Control Engineering SeriesEditor WilliamS.Levine DepartmentofElectricalandComputerEngineering UniversityofMaryland CollegePark,MD USA EditorialAdvisoryBoard RichardBraatz MarkSpong MassachusettsInstituteofTechnology UniversityofTexasatDallas Cambridge,MA Dallas,TX USA USA GrahamGoodwin MaartenSteinbuch UniversityofNewcastle TechnischeUniversiteitEindhoven Australia Eindhoven,TheNetherlands DavorHrovat MathukumalliVidyasagar FordMotorCompany UniversityofTexasatDallas Dearborn,MI Dallas,TX USA USA ZongliLin YutakaYamamoto UniversityofVirginia KyotoUniversity Charlottesville,VA Kyoto,Japan USA Forfurthervolumes: http://www.springer.com/series/4988 Ashish Tewari Aeroservoelasticity Modeling and Control AshishTewari DepartmentofAerospaceEngineering IndianInstituteofTechnology,Kanpur Kanpur UttarPradesh India ISSN2373-7719 ISSN2373-7727(electronic) ControlEngineering ISBN978-1-4939-2367-0 ISBN978-1-4939-2368-7(eBook) DOI10.1007/978-1-4939-2368-7 MathematicsSubjectClassification(2010): 70Q05, 74H10, 74H15, 74F10, 74S05, 74S25, 76G25, 76H05, 76J20, 76M22, 76M23, 76M25, 76M40, 93B15, 93B30, 93B36, 93B52, 93B55,93C05,93C10,93C15,93C35,93C40,93C80,93D05,93D20,32D99 LibraryofCongressControlNumber:2015933384 SpringerNewYorkHeidelbergDordrechtLondon © SpringerScience+BusinessMedia,LLC2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsorthe editorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforanyerrors oromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Totheorderandbeautyofnature, anditsfinelycraftedcontrollaws. Preface This book presents aeroservoelasticity (ASE) as a well-formed discipline with a systematic framework. While many research articles have appeared on the special applications of ASE–such as active flutter suppression and gust load alleviation– therearenotextbooksandmonographsonthegeneralandsystematicprocedureto befollowedinthemodelingandanalysisofaeroservoelasticsystems.Thisbookis afirststepintryingtofillthisimportantgapintheaerospaceengineeringliterature. This book introduces the basic math modeling concepts and highlights important developments involved in structural dynamics, unsteady aerodynamics, and con- trol systems. It also attempts to evolve a generic procedure to be applied forASE systemsynthesis.Thetreatmentincludesfinite-elementstructuralmodelingandde- tailedunsteadyaerodynamicmodelingatvariousspeedsforderivingthenecessary aeroelasticplants,withsamplecontrolapplicationstoactivefluttersuppression,load alleviation,andadverseASEcoupling. A general aeroelastic plant is derived via the finite-element structural dynamic model,unsteadyaerodynamicmodelsforvariousregimesinthefrequencydomain, andtheassociatedstate-spacemodelbyrationalfunctionapproximations.Formore advancedmodels,thefull-potential,Euler,andNavier–Stokesmethodsfortreating transonic and separated flows are also briefly addressed. EssentialASE controller designandanalysistechniquesareintroducedtothereader. Introductiontorobust control-law design methods of LQG/LTR and H2/H∞ synthesis is followed by a briefcoverageofnonlinearcontroltechniquesofdescribingfunctionsandLyapunov functions. The fundamental concepts are presented in such a way that the most important featurescanbeeasilydeduced.Thebreadthofcoverageissufficientforathorough understandingofASE. The focus of this book is on aeroservoelastic modeling, including a brief pre- sentation on robust and optimal control methods that can be applied to important aeroservoelastic design problems. It is not possible to give a more comprehensive ASEtreatmentinasinglebook,anditisenvisagedthatafuturebookcanbedevoted tomoreadvancedtopicssuchasadaptiveandnonlinearcontroldesigntechniques. Thisbookisaimedatgraduatestudentsandadvancedresearchersinaerospaceen- gineering,aswellasprofessionalengineers,technicians,andtestpilotsintheaircraft vii viii Preface industryandlaboratories.Thereaderisassumedtohavetakenbasicundergraduate coursesinmathematicsandphysics—particularlycalculus,complexvariables,lin- earalgebra,andfundamentaldynamics—andisencouragedtoreviewtheseconcepts atseveralplacesinthetext. AbookonASEisdifficulttowriteduetothebreadthoftopicsitmustnecessarily address.WhilethisbookcoverstheessentialsofmodelingaspectsofASEwithsome controlapplications,itgivessufficientmotivationtoareaderwithspecificresearch intereststofurtherexploretherelevanttopics.Furthermore,thetreatmentoftopicsis suchthatanovicecanquicklybuilduphis/herunderstandingofASEwithoutmuch difficulty.Referencesareselectedkeepingboththetypesofreadersinmind. This book has been long in writing, with the intention first having occurred to theauthorabout15yearsago. Nothavingaccesstotheindustrialcodesforfinite- elementandunsteadyaerodynamicsnecessaryforbuildingsuchanexposition,and notfindingthetimetowriteone’sowncodes, theprojectcontinuedtobedelayed untilabout2yearsago,whencouragewasfinallygatheredforthispurpose.Testing andvalidatingthecodesforthemanyexamplesinthebookwasitselfaformidable task,whichrequiredmanyhoursofpatientprogramming. I would like to thank Walt Eversman for his course on aeroelasticity, and for advisingmeinmygraduatestudies.TheeditorialandproductionstaffofBirkhäuser haveofferedmanyconstructiveinputsduringthepreparationofthemanuscript,for whichIamindebtedtothem.IamalsogratefultoTheMathWorks,Inc.forproviding thelatestMATLAB/Simulinkversionutilizedfortheexamplesthroughoutthebook. November2014 AshishTewari Contents 1 Aeroservoelasticity ............................................. 1 1.1 Introduction ............................................... 1 1.2 AnIllustrativeExample ..................................... 7 2 StructuralModeling............................................ 13 2.1 Introduction ............................................... 13 2.2 StaticLoadDeflectionModel ................................ 14 2.3 Beam-ShaftIdealization ..................................... 18 2.4 Dynamics ................................................. 22 2.5 LumpedParametersMethod ................................. 27 2.6 Rayleigh-RitzMethod....................................... 31 2.7 Finite-ElementMethod...................................... 33 2.7.1 WeakFormulationandGalerkin’sApproximation ......... 33 2.7.2 Euler-BernoulliBeamandShaftElements ............... 38 2.7.3 IllustrativeExample .................................. 47 2.7.4 PlateBendingElements............................... 51 3 UnsteadyAerodynamicModeling ................................ 59 3.1 Introduction ............................................... 59 3.2 GoverningEquations........................................ 61 3.2.1 ViscousFlow........................................ 61 3.2.2 InviscidFlow........................................ 64 3.2.3 PotentialFlow....................................... 65 3.2.4 TransonicSmall-DisturbanceFlow...................... 69 3.3 LinearizedSubsonicandSupersonicFlow...................... 71 3.4 IncompressibleFlowSolution ................................ 74 3.4.1 UnsteadyVortex-LatticeMethod........................ 80 3.4.2 ClassicalAnalyticalSolution........................... 91 3.5 IntegralEquationforLinearCompressibleFlow................. 102 3.5.1 VelocityPotentialFormulationbyGreen’sTheorem ....... 102 3.5.2 AccelerationPotentialFormulation ..................... 112 ix x Contents 3.6 SubsonicKernelFunctionandtheDoublet-LatticeMethod........ 116 3.6.1 NumericalResults.................................... 128 3.7 SupersonicLiftingSurfaceMethods........................... 136 3.7.1 Mach-BoxMethod ................................... 139 3.7.2 Doublet-PointMethod ................................ 141 3.8 TransonicSmall-DisturbanceSolutionbyGreen’sFunctionMethod 144 3.8.1 TransonicGreen’sIntegralEquation .................... 144 3.8.2 TransonicDoublet-LatticeMethod...................... 149 4 Finite-StateAeroelasticModeling ................................ 155 4.1 Finite-StateUnsteadyAerodynamicsModel .................... 155 4.1.1 TraditionalFlutterAnalysis ............................ 156 4.1.2 UnsteadyAerodynamicsinTimeDomain ................ 157 4.2 TransientAerodynamicsinTwo-Dimensions.................... 160 4.2.1 RationalFunctionApproximation....................... 161 4.2.2 IndicialAdmittancebyDuhamel’sIntegral ............... 166 4.2.3 TransientAerodynamicsinThree-Dimensions ............ 167 4.2.4 AlternativeMethodsfor3DTransientAerodynamics....... 179 4.2.5 DirectIntegrationofGoverningEquations ............... 182 4.3 State-SpaceRepresentation .................................. 183 4.3.1 TypicalSectionModel ................................ 185 4.3.2 Three-DimensionalWingModel........................ 190 4.3.3 IllustrativeExample .................................. 195 5 LinearAeroelasticControl ...................................... 207 5.1 Introduction ............................................... 207 5.2 LinearFeedbackStabilization ................................ 208 5.2.1 Servo-Actuators ..................................... 210 5.3 OptimalControl............................................ 214 5.3.1 Hamilton-Jacobi-BellmanEquation ..................... 215 5.3.2 LinearSystemswithQuadraticPerformanceIndex ........ 216 5.4 KalmanFilter.............................................. 219 5.5 Infinite-HorizonLinearOptimalControl ....................... 223 5.6 AdverseAereoservoelasticInteraction ......................... 225 5.6.1 Closed-LoopStabilizationoftheASESystem ............ 233 5.6.2 ActiveManeuverLoadAlleviation ...................... 235 5.7 RobustControlofLinearTime-InvariantSystems................ 239 5.7.1 LQG/LTRMethod ................................... 243 5.7.2 H2/H∞Control ..................................... 246 5.8 ActiveFlutterSuppression ................................... 250 6 NonlinearAeroservoelasticApplications .......................... 257 6.1 NonlinearAeroservoelasticity ................................ 257 6.2 DescribingFunctionsforNonlinearASE ....................... 257 6.3 Flapping-WingFlight ....................................... 260 6.3.1 LiftandThrustforFlappingFlight...................... 263 Contents xi 6.4 Shock-InducedBuffet....................................... 271 6.5 TransonicFlutter ........................................... 272 6.5.1 AdaptiveSuppressionofTransonicLCO:IllustrativeExample 273 AppendixA....................................................... 279 Appendix-B....................................................... 285 Appendix-C....................................................... 289 Appendix-D....................................................... 301 References........................................................ 303 Index ............................................................ 311