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Prelims-H6927.tex 11/7/2007 11:13 Pagei Flight Dynamics Principles This page intentionally left blank Prelims-H6927.tex 11/7/2007 11:13 Pageiii Flight Dynamics Principles M.V. Cook BSc, MSc, CEng, FRAeS, CMath, FIMA Senior Lecturer in the School of Engineering Cranfield University AMSTERDAM•BOSTON•HEIDELBERG•LONDON•NEWYORK•OXFORD PARIS•SANDIEGO•SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO Butterworth-HeinemannisanimprintofElsevier Prelims-H6927.tex 11/7/2007 11:13 Pageiv Butterworth-HeinemannisanimprintofElsevier LinacreHouse,JordanHill,OxfordOX28DP 30CorporateDrive,Suite400,Burlington,MA01803,USA Firstedition1997 Secondedition2007 Copyright©2007,M.V.Cook.PublishedbyElsevierLtd.Allrightsreserved TherightofMichaelCooktobeidentifiedastheauthorofthisworkhasbeenasserted inaccordancewiththeCopyright,DesignsandPatentsAct1988 Nopartofthispublicationmaybereproduced,storedinaretrievalsystemortransmittedinany formorbyanymeanselectronic,mechanical,photocopying,recordingorotherwisewithout thepriorwrittenpermissionofthepublisher PermissionsmaybesoughtdirectlyfromElsevier’sScience&TechnologyRights DepartmentinOxford,UK:phone(+44)(0)1865843830;fax(+44)(0)1865853333; email:permissions@elsevier.com.Alternativelyyoucansubmityourrequestonlineby visitingtheElsevierwebsiteathttp://elsevier.com/locate/permissions,andselecting ObtainingpermissiontouseElseviermaterial Notice Noresponsibilityisassumedbythepublisherforanyinjuryand/ordamagetopersons orpropertyasamatterofproductsliability,negligenceorotherwise,orfromanyuseor operationofanymethods,products,instructionsorideascontainedinthematerialherein. BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheLibraryofCongress ISBN:978-0-7506-6927-6 ForinformationonallButterworth-Heinemannpublicationsvisit ourwebsiteathttp://books.elsevier.com TypesetbyCharontecLtd(AMacmillanCompany),Chennai,India www.charontec.com PrintedandboundinGreatBritain 07080910 10987654321 Prelims-H6927.tex 11/7/2007 11:13 Pagev Contents Prefacetothefirstedition ix Prefacetothesecondedition xi Acknowledgements xiii Nomenclature xv 1. Introduction 1 1.1 Overview 1 1.2 Flyingandhandlingqualities 3 1.3 Generalconsiderations 4 1.4 Aircraftequationsofmotion 7 1.5 Aerodynamics 7 1.6 Computers 8 1.7 Summary 10 References 11 2. Systemsofaxesandnotation 12 2.1 Earthaxes 12 2.2 Aircraftbodyfixedaxes 13 2.3 Euleranglesandaircraftattitude 18 2.4 Axestransformations 18 2.5 Aircraftreferencegeometry 24 2.6 Controlsnotation 27 2.7 Aerodynamicreferencecentres 28 References 30 Problems 30 3. Staticequilibriumandtrim 32 3.1 Trimequilibrium 32 3.2 Thepitchingmomentequation 40 3.3 Longitudinalstaticstability 44 3.4 Lateralstaticstability 53 3.5 Directionalstaticstability 54 3.6 Calculationofaircrafttrimcondition 57 References 64 Problems 64 4. Theequationsofmotion 66 4.1Theequationsofmotionofarigidsymmetricaircraft 66 4.2Thelinearisedequationsofmotion 73 v Prelims-H6927.tex 11/7/2007 11:13 Pagevi vi Contents 4.3Thedecoupledequationsofmotion 79 4.4Alternativeformsoftheequationsofmotion 82 References 95 Problems 96 5. Thesolutionoftheequationsofmotion 98 5.1 Methodsofsolution 98 5.2 Cramer’srule 99 5.3 Aircraftresponsetransferfunctions 101 5.4 Responsetocontrols 108 5.5 Accelerationresponsetransferfunctions 112 5.6 Thestatespacemethod 114 5.7 Statespacemodelaugmentation 128 References 134 Problems 134 6. Longitudinaldynamics 138 6.1 Responsetocontrols 138 6.2 Thedynamicstabilitymodes 144 6.3 Reducedordermodels 147 6.4 Frequencyresponse 158 6.5 Flyingandhandlingqualities 165 6.6 Modeexcitation 167 References 170 Problems 171 7. Lateral–directionaldynamics 174 7.1 Responsetocontrols 174 7.2 Thedynamicstabilitymodes 183 7.3 Reducedordermodels 188 7.4 Frequencyresponse 195 7.5 Flyingandhandlingqualities 200 7.6 Modeexcitation 202 References 206 Problems 206 8. Manoeuvrability 210 8.1 Introduction 210 8.2 Thesteadypull-upmanoeuvre 212 8.3 Thepitchingmomentequation 214 8.4 Longitudinalmanoeuvrestability 216 8.5 Aircraftdynamicsandmanoeuvrability 222 References 223 9. Stability 224 9.1 Introduction 224 9.2 Thecharacteristicequation 227 9.3 TheRouth–Hurwitzstabilitycriterion 227 Prelims-H6927.tex 11/7/2007 11:13 Pagevii Contents vii 9.4 Thestabilityquartic 231 9.5 Graphicalinterpretationofstability 234 References 238 Problems 238 10. Flyingandhandlingqualities 240 10.1 Introduction 240 10.2 Shorttermdynamicmodels 241 10.3 Flyingqualitiesrequirements 249 10.4 Aircraftrole 251 10.5 Pilotopinionrating 255 10.6 Longitudinalflyingqualitiesrequirements 256 10.7 Controlanticipationparameter 260 10.8 Lateral–directionalflyingqualitiesrequirements 263 10.9 Flyingqualitiesrequirementsonthes-plane 266 References 271 Problems 272 11. Stabilityaugmentation 274 11.1 Introduction 274 11.2 Augmentationsystemdesign 280 11.3 Closedloopsystemanalysis 283 11.4 Therootlocusplot 287 11.5 Longitudinalstabilityaugmentation 293 11.6 Lateral–directionalstabilityaugmentation 300 11.7 Thepoleplacementmethod 311 References 316 Problems 316 12. Aerodynamicmodelling 320 12.1 Introduction 320 12.2 Quasi-staticderivatives 321 12.3 Derivativeestimation 323 12.4 Theeffectsofcompressibility 327 12.5 Limitationsofaerodynamicmodelling 335 References 336 13. Aerodynamicstabilityandcontrolderivatives 337 13.1 Introduction 337 13.2 Longitudinalaerodynamicstabilityderivatives 337 13.3 Lateral–directionalaerodynamicstabilityderivatives 350 13.4 Aerodynamiccontrolderivatives 371 13.5 NorthAmericanderivativecoefficientnotation 377 References 385 Problems 385 Prelims-H6927.tex 11/7/2007 11:13 Pageviii viii Contents 14. CourseworkStudies 390 14.1 Introduction 390 14.2 Workingtheassignments 390 14.3 Reporting 390 Assignment1. StabilityaugmentationoftheNorthAmericanX-15 hypersonicresearchaeroplane 391 Assignment2. Thestabilityandcontrolcharacteristicsofaciviltransport aeroplanewithrelaxedlongitudinalstaticstability 392 Assignment3. Lateral–directionalhandlingqualitiesdesignforthe LockheedF-104Starfighteraircraft. 396 Assignment4. AnalysisoftheeffectsofMachnumberonthelongitudinal stabilityandcontrolcharacteristicsoftheLTVA7-A Corsairaircraft 401 Appendices 1 AeroTrim–ASymmetricTrimCalculatorforSubsonic FlightConditions 405 2 DefinitionsofAerodynamicStabilityandControlDerivatives 412 3 AircraftResponseTransferFunctionsReferredtoAircraftBodyAxes 419 4 Units,ConversionsandConstants 425 5 AVeryShortTableofLaplaceTransforms 426 6 TheDynamicsofaLinearSecondOrderSystem 427 7 NorthAmericanAerodynamicDerivativeNotation 431 8 ApproximateExpressionsfortheDimensionless AerodynamicStabilityandControlDerivatives 434 9 TheTransformationofAerodynamicStabilityDerivativesfroma BodyAxesReferencetoaWindAxesReference 438 10 TheTransformationoftheMomentsandProductsofInertiafrom aBodyAxesReferencetoaWindAxesReference 448 11 TheRootLocusPlot 451 Index 457 Prelims-H6927.tex 11/7/2007 11:13 Pageix Preface to the first edition WhenIjoinedthestaffoftheCollegeofAeronauticssomeyearsagoIwaspresented withawellworncollectionoflecturenotesandinvitedtoteachAircraftStabilityand Controltopostgraduatestudents.Inspectionofthenotesrevealedtheunmistakable signsthattheirrootsreachedbacktotheworkofW.J.Duncan,whichisperhapsnot surprisingsinceDuncanwasthefirstProfessorofAerodynamicsatCranfieldsome 50yearsago.Itisundoubtedlyaprivilegeand,atfirst,wasverydauntingtobegiven the opportunity to follow in the footsteps of such a distinguished academic. From thathumblebeginningmyinterpretationofthesubjecthascontinuouslyevolvedto itspresentformwhichprovidedthebasisforthisbook. Theclassicallinearisedtheoryofthestabilityandcontrolofaircraftistimeless,it isbrilliantinitsrelativesimplicityanditisverysecurelyanchoredinthedomainof theaerodynamicist.Sowhatisnew?Theshortansweris;notagreatdeal.However, todaythematerialisusedandappliedinwaysthathavechangedconsiderably, due largelytotheadventofthedigitalcomputer.Thecomputerisusedastheprincipal toolforanalysisanddesign,anditisalsotheessentialcomponentofthemodernflight controlsystemonwhichalladvancedtechnologyaeroplanesdepend.Itisthelatter developmentinparticularwhichhashad, andcontinuestohave, amajorinfluence onthewayinwhichthematerialofthesubjectisnowused.Itisnolongerpossible toguaranteegoodflyingandhandlingqualitiessimplybytailoringthestabilityand controlcharacteristicsofanadvancedtechnologyaeroplanebyaerodynamicdesign alone.Flightcontrolsystemsnowplayanequallyimportantpartindeterminingthe flyingandhandlingqualitiesofanaeroplanebyaugmentingthestabilityandcontrol characteristicsoftheairframeinabeneficialway.Thereforethesubjecthashadto evolveinordertofacilitateintegrationwithflightcontroland,today,theintegrated subjectismuchbroaderinscopeandismorefrequentlyreferredtoasFlightDynamics. Thetreatmentofthematerialinthisbookreflectsmypersonalexperienceofusing, applyingandteachingitoveraperiodofmanyyears.Myformativeexperiencewas gained as a Systems Engineer in the avionics industry where the emphasis was on thedesignofflightcontrolsystems. Inmorerecentyears, inadditiontoteachinga formalcourseinthesubject, Ihavebeenprivilegedtohavespentverymanyhours teaching the classical material in the College of Aeronautics airborne laboratory aircraft.Thisexperiencehasenabledmetodevelopthematerialfromtheclassical treatmentintroducedbyDuncanintheearliestdaysoftheCollegeofAeronauticsto thepresenttreatment,whichisbiasedtowardsmodernsystemsapplications.However, thevitallyimportantaerodynamicoriginsofthematerialremainclearandforwhich Icantakenocredit. Modern flight dynamics tends be concerned with the wider issues of flying and handlingqualitiesratherthanwiththetraditional,andmorelimited,issuesofstability ix