1 J I H Dynamics of Flight Stability and Control &mIl I"EINYESTmACttl, AV,ua A ttl In L p,N . LIOTEOA ElECrJlUCI THIRDEDITION Dynamics of Flight Stability and Control BERNARD ETKIN University Professor Emeritus Institute for Aerospace Studies University of Toronto LLOYD DUFF REID Professor Institute for Aerospace Studies University of Toronto CINVSTAV 'PH ADQUISICION DELIBROS GUTt' 1t4'fESTl'tAt*l,., ESTUf!JleSAVHaAUt'IH LP.N . IIlLIOTEOA ELECTP'tICI JOHN WILEY &SONS, INC. ACQUISITIONS EDITORCliff Robichaud ASSISTANT EDITORCatherine Beckham SENIOR PRODUCTION EDITORCathy Ronda COVER DESIGNERLynn Rogan MANUFACTURING MANAGER ILLUSTRATION COORDINATOR Susan Stetzer Gene Aiello This book was set in Times Roman by General Graphic Services, and printed and bound by Hamilton Printing Company. The cover was printed by Hamilton Printing Company. Recognizing the importance of preserving what has been written, it isa policy of John Wiley & Sons, Inc. to have books of enduring value published in the United States printed on acid-free paper, and we exert our best efforts to that end. The paper in this book was manufactured by a mill whose forest management programs include sustained yield harvesting of its timberlands.Sustained yield harvesting principles ensure that the number of trees cut each year does not exceed the amount of new growth. CLASIF.: ADQUIS.:iFECHA:B - "T - .2 00'6--PROCED..('r:;;;.""Ip

$ Copyright 1996, by John Wiley &Sons; Inc. AHrights reserved. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic,photocopying, scanning or otherwise, except aspermitted under Sections107or 108 of the 1976 United States Copyright Act, without either the prior writtenoflhe Piibllsher; or authorization through payment of the appropriate per-copy feeto the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600.Requeststo the Publisher for permission should be addressed to the Permissions Department, John Wiley &Sons, Inc.,111River Street, Hoboken, NJ 07030, (201) 748-6011, fax(201) 748-6008. To order books or for customer service please, call1(800)-CALL-WILEY (225-5945). Library of Congress Cataloging-in-Publication Data Etkin, Bernard. Dynamics of flight:stability and control! Bernard Etkin, Lloyd Duff Reid.-3rd ed. p.cm. Includes bibliographical references (p. ). ISBN 0-471-03418-5(cloth: alk. paper) 1.Aerodynamics.2.Stability of airplanes.I.Reid, Lloyd D. II. Title. TL570.E751995 629. 132'3--dc20 Printed in the United States of America 10987 95-20395 CIP Tothe men and women of science and engineering whose contributions to aviation have made it a dominant forcein shaping the destiny of mankind,and who,with sensitivity and concern,develop and apply their technological arts toward bettering the future. eUTI' INYfSHIACM,Il IlL L P.N . ;IILIOTECJA l\\lf'!fNtEftIAELECTRICI PREFACE The first edition of thisbook appeared in1959-indeed before most students reading thiswere born.It waswell received both bystudentsandpracticingaeronauticalen-gineersof thatera.Thepaceof developmentinaerospaceengineeringduringthe decadethatfollowedwasextremelyrapid,andthiswasreflectedinthesubjectof flightmechanics.Thefirstauthorthereforesawtheneedatthetimeforamoread-vancedtreatmentof thesubjectthatincludedtherealityof theroundrotatingEarth and the real unsteady atmosphere,and hypersonic flight,and that reflected the explo-sive growth in computing power that wasthen taking place(and hasnot yet ended!). The result wasthe1972 volumeentitled Dynamics of Atmospheric Flight.That treat-mentmadenoconcessionstotheneedsof undergraduatestudents,butattempted rathertoportraythestateof theartof flightmechanicsasitwasthen.Tomeetthe needs of students,a second edition of the1959 book was later published in1982. It is thatvolumethatwehaverevisedin thepresent edition,although ina numberof de-tailswe have preferred the1972 treatment,and used it instead. Wehaveretained thesamephilosophyasin thetwopreceding editions.That is, we have emphasized basic principles rooted inthe physics offlight,essential analyti-caltechniques,andtypicalstabilityand controlrealities.Wecontinuetobelieve,as stated in the preface to the1959 edition, that thisisthe preparation that students need tobecomeaeronauticalengineerswhocanfacenewandchallengingsituationswith confidence. Thisedition improveson itspredecessors inseveral ways. It usesa real jet trans-port(theBoeing747)formanynumericalexamplesandincludesexercisesforstu-dentstowork inmostchapters.Welearned froma surveyof teachersof thissubject that the latter wasa sine qua non. Workingout these exercisesisanimportant part of acquiringskillinthesubject.Moreover,somedetailsinthetheoreticaldevelopment have been moved totheexercises,and it isgood practice inanalysisforthestudents todothese. Studentstakingacourseinthissubject areassumedtohaveagood background in mathematics,mechanics,andaerodynamics,typicalof a modem universitycourse inaeronauticaloraerospaceengineering.Consequently,mostof thisbasicmaterial has been moved toappendicesso asnot to interrupt the flowof the text. Thecontentof Chapters1 through3 isverysimilar tothatof thepreviousedi-tion.Chapter4,however,dealingwiththeequationsof motion,containstwovery significant changes.Wehavenot presented thenondimensional equationsof motion, but have left them in dimensional form to conform with current practice,and we have expressed theequationsinthestatevectorformnowcommonlyused.Chapter 5,on stabilityderivatives,isalmostunchangedfromthesecondedition,andChapters6 and7dealingwithstabilityandopenloopresponse,respectively,differfromtheir predecessorsmainlyintheuseof theB747asexampleand intheuseof thedimen-sionalequations.Chapter8,ontheother hand,onclosed loopcontrol,isverymuch expanded and almost entirely new.This isconsistent with themuch enhanced impor-tanceof automaticflightcontrolsystemsinmodemairplanes.Webelievethatthe vii d viiiPreface student who works through this chapter and does the exercises will have a good grasp of the basics of thissubject. Theappendicesof aerodynamicdatahavebeenretainedasusefulmaterialfor teachersand students.Thesame caveats applyasformerly.The data arenot intended for design,but only toillustrate ordersof magnitude and trends. They are provided to give students and teachers ready access tosome data to use in problems and projects. Weacknowledge with thanks the assistance of our colleague,Dr.J.H.de Leeuw, whoreviewedthemanuscriptof Chapter8andmadeanumberof helpfulsugges-tions. Ona personal note-as the firstauthor isnowin the11 thyear of hisretirement, thiswork would not havebeen undertaken had LloydReid not agreedtocollaborate in the task,and if Maya Etkin had not encouraged her husband totake it on andsup-ported him in carrying it out. Intum,thesecondauthor,havingusedthe1959editionasastudent(withthe firstauthorassupervisor),the1972textasaresearcher,andthe1982textasa teacher,wasboth pleased andhonored towork with Bernard Etkin inproducingthis most recent version of the book. Toronto December,1994 Bernard Etkin Lloyd Duff Reid CONTENTS CHAPTER 1Introduction 1.1The Subject Matter of Dynamics of Flight1 1.2The Tools of Flight Dynamicists5 1.3Stability, Control, and Equilibrium6 1.4The Human Pilot8 1.5Handling Qualities Requirement&"11 1.6Axes and Notation15 CHAPTER 2Static Stability and Control-Part 1 2.1General Remarks18 2.2Synthesis of Lift and Pitching Moment23 2.3Total Pitching Moment and Neutral Point29 2.4Longitudinal Control33 2.5The Control Hinge Moment41 2.6Influence of a Free Elevator on Lift and Moment44 2.7The Use of Tabs47 2.8Control Force to Trim48 2.9Control Force Gradient51 2.10Exercises52 2.11Additional Symbols Introduced in Chapter 257 CHAPTER 3Static Stability and Control-Part 2 3.1Maneuverability-Elevator Angle per g60 3.2Control Force per g63 3.3Influence of High-Lift Devices on Trim and Pitch Stiffness64 3.4Influence of the Propulsive System on Trim and Pitch Stiffness66 3.5Effect of Structural Flexibility72 3.6Ground Effect74 3.7CG Limits74 3.8Lateral Aerodynamics76 3.9Weathercock Stability (YawStiffness)77 3.10YawControl80 3.11Roll Stiffness81 3.12The Derivative C1/l83 3.13Roll Control86 3.14Exercises89 3.15Additional Symbols Introduced in Chapter 391 CHAPTER 4General Equations of Unsteady Motion 4.1General Remarks93 4.2The Rigid-Body Equations93 ix---_...- . ~ . - - - ..- . - - - . ~ - - - - - - - - - - - - - -xContents 4.3Evaluation of the Angular Momentum h96 4.4Orientation and Position of the Airplane98 4.5Euler's Equations of Motion100 4.6Effect of Spinning Rotors on the Euler Equations103 4.7The Equations Collected103 4.8Discussion of the Equations104 4.9The Small-Disturbance Theory107 4.10The Nondimensional System115 4.11Dimensional Stability Derivatives118 4.12Elastic Degrees of Freedom120 4.13Exercises126 4.14Additional Symbols Introduced in Chapter 4127 CHAPTER 5TheStability Derivatives 5.1General Remarks129 5.2The IXDerivatives129 5.3The u Derivatives131 5.4The q Derivatives135 5.5The eXDerivatives141 5.6The f3Derivatives148 5.7The pDerivatives149 5.8The r Derivatives153 5.9Summary of the Formulas154 5.10Aeroelastic Derivatives156 5.11Exercises159 5.12Additional Symbols Introduced in Chapter 5160 CHAPTER 6Stability of Uncontrolled Motion 6.1Form of Solution of Small-Disturbance Equations161 6.2Longitudinal Modes of a Jet Transport165 6.3Approximate Equations for the Longitudinal Modes171 6.4General Theory of Static Longitudinal Stability175 6.5Effect of Flight Condition on the Longitudinal Modes of a Subsonic Jet Transport177 6.6Longitudinal Characteristics of a STOL Airplane184 6.7Lateral Modes of a Jet Transport187 6.8Approximate Equations for the Lateral Modes193 6.9Effects of Wind196 6.10Exercises201 6.11Additional Symbols Introduced in Chapter 6203 CHAPTER 7Response to Actuation of the Controls-Open Loop 7.1General Remarks204 7.2Response of LinearlInvariant Systems207 7.3Impulse Response210 \ tit Contentsxi 7.4Step-Function Response213 7.5Frequency Response214 7.6Longitudinal Response228 7.7Responses to Elevator and Throttle229 7.8Lateral Steady States237 7.9Lateral Frequency Response243 7.10Approximate Lateral Transfer Functions247 7.11Transient Response to Aileron, and Rudder252 7.12Inertial Coupling in Rapid Maneuvers256 7.13Exercises256 7.14Additional Symbols Introduced in Chapter 7258 CHAPTER 8Closed-Loop Control 8.1General Remarks259 8.2Stability of Closed Loop Systems264 8.3Phugoid Suppression: Pitch Attitude Controller266 8.4Speed Controller270 8.5Altitude and Glide Path Control275 8.6Lateral Control280 8.7Yaw Damper287 8.8Roll Controller290 8.9Gust Alleviation295 8.10Exercises300 8.11Additional Symbols Introduced in Chapter 8301 APPENDIX AAnalytical Tools AlLinear Algebra303 A2The Laplace Transform304 A3The Convolution Integral309 A.4Coordinate Transformations310 ASComputation of Eigenvalues and Eigenvectors315 A6Velocity and Acceleration in an Arbitrarily Moving Frame316 APPENDIX BData for Estimating Aerodynamic Derivatives319 APPENDIX CMean Aerodynamic Chord,Mean Aerodynamic Center, and C357 macw APPENDIX DThe Standard Atmosphere and Other Data364 APPENDIX EData For the Boeing 747-100369 References372 Index377 Tn CHAPTER1 Introduction 1.1TheSubject Matter of Dynamics of Flight Thisbook isaboutthemotionof vehiclesthat flyintheatmosphere.Assuchit be-longstothebranch of engineeringscience called appliedmechanics.The threeitali-cized wordsabove warrant further discussion. Tobegin with fly-the dictionary defi-nitionisnotveryrestrictive,althoughit impliesmotionthroughtheair,theearliest applicationbeingof course tobirds.However,wealsosay"a stoneflies"or "an ar-rowflies,"sothenotionof sustention(lift)isnotnecessarilyimplied.Eventheat-mospheric medium islost in "the flight of angels." Wepropose asa logicalscientific definitionthatflyingbedefinedasmotionthrougha fluidmediumor emptyspace. Thusa satellite "flies" through space and a submarine "flies" through the water.Note thata dirigible in theair anda submarine in thewater are thesame froma mechani-calstandpoint-the weight in each instanceisbalanced bybuoyancy.Theyaresim-ply separated by three ordersof magnitude in density.By vehicle ismeant anyflying object that ismade upof anarbitrarysystem of deformable bodies that aresomehow joinedtogether.Toillustratewithsomeexamples:(1)Ariflebulletisthesimplest kind,whichcan bethoughtof asasingleideallyrigidbody.(2)A jet transportisa morecomplicatedvehicle,comprisingamainelasticbody(theairframeandallthe parts attached toit),rotatingsubsystems (the jet engines), articulated subsystems (the aerodynamic controls)and fluidsubsystems (fuel in tanks).(3) An astronaut attached tohis orbiting spacecraft by a long flexible cable isa further complex example of this general kind of system.Note that by the above definition a vehicle doesnot necessar-ily have tocarry goodsor passengers,although it usuallydoes.The logic of the defi-nitionsissimplythat theunderlyingengineeringscience iscommon toalltheseex-amples,and themethods of formulatingand solving problems concerning the motion are fundamentally the same. Asisusual with definitions,wecan findexamplesthat don't fitverywell.There arespecial casesof motionataninterfacewhichwemayor maynot include in fly-ing-for example,ships,hydrofoilcraftandair-cushionvehicles(ACV's).Inthis connectionitisworthnotingthatdevelopmentsof hydrofoilsandACV'sarefre-quentlyassociatedwiththeAerospaceindustry.Themaindifferencebetweenthese cases,andthoseof "true"flight,isthatthelatterisessentiallythree-dimensional, whereas' the interface vehicles mentioned (aswellascars,trains,etc.)moveapproxi-matelyinatwo-dimensionalfield.Theunderlyingprinciplesandmethodsarestill thesamehowever,withcertainmodificationsindetailbeingneededtotreatthese "surface" vehicles. Now having defined vehiclesand flying,we goon to look more carefully at what we mean by motion.It isconvenient tosubdivide it into several parts: 1 2Chapter 1.Introduction Aerodynamics Mechanics of ---Vehicle rigidbodiesdesign Mechanicsof ~ FLIGHTVehicle elastic structuresDYNAMICSoperation Humanpilot ~ Pilot dynamicstraining Appliedmathematics, machinecomputation ~~ PerformanceStability andAeroelasticity Navigationand (trajectory,control(handling(control,structural guidance maneuverability)qualities,airloads)integrity) Figure 1.1Block diagram of disciplines. Gross Motion: 1.Trajectory of the vehicle mass center. 1 2.''Attitude'' motion, or rotations of the vehicle "as a whole." Fine Motion: 3.Relativemotionof rotatingor articulatedsubsystems,suchasengines,gyro-scopes, or aerodynamic control surfaces. 4.Distortional motion of deformable structures, such as wing bending and twist-ing. 5.Liquid sloshing. This subdivision is helpful both from the standpoint of the technical problems as-sociatedwiththedifferentmotions,andof theformulationof theiranalysis.It is surely self-evident that studies of these motions must be central to the design and op-eration of aircraft,spacecraft, rockets, missiles, etc. To be able to formulateand solve therelevantproblems,wemustdrawonseveralbasicdisciplinesfromengineering science.The relationshipsareshown on Fig.1.1.It isquiteevident fromthisfigure that the practicing flight dynamicist requires intensive training in several branches of engineeringscience,anda broad outlook insofar asthe practical ramificationsof his work are concerned. Intheclassesof vehicles,inthetypesof motions,and in themedium of flight, this book treatsa very restricted set of all possible cases. It dealsonly with the flight lIt is assumed that gravity is uniform,and hence that the mass center and center of gravity (CG)are the same point. 1.1The Subject Matter ofDynamics of Flight3 of airplanes in the atmosphere. The general equations derived, and the methods of so-lutionpresented,arehoweverreadilymodifiedandextendedtotreatmanyof the other situations that are embraced by the general problem. All the fundamentalscience and mathematicsneeded todevelopthissubject ex-isted in theliterature by the time the Wright brothers flew.Newton,and other giants of the18th and19th centuries,such asBernoulli, Euler,Lagrange,and Laplace,pro-vided the building blocks in solid mechanics,fluidmechanics,and mathematics. The needed applications toaeronautics were made mostlyafter1900 by workers in many countries,of whomspecialreferenceshouldbemadetothe Wright brothers,G.H. Bryan, F.W.Lanchester, J.C.Hunsaker, H. B.Glauert, B.M.Jones,and S.B.Gates. Thesepioneersintroducedandextendedthebasisforanalysisandexperimentthat underlies all modern practice? This body of knowledge is well documented in several textsof thatperiod,forexample,Bairstow(1939).Concurrently,principallyinthe United States of America and Britain, a large body of aerodynamic data was accumu-lated,serving asa basis for practical design. .Newton'slawsof motionprovidetheconnectionbetweenenvironmentalforces and resulting motion for all but relativistic and quantum-dynamical processes, includ-ing all of "ordinary" and much of celestial mechanics. What then distinguishesflight dynamicsfromother branchesof appliedmechanics?Primarilyitisthespecialna-ture of the forcefieldswith which we have to be concerned,the absence of the kine-matical constraints central to machines and mechanisms,and the nature of the control systems used in flight.The external force fieldsmay be identified asfollows: "Strong" Fields: 1.Gravity 2.Aerodynamic 3.Buoyancy "Weak" Fields: 4.Magnetic 5.Solar radiation Weshould observe that two of these fields,aerodynamic and solar radiation, pro-duce important heat transfer tothe vehicle in addition tomomentum transfer (force). Sometimeswecannotseparatethethermalandmechanicalproblems(Etkinand Hughes,1967).Of thesefieldsonlythestrongonesareof interestforatmospheric andoceanicflight,theweakfieldsbeingimportantonlyinspace.It shouldbere-markedthatevenin atmosphericflightthegravityforcecan notalwaysbeapproxi-mated asa constant vector in aninertial frame.Rotationsassociatedwith Earth cur-vature,and theinversesquare law,becomeimportant incertain casesof high-speed and high-altitude flight (Etkin,1972). Theprediction,measurementandrepresentationof aerodynamicforcesarethe principal distinguishing featuresof flight dynamics. The size of this task is illustrated 2Anexcellentaccountof theearlyhistoryisgiveninthe1970vonKarmanLecturebyPerkins (1970). 4Chapter 1.Introduction Parametersofwing aerodynamics SHAPE: o SPEED:I Sections Subsonic 1.0 I IncompressibleTransonic MOTION:Constant velocity [u,v,w,p, q,r) = const Wings ~ ~ 5.0 I>M SupersonicHypersonic Variablevelocity [u(t),v(t),w(t),p(t),q(t),r(t)) ATMOSPHERE:""1------rl------,I ContinuumSlipFree-molecule I Uniform and at rest I Nonuniform and at rest (reentry) I Uniform and inmotion (gusts) I Nonuniform and inmotion Figure 1.2Spectrum of aerodynamic problems for wings. by Fig.1.2,which showstheenormous rangeof variablesthat need tobe considered inconnectionwithwingsalone.Tobeadded,of course,arethecomplicationsof propulsion systems (propellers, jets, rockets), compound geometries (wing+ body+ tail), and variable geometry (wing sweep, camber). Asremarkedabove,Newton'slawsstatetheconnectionbetweenforceandmo-tion.Thecommonestproblemconsistsof findingthemotionwhenthelawsforthe forcesaregiven(allthenumericalexamplesgiveninthisbookareof thiskind). However, we must be aware of certain important variations: 1.Inverseproblemsof firstkind-the systemandthemotionaregivenandthe forceshave to be calculated. 2.Inverse problems of the second kind-the forcesand the motion aregiven and thesystem parameters have tobe found. 3.Mixedproblems-theunknownsareamixtureof variablesfromtheforce, system,and motion. Examplesof theseinverseandmixedproblemsoftentum upinresearch,when one istryingtodeduceaerodynamicforcesfrom theobserved motion of a vehicle in flight or of a model in a wind tunnel. Another example isthe deduction of harmonics of theEarth'sgravityfieldfromobservedperturbationsof satelliteorbits.These problems are closely related to the "plant identification" or "parameter identification" problemof systemtheory.[InverseproblemsweretreatedinChap.11of Etkin (1959)]. 1.2TheToolsof Flight Dynamicists5 TYPES OF PROBLEMS The main types of flight dynamics problem that occur in engineering practice are: 1.Calculationof "performance"quantities,suchasspeed,height,range,and fuel consumption. 2.Calculation of trajectories, such aslaunch, reentry,orbital and landing. 3.Stability of motion. 4.Response of vehicle to control actuation and to propulsive changes. 5.Response toatmospheric turbulence, and how to control it. 6.Aeroelastic oscillations (flutter). 7.Assessment of human-pilot/machine combination (handling qualities). It takeslittleimaginationtoappreciatethat,inviewof themanyvehicletypes that havetobedealt with,a numberof subspecialtiesexist within theranksof flight dynamicists, related tosome extent tothe above problem categories.In the context of themodemaerospaceindustrytheseproblemsareseldomsimpleor routine.On the contrary they present great challenges in analysis,computation, and experiment. 1.2TheToolsof Flight Dynamicists The toolsused by flightdynamiciststosolve thedesignand operational problemsof vehicles are of three kinds: 1.Analytical 2.Computational 3.Experimental Theanalyticaltoolsareessentiallythesameasthoseusedinotherbranchesof mechanics,that isthemethodsof applied mathematics.One important branch of ap-plied mathematicsiswhatisnowknownassystemtheory,includingstability,auto-matic control,stochastic processes and optimization.Stability of the uncontrolled ve-hicle isneither a necessary nor a sufficient condition forsuccessful controlled flight. Goodairplaneshavehadslightlyunstable modesinsomepart of their flightregime, and on the other hand,a completely stable vehicle may have quite unacceptable han-dlingqualities.It isdynamic peiformancecriteriathatreallymatter,sotoexpenda greatdealof analyticalandcomputationaleffortonfindingstabilityboundariesof nonlinear and time-varying systems may not be really worthwhile.On the other hand, the computation of stability of small disturbances froma steady state,that is,the lin-eareigenvalueproblemthatisnormallypartof thesystemstudy,isveryusefulin-deed,andmaywellprovideenoughinformationaboutstabilityfromapractical standpoint. Onthecomputationside,themostimportantfactisthattheavailabilityof ma-chinecomputationhasrevolutionizedpracticeinthissubjectoverthepastfew decades.Problemsof systemperformance,systemdesign,andoptimizationthat 6Chapter 1.Introduction could not havebeen tackledat allin thepast arenowhandled ona moreor less rou-tine basis. The experimental tools of the flight dynamicist are generally unique to this field. First,therearethosethatareusedtofindtheaerodynamicinputs.Wind tunnelsand shock tubesthatcovermostof thespectrum of atmosphericflightarenowavailable inthemajoraerodynamiclaboratoriesof theworld.Inadditiontofixedlaboratory equipment,thereareaero ballisticrangesfordynamicinvestigations,aswellas rocket-boosted and gun-launched free-flight model techniques. Hand in hand with the developmentof thesegeneralfacilitieshasgonethat of amyriadof sensorsandin-struments,mainly electronic, formeasuring forces,pressures, temperatures,accelera-tion,angularvelocity,andsoforth.Theevolutionof computational fluiddynamics (CFD) hassharply reduced thedependence of aerodynamicistson experiment.Many resultsthatwereformerlyobtainedinwindtunneltestsarenowroutinelyprovided by CFD analyses. The CFD codes themselves, of course, must be verified by compar-ison with experiment. Second,wemustmentiontheflightsimulatorasanexperimentaltooluseddi-rectly by the flight dynamicist.In it hestudies mainly the matching of the pilot tothe machine. This isan essential step for radically new flight situations. The ability of the pilot tocontrol the vehicle must be assured long before the prototype stage. This can-not yet be done without test,although limited progress in this direction isbeing made throughstudiesof mathematical models of human pilots.Special simulators, built for most newmajoraircraft types,provide bothefficient meansforpilot training,anda researchtoolforstudyinghandlingqualitiesof vehiclesanddynamicsof humanpi-lots.Thedevelopmentof high-fidelitysimulatorshasmadeit possibletogreatlyre-duce the time and cost of training pilots to flynew types of airplanes. 1.3Stability,Control,and Equilibrium It isappropriate here to define what is meant by the terms stability and control.Todo so requires that we begin with the concept of equilibrium. A body is in equilibrium when it isat rest or in uniform motion (i.e., has constant linearandangularmomenta).Themostfamiliarexamplesof equilibriumarethe static ones;that is, bodies at rest. The equilibrium of anairplane in flight,however, is of thesecond kind;that is,uniform motion.Because theaerodynamicforcesarede-pendentontheangularorientationof theairplanerelativetoitsflightpath,and be-causetheresultantof themmust exactlybalanceitsweight,theequilibriumstateis without rotation; that is,it isa motion of rectilinear translation. Stability,or the lack of it,isa propertyof anequilibrium state.3 The equilibrium isstable if,when the body is slightly disturbed in anyof its degreesof freedom,it re-turnsultimatelytoitsinitialstate.ThisisillustratedinFig.1.3a.Theremaining sketchesof Fig.1.3showneutralandunstableequilibrium.ThatinFig.1.3disa more complex kind than that in Fig.1.3b in that the ball isstable with respect todis-placement in the y direction, but unstable with respect to xdisplacements. This has its counterpart intheairplane,whichmaybestablewithrespecttoonedegreeof free-domandunstablewithrespecttoanother.Twokindsof instabilityareof interest in 3It is also possible to speak of the stability of a transient with prescribed initial condition. _o__o_____~ ~ ~ ~___~______~___~____"\II 1.3Stability,Control,and Equilibrium7 Figure 1.3(a)Ball in a bowl-stable equilibrium.(b)Ball ona hill-unstable equilibrium.(c) Ball on a plane-neutral equilibrium. (d)Ball on a saddle surface-unstable equilibrium. airplane dynamics.In thefirst,called static instability,the bodydepartscontinuously from itsequilibrium condition.That ishowtheball in Fig.1.3b would behave if dis-turbed. The second,called dynamic instability,isa more complicated phenomenon in which the body oscillates about itsequilibrium condition with ever-increasing ampli-tude. Whenapplyingtheconceptof stabilitytoairplanes,therearetwoclassesthat mustbeconsidered-inherent stabilityandsyntheticstability.Thediscussionof the previousparagraphimplicitlydealtwithinherentstability,whichisapropertyof thebasicairframewitheither fixedor freecontrols,thatis,control-fixed stability or control-freestability.On the other hand,syntheticstability isthat provided by an au-tomatic flightcontrolsystem(AFCS)andvanishesif thecontrolsystemfails.Such automaticcontrolsystemsarecapableof stabilizinganinherentlyunstableairplane, orsimplyimprovingitsstabilitywithwhatisknownasstabilityaugmentationsys-tems(SAS).The questionof howmuch torelyonsuchsystemstomakeanairplane flyableentailsatrade-offamongweight,cost,reliability,andsafety.If theSAS worksmost of thetime,andif theairplanecan be controlledandlanded afterit has failed,albeitwithdiminished handlingqualities,then poor inherentstabilitymaybe acceptable.Current aviation technology shows an increasing acceptance of SAS in all classes of airplanes. If theairplane iscontrolled bya human pilot,some mild inherent instability can betolerated,if it issomethingthepilot can control,suchasa slowdivergence.(Un-stablebicycleshavelongbeenriddenbyhumans!).Ontheotherhand,thereisno , 1,18Chapter 1.Introduction marginforerrorwhentheairplaneisunderthecontrolof anautopilot,forthenthe closed loopsystem must bestable in itsresponse toatmosphericdisturbancesandto commands that come from a navigation system. Inaddition totherolecontrolsplay instabilizinganairplane,therearetwooth-ersthat are important. The first is to fixor to change the equilibrium condition (speed orangleof climb).Anadequatecontrolmustbepowerfulenoughtoproducethe wholerangeof equilibriumstatesof whichtheairplaneiscapablefromaperfor-mancestandpoint.Thedynamicsof thetransitionfromoneequilibriumstatetoan-otherareof interestandarecloselyrelatedtostability.Thesecondfunctionof the controlistoproducenonequilibrium,oracceleratedmotions;thatis,maneuvers. Thesemaybesteadystatesinwhichtheforcesandaccelerationsareconstant when viewedfroma referenceframefixedtotheairplane(forexample,asteadyturn),or theymaybetransientstates.Investigationsof thetransitionfromequilibriumtoa nonequilibriumsteadystate,orfromonemaneuveringsteadystatetoanother,form part of thesubject matter of airplane control.Verylargeaerodynamicforcesmayact ontheairplanewhenit maneuvers-a knowledgeof theseforcesisrequiredforthe proper design of the structure. RESPONSE TO ATMOSPHERIC TURBULENCE A topic that belongs in dynamicsof flight and that isclosely related tostability isthe responseof theairplane towind gradientsandatmosphericturbulence(Etkin,1981). This response isimportant fromseveral points of view.It hasa strong bearing on the adequacyof thestructure,onthesafetyof landingandtake-off,ontheacceptability of the airplane asa passenger transport,and on its accuracy asa gun or bombing plat-form. 1.4The HumanPilot Althoughtheanalysisandunderstandingof thedynamicsof theairplaneasaniso-latedunitisextremelyimportant,onemustbecarefulnottoforgetthatformany flightsituations it isthe response of the total.system, made upof the human pilot and theaircraft,that must beconsidered.It isforthisreason that thedesignersof aircraft shouldapplythefindingsof studiesintothehumanfactorsinvolvedinordertoen-sure that the completed system iswell suited to the pilots who must flyit. Some of the areas of consideration include: 1.Cockpitenvironment;theoccupantsof thevehiclemustbeprovidedwith oxygen, warmth, light,and so forth,tosustain them comfortably. 2.Instrument displays;instrumentsmust bedesignedandpositioned toprovide a useful and unambiguous flowof information tothe pilot. 3.Controlsandswitches;thecontrolforcesandcontrolsystemdynamicsmust be acceptable to the pilot, and switches must be so positioned and designed as toprevent accidental operation. Tables1.1to1.3present some pilot data con-cerning control forces. 4.Pilot workload;theworkload of the pilot can often be reduced through proper planning and the introduction of automatic equipment. b 1.4The Human Pilot9 Table 1.1 Estimates of the Maximum Rudder Forces that Can Be Exerted for Various Positions of the Rudder Pedal (BuAer, 1954) Rudder Pedal PositionDistance from Back of SeatPedal Force (in)(cm)(tb)(N) Back31.0078.742461,094 Neutral34.7588.274241,886 Forward38.5097.793341,486 Table 1.2 Hand-Operated Control Forces (From Flight Safety Foundation Human Engineering Bulletin 56-5H) (see figure in Table 1.3) Direction ofMovement1801501209060 Rt.hand525642 37c ~ ~ 24 (231)(249)(187)(165).(107) Pull Lft. hand5042343226 (222)(187)(151)(142)(116) Rt.hand5042363634 (222)(187)(160)(160)(151) Push Lft. hand4230262222 (187)(133)(116)(98)(98) Rt.hand1418242020Values given (62)(80)(107)(89)(89)represent Up maximum Lft. hand915171715 exertable (40)(67)(76)(76)(67) force in Rt. hand1720262620 pounds (76)(89)(116)(116)(89) (Newtons) Down by the 5 Lft. hand1318212118 percentile (58)(80)(93)(93)(80) man. Rt.hand1415151617 (62)(67)(67)(71)(76) Outboard Lft. hand88101012 (36)(36)(44)(44)(53) Rt.hand2020221820 (89)(89)(98)(80)(89) Inboard Lft. hand1315201617 (58)(67)(89)(71)(76) Note:The above results are those obtained from unrestricted movement of the subject. Any force required to overcome garment restriction would reduce the effective forces by the same amount.---... ----------- -. 10Chapter 1.Introduction DIRECTIONOFMOVEMENT Vert.ref.line 180 90 Outboard t Inboard t Outboard Table 1.3 Rates of Stick Movement in Flight Test Pull-ups Under Various Loads (BuAer, 1954) Pull-up 2 3 4 Maximum Stick Load (lb)(N) 35156 74329 77343 97431 Average Rate of StickTime for Full MotionDeflection (inls)(cm/s)(s) 51.85131.700.162 15.5839.570.475 11.0027.940.600 10.2726.090.750 b 1.5Handling Qualities Requirements11 Thecareexercisedinconsideringthehumanelement intheclosed-loopsystem madeupof pilot andaircraftcan determinethesuccessor failureof agivenaircraft design to complete its mission in a safe and efficient manner. Manycriticaltasksperformedbypilotsinvolvetheminactivitiesthatresemble thoseof aservocontrolsystem.Forexample,theexecutionof alandingapproach through turbulent air requires the pilot tomonitor the aircraft's altitude, position, atti-tude,andairspeedandtomaintainthesevariablesneartheirdesiredvaluesthrough theactuation of the control system.It hasbeen foundinthistype of controlsituation that the pilot can be modeled by a linear control system based either on classical con-troltheoryoroptimalcontroltheory(Etkin,1972;KleinmanetaI.,1970;McRuer and Krendel,1973). 1.5Handling Qualities Requirements Asaresultof theinabilitytocarryoutcompletelyrationaldesignof thepilot-machinecombination,itiscustomaryforthegovernmentagenciesresponsiblefor the procurement of military airplanes,or forlicensing civil airplanes,tospecify com-pliancewithcertain"handling(or flying)qualitiesrequirements"(e.g.,ICAO,1991; USAF,1980;USAF,1990).Handlingqualities referstothosequalitiesor character-isticsof anaircraftthatgoverntheeaseandprecision withwhichapilotisableto perform the tasks required in support of an aircraft role (Cooper and Harper,1969). Theserequirementshavebeendevelopedfromextensiveandcontinuingflight research.Inthefinalanalysistheyarebasedontheopinionsof researchtestpilots, substantiated by careful instrumentation. They vary from country to country and from agency toagency,and,of course, are different for different typesof aircraft.They are subject tocontinuousstudyand modification in order to keep them abreast of the lat-est research and design information.Because of these circumstances, it is not feasible topresent a detailed description of such requirements here. The following isintended .toshow the nature, not the detail, of typical handling qualities requirements.4 Most of the specific requirements can be classified under one of the following headings. CONTROL POWER Thetermcontrol power isusedtodescribetheefficacyof acontrolinproducinga rangeof steady equilibrium or maneuveringstates.For example,an elevator control, whichbytakingpositionsbetweenfullupandfulldowncanholdtheairplanein equilibrium at allspeedsin itsspeed range,forallconfigurations5 and CG positions, isa powerful control.On the other hand,a rudder that isnot capable at fulldeflection of maintaining equilibrium of yawingmomentsin a condition of one engineout and negligiblesideslipisnot powerfulenough.Thehandling qualitiesrequirementsnor-mally specify thespecificspeed ranges that must beachievable with fullelevator de-4Por a more complete discussion, see AGARD (1959); Stevens and Lewis (1992). 5Thisword describesthepositionof movableelementsof theairplane-for example,landingcon-figurationmeansthat landing flapsandundercarriagearedown,climb configuration meansthat landing gear is up,and flapsare at take-off position, and so forth. 12Chapter 1.Introduction flectioninthevariousimportant configurationsandtheasymmetricpower condition that the rudder must balance. They may also contain references tothe elevator angles requiredtoachievepositiveloadfactors,asinsteadyturnsandpull-upmaneuvers (see "elevator angle per g,"Sec.3.1). CONTROL FORCES The requirements invariablyspecify limitson thecontrol forcesthat must beexerted by thepilot inorder to effect specific changes froma given trimmed condition,or to maintainthetrimspeedfollowingasuddenchangeinconfigurationorthrottleset-ting.They frequentlyalsoincluderequirementsonthecontrolforcesin pull-upma-neuvers(see"control forceper g,"Sec.3.1).Inthecaseof light aircraft,thecontrol forcescan result directly frommechanical linkages between theaerodynamic control surfacesandthepilot'sflightcontrols.Inthiscasethehingemomentsof Sec.2.5 playa direct role in generating these forces.Inheavyaircraft,systemssuch aspartial or total hydraulic boost areused tocounteract the aerodynamic hinge momentsand a relatedorindependentsubsystemisusedtocreatethecontrolforcesonthepilot's flight controls. STATIC STABILITY Therequirementforstaticlongitudinalstability(seeChap.2)isusuallystatedin termsof theneutral point.The neutral point,defined more precisely inSec.2.3,isa special location of thecenter of gravity(CG)of theairplane.Inalimitedsenseit is the boundary between stable and unstable CG positions. It isusually required that the relevantneutralpoint(stickfreeorstickfixed)shallliesomedistance(e.g.,5%of themeanaerodynamicchord)behind themostaftpositionof theCG.Thisensures that the airplane will tend to flyat a constant speed and angle of attack aslongasthe controls are not moved. Therequirementonstaticlateralstabilityisusuallymild.It issimplythatthe spiralmode(seeChap.6)if divergentshall havea timetodoublegreater thansome stated minimum (e.g., 4s). DYNAMIC STABILITY The requirement ondynamicstability istypicallyexpressed intermsof the damping andfrequencyof anaturalmode.ThustheUSAF(1980)requiresthedampingand frequencyof thelateraloscillationforvariousflightphasesandstabilitylevelsto conform tothe values in Table1.4. STALLING AND SPINNING Finally,most requirementsspecifythat theairplane'sbehavior followinga stallor in aspinshallnotincludeanydangerouscharacteristics,andthat thecontrolsmust re-tain enough effectiveness to ensure a safe recovery to normal flight. I L 1.5Handling Qualities Requirements13 Table 1.41 Minimum Dutch RoD Frequency and Damping Flight Phase Min'dwnd' * Minwnd' LevelCategoryClass Min'd* radlsradls AI, IV0.190.351.0 II, III0.190.350.4 1BAll0.080.150.4 CI, II-C, IV0.080.151.0 II-L, III0.080.100.4 2AllAll0.020.050.4 3AllAll0- 0.4 iLevel,Phase and Class are defined in USAF.1980. *Note:The damping coefficientand the undanlped natural frequencyWnare defined in Chap. 6. RATING OF HANDLING QUALITIES Tobe able toassessaircraft handling qualitiesonemust havea measuring technique with which any given vehicle's characteristics can beIn the early days of avia-tion,thiswasdone by solicitingthecomments of pilotsafter they had flowntheair-craft.However,it wassoon found that a communications problem existed with pilots usingdifferentadjectivestodescribethesameflightcharacteristics.Theseambigui-ties have been alleviated considerably by the introduction of a uniform set of descrip-tive phrases by workers in the field.The most widely accepted set is referred to asthe "Cooper-Harper Scale," where a numerical rating scale is utilized in conjunction with aset of descriptivephrases.Thisscaleispresented in Fig.1.4.Toapplythisrating technique it is necessary to describe accurately the conditions under which the results were obtained.In addition it should be realized that the numerical pilot rating (1-10) is merelyashorthand notation forthe descriptive phrasesandassuch nomathemati-cal operations can be carried out on them in a rigoroussense.For example, a vehicle configuration rated as6 should not be thought to be "twice asbad"asone rated at 3. The commentsfromevaluation pilotsareextremely usefuland thisinformation will provide the detailed reasons for the choice of a rating. Other techniques have been applied to the rating of handlingFor exam-ple,attemptshavebeen madetousetheoverallsystem performanceasaratingpa-rameter.However,due tothepilot'sadaptivecapability,quite often he can cause the overallsystem responseof a bad vehicletoapproach that of a good vehicle,leading tothesame performance but vastlydiffering pilot ratings.Consequentlysystem per-formancehasnot proved tobe agood rating parameter. Amore promising approach involvesthemeasurement of thepilot'sphysiologicalandpsychologicalstate.Such methodslead toobjectiveassessmentsof howthesystemisinfluencingthehuman controller.The measurement of human pilot describing functionsis part of this tech-nique (Kleinman et al.,1970; McRuer and Krendel,1973; Reid,1969). Researchintoaircrafthandlingqualitiesisaimedinpartatascertainingwhich vehicle parameters influence pilot acceptance. It isobvious that the number of possi-.... ,J;. ( "-ADEOUACY FORSELECTEDTASK OR (AIRCRAFT CHARACTERISTICS DEMANDSONTHEPILOT REQUIREDOPERATION' INSELECTED TASKORREQUIREDOPERATION* Excellent Pilot compensationnot a factor for Highly desirabledesiredperformance Good Pilot compensationnot a factor for Negligible deficienciesdesiredperformance Fair - Some mildly Minimal pilot compensationrequired for unpleasant deficienciesdesired performance Yes Minor but annoying Desiredperformance requiresmoderate deficienciespilot compensation Isit No Deficiencies satisfactory without warrant-Moderately objectionable Adequate performancerequiresconsiderable improvement?deficienciespilot compensation improvement Very objectionable but Adequate performancerequires extensive tolerabledeficienciespilot compensation Voo Adequate performance not attainable with Major deficiencies maximum tolerable pilot compension. Deficiencies Controllability not inquestion. No / require- Considerable pilot compensationisrequired Major deficiencies improvement for control Major deficiencies Intense pilot compensationisrequired to retaincontrol No ./ Improvement Major deficiencies Control willbelost during some portion of -::::mandatory requiredoperation. Figure 1.4Handling qualities ratingscale; Cooper/Harper scale (Cooper and Harper,1969). PILOT RATING 1 2 3 4 5 6 7 8 9 10 - - - - - - - ~ - - - - - - - - - - ~ - ~ - - - - - - - - - - - - - - - - - - . - - - - .. ---6.0 1.0 Initialresponsefast, tendency to oscillationand toovershoot loads Slowinitially, thenoscillatory, tendencytooven:ontol Unacceptable Initialresponsefast, oversensitive,light slickforces ....... --...., Acceptable\ Veryslowresponse, largecontrolmotion tomaneuver, difficult totrim 1.6Axes and Notation15 Sluggish,large slickmotion andforces O ~ - - ~ - - - - - - ~ ~ - - ~ ~ - - ~ ~ ~ ~ ~ 0.10.5.1.02.03.04.0 Damping ratio, r Figure 1.5Longitudinal short-period oscillation-pilot opinion contours (O'Hara,1967). hIe combinations of parameters isstaggering,and consequently attemptsare made to studyone particular aspect of the vehiclewhile maintainingall othersina "satisfac-tory"configuration.Thusthetaskisformulatedinafashionthatisamenableto study.Therisk involved in thistechniqueisthat important interaction effectscan be overlooked.For example,it isfoundthat the degree of difficultya pilot findsin con-trollingan aircraft's lateral-directional mode influences his ratingof the longitudinal dynamics.Suchfactsmust betakenintoaccountwheninterpretingtest results.An-otherpossiblebiasexistsinhandlingqualitiesresultsobtainedinthepastbecause mostof theworkhasbeendoneinconjunctionwithfighteraircraft.Thefindings fromsuch research can often be presented as"isorating"curvessuch asthoseshown in Fig.1.5. 1.6Axes and Notation In this book the Earth is regarded asflatand stationary in inertial space. Anycoordi-natesystem,or frameof reference,attached totheEarth isthereforeaninertialsys-tem,oneinwhichNewton'slawsarevalid.Clearlyweshallneedsuchareference frame when we come to formulate the equations of motion of a flight vehicle. Wede-note that frameby FE(OE,XE'YE,ZE)'Itsorigin isarbitrarily located tosuit thecircum-stancesof the problem,the axis0EZE points vertically downward,and theaxis0EXE, whichishorizontal,ischosentopointinanyconvenientdirection,forexample, North,or alonga runway,or insome reference flightdirection.It isadditionallyas-sumed that gravityisuniform,andhencethatthemasscenterandcenter of gravity (CO)are the same point. The location of the CO is given by its Cartesian coordinates relativetoFE'ItsvelocityrelativetoFEisdenoted VE andisfrequentlytermedthe groundspeed. 16Chapter 1.Introduction tN,r Z,W II Figure 1.6Notation for body axes. L= rolling moment, M= pitching moment, N= yawing moment, p= rate of roll, q= rate of pitch,r = rate of yaw.[X,Y,Z]= components of resultant aerodynamic force.[u,v, w]=components of velocity of C relative to atmosphere. Aerodynamic forces,on the other hand, depend not on the velocity relative to FE' but ratheron thevelocityrelativetothesurroundingairmass(theairspeed),which will differ from the groundspeed whenever there isa wind.If we denote the wind ve-locityvectorrelativetoFEbyW,andthatof theCG relativetotheairbyVthen clearly VE=V+W The components of Win frame FE'that is, relative to Earth, are given by WE= [Wx Wywzf (1.6,1) (1.6,2) Vrepresentsthemagnitudeof theairspeed(thusretainingtheusualaerodynamics meaning of thissymbol). For the most part we will have W=0,making the airspeed the same as the inertial velocity. Asecond frameof reference will be needed in the development of theequations of motion. This frame is fixed to the airplane and moves with it,having its origin C at the CG,(see Fig.1.6). It is denoted FBand is commonly called body axes.Cxz is the planeof symmetryof thevehicle.Thecomponentsof theaerodynamicforcesand momentsthatactontheairplane,andof itslinear andangularvelocitiesrelativeto y v ----Projection of V on XIIplane ~ - . = - - - -(a) Figure 1.7(a)Definition of ax.(b)View in plane ofy and V, definition of {3. 1.6Axes and Notation17 theairaredenotedbythesymbolsgiveninthefigure.In thenotationof Appendix A.I, this means, for example, that (1.6,3) The vector Vdoesnot in general lie in anyof thecoordinate planes.Itsorienta-tion is defined by the two angles shown in Fig.1.7: Angle of attack, Angle of sideslip, u {3=sin-l -V (1.6,4) With these definitions,thesideslip angle {3is not dependent on the direction of ex in the plane of symmetry. The symbols used throughout the text correspond generally to current usageand are mainly used in a consistent manner. / CHAPTER2 Static Stability and Control-Part 1 2.1General Remarks Ageneraltreatmentof thestabilityandcontrolof airplanesrequiresastudyof the dynamicsof flight,and thisapproach istaken in later chapters.Much useful informa-tioncan beobtained,however,fromamorelimitedview,inwhichweconsidernot themotionof theairplane,butonlyitsequilibriumstates.Thisistheapproachin what iscommonly known asstatic stability and control analysis. The unsteady motions of an airplane can frequentlybe separated forconvenience into two parts.One of these consists of the longitudinal or symmetric motions; that is, those in which thewings remain level,and in which thecenter of gravitymovesin a verticalplane.Theotherconsistsof thelateralorasymmetricmotions;thatis, rolling,yawing,andsideslipping,whiletheangleof attack,thespeed,andtheangle of elevation of the xaxis remains constant. Thisseparation can be made forboth dynamicandstaticanalyses.However,the results of greatest importance forstatic stability are those associated with thelongitu-dinalanalysis.Thustheprincipalsubjectmatterof thisandthefollowingchapter is staticlongitudinalstabilityand control. A brief discussionof thestaticaspectsof di-rectional and rolling motions iscontained in Secs.3.9 and 3.11. Weshall be concerned with twoaspects of the equilibrium state.Under the head-ing stability weshall consider thepitching moment that actson the airplane when its angle of attack ischanged from the equilibrium value,asby a vertical gust.Wefocus our attention on whether or not thismoment acts in such a sense astorestore the air-plane to its original angle of attack. Under the heading control we discuss the use of a longitudinal control (elevator) to change the equilibrium value of the angle of attack. Therestrictiontoangleof attackdisturbanceswhendealingwithstabilitymust benoted,sincetheapplicabilityof theresultsistherebylimited.Whentheaerody-namiccharacteristicsof anairplanechangewithspeed,owingtocompressibilityef-fects,structural distortion,or the influence of the propulsive system, then the airplane maybeunstablewithrespecttodisturbancesinspeed.Suchinstabilityisnotpre-dictedbyaconsiderationof angleof attackdisturbancesonly.(SeeFig.1.3d,and identifyspeed with x,angleof attack with y.) A more general point of viewthan that adoptedinthischapterisrequiredtoassessthataspectof airplanestability.Sucha viewpoint istaken in Chap.6.Todistinguish between true general staticstability and themore limited version represented byemvs.ct,weuse theterm pitchstiffnessfor the latter. 18 - - ~ - - - - ~ - - - - - - - - - - - - -2.1General Remarks19 Althoughthemajor portionof thisandthefollowingchaptertreatsarigidair-plane,an introduction to the effects of airframe distortion is contained in Sec.3.5. THE BASIC LONGITUDINAL FORCES The basic flight condition for most vehicles issymmetric steady flight.In thiscondi-tion the velocity and force vectors are as illustrated in Fig.2.1. All the nonzero forces and motion variablesaremembersof the set defined as"longitudinal." The two main aerodynamic parameters of this condition are V and a. Nothing can be said in general about the way the thrust vector varieswith Vand a,sinceit isso dependent onthe typeof propulsion unit-rockets, jet,propeller,or turboprop. Two particular idealizations are of interest, however, 1.T independent of V,that iS,constant thrust;anapproximation forrocketsand pure jets. 2.TV independent of V,that is, constant power;an approximation for reciprocat-ing engines with constant-speed propellers. Thevariationof steady-stateliftanddragwitha forsubsonicandsupersonic Machnumbers(M 0,and the curvature of theCL vs.arelationmaybeanimportantconsiderationforflightathighCv When the vehicle isa "slender body," for example, a slender delta, or a slim wingless L w Figure 2.1Steady symmetric flight. 20Chapter 2.Static Stability and Control-Part 1 - - ~ - - - - - - - - - - - - - - - a Figure 2.2Lift and drag for subsonic and supersonic speeds. body,theCLcurve may havea characteristic upward curvature even at small a(Flax and Lawrence,1951). The upward curvature of CL at small aisinherently present at hypersonic Mach numbers (Truitt,1959). For the nonlinear cases, a suitable formu1a-.tion for CL is(USAF,1978) (2.1,3) whereCN"andCN""arecoefficients(independentof a)thatdependontheMach numberandconfiguration.[ActuallyCN hereisthecoefficientof theaerodynamic forcecomponent normal tothewingchord,andCN"isthe value of CL"ata=0,as can easilybeseen by linearizing(2.1,3)with respect toa.]Equation(2.1,2)forthe dragcoefficientcanservequitewellforflightdynamicsapplications. uptohyper-sonicspeeds(M>5)atwhichtheoryindicatesthattheexponentof CL decreases from2to~ .Miele(1962)presentsinChap.6averyusefulandinstructivesetof typicallift anddragdata forawiderangeof vehicletypes,fromsubsonictohyper-sonic. Balance, or Equilibrium Anairplanecan continue insteadyunaccelerated flightonlywhentheresultant external forceand moment about the CG both vanish.In particular,thisrequires that thepitchingmomentbezero.Thisistheconditionof longitudinalbalance.If the pitching moment were not zero,theairplane would experience a rotational accelera-tion component in the direction of the unbalanced moment.Figure 2.3showsa typi-cal graph of the pitching-moment coefficient about the CG1 versus the angle of attack foranairplane with a fixedelevator (curve a).The angle of attack ismeasured from thezero-liftlineof theairplane.Thegraphisastraightlineexceptnearthestall. Since zeroCm isrequired for balance,theairplane can flyonly at theangle of attack marked A, for the given elevator angle. lUnless otherwise specified, em always refers to moment about the eG. Nose up Nose down Balanced andpositive stiffness A_-Ii ---"\----Balancedbut negative stiffness 2.1General Remarks21 ...,.- __ ....----....b a Figure 2.3Pitching moment of an airplane about the CG. Pitch Stiffness Suppose that the airplane of curve aon Fig.2.3isdisturbed from itsequilibrium attitude,theangle of attack being increased tothat at Bwhile itsspeed remains unal-tered. It is now subject to a negative, or nose-down, moment, whose magnitude corre-spondstoBe.Thismomenttendstoreducetheangleof attacktoitsequilibrium value,andhenceisarestoringmoment.Inthiscase,theairplanehaspositivepitch stiffness, obviously a desirable characteristic. On the other hand,if Cm were given by the curve b,the moment acting when dis-turbedwould be positive,or nose-up,andwouldtendtorotatetheairplanestill far-ther from itsequilibrium attitude.Wesee that the pitch stiffness isdetermined by the signandmagnitude of theslope CJCmICJa.If the pitch stiffnessistobe positive at the equilibrium a,Cm must bezero,and ar;rrlda must benegative.It will be appreciated fromFig.2.3that analternativestatement is "Cmo must be positive,and aCm/daneg-ativeif theairplaneistomeetthis(limited)conditionforstableequilibrium."The variouspossibilitiescorrespondingtothepossiblesignsofCmo andCJCmlCJaare shown in Figs.2.3and 2.4. ~ - - " ' b Positive stiffness but unbalanced a Figure 2.4Other possibilities. 22Chapter 2.Static Stability and Control-Part 1 c__> ~ - - - ~ Positive camber Cmo negative Figure 2.5 Possible Configurations Zero camber Cmo =0 Cmo of airfoil sections. Negative camber Cmo positive The possible solutions fora suitable configuration arereadily discussed interms of the requirementsonCmo and aCm/aa.Westate here without proof (thisisgiven in Sec.2.3)thatacm/aa canbemadenegativeforvirtuallyanycombinationof lifting surfaces and bodies by placing the center of gravity far enough forward.Thus it is not the stiffness requirement, taken by itself, that restricts the possible configurations, but rathertherequirementthattheairplanemustbesimultaneouslybalancedandhave positivepitchstiffness.Sincea proper choice of theCG location canensurea nega-tiveaCm/aa,thenany configuration with a positive Cmo can satisfy the(limited)con-ditions for balanced and stable flight. Figure2.5showstheCmo of conventionalairfoilsections.If anairplanewereto consist of a straight wing alone (flying wing),then the wing camber would determine the airplane characteristics asfollows: Negative camber-flight possible at a> 0;i.e.,CL> 0 (Fig.2.3a). Zero camber-flight possible only ata=0,or CL =O. Positive camber-flight not possible at any positiveaor Cv For straight-winged taillessairplanes,only thenegative camber satisfiesthecon-ditions forstable,balanced flight.Effectively thesame result isattained if a flap,de-flectedupward,isincorporatedatthetrailingedgeof asymmetricalairfoil.Acon-ventionallow-speedairplane,withessentiallystraightwingsandpositivecamber, couldflyupsidedownwithoutatail,providedtheCGwerefarenoughforward (aheadof thewingmeanaerodynamiccenter).Flyingwingairplanesbasedona straight wing with negative camber are not in general use for three main reasons: + Camberedwingat CL =0Tailwith CL negative CG (a) CG S6ii=~ Tailwith CL positive+ Camberedwingat CL =0 (bl Figure 2.6Wing-tail arrangements with positive Cmo'(a)Conventional arrangement. (b)Tail-first or canard arrangement. - ~ 2.2Synthesis of Lift and Pitching Moment23 + Lift - Lift Figure 2.7Swept-back wing with twisted tips. 1.The dynamic characteristics tend to be unsatisfactory. 2.The permissible CG range is too small. 3.The drag and CLmax characteristics are not good. Thepositivelycamberedstraightwingcan beusedonlyin conjunctionwithan auxiliarydevicethatprovidesthepositiveCmo'Thesolutionadoptedbyexperi-mentersasfarbackasSamuelHenson(1842)andJohnStringfellow(1848)wasto add a tail behind thewing.The Wright brothers(1903)useda tailahead of thewing (canard configuration).Either of these alternatives can supplya positive Cmo'asillus-tratedinFig.2.6.Whenthewingisatzerolift,theauxiliarysurfacemust providea nose-upmoment.Theconventionaltailmustthereforebeatanegativeangleof at-tack, and the canard tail at a positive angle. An alternativetothewing-tail combination istheswept-back wingwithtwisted tips(Fig.2.7). When the net lift iszero,the forwardpart of thewinghas positive lift, and the rear part negative. The result isa positive couple, asdesired. Avariantof theswept-backwingisthedeltawing.ThepositiveCmo canbe achieved with such planforms by twisting thetips,byemploying negative camber,or by incorporating an upturned tailing edge flap. 2.2Synthesis of Lift and Pitching Moment The total lift and pitching moment of an airplane are, in general, functions of angle of attack,control-surfaceangle(s),Mach number,Reynoldsnumber,thrustcoefficient, anddynamicpressure?(Thelast-namedquantityentersbecauseof aeroelasticef-fects.Changes in the dynamic pressure (ipv2), when all the other parameters are con-stant,may induce enough distortion of thestructure toalter Cm significantly.) Anac-curatedeterminationof theliftand' pitchingmomentisoneof themajortasksina staticstability analysis.Extensive use ismade of wind-tunnel tests,supplemented by aerodynamic and aeroelastic analyses. 2When partial derivatives are taken in the followingequations with respect toone of these variables, for example, dC,jdOl,it is to be understood that all the others are held constant. 24Chapter 2.Static Stability and Control-Part 1 Forpurposesof estimation,thetotalliftandpitchingmomentmaybesynthe-sized fromthecontributionsof thevariouspartsof theairplane,that is,wing,body, nacelles,propulsivesystem,andtail,andtheirmutualinterferences.Somedatafor estimatingthevariousaerodynamicparametersinvolvedarecontainedinAppendix B,whilethegeneralformulationof theequations,intermsof theseparameters,fol-lowshere.In thischapter aeroelasticeffectsarenot included.Hence theanalysisap-plies to a rigid airplane. LIFT AND PITCHING MOMENT OF THE WING Theaerodynamicforcesonanyliftingsurfacecanberepresentedasaliftanddrag actingat themeanaerodynamiccenter,together witha pitching coupleindependent of theangleof attack (Fig.2.8).The pitching moment of thisforcesystem about the CG is given by (Fig.2.9)3 Mw=Macw + (Lwcoslrw+ Dwsinlrw)(h- hn)c + (Lwsinlrw- Dwcoslrw)z(2.2,1) Weassume that the angle of attack is sufficiently small to justify the approximations coslrw=1, andthe equation ismadenondimensional by dividingthrough by !pV2Sc.It then be-comes (2.2,2) Althoughit mayoccasionallybenecessarytoretainallthetermsin(2.2,2),experi-ence hasshown that the last term is frequentlynegligible,and that CDwlrwmay be ne-glected in comparison with CLw With these simplifications, we obtain Meanaerodynamic center Figure 2.8Aerodynamic forceson the wing. 3The notation hnwindicates that the mean aerodynamic center of the wing isalso the neutral point of the wing. Neutral point is defined in Sec.2.3. l 2.2Synthesis of Lift (lnd Pitching Moment25 Wing zero I ift direction Mean aerodynamic chord I-o0 forastableairplane.Inthesecond case,forexample exactly horizontal flight,8p =8p(V)and the8p term on the right-hand side of (2.4,23) remains.For such casesthegradient (d8etrjdV) isnot necessarily related tostability. Forpurposesof calculatingthepropulsioncontributions,thetermsCL8p d8p and Cm.d8p in (2.4,23) would be evaluated asdCLand dCm [see the notation of (2.3,1)]. Up'Pp These contributions tothe lift and moment are discussed in Sec.3.4. ThederivativesCLv andCmv maybequitelargeowingtoslipstreameffectson STOLairplanes,aeroelasticeffects,orMachnumbereffectsneartransonicspeeds. ThesevariationswithMcan resultinreversalof theslopeof 8etrim asillustratedon Fig.2.20.Thenegativeslopeat A,accordingtothestabilitycriterionreferredto above,indicates that the airplane is unstable at A. This can be seen asfollows.Let the airplane be in equilibrium flight at the point A, and be subsequently perturbed sothat itsspeed increases tothat of Bwith nochange inexor 8e. Now at Bthe elevator angle istoopositive fortrim:that isthereisanunbalanced nose-down moment on the air-plane. This puts the airplane into a dive and increases its speed still further.The speed willcontinuetoincreaseuntilpointCisreached,whenthe8e isagainthecorrect value fortrim,but here theslope ispositive and there isnotendency forthespeed to change any further. oV Figure 2.20Reversal of 8etrim slope at transonic speeds,8p =const. 40Chapter 2.Static Stability and Control-Part 1 STATIC STABILITY LIMIT, hs ThecriticalCGposition forzeroelevator trimslope(i.e.forstability)canbefound by setting (2.4,24) equal to zero.Recalling that Cm"=CL,,(h- hn),this yields or where Cmv hs=hn + -----'---CLv + 2CLe (2.4,25) (2.4,26) Dependingonthesignof Cmv'hsmaybegreaterorlessthanhn- Intermsof hs, (2.4,24) can be rewritten as ( dOe':}m)=CL"(C+ 2C)(h- h) dVsdetLvLes p (2.4,27) (h- hs)is the "stability margin," which may be greater or less than the static margin. FLIGHT DETERMINATION OF hn AND hs For thegeneralcase,(2.3,5)suggeststhatthemeasurementof hn requiresthemea-surementof Cm"andCL",Flightmeasurementsof aerodynamicderivativessuchas thesecanbemadebydynamictechniques.However,inthesimplercasewhenthe complicationspresentedbypropulsive,compressibility,or aeroelasticeffectsareab-sent,thentherelationsimplicit inFigs.2.18and2.19leadtoameansof findinghn fromtheelevatortrimcurves.Inthatcaseallthecoefficientsof (2.4,13)arecon-stants,and (2.4,28) dOeCL ~=- _" (h-h) d C ~ mdetn or(2.4,29) Thusmeasurementsof theslope of 0etrimvs.C ~ matvariousCG positionsproducea curve like that of Fig. 2.21, in which the intercept on the h axis is the required NP. When speed effects are present, it isclear from (2.4,27) that a plot of (doetriJdV)sp against h will determine hsasthe point where the curve crosses the h axis. h Figure 2.21Determination of stick-fixed neutral point from flight test.- - - - - - ~ - - - - - - - - - - - - - - - - ~ - - - - ~ - - - - - - - - -- ------------------------------2.5TheControl Hinge Moment41 2.5TheControl Hinge Moment Torotateanyof theaerodynamiccontrolsurfaces,elevator,aileron,or rudder,about its hinge,it isnecessary toapply a forcetoit toovercome the aerodynamic pressures that resist the motion. This forcemay be supplied entirely by a human pilot through a mechanical system of cables, pulleys, rods,and levers;it may be provided partly by a powered actuator;or the pilot may be altogether mechanically disconnected fromthe controlsurface("fly-by-wire"or "fly-by-light").Inanycase,the forcethat has tobe applied to the control surface must be known with precision if the control system that connectstheprimarycontrolsinthecockpit totheaerodynamicsurfaceistobede-signed correctly.The range of control system optionsissogreat that it isnot feasible inthistext topresenta comprehensivecoverageof them.Wehavethereforelimited ourselvesinthisandthefollowingchapter tosomematerialrelatedtoelevator con-trol forceswhen the human pilot supplies all of the actuation,or when a power assist relievesthepilot of a fixedfractionof theforcerequired.Thistreatment necessarily beginswitha discussionof theaerodynamics;that is,of theaerodynamichinge mo-ment. Theaerodynamicforcesonanycontrolsurfaceproduceamomentaboutthe hinge.Figure2.22showsatypicaltailsurfaceincorporatinganelevatorwithatab. Thetabusuallyexertsanegligibleeffectontheliftof theaerodynamicsurfaceto which it isattached,although its influence on the hinge moment islarge. Hinge line Elevator -t----_ c--- eTab L---r---L._.lTab hinge (a) Elevator hinge ~ - - - - - - - - - - - - - - C t - - - - - - - - ~ (b) ~ I Figure 2.22Elevator and tab geometry. (a)Plan view.(b)Section A-A. 42Chapter 2.Static Stability and Control-Part 1 The coefficient of elevator hinge moment is defined by He Cke=12-2PVSece Here Heisthemoment,abouttheelevator hingeline,of theaerodynamicforceson the elevator and tab,Seis the area of that portion of the elevator and tab that lies aft of the elevator hinge line,and ce is a mean chord of the same portion of the elevator and tab.Sometimesce istaken to be thegeometricmeanvalue,that is,ce =Sj2se, and other timesit istheroot-meansquareof CeoThetaper of elevatorsisusuallyslight, and the difference between the twovalues is generally small.The reader is cautioned tonotewhichdefinitionisemployedwhenusingreportsonexperimentalmeasure-ments of Che Of alltheaerodynamic parameters required instabilityand control analysis,the hinge-momentcoefficientsaremostdifficulttodeterminewithprecision.Alarge number of geometricalparametersinfluence thesecoefficients,andtherangeof de-signconfigurationsiswide.Scaleeffectstendtobelargerthanformanyotherpa-rameters,owingtothesensitivityof thehingemomenttothestateof theboundary layeratthetrailingedge.Two-dimensionalairfoiltheoryshowsthatthehingemo-ment of simple flapcontrolsislinearwithangleof attackandcontrolanglein both subsonic and supersonic flow. Thenormal-force distributionstypicalof subsonic flowassociated withchanges in a and5e are shown qualitatively in Fig.2.23. The force acting on the movable flap hasamomentaboutthehingethatisquitesensitivetoitslocation.Ordinarilythe hinge moments in both cases (a) and (b)shown are negative. In many practical casesit isasatisfactory engineeringapproximation toassume thatforfinitesurfacesChe isalinearfunctionof as,5e,and5tThereadershould notehoweverthatthereareimportantexceptionsinwhichstrongnonlinearitiesare present. We assume therefore that Che is linear, as follows, (2.5,1) where as istheangle of attack of the surface towhich the control isattached (wing or tail), and5t istheangle of deflection of thetab(positivedown).The determinationof the hingemomentthenresolvesitself intothedeterminationof bo,bI,b2,andb3The geometrical variables that enter are elevator chord ratio cjct,balance ratio cJce, nose shape, hinge location, gap, trailing-edge angle,and planform. When a set-back hinge isused,someof thepressureactsaheadof thehinge,andthe hingemoment isless than that of asimple flapwitha hingeat itsleading edge.The forcethat the control system must exert tohold theelevator at thedesiredangleisin direct proportionto the hinge moment. l :-r-- -~ V' 2.5The Control Hinge Moment43 (a) Figure 2.23Nonnal-force distribution over control surface at subsonic speed. (a)Force distribution over control associated with Citat 8e =O.(b) Force distribution over control associated with 8e at zero Cit. Weshall find it convenient subsequently tohavean equation like(2.5,1)witha instead of as.For taillessaircraft,asis equal toa, but for aircraft with tails,as=at. Let us write for both types (2.5,2) where for tailless aircraft Cheo = bo, Che"= bl.For aircraft with tails, the relation be-tween aand at is derived from (2.3,12) and (2.3,19), that is, (2.5,3) whence it followsthat for tailed aircraft,with symmetrical airfoilsections in the tail, for which bo =0, (a) (2.5,4) (b) 44Chapter 2.Static Stability and Control-Part 1 2.6Influence of a Free Elevator on Lift and Moment In Sec.2.3we have dealt with the pitch stiffnessof anairplane the controls of which arefixedinposition.Evenwithacompletelyrigidstructure,whichneverexists,a manually operated control cannot be regarded asfixed.A human pilot is incapable of supplying anideal rigid constraint. When irreversible power controlsare fitted,how-ever,thestick-fixedconditioniscloselyapproximated.Acharacteristicof interest fromthepoint of viewof handlingqualitiesisthestabilityof theairplanewhenthe elevator iscompletely freetorotate about its hinge under the influence of the aerody-namic pressuresthat act uponit.Normally,thestability in thecontrol-free condition islessthanwithfixedcontrols.It isdesirablethatthisdifferenceshouldbesmall. Sincefrictionisalwayspresent inthecontrolsystem,thefreecontrol isnever real-ized in practice either.However,the two ideal conditions, freecontrol and fixedcon-trol,represent the possible extremes. When the control is free,then Che =0,so that from (2.5,2) 1 Defree =- b2 (Cheo + Cheaa+ b3Dt)(2.6,1) Thetypical' upwarddeflectionof afree-elevatorona tailisshownin Fig.2.24.The corresponding lift and moment are CLt,ee=CLaa+Cmf,ee=C mo+ Cmaa+ CmSe Def,ee After substituting (2.6,1) into (2.6,2),we get where CLt,ee= + C La' a Cmf,ee=C:"o+ C:"aa CC a'=C'=C_LSehea LaLab 2 (2.6,2) (a) (2.6,3) (b) (a) (2.6,4) (b) (a) (2.6,5) (b) Whendueconsideration isgiven totheusualsignsof thecoefficients intheseequa-tions,weseethatthetwoimportantgradientsCLaandCma arereducedinabsolute Figure 2.24Elevator floating angle. 2.6Influence of a Free Elevator on Lift and Moment45 magnitudewhenthecontrol isreleased.Thisleads,broadlyspeaking,toa reduction of stability. FREE-ELEVATOR FACTOR For a taillessaircraft with a free elevator, the lift-curve slope is (cf 2.6,4b) I_(1CLaebl ) a-a---ab2 (2.6,6) Thefactorinparenthesesisthe freeelevator factor,andnormallyhasavalueless than unity. When the elevator is part of the tail,the floatingangle can be related toat, viz for bo =0 Che=biat + b20efree+ b30t=0 1 or oefree=- b2 (blat+ b3 0t ) (2.6,7) and the tail lift coefficient is (2.6,8) The effective lift-curve slope is (2.6,9) where F=(1- ae !!..2.)isthe freeelevator factor foratail.If Fatisused in place atb2 of atanda'inplaceof a,thenalltheequationsgiveninSec.2.3holdforaircraft with a free elevator. ELEVATOR-FREE NEUTRAL POINT It isevident from the preceding comment that the NP of a tailed aircraft when the el-evator is freeis given by (2.3,23) as . Fat_(aE)1aCm h ~=hnb+ -, VH1 - -a- ---;-aP Waaaa (2.6,10) Alternatively,we can derive the NP location from (2.6,5b),for weknow from(2.3,5) that C;""=CUh - h ~ ) (a) or C'1(CC) (h - hi) =~=- C_maehe" nC' a'm"b L"2 (2.6,11) 1[CC] =da(h- hn) - m ~ 2he"(b) 46Chapter 2.Static Stability and Control-Part 1 SinceCmSe isof differentformforthetwomaintypesof aircraft,weproceedsepa- rately below. TAILED AIRCRAFT CmSe isgiven by (2.4,8),so(2.6,11) becomes for this case Using (2.6,4b) this becomes h- hi= h- - ah-1(_C_h--.::e",-C-,LS",,-eh)+C;he"aV-na'nb2 nwba'b2 eH C heC Lo -hi= h+"Ve(hh)V nnIn- nwb- -a'b2H b2a (2.6,12) Finally, using (2.4,8) for CLSe'and (2.5,4) for Che",we get hi=h- - - 1 - - -(h- h)- + V ae bl (dE) (St_) nna' b2 dannwbSH (2.6,13) TAILLESS AIRCRAFT CmSe is given by (2.4,9) and Che"=bl.When these are substituted into (2.6,11) the re-sult is By virtue of (2.6,6) this becomes bdC h- hi= h- h__1_ nna'b2 dBe or bdC hi=h+ _1_ nna'b2 dBe (2.6,14) The difference - h) iscalled the control-free Btatic margin,When representa-tivenumericalvaluesareusedin(2.6,13)onefindsthathn -maybetypically about0.08.Thisrepresentsasubstantialforwardmovementof theNP,withconse-quent reduction of static margin, pitch stiffness, and stability. 1 2.7TheUse of Tabs47 2.7TheUseof Tabs TRIM TABS In order to flyat a given speed,or CL>it has been shown in Sec.2.4 that a certain ele-vatorangleSetrimisrequired.Whenthisdiffersfromthefree-floatingangleSef h'",must be interpreted cor-64Chapter 3.Static Stability and Control-Part 2 ~ ~ - - - - - - ~ ~ ~ ~ ~ - - - - - - - - - - ~ k o.... ....CGposition Controlfree neutral point-pointof zerogradient of controlforce at handsoffspeed Figure 3.3 ........ Controlfree maneuver point-point of zerocontrol forceper g Control force per g. rectly.Inthefirstplacethisdoesnotnecessarilymeanareversalof controlmove-mentper g,forthisisgovernedbytheelevatorangleper g.If h'".0, and would tend to keep the wings level. If rolling occurs about the wind vector, the stiffness iszero and the vehi-cle has no preferred roll angle. If ax < 0,then the stiffness is negative and .the vehicle would roll to the position cp=180, at which point Cz =0 and Cz",< O. The above discussion applies to a vehicle constrained, asstated,to one degree of freedom.It doesnot,byanymeans,givethefullanswerforanunconstrainedair-planetothequestion:"What happenswhentheairplanerollsawayfromawings-level attitude-does it tend to come back or not?" That answer can only be provided by a fulldynamic analysis like the kind given in Chaps. 6 and 7. The roll stiffness ar-gument given above, however,does help in understanding the behavior of slender air-planes, ones with very low aspect ratio and hence small roll inertia. These tend, in re-sponse to aileron deflection when at angle of attack, to rotate about the xaxis, not the velocityvector,and hence experience the rollstiffness effect at the beginning of the response. Eventhoughairplaneshavenofirst-orderaerodynamicrollstiffness,stableair-planesdohaveaninherent tendencytoflywithwingslevel.Theydosobecauseof what is known asthe dihedraleffect.This isa complex pattern involving gravityand thederivativeCZf3'whichowesitsexistencelargelytothewingdihedral(seeSec. 3.12). When rolled toananglecp,thereisa weight component\mgsin cpin the y di-rection (Fig.3.14). This induces a sideslip velocity to the right,with consequent f3> 0,andarollingmomentCzi3thattendstobringthewingslevel.Therollingand yawing motions that result fromsuch an initial condition are,however,strongly cou-pled,sonosignificant conclusionscan be drawnabout the behavior except by a dy-namic analysis (see Chap. 6). 3.12The Derivative C1(3 The derivativeCZf3 isof paramount importance.Wehavealreadynoted itsrelation to rollstiffnessandtothetendencyof airplanestoflywithwingslevel.Theprimary contributiontoCZf3 isfromthewing-its dihedralangle,aspect ratio,andsweepall being important parameters. The effect of wing dihedral isillustrated in Fig.3.15. With the coordinate system shown,the normal velocity component Vnon the right wing panel (R)is,for small di-hedral angle f, Vn=Wcos f+ v sin f ==w+vf and that on theother panel is w- vf. The termsvflV =f3f represent opposite changesintheangleof attackof thetwopanelsresultingfromsideslip.The"up-wind" panel has its angle of attack and therefore its lift increased, and vice versa. The resultisarollingmomentapproximatelylinear in bothf3andf, andhenceafixed 84Chapter 3.Static Stability and Control-Part 2 z Figure 3.15Dihedral effect.Vn= nonnal velocity of panel R= w cos r + Dsin r W+ Dr :. Drv[3r of R due to V= V= [3r. value of C1p for a given r. This part of C1p is essentially independent of wing angle of attack so long as the flow remains attached. Even in the absence of dihedral,a flat lifting wing panel has a C1p proportional to CvConsider thecase of Fig.3.16.The verticalinduced velocity(downwash)of the vortexwakeisgreater atLthanat Rsimplybyvirtueof thegeometryof thewake. Hence the local wing angle of attack and lift are less at L than at R,and a negative Cl results.Sincethiseffectdepends,essentiallylinearly,onthestrengthof thevortex wake, which is itself proportional to the wing Cv then the result is ilClp ocCv \

\ \ \ V(relative wind) Figure 3.16Yawed lifting wing. 3.12The Derivative C1p 85 --'C---Low wins:! Figure 3.17Influence of body on C l ~ ' INFLUENCE OF FUSELAGE ON C'p The flowfieldof thebodyinteractswith thewingin such a wayastomodifyitsdi-hedral effect. Toillustrate this,consider a long cylindrical body,of circular cross sec-tion,yawed with respect tothe main stream.Consider only the cross-flow component of thestream,of magnitudeV{3,andtheflowpatternwhichitproducesaboutthe body.Thisisillustrated inFig.3.17.It isclearlyseenthatthebodyinducesvertical velocitieswhich,whencombined withthemainstreamvelocity,alter thelocalangle of attackof thewing.Whenthewingisatthetopof thebody(high-wing),then the angleof attack distributionissuchastoproduceanegativerollingmoment;thatis, the dihedral effect isenhanced. Conversely,when the airplane has a lowwing,the di-hedraleffect isdiminished bythefuselageinterference.Themagnitude of theeffect isdependent upon the fuselage length ahead of the wing,itscross-section shape,and theplanformandlocationof thewing.Generally,thisexplainswhyhigh-wingair-planes usually have less wing dihedral than low-wing airplanes. INFLUENCE OF SWEEP ON C'p Wing sweep isa major parameter affectingC I ~ 'Consider the swept yawed wing illus-trated in Fig.3.18. According tosimple sweep theory it is the velocityVnnormal to a , wingreferenceline( ~chord lineforsubsonic,leadingedgeforsupersonic)thatde-terminesthelift.It followsobviouslythattheliftisgreaterontheright half of the Figure 3.18Dihedral effect of a swept wing. 86Chapter 3.Static Stability and Control-Part 2 v... iE;--_____ ... ______ ...1-_--, CG Figure 3.19Dihedral effect of the vertical tail. wingshownthanontheleft half,andhencethatthereisanegative rollingmoment. The rolling moment would be expected for small f3to be proportional to - =CLV2[COS2 (A- f3)- cos2 (A+ f3)] ==2CLf3V2 sin 2A Theproportionality withCL andf3iscorrect,but thesin 2Afactorisnotagoodap-proximationtothevariationwithA.The result isaCZ{3ocCL>thatcan be calculated by the methods of linear wing theory. INFLUENCE OF FIN ON CZp Thesideslippingairplanegivesrisetoa sideforceon theverticaltail(seeSec.3.9). When themeanaerodynamiccenter of theverticalsurface isappreciablyoffset from therollingaxis(seeFig.3.19)thenthisforcemayproduceasignificant rollingmo-ment. Wecan calculate thismoment from(3.9,3). When the rudder angle iszero, that is,Or=0,the moment is found to be thus and LlC=-a1 - - -- --,(oa) SFZF(VF)2 1{3Fof3SbV (3.12,1) 3.13Roll Control Theangleof bank of theairplane iscontrolled by theailerons.The primary function of these controls is to produce a rolling moment,although they frequently introduce a yawingmomentaswell.Theeffectivenessof theaileronsinproducingrollingand yawing moments isdescribed by the two control derivatives oC/ooa and oCJooa. The aileronangleoaisdefined asthemean value of the angular displacementsof thetwo ailerons.It ispositivewhen the right aileron movement isdownward (see Fig.3.20). ThederivativeoC/ooa isnormallynegative,rightailerondownproducingarollto the left. 1 3.13Roll Control87 Figure 3.20Aileron angle. Forsimpleflap-typeailerons,theincreaseinliftontherightsideandthede-creaseontheleftsideproduceadragdifferentialthatgivesapositive(nose-right) yawing moment.Since the normal reason for moving the right aileron down istoini-tiatea tum tothe left,then theyawingmoment isseen tobe in just thewrongdirec-tion.It isthereforecalledaileronadverseyaw.Onhigh-aspect-ratioairplanesthis tendencymayintroducedecideddifficultiesinlateralcontrol.Meansforavoiding thisparticular difficulty include theuse of spoilersand Frise ailerons.Spoilers areil-lustrated in Fig.3.21. They achieve the desired result by reducing the lift and increas-ing the dragon theside where thespoiler israised.Thus the rollingand yawingmo-mentsdevelopedaremutuallycomplementary with respect toturning.Friseailerons diminish the adverse yaw or eliminate it entirely by increasing the drag on the side of theup goingaileron.Thisisachievedbytheshapingof theaileronnoseandthe choiceof hingelocation.Whendeflectedupward,thegapbetweenthecontrolsur-faceandthewingisincreased,andtherelativelysharpnoseprotrudesintothe stream. Both these geometrical factorsproduce a drag increase. Section through spoiler Figure 3.21Spoilers. 88Chapter 3.Static Stability and Control-Part 2 The deflectionof theaileronsleadstostill additionalyawingmomentsoncethe airplanestartstoroll.Thesearecausedbythealteredflowaboutthewingandtail. TheseeffectsarediscussedinSec.5.7(Cn),andareillustratedinFigs.5.12and p 5.15. Afinalremarkaboutaileroncontrolsisinorder.Theyarefunctionallydistinct fromtheother twocontrolsin that theyareratecontrols.If theairplane isrestricted onlytorotationaboutthe xaxis,thentheapplicationof a constantaileronanglere-sults in a steadyrateof roll. The elevator and rudder,on the other hand,are displace-ment controls.When theairplaneisconstrained totherelevantsingleaxisdegreeof freedom,a constant deflection of these controls produces a constant angular displace-ment of theairplane.It appearsthatbothrateanddisplacement controlsareaccept-able to pilots. AILERON REVERSAL Thereisanimportantaeroelasticeffect on rollcontrolbyaileronsthat issignificant onmostconventionalairplanesatbothsubsonicandsupersonicspeeds.Thisresults from the elastic distortion of the wing structure associated with the aerodynamic load incrementproducedbythecontrol.AsillustratedinFig.2.23,theincrementalload caused by deflecting a control flapat subsonic speeds hasa centroid somewhere near themiddleof thewingchord.Atsupersonicspeedsthecontrolloadactsmainlyon thedeflected surface itself,andhence hasitscentroid even farthertotherear.If this load centroid isbehind the elasticaxisof thewingstructure,thena nose-down twist of the main wing surface results. The reduction of angle of attack corresponding to8a >0causesareductionin liftonthesurfaceascomparedwiththerigidcase,anda consequentreductioninthecontroleffectiveness.Theformof thevariationof CIBa with ipV2 for example can be seen by considering an idealized model of the phenom-enon.Lettheaerodynamictorsionalmomentresultingfromequaldeflectionof the twoaileronsbeT(y)ocipV28a andletT(y)beanti symmetric,T(y)=-T(-y). The twistdistributioncorrespondingtoT(y)is8(y),alsoantisymmetric,suchthat8(y)is proportional toT at anyreferencestation,andhenceproportional toipV28aFinally, theanti symmetrictwistcausesanantisymmetricincrementintheliftdistribution, giving a proportional rolling moment increment !lCI =kipV28aThe total rolling mo-ment due to aileron deflection is then !lCI=(CIB)rigid8a+ kipV28a and the control effectiveness is (3.13,1) (3.13,2) Asnoted above,(CIB)rigidisnegative,and kispositive if thecentroidof theaileron-induced lift isaftof thewingelasticaxis,thecommon case.HenceIchJdiminishes withincreasingspeed,andvanishesatsomespeedVR,theaileronreversalspeed. Hence or o = (ClB)rigid+ kip k= (3.13,3) 3.14Exercises89 Substitution of (3.13,3) into (3.13,2) yields CIBa=(CIB)rigid(1- ~ ) (3.13,4) Thisresult,of course,appliesstrictlyonlyif thebasicaerodynamicsarenot Mach-number dependent, i.e.so long asVR is at a value of Mappreciably below1.0.Other-wisekand(CIB)rigidarebothfunctionsof M, andtheequationcorrespondingto (3.13,4) is (3.13,5) where MR is the reversal Mach number. It isevident from(3.13,4)that the torsional stiffnessof the wing hasto be great enough to keepVR appreciably higher than the maximum flight speed if unacceptable loss of aileron control is to be avoided. 3.14Exercises 3.1Derive (3.1,4). 3.2Derive an expression for the elevator angle per g in dimensional form.Denote the de-rivatives of Land Mwith respect toaand q by aUaq=Lq,and so on. There are two choices:(1) dothe derivation in dimensional form from the begilming, or (2) convert thenondimensionalresult(3.1,6)todimensionalform.Doit bothwaysandcheck that they agree. 3.3Calculatethevariationof thecontrolforcepergwithaltitudefromthefollowing data. Ignore propulsion effects. Geometric Data Weight, W Wing area, S Wing mean aerodynamic chord, c It Tail area, St Se Mean elevator chord, ce G Aerodynamic Data a 50,000 lb (222,500 N) 937.5 ft2(87.10 m2) 12.80 ft (3.90 m) 31.85 ft (9.71m) 230 ff (21.4 m2) 71.30 ff (6.62 m7) 2.21ft (0.674 m) 300/ft (98.4/m) 0.088/deg O.044/deg 0.064/deg o 90Chapter 3.Static Stability and Control-Part 2 bi -0. 17/rad -b2 -0.48/rad C heq -0.846 CLq 0 Cmq -22.9 (h- hn)-0.10 de\da0.30 Ot0 3.4A small manually controlled airplane hasanundesirable handling characteristic-the control force per g is too large. List some design changes that could reduce it,and de-scribe the other consequences that each such change would entail. 3.5The rangeof elevatormotiononanairplaneisfrom20downto30up.Use Table 1.3asa guide,a fellowstudent asa model,anda tapemeasure toarriveata reason-able value for the elevator gearing ratio G. 3.6Twoairplanesaresimilar,butoneisjet-propelled andtheother hasa pistonengine and propeller. The thrust line in each case is well below the CG with zplc= 0.4. The power-off pitching moment atoe=0 isCm =0.1- 0.2CL"The throttle isset togive levelflightwithCL =0.4andVD=12.Considerseveralsteadyrectilinearflight conditionshavingthesamethrottlesettingbutdifferentelevatorsettings,CL values and flight-path angles.Find dCm(a)ldCL (for oe= 0)for the two airplanes when pass-ingthroughthealtitudecorrespondingtothelevelflightconditions.Asindicatedin Sec.3.4 and (6.4,10) dCmldCL isan index of static longitudinal stability under certain conditions. Assuming that theseconditionsaremet in thisproblem,howwill thesta-ticlongitudinal stability of the twoaircraft change asthe aircraft slow down? dC 3.7Deriveanexpression forthe increment Lld{3nattributable toa jet engine.(Hint,re-fer to(3.4,15).) 3.8Supposethatasaresultof anaccidentinflighttherearfuselageof anairplaneis damaged,sothattheflexibilityparameter k in(3.5,1)etseq.issuddenlyincreased. Theeffectislargeenoughthatthepilotnoticesalossinlongitudinalstabilityand control.Bearinginmindthattheintegrityof thefuselagestructuredependsonthe tailload Lt andthestabilityandcontrolonthefactorinparenthesesin(3.5,4),ana-lyze how the situation changes asthe pilot slows down and descends to an emergency landing.Consider two cases; (1)CLI initially positive,(2)CLI initially negative. 3.9Derive(3.9,8).Explain clearly each step in thedevelopment and justify anyassump-tions you make. 3.10Use AppendixBtodeterminetheelevator hingemoment parametersbi and b2 fora NACA0009airfoil(asymmetricairfoilwithathickness-to-chordratioof tic= 0.09). The elevator hasanelliptic nose,a sealed gap and a balance ratio of 0.2.In us-ingthecurvesassumethattransitionisattheleadingedge;R=107;tan(;) = tan ( cP;E)= 0.12;F3= 1;cic = 0.325; A= 4.84; M= O. 3.15Additional Symbols Introduced in Chapter 391 3.15Additional Symbols Introduced inChapter 3 aF aCdaaF ar aCdaar bspan of airplane cr mean rudder chord Chr rudder hinge-moment coefficient CI rolling-moment coefficient,CI"aCla{3 CLF vertical-tail lift coefficient CIaCLlaq Cmdamping in pitch,Cn yawing-moment coefficient, Cn"aCja{3 Cn8,aCjaSr CT Cw ddiameter of propeller or jet gacceleration due to gravity hm maneuver point, stick fixed (see 3.1,8) h;"maneuver point, stick free (see 3.2,5) Lrolling moment LFvertical-tail lift IFsee Fig. 3.12 mairplane mass MqaMlaq nload factor, VW Nyawing moment qangular velocity in pitch, radls qqc/2V SFvertical-tail area Sppropeller disk area Srarea of rudder Tthrust of one propulsion unit V Feffective velocity vector at the fin Vvvertical-tail volume ratio, SJJSb wwing loading, W/S ZFsee Fig. 3.19 92Chapter 3.StaticStability and Control-Part 2 fXFeffective angle of attack of the fin,(see Fig.3.12) f3sideslip angle rdihedral angle Epupwash at propeller Saaileron angle (see Fig.3.20) Srrudder angle (see Fig.3.12) Asweepback angle (J'sidewash angle f.L2m/pSc 4>bank angle CHAPTER4 General Equations of Unsteady Motion 4.1General Remarks The basis for analysis,computation, or simulation of the unsteady motions of a flight vehicleisthemathematicalmodelof thevehicleanditssubsystems.Anairplanein flightisaverycomplicateddynamicsystem.It consistsof anaggregateof elastic bodiessoconnected that both rigid and elastic relative motionscan occur.For exam-ple,thepropelleror jet-enginerotorrotates,thecontrolsurfacesmoveabouttheir hinges,and bending and twistingof thevariousaerodynamic surfaces occur.The ex-ternalforcesthat actontheairplanearealsocomplicated functionsof itsshapeand itsmotion.It seemsclearthatrealisticanalysesof engineeringprecisionarenot likely tobeaccomplished witha verysimple mathematical model.The model that is developed in the followinghas been foundby aeronauticalengineers and researchers tobeveryusefulinpractise.Webeginbyfirsttreatingthevehicleasasinglerigid body with six degrees of freedom.This body isfreeto move in the atmosphere under theactionsof gravityandaerodynamicforces-it isprimarilythenatureandcom-plexityof aerodynamicforcesthatdistinguishflightvehiclesfromotherdynamic , systems.Nextweaddthegyroscopiceffectsof spinningrotorsandthencontinue with a discussion of structural distortion (aeroelastic effects). AswasnotedinChap.1,theEarthistreatedasflatandstationaryininertial space.Theseassumptionssimplifythemodelenormously,andarequiteacceptable fordealingwithmostproblemsof airplaneflight.Theeffectsof aroundrotating Earth are treated at some length in Etkin (1972). Extensiveuseismadeinthedevelopmentsthatfollowof linearalgebra,with whichthereader isassumedtobe familiar.Appendix A.lcontainsa brief reviewof some pertinent material. 4.2TheRigid-Body Equations Inthe interest of completeness, the rigid-body equations are derived from first princi-ples,thatistosay,weapplyNewton'slawstoanelementdmof theairplane,and thenintegrateoverallelements.Thevelocitiesandaccelerationsmustof coursebe relativetoaninertial,or Newtonian,frameof reference. Aswe noted in Sec.1.6the frameFE'fixedtothe Earth,isassumed tobe such a frame.Wealsonoted there that velocities relative toFEare identified bya superscript E.Inorder toavoidthecarry-ingof thecumbersomesuperscriptthroughoutthefollowingdevelopment,weshall 93 94Chapter 4.General Equations of Unsteady Motion temporarilyassume that W=0 in (1.6,1),sothat VE =V,and make an appropriate adjustment at the end. In the frame FB VB= [uvwf(4.2,1) The position vector of dmrelativetotheorigin of FEisrc+ r(seeFig.4.1).In the frame FE'. and in the frame F B rB =[xyzY The inertial velocity of dm is vE = (tcE + t E)= V E + tE The momentum of dm is vdm,and of the whole airplane is ZE Figure 4.1Axes. (4.2,2) (4.2,3) (4.2,4) (4.2,5) 4.2The Rigid-Body Equations95 Since C is the mass center, the last integral in (4.2,5) is zero and (4.2,6) where m is the total mass'ofthe airplane. Newton's second law applied to dm is dfE=vEdm (4.2,7) where dfE is the resultant of all forces acting on dm. The integral of (4.2,7) is or,from (4.2,6) (4.2,8) The quantity f dfE isa summation of all the forcesthat act upon all the elements. The internalforces,that is,thoseexerted by oneelement uponanother,alloccur in equal and opposite pairs,by Newton's third law,and hence contribute nothing to the summation. Thus fE is the resultant external force acting upon the airplane. This equation relatesthe external forceon theairplane tothemotion of the CG. Weneedalsotherelationbetweentheexternalmomentandtherotationof theair-plane. It is obtained from a consideration of the moment of momentum. The moment of momentum of dm with respect to C is by definition db =rXvdm. It is convenient inthefollowingtousethematrixformof thecross' product(see Appendix A.l)so that Consider Now from (4.2,4), and the moment of df about C is dG=rXdf so that from (4.2,7) dGE =i'EdfE=i'EvEdm Equation (4.2,9) then becomes d_ dGE =- (dbE) - (vE - V E)vEdm dt Since v