixA.2,forwhichitcanbeshownthat therealpartsareallnegativeif ^ < 0 . ThematrixinEquation(2.16)hastheform A = 1 2 ^ I + B º .Eigenvaluessatisfy j B ^ I I j = j B I j = 0 ,where = ^ + .Since B = B T ,itseigenvalues = i ! areimaginary,andhence dynamicalsystemeigenvalues = ^ i ! indicatestability. 2.3.2SuperharmonicResonance( 2 ! 1 ˇ 1 or 2 ˇ ! n 2 ), 2 ! 1 = 1 + ˙ A˝rst-ordermultiplescalesanalysisofthesymmetriccase[43]showedtheexistenceofsuper- harmonicresonanceathalfthesecondnaturalfrequencyofthesystem.RamakrishnanandFeeny [21]haveshownthatasecond-orderperturbationanalysisofasingle-blademodelalsoreveals superharmonicresonanceswithorderof 3 . Weeliminatecoe˚cientsofseculartermsinEquation(2.13)thatareleadingtoanunbounded solutionand˝ndthesolvabilityconditionfor 2 ! 1 ˇ 1 as 2i A 0 j i ^ A j v j A j i B 2 e i ¹ ˙ 1 + 4 ˇ 3 j º ~ e 2 3 Õ k = 1 A k = 0 : Weinsert A j = j e i ¹ ˙ 1 + 4 ˇ 3 j º with j = X j + i Y j intoaboveequation,andthenwecansplit uptherealandimaginarypartsasbelow. 20 Figure2.4:Steady-statesuperharmonicresonancebladeresponseamplitudesversuselasticsti˙ness mistuningparameter v for ~ e = 0 : 2 , ˙ = 0 : 0064 , = 0 : 1 , j B j = 1 , ^ = 0 : 0445 .Thethirdblade isthemistunedblade. Realpart: Y 0 j = ˙ X j ^ 2 Y j + v j 2 X j + ~ e 2 2 3 Õ k = 1 X k cos 2 ˇ 3 ¹ k j º Y k sin 2 ˇ 3 ¹ k j º ; (2.17) Imaginarypart: X 0 j = ˙ Y j ^ 2 X j v j 2 Y j B 4 ~ e 2 2 3 Õ k = 1 X k sin 2 ˇ 3 ¹ k j º + Y k cos 2 ˇ 3 ¹ k j º : (2.18) Thesteady-stateresponseamplitudeis A j = q X 2 j + Y 2 j ,andsousing X j and Y j wecandetermine A j .Figure2.4showstheamplitudesofeachbladeasthemistuningparameter v isvaried,forthe casewhen ˙ = 0 : 0064 ; ~ e = 0 : 2 ; = 0 : 1 ; ^ = 0 : 0445 and j B j = 1 .When v = 0 ,eachbladehas thesameamplitude,consistentwith[43].Thisamplitudematcheswiththeamplitudesoftheblades atresonanceforthesymmetriccase( v = 0 )forsamesetofparameters.Forbothsmallnegative andpositivemistuningsshowninFigure2.4,thebladeamplitudesgothroughvariations.For largermagnitudemistuning,themistunedblade(thirdblade)amplitudeisdecreased.Verysmall mistuningsaretheworstinthesensethatonebladeundergoeslargervibrationamplitudesthanin thesymmetriccase.SinceFigure2.5standsfortunedcase,Figure2.6,Figure2.7andFigure2.8 21 showthesuperharmonicresonanceamplitudesasfunctionsofthedetuningparameter(excitation orderorfrequency),forvariousmistunings.Themistuningscausethesingleresonancepeakof thesymmetriccase[43]tosplitup,therebybroadeningthebandwidthofthesystemresonance. Inthese˝gures,twoofthethreepeakscanbeidenti˝edassuperharmonicresonancesofmodal frequencies ! n 2 and ! n 3 . IncomparisontothesymmetriccaseinFigure2.5,theexampleinFigure2.6showsthatthe mistuningcanalsocauseanincreasedvibrationamplitudeinoneofthebladesdependingonthe valueof v .Additionally,amplitudeversusfrequencyplotsareshowninFigure2.9andFigure2.10 foreachbladewithaspeci˝csetofparameters.When ^ decreases,theresonanceamplitudegets sharpenedandwhen B increasestheamplitudecurveisraisedaswell.Inall˝gures,theparameter ^ relatestothedampingfactor through ^ = 2 . Thesolutioninthe ˚ domainisreorganizedas s j = F j 0 + 2 B sin ¹ ˚ + 2 ˇ 3 j º + a j cos ¹ p 1 ˚ + j + 4 ˇ 3 j º (2.19) where a j = 2 q X j 2 + Y j 2 and j = tan 1 ¹ Y j X j º .Toanalyzetherotordynamics,the s j areinserted intoEquation(2.6)toobtain d 1 d ˚ = ^ Q ˚ ^ c r Í 3 k = 1 ^ d cos ˚ F k 0 + 2 B sin ¹ ˚ + 2 ˇ 3 k º + a k cos ¹ p 1 ˚ + k + 4 ˇ 3 k º ~ e 2 B sin ¹ ˚ + 2 ˇ 3 k º + a k p 2 1 cos ¹ p 1 ˚ + k + 4 ˇ 3 k º (2.20) Forall k ,thenondimensionalizedmeanaerodynamicforceterms F k 0 = Q k 0 š ^ k 0 arethesame. Byletting ˇ d ˚ dt andusing = 1 + 1 ,therotoraccelerationis d dt = d 1 d ˚ .Using ˚ ˇ t and p 1 ˚ ˇ ! n 2 t ,wecanthereforeexpress d dt inthetimedomainas 22 Figure2.5:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuningparameter for = 0 : 1 , ~ e = 0 : 2 , v = 0 , j B j = 1 , ^ = 0 : 005 (a1:˝rstblade,a2:secondblade,a3:3rdblade) d dt = ^ Q ˚ ^ c r Í 3 k = 1 ^ d cos t F k 0 + 2 B sin ¹ t + 2 ˇ 3 k º + a k cos ¹ ! n 2 t + k + 4 ˇ 3 k º ~ e 2 B sin ¹ t + 2 ˇ 3 k º + a k ¹ ! n 2 º 2 cos ¹ ! n 2 t + k + 4 ˇ 3 k º (2.21) Equation(2.21)has cos ¹ t º sin ¹ t + 2 ˇ 3 k º , cos ¹ t º cos ¹ ! n 2 t + k + 4 ˇ 3 k º , sin ¹ t + 2 ˇ 3 k º and cos ¹ ! n 2 t + k + 4 ˇ 3 k º terms,andthereforetherotorhastermsoffrequencies ; 2 ,and 3 . Thebladeresponsecanbeapproximatedinthetimedomain.Assuming s j = s j 0 andplugging Equation(2.12)intothe q j = s j ,wehave q j = ¹ F j 0 + 2 B sin ¹ t + 2 ˇ 3 j º + a j cos ¹ ! n 2 t + j + 4 ˇ 3 j ºº + O ¹ 4 º : (2.22) 2.3.3PrimaryResonance( ! 1 ˇ 1 or ˇ ! n 2 ) ! 1 = 1 + ˙ 23 Figure2.6:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuningparameter for = 0 : 1 , ~ e = 0 : 2 , v = 0 : 006 , j B j = 1 , ^ = 0 : 005 (a1:˝rstblade,a2:secondblade,a3:3rd blade) Figure2.7:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuningparameter for = 0 : 1 , ~ e = 0 : 2 , v = 0 : 1 , j B j = 1 , ^ = 0 : 005 (a1:˝rstblade,a2:secondblade,a3:3rdblade) Primaryresonanceisfarfromthedesignedoperationfrequenciesofwindturbines,andisnot likelytohappenunlessthereisarunawaysituation.However,itisofinteresttostudythedynamical phenomenon.Theharmonicforcingismodeledaseakforcing"in[43]toexaminetheprimary resonanceresponse.Thuswelet = ^ .Correspondingtothescalingof isthescalingof ^ d = ^ ^ d 24 Figure2.8:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuningparameter for = 0 : 1 , ~ e = 0 : 2 , v = 1 , j B j = 1 , ^ = 0 : 005 (a1:˝rstblade,a2:secondblade,a3:3rdblade) Figure2.9:Steady-statesuperharmonicresonancebladeresponseamplitudesversusfrequencyfor = 0 : 1 , ~ e = 0 : 2 , v = 0 : 1 , j B j = 1 , ^ = 0 : 005 ; 0 : 008 ; 0 : 01 ; 0 : 015 inEquation(2.6). Pluggingexpression ^ sin ! 1 0 + 2 ˇ 3 j inplaceof sin ! 1 0 + 2 ˇ 3 j intotheEqua- tion(2.9),weobtain s j 0 = F j 2 + A j e i 0 + c : c : (2.23) 25 Figure2.10:Steady-statesuperharmonicresonancebladeresponseamplitudesversusfrequency for = 0 : 1 , ~ e = 0 : 2 , v = 0 : 1 , j B j = 1 ; 2 ; 3 ; 4 , ^ = 0 : 05 and 2 D 0 D 1 s j 0 + D 2 0 s j 1 + s j 1 = F j 1 cos ! 1 0 + 2 ˇ 3 j ^ sin ! 1 0 + 2 ˇ 3 j + f ^ D 0 s j 0 + v j s j 0 cos ! 1 0 + 2 ˇ 3 j s j 0 + ~ e 2 Í 3 k = 1 D 2 0 s k 0 : (2.24) Weequatethecoe˚cientsofseculartermstozeroand˝ndthesolvabilityconditionfor ! 1 ˇ 1 as 2i A 0 j i ^ A j v j A j + i B v j + F j 1 2 F j 2 + i ^ 2 e i ¹ ˙ 1 + 2 ˇ 3 j º ~ e 2 3 Õ k = 1 A k = 0 : Plugging A j = j e i ¹ ˙ 1 + 2 ˇ 3 j º with j = X j + i Y j intoaboveequation,wethenseparatereal andimaginaryparts. Realpart: Y 0 j = ˙ + v j 2 º X j ^ 2 Y j c 1 + ~ e 2 2 3 Õ k = 1 X k cos 2 ˇ 3 ¹ k j º Y k sin 2 ˇ 3 ¹ k j º ; (2.25) Imaginarypart: X 0 j = ¹ ˙ v j 2 º Y j ^ 2 X j + v j 2 B + ^ 4 ~ e 2 2 3 Õ k = 1 X k sin 2 ˇ 3 ¹ k j º + Y k cos 2 ˇ 3 ¹ k j º ; (2.26) 26 Figure2.11:Steadystateprimaryresonanceresponseamplitudesversuselasticsti˙nessmistuning parameter( v )for = 0 : 1 , ˙ = 0 , ~ e = 0 : 2 , F j 0 = 0 , F j 1 = 0 : 001 , ^ = 0 : 005 , t = 1 , j B j = 1 (a1: ˝rstblade,a2:secondblade,a3:3rdblade) where c 1 = F j 1 4 F j 4 . Similartosuperharmonicresonancecase,wehavecomplicatedequationsfor X j , Y j andampli- tude A j asfunctionsofallitsparameters.Thecoe˚cientmatrixhasthesameeigenvaluesasthe superharmoniccase.Figure2.11showsanumericalplotofthevariationofamplitudewithrespect tomistuningsti˙nessparameter.Therelationbetweensteadystateprimaryresonanceresponse amplitudeandfrequencyisshowninFigure2.12for v = 0 : 005 whichreferstothepeakamplitude valueinFigure2.11.Likethesuperharmoniccase,theprimaryresonanceamplitudeofunison responsesofthesymmetricsystemmatchesthevalueat v = 0 inFigure2.11. Additionally,Figure2.13andFigure2.14presentthechangingresponseamplitudewithrespect tofrequencywhichisthecorrespondsto v = 0 : 1 and v = 1 .Observedphenomenaarevery similartosuperharmoniccase. Weanalyzetherotordynamicsbyrewritingthesolutionthein ˚ domainas s j = F j + 2 B sin ¹ ˚ + 2 ˇ 3 j º + a j cos ¹ p 1 ˚ + j + 4 ˇ 3 j º andpluggedintoEquation(2.6),where 27 Figure2.12:Steadystateprimaryresonanceresponseamplitudesversusdetuningparameterfor = 0 : 1 , v = 0 : 005 , ~ e = 0 : 2 , F j = 0 , F j 1 = 0 : 001 , ^ = 0 : 005 , t = 1 , j B j = 1 ,(a1:˝rstblade,a2: secondblade,a3:3rdblade) a j = 2 q X 2 j + Y 2 j and j = tan 1 ¹ Y j X j º . Thisgives d 1 d ˚ = ^ Q ˚ ^ c r Í 3 k = 1 ^ d cos ˚ F k 0 + 2 B sin ¹ ˚ + 2 ˇ 3 k º + a k cos ¹ p 1 ˚ + k + 4 ˇ 3 k º ~ e 2 B sin ¹ ˚ + 2 ˇ 3 k º + a k p 2 1 cos ¹ p 1 ˚ + k + 4 ˇ 3 k º (2.27) 2.3.4SubharmonicResonance( ! 1 ˇ 2 or ˇ 2 ! n 2 ) ! 1 = 2 + ˙ Subharmonicresonanceisnotlikelyinwindturbinessincewindturbinesusuallyperformat lowspeeds(i.e. 0 ,foratleastone .Figure2.17 showstherelationshipbetweenrealpartoftheeigenvaluesofsteadystatesystemversuselastic sti˙nessvariation v . Figure2.18andFigure2.19demonstratetheconnectionbetweenthefrequencyandrealpartof eigenvalues. 2.3.5ExistenceofResonanceConditions Theresonanceconditions ! n 2 ˇ š 2 , , 2 ,areobtainedfromEquation(2.13)where ! n 2 = s k 0 + k 1 2 m b .Theparametersinvolvedinthemodalfrequencya˙ectstheexistenceoftheresonance condition.When = 2 superharmonic, = 1 primary,and = 1 2 subharmonicresonancesoccur withrespectto ! n 2 = , k 0 + k 1 2 m b = 2 2 ; (2.30) 30 Figure2.15:Campbelldiagramshowing ! n 2 asafunctionof ,for k 0 = 1 , k 1 = 0 : 5 , m b = 1 . Intersectionwiththe 2 , and 2 linesindicatepossibleresonances. whichbringouttheexcitationfrequencycondition = s k 0 m b 2 k 1 : (2.31) InFigure2.15,anexampleCampbelldiagramcanbeseenforspeci˝cparametersthatwere usedto˝ndsystemeigenvalues.Itindicatesthat ! n 2 asafunctionof .Theintersectionpointsof linesand ! n 2 plotgivesustheresonancefrequencies.Althoughprimaryandsuperharmonic resonancesconditionsappearinFigure2.15,the 2 linedoesnotintersectwiththe ! n 2 curve, meaningthatbasically,thesubharmonicresonancewillnotoccurinthesystemwithsimilar parameterscaling.Thesystemsparametersintimedomainwerecalculatedforfourdi˙erent windturbinemodels.Turbinesstructuralpropertieswereobtainedfromthetechnicalreportof NationalRenewableEnergyLaboratory[58].TheseparameterscanbefoundinAppendixB. Basedontheresultfromtheparametersin[58], ! n 2 versus isnearly˛atandsoFigure2.16 31 Figure2.16:Campbelldiagramshowing ! n 2 asafunctionof ,for k 0 = 1 , k 1 = 0 : 1 , m b = 1 . Intersectionwiththe 2 , and 2 linesindicatepossibleresonances. qualitativelyrepresentstheexpectedresonanceconditions.FortheNREL5MW64meterblade, campbelldiagramisdrawnwithparameters k 0 = 603670 , k 1 = 4479 : 3 , mb = 14441 asshownin Figure2.20. 2.4NumericalSimulation Wewouldliketoperformnumericalsimulationstovalidatetheanalyticalresultsonthesu- perharmonicresonances.Thesystemequationsofmotion(3.1)and(2.2)inthetimedomainare simulatedwithMATLAB'sODEsolverThen,thefastFouriertransform(FFT)ofthe sampledanalyticalsolutionandthesimulationwerecompared,for q 1 , q 2 and q 3 ,asshownin Figures.2.22,2.23and2.24foraspeci˝csetofparameters.Theconstantrotorload, Q ˚ 0 ,and rotordampingvalues, c r ,werechosentoresultinameanrotorspeed( )equalto ! n 2 š 2 for theparameterset.Aselectedvalueof Q ˚ 0 ,giventheotherparameters,resultsinasteady-state rotorspeed, .Thesecondmodalfrequency, ! n 2 dependson viacentrifugalsti˙ening.A 32 Figure2.17:RealpartofeigenvaluesofXandYequationsystemversuselasticsti˙nessmistuning forsubharmoniccasefor = 0 : 1 , ~ e = 0 : 2 , ^ = 0 : 005 , ˙ = 0 : 0005 , plotof ! n 2 versustheachieved actual inoursimulationsisshowninFigure2.21.Thereisa gapintheachievedvaluesof actual ,tobediscussedshortly.Asweseefromthecomponents ofEquation(2.22),vibrationamplitudesatfrequencies and ! n 2 predictedbytheperturbation analysiscanbeobtained.Moreover,theexpressionsofmethodofmultiplescalessolutionsatthe meanspeedachievedareevaluatedduringsimulationtocomparepredictedresponseamplitudes. When actual ˇ ! n 2 š 2 ,thereisapotentialforasuperharmonicresonance.Thesystemexcitation frequency = 1 : 1225 rad/seccausesasuperharmonicresonance,asshowninFigures2.22,2.23 and2.24forthe˝rst,secondandthirdwindturbinebladesrespectivelyfortheselectedparameters. Thefrequencyofthesimulatedrotoraccelerationiscomparedtotheanalyticalsolutionof Equation(2.21),asshowninFigure2.25.TheanalyticalrotorsolutioninEquation(2.21)near superharmonicresonanceindicatesfrequencycomponentsof ; 2 ; and 3 .TheFFTpeaksof therotorsimulationalsooccuratthesefrequencies.Theanalyticalsolutionsshowgoodagreement withthesimulationsforallthreebladeandtherotorresponses. IfweseektovalidateFigures2.7,2.9and2.10,wewouldwanttoaimfordesiredvaluesin theparameter, ˙ desired .Insimulation,weachieveameanspeed ,fromwhichweobtainthe 33 Figure2.18:RealpartofeigenvaluesofXandYequationsystemversusdetuningparameter(rotor frequency)forsubharmoniccasefor = 0 : 1 , ~ e = 0 : 2 , ^ = 0 : 005 , v = 0 : 1 , actual ˙ .Whiletherotorloadrises,theactualrotorspeedincreasesaswellinFigure2.26and Figure2.27.Astherotorforcegrows,thereisanintervalinwhichrotorspeedstaysthesame.This isillustratedintermsof ˙ and inthe˝gures.Thisintervalislargerinnegativemistuningthan positivemistuning.Atsomepoint,thespeedhasasuddenjump,thenitcontinuestorise.Ifthe initialconditionfortherotorspeedishigherthanrotorfrequencyforthesuperharmoniccase,there isnojump.Thee˙ectofmistuningontherelationshipbetweenthedesireddetuningparameter ˙ desired andtheactualdetuningfrequency ˙ actual isshowninFigure2.28.Theinitialcondition a˙ectsalsocanbeseeninFigure2.29. Whileweincreasetherotorloading Q ˚ 0 inFigure2.30,whichusestherotorinitialconditionof supH š 2 ,allthreebladeamplitudesincreaseuntilapoint.Then,thereisajumpintheamplitudes andamplitudevaluesdecreasedramatically.Thishappensasaconsequenceofthejumpphenomena inFigure2.26andFigure2.28coordinatedwiththeresonancefeaturesinFigures2.7,2.9,2.10 and2.22. 34 Figure2.19:RealpartofeigenvaluesofXandYequationsystemversusdetuningparameter(rotor frequency)forsubharmoniccasefor = 0 : 1 , ~ e = 0 : 2 , ^ = 0 : 005 , v = 0 : 35 , 2.5SpeedLockingAnalysisforHAWTswithTunedBlades Inthepreviouswork,theblade-rotorsystemwithparametricexcitation[22,59,60]wasanalyzed bymultiplescalesmethodtodescribeprimaryandsuperharmonicresonances.Whileprimary resonancewaslessrelevanttowindturbines,itwasnotedthatthefeaturesofbothresonanceswere similar. Insection2.4andreference[61],theblade-rotorsystemequationsofmotioninthetimedomain weresimulatedwithMATLAB'sODEsolvertoobservethesuperharmonicresonances. Aspeed-lockingphenomenonwasalsoobserved,inwhichthesuperharmonicspeedwaslocked inforanintervalofappliedmeanloads.Inthesimulations,weachievedameanspeed by settingthemeanrotorload.Themeanspeedincreasedwithincreasingrotorload,asshownin Figure(2.26).However,astherotormomentgrewthrougharangewhichproducedrotorspeeds causingasuperharmonicresonanceintheblades,therewasanintervalinwhichrotorspeedstayed thesame.Thisintervalwaslargerinthepresenceofnegativeorpositivemistuning.Atacritical point,thespeedjumpedbacktotherisingtrend.Iftheinitialconditionfortherotorspeedwas 35 Figure2.20:CampbelldiagramforNREL5MW64meterbladeshowing ! n 2 asafunctionof , withparameters k 0 = 603670 N : m , k 1 = 4479 : 3 k g : m 2 , mb = 14441 k g : m 2 .Intersectionwiththe 2 , 3 , and 2 linesindicatepossibleresonancesatthecorresponding . higherthanrotorfrequencyforthesuperharmoniccase,thejumpwasgreatlyreduced. Inthissection,weseektoexplainthemechanismofthisphenomenononasingleblade-rotor system(forwhichtherotorsystemisbalanced).Althoughprimaryresonanceislessrelevantinthe motivationalwind-turbinesystem,itissimplertoanalyze,andthereishopethat,sincetheblade resonancesaresimilartothesuperharmonicresonances,theprimaryspeedlockingmightoccur withasimilarmechanismastheobservedsuperharmonicspeedlocking.Also,primaryresonance providesinsightforanalyzingthemorecomplicatedsuperharmoniccase. Below,weanalyzethespeedlockingusingtheharmonic-balancemethod. 2.5.1PrimaryResonanceCase Theapproximate(small-de˛ection)equationsofmotionofabalancedsinglebladeandrotorsystem are Ü q + 2 Û q + ¹ ! n 2 + cos ˚ º q + ^ d sin ˚ + ~ e Ü ˚ = Q (2.32) Ü ˚ + 2 2 Û ˚ + 2 ^ d cos ˚ q + 2 e Ü q = Q ˚ (2.33) 36 Figure2.21: ! n 2 versus actual graphwith IC = 0 : 5 supH , m b = 1 , e = 0 : 2 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , d = 0 : 2 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 where ~ e = e .Here, q isthesingle-modebladedisplacement, ˚ istherotorangle, ! n isthe blade-onlymodalfrequency, isasmallparameter, pertainstobladedamping, isthestrength ofparametricexcitation(speci˝callyherethe( q cos ˚ )term), ^ d isthestrengthofdirectexcitation ontheblades,andpertainstotheparametricexcitationoftherotor, 2 istherotordamping, ~ e isan inertialcouplingterm,and Q and Q ˚ arethemeanloadsonthebladesandrotorset.Bookkeeping with isnotneededforharmonicbalance, isretainedhereforinsight. HarmonicbalancemethodwillbeappliedEquation(2.32)andEquation(2.33).Weassumea steady-statesolutionoftheform q = D + A cos t + B sin t ˚ = t (2.34) AfterweinserttheseequationsintoEquation(2.32)andEquation(2.33)andtheconstant, 37 Figure2.22:FFTplotof q 1 fortheanalyticalsolutionandsimulationfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 , Q ˚ 0 ˇ 0 : 0198 , = 1 : 225 , ! n 2 = 2 : 2451 cos t and sin t termsarebalancedtozero.Herearethecoe˚cientsfromEquation(2.32): Constant : D ! n 2 + A 2 = Q cos t : A 2 + 2 B + ! n 2 + D = 0 sin t : B 2 2 ¹ A º + ! n 2 B + ^ d = 0 (2.35) wherewede˝ne ˇ , ˇ and ^ d ˇ d , Theconstantcoe˚cientfromEquation(2.33): Constant :2 2 + 2 ^ d A 2 = Q ˚ (2.36) 38 Figure2.23:FFTplotof q 2 fortheanalyticalsolutionandsimulationfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 , Q ˚ 0 ˇ 0 : 0198 , = 1 : 225 , ! n 2 = 2 : 2451 AfterEquation(2.35)andEquation(2.36)arereorganized,wehave D ! n 2 + A 2 = Q (2.37) A ¹ ! n 2 2 º + 2 B + D = 0 (2.38) 2 A + ¹ ! n 2 2 º B = d (2.39) d A 2 + 2 2 = Q ˚ (2.40) Equation(2.38)ismultipledby A andEquation(2.39)ismultipleby B ,thentheyareaddedup. ¹ ! n 2 2 º R 2 + DA = dB (2.41) where R 2 isde˝nedas R 2 = ¹ A 2 + B 2 º .Similarly,Equation(2.38)ismultipliedby B and Equation(2.39)ismultipleby A ,thentheyaresubtracted,resultingin 2 R 2 + DB = dA (2.42) 39 Figure2.24:FFTplotof q 3 fortheanalyticalsolutionandsimulationfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 , Q ˚ 0 ˇ 0 : 0198 , = 1 : 225 , ! n 2 = 2 : 2451 Weassumed D ˇ 0 andEquation(2.41)andEquation(2.42)aresolvedtogethertoobtain R . EquationsaremanipulatedasEqn. ¹ 2 : 41 º 2 + Equation ¹ 2 : 42 º 2 and R 2 isobtainedas R 2 ¹¹ ! 2 n 2 º 2 + ¹ 2 º 2 º = d 2 R = d q ¹ ! 2 n 2 º 2 + ¹ 2 º 2 (2.43) AfterEquation(2.40)andEquation(2.42)arecombined,wehavefound 2 2 + Q ˚ = R 2 (2.44) R isinsertedintoEquation(2.44),then Q ˚ isachievedas Q ˚ = 2 2 + d 2 ¹ ! 2 n 2 º 2 ¹ 2 º 2 (2.45) Equation(2.45)isplottedasrotorforcingversusrotorspeedandtheninvertedinFigure2.31.In 40 Figure2.25:FFTplotof Û ˚ fortheanalyticalsolutionandsimulationfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 , Q ˚ 0 ˇ 0 : 0198 , = 1 : 225 , ! n 2 = 2 : 2451 comparisonwithFigure2.26,thespeedlockingtrendcanbeseen,howevergreatlyexaggerated. Wenotethatitisprimaryresonanceinsteadofsuperharmonicresonance. ReferringbacktoEquations(2.32)and(2.33),itisapparentthattheinertialcouplingisnot neededtoproducespeedlocking,norarethemeanbladeload Q andmeanbladeresponse D .The rotorparametricterm ^ d andmeanrotorload Q ˚ areimportantcontributors.Theparametersin Figure2.31aregeneric,andaremeanttoshowthephenomenon. 2.5.2SuperharmonicResonanceCaseat 2 ˇ ! n 2 Secondaryresonancesshowessentiallythesamespeed-lockingphenomenonasprimaryresonance althoughatasmallerscale,notsurprisingly.Theequationsofmotionscaledforsecondary resonancesare Ü q + 2 Û q + ¹ ! n 2 + cos ˚ º q + d sin ˚ + ~ e Ü ˚ = Q (2.46) Ü ˚ + 2 2 Û ˚ + d cos ˚ q + 2 e Ü q = Q ˚ (2.47) 41 Figure2.26: actual versus Q ˚ Graphwithnegativeandpositivemistuningcasesandtunedcase for m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 Theharmonic-balancemethodisappliedtoEquations(2.46)and(2.47).Forsuperharmonic responses,weassume q = D + A cos2 t + B sin2 t + C cos t + E sin t ˚ = t + f cos t + g sin t (2.48) Calculationsawayfromprimaryresonanceshowthat C issmallerthan E .Weassumethat C , D and Q arenegligibleinEquations(2.46)and(2.47),andtreat f and g assmall. AfterweinsertEquation(2.48)intoEquation(2.46)andEquation(2.47),wehavebalancedthe constant, sin t , cos t cos2 t , sin2 t termstozero.Aftermanymanipulationsofthebalance equations,˝nallywefoundsuperharmonicresonanceamplitudeatordertwo,where 2 ˇ ! n ,is R 2 2 = ¹ º 2 ¹ 4 d 3 ! n 2 º 2 4 »¹¹ ! n 2 4 2 º + 1 2 g º 2 + ¹ 4 º 2 ¼ 4 ¹ d 2 ! n 2 º 2 (2.49) 42 Figure2.27: actual versus Q ˚ graphfordi˙erentinitialconditionfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 where R 2 2 = A 2 + B 2 . ByusingEquation(2.49),andtheconstantbalancefromEquation(2.47),wehavefoundthe rotorforceequationas Q ˚ = 2 2 + 2 2 ¹ d º 2 2 R 2 2 (2.50) where = 4 » 4 + ¹ 2 2 º 2 ¼ .Noticethat R 2 2 showsaresonancepeakatthesuperharmonic frequency, 2 ˇ ! n ,butscaledby 2 ,andthus Q ˚ shouldhaveasmallpeakwhenplottedversus .Invertingthisplotsuggestsapossible Q ˚ intervalofspeedlocking. Wefoundthepeakvalueofforcinglevelat ˇ ! peak where ¹¹ ! 2 n 4 2 º + 1 2 g º 2 = 0 as Q ˚ peak = 2 2 + 8 2 2 ¹ d 2 3 ! n º 2 ¹ 4 º 4 2 2 + ¹ d 2 ! n º 2 (2.51) 43 Figure2.28: ˙ desired versus ˙ actual withnegativeandpositivemistuningcasesandtunedcasefor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 Similarly,inFigure2.32thespeedlockingtrendcanbeseenwithsuperharmoniccaseatorder twoincomparisonwithFigure2.26.TheparametersthatusedtocreateFigure2.32aregenericto showthespeedlockingphenomenon. 2.6Conclusions In-planevibrationsofamistunedthree-bladewindturbinewerestudied.Byusingasimpli˝ed model,in-planevibrationsweretakenintoconsideration.Thebladesandrotorequationswere writteninthe ˚ domaintodecouplethebladeequationsfromtherotorequation. Afterdecoupling,thedi˙erentialbladeequationscontainedparametricanddirectexcitation. Welookedatresonancesduetocyclicgravitationalandaerodynamicloading.Itwasobserved thatslightmistuninginasinglebladecouldcausethesingleresonancepeakofthetunedcaseto splitintotwoorthreeresonancepeaks,inboththesuperharmonicandprimaryresonancesand 44 Figure2.29: ˙ desired versus ˙ actual graphfordi˙erentinitialconditionfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 thebladestakeondi˙erentsteadystateamplitudes.Theresonantamplitudeofatleastoneofthe bladesislargerthanintheperfectlytunedcase.Ontheotherhand,subharmonicresonancewillnot occurinarotatingsystemwithsimilarparameterscaling.Subharmonicresonancesmayormay notbepossibleinthisdynamicalsystem,dependingonparameters,andatrotorspeedswelloutside theexpectedoperatingconditionsofwindturbines.Subharmonicresonancesinvolveinstabilities similartothoseoftheMathieuequation,butmorecomplicated. Speci˝cally,superharmonicresonancesisfocusedon,sincetheyaremorelikelytobeprob- lematicinthelow-frequencyoperatingrangesofwindturbines.Thee˙ectsofparametersonthe superharmonicresonancesareexamined,andanumericalstudyisconductedaimedatverifying theanalyticalresults.Wealsolookedattherotordynamicsduringresonance. Atthesuperharmonicresonantfrequency,positivemistuningleadstoanincreasedresonant amplitudeofoneoftheblades.However,veryslightnegativemistuningcanleadtoaslight amplitudeincreaseintwooftheblades,whilemorenegativemistuningsleadtoadecreased 45 Figure2.30: Q ˚ versusperturbationsolutionamplitudegraphfor m b = 1 , J r = 10 , c b = 0 : 01 , c r = 0 : 01 , k 0 1 ; 2 = 4 : 7 , k 0 3 = 4 : 75 , k v = 0 : 5 , k 1 = 0 : 27 , k 2 = 0 : 05 , d = 0 : 2 , e = 0 : 2 amplitudeintwooftheblades.Thee˙ectofdampingissimilartothatoflinearoscillators,in whichtheresonancespeakssharpenwithdecreaseddamping.Increaseddampingwasobservedto quenchoneofthethreesuperharmonicpeaksinthemistunedblade.Increasingtheforcinglevel increasestheresonancepro˝leuniformly. TheFFTsofnumericalsimulationsofbladesata˝xedsetofparametersinsuperharmonic resonanceagreedverywellwiththoseofthesampledanalyticalresponses.Therotordynamics werealsoexpressedanalytically.Inthesuperharmonicresonance,theanalyticalrotorresponsewas showntohavethreefrequencies, ; 2 ; and 3 ,duetononlineare˙ects.Thenumericalsolutions capturedthesethreefrequencycomponentsandeachagreedverywellinamplitude. Weaimedtonumericallycon˝rmthebladeresponseamplitudesatvariousrotorspeedsnear resonance.Therotorspeedisdeterminedbytheinputloadtotherotor.Simulationsshowedthat themeanrotorspeedcanlockintothesuperharmonicspeedoverarangeofmeanrotorloads,and, withfurtherincreaseoftherotorload,experienceajumpoutofthesuperharmonicspeed.Thus,it 46 Figure2.31:Speed-lockinggraph versus Q ˚ forprimaryresonancefor = 0 : 1 ; 2 = 0 : 5 ;! n 2 = 2 : 23 ; d = 1 : 871 ispossiblethatsomedetuningparametersinFigures2.7,2.9and2.10cannotbeachieved,aswell asresonantbladeresponsesinsuchrangesofdetuningparameters.Weinvestigatedthee˙ectsof mistuninglevelandinitialconditionsontherelationshipbetweentherotorloadandmeanspeed.In addition,welookedintotherelationshipbetweenrotorloadandthebladeamplitudes.Weobserved allthreebladeamplitudesjumpeddownsuddenlyatacertainvalueoftheincreasingrotorload. Lastly,ourpurposewastoanalyticallycon˝rmthebladeresponseamplitudesatvariousrotor speedsnearresonanceandverifyspeedlockingphenomenon.Wefoundthatspeed-lockingisdue totheaverageinteractionbetweenthebladevibrationandrotormotionintherotorequation,andits balanceagainstthemeanrotormoment.Thephenomenonwasexaminedforane˙ective(balanced) singleblade-rotorsystembyapplyingharmonicbalancemethod. 47 Figure2.32:Speed-lockinggraph versus Q ˚ forsuperharmonicresonanceordertwofor g = 1 ; = 0 : 01 ; 2 = 0 : 0135 ;! n 2 = 2 : 23 ; d = 1 : 871 ; = 0 : 5 48 CHAPTER3 SECOND-ORDERPERTURBATIONANALYSISOFIN-PLANEBLADE-HUB DYNAMICSOFHORIZONTAL-AXISWINDTURBINES 3.1Introduction Thisworktakesinterestinthecoupledblade-rotordynamicswithandwithoutmistuning. MoregeneralmistunedrotationalsystemshavebeenanalyzedbymanyresearchersThe e˙ectsofmistuningontheturbinebladeswithmechanicalcouplingwasanalyzedbyWhitehead [51].Ewins[52]andChaandSinha[53]workedonbrokensymmetryinbladeddisk.Also,the e˙ectsofmistuningonfour-degree-of-freedomsystemswithparametricsti˙nesswasstudied[55]. MistuningonSinhaandGri˚n[45]andDick etal. [46]workedonnonlinearityinrotorstoshow thedeviationfromthesymmetricresponse.Thewind-turbinerotor-bladessystemwithtuned[43] andweaklymistunedblades[60]wereanalyzedto˝ndsteadystatedynamicsbyusinga˝rst-order perturbationmethod.Theanalysisshowedthatdirectandparametricexcitationcombinetocausea superharmonicresonanceathalfthe˝rstmodalfrequency,aprimaryresonance,andsubharmonic resonanceattwicethenaturalfrequencyofthesystem.Inthetunedcase,eachbladehadthe sameamplitudebecausethesystemwassymmetricandlinear,whileinthemistunedcasevibration localizationcouldoccur. Inthisstudy,weapplyasecond-ordermethodofmultiplescalestotheequationsofmotionfor in-planetunedandweaklymistunedbladesofhorizontalaxiswindturbinetodeterminethesteady statedynamics,withfocusonthesuperharmonicresonanceoforderthreeforthelinearsystemwith hardforcing. 3.2ThreeBladeTurbineEquations Inthischapter,weanalyzebladesEquation(2.1)androtorEquation(2.2)asstudiedinSec- tion2.2.Rewrittingtherotorand j th bladelinearizedequationsofmotioninthetimedomainfor 49 j = 1 , 2 and 3 : m b Ü q j + c b Û q j + ¹ k 0 j + k 1 Û ˚ 2 + k 2 cos ˚ j º q j + d sin ˚ j + e Ü ˚ = Q j ; (3.1) J r Ü ˚ + c r Û ˚ + 3 Õ k = 1 ¹ d cos ˚ k q k + e Ü q k º = Q ˚ ; (3.2) where k 0 1 = k 0 2 = k 0 , k 0 3 = k 0 + k v where k 0 isblade'smodalelasticsti˙ness, isasmall parameterand k v istheelasticsti˙nessvariationofthemistunedblade, m b istheinertiaofa singleblade, J r isthetotalinertiaofblade-hubsystem, e isthecouplingterm, q j istheassumed modalcoordinatefor j th blade, ˚ istherotorangle, ˚ j = ˚ + 2 ˇ 3 j isthebladeangle, k 1 and k 2 aresti˙nesscontributionsofcentrifugalandgravitationale˙ects,respectively, Q j and Q ˚ are generalizedforcingtermsduetoaeroealsticloadingand c b and c r aregenericdampingcoe˚cients [60].De˝nitionsoftheseparameterscanbefoundintheAppendixA.1.Towermotionisneglected andonlyin-planevibrationistakenintoconsiderationbyusingsimpli˝edmodelasshownin Figure2.1.Forthezerogravitysystem,Figure3.1showsthehownaturalfrequencieschange withrespecttomistuningsti˙nessparameter.Inthesymmetriccase, k v = 0 ,thesecondandthird naturalfrequenciesareequal, ! n 3 = ! n 2 . Following[60]asinChapter2,theindepedentvariabletimewaschangedtorotorangle ˚ and Equations(3.1)and(3.2)aretransformedtoanewform 2 q 00 j + 0 q 0 j + ~ c b q 0 j + ¹ ~ k 0 j + ~ k 1 2 + ~ k 2 cos ˚ j º q j + ~ d sin ˚ j + ~ e 0 = ~ Q j ; (3.3) 0 + ~ c r + ˜ 3 Õ k = 1 h ~ d cos ˚ k q k + ~ e ¹ 2 q 00 k + 0 q 0 k º i = ~ Q ˚ ; (3.4) where ¹º 0 = d ¹º d ˚ ,and ~ k 0 1 = ~ k 0 2 = ~ k 0 and ~ k 0 3 = ~ k 0 + ~ k v andwhere ~ c b = c b m b , ~ k 0 = k 0 m b 2 , ~ k v = k v m b 2 , ~ k 1 = k 1 m b , ~ k 2 = k 2 m b 2 , ~ d = d m b 2 , ˜ = m b J r , ~ Q j = Q j m b 2 , ~ e = e m b , 50 Figure3.1:Eigenvaluesversuselasticsti˙nessmistuningparameter( k v )plotfor k 0 = 5 , k 1 = 0 : 5 , Û = 0 , m = 1 , J = 1 , e = 0 : 2 , = 0 : 1 (e1:˝rstblade,e2:secondblade,e3:thirdblade) ~ c r = c r J r , ~ Q ˚ = Q ˚ J r 2 Thesmallparameterwasde˝nedas = m b š J r since J r ismuchlargerthan m b .Todecouple therotorequationfromtheleading-orderbladeequations,anondimensionalizationwasapplied andnewscalingwasdoneonEquations(3.3)and(3.4): = 1 + 2 1 ; ~ c b = ^ c b ; ~ k 2 = ^ k 2 ; ~ d = ^ d ; ~ c r = 2 ^ c r ;˜ = ; q j = s j ; ~ Q j = ^ Q j ; ~ Q ˚ = 2 ^ Q ˚ : Theequationsarerewrittenintermsofnewscaledbladecoordinates s j andhubcoordinate v 1 as s 00 j + ^ c b s 0 j + ¹ ~ k 0 j + ~ k 1 + ^ k 2 cos ˚ j º s j + ^ d sin ˚ j + ~ e 0 1 = ^ Q j + H : O : T :; (3.5) 0 1 + ^ c r + 3 Õ k = 1 ¹ ^ d cos ˚ k s k + ~ es 00 k º = ^ Q ˚ + H : O : T : (3.6) whereH.O.T.meanshigher-orderterms. 51 Detailedstepsandassumptionsarenotgivenhereandcanbefoundin[60].Afterrearranging theterms,thedecoupledbladeequationswereobtainedas s 00 j + ^ k 0 s j = Q j 0 ^ d sin ˚ + 2 ˇ 3 j + h Q j 1 cos ˚ + 2 ˇ 3 j ~ eQ ˚ 0 + ~ e ^ c r ^ c b s 0 j ~ k v j s j ^ k 2 cos ˚ + 2 ˇ 3 j s j + ~ e 2 Í 3 k = 1 s 00 k i ; (3.7) 3.3Second-OrderMethodofMultipleScales WereorganizeEquation(3.7)onthebasisofnewindependentvariable = p 1 ˚ ,where p 1 = q ^ k 0 = r k 0 š 2 + k 1 m b .Theequationofmotioninthe domainbecomes s 00 j + s j = F j F j 0 sin ! 1 + 2 ˇ 3 j + h F j 1 cos ! 1 + 2 ˇ 3 j + f s 0 j v j s j + ~ e 2 Í 3 k = 1 s 00 k cos ! 1 + 2 ˇ 3 j s j i ; (3.8) wherenow ¹º 0 = d d and F j = Q j 0 ^ k 0 ; F j 0 = ^ d ^ k 0 ;! 1 = 1 p 1 ; F j 1 = Q j 1 ^ k 0 ; f = ~ e ¹ ^ c r Q ˚ 0 º ^ k 0 ; = ^ c b p 1 ; = ^ k 2 ^ k 0 ; v j = ~ k v j ^ k 0 Accordingtothe˝rst-orderperturbationanalysisonlineartunedandmistuned3-bladehorizantal axiswindturbine,superharmonicresonanceexistsatordertwoatthesystemnaturalfrequency [60].Byapplyingthesecond-orderperturbationanalysis[26]toEquation(3.8)weincludethree scales ¹ 0 ; 1 ; 2 º and s j isseparatedintodominantsolution s j 0 andsmallvariations s j 1 and s j 2 suchthat s j = s j 0 ¹ 0 ; 1 ; 2 º + s j 1 ¹ 0 ; 1 ; 2 º + 2 s j 2 ¹ 0 ; 1 ; 2 º (3.9) 52 where i = i 0 . Then d d = D 0 + D 1 + 2 D 2 , d 2 d 2 = D 2 0 + ¹ 2 D 0 D 1 º + 2 ¹ D 2 1 + 2 D 0 D 2 º ; where D i = @ @ i TheseformulationsarepluggedintoEquation(3.8)andthenwebalancethecoe˚cientsof 0 , 1 and 2 : O ¹ 1 º : D 2 0 s j 0 + s j 0 = F j ~ F j 0 sin ! 1 0 + 2 ˇ 3 j (3.10) where ~ F j 0 sin ! 1 0 + 2 ˇ 3 j = F j 0 sin ! 1 0 + 2 ˇ 3 j + f . O ¹ º : D 2 0 s j 1 + s j 1 = 2 D 0 D 1 s j 0 D 0 s j 0 v j s j 0 + ~ e 2 Í 3 k = 1 D 2 0 s k 0 cos ! 1 0 + 2 ˇ 3 j s j 0 + F j 1 cos ! 1 0 + 2 ˇ 3 j O ¹ 2 º : D 2 0 s j 2 + s j 2 = 2 D 0 D 1 s j 1 2 D 0 D 2 s j 0 D 2 1 s j 0 D 0 s j 1 D 1 s j 0 v j s j 1 cos ¹ ! 1 0 + ˚ j º s j 1 + ~ e 2 Í 3 k = 1 ¹ 2 D 0 D 1 s k 0 + D 2 0 s k 1 º (3.11) Solvingthe O ¹ 1 º Equation(3.10), s j 0 isdeterminedas s j 0 = F j 2 + A j e i 0 i j e i ¹ ! 1 0 º + c : c : (3.12) 53 where A j = ¹ X j + iY j º e i ˙ 0 and j = F j 0 2 ¹ 1 ! 2 1 º e i ˚ j and c : c referstocomplexconjugate. X and Y arefunctionsof 1 and 2 .The s j 0 Equation(3.12)ispluggedintothe O ¹ º Equation(3.11),and thenwerewrite O ¹ º equation D 2 0 s j 1 + s j 1 = 2 iD 1 A j e i 0 j ! 1 e i ! 1 0 + iA j e i 0 + c : c : 1 2 e i ˚ j e i 0 ! 1 + c : c : F j i j e i 0 ! 1 + A j e i 0 + i j e i 0 ! 1 + A j e i 0 + 1 2 F j 1 e i ˚ j e i 0 ! 1 + F j 1 e i ˚ j e i 0 ! 1 v j F j 2 + A j e i 0 i j e i ! 1 0 + c : c : + ~ e 2 Í 3 k = 1 i ! 2 1 A k e i ! 1 0 A k e i 0 + c : c (3.13) Fourdi˙erentresonanceconditionscanbeidenti˝edfromEquation(3.13): 1. Nospeci˝crelationbetween ! 1 andthenaturalorder= 1 2. ! 1 ˇ 1 3. ! 1 ˇ 2 4. ! 1 ˇ 1 š 2 Thecaseof ! 1 ˇ 1 isproperlytreatedwithsoftexcitation. ! 1 ˇ 2 and ! 1 ˇ 1 š 2 werestudied in[60]. 3.3.1NonresonantCaseat O ¹ º Weconcentrateonthe˝rstcasewherethereisnospeci˝crelationshipbetweenforcingfrequency andnaturalfrequency.ThenonresonantsolvabilityconditionforEquation(3.13)isfoundby eliminatingthecoe˚cientofsecularterms e i 0 : 54 2 iD 1 A j iA j ~ e 2 3 Õ k = 1 A k v j A j = 0 (3.14) WesolvetherestofEquation(3.13)whichcorrespondstononresonanttermstoobtainthe particularsolution.Since A j isnotafunctionofindependentvariable 0 ,wetreat A j asconstant. Then s j 1 = Q j 2 + U j e i ! 1 0 + V j e i ¹ ! 1 1 º 0 + W j e i ¹ ! 1 + 1 º 0 + L j e i 2 ! 1 0 + c : c : (3.15) where Q j = i e i ˚ j j + i e i ˚ j j , U j = 1 1 ! 2 1 1 2 F j 1 e i ˚ j j ! 1 2 e i ˚ j F j + ~ e 2 Í 3 k = 1 i ! 2 1 j , V j = 1 1 ¹ ! 1 1 º 2 2 e i ˚ j A j , W j = 1 1 ¹ ! 1 + 1 º 2 2 e i ˚ j A j , L j = i j 2 ¹ 1 4 ! 2 1 º Then,weplug s j 0 and s j 1 fromEquation(3.12)andEquation(3.15)intoEquation(3.11).After wereorganizethe O ¹ 2 º equation,wehave 55 D 2 0 s j 2 + s j 2 = D 2 1 ¹ A j º e i 0 2 iD 2 ¹ A j º e i 0 D 1 ¹ A j º e i 0 + v j Q j 2 + U j e i ! 1 0 + V j e i ¹ ! 1 1 º 0 + W j e i ¹ ! 1 + 1 º 0 i ¹ ! 1 1 º V j e i ¹ ! 1 1 º 0 + i ¹ ! 1 + 1 º W j e i ¹ ! 1 + 1 º 0 + i ! 1 U j e i ! 1 0 + 2 i ! 1 L j e 2 i ! 1 0 1 2 e i ˚ j e i 0 ! 1 + c : c Q j 2 + U j e i ! 1 0 + V j e i ¹ ! 1 1 º 0 + W j e i ¹ ! 1 + 1 º 0 + L j e i 2 ! 1 0 + c : c : + ~ e 2 Í 3 k = 1 ! 1 + 1 º 2 W k e i ¹ ! 1 + 1 º 0 4 ! 2 1 L k e 2 i ! 1 0 ! 2 1 U k e i ! 1 0 ¹ ! 1 1 º 2 V k e i ¹ ! 1 1 º 0 + 2 iD 1 A k e i 0 2 i ¹ ! 1 1 º e i ˚ j D 1 A j 2 ¹ 1 ! 1 1 º 2 º e i ¹ ! 1 1 º ˚ j i ¹ ! 1 + 1 º e i ˚ j D 1 A j 2 ¹ 1 ! 1 + 1 º 2 º e i ¹ ! 1 + 1 º ˚ j + c : c (3.16) ByexaminingtheEquation(3.16),weobservetheresonanceconditionsasfollows. 1. Nospeci˝crelationbetween ! 1 andthenaturalorder= 1 2. ! 1 ˇ 1 3. ! 1 ˇ 2 4. ! 1 ˇ 1 š 2 5. ! 1 ˇ 1 š 3 Thecases ! 1 ˇ 1 , ! 1 ˇ 2 and ! 1 ˇ 1 š 2 ,ifproperlytreated,wouldhaveaddedasecularterm totheEquation(3.14),removedaseculartermfromEquation(3.15),andhenceEquation(3.16) wouldneedtobeadjusted. 56 3.3.2SuperharmonicCaseatOrder 3 at O ¹ 2 º Inourpaper,wespeci˝callyfocusonthesuperharmonicresonancecaseat ! 1 ˇ 1 š 3 .Inthetime domain,thismeansthat ˇ ! n š 3 ; where ! n isthelowestoscillatorymodalfrequency.Aswe followthesamestepsin[26]and[29],weobtainthesolvabilityconditionforsuperharmaniccase fromEquation(3.16),using 3 ! 1 = 1 + ˙ ,as 2 iD 2 ¹ A j º D 1 ¹ A j º D 2 1 A j 2 L j e i ˚ j e i ˙ 0 + 2 i ~ e 2 3 Õ k = 1 D 1 ¹ A k º = 0 (3.17) FromtheEquation(3.14), D 1 A j = 2 A j + 1 2 i ~ e 2 3 Õ k = 1 A k + 1 2 i v j A j Wecomputedthe D 2 1 A j equationbydi˙erentiatingtheexpressionof D 1 A j .Insertingthe undi˙erentiated D 1 A j equationintoit,toobtain D 2 1 A j = 2 4 i v j 2 1 4 2 v j ! A j + ~ e 2 3 Õ k = 1 A k 3~ e 2 4 i 2 v j 2 ! InsertingtheequationaboveintotheEquation(3.17),weendupwiththe D 2 A j equationas D 2 A j = 1 8 i 2 1 8 i 2 v j A j 1 4 i L j e ¹ ˚ j + ˙ 0 º i + 9 8 i ~ e 4 + 1 4 i v j ~ e 2 1 2 ~ e 2 3 Õ k = 1 A k (3.18) Forthepurposeofobtaininganexpressionfor A j ,solvabilityconditionequations D 1 A j and D 2 A j needtobesolvedtogether.Werecombinethe and 1 ; 2 scalesinaprocess.Thisprocess iscalledastitution[26]and[29].Thereconstitutionequationis dA j d = D 1 A j + 2 D 2 A j (3.19) After D 1 A j and D 2 A j equationsareinsertedintotheEquation(3.19),weobtainthereconstituted di˙erentialequationas 57 dA j d = A j 2 + i ~ e 2 2 Í 3 k = 1 A k + i v j 2 A j + 2 i 2 8 i 2 v j 8 ! A j 2 2 j 8 ¹ 1 4 ! 2 1 º e ¹ ˚ j + ˙ 0 º i ! + 2 ~ e 2 9 8 i ~ e 2 2 + i v j 4 Í 3 k = 1 A k (3.20) WelookforasolutionintheCartesiancoordinateform A = ¹ X + iY º e ¹ i ˙ 0 º . A j and j = F j 0 2 ¹ 1 ! 2 1 º e i ˚ j equationsareinsertedintotheEquation(3.20),andeachsideof equationdividedby e ¹ i ˙ 0 º .Thenwesplittheequationintorealandimaginaryparts.After simpli˝cation,weobtain Realpart: Û X j = z 1 X j + z 2 3 Õ k = 1 X k + z 3 Y j + z 4 3 Õ k = 1 Y k + z 5 cos ¹ 2 ˚ j º (3.21) Imaginarypart: Û Y j = z 3 X j z 4 3 Õ k = 1 X k + z 1 Y j + z 2 3 Õ k = 1 Y k + z 5 sin ¹ 2 ˚ j º (3.22) where z 1 ; z 2 ; z 3 ; z 4 ; z 5 canbefoundintheAppendixC.1.Forsteadystatebehavior, Û X j = 0 and Û Y j = 0 .Wecan˝ndapolarform A j = 1 2 a j e i byusing X j and Y j .Thentheresponseamplitudeis givenas a j = 2 q X 2 j + Y 2 j . 3.4Results Inthissection,weanalyzethesecond-ordersuperharmonicresonancebehaviorfortunedand mistunedblades.Table3.1liststhefrequencyratiosatwhichresonanceshavebeenidenti˝edby ˝rstorsecond-ordermethodofmultiplescalesexpansion.AsseeninTable3.1,second-order perturbationanalysisrevealsthesuperharmonicresonancesatorder 3 fortunedandmistunedcase. 58 Table3.1:ResonanceChart. R 1 :Resonanceidenti˝edat˝rst-orderofMMSexpansion. R 2 : Resonanceidenti˝edatsecond-orderofMMSexpansion. :Knownresonancecase/Instability notuncovereduptotwoordersofexpansion Forcing( ! 1 ) Tuned Mistuned 1 R 1 R 1 2 R 1 R 1 2 š 3 - - 1 š 2 R 1 R 1 1 š 3 R 2 R 2 3.4.1TunedBladeCase( v = 0 ) Here,weassumethatthereisnomistuningontheturbineblades ¹ v j = 0 º .Thenthemagnitudeof steadystateresponseamplitude a isthesameforeachblade,andisgivenas a = 2 E j s 2 2 4 + ˙ + 2 2 8 2 = E s 2 4 + ˙ + 2 8 2 (3.23) where E = j E j j = 2 j 8 ¹ 1 4 ! 2 1 º ,noting j areequalforall j . Forconvenience,wede˝ned j = j 8 ¹ 1 4 ! 2 1 º so E j and becomeas E = j E j j = 2 = F j 0 16 ¹ 1 ! 2 1 º¹ 1 4 ! 2 1 º (3.24) FromEquation(3.23),weobtain a max and ˙ max as a max = 2 E j ˙ max = 2 8 (3.25) Figure3.2demonstratesanumericalplotforallthreebladeamplitudeswithrespecttodetuning parameter.Itstatesthatthreeofthebladeamplitudesaresamebecauseofthesymmetry.Forsame 59 Figure3.2:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuningparameter for ~ e = 0 : 2 , = 0 : 1 , j j = 10 , ˚ = 0 , = 0 : 1 , = 0 : 005 (a1:˝rstblade,a2:secondblade,a3: thirdblade) setofparametersthatareusedforplotgive a max = 4 at ˙ = 0 : 000375 fromtheEquation(3.25). ThesevaluesareconsistentwiththemaximumvalueinFigure3.2. AmplitudeversusfrequencyplotsareshowninFigure3.3andFigure3.4.When decreases theresonanceamplitudegetsharpenedandwhen j j increasesamplitudeincreasesaswell,where isde˝nedintheAppendixC.1.Inall˝gures,dampingfactoractas ^ whichis ^ = 2 . 3.4.2MistunedBladeCase( v , 0 ) Inthissectionweapplymistuning v toasinglebladeandobserveitse˙ectontheamplitudesof thethreeblades.Figure3.5presentstheamplitudesofeachbladeforaspeci˝csetofparameters as v varies. Forthesymmetriccasewhen v = 0 ,allbladeshavethesameamplitude.When v isvery small,amplitudesaresigni˝cantlya˙ected.Asseenfromtheplot,whenmistuninggrows,the thirdblade(mistunedblade)amplitudedecreases.Thesecondblade'samplitude,withverysmall mistuning,getslargerthaninthesymmetriccase.Forvariousvaluesofmistuningparameter v ,the 60 Figure3.3:Steadystatesuperharmonicresonanceresponseamplitudesversusfrequencyfor ~ e = 0 : 2 , = 0 : 1 , j j = 10 , ˚ = 0 , = 0 : 1 , = 0 : 005 ; 0 : 01 ; 0 : 03 Figure3.4:Steadystatesuperharmonicresonanceresponseamplitudesversusfrequencyfor ~ e = 0 : 2 , ˚ = 0 , = 0 : 1 , = 0 : 005 , = 1 ; 2 ; 3 superharmonicresonanceamplitudechangeswithrespecttodetuningparameter ˙ ,asillustrated inFigure3.6,Figure3.7,Figure3.8andFigure3.9.Thesingleresonancepeakwhichisseenin Figure3.2issplitupbymistuningwhichcanbeseenonmistunedcaseplots.Comparingtothe symmetriccaseamplitudeinFigure3.2andmistunedcaseamplitudefor v = 0 : 006 inFigure3.6, 61 theresonanceamplitudeforonebladeisincreasedforthesamesetofparameters,whiletheother bladeamplitudeshavedecreased.Thispointstothepossibilityofvibrationlocalization. Figure3.5:Steadystatesuperharmonicresonanceresponseamplitudesversuselasticsti˙ness mistuningparameter v for ~ e = 0 : 2 , = 0 : 1 , j j = 1 , ˚ = 0 , = 0 : 1 , = 0 : 005 , ˙ = 0 (a1:˝rst blade,a2:secondblade,a3:thirdblade) 3.5RemainingTask Weexpectthatconstantloadcancausesuperharmonicresonancewhichwasnotrevealedon ˝rst-orderperturbationanalysis.Here,weapplysecond-orderperturbationanalysisfocusingon constantloading. 3.5.1SuperharmonicCaseatOrder 2 at O ¹ º Asasecondcasewhere ! 1 ˇ 1 š 2 isstudiedatorderof O ¹ º .Thesuperharmonicsolvability conditionatorder2forEquation(3.13)isobtainedbycancellingthecoe˚cientofsecularterms e i 0 and e i 2 ! 1 0 out. 2 iD 1 A j iA j ~ e 2 3 Õ k = 1 A k v j A j + 1 2 i j e i ˚ j e i ˙ 0 = 0 (3.26) 62 Figure3.6:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuningparameter for ~ e = 0 : 2 , = 0 : 1 , j j = 10 , ˚ = 0 , = 0 : 1 , = 0 : 005 (a1:˝rstblade,a2:secondblade,a3: thirdblade) AfterweremovethetermsthatEquation(3.26)havefromEquation(3.13),restoftheequation issolvedtogettheparticularsolution.Equation(3.13)becomes D 2 0 s j 1 + s j 1 = Q j 2 + P j e i ! 1 0 + R j e i ¹ ! 1 1 º 0 + S j e i ¹ ! 1 + 1 º 0 + c : c : (3.27) where Q j = i e i ˚ j j + i e i ˚ j j v j F j , P j = 1 2 F j 1 e i ˚ j j ! 2 e i ˚ j F j + ~ e 2 Í 3 k = 1 i ! 2 A k , R j = 2 e i ˚ j A j , S j = 2 e i ˚ j A j SolvingtheEquation(3.27)to˝ndthe s j 1 s j 1 = Q j 2 + U j e i ! 1 0 + V j e i ¹ ! 1 1 º 0 + W j e i ¹ ! 1 + 1 º 0 + c : c : (3.28) where U j = P j 1 ! 2 1 , V j = R j 1 ¹ ! 1 1 º 2 , W j = S j 1 ¹ ! 1 + 1 º 2 63 Figure3.7:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuningparameter for ~ e = 0 : 2 , = 0 : 1 , j j = 10 , ˚ = 0 , = 0 : 1 , = 0 : 005 (a1:˝rstblade,a2:secondblade,a3: thirdblade) InsertingtheEquations(3.12)and(3.28)intoEquation(3.11)givesus D 2 0 s j 2 + s j 2 = D 2 1 ¹ A j º e i 0 2 iD 2 ¹ A j º e i 0 D 1 ¹ A j º e i 0 + v j Q j 2 + U j e i ! 1 0 + V j e i ¹ ! 1 1 º 0 + W j e i ¹ ! 1 + 1 º 0 + c : c i ! 1 U j e i ! 1 0 + i ¹ ! 1 1 º V j e i ¹ ! 1 1 º 0 + i ¹ ! 1 + 1 º W j e i ¹ ! 1 + 1 º 0 + c : c : 1 2 e i ˚ j e i 0 ! 1 + c : c Q j 2 + U j e i ! 1 0 + V j e i ¹ ! 1 1 º 0 + W j e i ¹ ! 1 + 1 º 0 + c : c : + ~ e 2 Í 3 k = 1 ! 1 1 º 2 V k e i ¹ ! 1 1 º 0 ¹ ! 1 + 1 º 2 W k e i ¹ ! 1 + 1 º 0 ! 2 1 U k e i ! 1 0 + 2 iD 1 A k e i 0 2 © « i ¹ ! 1 1 º e i ˚ j D 1 A j 2 ¹ 1 ! 1 1 º 2 º e i ¹ ! 1 1 º ˚ j i ¹ ! 1 + 1 º e i ˚ j D 1 A j 2 ¹ 1 ! 1 + 1 º 2 º e i ¹ ! 1 + 1 º ˚ j ª ® ¬ + c : c (3.29) ExaminingtheEquation(3.29)forthesuperharmoniccaseatorder 2 ,wearriveatthefollowing resonancecases: 64 Figure3.8:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuningparameter for ~ e = 0 : 2 , = 0 : 1 , j j = 10 , ˚ = 0 , = 0 : 1 , = 0 : 005 (a1:˝rstblade,a2:secondblade,a3: thirdblade) 1. Nospeci˝crelationbetween ! 1 andthenaturalorder= 1 2. ! 1 ˇ 1 3. ! 1 ˇ 2 4. ! 1 ˇ 1 š 2 Ifweredotheanalysesfor ! 1 ˇ 1 and ! 1 ˇ 2 casesproperly,Equation(3.29)willneedtobe adjusted. 3.5.2SuperharmonicCaseatOrder 2 at O ¹ 2 º Wewillfocusonspeci˝callysuperharmonicresonancecaseatorder2forsecondexpansion.As wefollowthesamestepsinSection3.3.2,wereceivethesolvabilityconditionforsuperharmonic casefromEquation(3.29),using 2 ! 1 = 1 + ˙ ,as 2 iD 2 A j D 1 A j D 2 1 A j 2 U j e i ˚ j e i ˙ 0 + 2 i ~ e 2 3 Õ k = 1 D 1 ¹ A k º (3.30) 65 Figure3.9:Steadystatesuperharmonicresonanceresponseamplitudesversusdetuningparameter for ~ e = 0 : 2 , = 0 : 1 , j j = 10 , ˚ = 0 , = 0 : 1 , = 0 : 005 (a1:˝rstblade,a2:secondblade,a3: thirdblade) FromtheEquation(3.26), D 1 A j = 2 A j + 1 2 i ~ e 2 3 Õ k = 1 A k + 1 2 i v j A j + 1 4 j e i ˚ j e i ˙ 0 (3.31) UsingEquation(3.31), D 2 1 A j isderivedas D 2 1 A j = 2 D 1 A j + i ~ e 2 2 3 Õ k = 1 D 1 A k + i v j 2 D 1 A j Insertingthe D 2 1 A j and D 1 A j expressionsintotheEquation(3.30),thenweobtainthe D 2 A j equationas D 2 A j = 1 8 i 2 1 8 i 2 v j A j + 9 8 i ~ e 4 + 1 4 i v j ~ e 2 1 2 ~ e 2 3 Õ k = 1 A k + j 16 ¹ i v j º e i ˚ j e i ˙ 0 + 3~ e 2 16 3 Õ k = 1 k e ¹ i ˚ k º e ¹ i ˙ 0 º + i F j 1 e i 2 ˚ j 8 ¹ 1 ! 2 1 º i j ! 1 e i ˚ j 8 ¹ 1 ! 2 1 º i 2 F j e i 2 ˚ j 16 ¹ 1 ! 2 1 º ~ e 2 ! 2 1 e i ˚ j 8 ¹ 1 ! 2 1 º 3 Õ k = 1 A k e i ˙ 0 (3.32) 66 Solvabilityconditions D 1 A j and D 2 A j willbesolvedtogethertodeterminethe A j equation. Recombiningthetimescales ¹ 1 ; 2 ; 3 º inprocessallowustodeterminethereconstitution. dA j d = D 1 A j + 2 D 2 A j (3.33) Equations D 1 A j and D 2 A j areinsertedintotheEquation(3.33)togetthereconstituteddi˙er- entialequationas dA j d = 2 A j + 1 2 i ~ e 2 3 Õ k = 1 A k + 1 2 i v j A j + 1 4 j e i ˚ j e i ˙ 0 + 2 1 8 i 2 1 8 i 2 v j A j + 9 8 i ~ e 4 + 1 4 i v j ~ e 2 1 2 ~ e 2 Í 3 k = 1 A k + j 16 ¹ i v j º e i ˚ j e i ˙ 0 + 3~ e 2 16 Í 3 k = 1 k e ¹ i ˚ k º e i ˙ 0 + i F j 1 e i 2 ˚ j 8 ¹ 1 ! 2 1 º i j ! 1 e i ˚ j 8 ¹ 1 ! 2 1 º i 2 F j e i 2 ˚ j 16 ¹ 1 ! 2 1 º ~ e 2 ! 2 1 e i ˚ j 8 ¹ 1 ! 2 1 º Í 3 k = 1 A k e i ˙ 0 (3.34) A = ¹ X + iY º e ¹ i ˙ 0 º and j = F j 0 2 ¹ 1 ! 2 1 º e i ˚ j termsareinsertedintoEquation(3.34)andlater dA j d equationisdividedby e i ˙ 0 .Subsequently,we˝ndoutrealandimaginarypartsas Realpart: Û X j = C 1 X j + C 2 3 Õ k = 1 X k + C 3 Y j + C 4 3 Õ k = 1 Y k + C 5 cos ¹ ˚ j º 3 Õ k = 1 X k + C 6 sin ¹ 2 ˚ j º + C 7 cos ¹ 2 ˚ j º + C 8 3 Õ k = 1 cos ¹ 2 ˚ k º (3.35) Imaginarypart: Û Y j = C 3 X j C 4 3 Õ k = 1 X k + C 1 Y j + C 2 3 Õ k = 1 Y k + C 5 sin ¹ ˚ j º 3 Õ k = 1 X k C 6 cos ¹ 2 ˚ j º + C 7 sin ¹ 2 ˚ j º + C 8 3 Õ k = 1 sin ¹ 2 ˚ k º (3.36) where C 1 ; C 2 ; C 3 ; C 4 ; C 5 ; C 6 ; C 7 ; C 8 canbefoundintheAppendixC.2.1. 67 Û X j and Û Y j areexpressedinmatrixformas 2 6 6 6 6 6 4 Û X j Û Y j 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 C 1 C 3 C 3 C 1 3 7 7 7 7 7 5 2 6 6 6 6 6 4 X j Y j 3 7 7 7 7 7 5 + 2 6 6 6 6 6 4 C 2 + C 5 cos ¹ 2 ˚ j º C 4 C 4 C 2 + C 5 cos ¹ 2 ˚ j º 3 7 7 7 7 7 5 2 6 6 6 6 6 4 Í 3 k = 1 X k Í 3 k = 1 Y k 3 7 7 7 7 7 5 + 2 6 6 6 6 6 4 C 6 sin ¹ 2 ˚ j º + C 7 cos ¹ 2 ˚ j º + C 8 Í 3 k = 1 cos ¹ 2 ˚ k º C 6 cos ¹ 2 ˚ j º + C 7 sin ¹ 2 ˚ j º + C 8 Í 3 k = 1 sin ¹ 2 ˚ k º : 3 7 7 7 7 7 5 Speci˝callyforthetunedcase,letting v j = 0 ,Equations(3.35)and(3.35)becomes Realpart: Û X j = T 1 X j + T 2 3 Õ k = 1 X k + T 3 Y j + T 4 3 Õ k = 1 Y k + T 5 cos ¹ ˚ j º 3 Õ k = 1 X k + T 6 sin ¹ 2 ˚ j º + T 7 cos ¹ 2 ˚ j º + T 8 3 Õ k = 1 cos ¹ 2 ˚ k º (3.37) Imaginarypart: Û Y j = T 3 X j T 4 3 Õ k = 1 X k + T 1 Y j + T 2 3 Õ k = 1 Y k + T 5 sin ¹ ˚ j º 3 Õ k = 1 X k T 6 cos ¹ 2 ˚ j º + T 7 sin ¹ 2 ˚ j º + T 8 3 Õ k = 1 sin ¹ 2 ˚ k º (3.38) where T 1 ; T 2 ; T 3 ; T 4 ; T 5 ; T 6 ; T 7 ; T 8 canbefoundintheAppendixC.2.2 Û X j and Û Y j arerepresentedinmatrixformas 2 6 6 6 6 6 4 Û X j Û Y j 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 T 1 T 3 T 3 T 1 3 7 7 7 7 7 5 2 6 6 6 6 6 4 X j Y j 3 7 7 7 7 7 5 + 2 6 6 6 6 6 4 T 2 + T 5 cos ¹ 2 ˚ j º T 4 T 4 T 2 + T 5 cos ¹ 2 ˚ j º 3 7 7 7 7 7 5 2 6 6 6 6 6 4 Í 3 k = 1 X k Í 3 k = 1 Y k 3 7 7 7 7 7 5 + 2 6 6 6 6 6 4 T 6 sin ¹ 2 ˚ j º + T 7 cos ¹ 2 ˚ j º + T 8 Í 3 k = 1 ¹ cos2 ˚ k º T 6 cos ¹ 2 ˚ j º + T 7 sin ¹ 2 ˚ j º + Z 8 Í 3 k = 1 ¹ sin2 ˚ k º 3 7 7 7 7 7 5 68 Magnitudeofsteadystateresponseamplitudewillbefoundandamplitudeversusfrequency plotswillbeobtainedforsuperharmonicresonancecaseatorder 2 .Moreover,wewouldlike toperformnumericalsimulationstovalidatetheanalyticalresultsonthebothsuperharmonic resonancesatorder 2 and 3 . 3.6Conclusions In-planevibrationofacoupledthree-bladewindturbineswerestudied.Equationsofmotion werederivedpreviously.Afterdecouplingthebladeequations,asecond-orderpertubationanalysis wasappliedonbothtunedandmistunedthree-bladewindturbines. Theanalysisbroughtoutthesuperharmanicresonancesatone-thirdnaturalfrequency.This resonancecasecouldnotcapturedwitha˝rst-ordermethodofmultiplescalesanalysis.With mistuning,thesuperharmonicresonancesplitsasingleresonancepeakonthetunedcaseinto multipleresonancepeaks.Forsteady-statesuperharmonicresonanceresponse,amplitudeinceases whiledampingfactor decreases.Ontheotherhand,theresponseamplitudegetslargerwhen j j grows.Observationofmistuningindicatesthatamplitudeofonebladeincreasescomparedtothe tunedsystemfor 0 < v < 0 : 03 . Ongoingandfutureworkwilladdressaformulationoftheresultingdynamicsoftherotor, andthee˙ectsofconstantloading.Steadystateresponseamplitudewillbedetermeninedfor superharmoniccaseatordertwoandwilldonumericalsimulationtocon˝rmanalyticalsolutionsat ordertwoandthree.Fordynamicalinterest,wecanstudytheprimaryandsubharmonicresonances aswell.Thee˙ectofnonlinearityontheresonancesisalsoofinterest. 69 CHAPTER4 APPROXIMATEGENERALRESPONSESOFTUNEDANDMISTUNED 4-DEGREE-OF-FREEDOMSYSTEMSWITHPARAMETRICSTIFFNESS 4.1Introduction Wehavestudiedsystemswithcyclicsti˙nessanddirectexcitation.Whenlinearized,these systemshavetransientandsteady-statesolutions.Wehavefocusedonsteady-statebehaviorwith theperturbationanalyses.Wecanconsiderthetransientbehaviorastheresponsetothesystemwith cyclicsti˙ness,butwithoutthedirectforcing.Tothisend,weconsideraFloquet-basedapproach tomulti-degree-of-freedomlinearsystemswithcyclicsti˙ness. Inthiswork,generalsolutionstoMathieu-typemulti-degree-of-freedomsystemsoftheform M Ü x + K ¹ t º x = 0 ; (4.1) areexaminedindetail,where x isa d 1 vectorofcoordinatedisplacements,where d isthenumber ofdegreesoffreedom,and M and K ¹ t º arethemassandtime-varyingsti˙nessmatrices. Ageneralinitialconditionresponseaswellasthestabilitycharacteristicsofthesystemis sought.Intendingtoproducethatresult,inplaceofassumingaperiodicsolution,aFloquet-type solutionisassumedas x j ¹ r º = e i r t n Õ k = n c ¹ r º j ; k e ik ! t ; (4.2) wheretheindex r distinguishesbetween 2 d independentFloquetsolutiontermsfora d -degree- of-freedomsystem, j presentscoordinatesand k refersharmonics.Asfollowinguptheworkon approximategeneralresponseofsymmetrictwoandthreeDOFsytemswithparametricsti˙ness, theassumedsolutionispluggedintotheequationsofmotion,andbyapplyingharmonicbalance, thecharacteristicexponents, r andassociatedFouriercoe˚cients, c ¹ r º ,aredetermined.Then,by 70 usingaprocedurethatissimilartotheoneusedin[32],theresponsetoanarbitraryinitialcondition canbedeterminedbyconsideringalinearcombinationofthe x r . 4.2Analysis TheresponseanalysisprocedurecanbeexplainedbystudyingonexampleMDOFsystems, namelytunedandmistuned4DOFsystems.Soastoachievethegoal,amass-springchainas showninFigure4.1,withperiodicsti˙nessisstudied. Figure4.1:AfourDOFspring-masschain. Theequationsofmotionare Ü X + ¹ 3 + + º X ¹ 1 + cos » ! t + 2 ˇ 3 ¼º x 2 ¹ 1 + cos » ! t + 4 ˇ 3 ¼º x 3 ¹ 1 + + cos » ! t ¼º x 4 = 0 Ü x 2 + ¹ 1 + cos » ! t + 2 ˇ 3 ¼º¹ x 2 X º = 0 Ü x 3 + ¹ 1 + cos » ! t + 4 ˇ 3 ¼º¹ x 3 X º = 0 Ü x 4 + ¹ 1 + + cos » ! t ¼º¹ x 4 X º = 0 : (4.3) where m 2 = m 3 = m 4 = 1 , M = m 2 , K = and k i ¹ t º = ¹ 1 + i + cos » ! t + ¹ 2 ˇ š 3 º¹ i º¼º ,where i = 2 ; 3 ; 4 , 2 = 3 = 0 and 4 = : Forspeci˝cparametervalues,theeigenvaluesareplottedasa functionofsti˙nessmistuningparameter( )asshowninFigure4.2.When = 0 andthesystem istuned ¹ = 0 º modalfrequenciesare ! 1 = 0 : 3047 , ! 2 = 1 , ! 3 = 1 and ! 4 = 2 : 075 .Figure 71 4.3showsbothtuned( = 0 )andmistunedsystemsspace( = 1 )modeshapesinsameplotto demonstratethee˙ectofdetuningparameteronthemodeshapes. Figure4.2:Eigenvaluesversussti˙nessmistuningparameter for = 0 , ! = 1 : 6 , m 2 = m 3 = m 4 = = 1 , = 0 : 4 Figure4.3:Modeshapesofthetuned( = 0 )andmistuned( = 1 )systems 72 4.2.1TunedFour-Degree-of-FreedomExample Followingthereference[32],theresponseisfoundbyassumingaFloquettypesolutionwith ˝niteharmonics,asgiveninEquation(4.2)andinsertingintothesystemequationsofmotion. Particularly,inthismodel,weseekfor x 1 ¹ t º = e i t n Õ k = n c 1 ; k e ik ! t x 2 ¹ t º = e i t n Õ k = n c 2 ; k e ik ! t x 3 ¹ t º = e i t n Õ k = n c 3 ; k e ik ! t x 4 ¹ t º = e i t n Õ k = n c 4 ; k e ik ! t : Governingequationsfor c j ; k 'saredeterminedbyusingharmonicbalanceandwritingtheharmonic balanceequationinmatrixform, A c = 2 6 6 6 6 6 6 6 6 6 6 6 4 A 11 A 12 A 13 A 14 A 21 A 22 A 23 A 24 A 31 A 32 A 33 A 34 A 41 A 42 A 43 A 44 3 7 7 7 7 7 7 7 7 7 7 7 5 © « c 1 c 2 c 3 c 4 ª ® ® ® ® ® ® ® ® ¬ = © « 0 0 0 0 ª ® ® ® ® ® ® ® ® ¬ ; (4.4) where c j = » c j ; n ::: c j ; 1 c j ; 0 c j ; 1 ::: c j ; n ¼ T ,and A ij 'scorrespondto ¹ 2 n + 1 º¹ 2 n + 1 º block matrices. Thedeterminantofthecoe˚cientmatrix, A ¹ º ,mustbeequaltozerotoownanonzero c solution.Wecan˝ndthecharacteristicequationfor intermsof , , ! and m 1 = m 2 = m 3 = . Then,foreach ,wecanobtainthe c vector,bysolving A ¹ º c = 0 .Thecharacteristicequation produceroots q ,where q isfrom 1 to 2 d ¹ 2 n + 1 º and d isnumberofdegreesoffreedom, n is thenumberofassumedharmonics.Nevertheless,thereareessentially 2 d principalroots,andthe other 0 s arelinkedtotheprincipalrootsbytherelation ^ s = ^ r k ! i .Sincethecorresponding exponentialpartcanbewrittenas e i r t e ik ! t andthesecondpartcanbeinsertedintotheperiodic part,itisobservedthattheserootsdonotconducetoextrasolutions. 73 Asfollowinguptheworkon2-DOFand3-DOFmass-springchainexamples[32], q sare pluggedintothecoe˚cientmatrix,thennullspaceof A ¹ q º providesthesolutionof c ¹ q º 's. Theideaaboutthestabilityandthefrequencysubjectofthesolutioncanbeobtainedbythe rootsofthecharacteristicequation.Evenifonlyoneoftherootshasanegativeimaginarypart,the solutionwillgrowunstablebecauseoftheexponentialpart.Ifallrootshaveapositiveimaginary part,thesolutionisbounded.Particularly,iftherootsarereal,thesolutioniseitherperiodicor quasi-periodic.InFigure4.4,thestabilityregionsforthe4DOFmass-springchainaredrawnby evaluatingtheimaginarypartsofthecharacteristicrootsfor = 1 , = 1 . Thefrequencyvaluescanbedeterminedwithcombinationoftheexponentialandtheperiodic partsofthefrequenciesas j Re ¹ r º k ! j .In4DOFsystem,thereareeightprincipalcharacteristic rootsandthegeneralresponsesolutioniswrittenintermsof"modalcomponents"as x ¹ t º = 8 Õ r = 1 a r x r ¹ t º ; (4.5) where x r = 2 6 6 6 6 6 6 6 6 6 6 6 4 x ¹ r º 1 x ¹ r º 2 x ¹ r º 3 x ¹ r º 4 3 7 7 7 7 7 7 7 7 7 7 7 5 (4.6) a r 'saretobedeterminedfromtheinitialconditions.Arbitraryinitialconditionscanbede˝nedas aclassoflinearequationsintermsoftheconstants a r ,as 2 6 6 6 6 6 4 x 0 Û x 0 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 x 1 ¹ 0 º ::: x 8 ¹ 0 º Û x 1 ¹ 0 º ::: Û x 8 ¹ 0 º 3 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 4 a 1 : : : a 8 3 7 7 7 7 7 7 7 7 5 : (4.7) Asstatedtheinpreviousstudy[32],theinitialcondition h x T 0 Û x T 0 i T = x s ¹ 0 º T Û x s ¹ 0 º T T results in a s = 1 and a r , s = 0 .Consequently,thedi˙erentmodalfunctionscanbeobtainedifascalar multipleofeachcolumncanbeusedasaninitialcondition.Accordingly,theresponsecanbe 74 determinedasthesamelinearcombinationofthecorrespondingmodalfunctionsbywritingan initialconditionasalinearcombinationof x s ¹ 0 º T Û x s ¹ 0 º T T 's. Figure4.4:Stabilityregionsforthetuned4DOFmass-springchainfor n = 2 , = 1 and = 0 : 4 Theresponsefrequencyisacombinationoffrequenciesoftheexponentialandperiodicparts as j Re ¹ r º k ! j .Forasetofparameters,theresponsefrequenciesasafunctionofexcitation frequencyaregiveninFigure4.5foroneharmonic.Twofrequencybranchescollideandconstitute onebranchwithvariationsin ! .Thiscollisionsstandforstabilitytransitions.Also,thebranches areoverlapbecauseoftherepeatedfrequenciesonthetunedcase.Thetopbranchdoesnotmerge withanybranchinFigure4.5.Wecanpredicttoseemorebranchesinresponsefrequenciesplot withhigherharmonics,thusthetopbranchcancollide. The4DOFspring-masschainwasstudiedwith n = 2 harmonics,forvarioussetsofparameters andinitialconditions.Theresultswerecomparedtothosedeterminedfromanumericalwork,and theinitialconditionresponseandFFTplotsweregiveninFigure4.6andFigure4.7for = 0 : 2 andinFigure4.8andFigure4.9for = 0 : 6 . 75 Figure4.5:Responsefrequenciesplotasafunctionofexcitationfrequencywithparameters n = 1 , = 0 , = 0 : 4 , = 1 and = 0 : 6 . 4.2.2MistunedFour-Degree-of-FreedomExample Insteadofassumingaperfectlytunedmodel,thesti˙nessterm k 4 ¹ t º isassumedtohaveavariation ,suchthat k 4 ¹ t º = ¹ 1 + + cos » ! t + ¹ 2 ˇ º¼º . Followinguptheworkontunedsystem,generalsolutionEquation(4.2)isinsertedintoEqua- tion(4.3)andsamestepsareapplied.Then,characteristicexponentsandcorrespondingeigenvec- torsareobtained.Thestabilitywedgesofthe4DOFsystemareplottedbyexaminingtheimaginary partsofthecharacteristicexponentsforoneharmonicand = 0 : 4 , = 1 , = 0 : 2 inFigure4.10 andfortwoharmonicsand = 0 : 4 , = 1 , = 1 inFigure4.11. Figure4.12showstheresponsefrequenciesasfunctionof ! formistunedsystem.Repsonse frequencyplotslookverysimilarfortunedandmistunedcases,butwecangetsome˝nedetails,the placeswhererepeatedfrequencycomponentsaresplituplittlebit.Symmetrybreakingseperates repeatedfrequenciesandcombinationoffrequenciesgetlittlebitmorecomplicatedondetuned case. FFTplotsandtheinitialconditionresponseswerefoundassuming n = 2 harmonicsand detuningterm = 0 : 2 ,fordi˙erentsetsofparameters,andwerecomparedtothoseobtainedfrom anumericalstudy,asgiveninFigures4.13and4.14for = 0 : 2 andFigures4.15and4.16for 76 Figure4.6:Responseplotsfor n = 2 , ! = 0 : 75 , = 0 : 2 , = 0 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T . = 0 : 6 . 4.3Discussion TheinstabilitywedgesinFigure4.4maybebasedat ! = ¹ ! i + ! j ºš N ; where N isapositive integer.Inthetuned4DOFsystemfor = 1 and = 0 : 4 ,the = 0 naturalfrequencieswere ! 1 = 0 : 3047 , ! 2 = ! 3 = 1 and ! 4 = 2 : 075 .Weobservethetwomainsubharmonicinstability wedgesstartingatfrequencies ! ˙ ! 2 + ! 3 = 2 , ! ˙ ! 1 + ! 2 ; 3 = 1 : 305 and ! ˙ ! 2 ; 3 + ! 4 = 3 : 07 . Additionally,ifthereisawedgeat 2 ! 4 = 4 : 15 ,itisnotinthedomainofFigure4.4.Someprimary andsuperharmonicwedgesoriginateatfrequenciesthat˝tintothepattern,suchas ! 2 = ! 3 = 1 , ¹ ! 1 + ! 2 ºš 2 = ¹ ! 1 + ! 3 ºš 2 = 0 : 65 , ! 1 = 0 : 304 , ¹ ! 2 + ! 4 ºš 2 = ¹ ! 3 + ! 4 ºš 2 = 1 : 537 and ! 4 š 2 = 1 : 035 .Simulationsweredonefordi˙erentparametervaluestocon˝rmthecompatibility. Forinstance,at = 0 : 8 and ! = 1 : 1 thesimulationwasunstable,althoughat ! = 1 : 6 thesimulation 77 Figure4.7:FFTplotsfor n = 2 , ! = 0 : 75 , = 0 : 2 , = 0 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T . wasstable,inagreementwiththestabilityregioninFigure4.4.Aspuriousinstabilityfeaturenear ! = 0 : 15 wasdisplayedinFigure4.4.Similartrendswereobservedfromsimulationsofthe four-DOFsystemwithdi˙erentvaluesof whichisnotshownhere. Thedetuned4DOFsystemfor = 1 , = 1 and = 0 : 4 ,thenaturalfrequencieswere ! 1 = 0 : 3065 , ! 2 = 1 ! 3 = 1 : 206 and ! 4 = 2 : 4188 .Weseethewedgesat 2 ! 1 ˙ 0 : 604 , 2 ! 2 ˙ 2 , 2 ! 3 ˙ 2 : 41 , ! 1 + ! 2 ˙ 1 : 306 , ! 1 + ! 3 ˙ 1 : 512 , ! 2 + ! 3 ˙ 2 : 206 , ! 2 + ! 4 ˙ 3 : 418 , ! 1 + ! 4 ˙ 2 : 713 , ! 3 + ! 4 ˙ 3 : 616 and 2 ! 4 ˙ 4 : 84 showingthesubharmonicinstabilitiesofeach Someprimaryandsuperharmonicwedgesarealsobasedatfrequencieswhichmatchthe pattern,suchas ! 2 = 1 , ! 3 = 1 : 206 , ! 4 = 2 : 418 , ¹ ! 2 + ! 3 ºš 2 = 1 : 103 , ¹ ! 1 + ! 2 ºš 2 = 0 : 605 , ¹ ! 1 + ! 4 ºš 2 = 1 : 36 , ¹ ! 1 + ! 3 ºš 2 = 0 : 751 , ¹ ! 2 + ! 4 ºš 2 = 1 : 712 and ¹ ! 3 + ! 4 ºš 2 = 1 : 81 .In Table4.1,someinstabilitywedgesfortunedcasewithoneharmonicandmistunedbladecasewith 78 Figure4.8:Responseplotsfor n = 2 , ! = 0 : 75 , = 0 : 6 , = 0 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T . oneandtwoharmonicsweredemonstrated.Comparingthetunedandmistunedstabilityplots,the mistunedsystemhasmanymoreinstabilitywedges,andgenerallydepictsagreatertendencyto destabilize. AccordingtotheFloquetsolution,whichinvolvesalinearcombinationofterms e i t p ¹ t º ,itis clearthatinitialconditionscanbespeci˝edsuchthatonlyoneofthethesetermsisactive.Inthis respect,wenamethemresponses".Analyticalandnumericalfreeresponsesarepointedout inFigures4.13-4.16,for n = 2 ,forthechosenparametersandinitialconditions.Thetimeresponses showedgoodagreement.Around8-10responseharmonicswerepredicted,mostlyaccurately,with acoupleinstancesoflowamplitudespuriousharmonics.Somehigherfrequencyharmonicswere notcapturedanalyticallywith n = 2 . Theworkshownhereinvolvesthesolutiontoanonstandardeigenvalueproblemintheform 79 Figure4.9:FFTplotsfor n = 2 , ! = 0 : 75 , = 0 : 6 , = 0 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T . ofEquation(4.4)inwhichmatrix A haselementswith 2 terms.Thecharacteristicequationis polynomialofdegree 2 d ¹ 2 n + 1 º in .Asthenumberofharmonics, n ,increasestheremaybe computationallimitson˝ndingsymbolicsolutionsforthe .Despitethis, n = 2 harmonicswere abletoe˙ectivelypredicttheresponseforthesystemsstudiedinthisstudy. 4.4Conclusions Generalresponsesofa4DOFmass-springsystemwithparametricexcitationwerestudied. AssumingaFloquet-typesolution,andapplyingtheharmonicbalancemethod,thefrequency contentandstabilityofthesolutionwerefound.Later,theanalysiswasextendedtoasystem withmistunedparameters,andthee˙ectofsymmetrybreakingonsystemresponsewasstudied. Inaddition,thetimeresponseandFFTplotswereproducedforvariousparametersandinitial conditionsinthebothtunedandmistunedcase.Then,analyticalresultswerecon˝rmedby 80 Figure4.10:Stabilityplotofthemistuned4DOFsystemfor n = 1 , = 0 : 2 , = 1 and = 0 : 4 Figure4.11:Stabilityplotforthemistuned4DOFsystemfor n = 2 , = 1 , = 1 and = 0 : 4 numericalsimulations.Infuturework,thismethodwillbeappliedonmistunedthree-bladewind turbinemodelsto˝ndtheresponsecharacteristics. 81 Figure4.12:Responsefrequencyplotasafunctionofexcitationfrequencyformistunedcasewith parameters n = 1 , = 0 : 2 , = 0 : 4 , = 1 and = 0 : 6 . Figure4.13:Responseplotsofdetunedsystemfor n = 2 , = 0 : 2 , ! = 0 : 8 , = 0 : 2 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T . 82 Figure4.14:FFTplotsofdetunedsystemfor n = 2 , = 0 : 2 , ! = 0 : 8 , = 0 : 2 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T . Figure4.15:Responseplotsofdetunedsystemfor n = 2 , = 0 : 2 , ! = 0 : 8 , = 0 : 6 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T . 83 Figure4.16:FFTplotsofdetunedsystemfor n = 2 , = 0 : 2 , ! = 0 : 8 , = 0 : 6 , = 0 : 4 , x ¹ 0 º = » 0000 ¼ T and Û x ¹ 0 º = » 1111 ¼ T . 84 Table4.1:Primary,superharmonicandsubharmonicinstabilitywedgesbasedat ! i + ! j N fortuned andmistunedcases Primary Superharmonic Subharmonic TunedCase ¹ n = 2 ; = 0 º ! 1 = 0 : 3047 , ! 2 ; 3 = 1 , ! 4 = 2 : 075 ! 1 ! 2 ! 3 ! 1 + ! 2 2 , ! 1 + ! 3 2 ! 2 + ! 4 2 , ! 3 + ! 4 2 , ! 4 2 ¹ ! 1 + ! 2 º , ¹ ! 1 + ! 3 º ¹ ! 2 + ! 3 º , ¹ ! 2 + ! 4 º ¹ ! 3 + ! 4 º , 2 ! 4 MistunedCase ¹ n = 2 ; = 1 º ! 1 = 0 : 3065 , ! 2 = 1 , ! 3 = 1 : 206 , ! 4 = 2 : 4188 ! 2 ! 3 ! 4 ! 1 + ! 2 2 , ! 1 + ! 3 2 , ! 2 + ! 4 2 ! 3 + ! 4 2 , ! 1 + ! 4 2 , ! 2 + ! 3 2 2 ! 1 , 2 ! 2 , 2 ! 3 2 ! 4 ¹ ! 1 + ! 2 º , ¹ ! 1 + ! 3 º ¹ ! 2 + ! 3 º , ¹ ! 2 + ! 4 º ¹ ! 3 + ! 4 º , ¹ ! 1 + ! 4 º MistunedCase ¹ n = 1 ; = 0 : 2 º ! 1 = 0 : 3053 , ! 2 = 1 , ! 3 = 1 : 059 , ! 4 = 2 : 1409 ! 1 ! 2 ! 3 ! 4 ! 1 + ! 2 2 , ! 1 + ! 3 2 , ! 2 + ! 4 2 ! 3 + ! 4 2 , ! 2 + ! 3 2 , ! 4 2 2 ! 1 , 2 ! 2 , 2 ! 3 , 2 ! 4 ¹ ! 1 + ! 2 º , ¹ ! 1 + ! 3 º ¹ ! 2 + ! 3 º , ¹ ! 2 + ! 4 º ¹ ! 3 + ! 4 º , ¹ ! 1 + ! 4 º 85 CHAPTER5 ONGOINGWORK 5.1ParametricIdenti˝cationofTheMathieuEquationwithaConstant Load 5.1.1IntroductionandObjective Studiesonparametricallyexcitedsystemshavemanyapplications,includingshipdynamics,wind- turbine-bladevibration,andmicroresonators.ThisstudyregardsalinearMathieuequationwitha constantforceterm: Ü q + 2 Û q + ¹ ! 2 + cos t º q = F 0 (5.1) Thewindturbineblademodelsincludecyclicsti˙nessandconstant-plus-cyclicdirectloading, intheformofaforcedMathieuequation.Itwouldbeinterestingtoconsiderparameterestimationin theMathieusystem.Theunforced,dampedMathieusystemhasazerosolutionthatiseitherstable orunstable,whiletheforcedMathieusystemshavenon-zerosteady-stateoscillations,whichcan bestableorunstable.Weconsiderexploitingthesteady-stateresponseofaforcedMathieusystem toestimatetheparameters,namelythestrengthofparametricexcitation,damping,andexcitation level.MuchattentionhasbeengiventotheresponseandstabilityanalysisoftheMathieuequation, forexamplebyperturbationtechniquessuchasthemethodofmultiplescales.Inthisworkweaim toestimateparameters , and F 0 fromresponses. 5.1.2Background ThesolutionoftheforedlinearMathieuequationcanbeexpressedasasumofhomogeneousand particularsolutions,givenas q = q h + q p ,where q h satis˝es Ü q h + 2 Û q h + ¹ ! 2 + cos t º q h = 0 and q p satis˝es Ü q p + 2 Û q p + ¹ ! 2 + cos t º q p = F 0 .FromtheFloquettheory,the q h term hasthevariousinstabilitiesthatoriginateduetoparametricexcitation.The q p parthasbeen 86 Figure5.1:AmplitudesofsimulatedresponsesofEquation(5.1)showingprimaryandsuper- harmonicresonancesandanunstableresponseatsubharmonicresonanceduetoincreaseofthe parametricforcingamplitude; = 0 : 1 ; = 0 : 25 ; F 0 = 2 .Di˙erentcurvesdepict = 0 : 5 and 1 . (Figuretakenfrom[1]). approximatedbyperturbationtechniques,andcanhaveresonanceconditionsaswellasinstabilities [1].Figure5.1showssteady-stateforced-responseamplitudesasafunctionoftheparametric frequency ,andillustratesthatthesystemofEquation(5.1)canhaveprimaryandsecondary resonancesandinstabilities. In[1],asecond-ordermultiplescalesanalysiswasappliedto˝ndexpressionsforresponse amplitudesatprimaryandsuperharmonicresonanceconditionsasafunctionofsystemparameters. Perturbationanalysisisalid"for .Predictionsdeterioratewhen becomesge". Specialsteady-statesolutionsareapproximatedbelow. 1. Lowfrequencylimit:aquasi-staticapproximationismadebyletting Û q ˙ 0 and Ü q ˙ 0 in 87 Equation(5.1)andsolvingfor q : q = F 0 ! n 2 + cos t ˙ F 0 ! n 2 F 0 ! n 2 cos t whichoscillateswithamean,maximum,andminimumvaluesof q ˙ F 0 ! n 2 , q max ˙ F 0 ! n 2 , q min ˙ F 0 ! n 2 + . 2. Superharmonicresonanceoforder2:If ˙ ! n š 2 ,thentheresponseis q ˙ F 0 ! n 2 + a cos ¹ 2 t ˚ º F 0 ! 2 n ¹ ! 2 n 2 º cos ¹ t º + O ¹ 2 º withapeakvalueof a as a max s = F 0 2 3 ! 5 n . 3. Primaryresonance:If ˙ ! n ,thentheresponseis q ˙ F 0 ! n 2 + a cos ¹ t ˚ º + a cos ¹ ! n + º t + ¼ 2 ¹ 2 + 2 ! n º + a ! 2 n cos ¹ ˙ t + º withapeakvalueofas a max p = F 0 2 ! 3 n 5.1.3ParameterEstimationProcedure WefollowanideabyNayfeh[62],inwhichexpressionsforresonantresponsesandbifurcation pointsfromperturbationanalyseswerecomparedtosimulatedorexperimentalmeasurementsof theseeventsinordertoestimateparametersinadi˙erentialequationmodel.Wedoasfollows. 1. Weestimate ! n fromthepeaksofafrequencysweepsuchasthatinFigure5.1. 88 Figure5.2:Examplecalculationwiththesuperharmonicresonance 2. Weexcitesuperharmonicresonanceat ˇ ! n š 2 , x = X s 0 + X s 1 cos ¹ ! n t + º + X s 1 š 2 cos ¹ ! n 2 t º , where X s 0 = F 0 ! 2 n , X s 1 = a max s = F 0 2 3 ! 5 n , X s 1 š 2 = 4 F 0 3 ! 4 n , atthepeakresponse,andweobtainvaluesof X s 0 , X s 1 and X s 1 š 2 withthehelpofafastFourier transform(FFT)asseeninFigure5.2. 3. Weexciteprimaryresonanceat ˇ ! n , x = X p 0 + X p 1 cos ¹ ! n t + º 89 Figure5.3:Examplecalculationwiththeprimaryresonance where X p 0 = F 0 ! 2 n , X p 1 = a max p = F 0 2 ! 3 n atthepeak.We˝ndthevaluesof X p 0 and X p 1 usingtheFFTplotsasshowninFigure5.3. Thevaluesof X s 0 and X p 0 giveus F 0 .Figure5.4showstheestimatedforcecomparedtotrue valuesof F 0 withrespecttodi˙erent values.From X s 1 X p 1 = 2 3 ! 2 n ,weextractthevalueof . Next, isinsertedintotheexpression X s 1 = F 0 ¹ º 2 3 ¹ º ! 2 n toobtain .Dampingratioisalso foundfrom 2 ! n = 2 as = ! n . 5.1.4ResultsandDiscussion Wesimulatedresponseswith ! n = 1 , = 0 : 5 , = 0 : 1 ,and F 0 = 2 ; andvariousvaluesof .We estimatedparametersbycomparingresonantamplitudestotheiranalyticalexpressions.Figure5.5 90 Figure5.4:Estimatedforce F demonstratestheestimationof and ( 8 isplotted)forvarioustruevaluesof . Theperturbationsolutiondegradesas increases,butweseegoodestimatesof withina limitedrange.Damping isestimatedwithinabout25%.Dampingcanbedi˚culttoobtain accuratelyinvibrationsystems. 5.1.5ProposedWork Weestimatedparametersinalinearparametricallyexcitedvibrationsystem.Accuracydependson theasymptoticsofperturbationsolutionswhichunderlietheapproach.Futureworkcaninvolve sensitivityanalysis,erroranalysis,andexperiments.Also,thisapproachcanbeformulatedforthe caseof Ü q + 2 Û q + ¹ ! 2 + cos t º q = F 1 sin t Then,wecancombine(superpose)forcaseof F 0 + F 1 sin t ,aswind-turbinemodels. 91 Figure5.5:Estimated 'sand 's.(Thevaluesof 8 and 8 est areplotted.) 5.2Second-OrderPerturbationAnalysisofForcedNonlinearMathieuEqua- tion 5.2.1Introduction Ourinterestinstudyingthein-planedynamicsofwindturbinebladedirectsustothisstudy.There hasbeenvariousreseachonparametricallyexcitedsystemsthatsuitsspeci˝cminorvariationsof theMathieuequation.Newman etal. [63]studieddynamicsofanonlinearparametricallyexcited partialdi˙erentialequation.BifurcationsinaMathieuequationwithcubicnonlineratieswere analyzedin[64]andstabilityanalysisofaparametricallyexcitedrotatingsystemwasdonein [65].FrequencylockinginaforcedMathieu-vanderPol-Du˚ngsystemwasstudiedbyPandey etal. [66]andbifurcationofsubharmonicresonancesintheMathieuequationwasanalyzedin [67].Higher-orderperturbationanalysis[68]andFloquettheorywithharmonicbalancesolution [34]havebeenappliedtostudythestabilitywedgesoftheMathieuequation.SincetheMathieu 92 equationalignswellwithhigher-ordermultiplescalesanalysisto˝gureoutstabilitycharacteristics out,weworkwithsecond-ordermethodofmultiplescales. Inthiswork,welookforsuperharmonicresonanceatorderthreeforthenonlinearMathieusys- temwithhardforcingbyapplyingasecond-orderperturbationexpansion.Sayed etal. [27]applied higher-orderexpansionstoanalysestabilityandresponseofanonlineardynamicalsystem.Onthe otherhand,Romero etal. [69]employeddi˙erentscalingtechniquestoanalyzethequasiperiodic dampedMathieuequation. 5.2.2ANonlinearMathieuEquationwithHardExcitation Inthisanalysis,weconsiderhardexcitation,i.e.directforcing F whichisoforder1.Theequation withnonlinearityis Ü q + 2 Û q + ¹ ! 2 + cos t º q + q 3 = F sin ¹ t + º ; (5.2) Applyingthemethodofmultiplescales,weworkwiththreetimescales ¹ T 0 ; T 1 ; T 2 º and q 0 asthe dominantsolution,with q 1 and q 2 areslowvariationsofthatsolution.Speci˝cally, q = q 0 ¹ T 0 ; T 1 ; T 2 º + q 1 ¹ T 0 ; T 1 ; T 2 º + 2 q 2 ¹ T 0 ; T 1 ; T 2 º + ::: (5.3) where T i = i T 0 .Then d dt = D 0 + D 1 + 2 D 2 andwhere D i = @ @ T i : Weinserttheseexpressionsintoourordinarydi˙erentialEquation(5.2)andthensimplifyand theequationsfor 0 ; 1 ; 2 coe˚cientsarepulledoutas O ¹ 1 º : D 2 0 q 0 + ! 2 q 0 = F ¹ sin T 0 + º O ¹ º : D 2 0 q 1 + ! 2 q 1 = 2 D 0 q 0 2 D 0 D 1 q 0 q 0 cos T 0 q 3 0 O ¹ 2 º : D 2 0 q 2 + ! 2 q 2 = 2 D 0 D 1 q 1 ¹ D 2 1 + 2 D 0 D 2 º q 0 2 ¹ D 0 q 1 + D 1 q 0 º q 1 cos T 0 3 q 2 0 q 1 (5.4) 93 Bysolvingthe O ¹ 1 º equation,weobtainthe q 0 solutionas q 0 = Ae i ! T 0 i e i T 0 + c : c : (5.5) where = F 2 ¹ ! 2 2 º ,and A = Be i ˙ t ,with B = X + iY Thecoe˚cient A ,andhence X and Y arefunctionsof T 1 and T 2 .PluggingEquation(5.5)intothe O ¹ º expression,weget D 2 0 q 1 + ! 2 q 1 = 2 ¹ Ai ! e i ! T 0 + e i T 0 º 2 D 1 Ai ! e i ! T 0 2 Ae i ¹ ! + º T 0 + Ae i ¹ ! º T 0 i e 2 i T 0 i A 3 e 3 i ! T 0 + i 3 e 3 i T 0 3 A 2 i e i ¹ 2 ! + º T 0 3 2 Ae i ¹ ! + 2 º T 0 + 3 iA 2 e i ¹ 2 ! º T 0 + 3 A 2 Ae i ! T 0 3 i 2 e i T 0 3 i 2 Ae i ¹ 2 ! º T 0 + 6 A e i ! T 0 6 iA A e i T 0 + c : c (5.6) ForEquation(5.6),thereare˝vepossiblecasesthatleaduptoresonanceconditions. 1. Nospeci˝crelationshipbetween and ! at O ¹ º 2. ˇ ! 3. ˇ 2 ! 4. ˇ ! š 2 5. ˇ ! š 3 94 5.2.3Case1:NoResonanceat O ¹ º Betweenthenaturalfrequency ! andtheforcingfrequency ,thereisnospeci˝crelationship.We extractseculartermsfromEquation(5.6)andequatethemtozero,suchthat 2 i ! D 1 A 2 i ! A 3 A 2 A + 6 A = 0 : Then,solvingtherestoftheODEinEquation(5.6),weobtaintheparticularsolutionfor q 1 as q 1 = 2 e i T 0 2 ! 2 + A 2 ¹ ! + º 2 ! 2 e i ¹ ! + º T 0 + A 2 ¹ ! º 2 ! 2 e i ¹ ! º T 0 + i 2 ¹ ! 2 4 2 º e 2 i T 0 + A 3 8 ! 2 e 3 i ! T 0 + i 3 9 2 ! 2 e 3 i T 0 + 3 iA 2 ! 2 2 ! + º 2 e i ¹ 2 ! + º T 0 + 3 2 A ! 2 ! + 2 º 2 e i ¹ ! + 2 º T 0 + 3 i A 2 ¹ 2 ! º 2 ! 2 e i ¹ 2 ! º T 0 + 3 i 2 ! 2 2 e i T 0 + 3 i 2 A ! 2 2 ! º 2 e i ¹ 2 ! º T 0 + 6 i A A ! 2 2 e i T 0 + i 2 ! 2 + c : c (5.7) 5.2.4Case2:PrimaryResonanceat O ¹ º When ˇ ! ,i.e.letting = ! + ˙ ( ˙ isdetuningparameter)inEquation(5.6),thesolvability conditionis 2 i ! D 1 A 2 i ! A 3 A 2 A + 6 A e i ˙ T 1 2 + 3 iA 2 + 3 i 2 + 3 i 2 A + 6 iA A = 0 Then,theparticularsolutionfor q 1 turnsouttobe q 1 = A 2 ¹ ! + º 2 ! 2 e i ¹ ! + º T 0 + A 2 ¹ ! º 2 ! 2 e i ¹ ! º T 0 + i 2 ¹ ! 2 4 2 º e 2 i T 0 + A 3 8 ! 2 e 3 i ! T 0 + i 3 9 2 ! 2 e 3 i T 0 + 3 iA 2 ! 2 ¹ 2 ! + º 2 e i ¹ 2 ! + º T 0 + 3 2 A ! 2 ¹ ! + 2 º 2 e i ¹ ! + 2 º T 0 + i 2 ! 2 + c : c (5.8) 95 5.2.5Case3:SubharmonicResonanceofOrder 1 š 2 at O ¹ º When ˇ 2 ! ,i.e. = 2 ! + ˙ ,thesolvabilityconditionforEquation(5.6)is 2 i ! D 1 A 2 i ! A 3 A 2 A + 6 A A 2 e i ˙ T 1 = 0 TheparticularsolutionofEquation(5.6)for q 1 isobtainedas q 1 = 2 e i T 0 2 ! 2 + A 2 ¹ ! + º 2 ! 2 e i ¹ ! + º T 0 + i 2 ¹ ! 2 4 2 º e 2 i T 0 + A 3 8 ! 2 e 3 i ! T 0 + i 2 ! 2 + i 3 9 2 ! 2 e 3 i T 0 + 3 iA 2 ! 2 ¹ 2 ! + º 2 e i ¹ 2 ! + º T 0 + 3 2 A ! 2 ¹ ! + 2 º 2 e i ¹ ! + 2 º T 0 + 3 i 2 ! 2 2 e i T 0 + 3 i A 2 ¹ 2 ! º 2 ! 2 e i ¹ 2 ! º T 0 + 3 i 2 A ! 2 ¹ 2 ! º 2 e i ¹ 2 ! º T 0 + 6 i A A ! 2 2 e i T 0 + c : c (5.9) 5.2.6Case4:SuperharmonicResonanceofOrder 2 at O ¹ º When ˇ ! š 2 ,i.e. 2 = ! + ˙ ,thesolvabilityconditionforEquation(5.6)is 2 i ! D 1 A 2 i ! A 3 A 2 A + 6 A + i 2 e i ˙ T 1 = 0 TheparticularsolutionofEquationequation(5.6)for q 1 isfoundas q 1 = 2 e i T 0 2 ! 2 + A 2 ¹ ! + º 2 ! 2 e i ¹ ! + º T 0 + A 2 ¹ ! º 2 ! 2 e i ¹ ! º T 0 + A 3 8 ! 2 e 3 i ! T 0 + 3 i 2 ! 2 2 e i T 0 + i 3 9 2 ! 2 e 3 i T 0 + 3 iA 2 ! 2 ¹ 2 ! + º 2 e i ¹ 2 ! + º T 0 + 3 2 A ! 2 ¹ ! + 2 º 2 e i ¹ ! + 2 º T 0 + 3 i A 2 ¹ 2 ! º 2 ! 2 e i ¹ 2 ! º T 0 + 3 i 2 A ! 2 ¹ 2 ! º 2 e i ¹ 2 ! º T 0 + 6 i A A ! 2 2 e i T 0 + i 2 ! 2 + c : c (5.10) 96 5.2.7Case5:SuperharmonicResonanceofOrder 3 at O ¹ º When ˇ ! š 3 ,i.e. 3 = ! + ˙ ,thesolvabilityconditionforEquation(5.6)is 2 i ! D 1 A 2 i ! A 3 A 2 A + 6 A i 3 e i ˙ T 1 = 0 (5.11) Theparticularsolutionfor q 1 thenbecomes q 1 = 2 e i T 0 2 ! 2 + A 2 ¹ ! + º 2 ! 2 e i ¹ ! + º T 0 + A 2 ¹ ! º 2 ! 2 e i ¹ ! º T 0 + A 3 8 ! 2 e 3 i ! T 0 + i 2 ¹ ! 2 4 2 º e 2 i T 0 + 3 iA 2 ! 2 ¹ 2 ! + º 2 e i ¹ 2 ! + º T 0 + 3 2 A ! 2 ¹ ! + 2 º 2 e i ¹ ! + 2 º T 0 + i 2 + 3 i A 2 ¹ 2 ! º 2 ! 2 e i ¹ 2 ! º T 0 + 3 i 2 ! 2 2 e i T 0 + 3 i 2 A ! 2 ¹ 2 ! º 2 e i ¹ 2 ! º T 0 + 6 i A A ! 2 2 e i T 0 + c : c (5.12) Tofollowthenextstepssmoothly,theexpressionofEquation(5.12)issimpli˝edasfollows: q 1 = N ! 2 2 e i T 0 + M ! 2 4 2 e 2 i T 0 + P ! 2 ¹ + ! º 2 e i ¹ + ! º T 0 + Q ! 2 + R ! 2 ¹ ! º 2 e i ¹ ! º T 0 + S ! 2 9 ! 2 e 3 i ! T 0 + T ! 2 ¹ + 2 ! º 2 e i ¹ + 2 ! º T 0 + U ! 2 ¹ 2 + ! º 2 e i ¹ 2 + ! º T 0 + V ! 2 ¹ 2 ! º 2 e i ¹ 2 ! º T 0 + W ! 2 ¹ 2 ! º 2 e i ¹ 2 ! º T 0 + c : c (5.13) where N = 2 + 3 i 2 + 6 i A A , M = i 2 , P = A 2 , Q = i 2 , R = A 2 , S = A 3 , T = 3 A 2 i , U = 3 2 A , V = 3 i A 2 , W = 3 2 A 97 Wearespeci˝callyfocusingonsuperharmonicresonancecaseof O ¹ 2 º atorder3todetermine termsthatcane˙ecttheresonancecondition. Substitutingsolutionsfor q 0 fromEquation(5.5)and q 1 fromEquation(5.13)intoEquation(5.4) at O ¹ 2 º ,forsuperharmoniccasewhen ˇ ! š 3 ,i.e. 3 = ! + ˙ ,theequationat O ¹ 2 º becomes D 2 0 q 2 + ! 2 q 2 = 2 ! iD 2 A D 2 1 A 2 D 1 A P 2 ! 2 ¹ + ! º 2 R 2 ! 2 ¹ ! º 2 + 3 Ai N ! 2 2 3 Ai N ! 2 2 + 3 i A V ! 2 ¹ 2 ! º 2 3 A 2 S ! 2 9 2 3 i AT ! 2 ¹ 2 ! + º 2 + 3 2 U ! 2 ¹ 2 + ! º 2 + 3 2 W ! 2 ¹ 2 ! º 2 e i ! T 0 M 2 ! 2 4 2 + 3 2 N ! 2 2 + 3 Ui A ! 2 ¹ 2 + ! º 2 + Ai W ! 2 ¹ 2 ! º 2 e 3 i T 0 + N : S : T : (5.14) where N : S : T : refersnonsecularterms. Equatingtheexpressedseculartermstozero,letting 3 ˇ ! + ˙ ,providesthesolvabilitycondition at O ¹ 2 º .Thiscolvabilityconditionandthesolvabilityconditionfrom O ¹ º inEquation(5.11)are nowlistedtogether,followingtheanalysisdonein[68]: O ¹ º : 2 i ! D 1 A 2 i ! A 3 A 2 A + 6 A i 3 e i ˙ T 1 = 0 O ¹ 2 º : 2 i ! D 2 A D 2 1 A 2 D 1 A + ¹ K 1 + K 2 A + K 3 A º e i ˙ T 1 + K 4 + K 5 A + K 6 A + K 7 A 2 = 0 (5.15) where K 1 = M 2 ¹ ! 2 4 2 º + 3 2 N ! 2 2 , K 2 = 3 i W ! 2 ¹ 2 ! º 2 , K 3 = 3 i U ! 2 ¹ 2 + ! º 2 , K 4 = 2 P ! 2 ¹ + ! º 2 + R ! 2 ¹ ! º 2 + 3 2 U ! 2 ¹ 2 + ! º 2 + 3 2 W ! 2 ¹ 2 ! º 2 , K 5 = 3 i N ! 2 2 3 i N ! 2 2 , K 6 = 3 i V ! 2 ¹ 2 ! º 2 3 i T ! 2 ¹ + 2 ! º 2 , K 7 = 3 S ! 2 9 ! 2 98 Weunfoldthesuperharmonicresonanceatorder3byusinghardforcinginthisanalysis,however itwasnotcapturedwithweakforcingin[21].Inaprocesscalledreconstitution,weputtheterms at O ¹ º and O ¹ 2 º togetherintoasingleordinarydi˙erentialequationandlookforsolutions. Fromexpressionof O ¹ º inEquation(5.15),weextract D 1 A ,thenobtain D 1 A as D 1 A = A + 3 iA 2 A 2 ! + 6 iA 2 ! 3 e i ˙ T 1 2 ! and D 1 A = A 3 i A 2 A 2 ! 6 i A 2 ! 3 e i ˙ T 1 2 ! : Wecomputetheexpressionof D 2 1 A using D 1 A and D 1 A as D 2 1 A = 2 A 6 i A 2 A ! 6 i A ! + 3 e i ˙ T 1 2 ! 9 2 A 3 A 2 4 ! 2 36 2 A 2 A 4 ! 2 + 36 2 A 2 2 4 ! 2 3 i 2 A 2 3 e i ˙ T 1 4 ! 2 3 i 2 3 A Ae i ˙ T 1 4 ! 2 6 i 2 4 e i ˙ T 1 4 ! 2 i ˙ 3 e i ˙ T 1 2 ! Then,weinsertthe D 1 A and D 2 1 A intothe O ¹ 2 º expressionintheEquation(5.15)toobtain D 2 A . 2 i ! D 2 A = 2 A 6 i A 2 A ! 6 i A ! + 3 e i ˙ T 1 2 ! 9 2 A 3 A 2 4 ! 2 36 2 A 2 A 4 ! 2 + 36 2 A 2 2 4 ! 2 3 i 2 A 2 3 e i ˙ T 1 4 ! 2 3 i 2 3 A Ae i ˙ T 1 4 ! 2 6 i 2 4 e i ˙ T 1 4 ! 2 i ˙ 3 e i ˙ T 1 2 ! + 2 A + 3 iA 2 A 2 ! + 6 iA 2 ! 3 e i ˙ T 1 2 ! ¹ K 1 + K 2 A + K 3 A º e i ˙ T 1 K 4 K 5 A K 6 A K 7 A 2 (5.16) Notingthat dA dt = ¹ D 0 + D 1 + D 2 º A = D 1 A + 2 D 2 A thesolvabilityconditionequations D 1 A and D 2 A areworkedtogetherouttogetasingleODEfor A .(AswementionedinChapter3.3.2, recombiningthetimescales ¹ T 0 ; T 1 ; T 3 º isaprocesscalledtitution[26]and[29]).The 99 resultingreconstitutedequationis 2 i ! dA dt + 2 ! i D 1 A + 2 ! i 2 D 2 A = 0 (5.17) Subsequently, D 1 A and D 2 A expressionsareplacedintotheequation(5.17)togettherecon- stituteddi˙erentialequationas 2 ! i dA dt + 2 i ! A 3 A 2 A 6 A i 3 e i ˙ T 1 2 2 A 6 i A 2 A ! 6 i A ! + 3 e i ˙ T 1 2 ! 9 2 A 3 A 2 4 ! 2 36 2 A 2 A 4 ! 2 + 36 2 A 2 2 4 ! 2 3 i 2 A 2 3 e i ˙ T 1 4 ! 2 3 i 2 3 A Ae i ˙ T 1 4 ! 2 6 i 2 4 e i ˙ T 1 4 ! 2 i ˙ 3 e i ˙ T 1 2 ! + 2 A + 3 iA 2 A 2 ! + 6 iA 2 ! 3 e i ˙ T 1 2 ! K 1 + K 2 A + K 3 A º e i ˙ T 1 K 4 K 5 A K 6 A K 7 A 2 = 0 (5.18) Followingtheprocedurein[68],weseekasolutionintheCartesiancoordinateform A = ¹ X + iY º e i ˙ t ,withreal X and Y .Expressions A and areinsertedintotheequation(5.18), thentheequationisdividedbycommonexponentialterm e i ˙ t andrealandimaginarypartsare separated. Imaginarypart: 2 ! Û X + Z 0 sin3 Z 1 cos3 Z 2 X 2 cos3 Z 3 Y 2 cos3 + Z 4 X 2 cos3 + Z 5 Y 2 cos3 + Z 6 XY sin3 + Z 7 X + Z 9 Y + Z 11 XY 2 + Z 14 X 2 Y + Z 15 X 2 Y 3 + Z 17 X 4 Y + Z 19 Y 3 + Z 21 Y 5 + Z 22 cos + Z 23 X 3 = 0 (5.19) 100 RealPart: 2 ! Û Y + Z 0 cos3 Z 1 sin3 Z 2 X 2 sin3 Z 3 Y 2 sin3 + Z 4 X 2 sin3 + Z 5 Y 2 sin3 + Z 6 XY cos3 + Z 8 X + Z 10 Y + Z 12 XY 2 + Z 13 XY 4 + Z 16 X 3 Y 2 + Z 18 X 3 + Z 20 X 5 + Z 22 sin + Z 24 X 2 Y + Z 25 Y 3 = 0 (5.20) where Z 0 ; Z 1 ; Z 2 ; Z 3 ; Z 4 ; Z 5 ; Z 6 ; Z 7 ; Z 8 ; Z 9 ; Z 10 ; Z 11 ; Z 12 ; Z 13 ; Z 14 ; Z 15 ; Z 16 ; Z 17 ; Z 18 ; Z 19 ; Z 20 ; Z 21 ; Z 22 ; Z 23 ; Z 24 ; Z 25 canbefoundinAppendixD. Equations(5.19)and(5.20)aredemonstratedinmatrixformas 2 ! 2 6 6 6 6 6 4 Û X Û Y 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 Z 7 Z 8 Z 8 Z 7 3 7 7 7 7 7 5 2 6 6 6 6 6 4 X Y 3 7 7 7 7 7 5 + ¹ X 2 + Y 2 º 2 6 6 6 6 6 4 Z 11 Z 12 Z 12 Z 11 3 7 7 7 7 7 5 2 6 6 6 6 6 4 X Y 3 7 7 7 7 7 5 + ¹ X 2 + Y 2 º 2 2 6 6 6 6 6 4 0 Z 13 Z 13 0 3 7 7 7 7 7 5 2 6 6 6 6 6 4 X Y 3 7 7 7 7 7 5 + 2 6 6 6 6 6 4 ¹ Z 2 + Z 4 º X 2 cos ¹ 3 º Z 4 ¹ 2 XY º sin ¹ 3 º + ¹ Z 2 Z 4 º Y 2 cos ¹ 3 º ¹ Z 2 Z 4 º X 2 sin ¹ 3 º + Z 4 ¹ 2 XY º cos ¹ 3 º + ¹ Z 2 + Z 4 º Y 2 sin ¹ 3 º 3 7 7 7 7 7 5 + 2 6 6 6 6 6 4 Z 0 sin ¹ 3 º + ¹ Z 1 + Z 22 º cos ¹ 3 º Z 0 cos ¹ 3 º + ¹ Z 1 + Z 22 º sin ¹ 3 º 3 7 7 7 7 7 5 (5.21) Inthenextstagesofthiswork,wewillseeksteady-statesolutionsof X and Y inEquation(5.21), analyzethestability,andthenmakeinterpretations. 5.2.8Conclusion Thehard-forced,nonlinearMathieuequationwasanalyzed.Second-orderperturbationexpansion wasappliedtodeterminestabilitycharacteristicbasedonsystemparameters, ;;; F ;˙ and ! .It revealsaprimaryresonance,superhamonicsatorders2and3andalsoasubharmonicatorder 1 š 2 . Sincewehaveachallengetosolvethenonlinearordinarydi˙erentialequations(5.21)dueto the˝fthdegreeterminthe Û X and Û Y expressions,thisanalysisisnotyetcompleted,andwewill focusonsteady-statesolutionsandstabilitiesinthefuture. 101 5.2.9ProposedWork Wewouldliketosolvethenonlinearordinarydi˙erentialequations(5.21)analyticallyandnu- merically,andthenwewillperformnumericalsimulationstovalidatetheanalyticalresultsonthe resonances. 102 CHAPTER6 CONCLUSIONANDFUTUREWORK 6.1Conclusion Thepurposeofthisthesiswastoanalyzethecharacteristicsofthedynamicsofhorizontalaxis windturbines.Cylicloadsonwindturbinebladesandtheire˙ectsonthebladevibrationwere considered.Ananalysisofin-planecoupledblade-hubdynamicsofthree-bladewindturbineswith tunedandmistunedbladeswasstudied.Four-degree-of-freedomsystemswithparametricsti˙ness wereexaminedbecausethissystemmatchedthemotivationofthree-bladewindturbineandrotor system. InChapter2,byusingasimpli˝edmodel,in-planevibrationsweretakenintoconsideration foradetunedthree-bladewindturbine.Coupledbladesandrotorequationofmotionsintime domainweretransformedintothe ˚ (rotor-angle)domaintodecouplethebladeequationsfromthe rotorequation.Sinceweassumedvariationsintherotorspeedweresmallandappliedanondi- mensionalizationandparameterscaling,bladeequationsweredecoupledfromrotorequation.The ˝rst-ordermethodofmultiplescaleswasemployedtoanalysetheuncoupledbladeequations.After decouplingsteps,thebladeequationsincludedparametricanddirectexcitation.Superharmonicand subharmonicresonanceswerecausedbytheseexcitations.Weexaminedresonancesduetocyclic gravitationalandaerodynamicloading.Superharmonicresonancewasobservednear 2 ! 1 = 1 + ˙ where ! 1 isascaledexcitationorder.Forthesmallpositivemistuning,thebladeamplitudesgo throughvariations.Whenthemistuningwaslarger,themistunedbladeamplitudesdiminished. Verysmallmistuningwastheworst,meaningonebladeexperiencedlargervibrationamplitude thaninthesymmetriccase.Wehaveobservedthatsuperharmonicresonancesthatwereseeninthe symmetriccasecouldbesplitintomultipleresonancepeaks,andthatthebladescouldthentakeon di˙erentsteadystateamplitudes.Likewise,thesamewastrueforprimaryresonance.Thee˙ects ofparametersonthesuperharmonicresonancesarestudied,andanumericalstudyiscarriedout 103 aimedatverifyingtheanalyticalresults.Moreover,therotordynamicsduringresonancewasinves- tigated.Additiontoillustratingthee˙ectsofdampingandforcinglevel,the˝rst-orderperturbation solutionsareveri˝edwithcomparisonstonumericalsimulationsatsuperharmonicresonanceof ordertwo.Thesimulationpointtospeed-lockingphenomenon,inwhichthesuperharmonicspeed islockedinforanintervalofappliedmeanloads.Later,thee˙ectofrotorloadingontherotorspeed andbladeamplitudesisinvestigatedfordi˙erentinitialconditionsandmistuningcases.Lastly,we aimtoanalyticallycon˝rmthebladeresponseamplitudesatvariousrotorspeedsnearresonance andverifyspeedlockingphenomenonbyapplyingmethodofharmonicbalance.Fromanother pointofview,subharmonicresonancewouldnotoccurinarotatingsystemwithsimilarparameter scaling.Subharmonicresonancesmayormaynotbepossibleinthisdynamicalsystem,depending onparameters,andatrotorspeedswelloutsidetheexpectedoperatingconditionsofwindturbines. SubharmonicresonancesinvolveinstabilitiessimilartothoseoftheMathieuequation,butmore complicated.weaimedtoanalyticallycon˝rmthebladeresponseamplitudesatvariousrotor speedsnearresonanceandverifyspeedlockingphenomenon. Furthermore,weextendedthepreviousstudytohigher-orderperturbationexpansionanalysis onthecoupledblade-rotordynamicsofhorizontal-axiswindturbineswithandwithoutmistuning. Second-ordermethodofmultiplescalesanalysisrevealedthesuperharmonicresonancecaseat orderthree.However,thisresonancecouldnotbeobservedinthelinearizedsystemwitha ˝rst-orderexpansion.Thesuperharmonicresonancesplitasingleresonancepeakonthetuned caseintoseveralresonancepeaksduetoweakblademistuningforsteady-statesuperharmonic resonanceresponse.Theamplitudeinceasedwhilethedampingfactor decreased.Also,the responseamplitudegetslargerwhen j j increases.Theresonanceamplitudeofonebladeforthe symmetry-brokensystemincreasedcomparedtothetunedcasefor 0 < v < 0 : 03 ,whiletheother twoblades'amplitudesdiminished.Thisimpliestothepossibilityofvibrationlocalization. InChapter4,generalresponsesofatunedfour-degree-of-freedommass-springsystemwith parametricexcitationwerestudied.Thefrequencycontentandthestabilitycharacteristicsofthe generalsolutionwereobtainedbyassumingamulti-degree-of-freedomFloquet-typesolutionand 104 applyingtheharmonicbalancemethod.Lateron,thisworkwasextendedtothemistunedfour- degree-of-freedomsystemtofocusonthee˙ectofsymmetrybreakingonsystemresponse.When wecomparedthetunedandmistunedstabilitygraphs,numerousinstabilitywedgeswereobserved inthedetunedcaseandshowedmoretendencytodestabilizeingeneral.Theinstabilitywedges werebasedat ! = ¹ ! i + ! j ºš N ,where i = 1 ;:::; d ,and j = 1 ;:::; d ,where d isthedegreeof freedom,and N isthepositiveinteger.Additionaly,FFTandtimeresponsegraphsweregenerated fordi˙erentsetsofparametersandinitialconditionsforboththetunedandmistunedcases.Then, theseresultsshowedgoodagreementwiththenumericalsimulationsdonebytheODEsolveron MATLAB. Inthe˝rstpartofChapter5,alinearMathieueqauationwithaconstantloadwasstudiedto estimateparametersinalinearparametricallyexcitedvibrationsystem.Parameterswerecalculated approximatelybycomparingresonantamplitudestotheiranalyticalexpressions.Wehaveseen goodestimatesof withinalimitedrange,sincetheperturbationsolutiondegradesas increases. Although,dampingcanbedi˙uculttodetermineproperlyinvibrationsystems, wasestimated withinabout25%.Accuracydependsontheasymptoticsofperturbationsolutionswhichconstitute theapproach. ThesecondpartofChapter5wasonthesecond-orderperturbationanalysisofaforcednonlinear Mathieuequation.Aperturbationexpansionoftheequationshowedtheexistenceofmultiple subharmonicandsuperharmonicresonancecases.Ananalyticalframeworkwasproducedbya second-orderperturbationexpansionmethodtoperceivethesystembehaviormoree˙ectively. Themethodwasappliedtodeterminethestabilitycharacteristicsbasedonsystemparameters, ;;; F ;˙ and ! .Primaryresonance,superhamonicsatorders2and3andalsosubharmonicsat order 1 š 2 wereobtained.However,wespeci˝callyfocusedonsuperharmonicresonancecaseat O ¹ 2 º atorder 3 to˝ndthetermsthatcanin˛uencetheresonanceconditionforthehard-forced nonlinearMathieueqaution.Sincewindturbineswereinventedtoperformbelowthelowestnatural frequencylevel,theexistenceofsuperharmonicresonancesmaybeconsiderablyimportant. Overall,thisthesispresented˝rstandsecond-orderperturbationanalysesonbothtunedand 105 detunedthree-bladewindturbines.Four-degree-of-freedomsystemswithparametricexcitation analysiswerealsostudied. 6.2FutureWork Thestudydoneinthisthesiscanbeextendedtothefollowing. InChapter3,futureworkcanaddressaformulationoftheresultingdynamicsoftherotor andthee˙ectsoftheconstantloading.Steady-stateresponseamplitudescanbeobtainedfor superharmonicresonancecaseatordertwo.Later,numericalplotscanbedrawnforallthree blades'amplitudeswithrespecttothedetuningparameter,andnumericalsimulationscanbe donetoverifyanalyticalsolutionsatordertwoandthree.Thee˙ectofnonlinearityonthe resonanceisalsoofinterest. TheFloquet-basedanalysisoffour-degree-of-freedomsystemswithparametricexcitation canbeappliedtomistunedthree-bladewind-turbineequationstoobservetheparametric instabilitiesbehaviorinthetransientdynamicsofcoupledblade-hubturbinemodels. Thestudyofparametricidenti˝cationoftheMathieuequationwithaconstantloadcanfollow thesensitivityanalysis,erroranalysis,andexperiments. Second-orderperturbationanalysisoftheforcednonlinearMathieuequationcanbesolved analyticallyand/ornumerically,andthennumericalsimulationscanbeperformedtoverify theanalyticaloutcomes. 106 APPENDICES 107 APPENDIXA IN-PLANETHREE-BLADEMISTUNEDTURBINEEQUATIONS A.1ParametersusedintheEquationsofMotion ExpressionsfortheparametersinEquationsequation(3.1)andequation(2.2)aregivenbelow: m b = ¹ L 0 m ¹ x º v ¹ x º 2 + J ¹ x º 0 v ¹ x º 2 dx ; k 0 = ¹ L 0 EI ¹ x º 00 v ¹ x º 2 dx ; k 1 = ¹ L 0 xm ¹ x º ¹ x 0 0 v ¹ ˘ º 2 d ˘ m ¹ x º v ¹ x º 2 J ¹ x º 0 v ¹ x º 2 dx ; k 2 = ¹ L 0 g m ¹ x º ¹ x 0 0 v ¹ ˘ º 2 d ˘ dx ; d = ¹ L 0 g m ¹ x º v ¹ x º dx ; e = ¹ L 0 ¹ xm ¹ x º v ¹ x º + J ¹ x º 0 v ¹ x ºº dx ; J r = J hub + 3 ¹ L 0 ¹ x 2 m ¹ x º + J ¹ x ºº dx ; Q j = ¹ L 0 f j ¹ x º v ¹ x º dx ; Q ˚ = 3 Õ j = 1 ¹ L 0 xf j ¹ x º dx : where x istheaxisalongthelengthoftheundeformedblade, m ¹ x º ismassperunitlength, EI and J arethein-planebendingsti˙nessandmassmomentofinertiaperlengthabouttheneutral axis, J hub isthehubinertia, v istheassumedmodalfunction,whichisthe˝rstuniformcantilever beammode,and f j ¹ x º accountsforthedistributedaerodynamicloadsonthe j th blade.Inthese expressions, ¹º 0 = d ¹ºš dx . The˛owisassumedtobesteadyforasimpli˝edmodelin[43],andthewindspeedisassumed tobeslightlyincreasinglinearlywithheight h as u w ind = u 0 + hu 1 = u 0 x cos ˚ j u 1 .Though contributionofstatevariationsontheangleofattackisneglected,theliftforceisproportionalto 108 j ® u rel j 2 ,where ® u blade = x Û ˚ ^ y j ® u rel ® u rel = ® u w ind ® u blade : f j ¹ x º and ® u rel arede˝nedas ® u rel = ¹ u 0 x cos ˚ j u 1 º ^ z x Û ˚ ^ y j ; f j ¹ x º = c p h ¹ u 0 x cos ˚ j u 1 º 2 + ¹ x Û ˚ º 2 i ; where c p isaconstantwhichisrelatedtheairdensity,liftcoe˚cient,andothergeometricparameters, ^ x j and ^ y j aretheaxialandthein-planebendingdirectionsofthe j th blade,and z istheout-ofplane direction. Inserting f j ¹ x º intothe Q j and Q ˚ expressions,weget Q j = ¹ L 0 c p u 2 0 2 u 0 u 1 x cos ˚ j + Û ˚ 2 x 2 + O ¹ 2 º ¹ x º dx ; Q ˚ = 3 Õ j = 1 c p u 2 0 L 2 2 + Û ˚ 2 L 4 4 2 u 0 u 1 cos ˚ j L 3 3 + O ¹ 2 º ! : Since Í 3 j = 1 cos ˚ j = 0 , Q ˚ canbeexpressedas Q ˚ = ^ Q ˚ 0 + ^ Q ˚ 1 Û ˚ 2 .Then, Q j isassumedto havetheformof Q j = ^ Q j 0 + ^ Q j 1 cos ˚ j + ^ Q j 2 Û ˚ 2 forsmall .After Û ˚ = = ¹ 1 + 2 1 º is pluggedinto Q j and Q ˚ expressions,thentheybecomeasfollow Q ˚ = ^ Q ˚ 0 + ^ Q ˚ 1 2 + O ¹ 2 º (A.1) Q j = ^ Q j 0 + ^ Q j 2 2 + ^ Q j 1 cos ˚ j + O ¹ 2 º : Although, isconstant, Q ˚ and Q ˚ canbewrittenas Q ˚ = Q ˚ 0 + O ¹ 2 º (A.2) Q j = Q j 0 + Q j 1 cos ˚ j + O ¹ 2 º : 109 A.2ModalFrequencyEquationsandRootsofCoe˚cientMatrix Asmentionedinsection2.2,thirdandfourthmodalfrequencyequationsaregivenbelow: ! 2 n 3 = m b J r k v + 2 k 1 Û ˚ 2 m b J r + 2 k 0 m b J r 2 e 2 k v 3 e 2 k 1 Û ˚ 2 3 e 2 k 0 2 m 2 b J r 3 e 2 m b " m b J r k v 2 k 1 Û ˚ 2 m b J r 2 k 0 m b J r + 2 e 2 k v + 3 e 2 k 1 Û ˚ 2 + 3 e 2 k 0 2 2 m 2 b J r 3 e 2 m b 4 m 2 b J r 3 e 2 m b k 1 Û ˚ 2 J r k v + k 0 J r k v + k 2 1 Û ˚ 4 J r + 2 k 0 k 1 Û ˚ 2 J r + k 2 0 J r 2 m 2 b J r 3 e 2 m b # 1 š 2 ! 2 n 4 = m b J r k v + 2 k 1 Û ˚ 2 m b J r + 2 k 0 m b J r 2 e 2 k v 3 e 2 k 1 Û ˚ 2 3 e 2 k 0 2 m 2 b J r 3 e 2 m b + " m b J r k v 2 k 1 Û ˚ 2 m b J r 2 k 0 m b J r + 2 e 2 k v + 3 e 2 k 1 Û ˚ 2 + 3 e 2 k 0 2 2 m 2 b J r 3 e 2 m b 4 m 2 b J r 3 e 2 m b k 1 Û ˚ 2 J r k v + k 0 J r k v + k 2 1 Û ˚ 4 J r + 2 k 0 k 1 Û ˚ 2 J r + k 2 0 J r 2 m 2 b J r 3 e 2 m b # 1 š 2 InEquation2.16,wehaverepresantationoftherealandimaginarypartinmatrixform.At steadystatesolution,therootsofcoe˚cientmatrixfornonresonantcaseare: 3 = " ~ e 2 3 q 2~ e 2 v + 9~ e 4 + 2 v + 2 v v q 2~ e 2 v + 9~ e 4 + 2 v + v 9~ e 4 2 # 1 š 2 110 4 = + " ~ e 2 3 q 2~ e 2 v + 9~ e 4 + 2 v + 2 v v q 2~ e 2 v + 9~ e 4 + 2 v + v 9~ e 4 2 # 1 š 2 5 = " ~ e 2 3 q 2~ e 2 v + 9~ e 4 + 2 v 2 v v q 2~ e 2 v + 9~ e 4 + 2 v v 9~ e 4 2 # 1 š 2 6 = + " ~ e 2 3 q 2~ e 2 v + 9~ e 4 + 2 v 2 v + v q 2~ e 2 v + 9~ e 4 + 2 v v 9~ e 4 2 # 1 š 2 111 APPENDIXB WINDTURBINESPARAMETERS B.1Blade'sModalParameters Systemmodalparametersforfourbaselinewindturbinemodelsarecalculatedbyusingthedata fromNationalRenewableEnergyLaboratory'stechnicalreportbyRinkerandDykes[58].The modelshavefourdi˙erentratedpowersas750kilowatts[kW],1.5megawatts[MW],3.0MW,and 5.0MW.Eachbaselinemodelhasthreeidenticalbladesthatareplacedinanupwindcon˝guration andallfourbaselineblademodelswereassumedtoconsistofa1.78-mm-thick˝berglassskinof largelytriaxialmaterialsandwichingabalsacoreforstability[70].Thehubismadeofaductile ironcastingintheshapeofaspherewithopeningsforthebladesandfortheshaftconnections. Thehubismodeledasaspheremadeofductileironwithopeningsforthebladesandfortheshaft connections.Thehubheightis20%largerthantherotordiameterandthehubouterdiameteris0.05 timestherotordiameter.Thelongerbladestendtooperateatlowerrotorspeedsthantheshorter blades.Thewindturbinepartsstructuralpropertiesforthefourbaselinemodelsaresummarized inTableB.1.Thebladeandtowerstructuraldampingvalueswerenotprovidedintheoriginal WindPACTdesign˝les,thereforethedampingvaluesobtainedintheWindPact1.5MWmodel wereprovided.Weshouldnotethatthesedampingvaluesaresubstantiallyhigherthanthoseused intheNREL5MWmodel[58].Jonkman etal. [8]providestructuraldampingratioas0.4774% ofthecriticaldampingfortheNRELo˙shore5-MWbaselinewindturbine(61.5meters). ByusingthedistributedbladestructuralpropertiesdatafromNationalRenewableEnergyLab- oratory'stechnicalreport[58],wehavecalculatedthemodalparametersde˝nedinAppendixA.1. Thedatafor5MWwindturbinemodelisprovidedatTableB.2asareference.The750kW,1.5 MWand3MWmodelsblade'sdistributedparametersaretabulatedinNREL'stechnicalreport [58].Massmomentofinertiaperlengthabouttheneutralaxis, J ,isnottakenintoaccount. 112 TableB.1:StructuralPropertiesfortheBaselineWindTurbineModels Parameters 750KW 1.5MW 3MW 5MW BladeLength( m ) 23.75 33.25 47.025 60.8 BladeMass( k g ) 1,941 4,336 13,238 27,854 HubInertiaaboutLSS( k g : m 2 ) 5,160 29,975 197,987 668,485 HubHeight( m ) 60 84 119 154 HubDiameter( m ) 2.5 3.50 4.95 6.40 RotorDiameter( m ) 50 70 99 128 RotorMass( k g ) 12,381 32,167 101,319 209,407 TowerMass( k g ) 53,776 125,364 351,798 775,094 NacelleMass( k g ) 20,950 52,839 132,598 270,669 RatedRotorSpeed ( rpm ) 28.648 20.463 14.469 11.191 Blade˝rstMassMomentofInertia( k g : m ) 14,605 46,497 207,135 563,188 BladeSecondMassMomentofInertia( k g : m 2 ) 180,640 798,506 5,012,212 17,475,408 BladeFlapwiseStructuralDamping 3.882%ofcritical BladeEdgewiseStructuralDamping 5.900%ofcritical TowerStructuralDamping,AllModes 3.435%ofcritical Themid-pointnumericalintegrationruleisappliedtodistributedbladestructuralpropertydata tabulatedin[58]andtheresultiscomparedwiththeprovidedblademass.Theestimationsare abletopredictthetotalmasswithover99.7%accuracyasseeninTableB.3.Thesystemmodal parametersintheequationsofmotionintimedomaincanbefoundinTableB.4.Onlyedgewise naturalfrequenciesforbaselinemodels, ! n 2 areprovidedinthetable. Edgewiseand˛apwisemodalnaturalfrequenciesfor61.5meterbladearefoundbymethodof assumedmode.ThesevaluesarecomparedwithNRELandSandia'sresultsinTableB.5.National 113 laboratorieshaveuseddi˙erenttypeofmethodsto˝ndbladenaturalfrequencies.In[71],[72] and[73],wehavefoundthereferencedatatocompare.Jonkman et.al [71]fromNRELhaveused FASTandADAMS,althoughResor et.al fromSandiahaveusedANSYS[72]andBMODES [73]softwarestocalculatetheblademodalfrequencies.Totalmassofthebladeis17740kg,rated rotorspeedis12.1rpmandbladetipspeedis80m/s.Sincesingleassumedmodeistakeninto accountinourcalculation,˛apwiseandedgewisemodalfrequenciesareslightlyhigherthanNREL andSandia.Whenthenumberofassumedmodesisincreased,themodalnaturalfrequencieswill approachconvergencetotheirreferencevalues. B.2TheRelationBetweentheBladeSizeandtheParametricE˙ects Theparametrice˙ectsbecomemoresigni˝cantwhenhorizontalaxiswindturbinebladesget longer.Aswindturbinebladesaredesignedlonger,theirthicknessisnotnecessarilychangedinthe sameproportion,andismorelikelytochangeinasmallerproportionsuchthatthebladesbecome relativelymoreslenderwithincreasingsize.ThelengthofNREL's23.75m,33.25m,47.025m and60.8mbaselinemodelsarescaledwhiletheotherdimensionsarekeptthesametoinvestigate thee˙ectofbladesizeontheparametrice˙ects.Inaworst-casescenario,thelengthoftheblades inTableB.4ismultipliedbytwoandthree,whilethecrosssectionalareaiskeptthesame. Whileparametricsti˙nessratiosforthe5.0MWmodelis0.0623forthedoubledlengthblade, itis0.21forthe3timesscaledupbladewhichisabout27timeshigherthantheoriginalblade's ratioasseeninTableB.6.Scalingonlythelengthcauseddramaticchangesintheparametric e˙ects.Sincewehaveonlyscaledthelengthofthebladesinsteadofscalingthewholevolume, thesevaluesdrawtheworstscenariointhesenseofparametrice˙ectsonthesystems.Scalingthe wholebladewouldresultinalinear-likeincreaseintheparametrice˙ectswithincreasinglength. Then,weexpectarealisticbladetrendtobesomewherebetweenthescaledlengthandvolume ratios.Thereforewecanconcludethatasthebladesgetlongerinlength,theparametrice˙ects becomemoremeaningful. AsseenfromFigureB.1,theratiooftheparametricandelasticmodalsti˙nessesisestimated 114 forthescaledversionsoftheNREL'sbladesforfourmodelstopresenttherelationbetweenthe bladesizeandtheparametrice˙ects. FigureB.1:Parametricsti˙nessratiosforscaledblademodelsandactualblademodels Naturalfrequenciesarefoundatdi˙erentrotationanglestoshowthesigni˝canceofthegrav- ity'sparametrice˙ect.Thebladenaturalfrequenciesalterwiththerotorangle, ˚ ,duethegravity hassti˙eningandsofteninge˙ects.Whenthebladeisupright, ˚ = ˇ ,thegravitationalforce compressesit,andmakesthebladelesssti˙inbending.Whentherotorangleis ˚ = 0 ,the gravitationalforcepullsthebladeandincreasestheblade'sbendingsti˙ness.Thesevariationsin sti˙nesscanbeestimatedfromthenaturalfrequencies.Forabladewithmodalmass m b ,elastic sti˙ness k 0 andparametricsti˙ness, k 2 ,duetogravity,thesystemnaturalfrequencyinupright, horizontalanddownwardpositionscanbecalculatedrespectivelyasfollow: ! u = r k 0 k 2 m b ;! h = r k 0 m b ;! d = r k 0 + k 2 m b . Theratioofthefrequenciesofdownwardandhorizontalorientationsis 115 ! d ! h = r k 0 + k 2 k 0 where ! d and ! h aredownwardandhorizontalbladefrequencies,respectively.Thereforetheratio oftheparametricsti˙nesstoelasticsti˙nesscanbefoundas k 2 k 0 = ! d ! h 2 1 . IntheTableB.7,modalnaturalfrequenciesofbaselineblademodelsareprovidedfordi˙erent positions. Theparametrice˙ectisestimatedforscaledversionsoftheNREL'sbladesbyusingexpression k 2 š k 0 = ! 2 d š ! 2 h 1 .ModalfrequenciesforscaledblademodelsofNREL60.8meterwind turbineareprovidedfordi˙erentpositiontoshowthee˙ectofparametricsti˙nessduetogravity inTableB.8.Additionaly,TableB.9presenthowtheparametrice˙ectbecomesmoresigni˝cant byincreasingthelengthofblades. Actualandscaled-lengthbladetipplacementsareobtainedforsuperharmonicresonancesat order2and3inFigureB.2.Redandbluelinesshowthescaledlengthbladetipdisplacementfor superharmonicresonancesatorder2and3butthesetipdisplacementsarestandforupperboundof themodalcoordinateamplitude.Dampingratioistaken = 5 : 9 forblades L < 64 ,and = 0 : 477 forblades L 64 .JasonJonkman etal. [8]providedampingfactoras = 0 : 477 for64-meter 5-mwreferencewindturbine.AlthoughDykesandRinker[58]statedthat = 5 : 9 isfor35-meter blade,theyhaveusedsamedampingfactorfor25,35,49and64-meterblades.Therefore,wewere notableto˝ndthedampingratiosforlongerblades,weonlyappliedthesetwodampingratiosin outplots. Parametrice˙ectonactualandscaled-lengthbladetipplacementsareinvestigatedforsuper- harmonicresonancesatorder2inFigureB.3.Whenbladeisshorter,parametrice˙ectisnot powerful,butwhenbladegetslonger,itbecomesmeaningful.Sinceweonlyscaledtheblades lenghtwise,thesetipdisplacementvaluesgivetheworst-casescenariointermsoftheparametric sti˙nesse˙ect. 116 FigureB.2:Superharmonicresonanceordere˙ectonactualandscaledbladetipdisplacement FigureB.3:Parametrice˙ectonactualandscaledbladetipdisplacementforsuperharmanic resonanceatorder2 117 TableB.2:Distributedbladestructuralpropertiesforthe5MWmodeltakenfromNREL Radius( m ) r MassDenstiy( k g š m ) EdgewiseSti˙nessEI( N : m 2 ) 3.2 3.2 3708.41 6.37E+10 4.5 1.3 622.32 1.40E+10 6.4 1.9 632.67 1.38E+10 9.6 3.2 649.91 1.34E+10 12.8 3.2 667.16 1.30E+10 16 3.2 684.4 1.26E+10 19.2 3.2 650.77 1.06E+10 22.4 3.2 617.15 8.58E+09 25.6 3.2 583.52 6.57E+09 28.8 3.2 549.9 4.55E+09 32 3.2 516.27 2.54E+09 35.2 3.2 458.05 2.15E+09 38.4 3.2 399.83 1.77E+09 41.6 3.2 341.6 1.38E+09 44.8 3.2 283.38 9.96E+08 48 3.2 225.16 6.10E+08 51.2 3.2 184.52 4.99E+08 54.4 3.2 143.89 3.88E+08 57.6 3.2 103.25 2.76E+08 60.8 3.2 62.62 1.65E+08 64 3.2 21.99 5.36E+07 118 TableB.3:Blademasscomparisonforbaselinemodelandmid-pointnumericalintegrationmethod (Mid-pointI.M.) BaselineModels 750kW 1.5MW 3.0MW 5.0MW ProvidedBladeMass(kg) 1,941 4,336 13,238 27854 BladeMassbyMid-pointI.M.(kg) 1935.8 4326.6 13216 27843 AccuracyRate(%) 99.73209686 99.78321033 99.83381175 99.96050837 TableB.4:Blademodalparametersforbaselinewindturbinemodels BaselineModels 750kW 1.5MW 3.0MW 5.0MW Lengthfromcenterm 25 35 49.5 64 BladeLength( m ) 23.75 33.25 47.025 60.8 ModalMass m b ( k g m 2 ) 997.0195 1904.3 6963.9 14441 Elasticsti˙ness k 0 ( Nm ) 224610 246280 453990 603670 CentrifugalSti˙ness k 1 ( k g m 2 ) 303.5731 598.3964 2149.8 4479.3 GravitationalSti˙ness k 2 ( Nm ) 812.2177 1118 2914.3 4706.5 DirectGravitationalterm d ( Nm ) 1340.2 2010.9 4816 7861.8 CouplingTerm e ( k g m 2 ) 14449 41953 201430 541840 TotalInertia J r ( k g m 2 ) 666600 2957400 18578000 64311000 = m b š J r 0.001495679 0.00064391 0.000374847 0.000224549 k 0 š k 1 739.887691 411.5666471 211.177784 134.7688255 ! n 2 = s ¹ k 0 + k 1 2 º m b (Hz) 2.404520805 1.820943987 1.292662632 1.034768525 k 2 š k 0 0.003616124 0.004539548 0.006419304 0.007796478 119 TableB.5:Blademodalfrequencies(Hz)comparisonwiththedatafromNRELandSANDIA NREL SANDIA Calculated Method FAST ADAMS ANSYS BMODES AssumedModes Flapwise 0.69 0.70 0.87 0.95 1.09 Edgewise 1.089 1.087 1.060 1.240 1.240 TableB.6:Parametrice˙ectforscaledbladesize BaselineModels 750kW 1.5MW 3.0MW 5.0MW k 2 š k 0 forL 0.003616124 0.004539548 0.006419304 0.007796478 k 2 š k 0 for2L 0.028928222 0.035818937 0.051354209 0.062372446 k 2 š k 0 for3L 0.097634055 0.120885609 0.173315492 0.210506306 TableB.7:Modalfrequencies(Hz)forthebaselineblademodels BaselineModels 750kW 1.5MW 3.0MW 5.0MW ! d ¹ ˚ = 0 º 2.394345889 1.814975491 1.289812899 1.033541542 ! h ¹ ˚ = ˇ š 2 º 2.390028468 1.810869879 1.285692873 1.029535957 ! u ¹ ˚ = ˇ º 2.385703234 1.806754938 1.281559603 1.025514726 TableB.8:Modalfrequencies(Hz)forthescaledblademodelsofNREL60.8m BladeLength L=60.8m 2L=2*60.8 3L=3*60.8 ! d ¹ ˚ = 0 º 1.033541542 0.265283507 0.125856627 ! h ¹ ˚ = ˇ š 2 º 1.029535957 0.257378254 0.114391185 ! u ¹ ˚ = ˇ º 1.025514726 0.249222376 0.101640523 120 TableB.9:Parametrice˙ect ¹ k 2 š k 0 º forthescaledbladeofNRELbaselinemodels ScaledLength ¹ k 2 š k 0 º forL ¹ k 2 š k 0 º for2L ¹ k 2 š k 0 º for3L L=23.75m 0.003616124 0.028928222 0.097634055 L=33.25m 0.004539548 0.035818937 0.120885609 L=49.5m 0.006419304 0.051354209 0.173315492 L=60.8m 0.007796478 0.062372446 0.210506306 121 APPENDIXC SECOND-ORDERPERTURBATIONANALYSISOFIN-PLANETHREE-BLADE TUNEDANDMISTUNEDTURBINES C.1SuperharmonicCaseatOrder 3 at O ¹ 2 º Coe˚cientsof X and Y termsinEquation(3.21)andEquation(3.22)aregivenbelow z 1 = 1 2 ¹ º z 2 = 2 ~ e 2 2 z 3 = 1 2 v j + 1 8 2 2 + 1 8 2 2 v j + ˙ z 4 = 2 ~ e 2 v j 4 9 2 ~ e 4 8 ~ e 2 2 z 5 = 2 2 where = F j 0 16 ¹ 1 ! 2 1 º¹ 1 4 ! 2 1 º 122 C.2SuperharmonicCaseatOrder 2 at O ¹ 2 º C.2.1Mistunedcasecoe˚cients Coe˚cientsofreconstituteddi˙erentialequation'srealpartfromEquation(3.35),andimaginary partfromEquation(3.36)are C 1 = 1 2 ¹ º C 2 = 2 ~ e 2 2 C 3 = 1 2 v j + 1 8 2 2 + 1 8 2 2 v j + ˙ C 4 = 2 ~ e 2 v j 4 9 2 ~ e 4 8 ~ e 2 2 C 5 = 2 ~ e 2 ! 2 1 8 ¹ 1 ! 2 1 º C 6 = 2 F j 0 32 ¹ 1 ! 2 1 º 2 F j 1 8 ¹ 1 ! 2 1 º + 2 F j 0 ! 1 16 ¹ 1 ! 2 1 º 2 + 2 2 F j 16 ¹ 1 ! 2 1 º C 7 = F j 0 8 ¹ 1 ! 2 1 º 2 F j 0 v j 32 ¹ 1 ! 2 1 º C 8 = 2 3~ e 2 F j 0 32 ¹ 1 ! 2 1 º 123 C.2.2Tunedcasecoe˚cients Coe˚cientsofreal(3.37)andimaginary(3.38)partsofreconstituteddi˙erentialequationfortuned systemare T 1 = 1 2 ¹ º T 2 = 2 ~ e 2 2 T 3 = 1 8 2 2 + ˙ T 4 = 9 2 ~ e 4 8 ~ e 2 2 T 5 = 2 ~ e 2 ! 2 1 8 ¹ 1 ! 2 1 º T 6 = 2 F j 0 32 ¹ 1 ! 2 1 º 2 F j 1 8 ¹ 1 ! 2 1 º + 2 F j 0 ! 1 16 ¹ 1 ! 2 1 º 2 + 2 2 F j 16 ¹ 1 ! 2 1 º T 7 = F j 0 8 ¹ 1 ! 2 1 º T 8 = 2 3 ~ e 2 F j 0 32 ¹ 1 ! 2 1 º 124 APPENDIXD SECOND-ORDERPERTURBATIONANALYSISOFNONLINEARMATHIEU EQUATIONWITHHARDEXCITATION Z 0 s termsfromEquations(5.20)and(5.19)insection5.2.7are Z 0 = 6 2 F 3 8 ¹ ! 2 2 º 4 + 6 2 2 F 5 128 ! 2 ¹ ! 2 2 º 5 + 2 F 3 16 ! ¹ ! 2 2 º 3 Z 1 = 2 ˙ F 3 16 ! ¹ ! 2 2 º 3 + 9 2 2 F 3 32 ¹ ! 2 2 º 6 F 3 8 ¹ ! 2 2 º 3 Z 2 = Z 3 = 3 2 2 F 3 32 ! 2 ¹ ! 2 2 º 3 + 18 2 2 F 3 8 ¹ ! 2 2 º 4 + 9 2 2 F 3 8 ¹ ! 2 ¹ 2 ! º 2 º¹ ! 2 2 º 3 + 9 2 2 F 3 8 ¹ ! 2 ¹ 2 + ! º 2 º¹ ! 2 2 º 3 Z 6 = 2 Z 5 = 2 Z 4 = 6 2 2 F 3 32 ! 2 ! ¹ ! 2 2 º 3 Z 7 = Z 10 = 2 ! 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